Central Bibliography

abstract:
We study how coherent Stokes and anti-Stokes Raman scattering influence coherent population trapping. In an experiment using an atomic sodium vapor cell we observe induced transparency, induced absorption, and gain features, all of subnatural linewidth. The electromagnetically induced resonance is a peak or a dip depending on which side of the optical transition the fields are tuned to, and thus whether coherent anti-Stokes Raman scattering or coherent Stokes Raman scattering is the dominant process.

abstract:
Positive and negative subnatural-width resonances (SNWR) were observed in the absorption and fluorescence of rubidium vapor under excitation by two copropagating optical waves with variable frequency offset. The two optical fields resonantly couple Zeeman sublevels, belonging to the same ground-state hyperfine level (GSHL), to an intermediate excited state. The SNWR present opposite signs depending on which GSHL participates in the interaction with the two optical waves. For both Rb isotopes an increase in the transparency with reduced fluorescence occurs for the lower GSHL while the absorption and fluorescence are increased for the upper GSHL. The influence of external magnetic field, polarization, and intensity of applied optical fields on the SNWR is examined. The narrowest observed resonance has a width of 10 kHz (full width at half maximum). The origin of the SNWR is discussed in terms of coherent processes involving ground-state Zeeman sublevels.

abstract:
We report the first demonstration of a technique by which an optically thick medium may be rendered transparent. The transparency results from a destructive interference of two dressed states which are created by applying a temporally smooth coupling laser between a bound state of an atom and the upper state of the transition which is to be made transparent. The transmittance of an autoionizing (ultraviolet) transition in Sr is changed from exp(-20) without a coupling laser present to exp(-1) in the presence of a coupling laser.

abstract:
The phenomenon of coherent population trapping in alkali-metal atoms is analyzed by means of a perturbation approach. Closed form transparent solutions are obtained for the coherences existing within the system and the populations of the ground levels and of the excited state. The presence of dark lines and coherent microwave emission from the ground state are made explicit. Experimental results that confirm the theoretical calculations are reported for the case of cesium in a buffer gas. Conclusions are drawn in connection with the application of coherent population trapping to the field of atomic frequency standards.

abstract:
We report on the atom optical manipulation of an atom laser beam. Reflection, focusing, and its storage in a resonator are demonstrated. Precise and versatile mechanical control over an atom laser beam propagating in an inhomogeneous magnetic field is achieved by optically inducing spin flips between atomic ground states with different magnetic moment. The magnetic force acting on the atoms can thereby be effectively switched on and off. The surface of the atom optical element is determined by the resonance condition for the spin flip in the inhomogeneous magnetic field. More than 98\% of the incident atom laser beam is reflected specularly.

abstract:
We have demonstrated a detection scheme for atom laser beams that allows for a continuous measurement of the atom density and readout of the data in real time. The atoms in the atom laser beam are transferred locally from the lower to the upper hyperfine ground state of $^{87}$Rb by coherent coupling and are subsequently detected by an absorption measurement. The detection does not affect the Bose-Einstein condensate in the magnetic trap or the atom laser beam outside the detection region.

abstract:
We use Bloch oscillations to accelerate coherently rubidium atoms. The variation of the velocity induced by this acceleration is an integer number times the recoil velocity due to the absorption of one photon. The measurement of the velocity variation is achieved using two velocity selective Raman $\pi$-pulses: the first pulse transfers atoms from the hyperfine state $5\rm S_{1/2}$, $\left\vert F=2, m_F = 0\right>$ to $5\rm S_{1/2}$, $\left\vert F=1,m_F = 0\right>$ into a narrow velocity class. After the acceleration of this selected atomic slice, we apply the second Raman pulse to bring the resonant atoms back to the initial state $5\rm S_{1/2}$, $\left\vert F=2, m_F = 0\right>$. The populations in ($F=1$ and $F=2$) are measured separately by using a one-dimensional time-of-flight technique. To plot the final velocity distribution we repeat this procedure by scanning the Raman beam frequency of the second pulse. This two $\pi$-pulses system constitutes then a velocity sensor. Any noise in the relative phase shift of the Raman beams induces an error in the measured velocity. In this paper we present a theoretical and an experimental analysis of this velocity sensor, which take into account the phase fluctuations during the Raman pulses.

abstract:
We show how one can perform arbitrary rotation of any qubit, using delayed laser pulses through nonadiabatic evolution, i.e., via transitions among the adiabatic states. We use a double-$\Lambda$ scheme and use a set of control parameters such as detuning, ratio of pulse amplitudes, time-separation of two pulses for realizing different rotations of the qubit. We also investigate the effect of different kinds of chirping, namely linear chirping and hyperbolic tangent chirping. Our work using nonadiabatic evolution adds to the flexibility in the implementation of logic gate operations and show how to achieve control of quantum systems by using different types of pulses.

abstract:
The geometric phase has recently been suggested and demonstrated experimentally in nuclear magnetic resonance (NMR) as a tool to achieve quantum computation that is resilient to certain types of errors. These analyses has been mainly concerned with cyclic quantum evolutions such as in the case where the computational basis is adiabatically taken around a loop in the parameter space of the Hamiltonian. In this thesis, quantum computation based on the cyclic geometric phase is extended to the noncyclic case. Adiabatic NMR implementations of geometric phase shift gates at any precession angle and their concomitant fault-tolerance with respect to the amplitude of the oscillating field are analysed. Nonadiabatic single-qubit geometric quantum computation is considered from the noncyclic perspective, with particular emphasis on errors in precession angles.

abstract:
While Nuclear Magnetic Resonance (NMR) techniques are unlikely to lead to a large scale quantum computer they are well suited to investigating basic phenomena and developing new techniques. Indeed it is likely that many existing NMR techniques will find uses in quantum information processing. Here I describe how the composite rotation (composite pulse) method can be used to develop quantum logic gates which are robust against systematic errors.

abstract:
I describe the use of techniques based on composite rotations to develop controlled phase gates in which the effects of weak Ising couplings are suppressed. A tailored composite phase gate is described which both suppresses weak couplings and is relatively insensitive to systematic errors in the size of strong couplings.

abstract:
Quantum mechanics permits an entity, such as an atom, to exist in a superposition of multiple states simultaneously. Quantum information processing (QIP) harnesses this profound phenomenon to manipulate information in radically new ways. A fundamental challenge in all QIP technologies is the corruption of superposition in a quantum bit (qubit) through interaction with its environment. Quantum \emph{bang-bang} control provides a solution by repeatedly applying `kicks' to a qubit, thus disrupting an environmental interaction. However, the speed and precision required for the kick operations has presented an obstacle to experimental realization. Here we demonstrate a phase gate of unprecedented speed on a nuclear spin qubit in a fullerene molecule (N@C60), and use it to bang-bang decouple the qubit from a strong environmental interaction. We can thus trap the qubit in closed cycles on the Bloch sphere, or lock it in a given state for an arbitrary period. Our procedure uses operations on a second qubit, an electron spin, in order to generate an arbitrary phase on the nuclear qubit. We anticipate the approach will be vital for QIP technologies, especially at the molecular scale where other strategies, such as electrode switching, are unfeasible.

abstract:
This thesis reports on the design and setup of a new atom trap apparatus, which is developed to confine few rubidium atoms in ultrahigh vacuum and make them available for controlled manipulations. To maintain low background pressure, atoms of a vapour cell are transferred into a cold atomic beam by laser cooling techniques, and accumulated by a magneto-optic trap (MOT) in a separate part of the vacuum system. The laser cooled atoms are then transferred into dipole traps made of focused far-off-resonant laser fields in single- or crossed-beam geometry, which are superimposed with the center of the MOT. Gaussian as well as hollow Laguerre-Gaussian (LG$_{01}$ beam profiles are used with red-detuned or blue-detuned light, respectively. Microfabricated dielectric phase objects allow efficient and robust mode conversion of Gaussian into Laguerre-Gaussian laser beams. Trap geometries can easily be changed due to the highly flexible experimental setup. The dipole trap laser beams are focused to below 10~microns at a power of several hundred milliwatts. Typical trap parameters, at a detuning of several ten nanometers from the atomic resonance, are trap depths of few millikelvin, trap frequencies near 30~kHz, trap light scattering rates of few hundred photons per atom and second, and lifetimes of several seconds. The number of dipole-trapped atoms ranges from more than ten thousand to below ten. The dipole-trapped atoms are detected either by a photon counting system with very efficient straylight discrimination, or by recapture into the MOT, which is imaged onto a sensitive photodiode and a CCD-camera. Due to the strong AC-Stark shift imposed by the high intensity trapping light, energy-selective resonant excitation and detection of the atoms is possible. The measured energy distribution is consistent with a harmonic potential shape and allows the determination of temperatures and heating rates. In first measurements, the thermal energy is found to be about 10\% of the trap depth. In a crossed beam geometry with red-detuned laser light, efficient transfer of atoms between the beams is observed. Optimum transfer occurs when the two beams cross at a radial offset, which can be qualitatively understood when the particle energy and geometrical properties of the two-beam trapping potential are considered. Numerical simulations reproduce the general features of the measured transfer efficiency vs. radial beam offset. Atoms have been radially confined in a blue-detuned hollow beam. This configuration is currently extended to a three-dimensionally confining blue-detuned dipole trap. For advanced laser cooling, state manipulation and spectroscopy, a double-diode laser system has been set up which is phase-locked with a difference frequency near 6.834~GHz to drive Raman transitions between the hyperfine-split ground states of $^{87}$Rb atoms. Dark resonances with linewidths below 100~Hz have been observed in a buffer gas loaded rubidium vapour cell.

abstract:
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

abstract:
Recent experimental developments in electronic and optical technology have made it possible to experimentally realize in space and time well localized single photon quantum- mechanical states. In these lectures we will first remind ourselves about some basic quantum mechanics and then discuss in what sense quantum-mechanical single-photon interference has been observed experimentally. A relativistic quantum-mechanical description of single-photon states will then be outlined. Within such a single-photon scheme a derivation of the Berry-phase for photons will given. In the second set of lectures we will discuss the highly idealized system of a single two-level atom interacting with a single-mode of the second quantized electro-magnetic field as e.g.~realized in terms of the micromaser system. This system possesses a variety of dynamical phase transitions parameterized by the flux of atoms and the time-of-flight of the atom within the cavity as well as other parameters of the system. These phases may be revealed to an observer outside the cavity using the long-time correlation length in the atomic beam. It is explained that some of the phase transitions are not reflected in the average excitation level of the outgoing atom, which is one of the commonly used observable. The correlation length is directly related to the leading eigenvalue of a certain probability conserving time-evolution operator, which one can study in order to elucidate the phase structure. It is found that as a function of the time-of-flight the transition from the thermal to the maser phase is characterized by a sharp peak in the correlation length. For longer times-of-flight there is a transition to a phase where the correlation length grows exponentially with the atomic flux. Finally, we present a detailed numerical and analytical treatment of the different phases and discuss the physics behind them in terms of the physical parameters at hand.

abstract:
Fifty years of developments in nuclear magnetic resonance (NMR) have resulted in an unrivaled degree of control of the dynamics of coupled two-level quantum systems. This coherent control of nuclear spin dynamics has recently been taken to a new level, motivated by the interest in quantum information processing. NMR has been the workhorse for the experimental implementation of quantum protocols, allowing exquisite control of systems up to seven qubits in size. Here, we survey and summarize a broad variety of pulse control and tomographic techniques which have been developed for and used in NMR quantum computation. Many of these will be useful in other quantum systems now being considered for implementation of quantum information processing tasks.

abstract:

abstract:
In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I define quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor's factorization algorithm and Grover's algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and finite precision. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics.

abstract:
We investigate the behavior of $^{85}$Rb atoms in the field of a sequence of frequency-chirped short laser pulses for coherent manipulation of the Rb atomic beam. The analysis is based on numerical solution of equations for the density matrix elements of the hyperfine levels of the 5S$_{1/2}$ -- 5P$_{3/2}$ transition in $^{85}$Rb atom. We analyze two different regimes of interaction including relatively short laser pulses (when the width of the pulse envelope spectrum is of order or exceeds frequency interval between the hyperfine levels resulting in effective mixing of them) and relatively long ones (when the ground hyperfine levels are resolved but the excited ones are not resolved). We show that in all regimes considered, the interaction of a frequency-chirped laser pulse with the multilevel $^{85}$Rb system is similar to the interaction with an effective two-level atom at sufficiently large peak intensities of the pulses. It allows us to perform efficient excitation of the multilevel atom by transferring populations of two hyperfine ground states to the excited ones and back to the ground states using a pair of frequency-chirped laser pulses. We present experimental results of utilizing this scheme of population transfer for coherent manipulation of an ensemble of $^{85}$Rb atoms in a magneto-optical trap.

abstract:
This paper reviews the use of adiabatic approximations in quantum optics.The general principle is explained in terms of the Landau-Zener model and the recently developed stimulated Raman adiabatic passage scheme. The features characteristic of adiabatic evolution are extracted from these examples. Our recent work on adiabatic level preparation and cavity mode transfer of excitation is presented and discussed.

abstract:
Examination of graphical displays of solutions to the time-dependent Schrödinger equation modeling a laser-excited three-level atom suggests that an energy level may be regarded as virtual when it is detuned from resonance by more than two Rabi frequencies.

abstract:
We present an apparatus for generating a multi-frequency laser field to coherently couple the $F=3$ and $F=4$ ground state of trapped cesium atoms through Raman transitions. We use a single frequency diode laser and generate sidebands by means of a 9.2~GHz electro-optic modulator. With an interferometer, we separated the sidebands and carrier, sending them to the trapped atoms in opposite directions. The Rabi oscillation of the populations of $F=3$ and $F=4$ is monitored. We find that due to destructive quantum interference of two simultaneous Raman transitions the expected Rabi frequency is reduced by a factor that is in quantitative agreement with theoretical expectations. It is demonstrated how this interference can be suppressed experimentally. Besides, we demonstrate the application of the setup for Raman spectroscopy of Zeeman sublevels and of the vibrational states of a small number of trapped atoms.

abstract:
A new method is presented for the deflection of atomic and molecular beams by momentum transfer from resonantly absorbed and reemitted electromagnetic radiation. The method, which uses adiabatic rapid passage by chirped optical pulses, is capable of high energy efficiency. It can be used for isotope separation.

abstract:
Experimental verification is presented of recent theoretical predictions of spin inversion schemes that rely on temporal interferences during the sweep. The experiments were carried out using liquid state NMR. It is shown how under non-adiabatic conditions, both complete inversion and complete non-inversion can be achieved, due to slight but easily reproducible alteration of one control parameter. The implications of this scheme for quantum computing are discussed.

abstract:
We develop an adiabatic two-mode Floquet theory to analyse multiphoton coherent population transfer in $N$-level systems by two delayed laser pulses, which is a generalization of the three-state stimulated Raman adiabatic passage (STIRAP). The main point is that, under conditions of non-crossing and adiabaticity, the outcome and feasibility of a STIRAP process can be determined by the analysis of two features: (i) the lifting of degeneracy of dressed states at the beginning and at the end of the laser pulses, and (ii) the connectivity of these degeneracy-lifted branches in the quasienergy diagram. Both features can be determined by stationnary perturbation theory in the Floquet representation. As an illustration, we study the corrections to the RWA of the (1+1) STIRAP in strong fields and for large detunings. We analyse the possible breakdown of connectivity. In strong fields, the complete transfer is achieved, but the intermediate state, unpopulated within the RWA, can become populated during the process. In the (2+1) STIRAP, we show a residual degeneracy in a four-level system, that can be lifted by additional Stark shifts. The complete transfer is achieved under conditions of connectivity.

abstract:
We present a general theory of adiabatic rapid passage (ARP) with intense, linearly chirped laser pulses. For pulses with a Gaussian profile and a fixed bandwidth, we derive a rigorous formula for the maximum temporal chirp rate that can be sustained by the pulse. A modified Landau-Zener formula displays clearly the relationships among the pulse parameters. This formula is used to derive the optimal conditions for efficient, robust population transfer. As illustrations of the theory, we present results for two- and four-level systems, and selective vibronic excitation in the I$_{2}$ molecule. We demonstrate that population transfer with chirped pulses is more robust and more selective than population transfer with transform-limited pulses.

abstract:
The deflection of Ne atoms in the metastable state $^{3}$P$_{2}$ by coherent transfer of the momenta of 4 or 8 photons is demonstrated, based on the technique developed for coherent population transfer with delayed pulses (STIRAP). After deflection the intensity profile of the isotope of mass 20 is fully seperated from that for the undeflected atoms of mass 22. It is furthermore shown, how the interplay of Larmor precession of the electronic magnetic moment and the sequential deflection in two spatially separated zones can be used to measure the magnetic field, integrated over the flight-path between the transfer zones.

abstract:
We describe the operation of a laser-cooled rubidium $^{87}$Rb frequency standard. We present a new measurement of the $^{87}$Rb hyperfine frequency with a $1.3 \times 10^{-14}$ relative accuracy, by comparison with a Cs fountain primary standard. The measured $^{87}$Rb ground-state hyperfine splitting is $\nu_{^{87}\mbox{\scriptsize Rb}}=6~834~682~610.90429(9)$~Hz. This value differs from previously published values (see ESSEN L., HOPE E. G. AND SUTCLIFFE D., Nature, \textbf{189} 1961 298; PENSELIN S., MORAN T., COHEN W. AND WINKLER G., Phys. Rev., \textbf{127} 1962 524; ARDITI M. AND CEREZ P., \emph{IEEE Trans. Instrum. Meas}., \textbf{IM-21} 1972 391) by about $2\mbox{-}3$~Hz and is 104 times more accurate. Because of the low collisional shift in $^{87}$Rb, future improvements may lead to a stability of $1 \times 10^{-14}\tau^{-1/2}$ and a relative accuracy in the $10^{-17}$ range.

abstract:
We report the realization of a hydrogen atom interferometer experiment using light as the atomic beam splitter. The wave packets of hydrogen atoms excited to the metastable 2S state are coherently split up and later recombined with the help of intense nanosecond light pulses. The pulses are generated by a novel phase-coherent source. These experiments can be seen as a step towards a precision measurement of the recoil energy of a hydrogen atom when absorbing a photon and thus of $\hbar/m_\ab{hydrogen}$.

abstract:
A simple, rigorous geometrical representation for the Schrödinger equation is developed to describe the behavior of an ensemble of two quantum-level, noninteracting systems which are under the influence of a perturbation. In this case the Schrödinger equation may be written, after a suitable transformation, in the form of the real three-dimensional vector equation $dr/dt= \omega \times r$, where the components of the vector r uniquely determine psi of a given system and the components of omega represent the perturbation. When magnetic interaction with a spin $1/2$ system is under consideration, ``r'' space reduces to physical space. By analogy the techniques developed for analyzing the magnetic resonance precession model can be adapted for use in any two-level problems. The quantum-mechanical behavior of the state of a system under various different conditions is easily visualized by simply observing how $r$ varies under the action of different types of $\omega$. Such a picture can be used to advantage in analyzing various MASER-type devices such as amplifiers and oscillators. In the two illustrative examples given (the beam-type MASER and radiation damping) the application of the picture in determining the effect of the perturbing field on the molecules is shown and its interpretation for use in the complex Maxwell's equations to determine the reaction of the molecules back on the field is given.

abstract:
Theoretical progress in the cooling of internal degrees of freedom of molecules using shaped laser pulses is reported. The emphasis is on general concepts and universal constraints. Several alternative definitions of cooling are considered, including reduction of the von Neumann entropy, $-tr(\hat{\rho} log \hat{\rho})$ and increase of the Renyi entropy, $tr(\hat{\rho}^{2})$. A distinction between intensive and extensive considerations is used to analyse the cooling process in open systems. It is shown that the Renyi entropy increase is consistent with an increase in the system phase space density and an increase in the absolute population in the ground state. The limitations on cooling processes imposed by Hamiltonian generated unitary transformations are analyzed. For a single mode system with a ground and excited electronic surfaces driven by an external field it is shown that it is impossible to increase the ground state population beyond its initial value. A numerical example based on optimal control theory demonstrates this result. For this model only intensive cooling is possible which can be classified as evaporative cooling. To overcome this constraint, a single bath degree of freedom is added to the model. This allows a heat pump mechanism in which entropy is pumped by the radiation from the primary degree of freedom to the bath mode, resulting in extensive cooling.

abstract:
Laser cooling of the vibrational motion of a molecule is investigated. The scheme is demonstrated for cooling the vibrational motion on the ground electronic surface of HBr. The radiation drives the excess energy into the excited electronic surface serving as a heat sink. Thermodynamic analysis shows that this cooling mechanism is analogous to a synchronous heat pump where the radiation supplies the power required to extract the heat out of the system. In the demonstration the flow of energy and population from one surface to the other is analyzed and compared to the power consumption from the radiation field. The analysis of the flows shows that the phase of the radiation becomes the active control parameter which promotes the transfer of one quantity and stops the transfer of another. In the cooling process the transfer of energy is promoted simultaneously with the stopping population transfer. The cooling process is defined by the entropy reduction of the ensemble. An analysis based on the second law of thermodynamics shows that the entropy reduction on the ground surface is more than compensated for by the increase in the entropy in the excited surface. It is found that the rate of cooling reduces to zero when the state of the system approaches an energy eigenstate and is therefore a generalization of the third law of thermodynamics. The cooling process is modeled numerically for the HBr molecule by a direct solution of the Liouville von Neuman equation. The density operator is expanded using a Fourier basis. The propagation is done by a polynomial approximation of the evolution operator. A study of the influence of dissipation on the cooling process concludes that the loss of phase coherence between the ground and excited surface will stop the process. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

abstract:
Frequency swept pulses are used to invert population over a broad band of transition frequencies without the need for a precise calibration of the pulse amplitude. As long as the adiabatic approximation is valid, population is adiabatically inverted at each individual transition frequency of the swept range. Even though this picture fails to be true when the adiabatic approximation breaks down, population inversion can still be achieved for an appreciable range of transition frequencies and field amplitudes. We discuss population inversion in an ensemble of two-level systems by a frequency-swept pulse, the so-called constant adiabaticity pulse, without invoking the adiabatic approximation. The equations of motion are integrated by a seminumerical method to analyze population inversion in the regime where usually the adiabatic approximation is applied. The effects of resonance offset and variable pulse amplitude are described by an average Hamiltonian expansion to discuss pulse performance beyond the validity of the adiabatic approximation. As a function of the adiabaticity parameter (reciprocal of the pulse area), the inversion bandwidth increases in a stepwise fashion due to the consecutive cancellation of average Hamiltonians. The first inversion over a \emph{finite} range of transition frequencies and pulse amplitudes is shown to occur for an adiabaticity parameter of $1/\surd 15$.

abstract:
We demonstrate and analyze a novel scheme for complete transfer of atomic or molecular population between two bound states, by means of Stark-chirped rapid adiabatic passage (SCRAP). In this two-laser technique a delayed-pulse laser-induced Stark shift sweeps the transition frequency between two coupled states twice through resonance with the frequency of the population-transferring coupling laser. The delay of the Stark-shifting pulse with respect to the pulse of the coupling-laser Rabi frequency guarantees adiabatic passage of population at one of the two resonances while the evolution is diabatic at the other. The SCRAP method can give a population-transfer efficiency approaching unity. We discuss the general requirements on the intensity and timing of the pulses that produce the Rabi frequency and, independently, the Stark shift. We particularly stress extension to a double-SCRAP technique, a coherent variant of stimulated emission pumping in the limit of strong saturation. We demonstrate the success of the SCRAP method with experiments in metastable helium, where a two-photon transition provides the Rabi frequency.

abstract:
We present a new perturbation expansion for calculating the effects of arbitrary pulse shapes in two-level systems, even when the effects are grossly nonlinear. The first two terms have simple physical interpretations. This expansion converges rapidly for all values of resonance offset with simple shapes, and for any pulse shape far from resonance. We generate very simple, symmetric, single phase pulse shapes which produce uniform inversion or polarization and which can be combined into multiple pulse sequences. We also show that pulse shape modification is superior to construction of composite pulse sequences, since such sequences must become erratic far from resonance.

abstract:
There has recently been considerable interest in the use of nuclear magnetic resonance (NMR) as a technology for the implementation of small quantum computers. These computers operate by the laws of quantum mechanics, rather than classical mechanics and can be used to implement new quantum algorithms. Here we describe how NMR in principle can be used to implement all the elements required to build quantum computers, and draw comparisons between the pulse sequences involved and those of more conventional NMR experiments.

abstract:
In this article, we review a different approach to controlling quantum systems. We show that the quantum control problem can be greatly simplified by limiting the duration of the driving force to less than one characteristic period of the system. (For an atomic-electron Rydberg wavepacket this would be the Kepler period, for example, or the vibrational period in the case of a molecule.) If the target state is a bound state of the system, then for times less than the characteristic period, the particle does not have the opportunity to reach the system's boundary and acts essentially as a classical free particle. Such a restriction on the duration of the driving field allows an analytic solution to be found, even in the nonperturbative regime, helping clarify some of the differences between the perturbative and the nonperturbative regimes of excitation. We also show that our solution is nonunique, and the quantum controller has a multiplicity of solutions to chose from.

abstract:
We discuss the dynamics of a two-level model in a strong coupling regime through the analysis of the probability amplitudes. It is seen that the theory recovers Rabi flopping also for strong coupling. Population trapping can occur and a decay constant for the excited state is also computed. Then, spontaneous emission can be made to disappear or, otherwise, the emitted photon turns out to have a frequency being an odd harmonic of the frequency of the perturbation.

abstract:
We present a generalization to three qubits of the standard Bloch sphere representation for a single qubit and of the seven-dimensional sphere representation for two qubits presented in Mosseri et al (Mosseri R and Dandoloff R 2001 J. Phys. A: Math. Gen. 34 10243). The Hilbert space of the three-qubit system is the 15-dimensional sphere S $^{15}$, which allows for a natural (last) Hopf fibration with S $^{8}$ as base and S $^{7}$ as fibre. A striking feature is, as in the case of one and two qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling three qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the three-qubit state. Conjectures on the possibility of generalizing the construction for higher qubit states are also discussed.

abstract:
We discuss a generalization of the standard Bloch sphere representation for a single qubit to two qubits, in the framework of Hopf fibrations of high-dimensional spheres by lower dimensional spheres. The single-qubit Hilbert space is the three-dimensional sphere S ~$^{3}$. The S ~$^{2}$ base space of a suitably oriented S ~$^{3}$ Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the overall qubit phase degree of freedom. For the two-qubits case, the Hilbert space is a seven-dimensional sphere S ~$^{7}$, which also allows for a Hopf fibration, with S ~$^{3}$ fibres and a S ~$^{4}$ base. The most striking result is that suitably oriented S ~$^{7}$ Hopf fibrations are entanglement sensitive. The relation with the standard Schmidt decomposition is also discussed.

abstract:
Physical constraints such as positivity endow the set of quantum states with a rich geometry if the system dimension is greater than 2. To shed some light on the complicated structure of the set of quantum states, we consider a stratification with strata given by unitary orbit manifolds, which can be identified with flag manifolds. The results are applied to study the geometry of the coherence vector for n -level quantum systems. It is shown that the unitary orbits can be naturally identified with spheres in \bb{R}^{n^2-1} only for n = 2. In higher dimensions the coherence vector only defines a non-surjective embedding into a closed ball. A detailed analysis of the three-level case is presented. Finally, a refined stratification in terms of symplectic orbits is considered.

abstract:
A disentanglement of the rotation operator introduced by Santiago and Vaidya is used to prove the composition rule for the Euler - Rodrigues parameters in a representation-independent manner. It is also shown that one of the Santiago and Vaidya disentangling coefficients embodies the Darboux transformation of differential geometry, which is equivalent to stereographic projection of the Bloch sphere onto the complex plane.

abstract:
The authors have considered the problem of adiabatic rapid passage (ARP) in a dressed atom formalism. They find that the conditions for ARP can be understood as conditions for minimising the probability of dressed atomic state transitions, and that these probabilities can be used to obtain an estimate of the degree of population reversal. Furthermore, by considering ARP in the dressed atom formalism the similarity between ARP and the Landau-Zener non-adiabatic transition problem is made quite apparent.

abstract:
Adiabatic passage with optical pulses has been used as an efficient tool for the transfer of atomic populations between Zeeman sublevels. The composition of the final state after adiabatic passage is investigated in the context of applications to atomic fountain frequency standards. Theoretical and experimental studies on the influence of the purity of the initial state on the result of the transfer are presented. Similarly, the role of the quality of polarization of the laser pulses in the process of adiabatic passage is discussed.

abstract:
We consider an adiabatic population transfer process that resembles the well-established stimulated Raman adiabatic passage. In our system, the states have nonzero angular momentums J , therefore, the coupling laser fields induce transitions between the magnetic sublevels of the states. In particular, we discuss the possibility of creating coherent superposition states in a system with coupling pattern J \equal; 0 \Leftrightarrow; 1 and 1 \Leftrightarrow; 2 . Initially, the system is in the J \equal; 0 state. We show that by, applying two delayed overlapping laser pulses, it is possible to create any final superposition state of the magnetic sublevels \vert;2, \−2\rangle; , \vert;2, 0\rangle; , \vert;2, \plus;2\rangle; . Moreover, we find that the relative phases of the applied pulses influence not only the phases of the final superposition state but also the probability amplitudes. We show that if we fix the shape and the time delay between the pulses, the final state space can be entirely covered by varying the polarizations and relative phases of the two pulses. Performing numerical simulations we find that our transfer process is nearly adiabatic for the whole parameter set.

abstract:
We study the effect of initial states on the chirping excitation of NO molecules in order to interpret recent experimental data. The results show that excitation is efficient when the probability density localizes along the polarization direction of the external field. Therefore, the thermal effect on the density distribution must be taken into account for the experimental case of 15~K. Furthermore, as the adiabatic limit is fully satisfied, the interference pattern of excited state populations disappears. This becomes a limiting factor under experimental conditions. We also find that excitation can be enhanced significantly when the pulse intensity is increased to fit the adiabatic criterion.

abstract:
A simple method for measuring the parameters of a completely or partially coherent superposition of multiple atomic states is proposed. The method, which can be applied in principle to a superposition of arbitrary energy states, is described in the case of a superposition of magnetic sublevels of a degenerate ground level. It is based upon coupling the sublevels involved in the superposition to the sublevels of a degenerate excited level by using a single elliptically polarized laser pulse. Such an interaction maps the parameters of the initial superposition onto the populations of the excited sublevels. By measuring the subsequent fluorescence from the excited sublevels for several suitably chosen laser polarizations one can deduce unambiguously the parameters of the initial superposition.

abstract:
In this letter, we present the first experimental results of climbing a vibrational ladder by Raman chirped adiabatic rapid passage. We obtain the bandwidth necessary for this experiment by utilizing pulses from a unique dual-wavelength high-intensity laser. The vibrationally excited molecules are field ionized by an intense 180~fs laser pulse, enabling us to probe the extent of the vibrational excitation of the symmetric stretch based on an enhanced ionization calculation.

abstract:
Chirped pulses allow for non-adiabatic passage between dressed states of Raman processes. It is shown that using an appropriately chirped pulse, one can control the interference between resonant and non-resonant pathways in stimulated Raman processes. A three-level system is studied in detail and application to complete ro-vibrational level inversion to high vibration quantum numbers ($v > 10$) in H$_{2}$ via RCNAP (Raman chirped non-adiabatic passage) is presented.

abstract:
Methods for, and limitations to, the generation of entangled states of trapped atomic ions are examined. As much as possible, state manipulations are described in terms of quantum logic operations since the conditional dynamics implicit in quantum logic is central to the creation of entanglement. Keeping with current interest, some experimental issues in the proposal for trapped-ion quantum computation by J.~I.~Cirac and P.~Zoller (University of Innsbruck) are discussed. Several possible decoherence mechanisms are examined and what may be the more important of these are identified. Some potential applications for entangled states of trapped-ions which lie outside the immediate realm of quantum computation are also discussed.

abstract:
We examine interaction of a single frequency-chirped laser pulse with three-level atoms that have a configuration of levels. We show that it is possible to produce complete fast and robust population transfer of all atoms of the ensemble with Doppler-broadened transition lines from one ground state into the other ground state with negligibly small temporary population of the excited state by controlling the intensity of the laser pulse and the direction and speed of the frequency chirp.

abstract:
We propose and investigate theoretically a novel scheme for transient slowing and cooling of two-level quantum systems with narrow transition linewidths by a sequence of counterpropagating, short, linearly polarized laser pulses with special frequency chirping. Both internal degrees of freedom and the motion of the center of mass of quantum systems are considered quantum mechanically. Interaction with a large number of laser pulses during the decay time permits a drastic decrease in the cooling time of such systems.

abstract:
A theoretical and experimental study is presented of the effect of stimulated radiation pressure (SRP) under the conditions of strong saturation of the resonance transition when the interaction times between the atom and the field are short (shorter than the radiative lifetime). An experimental method for studying the scattering of an atomic beam by the field of a standing light wave is described. The measured dependence of the scattering efficiency on the electric field and the detuning from resonance are reported. Experimental relationships are compared with theoretical calculations based on the quasi-classicaldescription of the motion of atoms in the field of a light wave, using an effective potential. It is shown that the efficiency of scattering of atoms by SRP forces is of the order of unity in relatively weak (100--1000- V /cm) laser fields.

abstract:
Atoms in a thermal beam can be cooled, decelerated, and stopped using the radiation pressure from a nearly resonant laser beam. Several groups have already used this laser-cooling process on an atomic sodium beam. The techniques and results of the various experimental groups are reviewed, and applications of laser-cooled atoms, in particular the possibility of confining them in electromagnetic traps, are discussed.

abstract:
We present an analytical expression for the response of a two-level atom to a frequency-modulated optically coherent pulse train. The optical beam has sinusoidal frequency modulation and is chopped to have a square-wave envelope. We assume that the laser pulses are short compared with the atomic-decay time, the pulse-repetition time, and the modulation period. With this short-pulse assumption we are able to use a method similar to Temkin s J. Opt. Soc. Am. B \textbf{10} , 830 839 (1993) and solve the optical Bloch equations in closed form.

abstract:
A significant development in computing has been the discovery that the computational power of quantum computers exceeds that of Turing machines. Central to the experimental realization of quantum information processing is the construction of fault-tolerant quantum logic gates. Their operation requires conditional quantum dynamics, in which one sub-system undergoes a coherent evolution that depends on the quantum state of another sub-system; in particular, the evolving sub-system may acquire a conditional phase shift. Although conventionally dynamic in origin, phase shifts can also be geometric. Conditional geometric (or `Berry') phases depend only on the geometry of the path executed, and are therefore resilient to certain types of errors; this suggests the possibility of an intrinsically fault-tolerant way of performing quantum gate operations. Nuclear magnetic resonance techniques have already been used to demonstrate both simple quantum information processing and geometric phase shifts. Here we combine these ideas by performing a nuclear magnetic resonance experiment in which a conditional Berry phase is implemented, demonstrating a controlled phase shift gate.

abstract:
The rotating-wave approximation (RWA) is a formalism of great utility in the description of the coherent excitation of atoms and molecules by laser light. Not only does it give results in agreement with experiment, it also provides a simple framework allowing the Hamiltonian of a system to be written down from inspection of the state-linkage diagram. Recent interest in systems with a two-photon coupling prompted an investigation of the structure of two-photon terms in RWA Hamiltonians. In carrying through the derivation, an interaction with adiabatic elimination was discovered. It is shown that adiabatic elimination must be performed before application of the RWA, else terms are dropped that ought to be retained. RWA Hamiltonians for three-state systems with one and two two-photon linkages are displayed.

abstract:
We examine the possibility of transfer of atomic coherence via stimulated Raman adiabatic passage (STIRAP). As a case study, we consider a five-level scheme in which the four ground states can be coupled to the common excited state by four different laser pulses. We calculate the fidelity for the preparation of the atomic system in a superposition of two ground states starting from a superposition of the other two ground states. We prove the feasibility of transfer of coherences via STIRAP and derive the necessary conditions for the sequence of laser pulses.

abstract:
We describe a method for creating an arbitrary coherent superposition of two atomic states in a controlled and robust way by using a sequence of three pulses in a four-state system. The proposed technique is based on the existence of two degenerate dark states (i.e.~states having no component of the excited state) and their interaction. The mixing of the dark states can be controlled by changing the relative delay of the pulses, and thus an arbitrary superposition state can be generated. It is shown that the method is robust against small variations of parameters (e.g. the area of the pulses) and is insensitive to radiative decay from the intermediate excited state. A time reversed version of the technique makes possible the determination of phase occurring in a superposition of two atomic states.

abstract:
We show that the technique of Stark-chirped rapid adiabatic passage (SCRAP), hitherto used for complete population transfer between two quantum states, offers a simple and robust method for creating coherent superpositions of states. SCRAP uses two laser pulses: a strong far off-resonant pulse modifies the transition frequency by inducing ac Stark shifts in the energies of the two states, and an appropriately offset in time, near-resonant and moderately strong pump pulse drives the population between the states via one of the induced diabatic level crossings. The populations in the created superposition are controlled by the detuning of the pump laser from the transition frequency and are insensitive to variations in the intensities of the pump and Stark lasers, as long as these are sufficiently large to allow adiabatic evolution.

abstract:
The preparation of an optically dense ensemble of three- level systems in dark states of the interaction with coherent radiation is discussed. It is shown that methods involving spontaneous emissions of photons such as Raman optical pumping fail to work beyond a critical density due to multiple scattering and trapping of these photons and the associated decay of the dark state(s). In optically thick media coherent-state preparation is only possible by entirely coherent means such as stimulated Raman adiabatic passage (STIRAP). It is shown that STIRAP is the underlying physical mechanism for electromagnetically induced transparency (EIT).

abstract:
We present a perturbative analysis of Floquet eigenstates in the context of two delayed laser processes (STIRAP) in three level systems. We show the efficiency of a systematic perturbative development which can be applied as long as no non-linear resonances occur.

abstract:
The magnetic moments of nuclei in normal matter will result in a nuclear paramagnetic polarization upon establishment of equilibrium in a constant magnetic field. It is shown that a radiofrequency field at right angles to the constant field causes a forced precession of the total polarization around the constant field with decreasing latitude as the Larmor frequency approaches adiabatically the frequency of the r-f field. Thus there results a component of the nuclear polarization at right angles to both the constant and the r-f field and it is shown that under normal laboratory conditions this component can induce observable voltages. In Section 3 we discuss this nuclear induction, considering the effect of external fields only, while in Section 4 those modifications are described which originate from internal fields and finite relaxation times.

abstract:
A treatment of the magnetic resonance is given for a particle with spin $1/$ in a constant field $H_{0}$ and under the action of an arbitrary alternating field with circular frequency $\omega$ perpendicular to $H_{0}$. A method of finding a solution, valid at any time, is given which converges the better the smaller the deviations from a rotating field or the larger $H_{0}$. It is shown that in the lowest order correction the shape of the resonance curve is unchanged but that it is shifted by a percentage amount $H_{1}^{2}/16 H_{0}^{2}$ where $H_{1}$ is the effective amplitude of the oscillating field. This also involves a correction in the values of the magnetic moments thus obtained towards smaller values which however in all practical cases is negligibly small.

abstract:
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The $n$th order function expresses the correlation of values of the fields at $2n$ different points of space and time. Certain values of these functions are measurable by means of $n$-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.

abstract:
A study has been made of the ground states of $^{85}$Rb and $^{87}$Rb with a high-resolution atomic beam magnetic resonance apparatus. The hfs interactions have been measured to give $\Delta\nu_{85} = 3~035~732~439\pm5$~cps, $\Delta\nu_{87}=6~834~682~614\pm 3$~cps. The nuclear magnetic moment $\mu_{I}$ for $^{85}$Rb has been measured independently of $\Delta\nu$ by observing the separation between components of the doublet $(3, -1)$ \leftrightarrow (2, -2)$ and $(3, -2) \leftrightarrow (2, -1)$ near the frequency minimum. This frequency is given by $\delta\nu = 2g_{I}\mu_{0}H$. The value of $\mu_{I}$ is calculated to be 1.348 206(45)~nm without diamagnetic correction or 1.352 70(25)~nm with it. The mean frequency of the doublet, $2~611~882~320\pm 20$~cps, agrees with that calculated by the Breit-Rabi equation to within one part in $10^{8}$. The value of the hfs anomaly, $_{85}\Delta_{87}$, is calculated to be 0.003 513 5(17).

abstract:
The nonadiabatic transitions which a system with angular momentum $J$ makes in a magnetic field which is rotating about an axis inclined with respect to the field are calculated. It is shown that the effects depend on the sign of the magnetic moment of the system. We therefore have an absolute method for measuring the sign and magnitude of the moment of any system. Applications to the magnetic moment of the neutron, the rotational moment of molecules, and the nuclear moment of atoms with no extra-nuclear angular momentum are discussed.

abstract:
The problem of calculating nonadiabatic transition probabilities is considered. It is shown that the general Güttinger equations are incorrect and lead to erroneous results in any case other than that of the rotating magnetic field, which he considered. The corrected equations are applied in the calculation of the transition probabilities between the various magnetic states of a field precessing with constant angular velocity.

abstract:

abstract:
The theory of coherent anti-Stokes Raman scattering (CARS) is extended to include the effect of pump fluctuations. The intensities and spectra of lines in resonant CARS are calculated to all orders in fields assuming a phase-diffusion model for waves at the two pump frequencies. The bandwidth of the two lasers enters in a much more complicated way than following a simple scaling of $T_{1}$ or $T_{2}$. Various resonances in CARS spectra due to dynamic splitting of the energy levels are discussed for a range of detunings, field intensities, and bandwidths. In contrast to the usual spectra in strong fields, the Rabi sidebands appear as dispersion-shaped structures. The laser linewidth is shown to change dramatically the CARS line shape. The case of no saturation is also treated, thus allowing for the inclusion of more general line shapes and fluctuations of the pump waves and the nonlinear susceptibility tensor $\chi^{(3)}$. Gaussian statistics of the pump field are shown to lead to enhancement factors in CARS intensity, similar to those appearing in the context of multiphoton absorption processes.

abstract:
Excitation of a quantum ladder system by frequency-swept (-chirped) laser pulses has been investigated by studying an atomic model system, the 5s-5p-5d ladder of rubidium. The population transfer is significantly enhanced when the frequency is swept such that it follows the spacing of the ladder. At intermediate intensities we observe oscillations in the population transfer to the upper level (5d), which can be attributed to interference between two routes of excitation: sequential transfer and direct two-photon excitation via a virtual intermediate state. We determine simultaneously the population transfer to the upper level (5d) and the direct three-photon ionization, the latter corresponding to a three-step ladder with the final state being in the continuum: 5s-5p-5d-$\epsilon$p/$\epsilon$f. Experiments were performed, using 100~fs pulses at $\sim 780$~nm from a self-mode-locked Ti:Al$_{2}$O$_{3}$ oscillator amplified at 10~Hz.

abstract:
We show that a set of gates that consists of all one-bit quantum gates [U(2)] and the two-bit exclusive-OR gate [that maps Boolean values ($x$,$y$) to ($x$,$x\oplus y$)] is universal in the sense that all unitary operations on arbitrarily many bits n [U($2^{n}$)] can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical and of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for $n$-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary $n$-bit unitary operations.

abstract:
We describe a class of continuously phase-modulated radiation pulses that result in coherent population inversion on resonance as well as over a large range of transition frequencies and radiation field strengths. This is a population-inversion analogy to self-induced transparency. Simulations of the inversion properties of the modulated inversion pulse (MIP) are presented. It is shown that the inversion behavior can be explained by treating the MIP as a highly efficient adiabatic sweep. Criteria for establishing adiabaticity are discussed in detail. Finally, a method is presented for generating a sequence of phase-shifted radio-frequency pulses, from the continuously modulated pulse, which can be implemented on modern NMR and coherent optical spectrometers; experimental confirmation is given.

abstract:
We present an alternative method for laser cooling of atoms close to the one-dimensional recoil limit. This method is particularly suited for atoms not accessible to conventional sub-Doppler cooling methods. It is based on the repeated selection and accumulation of slow atoms from a precooled atomic cloud and on the repeated rethermalization of the remaining atoms. The prepared ensemble is used to measure atom interferences with increased visibility.

abstract:
We introduce the nonadiabatic, or Aharonov-Anandan, geometric phase as a tool for quantum computation and show how this phase on one qubit can be monitored by a second qubit without any dynamical contribution. We also discuss how this geometric phase could be implemented with superconducting charge qubits. While the nonadiabatic geometric phase may circumvent many of the drawbacks related to the adiabatic (Berry) version of geometric gates, we show that the effect of fluctuations of the control parameters on nonadiabatic phase gates is more severe than for the standard dynamic gates. Similarly, fluctuations also affect to a greater extent quantum gates that use the Berry phase instead of the dynamic phase.

abstract:
Systematic errors in quantum operations can be the dominating source of imperfection in achieving control over quantum systems. This problem, which has been well studied in nuclear magnetic resonance, can be addressed by replacing single operations with composite sequences of pulsed operations, which cause errors to cancel by symmetry. Remarkably, this can be achieved without knowledge of the amount of error epsilon. Independent of the initial state of the system, current techniques allow the error to be reduced to $O(\epsilon^{3})$. Here, we extend the composite pulse technique to cancel errors to $O(\epsilon^{n})$, for arbitrary $n$.

abstract:
A great deal of effort has been applied to understanding population dynamics within a variety of coherent excitation schemes. The goal in such studies has been to understand the conditions necessary for efficient transfer of population from one state to another. While many theoretical treatments include the effects of natural lifetimes that are present in any given system, some neglect this important aspect when considering specific cases. Adiabatic approximations are also widely made. Additionally, it is often difficult to envision how the different parameters controlling efficient population transfer are interrelated or even which parameters are the most critical, especially when the decay lifetimes are taken into account. This work describes a numerical study of coherent excitation applied to a $^{87}$Rb ladder system where spontaneous decay rates are included, and adiabaticity is not assumed. A useful method is introduced to explore the interdependence of various excitation parameters. The efficiency of population transfer as a function of several experimentally controllable parameters is explored, and other general trends are summarized.

abstract:
We present detailed calculations of a new scheme for the adiabatic transfer of population in molecular vibrational ladders and for controlling chemical reactions: Raman chirped adiabatic passage. The scheme makes use of far-off-resonant Raman transitions with one chirped and one monochromatic laser pulse. We show that this method can be used to climb vibrational ladders in molecular systems and, with sufficiently large chirping, can vibrationally dissociate diatomic molecules.

abstract:
We describe the use of composite rotations to combat systematic errors in single-qubit quantum logic gates and discuss three families of composite rotations which can be used to correct off-resonance and pulse length errors. Although developed and described within the context of nuclear magnetic resonance quantum computing, these sequences should be applicable to any implementation of quantum computation.

abstract:
By employing an analogy with a spin-$J$ system in a constant magnetic field the authors obtain a simple soluble model for stepwise laser excitation of an $N$-level system $(N=2J+1)$, including analytic treatment of Doppler detuning and of ionization loss. The solutions are periodic and hence permit complete population inversion. A simple graphical realization of the population dynamics, a generalization of the Feynman, Vernon, and Hellwarth vector model of the two-level Bloch equation, is then described.

abstract:
We derive an infinite-order correction to the adiabatic approximation for the polarization induced in a two-level system by nearly resonant laser irradiation in the low-Rabi-frequency limit for two general classes of field envelopes. We find that Rabi oscillations at the resonant sideband frequency are a general occurrence and study the influence of the pulse shape on the form of the asymptotically decreasing amplitude of the Rabi oscillations as a function of the detuning and time constant. We go beyond the low-Rabi-frequency limit by comparing the analytic solution with numerical solutions of Schrödinger's time-dependent equation. For symmetric laser-pulse envelopes, the numerical solutions predict eigenvalues of the pulse area at which the amplitude of the Rabi oscillations is zero. The phase of the temporal oscillations changes by $\pi$ at these eigenvalues. For the special case of a hyperbolic-secant envelope, these eigenvalues correspond to the $2n\pi$-area pulses of self-induced transparency. For large-area pulses, the central region of the polarization as a function of time contains additional oscillations, the number of oscillations being determined by the number of pulse-area eigenvalues. For a propagating pulse, these oscillations are impressed on the field and amplified, thereby initiating pulse breakup (nonresonant self-induced transparency).

abstract:
We investigate the behavior of a three-level atomic system with $\Lambda$ structure of levels in the field of a sequence of short frequency-chirped bichromatic laser pulses (BLP's) in the adiabatic passage regime of interaction. We show that the efficiency of the population transfer in this atomic system is sensitive to the relative phase of the pulses forming the BLP when the atom is prepared in a coherent superposition of its two ground states initially. We propose to use this sensitivity for coherent fast and robust writing and reading of optical phase information. The initial preparation of the atom in the ``dark'' superposition state is suggested through the action of a sequence of a few frequency-chirped BLP's with duration longer than the atomic relaxation time.

abstract:
We describe a method for loading paramagnetic atoms or molecules into a magnetic trap. A $^{3}$He buffer gas is employed to thermalize atoms or molecules to a temperature of approximately 240~mK, lower than the depth of the trap. A model is described that indicates an initial loading density of approximately $10^{13}$~cm$^{-3}$. Once loading has taken place the buffer gas is removed by cryopumping. Evaporative cooling can then be applied to further lower the temperature and increase the density of the trapped sample.

abstract:
We examine the role of Berry's geometrical phase [Proc. R. Soc. London, Ser. A \emph{392}, 45 (1984)] in the coherent excitation of atoms. The dynamics of a two-level atom prepared in a superposition of its eigenstates will be modified in a nontrivial fashion by the Berry phase. We extend Berry's formalism to treat the Liouville equation for the density matrix. This extension allows us to discuss statistical mixtures and to incorporate Berry's phase within the well-known Bloch-vector description of optical resonance. Furthermore, our density-matrix formalism provides the means of investigating Berry's phase in open, dissipative systems.

abstract:
Laser-cooled rubidium atoms have been trapped in a nondissipative optical lattice with a large lattice constant. The lattice potential is generated by the ac Stark shift in an infrared standing wave near 10.6~$\mu$m. The atoms are confined in steep potential wells with a periodicity of several micrometers. By modulating the potential depth we parametrically excite the atoms and measure their vibrational frequencies.

abstract:
In an adiabatic rapid passage experiment, the Bloch vector of a two-level system (qubit) is inverted by slowly inverting an external field to which it is coupled, and along which it is initially aligned. In twisted rapid passage, the external field is allowed to twist around its initial direction with azimuthal angle $\phi(t)$ at the same time that it is inverted. For polynomial twist, $\phi(t)\sim Bt^{n}$. We show that for $n \geq 3$, multiple avoided crossings can occur during the inversion of the external field, and that these crossings give rise to strong interference effects in the qubit transition probability. The transition probability is found to be a function of the twist strength B, which can be used to control the time separation of the avoided crossings, and hence the character of the interference. Constructive and destructive interference are possible. The interference effects are a consequence of the temporal phase coherence of the wave function. The ability to vary this coherence by varying the temporal separation of the avoided crossings renders twisted rapid passage with adjustable twist strength into a temporal interferometer through which qubit transitions can be greatly enhanced or suppressed. Possible application of this interference mechanism to construction of \emph{fast fault-tolerant} quantum controlled-NOT and NOT gates is discussed.

abstract:
The spectrum of two-photon transitions presents destructive interferences (dark resonances) that can be used to achieve continuous-wave Raman sub-recoil cooling, as shown in the present paper. A simple one-dimensional model is developed and used to show the ability of the method to perform subrecoil cooling.

abstract:
We present a theoretical and experimental study of the efficiency of adiabatic transfer between the Zeeman substates of the cesium ground level, using the D$_{1}$ $F=4 \rightarrow 4\prime$ transition. In order to understand the application of the adiabatic condition to such multistate systems, we examine the separation of their energy eigenstates as a function of the number of participating states. We present a systematic investigation of the physical factors affecting the efficiency of transfer in a multistate system and we see that velocity selection plays an important role in these calculations. We use the theory to compare the suitability of adiabatic transfer with $F=1 \rightarrow 1\prime$ and $F=4 \rightarrow 4\prime$ transitions for atom optics.

abstract:
Using the stimulated force exerted by counterpropagating ? pulses from a mode-locked Ti:sapphire laser we have focused a beam of laser-cooled cesium atoms along one dimension to about 57\% of its original width in the detection zone. We determined the force profile outside and inside the overlap region of the pulses and found agreement with an earlier theoretical prediction. The scheme does not require an effective two-level system and is therefore suitable for a large variety of elements.

abstract:
We show that, by virtue of the linewidth reduction provided by counter-propagating beams, it is possible to use the optical Stark shift to sweep a two-photon resonance through the sum frequency of the applied light while simultaneously satisfying the conditions required for optical adiabatic rapid passage. Consequently, the application of a near-resonant pulse of light can result in complete population inversion of a two-photon transition.

abstract:
We have demonstrated velocity-selective coherent population trapping (VSCPT) in a two-level system created by circularly polarized light driving the 2 $^{3}$S$_{1} \rightarrow 3^{3}$P$_{2}$ transition in metastable helium. It is quite different from the usual VSCPT because there need be no consideration of selection rules, polarization, or internal atomic states. This most primitive case elicits the simple nature of VSCPT as a special kind of quantum interference, and demonstrates the presence of VSCPT in a system that has only two internal levels. It is readily observed for this transition because the ratio of the recoil frequency to the natural linewidth is 0.22, two orders of magnitude larger than for most laser cooling experiments. Our trapped state is fed by Doppler cooling, which is unusually effective here because of the large recoil, and is totally absent in previously described VSCPT experiments.

abstract:
We present quantitative measurements of the spatial density profile of Bose-Einstein condensates of sodium atoms confined in a 4-Dee magnetic bottle. The condensates are imaged in transmission with near-resonant laser light. We demonstrate that the Thomas-Fermi surface of a condensate can be determined to better than 1\%. More generally, we obtain excellent agreement with mean-field theory. We conclude that precision measurements of atomic scattering lengths and interactions between phase-separated cold atoms in a harmonic trap can be performed with high precision using this method.

abstract:
We introduce a procedure for qubit rotation, alternative to the commonly used method of Rabi oscillations of controlled pulse area. It is based on the technique of stimulated Raman adiabatic passage and, therefore, it is robust against fluctuations of experimental parameters. Furthermore, our work shows that it is, in principle, possible to perform quantum logic operations via stimulated Raman adiabatic passage. This opens up the search for a completely new class of schemes to implement logic gates.

abstract:
We introduce the ``shaped-adiabatic-passage'' method that allows one to realize a complete population transfer from a given state $| 1\rangle$ to superpositions of energy eigenstates $|\Psi\rangle = \Sigma_{k}c_{k}e^{-i\omega kt}| k\rangle$ of any desired composition. This objective can be achieved by using a ``shaped'' (dump or Stokes) laser pulse, formed by simultaneous monochromatic pulses of the same smooth profile, resonantly coupling the states $\l k\rangle$ with a single intermediate state $| 2\rangle$, followed by a (pump or anti-Stokes) pulse, coupling this intermediate state with the initial state $| 1\rangle$. The pulse shapes can be obtained from first-order perturbation theory, so the method can be used for efficient information coding in the form of superpositions of atomic Rydberg states or of molecular vibrational states.

abstract:
Rabi oscillations reflect the existence of a coherent superposition of two molecular eigenstates during the interaction of molecules with a strong, resonant electromagnetic field. This phenomenon is not restricted to two-level systems. We demonstrate its occurrence in a three-level system where the radiation induces a two-photon absorption. The intermediate level is (slightly) detuned from resonance. For a theoretical description the dressed-state picture is adopted. The two-photon excitation is performed either by using one laser, yielding a one-color two-photon process, or by using two counterpropagating, spatially overlapping laser beams, yielding a two-color two-photon process. Rabi oscillations are produced by varying the fluence, i.e., by changing either the laser power or the molecule-laser interaction time. For the one-color case the two-photon transition dipole moment for an SF$_{6}$ transition has been determined from the relation between laser fluence and number of Rabi oscillations. For the two-color cases the detuning with the intermediate level is varied by tuning the respective laser frequencies while keeping the sum frequency constant. A dramatic change in the degree of excitation as a function of the intermediate detuning has been observed.

abstract:
A semiclassical model is used to investigate the possibility of selectively exciting one of two closely spaced, uncoupled Raman transitions. The duration of the intense pump pulse that creates the Raman coherence is shorter than the vibrational period of a molecule (impulsive regime of interaction). Pulse shapes are found that provide either enhancement or suppression of particular vibrational excitations.

abstract:
A method of entangled states preparation of two-qubit systems is proposed. The method combines the techniques of coherent control by manipulation of the relative phase between the pulses and adiabatic control by using time-delayed pulse sequences. Simple analytic expressions are given for the wave function of the system as a function of phase and pulse area which show that complete control of entanglement is possible with resonant pulses. The connection between the phase of the entanglement and the control parameters is clarified and the effects of different pulse sequences are analyzed to design the optimal schemes. An off-resonant scheme is proposed that simplifies the preparation of certain entangled states.

abstract:
We study atomic-beam deflection by adiabatic passage between Zeeman ground levels via Raman transitions induced by counterpropagating $\sigma\pm$$\pm$-polarized lasers. We show that complete population transfer between the ground states can be achieved, which corresponds to the scattering of the atomic wave packet into a \emph{single} final momentum state by absorption and induced emission of laser photons. Although the lasers can be resonant, the excited state(s) are never populated during the adiabatic transfer, which suppresses the effects of spontaneous emission and preserves the coherence of the atomic wave function. This scheme has attractive features as a beam splitter and mirror for atomic interferometry.

abstract:
Composite pulse sequences designed for nuclear magnetic resonance experiments are currently being applied in many quantum information processing technologies. We present an analysis of a family of composite pulse sequences used to address systematic pulse-length errors in the execution of quantum gates. It has been demonstrated by Cummins \emph{et al}. [Phys. Rev. A \textbf{67}, 042308 (2003)] that for this family of composite pulse sequences, the fidelity of the resulting unitary operation compared with the ideal unitary operation is $1--C\epsilon^{6}$, where $\epsilon$ is the fractional error in the length of the pulse. We derive an exact expression for the sixth-order coefficient $C$ and from this deduce conditions under which this sixth-order dependence is observed. We also present pulse sequences which achieve the same fidelity.

abstract:
A selective excitation technique based on light interference is proposed to control quantum systems by frequency-chirped laser fields. Interference of two identical, delayed and phase-shifted pulses is used to modulate the laser spectrum and project it onto the time domain. By adjusting the delay and phase shift, selected transitions can be brought into the ``holes'' of the spectrum and thus remain nonexcited. The possibility to selectively manipulate or even ``shut down'' resonant transitions, making the medium transparent to the field, is shown for the Rb atom.

abstract:
We propose a scheme for creating atomic coherent superposition states via the technique of stimulated Raman adiabatic passage in a $\Lambda$-type system where the final state has twofold levels. With the application of a control field, it is found that the presence of double dark states leads to two arbitrary coherent superposition states with equal amplitude but inverse relative phases, even though the condition of multiphoton resonance is not met. In particular, two orthogonal maximal coherent superposition states are created when the control field is resonant with the transition of the twofold levels. This scheme can also be extended to manifold $\Lambda$-type systems.

abstract:
In this paper we propose and analyze various operating regimes of a quantum phase gate built on two atoms trapped in two independent dipole traps. The gate operates when the atoms are excited using a two-photon transition from the hyperfine manifold of ground states up to Rydberg states with strong dipole-dipole interaction. Experimental requirements are discussed to reach a fast (microsecond) gate operation.

abstract:
We show that the technique of Stark-chirped rapid adiabatic passage (SCRAP), hitherto used for complete population transfer between two quantum states, offers a simple and robust method for complete population transfer amongst three states in atoms and molecules. In this case SCRAP uses three laser pulses: a strong far-off-resonant pulse modifies the transition frequencies by inducing dynamic Stark shifts and thereby creating time-dependent level crossings amongst the three diabatic states, while near-resonant and moderately strong pump and Stokes pulses, appropriately offset in time, drive the population between the initial and final states via adiabatic passage. The population transfer efficiency is robust to variations in the intensities of the lasers, as long as these intensities are sufficiently large to enforce adiabatic evolution. With suitable pulse timings the population in the (possibly decaying) intermediate state can be minimized, as with stimulated Raman adiabatic passage (STIRAP). This technique applies to one-photon as well as multiphoton transitions and it is also applicable to media exhibiting inhomogeneous broadening; these features represent clear advantages over STIRAP by overcoming the inevitable dynamical Stark shifts that accompany multiphoton transitions as well as unwanted detunings, e.g., induced by Doppler shifts.

abstract:
We examine the topology of eigenenergy surfaces associated to a three-state system driven by two quasi-resonant fields. We deduce mechanisms that allow us to generate various coherent superposition of two states using an additional field, far off resonances. We report the numerical validations in mercury atoms as a model system, creating the coherent superpositions of two excited states and of two states coupled by a Raman process.

abstract:
We examine similarities and differences in three schemes that are capable of producing complete population transfer in multistate systems: generalized ? pulses, adiabatic passage by pulse chirping, and counterintuitive pulses or stimulated Raman adiabatic passage. We use the picture of adiabatic following through avoided crossings of instantaneous eigenvalues to exhibit the essential differences between the latter two procedures.

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We present a theoretical description of a scheme in which coordinated laser pulses transfer population efficiently from any initially populated state (e.g., the ground state) to a multiply excited state of an atom or molecule without producing appreciable population in intermediate states. Our analytic results for multilevel excitation transfer provide a simple yet instructive extension to the counterintuitive pulse sequence studied previously for a three-state system.

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We propose a technique for robust and efficient navigation in the Hilbert space of entangled symmetric states of a multiparticle system with externally controllable linear and nonlinear collective interactions. A linearly changing external field applied along the quantization axis creates a network of well separated level crossings in the energy diagram of the collective states.One or more transverse pulsed fields applied at the times of specific level crossings induce adiabatic passage between these states. By choosing the timing of the pulsed field appropriately, one can transfer an initial product state of all $$N$ spins into (i) any symmetric state with $n$ spin excitations and (ii) the $N$-particle analog of the Greenberger-Horne-Zeilinger state. This technique, unlike techniques using pulses of specific area, does not require precise knowledge of the number of particles and is robust against variations in the interaction parameters. We discuss potential applications in two-component Bose condensates and ion-trap systems.

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We describe a technique for creating superpositions of degenerate quantum states, such as are needed for beam splitters used in matter-wave optics, by manipulating the timing of three orthogonally polarized laser beams through which moving atoms (or molecules) pass; motion across the laser beams produces pulses in the atomic rest frame. As illustrated with representative simulations for transitions in metastable neon, a single pass through three overlapping laser beams can produce superpositions (with preselected phase) of atomic beams differing by transverse momentum corresponding to the momentum of four photons. Like the two-photon momentum transfer of the tripod linkage pattern which it extends, the method relies on controlled adiabatic time evolution in the Hilbert subspace of two degenerate dark states. It is thus a generalization to multiple dark states (and larger transfers of linear momentum to the atomic beam) of the single dark state occurring with the stimulated Raman adiabatic passage (STIRAP) technique, and therefore it is potentially insensitive to decoherence due to spontaneous emission. By extending the tripod-linkage system to more numerous degenerate states, the technique not only increases the atomic beam deflections but, as we demonstrate, allows control over the superposition phase and amplitudes. LIke other techniques based on adiabatic time evolution, the technique is robust with respect to variations of the intensity, timing, and other characteristics of the laser fields. Unlike STIRAP, the same robust partial population transfer occurs for opposite timings of the pulse sequence, as is needed for such procedures as Hadamard gates.

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We present a simple method for creating a maximal coherent superposition of $N$ states (a superposition state with equal amplitudes) in a robust way. We show that in the adiabatic limit the robustness of the population transfer out of a single state to the superposition state is equivalent to stimulated Raman adiabatic passage. As is typical for schemes based upon population trapping the method is insensitive to radiative decay from excited states.

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We describe an efficient and robust method for producing an entangled state of a two-spin system, using a sequence of two pulse pairs. We show that the mixing angle of the entangled state has a purely geometrical origin, so it is insensitive to small variations of the time-integrated pulse amplitude.

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This paper analyzes the extension of the three-state process of stimulated Raman adiabatic passage to chainwise-connected multistate systems. A necessary condition for such a process is the existence of an \emph{adiabatic-transfer state}, which connects adiabatically the initial state of the chain to its final state. Various counterintuitive pulse sequences are examined, in all of which the pulse that drives the last transition of the chain precedes the pulse driving the first transition, while the pulses driving the intermediate transitions may have different timings. The paper demonstrates some important qualitative differences and similarities between the systems with odd and even number of states. In the \emph{on-resonance} case, an adiabatic-transfer state always exists for an odd number of states while it never exists for an even number of states. In the \emph{off-resonance} case, however, the two types of systems behave in a very similar manner and the condition for existence of an adiabatic-transfer state is essentially the same. This condition, which imposes certain limitations on the laser parameters, is derived in a simple and compact form. It is also shown analytically that, besides by large detunings, the populations of the intermediate states (which are generally nonzero during the transfer) can be damped by large intermediate couplings. It is concluded that for an odd number of states, the optimal case is the on-resonance one, with equal and large intermediate couplings. For an even number of states, the optimal (off-resonance) case is when the intermediate couplings and the detunings have similar values. Various numerical examples of success and failure of multistate population transfer confirm the analytic conclusions.

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We present a method, using adiabatic passage, for creating a maximally coherent superposition in a two-state atom. The method exploits the analogy between the two-state Bloch equations and the three-state stimulated Raman adiabatic passage (STIRAP) equations, and uses two sequential but overlapping pulsed fields, a driving pump pulse and a delayed but overlapped detuning pulse. Like STIRAP, this method is robustly insensitive to small changes in pulse properties.

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We demonstrate that various properties of population transfer by delayed pulses in three-level systems on two-photon resonance can be deduced analytically and for general pulse shapes. We use the fact that the three-level system reduces to effective two-level problems at large intermediate-level detuning $\Delta$, on resonance $(\Delta = 0)$ and for completely overlapping pulses. Special attention is paid to the effect of the pulse order on the population transfer efficiency. We show that on resonance the transfer efficiency depends substantially on the pulse order, while at large $\Delta$ it does not. We also find that under some natural restrictions on the symmetry of the problem, the population of the initial level does not depend on the pulse order at any $\Delta$. Furthermore, we demonstrate that the population transfer in the three-level system can be viewed as a level-crossing problem in an equivalent two-level system not only at large $\Delta$ (which is known) but also on resonance, $\Delta = 0$. The effective on-resonance two-level problem is interesting by itself as it shows that a level crossing and adiabatic evolution do not necessarily lead to complete population inversion. As examples throughout the paper, we present several analytically solvable models.

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The paper examines the effect of irreversible dissipation from the intermediate state on the efficiency of population transfer by partially overlapping delayed pulses in three-state systems. Several general approximations to the final-state population for both the intuitive and counterintuitive pulse sequences are derived. They show that the loss of transfer efficiency is much stronger for the intuitive pulse sequence, as then the intermediate state is significantly populated during the transfer. For the counterintuitive sequence, the damping of the final-state population is found to be exponential for small decay rates and polynomial for large ones; moreover, the range of decay rates, over which the transfer efficiency remains high, is proportional to the squared pulse area. The paper also presents an analytically solvable model, involving smooth delayed pulses, as well as numerical results and analytic approximations for Gaussian pulses.

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In this Rapid Communication we present a method of extending the harmonic generation plateau to high energies with very high conversion efficiencies using moderate intensities. Preparing the initial state as a coherent superposition of the ground state and an excited state we can induce dipole transitions between the continuum and the ground state via the intermediate (excited) state responsible for the ionization. We solve numerically the Schrödinger equation and show that the spectrum contains two distinct plateaus, with cutoffs at $U_{e}+3U_{p}$ and $U_{g}+3U_{p}$. Using an essential states method we demonstrate that the plateaus are due to recombination into the excited and ground states. We briefly discuss possible experimental realizations of the scheme we present.

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The multiple-linear-time-scale method is used to construct a perturbation theory for $N$-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\l H_{1}\l \ll \hbar\bar{\omega}$, where $\l H_{1}\l$ represents the inter-action strength and $\bar{\omega}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities $(I\sim 10^{14}$~W/cm$^{2}$) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\sim 10^{10}$~W/cm$^{2}$), the approximate solution is valid for many thousands of Rabi cycles.

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It is shown that a three-level atom in the $\Lambda$ configuration with arbitrary detunings can be exactly reduced to a two-level system with an effective Raman coupling, which depends nonlinearly on the intensity of the two radiation fields. This is done by exactly evaluating the unitary transformation introduced by Alexanian and Bose [Phys. Rev. A 52, 2218 (1995)] for a three-level atom coupled to two modes of the radiation field. We obtain an exact transformed Hamiltonian in which one of the three levels is decoupled for all values of the detunings. In particular, our result is then valid for any ratios of the coupling constants to detunings, even for zero detuning, in contrast to earlier work which requires that these ratios be small. We find the the eigenvalues of the exact transformed Hamiltonian and study its population dynamics.

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We propose the use of Stark chirped rapid-adiabatic-passage method, a technique in which the energy of a target state is swept through resonance by a slowly varying dynamic Stark shift to induce complete population transfer from the ground $1s$ state to the metastable $2s$ state of the hydrogen atom. Parasitic ionization processes are strongly reduced by using a two-color excitation scheme. Our detailed numerical calculations show that under judicious choice of pulse parameters, up to 98\% of the population can be found in the $2s$ state at the end of the process.

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We present a new laser-cooling scheme based on velocity-selective optical pumping of atoms into a nonabsorbing coherent superposition of states. This method has allowed us to achieve transverse cooling of metastable $^{4}$He atoms to a temperature of 2~$\mu$K, lower than both the usual Doppler cooling limit (23~$\mu$K) and the one-photon recoil energy (4~$\mu$K). The corresponding de Broglie wavelength (1.4~$\mu$m) is larger than the atomic-transition optical wavelength.

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We have created a Bose-Einstein condensate (BEC) of $^{87}$Rb atoms directly in an optical trap. We employ a quasielectrostatic dipole force trap formed by two crossed CO$_{2}$ laser beams. Loading directly from a sub-Doppler laser-cooled cloud of atoms results in initial phase space densities of $\sim 1/200$. Evaporatively cooling through the BEC transition is achieved by lowering the power in the trapping beams over $\sim 2$~s. The resulting condensates are $F = 1$ spinors with $3.5\times 10^{4}$ atoms distributed between the $m_{F} = (-1,0,1)$ states.

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We obtain pulse-driven Rabi oscillations guided by a generalization of the rotating-wave approximation to include, in the optical-Bloch equations, two-level systems with a time-varying transition energy. We achieve this by using chirped pulses with the central frequency given by the time-varying transition energy. Using this approach, we predict Rabi oscillations in intersubband transitions in a two-subband n-type modulation-doped quantum well by taking into account the time-dependent intersubband energy-gap renormalization due to depolarization-shift effects. We obtain Rabi oscillations for $j\pi (j = 0,1,2,\ellipsis)$ pulses in the presence of dephasing.

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We report the first observation of optically trapped atoms. Sodium atoms cooled below $10^{-3}$~K in ``optical molasses'' are captured by a dipole-force optical trap created by a single, strongly focused, Gaussian laser beam tuned several hundred gigahertz below the $D_{1}$ resonance transition. We estimate that about 500 atoms are confined in a volume of about $10^{3}$~$\mu$m$^{3}$ at a density of $10^{11}-10^{12}$~cm$^{-3}$. Trap lifetimes are limited by background pressure to several seconds. The observed trapping behavior is in good quantitative agreement with theoretical expectations.

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Using nuclear magnetic resonance techniques with a solution of chloroform molecules we implement Grover's search algorithm for a system with four states. By performing a tomographic reconstruction of the density matrix during the computation good agreement is seen between theory and experiment. This provides the first complete experimental demonstration of loading an initial state into a quantum computer, performing a computation requiring fewer steps than on a classical computer, and then reading out the final state.

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We describe a quantum computer based upon the coherent manipulation of two-level atoms between discrete one-dimensional momentum states. Combinations of short laser pulses with kinetic energy dependent free phase evolution can perform the logical invert, exchange, controlled-NOT, and Hadamard operations on any qubits in the binary representation of the momentum state, as well as conditional phase inversion. These allow a binary right rotation, which halves the momentum distribution in a single coherent process. Fields for the coherent control of atomic momenta may thus be designed as quantum algorithms.

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We report on the first observation of stimulated Raman scattering from a $\Lambda$-type three-level atom, where the stimulation is realized by the vacuum field of a high-finesse optical cavity. The scheme produces one intracavity photon by means of an adiabatic passage technique based on a counterintuitive interaction sequence between pump laser and cavity field. This photon leaves the cavity through the less-reflecting mirror. The emission rate shows a characteristic dependence on the cavity and pump detuning, and the observed spectra have a subnatural linewidth. The results are in excellent agreement with numerical simulations.

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A method of phase control of entanglement in two-qubit systems is proposed. We show that by changing a relative phase of the pulses that drive the transitions in a two-qubit system with closed-loop couplings, one can control entanglement at will. The method relies on adiabatic dynamics via time-delayed pulse sequences and can be implemented with both resonant and nonresonant transitions.

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A quantum information processor is proposed that combines experimental techniques and technology successfully demonstrated either in nuclear magnetic resonance experiments or with trapped ions. An additional inhomogenenous magnetic field applied to an ion trap (i) shifts individual ionic resonances (qubits), making them distinguishable by frequency, and (ii) mediates the coupling between internal and external degrees of freedom of trapped ions. This scheme permits one to individually address and coherently manipulate ions confined in an electrodynamic trap using radiation in the radiofrequency or microwave regime.

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We demonstrate the operation of a two-bit ``controlled-NOT'' quantum logic gate, which, in conjunction with simple single-bit operations, forms a universal quantum logic gate for quantum computation. The two quantum bits are stored in the internal and external degrees of freedom of a single trapped atom, which is first laser cooled to the zero-point energy. Decoherence effects are identified for the operation, and the possibility of extending the system to more qubits appears promising.

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We determine the energies of twelve vibrational levels lying within 20~GHz of the lowest dissociation limit of $^{85}$Rb$_{2}$ with two-color photoassociation spectroscopy of ultracold $^{85}$Rb atoms. The levels lie in an energy range for which singlet and triplet states are mixed by the hyperfine interaction. We carry out a coupled channels bound state analysis of the level energies, and derive accurate values for $^{85}$Rb$_{2}$ interaction parameters. The information obtained is sufficient to allow for quantitative calculations of arbitrary Rb ultracold collision properties.

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Some early studies, seen as anticipations and examples of the geometric phase, are discussed. An early work was about quantum systems forced round a cycle by a slow circuit of parameters that govern them. This work gave rise to a number of applications such as parallel transport, coiled light, polarization cycles, degeneracy, and curved surfaces. The first of these applications considers the geometric state as anholonomic for the parallel transport of quantum states which are represented mathematically by complex unit vectors in Hilbert space. The geometric state for the coiled light application is represented by a rotation of the direction of polarized light after it has travelled along a coiled optical fiber. In a different application to optics, called polarization circles, light is travelling in a fixed direction with a slowly changing state of polarization. The geometric phase can be considered as an expression of a simple property of families of matrices that depend on parameters. Their eigenvectors are not single-valued when parallel-transported via changes of parameters and they do not return to their original values (this is called degeneracy).

abstract:
The exponentially small probability of transition between two quantum states, induced by the slow change over infinite time of an analytic hamiltonian $\hat{H}$ = H($\delta $t)$\cdot \hat{\boldsymbol{S}}$ (where $\delta $ is a small adiabatic parameter and $\hat{\boldsymbol{S}}$ is the vector spin-$\frac{1}{2}$ operator), contains an additional factor exp {$\Gamma _{\text{g}}$} of purely geometric origin (that is, independent of $\delta $ and $\hslash $). For $\Gamma _{\text{g}}$ to be non-zero, $\hat{H}$ must be complex hermitian rather than real symmetric, and the hamiltonian curve H($\tau $) must not lie in a plane through the origin nor be a helix identical (up to rigid motion) with its time reverse. An expression is given for $\Gamma _{\text{g}}$, as an integral from the real t axis to the complex time of degeneracy of the two states. Explicit examples are given. The geometric effect could be observed in experiments with spinning particles.

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The way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation, for a 2-state quantum system. By transforming to a series of superadiabatic bases, clinging ever closer to the exact evolving state, it is shown that transition histories renormalize onto a universal one, in which the amplitude grows to its final value as an error function (rather than via large oscillations as in the ordinary adiabatic basis). The time for the universal transition is of order $\surd (\hslash /\delta)$ where $\delta $ is the small adiabatic (slowness) parameter. In perturbation theory the pre-exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value ${\textstyle\frac{1}{3}}\pi $ (for the ordinary adiabatic basis) to the correct value unity. The various histories could be observed in spin experiments.

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A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters $\mathbf{R}$ in its Hamiltonian $\hat{H}(\mathrm{R})$, will acquire a geometrical phase factor $\exp {i\gamma(\mathrm{C})}$ in addition to the familiar dynamical phase factor. An explicit general formula for $\gamma$(C) is derived in terms of the spectrum and eigenstates of $\hat{H}(\mathbf{R})$ over a surface spanning C. If C lies near a degeneracy of $\hat{H}, \gamma$(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration $\gamma$(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.

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The subject of quantum computing brings together ideas from classical information theory, computer science, and quantum physics. This review aims to summarize not just quantum computing, but the whole subject of quantum information theory. Information can be identified as the most general thing which must propagate from a cause to an effect. It therefore has a fundamentally important role in the science of physics. However, the mathematical treatment of information, especially information processing, is quite recent, dating from the mid-20th century. This has meant that the full significance of information as a basic concept in physics is only now being discovered. This is especially true in quantum mechanics. The theory of quantum information and computing puts this significance on a firm footing, and has led to some profound and exciting new insights into the natural world. Among these are the use of quantum states to permit the secure transmission of classical information (quantum cryptography), the use of quantum entanglement to permit reliable transmission of quantum states (teleportation), the possibility of preserving quantum coherence in the presence of irreversible noise processes (quantum error correction), and the use of controlled quantum evolution for efficient computation (quantum computation). The common theme of all these insights is the use of quantum entanglement as a computational resource. It turns out that information theory and quantum mechanics fit together very well. In order to explain their relationship, this review begins with an introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then outlined, and the Einstein, Podolsky and Rosen (EPR) experiment described. The EPR-Bell correlations, and quantum entanglement in general, form the essential new ingredient which distinguishes quantum from classical information theory and, arguably, quantum from classical physics. Basic quantum information ideas are next outlined, including qubits and data compression, quantum gates, the `no cloning' property and teleportation. Quantum cryptography is briefly sketched. The universal quantum computer (QC) is described, based on the Church-Turing principle and a network model of computation. Algorithms for such a computer are discussed, especially those for finding the period of a function, and searching a random list. Such algorithms prove that a QC of sufficiently precise construction is not only fundamentally different from any computer which can only manipulate classical information, but can compute a small class of functions with greater efficiency. This implies that some important computational tasks are impossible for any device apart from a QC. To build a universal QC is well beyond the abilities of current technology. However, the principles of quantum information physics can be tested on smaller devices. The current experimental situation is reviewed, with emphasis on the linear ion trap, high-Q optical cavities, and nuclear magnetic resonance methods. These allow coherent control in a Hilbert space of eight dimensions (three qubits) and should be extendable up to a thousand or more dimensions (10 qubits). Among other things, these systems will allow the feasibility of quantum computing to be assessed. In fact such experiments are so difficult that it seemed likely until recently that a practically useful QC (requiring, say, 1000 qubits) was actually ruled out by considerations of experimental imprecision and the unavoidable coupling between any system and its environment. However, a further fundamental part of quantum information physics provides a solution to this impasse. This is quantum error correction (QEC). An introduction to QEC is provided. The evolution of the QC is restricted to a carefully chosen subspace of its Hilbert space. Errors are almost certain to cause a departure from this subspace. QEC provides a means to detect and undo such departures without upsetting the quantum computation. This achieves the apparently impossible, since the computation preserves quantum coherence even though during its course all the qubits in the computer will have relaxed spontaneously many times. The review concludes with an outline of the main features of quantum information physics and avenues for future research.

abstract:
The measurements of the hyperfine structure of free, naturally occurring, alkali atoms are reviewed. The experimental methods are discussed, as are the relationships between hyperfine structure data and other atomic constants.

abstract:
The authors discuss the technique of stimulated Raman adiabatic passage (STIRAP), a method of using partially overlapping pulses (from pump and Stokes lasers) to produce complete population transfer between two quantum states of an atom or molecule. The procedure relies on the initial creation of a coherence (a population-trapping state) with subsequent adiabatic evolution. The authors present the basic theory, with some extensions, and then describe examples of experimental utilization. They note some applications of the technique not only to preparation of selected states for reaction studies, but also to quantum optics and atom optics.

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The use of a rotating coordinate system to solve magnetic resonance problems is described. On a coordinate system rotating with the applied rotating magnetic field the effective field is reduced by the Larmor field appropriate to the rotational frequency. However, on such a coordinate system problems can more readily be solved since there is no time variation of the field. The solution in a stationary frame of reference is then obtained by a transformation from the rotating to the stationary frame. This procedure is equally valid in classical and in quantum-mechanical problems. The method is applied both to the molecular beam magnetic resonance method and to resonance absorption and nuclear induction experiments.

abstract:
The treatment of surfaces by plasma immersion ion implantation requires pulsed power modulators to provide negative high voltage pulses. To achieve this requirement, we have developed three basic circuit configurations in our laboratory: pulse forming network (PFN), hard-tube pulser (HT) and Blumlein line. In his paper we discuss these three types of circuit topologies.As experimental results, first we present the voltage/current characteristic waveforms of PFN and HT pulsers in the PIII treatment of different materials (aluminum, silicon and stainless steel) as well we describe the surface characterization of the materials thereof treated. And finally,we show recent high voltage tests of a high voltage Blumlein pulser (150kV/300A/1$\mu$s) to be used in surface treatments of polymers and aluminum alloys.

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The authors describe the design, construction and operating characteristics of a fast-risetime high-voltage pulser for Pockels cell driving. The risetime is less than 0.5~ns and the jitter less than 1~ns while the trigger delay time is approximately 20~ns. As an application the authors show that the use of this pulser in a self-injected Nd:YAG laser allows the production of subnanosecond duration high peak power impulses with low jitter and good stability.

abstract:
A fast high voltage switch based on ten transformer-isolated power metal-oxide semiconductor field-effect transistors in series is described. This circuit can switch 8~kV to ground with a fall time of $\approx$230~ns, and has proven to be useful for beam potential re-referencing in pulsed ion beam experiments.

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Linear time-invariant (LTI) phase noise theories provide important qualitative design insights but are limited in their quantitative predictive power. Part of the difficulty is that device noise undergoes multiple frequency translations to become oscillator phase noise. A quantitative understanding of this process requires abandoning the principle of time invariance assumed in most older theories of phase noise. Fortunately, the noise-to-phase transfer function of oscillators is still linear, despite the existence of the nonlinearities necessary for amplitude stabilization. In addition to providing a quantitative reconciliation between theory and measurement, the time-varying phase noise model presented in this tutorial identifies the importance of symmetry in suppressing the upconversion of $1/f$ noise into close-in phase noise, and provides an explicit appreciation of cyclostationary effects and AM-PM conversion. These insights allow a reinterpretation of why the Colpitts oscillator exhibits good performance, and suggest new oscillator topologies. Tuned LC and ring oscillator circuit examples are presented to reinforce the theoretical considerations developed. Simulation issues and the accommodation of amplitude noise are considered in appendixes

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Controlling quantum dynamical processes is the key to practical applications of quantum physics, for example in quantum information science. The manipulation of light-matter interactions at the single-atom and single-photon level can be achieved in cavity quantum electrodynamics, in particular in the regime of strong coupling in which atom and cavity form a single entity. In the optical domain, this requires a single atom at rest inside a microcavity. Here we show that an orthogonal arrangement of a cooling laser, trapping laser and cavity vacuum gives rise to a unique combination of friction forces that act along all three directions. This combination of cooling forces is applied to catch and cool a single atom in a high-finesse cavity. The high cooling efficiency leads to a low temperature and an average single-atom trapping time of 17~s, during which the strongly coupled atom can be observed continuously.

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Simultaneous two-dimensional trapping of neutral dipolar molecules in low- and high-field seeking states is analyzed. A trapping potential of the order of 20~mK can be produced for molecules such as ND$_{3}$ with time-dependent electric fields. The analysis is in agreement with an experiment where slow molecules with longitudinal velocities of the order of 20~m/s are guided between four 50~cm long rods driven by an alternating electric potential at a frequency of a few kHz.

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The creation of a photon-atom bound state was first envisaged for the case of an atom in a long-lived excited state inside a high-quality microwave cavity. In practice, however, light forces in the microwave domain are insufficient to support an atom against gravity. Although optical photons can provide forces of the required magnitude, atomic decay rates and cavity losses are larger too, and so the atom-cavity system must be continually excited by an external laser. Such an approach also permits continuous observation of the atom's position, by monitoring the light transmitted through the cavity. The dual role of photons in this system distinguishes it from other single-atom experiments such as those using magneto-optical traps, ion traps or a far-off-resonance optical trap. Here we report high-finesse optical cavity experiments in which the change in transmission induced by a single slow atom approaching the cavity triggers an external feedback switch which traps the atom in a light field containing about one photon on average. The oscillatory motion of the trapped atom induces oscillations in the transmitted light intensity; we attribute periodic structure in intensity-correlation-function data to `long-distance' flights of the atom between different anti-nodes of the standing-wave in the cavity. The system should facilitate investigations of the dynamics of single quantum objects and may find future applications in quantum information processing

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When $N$ driven atoms emit in phase into a high-$Q$ cavity mode, the intracavity field generated by collective scattering interferes destructively with the pump driving the atoms. Hence atomic fluorescence is suppressed and cavity loss becomes the dominant decay channel for the whole ensemble. Microscopically, 3D light-intensity minima are formed in the vicinity of the atoms that prevent atomic excitation and form a regular lattice. The effect gets more pronounced for large atom numbers, when the sum of the atomic decay rates exceeds the rate of cavity losses and one would expect the opposite behavior. These results provide new insight into recent experiments on collective atomic dynamics in cavities.

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The coupling of individual atoms to a high-finesse optical cavity is precisely controlled and adjusted using a standing-wave dipole-force trap, a challenge for strong atom-cavity coupling. Ultracold Rubidium atoms are first loaded into potential minima of the dipole trap in the center of the cavity. Then we use the trap as a conveyor belt that we set into motion perpendicular to the cavity axis. This allows us to repetitively move atoms out of and back into the cavity mode with a repositioning precision of 135~nm. This makes it possible to either selectively address one atom of a string of atoms by the cavity, or to simultaneously couple two precisely separated atoms to a higher mode of the cavity.

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We present a semiclassical theory that can describe the centre-of-mass motion of a neutral atom in a strongly coupled cavity field. We show that a cavity-assisted cooling mechanism can lead to efficient loading of an atom into a wavelength size optical microtrap.

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We present a systematic semiclassical model for the simulation of the dynamics of a single two-level atom strongly coupled to a driven high-finesse optical cavity. From the Fokker-Planck equation of the combined atom-field Wigner function we derive stochastic differential equations for the atomic motion and the cavity field. The corresponding noise sources exhibit strong correlations between the atomic momentum fluctuations and the noise in the phase quadrature of the cavity field. The model provides an effective tool to investigate localization effects as well as cooling and trapping times. In addition, we can continuously study the transition from a few-photon quantum field to the classical limit of a large coherent field amplitude.

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We extend an earlier semiclassical model to describe the dissipative motion of $N$ atoms coupled to $M$ modes inside a coherently driven high-finesse cavity. The description includes momentum diffusion via spontaneous emission and cavity decay. Simple analytical formulas for the steady-state temperature and the cooling-time for a single atom are derived and show surprisingly good agreement with direct stochastic simulations of the semiclassical equations for $N$ atoms with properly scaled parameters. A thorough comparison with standard free-space Doppler cooling is performed and yields a lower temperature and a cooling-time enhancement by a factor of $M$ times the square of the ratio of the atom-field coupling constant to the cavity decay rate. Finally, it is shown that laser cooling with negligible spontaneous emission should indeed be possible, especially for relatively light particles in a strongly coupled field configuration.

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We present a time-dependent quantum calculation of the scattering of a few-photon pulse on a single atom. The photon wave packet is assumed to propagate in a transversely strongly confined geometry, which ensures strong atom-light coupling and allows a quasi-one-dimensional treatment. The amplitude and phase of the transmitted, reflected, and transversely scattered part of the wave packet strongly depend on the pulse length (bandwidth) and energy. For a transverse mode size of the order of $\lambda^{2}$, we find nonlinear behavior for a few photons already, or even for a single photon. In a second step we study the collision of two such wave packets at the atomic site and find striking differences between the Fock state and coherent state wave packets of the same photon number.

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Rubidium atoms are accumulated in a conventional MOT and transferred into a dipole trap produced by a 1W Ti:Sapphire laser which is tuned to 810~nm and focused to a waist of about 15~$\mu$m. We achieve trapping frequencies above the recoil frequency, such that the Lamb-Dicke regime of cooling can be reached, and we observe trap lifetimes of several seconds. We report about the experimental setup and the characterisation of the trap. Measurements of the trap loss show a strong wavelengthdependent quadratic contribution due to inelastic collisions, including photoassociation. Measured trap frequencies agree with the expectation. We also present the first steps towards a blue-detuned trap consisting of two crossed hollow laser beams.

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We investigate the optical detection of single atoms held in a microscopic atom trap close to a surface. Laser light is guided by optical fibers or optical microstructures via the atom to a photodetector. Our results suggest that with present-day technology microcavities can be built around the atom with sufficiently high finesse to permit unambiguous detection of a single atom in the trap with 10~$\mu$s of integration. We compare resonant and nonresonant detection schemes and discuss the requirements for detecting an atom without causing it to undergo spontaneous emission.

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A new method to track the motion of a single particle in the field of a high-finesse optical resonator is analyzed. It exploits sets of near-degenerate higher-order Gaussian cavity modes, whose symmetry is broken by the position dependent phase shifts induced by the particle. Observation of the spatial intensity distribution outside the cavity allows direct determination of the particle's position. This is demonstrated by numerically generating a realistic atomic trajectory using a semiclassical simulation and comparing it to the reconstructed path. The path reconstruction itself requires no knowledge about the forces on the particle. Experimental realization strategies are discussed.

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Following almost a century of debate on possible ``independent of measurement'' elements of reality, or ``induced'' elements of reality - originally invoked as an ad-hoc collapse postulate, we propose a novel line of interference experiments which may be able to examine the regime of induced elements of reality. At the basis of the proposed experiment, lies the hypothesis that models of ``induced'' elements of reality should exhibit symmetry breaking within quantum evolution. The described \emph{symmetry experiment} is thus aimed at being able to detect and resolve spatial symmetry breaking signatures. The proposed experiment stands at the edge of present day technological abilities and will be, so we believe, realizable in the near future.

abstract:
We review the modifications and implications of the effect of light forces on atoms when the field is enclosed in an optical resonator of high finesse. The systems considered range from a single atom strongly coupled to a single mode of a high-$Q$ microcavity to a large ensemble of atoms in a highly degenerate quasi-confocal resonator. We set up general models that allow us to obtain analytic expressions for the optical potential, friction, and diffusion. In the bad-cavity limit the modified cooling properties can be attributed to the spectral modifications of light absorption and spontaneous emission in a form of generalized and enhanced Doppler cooling. For the strong coupling regime in a good cavity, we identify the dynamical coupling between the light field intensity and the atomic motion as the central mechanism underlying the cavity-induced cooling. The dynamical cavity cooling, which does not rely on spontaneous emission, can be enhanced by multimode cavity geometries because of the effect of coherent photon redistribution between different modes. The model is then generalized to include several distinct frequencies to account for more general trap geometries. Finally we show that the field-induced buildup of correlations between the motion of different particles plays a central role in the scaling behavior of the system. Depending on the geometry and parameters, its effect ranges from strong destructive interference, slowing down the cooling process, to self-organized crystallization, implying atomic self-trapping and faster cooling to lower temperatures by cooperative coherent scattering.

abstract:
A single atom emitting single photons is a fundamental source of light. But the characteristics of this light depend strongly on the environment of the atom. For example, if an atom is placed between two mirrors, both the total rate and the spectral composition of the spontaneous emission can be modified. Such effects have been observed using various systems: molecules deposited on mirrors, dye molecules in an optical cavity, an atom beam traversing a two-mirror optical resonator, single atoms traversing a microwave cavity and a single trapped electron. A related and equally fundamental phenomenon is the optical interaction between two atoms of the same kind when their separation is comparable to their emission wavelength. In this situation, light emitted by one atom may be reabsorbed by the other, leading to cooperative processes in the emission. Here we observe these phenomena with high visibility by using one or two single atom(s), a collimating lens and a mirror, and by recording the individual photons scattered by the atom(s). Our experiments highlight the intimate connection between one-atom and two-atom effects, and allow their continuous observation using the same apparatus

abstract:
Cavity-mediated cooling of the center-of-mass motion of a transversally, coherently pumped atom along the axis of a high-$Q$ cavity is studied. The internal dynamics of the atomic dipole strongly coupled to the cavity field is treated by a non-perturbative quantum mechanical model, while the effect of the cavity on the external motion is described classically in terms of the analytically obtained linear friction and diffusion coefficients. Efficient cavity-induced damping is found which leads to steady-state temperatures well-below the Doppler limit. We reveal a mathematical symmetry between the results here and for a similar system where, instead of the atom, the cavity field is pumped. The cooling process is strongly enhanced in a degenerate multimode cavity. Both the temperature and the number of scattered photons during the characteristic cooling time exhibits a significant reduction with increasing number of modes involved in the dynamics. The residual number of spontaneous emissions in a cooling time for large mode degeneracy can reach and even drop below the limit of a single photon.

abstract:
Trapped and laser-cooled ions are increasingly used for a variety of modern high-precision experiments, for frequency standard applications, and for quantum information processing. Therefore laser cooling of trapped ions is reviewed, the current state of the art is reported, and several new cooling techniques are outlined. The principles of ion trapping and the basic concepts of laser cooling for trapped atoms are introduced. The underlying physical mechanisms are presented, and basic experiments are briefly sketched. Particular attention is paid to recent progress by elucidating several milestone experiments. In addition, a number of special cooling techniques pertaining to trapped ions are reviewed; open questions and future research lines are indicated.

abstract:
We report on trapping a single neutral atom in the standing-wave light field of a high-finesse optical cavity containing one photon on average, a single-photon optical trap, or SPOT for short. This trap has the novel feature that the light field is also used to observe the atom in real time. The oscillatory motion of the trapped atom induces well-resolved oscillations of the light intensity. Periodic structure is visible in the fourth-order intensity correlation function, attributed to long-distance flights of the atom along the standing wave. The finite duration of those flights provides evidence for cavity-mediated cooling of atoms. We discuss the various mechanisms determining the trapping time and compare the results with a quantum-jump Monte Carlo simulation to interpret the observed signals.

abstract:
A laser cooling method for trapped atoms is presented which achieves ground state cooling by exploiting quantum interference in a $\lambda$-shaped arrangement of atomic levels driven by two lasers. The scheme is technically simpler than existing methods of sideband cooling, yet it can be significantly more efficient, in particular when several motional modes are involved. We have applied the method to a single Calcium ion in a Paul trap, coupling a single laser to the Zeeman structure of its S$_{1/2} \rarrow \mbox{P}_{1/2}$ dipole transition at 397~nm. We have achieved more than 90\% ground-state occupation probability. By suitably tuning the laser parameters, we obtain simultaneous ground-state cooling of two oscillator modes. This is of great practical importance for the implementation of quantum logic schemes with trapped ions.

abstract:
Atoms coupled to optical fields strongly confined in two spatial dimensions, as in solid-state microstructures, can experience large Lamb shifts due to a spectrally strongly asymmetric mode density. We use the generic example of a quasi-one-dimensional waveguide structure driven close to cutoff frequency of a new transverse branch of propagating modes. We analytically find strong shifts of the atomic resonance frequency due to the modified vacuum, which can be an order of magnitude larger than the atomic linewidth. At the same time one gets significantly enhanced scattering of the guided light by the atom, which could be used as a tool to investigate these effects or to build non-destructive single-atom detectors.

abstract:
An excited-state atom whose emitted light is backreflected by a distant mirror can experience trapping forces, because the presence of the mirror modifies both the electromagnetic vacuum field and the atom's own radiation reaction field. We demonstrate this mechanical action using a single trapped barium ion. We observe the trapping conditions to be notably altered when the distant mirror is translated across an optical wavelength. The well-localized barium ion enables the spatial dependence of the forces to be measured explicitly. The experiment has implications for quantum information processing and may be regarded as the most elementary optical tweezers.

abstract:
We demonstrate feedback on the motion of a single neutral atom trapped in the light field of a high-finesse cavity. Information on the atomic motion is obtained from the transmittance of the cavity. This is used to implement a feedback loop in analog electronics that influences the atom's motion by controlling the optical dipole force exerted by the same light that is used to observe the atom. In spite of intrinsic limitations, the time the atom stays within the cavity could be extended by almost 30\% beyond that of a comparable constant-intensity dipole trap.

abstract:
Based on a real-time measurement of the motion of a single ion in a Paul trap, we demonstrate its electromechanical cooling below the Doppler limit by homodyne feedback control (cold damping). The feedback cooling results are well described by a model based on a quantum mechanical master equation.

abstract:
We give a snapshot of the rapidly developing field of ultracold polar molecules and walk the reader through the papers appearing in this Topical Issue.

abstract:
Ultracold molecules are associated from an atomic Bose-Einstein condensate by ramping a magnetic field across a Feshbach resonance. The reverse ramp dissociates the molecules. The kinetic energy released in the dissociation process is used to measure the widths of four Feshbach resonances in $^{87}$Rb. This method to determine the width works remarkably well for narrow resonances even in the presence of significant magnetic-field noise. In addition, a quasimonoenergetic atomic wave is created by jumping the magnetic field across the Feshbach resonance.

abstract:
The quadrupole S$_{1/2}$ -- D$_{5/2}$ optical transition of a single trapped Ca$^{+}$ ion, well suited for encoding a quantum bit of information, is coherently coupled to the standing wave field of a high finesse cavity. The coupling is verified by observing the ion's response to both spatial and temporal variations of the intracavity field. We also achieve deterministic coupling of the cavity mode to the ion's vibrational state by selectively exciting vibrational state-changing transitions and by controlling the position of the ion in the standing wave field with nanometer precision.

abstract:
We study the motion of two atoms trapped at distant positions in the field of a driven standing wave high-$Q$ optical resonator. Even without any direct atom-atom interaction the atoms are coupled through their position dependent influence on the intracavity field. For sufficiently good trapping and low cavity losses the atomic motion becomes significantly correlated and the two particles oscillate in their wells preferentially with a 90~degrees relative phase shift. The onset of correlations seriously limits cavity cooling efficiency, raising the achievable temperature to the Doppler limit. The physical origin of the correlation can be traced back to a cavity mediated cross-friction, i.e.~a friction force on one particle depending on the velocity of the second particle. Choosing appropriate operating conditions allows for engineering these long range correlations. In addition this cross-friction effect can provide a basis for sympathetic cooling of distant trapped clouds.

abstract:
We derive an equation for the cooling dynamics of the quantum motion of an atom trapped by an external potential inside an optical resonator. This equation has broad validity and allows us to identify novel regimes where the motion can be efficiently cooled to the potential ground state. Our result shows that the motion is critically affected by quantum correlations induced by the mechanical coupling with the resonator, which may lead to selective suppression of certain transitions for the appropriate parameters regimes, thereby increasing the cooling efficiency.

abstract:
We discuss a scheme to cool, trap and manipulate an ensemble of polarizable particles moving in the field of a multimode optical cavity via the correlated dynamics of the field and the particle motion. Using a large detuning between the atoms and the field, spontaneous emission plays a negligible role in the dynamics and the cooling scheme only requires a sufficiently large optical dipole moment. Increasing the particle number slows down the cooling process but it can be accelerated using an increasing number of field modes with higher pump amplitudes. For the special case of a two mode ring cavity and assuming small deviations of the particle positions from the potential minima, the frequencies and damping rates of the vibrational excitation modes can be explicitly calculated. We find a rapid damping of the centre-of-mass motion and relatively slow damping rates for the relative particle oscillations. These predictions agree quite well with a quantum treatment of the atomic motion as used for the excitations of a non-interacting Bose gas (at $T = 0$) inside the cavity field. Due to the position-dependent mode coupling, the cooling process in a multimode configuration in general happens much faster than for a standing wave geometry. These analytical results are confirmed by $N$-particle simulations of the semiclassical equations and show even enhanced damping due to the anharmonicity of the full potential.

abstract:
We investigate the external and internal dynamics of a two-level atom in a standing wave cavity. In the strong coupling regime, where the atom field coupling $g$ dominates the atomic and cavity decay rates ($\Gamma$,$\kappa$), a cooling mechanism entirely different from free-space Doppler cooling appears. Under suitable operating conditions, the cavity dynamics induces a Sisyphus type cooling, which is the dominant contribution to the total friction force acting on a moving atom. Simple equations describing the key properties of this effect are derived from a completely classical picture and confirmed by a semiclassical approach. The model is investigated in the bad and good cavity limits, and analytic expressions for the friction coefficient and the momentum diffusion for slow atoms are derived. Using a continued fractions expansion the cooling force for arbitrary velocities is evaluated numerically. The result is used to calculate the equilibrium temperatures of the atom of the order of $kT\approx \hbar\kappa$, which can be much lower than the free-space Doppler limit and agree well to those obtained by quantum wave-function simulations.

abstract:
A continuously operated electrostatic trap for polar molecules is demonstrated. The trap has a volume of $\approx 0.6$~cm$^{3}$ and holds molecules with a positive Stark shift. With deuterated ammonia from a quadrupole velocity filter, a trap density of $\approx 10^{8}$~cm$^{-3}$ is achieved with an average lifetime of 130~ms and a motional temperature of $\approx 300$~mK. The trap offers good starting conditions for high-precision measurements, and can be used as a first stage in cooling schemes for molecules and as a ``reaction vessel'' in cold chemistry.

abstract:
We solve the quantum Langevin equations of motion for $N$ point-like two-level atoms moving in an externally pumped cavity field. In the limit of the low saturation of the atoms, we obtain analytical expressions for the dipole force, the velocity-dependent force and the momentum diffusion coefficient for each atom in the presence of other atoms. The expressions show that in general the forces on each atom depend on the position and the velocity of all the other atoms.

abstract:
We study N coherently driven two-level atoms strongly coupled to a high finesse optical resonator. Destructive interference between the cavity-mode field and the pump field, coupled via the atoms, induces enhanced scattering into the cavity mode, which is orders of magnitude larger than the fluorescence signal, while 3D intensity minima in the vicinity of each atom are created, minimizing atomic excitation. Even for a cavity linewidth larger than the atomic linewidth, this phenomenon is established for growing atom number $N$, giving a coherent amplitude of the cavity field which is independent of $N$. The magnitude of this interference effect depends on the relative atomic positions and is strongest for a regular lattice of atoms. These results provide new insight into recent experiments on collective atomic dynamics in cavities.

abstract:
We study the quantum dynamics of $N$ coherently driven two-level atoms coupled to an optical resonator. In the strong coupling regime the cavity field generated by atomic scattering interferes destructively with the pump on the atoms. This suppresses atomic excitation and even for strong driving fields prevents atomic saturation, while the stationary intracavity field amplitude is almost independent of the atom number. The magnitude of the interference effect depends on the detuning between laser and cavity field and on the relative atomic positions and is strongest for a wavelength spaced lattice of atoms placed at the antinodes of the cavity mode. In this case three dimensional intensity minima are created in the vicinity of each atom. In this regime spontaneous emission is suppressed and the dominant loss channel is cavity decay. Even for a cavity linewidth larger than the atomic natural width, one regains strong interference through the cooperative action of a sufficiently large number of atoms. These results give a new key to understand recent experiments on collective cavity cooling and may allow to implement fast tailored atom-atom interactions as well as nonperturbative particle detection with very small energy transfer.

abstract:
All conventional methods to laser-cool atoms rely on repeated cycles of optical pumping and spontaneous emission of a photon by the atom. Spontaneous emission in a random direction provides the dissipative mechanism required to remove entropy from the atom. However, alternative cooling methods have been proposed for a single atom strongly coupled to a high-finesse cavity; the role of spontaneous emission is replaced by the escape of a photon from the cavity. Application of such cooling schemes would improve the performance of atom-cavity systems for quantum information processing. Furthermore, as cavity cooling does not rely on spontaneous emission, it can be applied to systems that cannot be laser-cooled by conventional methods; these include molecules (which do not have a closed transition) and collective excitations of Bose condensates, which are destroyed by randomly directed recoil kicks. Here we demonstrate cavity cooling of single rubidium atoms stored in an intracavity dipole trap. The cooling mechanism results in extended storage times and improved localization of atoms. We estimate that the observed cooling rate is at least five times larger than that produced by free-space cooling methods, for comparable excitation of the atom.

abstract:
We present a time dependent quantum calculation of the scattering of a few-photon pulse on a single atom. The photon wave packet is assumed to propagate in a transversely strongly confined geometry, which ensures strong atom-light coupling and allows a quasi 1D treatment. The amplitude and phase of the transmitted, reflected and transversely scattered part of the wave packet strongly depend on the pulse length (bandwidth) and energy. For a transverse mode size of the order of $\lambda^2$, we find nonlinear behavior for a few photons already, or even for a single photon. In a second step we study the collision of two such wave packets at the atomic site and find striking differences between Fock state and coherent state wave packets of the same photon number.

abstract:
An atom in a high-$Q$ cavity, which is coherently driven at the frequency of a cavity mode, exhibits strong suppression of fluorescence when the atomic decay rate exceeds the cavity linewidth. This effect is due to destructive interference of cavity and pump field, such that at the atomic position the total field intensity has a local minimum. For atomic ensembles the magnitude of the interference effect grows with atom number and depends on the relative atomic positions. It is strongest for a wavelength spaced array of atoms placed at the antinodes of the cavity mode. This suppresses fluorescence and enhanced collective scattering into the cavity mode. We analyze the mechanical forces in the regime where the interference condition is fulfilled. We show that the atomic pattern is mechanically stable whenever the driving frequency is red detuned with respect to the cavity frequency, irrespective of the atomic transition frequency. Hence atomic selforganization, as predicted in [6] can also occur in the parameter regime where superradiant scattering is suppressed by collective interference.

abstract:
A novel method of ground-state laser cooling of trapped atoms utilizes the absorption profile of a three- (or multi-) level system that is tailored by a quantum interference. With cooling rates comparable to conventional sideband cooling, lower final temperatures may be achieved. The method was experimentally implemented to cool a single Ca$^{+}$ ion to its vibrational ground state. Since a broad band of vibrational frequencies can be cooled simultaneously, the technique will be particularly useful for the cooling of larger ion strings, thereby being of great practical importance for initializing a quantum register based on trapped ions. We also discuss its application to different level schemes and for ground-state cooling of neutral atoms trapped by a far-detuned standing wave laser field.

abstract:
We study the zero-temperature dynamics of Bose-Einstein condensates in driven high-quality optical cavities in the limit of large atom-field detuning in a one-dimensional model. We calculate the stationary ground state and the spectrum of coupled atom and field mode excitations for standing-wave cavities as well as for travelling-wave cavities. Finite cavity response times lead to damping or controlled amplification of these excitations. Analytic solutions in the Lamb-Dicke expansion are in good agreement with numerical results for the full problem and show that oscillation frequencies and the corresponding damping rates are qualitatively different for the two cases.

abstract:
In this paper, we propose a machine learning approach to title extraction from general documents. By general documents, we mean documents that can belong to any one of a number of specific genres, including presentations, book chapters, technical papers, brochures, reports, and letters. Previously, methods have been proposed mainly for title extraction from research papers. It has not been clear whether it could be possible to conduct automatic title extraction from general documents. As a case study, we consider extraction from Office including Word and PowerPoint. In our approach, we annotate titles in sample documents (for Word and PowerPoint respectively) and take them as training data, train machine learning models, and perform title extraction using the trained models. Our method is unique in that we mainly utilize formatting information such as font size as features in the models. It turns out that the use of formatting information can lead to quite accurate extraction from general documents. Precision and recall for title extraction from Word are 0.810 and 0.837 respectively, and precision and recall for title extraction from PowerPoint are 0.875 and 0.895 respectively in an experiment on intranet data. Other important new findings in this work include that we can train models in one domain and apply them to other domains, and more surprisingly we can even train models in one language and apply them to other languages. Moreover, we can significantly improve search ranking results in document retrieval by using the extracted titles.

abstract:

abstract:
I study theoretically the responsivity of optical modulators. For the case of a linear response, by using perturbation theory I find an upper bound imposed on the responsivity. For the case of a two-mode modulator I find a lower bound imposed on the optical path required for achieving full modulation when the maximum birefringence strength is given.

abstract:
We present a scalable scheme to design optimized soft pulses and pulse sequences for coherent control of interacting quantum many-body systems. The scheme is based on the cluster expansion and the time-dependent perturbation theory implemented numerically. This approach offers a dramatic advantage in numerical efficiency, and it is also more convenient than the commonly used Magnus expansion, especially when dealing with higher-order terms. We illustrate the scheme by designing 2nd-order self-refocusing $\pi$-pulses and a 6th-order 8-pulse refocusing sequence for a chain of qubits with nearest-neighbor couplings. We also discuss the performance of soft-pulse refocusing sequences in suppressing decoherence due to low-frequency environment.

abstract:
We analyse the effect of the intermediate-level detuning on the efficiency of the stimulated Raman adiabatic passage process in three-level systems. We present an exactly solvable analytic model, involving smooth delayed pulses, as well as numerical results and an analytic approximation for Gaussian pulses. Both types of pulses demonstrate that for fixed pulse strengths, the transfer efficiency is adversely affected as the intermediate-level detuning increases because of the deteriorating adiabaticity and eventually vanishes for very large detuning. It is shown that the width of the detuning range, over which the transfer efficiency remains high, is proportional to the squared pulse area. A simple analysis suggests that this feature should be generally valid for any smooth delayed pulses.

abstract:
The Rabi oscillations of a two-level atom illuminated by a laser on resonance with the atomic transition may be suppressed by the atomic motion through averaging or filtering mechanisms. The optical analogs of these velocity effects are described. The two atomic levels correspond in the optical analogy to orthogonal polarizations of light and the Rabi oscillations to polarization oscillations in a medium which is optically active, naturally or due to a magnetic field. In the latter case, the two orthogonal polarizations could be selected by choosing the orientation of the magnetic field, and one of them be filtered out. It is argued that the time-dependent optical polarization oscillations or their suppression are observable with current technology.

abstract:
We present an analytically solvable model for stimulated Raman adiabatic passage (STIRAP) processes in three-level systems. It involves realistic separated pulses which vanish at infinite times, whose pulse areas are finite and whose envelopes are smooth functions of time. The solution is obtained using the correspondence between three-level systems on resonance and two-level systems. The analytic model confirms the breakdown of the Dykhne-Davis-Pechukas exponential dependence of the non-adiabatic transition probability on the adiabaticity parameter found numerically recently.

abstract:

abstract:
A pair of diode laser frequency standards stabilized to 778~nm two-photon rubidium transitions have been developed and characterized. Their sensitivity to vapour pressure, modulation depth and optical power have been investigated, along with their relative stability and day-to-day reproducibility. The Allan deviation of the standards locked to the $5^{2}S$_{1/2} (F_{g} = 3) - 5^{2}$D$_{5/2} (F_{e} = 5)$ transition of $^{85}$Rb is $9.3 \times 10^{-13}\tau^{-1/2} for 1~s $< \tau <$ 100~s, reaching a floor of $1 \times 10^{-13} at 100~s. The long-term repeatability of an individual system locked to this line is 1.2~kHz, corresponding to 3.1~parts in $10^{12}$. The system-to-system reproducibility is limited by cell impurities to 4.9~kHz and 4.5~kHz, respectively, for this line and the $5^{ }$S$_{1/2} (F_{g} = 2) - 5^{2}$D$_{5/2} (F_{e} = 4)$ transition in $^{87}$Rb. The mean observed frequencies for these transitions, averaged for the two systems and corrected for light shift and second-order relativistic Doppler effect, are 385 285 142 375.1 (4.9)~kHz and 385 284 566 374.2 (4.5)~kHz, respectively.

abstract:
In adiabatic rapid passage, the Bloch vector of a qubit is inverted by slowly inverting an external field to which it is coupled, and along which it is initially aligned. In \emph{non-adiabatic} twisted rapid passage, the external field is allowed to twist around its initial direction with azimuthal angle $\phi(t)$~at the same time that it is non-adiabatically inverted. For polynomial twist, $\phi(t)\sim Bt^{2}$. We show that, for $n \geq 3$, multiple qubit resonances can occur during a single inversion of the external field, producing strong interference effects in the qubit transition probability. The character of the interference is controllable through variation of the twist strength B. Both constructive and destructive interference are possible, allowing qubit transitions to be greatly enhanced or suppressed. Experimental confirmation of these controllable interference effects has already occurred. Application of this interference mechanism to the construction of fast fault-tolerant quantum controlled-NOT and NOT gates is discussed.

abstract:
The fundamentals of Doppler-free saturated absorption spectroscopy were explored for a rubidium sample. Two probe laser beams from a 780-nm diode laser were directed through the Rb cell, and one of them was crossed by a pump beam in the opposite direction. Because of the changes in the ground-state population caused by the pump beam, subtraction of the intensity signals from the two probe beams resulted in a non-Doppler-broadened signal. The three Doppler-free peaks corresponded to three of the ground state levels of Rb: the $F = 2$ energy level of $^{87}$Rb, the $F = 3$ energy level of $^{85}$Rb, and the $F = 2$ energy level of $^{85}$Rb. These three peaks were separated by 0.0575~ms and 0.145~ms in our data. The theoretical separations of 1.2~GHz and 3.1~GHz both give the same calibration: 21~GHz/ms. This means there is a 24\% error in our rough estimate of the calibration at 16~GHz/ms. Using the 21~GHz/ms calibration, the FWHM of the peaks were 525~MHz, 630~MHz, and~735 MHz, which are all above the Doppler limit of 500~MHz. Errors in this experiment included mistakes during the calibration, the improper use of AC coupling, and the wide bandwidth of the diode laser.

abstract:
A zero-area pulse produces no excitation in a two-state quantum system on exact resonance. We show analytically and numerically that off resonance the transition probability may acquire significant values, and as the Rabi frequency increases, it approaches steadily unity, a feature reminiscent of adiabatic passage. The nature of the transition is not, however, of adiabatic nature. We show that complete population transfer occurs as a result of a quasiresonant transition between the adiabatic states due to a $\delta$-function-type interaction with an area of $\pi$.

abstract:
The phase of an electronic wave function is shown to play an important role in coherent control experiments. By using a pulse shaping system with a femtosecond laser, we explore the phase relationships among resonant and off-resonant Raman transitions in Li$_{2}$ by measuring the phases of the resulting wave packets, or quantum beats. Specific pixels in a liquid-crystal spatial light modulator are used to isolate the resonant and off-resonant portions of the Raman transitions in Li$_{2}$. The off-resonant Raman transitions have an approximately 90$^{\circ}$ phase shift with respect to the resonant Raman transition, and there is an approximately 180$^{\circ}$ phase shift between the blue-detuned and the red-detuned off-resonant Raman transitions. Calculations using second-order time-dependent perturbation theory for the electronic transitions agree with the experimental results for the laser pulse intensities used here. Interferences between the off-resonant Raman transitions as a function of detuning are used to demonstrate coherent control of the Raman quantum wave packet.

abstract:
We have investigated the production of continuous coherent blue laser light by frequency upconversion using a five-level system in rubidium vapour. Two low-power lasers, at 780 and 776 nm, induce strong atomic coherence in the 5S-5P-5D states (see figure). The atoms decay to the 6P excited state, from which stimulated emission produces a coherent blue (420 nm) beam.

abstract:
We report new techniques for driving high-fidelity stimulated Raman transitions in trapped-ion qubits. An electro-optic modulator induces sidebands on an optical source, and interference between the sidebands allows coherent Rabi transitions to be efficiently driven between hyperfine ground states separated by 14.53~GHz in a single trapped $^{111}$Cd$^{+}$ ion.

abstract:
We discuss the dependence of the population transfer probability in the three-level $\Lambda$-scheme using stimulated Raman adiabatic passage on detuning from the two-photon resonance. For a large decay rate of the intermediate level we found that the two-photon lineshape is close to Gaussian with linewidth inversely proportional to the square root of the decay rate. The shift of the maximum of the two-photon line from exact two-photon resonance has been investigated.

abstract:
A new molecular beam resonance method using separated oscillating fields at the incident and emergent ends of the homogeneous field region is theoretically investigated in this paper. An expression is obtained for the quantum mechanical transition probability of a system between two states when the system is subjected to such separated oscillating fields. This is numerically averaged over the molecular velocity distribution and provides the theoretical shape of the resonance curves. It is found that resonances with such a technique have a theoretical half-width only 0.6 as great as those by conventional molecular beam resonance methods. In addition to producing sharper resonance minima, the new method has its resonances much less broadened by inhomogeneities of the fixed field, it makes possible resonance experiments in regions into which an oscillating field cannot be introduced, and it is more convenient and effective with short wave-length radiation.

abstract:
The use of continuous wave cavity ring-down spectroscopy (cw CRDS) with near infra-red diode lasers is demonstrated for quantitative detection of trace levels of unsaturated volatile organic compounds (VOCs) at wavelengths that avoid overlapping absorptions by more abundant atmospheric constituents such as H$_{2}$O and CO$_{2}$. The current detection limit, with due allowance for pressure broadening by 1~atmosphere of air, is 6~parts per billion by volume (ppbv) for ethyne at an air wavelength of 1519.670~nm, and is sufficient for direct atmospheric detection of this molecule in many urban environments. Detection limits for alkenes are inferior, and, without incorporating the consequences of pressure broadening, include 78~ppbv for ethene and 900~ppbv for 1,3-butadiene. While the CRDS detection method offers several advantages over established gas chromatographic techniques for monitoring of small VOCs such as ethyne, it appears to be less well suited to study of larger organic compounds. Methods are discussed for improving the instrument to reach the sensitivities required to monitor the various alkenes and other C--H containing molecules in the troposphere.

abstract:
Implementation of quantum logical gates for multilevel system is demonstrated through decoherence control under the quantum adiabatic method using simple phase modulated laser pulses. We make use of selective population inversion and Hamiltonian evolution with time to achieve such goals robustly instead of the standard unitary transformation language.

abstract:
The problem of achieving population inversion adiabatically in an $N$-level system using one or more laser fields whose detunings and/or amplitudes are continuously varied is studied analytically and numerically. The SU($N$) coherence vector picture is shown to suggest unexpected inversion procedures and also to give a generalized interpretation of adiabatic following. It is shown that the ($N^{2}-1$)-dimensional SU($N$) space contains an ($N-1$)-dimensional steady-state subspace $\Gamma(t)$ whose orthonormal basis vectors $\Gamma \rightarrow _{1}$,\ldots,$\Gamma \rightarrow _{N-1} are given explicitly in terms of the Hamiltonian matrix elements. The motion of the system can be interpreted as a ``generalized precession'' of S$\rightarrow$ about $\Gamma$. Multilevel adiabatic following occurs when the angle $\Chi(t)$ between the coherence vector S$\rightarrow$ and its projection onto $\Gamma$ is very small. The multiple dimension of $\Gamma$ is shown to provide a variety of paths for adiabatic inversion. The adiabatic solution is obtained by solving $N-1$ simple equations for the directional cosines of S$$\rightarrow$ on $\Gamma \rightarrow _{i}$. The adiabatic solution and time scale and the state taken up by the atomic variable are discussed analytically and numerically for a three-level system.

abstract:
The travelling periodic dipole potential of an accelerated optical lattice, created by high-intensity short-pulse lasers, can accelerate polarizable atoms and molecules that are initially at room temperature to hyperthermal velocities (10-100 km/s). We study the acceleration of trapped and untrapped ensembles of particles and show that a significant fraction (30\%) of uniformly distributed particles can be accelerated to high velocities over micron-size distances, within nanosecond and subnanosecond time scales.

abstract:
We propose to measure the electron's permanent electric dipole moment (EDM) using cesium atoms trapped in a sparsely populated, trichromatic, far blue-detuned three-dimensional (3D) optical lattice. In the proposed configuration, the atoms can be strongly localized near the nodes of the light field and isolated from each other, leading to a strong suppression of the detrimental effects of atom-atom and atom-field interactions. Three linearly polarized standing waves with different frequencies create an effectively linearly polarized 3D optical lattice and lead to a strong reduction of the tensor light shift, which remains a potential source of systematic error. Other systematics concerning external field instability and gradients and higher-order polarizabilities are discussed. Furthermore, auxiliary atoms can be loaded into the same lattices as effective ``comagnetometers'' to monitor various systematic effects, including magnetic-field fluctuations and imperfect electric-field reversal. We estimate that a sensitivity 100 times higher than the current upper bound for the electron's EDM of $4 \times 10^{-27}$~$e$cm can be achieved with the proposed technique.

abstract:
We launch Cs atoms using a moving three-dimensional (3D) optical lattice. Atoms are initially spin polarized and cooled to the ground state of the optical potential using 3D Raman sideband cooling and then accelerated in the lattice to velocities of up to 3~m/s. Subsequent adiabatic lowering of the potential releases the atoms from the lattice. Three-dimensional kinetic temperatures as low as 200~nK were achieved. The observed temperature of the launch is independent of the acceleration and final velocity of the atoms. In an alternative approach, we first accelerate the atoms to velocities of up to 5~m/s using moving molasses and then cool them with the lattice in the comoving frame of the atoms to temperatures as low as 150~nK.

abstract:
We trap $10^{7}$ cesium atoms in a far red detuned 1D optical lattice. With degenerate Raman sideband cooling we achieve a vibrational ground state population of 80\% for the steep trapping direction. Collisional coupling enables us to cool the spin-polarized gas in 3D without loss of atoms to a peak phase space density of 1/180 at a mean temperature of 2.8~$\mu$K and a density of $1.4 \times 10^{13}$~cm$^{-3}$.

abstract:
We observe and study the dynamic formation of cold Cs$_{2}$ molecules near collision Feshbach resonances in a cold cesium sample. The resonance Iinewidth is as low as $E/h = 5$~kHz, or equivalently, $10^{-11}$~ eV. We suggest that few-atom, interaction effects can be studied in a 3D optical lattice where several atoms can be confined and isolated in an optical cell, which allows exquisite control of the atomic density and the interaction cross section.

abstract:
We demonstrate a simple, general purpose method to cool neutral atoms. A sample containing $3 \times 10^{8}$ cesium atoms prepared in a magneto-optical trap is cooled and simultaneously spin polarized in 10~ms at a density of $1.1 \times 10^{11}$~cm$^{-3}$ to a phase space density $n \lambda_{dB}^{3} = 1/500, which is almost 3 orders of magnitude higher than attainable in free space with optical molasses. The technique is based on 3D degenerate Raman sideband cooling in optical lattices and remains efficient even at densities where the mean lattice site occupation is close to unity.

abstract:
We have simultaneously spin polarized and subrecoil cooled sodium atoms trapped in a blue detuned optical dipole trap. Temperatures of 1.5~$\mu$K and less have been achieved for atoms which have been polarized into any of three $F=1$ ground-state magnetic sublevels of sodium. The final phase-space density is comparable to the best achieved by an all-optical cooling technique.

abstract:
Laser-cooling of atoms and atom-trapping are finding increasing application in many areas of science. One important use of laser-cooled atoms is in atom interferometers. In these devices, an atom is placed into a superposition of two or more spatially separated atomic states; these states are each described by a quantum-mechanical phase term, which will interfere with one another if they are brought back together at a later time. Atom interferometers have been shown to be very precise inertial sensors for acceleration,, rotation and for the measurement of the fine structure constant. Here we use an atom interferometer based on a fountain of laser-cooled atoms to measure g, the acceleration of gravity. Through detailed investigation and elimination of systematic effects that may affect the accuracy ofthe measurement, we achieve an absolute uncertainty of $\Delta g/g$ \approx 3 \times 10^{-9}, representing a million-fold increase in absoluteaccuracy compared with previous atom-interferometer experiments. We also compare our measurement with the value of g obtained at the same laboratory site using a Michelson interferometer gravimeter (a modern equivalent of Galileo's `leaning tower' experiment in Pisa). We show that the macroscopic glass object used in this instrument falls with the same acceleration, to within 7 parts in $10^{9}$, as a quantum-mechanical caesium atom.

abstract:
Reflection of thermal atoms by a pulsed standing wave with a duration in the nanosecond range is studied. The momentum distribution of the reflected atoms is determined by calculations based on the adiabatic atom-photon interactions. It is shown that with a proper choice of the field intensity and the pulse duration the standing-wave pattern functions as a row of independent atom mirrors. At an optimum choice of the parameter values, the fraction of the elastically reflected atoms is more than 20\%. Furthermore, we show that the pulsed standing-wave mirror can be used to manipulate their final momentum distribution. When using laser pulses with an intensity of several tens of MW/cm$^{2}$, tens of thousands of atoms can be reflected by a single laser pulse.

abstract:
This review describes the methods of trapping cold atoms in electromagnetic fields and in the combined electromagnetic and gravity fields. We discuss first the basic types of the dipole radiation forces used for cooling and trapping atoms in the laser fields. We outline next the fundamentals of the laser cooling of atoms and classify the temperature limits for basic laser cooling processes. The main body of the review is devoted to discussion of atom traps based on the dipole radiation forces, dipole magnetic forces, combined dipole radiation-magnetic forces, and the forces combined of the dipole radiation-magnetic and gravity forces. Physical fundamentals of atom traps operating as waveguides and cavities for cold atoms are also considered. The review ends with the applications of cold and trapped atoms in atomic, molecular and optical physics.

abstract:
Laser-cooling techniques offer novel opportunities in the generation and manipulation of entangled quantum states. Atoms trapped in infrared mesoscopic optical lattices are presented as attractive candidates for both the production of highly entangled states and quantum logic experiments. Recent experimental work on the realization of such a lattice, based on atoms trapped in the antinodes of an infrared standing wave near 10.6~$\mu$m, is reviewed. This culminates in the individual lattice sites being resolved with an optical microscope. Finally, with the aim of achieving greater control over molecules, a matter-wave interferometer is proposed, in which the probability for absorption of photon momenta depends on the particle velocity, rather than on the absolute laser detuning from a particular optical transition. This scheme has prospects for the laser cooling of both atoms and molecules

abstract:
This paper presents an analytic study of the properties of a three-state system coupled to two external laser fields on two-photon resonance. The intermediate state is generally off single-photon resonance and can decay irreversibly out of the system. The two fields are assumed to have the same pulse-shaped time dependence but may have different strengths. Various general properties for arbitrary pulse shapes, detunings and decay rates are derived and illustrated with an exactly solvable analytic model. They include the conditions for complete population transfer and depletion, complete population loss or no loss, various limitations on the values of the populations, the effect of overdamping at large decay rates and others.

abstract:
We show that the lineshape and size of the detected signal in ultra-high-resolution stimulated Raman spectroscopy computed from a general expression of the modulated signal are in good agreement with the experimental results. Using the theoretical expression of the signal, we analyze the resolving power and the sensitivity of our Raman spectrometer. We show that a second harmonic detection in high-frequency transfer modulation generates a lineshape without modulation broadening. A 2.6-kHz-wide resonance has been obtained with iodine, mainly limited by both the transit time and collisions

abstract:
We report the first observation of translationally cold ($\sim 90$~$\mu$K) Rb$_{2}$ molecules. They are produced in a magneto-optical trap in their triplet ground state. The detection is performed by selective mass spectroscopy after two-photon ionization into Rb$^{2+}$, resonantly enhanced through the intermediate $a ^{3}\Sigma_{u}^{+} \rightarrow 2 ^{3}\Pi_{g}$ molecular band. The two rubidium isotopes present very different types of behavior that are interpreted in terms of their respective collisional properties.

abstract:
We report the first observation of optically trapped cold neutral molecules. Cesium dimers in the elect ronic ground state are produced directly in a magneto-optical trap and transferred to a dipole trap formed at the focus of a CO$_{2}$ laser beam ( $\lambda = 10.6$~$\mu$m). These neutral molecules were detected using photoionization and time-of-flight spectroscopy. Initial experiments indicate a cold molecule trap lifetime on the order of half a second.

abstract:
We study the dynamics of ultracold atoms in a periodic optical potential submitted to a constant external force. Bloch oscillations in the fundamental and first excited bands of the potential are observed. In addition to a solid-state analysis we give a quantum-optics interpretation of this effect in terms of photon exchanges between the atoms and the laser waves. The dynamics of Bloch oscillations are equivalent to a sequence of adiabatic rapid passages between momentum states and can be described using the dressed-atom approach. We demonstrate efficient and dissipation-free acceleration of atoms by coherent transfer of a large number of photon momenta ($\approx 100$). This technique produces atomic beams with a very small longitudinal velocity spread that may find applications in atom optics and precision measurements.

abstract:
We have observed transfer of momentum and ground state population in laser-cooled cesium by adiabatic following of a slowly evolving light field. In this new technique for mechanical manipulation of atoms, spontaneous emission is suppressed since the atoms evolve in a ``dark'' state that follows the light field. This means that the phase coherence of the atom is preserved so that this technique is useful in the realization of coherent atomic beam splitters and mirrors. Our experimental results are in good agreement with optical Bloch equation calculations.

abstract:
Sodium atoms have been cooled in two and three dimensions with stimulated Raman transitions. Atoms in 2D have been cooled to $V_{\mbox{rms}} = 1.2 \hbar k/M$, corresponding to an effective temperature of $T_{\mbox{eff}} = 1.7$~$\mu$K while atoms in 3D have been cooled to $V_{\mbox{rms}} = 2.3 \hbar k/M$, or $T_{\mbox{eff}} = 4.3$~$\mu$K.

abstract:
We report the confinement of large clouds of ultra-cold $^{85}$Rb atoms in a standing-wave dipole trap formed by the two counter-propagating modes of a high-$Q$ ring-cavity. Studying the properties of this trap we demonstrate loading of higher-order transverse cavity modes and excite recoil-induced resonances.

abstract:
We report on an experimental study of recoil-induced resonances as a method of velocimetry for cold atomic samples. We present a refined experimental method that greatly improves the sensitivity of the measurement over previous experiments. Using frequency-modulation (FM) spectroscopy techniques we achieve a sensitivity that approaches the shot noise limit. In addition, we present a novel approach to deriving the line shape of the observed signal, based on the concept of quantum transport and tunnelling in motional Bloch bands.

abstract:
Radiation at 308~nm has been obtained by frequency doubling the output of a commercial diode laser cooled to 165~K. A single pass through a crystal of LiIO$_{3}$ converted 1~mW of 616~nm radiation to 50~pW of UV, and this was used to detect the OH radical in absorption in a flow tube. Possible extensions of the method for detection of OH in the atmosphere are discussed.

abstract:
Enhancement cavities using off-axis spherical mirrors allow the elliptical Gaussian beams from semiconductor diode lasers to be matched directly to the optimum beam profile for frequency conversion in a nonlinear crystal. We give a general recipe for the design of such cavities.

abstract:
We analyze the resonant cavity enhancement of a hollow ``optical bottle beam'' for the dipole-force trapping of dark-field-seeking species. We first improve upon the basic bottle beam by adding further Laguerre-Gaussian components to deepen the confining potential. Each of these components itself corresponds to a superposition of transverse cavity modes, which are then enhanced simultaneously in a confocal cavity to produce a deep optical trap needing only a modest incident power. The response of the trapping field to displacement of the cavity mirrors offers an unusual form of mechanical amplifier in which the Gouy phase shift produces an optical Vernier scale between the Laguerre-Gaussian beam components.

abstract:
Dipole force traps for dark-field seeking states of atoms and molecules require regions of low intensity that are completely surrounded by a brighter optical field. Confocal cavities allow the resonant enhancement of these interesting transverse mode superpositions, and put deep far-off-resonance traps within reach of low-power diode lasers. In this paper, we show how an array of dark-field rings may be created simply using a single Gaussian beam. Such a geometry lends itself to the study of toroidal Bose-Einstein condensates.

abstract:
Using the stimulated force exerted by counterpropagating $pi$~pulses from a mode-locked Ti:sapphire laser we have focused a beam of laser-cooled cesium atoms along one dimension to about 57\% of its original width in the detection zone. We determined the force profile outside and inside the overlap region of the pulses and found agreement with an earlier theoretical prediction. The scheme does not require an effective two-level system and is therefore suitable for a large variety of elements.

abstract:
The force resulting from a position-dependent sequence of interactions with short counter-propagating $\pi$-pulses of laser radiation can propel atoms towards the small region where the pulses overlap. The optical trap thus formed may be combined with Doppler-cooling laser beams.

abstract:
We report experimental results at 1.057~GHz that demonstrate the ability of a planar left-handed lens, with a relative refractive index of $-1$, to form images that overcome the diffraction limit. The left-handed lens is a planar slab consisting of a grid of printed metallic strips over a ground plane, loaded with series capacitors ($C$) and shunt inductors ($L$). The measured half-power beamwidth of the point-source image formed by the left-handed lens is 0.21 effective wavelengths, which is significantly narrower than that of the diffraction-limited image corresponding to 0.36 wavelengths.

abstract:
A simple scheme for two-dimensional velocity selection is proposed and theoretically investigated. The scheme uses only two laser beams exciting stimulated Raman transition in a three-level atom. Parameters of velocity-selective $\pi$-pulse as well as velocity distribution of selected atoms are calculated. It is shown that the distribution width can be lower than the recoil velocity for the narrow transitions used. A feasible scheme for the accumulation of selected atoms in magnetic traps is discussed.

abstract:
Ultra-cold atoms trapped in an optical dipole trap and prepared in a coherent superposition of their hyperfine ground states, decohere as they interact with their environment. We demonstrate than the loss in coherence in an``echo'' experiment, which is caused by mechanisms such as Rayleigh scattering, can be suppressed by the use of a new pulse sequence. We also show that the coherence time is then limited by mixing to other vibrational levels in the trap and by the finite lifetime of the internal quantum states of the atoms.

abstract:
We study the dynamics of cold rubidium atoms interacting with a periodically pulsed standing wave which has a polarization gradient. Due to optical pumping, the atomic population is trapped in ground states which periodically evolve to dark states each time the standing wave is switched on. These recurring dark states are coherent superpositions of several momentum states. In the experiment we measure a comb-like momentum distribution with a peak width of about $0.3 \hbar k$. A high contrast is observed when the pulse frequency of the standing wave coincides with the recurrence frequency of the dark states.

abstract:
We experimentally study the motion of atoms interacting with a periodically pulsed near resonant standing wave with a polarization gradient. The atoms are cooled to a comblike momentum distribution. The peaks have widths of $\approx 0.3 \hbar k$ and a spacing which is an integer multiple of the recoil momentum $\hbar k$. The atomic population is accumulated in ground states which periodically evolve to dark states each time the standing wave is switched on. These light pulse synchronously recurring dark states exist for discrete pulse frequencies, even if no stationary dark states are present.

abstract:
This paper summarizes some important achievements of quantum information processing with trapped ions or neutral atoms. In particular, we describe the storage of information and realization of two-qubit gates with ions, as well as the creation of entanglement and quantum simulation with cold atoms in optical lattices.

abstract:
We discuss population transfer from the ground to the excited state of a three-level atom or molecule in a Raman configuration exposed to pump and Stokes lasers. The middle level is exactly ``eliminated'' by means of a unitary transformation of the Hamiltonian. The dynamics is assumed to obey the conditions of adiabatic passage. We find two distinct ways in which the transfer of population can be carried out.

abstract:
We have studied the absorption of a weak probe beam through cold rubidium atoms in a magneto-optic trap. The absorption spectrum shows two peaks with the smaller peak having linewidth as small as 28\% of the natural linewidth. The modification happens because the laser beams used for trapping also drive the atoms coherently between the ground and excited states. This creates ``dressed'' states whose energies are shifted depending on the strength of the drive. Linewidth narrowing occurs due to quantum coherence between the dressed states. The separation of the states increases with laser intensity and detuning, as expected from this model.

abstract:
This paper reviews some of our recent results in nonlinear atom optics. In addition to nonlinear wave-mixing between matter waves, we also discuss the dynamical interplay between optical and matter waves. This new paradigm, which is now within experimental reach, has the potential to impact a number of fields of physics, including the manipulation and applications of atomic coherence, and the preparation of quantum entanglement between microscopic and macroscopic systems. Possible applications include quantum information processing, matter-wave holography, and nanofabrication.

abstract:
Detailed theoretical analysis and numerical simulation indicate that nearly complete electronic population inversion of molecular systems can be achieved with intense positively chirped broadband laser pulses. To provide a simple physical picture, a two-level model is used to examine the condition for the so-called $\pi$-pulses and a four-level model is designed to demonstrate for molecular systems the correlation between the sign of the chirp and the excited state population. The proposed molecular $\pi$-pulse is the combined result of vibrational coherence in the femtosecond regime and adiabatic inversion in the picosecond regime. Numerical results for a displaced oscillator, for LiH and for I$_{2}$, show that the proposed molecular $\pi$-pulse scheme is robust with respect to changes in field parameters such as the linear positive chirp rate, field intensity, bandwidth, and carrier frequency, and is stable with respect to thermal and condensed phase conditions including molecular rotation, rovibronic coupling, and electronic dephasing.

abstract:
Theoretical analysis leads us to the intriguing conclusion that nearly complete electronic population inversion of molecules can be achieved with intense positively chirped broadband laser pulses, as a combined result of vibrational coherence and adiabatic inversion. Strong field quantum calculations demonstrate inversion probabilities of up to 99\%. The results are robust with respect to changes in light field parameters as well as to thermal and condensed phase conditions, are supported by experimental evidence, and have many potential applications in chemistry, biochemistry, biology, and physics.

abstract:
We review some basic techniques for laser-induced adiabatic population transfer between discrete quantum states in atoms and molecules.

abstract:
We show that the concept of $\pi$-pulses can be extended to multilevel systems. Generalized $\pi$-pulses selectively excite a target state via a mechanism that is closely related to the familiar excitation dynamics in a two-level system. The corresponding generalized area theorem does not refer to the ``area under the pulse envelope,'' but to an integral over a difference of instantaneous quasienergies. Nevertheless, there are the same possibilities of pulse shaping for the generalized pulses as for their two-level counterparts. A semiclassical interpretation of the resonance condition leads to an analytical approximation to the relevant quasienergies, and shows that the excitation mechanism is universal.

abstract:
Deflection of two-level atoms by a pulsed standing wave with a pulse duration of a few nanoseconds is studied by using the density matrix formalism. The effect of the limited coherence time of the pulsed laser field on the momentum distribution of the deflected atoms is investigated. In particular, we determine the coherence time at which the deflection by a pulsed standing wave differs significantly from the zero-relaxation case.

abstract:
The application of counterpropagating, short, chirped laser pulses for the deflection and splitting of an atomic beam is investigated. A simple model is proposed to describe the case in which the pulses induce population transfer in the adiabatic-passage regime and in which the effects of spontaneous emission are negligible during the action of a single laser pulse. Spontaneous emission between the pulses is not neglected, however, and it is shown that it has a significant effect on the evolution of the average transversal velocity of the atoms, as well as the dispersion of the transversal velocity.

abstract:
We discuss the quantum effect of the optical control field such as a $\pi$-pulse and a $\pi/2$-pulse on the decoherence of a two-level system to be controlled in the experiments such as atom interference and quantum computation. A general solution for the Jaynes-Cummings model is obtained for this purpose, and the reduced density operator of the controlled two-level atom after the interaction is calculated. Examining typical initial quantum states for the control field such as number state, large-amplitude coherent state and smallamplitude coherent state, we can compare these states in the scope of atomic coherence controllability. It became clear that a number state field mostly destroys the purity of the two-level atom, and that a large-amplitude coherent state is ideal (except that large energy is required to control the atomic state). Use of a coherent state with considerably small photon number will cause decoherence, but not too much.

abstract:
We analyze and compare three different strategies, all aimed at controlling and eventually halting decoherence. The first strategy hinges upon the quantum Zeno effect, the second makes use of frequent unitary interruptions (``bang-bang'' pulses and their generalization, quantum dynamical decoupling), and the third uses a strong, continuous coupling. Decoherence is shown to be suppressed only if the frequency $N$ of the measurements or pulses is large enough or if the coupling $K$ is sufficiently strong. Otherwise, if $N$ or $K$ is large, but not extremely large, all these control procedures accelerate decoherence. We investigate the problem in a general setting and then consider some practical examples, relevant for quantum computation.

abstract:
We investigate the behavior of $^{85}$Rb atoms in the field of a sequence of frequency-chirped short laser pulses. The analysis is based on a numerical solution of equations for the probability amplitudes of the hyperfine levels of the $5S_{1/2} - 5P_{3/2}$ transition in the $^{85}$Rb atom and the dressed-states analysis. We analyze different regimes of interaction, including relatively short laser pulses (when the width of the pulse envelope spectrum is of the order of or exceeds the frequency interval between the hyperfine levels resulting in effective mixing of them) and relatively long ones (when the ground hyperfine levels are resolved but the excited ones are not resolved). In the latter case dependence of the population transfer efficiency on the initial coherence of the ground states is analyzed. The case of long laser pulses when all working hyperfine levels are resolved is also discussed using numerical simulations and a dressed-states analysis. We show that in all regimes considered, the interaction of a frequency-chirped laser pulse with the multilevel $^{85}$Rb system is similar to the interaction with an effective two-level atom at sufficiently large peak intensities of the pulses. It allows us to perform efficient excitation of the multilevel atom by transferring populations of two hyperfine ground states to the excited ones and back to the ground states using a pair of frequency-chirped laser pulses. We propose to utilize this scheme of population transfer for the coherent manipulation of a beam of $^{85}$Rb atoms using sequences of counterpropagating frequency-chirped short laser pulses.

abstract:
A coherent mechanism of robust population inversion in atomic and molecular systems by a chirped field is presented. It is demonstrated that a field of sufficiently high chirp rate imposes a certain relative phase between a ground and excited state wavefunction of a two-level system. The value of the relative phase angle is thus restricted to be negative and close to 0 or $-\ pi$ for positive and negative chirp, respectively. This explains the unidirectionality of the population transfer from the ground to the excited state.In a molecular system composed of a ground and excited potential energy surface the symmetry between the action of a pulse with a large positive and negative chirp is broken. The same framwork of the coherent mechanism can explain the symmetry breaking and the population inversion due to a positive chirped field.

abstract:
We study light coherent transport in the weak localization regime using magneto-optically cooled strontium atoms. The coherent backscattering cone is measured in the four polarization channels using light resonant with a $J=0$ to $J=1$ transition of the Strontium atom. We find an enhancement factor close to 2 in the helicity preserving channel, in agreement with theoretical predictions. This observation confirms the effect of internal structure as the key mechanism for the contrast reduction observed with an Rubidium cold cloud (see: Labeyrie et al., PRL 83, 5266 (1999)). Experimental results are in good agreement with Monte-Carlo simulations taking into account geometry effects.

abstract:
We present a scheme of population transfer between two metastable (ground) states of the $\Lambda$ atom without considerable excitation of the atom using single frequency-chirped laser pulses. The physics of the process is generation of the ``trapped'' superposition of the ground states by the laser pulse at sufficiently high laser peak intensity. The main conditions for realization of this regime are the following: The width of the transform-limited laser pulse envelope frequency spectrum (without chirp) must be smaller and the peak Rabi frequency of the pulse must be larger than the frequency interval between the two ground states of the $\Lambda$ atom. During the frequency chirp, the laser pulse must first come into resonance with the transition from the initially occupied ground state to the excited state and after that with the transition between the excited and second initially empty ground states. In the case when the envelope frequency spectrum width (without chirp) of the pulse exceeds the frequency interval between the two ground states, we show a possibility of controllable generation of superposition of the ground states with a controllable excitation of the $\Lambda$ atom.

abstract:
We propose a new coherent atomic beam splitter that is based on the induced redistribution of photons in the intense field of two collinear standing waves with different frequencies. In the dressed-atom approach, we show that this scheme can provide a clear large-angle splitting into two components without being restricted to the Raman-Nath regime.

abstract:
Motivated by the production of strong optical forces on atoms, we have done numerical studies of adiabatic rapid passage in light that is both frequency and amplitude modulated in the form of chirped pulses, and we found unanticipated regularities. We have quantified these regularities in terms of a periodic Hamiltonian, and obtained a result similar to the Bloch theorem, but in the time domain. This led to further numerical studies and a quantitative description of the behavior of an atom under multiply repetitive sweeps.

abstract:
We have measured the deflection of a thermal sodium beam by a force existing in a field of two counterpropagating short laser pulses. This field was created by a $\Pi$-pulse train of a mode-locked (mode) (spacing 82~ MHz) dye laser beam, perpendicularly crossing a thermal sodium beam and retroreflected from a mirror behind it. Unlike the spontaneous force, this $\Pi$-pulse force is not limited by saturation effects and does not heat up the sodium beam transversally. Calculations yield a maximum $\Pi$-pulse force about three times larger than the spontaneous force, being in good agreement with our experimental results.

abstract:
Atomic mirrors and beam splitters based on adiabatic following by delayed laser pulses in a three-level Raman configuration are analyzed. We show that in a pure three-level system of two ground states and one excited state, laser pulses tuned precisely on resonance with the excited state do not produce any accumulated phase shifts due to ac Stark shifts even if the process is nonadiabatic. A numerical simulation suggests that multiple delayed laser pulses are attractive for precision interferometry since high transfer efficiencies (up to 98.7\% per pulse for the cesium $D_{1}$ line) and low ac Stark shifts are expected. A possible application to optical two-photon spectroscopy for an optical clock is discussed.

abstract:
We demonstrate an atomic interferometer based on the transfer of population between two ground states via adiabatic following using a nonabsorbing ``dark'' superposition state. This type of interferometer promises to be useful for precise measurement of the photon recoil energy and also for large area atomic interferometers since it allows transfer of a large number of photon recoil momenta to the atoms with high efficiency. In preliminary experiments, we have obtained a coherent transfer efficiency of 95\% with transfer of photon momenta and 98.4\% with no transfer of momenta. Over 140 photon momenta have been transferred coherently to the atoms.

abstract:
We use microwave-induced adiabatic passages for selective spin flips within a string of optically trapped individual neutral Cs atoms. We position-dependently shift the atomic transition frequency with a magnetic field gradient. To flip the spin of a selected atom, we optically measure its position and sweep the microwave frequency across its respective resonance frequency. We analyze the addressing resolution and the experimental robustness of this scheme. Furthermore, we show that adiabatic spin flips can also be induced with a fixed microwave frequency by deterministically transporting the atoms across the position of resonance.

abstract:
We show how a control of the dynamics of a decay process can be achieved by the application of a series of $2\pi$ pulses on an auxiliary transition. The $2\pi$ pulse changes the phase of the ground state by $\pi$ while leaving the phase of the excited state unaltered. This produces quantum interferences between the transition amplitudes for evolution in the short interval, just before and after the $2\pi$ pulse. Such an interference under suitable tailoring of the density-of-states of the bath and the time $\tau$ leads to accelerated decay.

abstract:
While complicated, unreliable alternatives to Doppler effect were proposed, an elementary optical light- matter interaction provides one which is commonly observed in the labs, but with a distortion due to the use of short, powerful laser pulses. It is generally assumed that Raman scattering in gases is incoherent. This assumption fails if the pressure is lowered enough to increase the relaxation times over the length of light pulses; the ``Impulsive Stimulated Raman Scattering'' (ISRS), generally used to study dense matter with ultrashort laser pulses, is adapted to the low energy pulses making the incoherent light beams; the usual light is redshifted by some very low pressure gases while it propagates. To produce this adapted ISRS called ``Incoherent Light Spatially Coherent Raman Scattering'' (ILSCRS), a molecule must have an hyperfine structure: polyatomic molecules must be heavy or have odd numbers of electrons; light atoms and the other molecules must be perturbed by a Stark or Zeeman effect. ILSCRS redshifts may be distinguished from Doppler redshifts using a very difficult to observe dispersion of ILSCRS redshifts. This dispersion may explain the discrepancies of the fine structures in the spectra of the quasars, presently attributed to a variation of the fine structure constant. While the present interpretation of the Lyman forest in the spectra of quasars requires clouds stressed, for instance, by sheets of dark matter, ILSCRS interpretation requires only usual physical concepts. It produces thermal radiations from short wavelengths, just as dust.

abstract:
We report the confinement and cooling of an optically dense cloud of neutral sodium atoms by radiation pressure. The trapping and damping forces were provided by three retroreflected laser beams propagating along orthogonal axes, with a weak magnetic field used to distinguish between the beams. We have trapped as many as $10^{7}$ atoms for 2~min at densities exceeding $10^{11}$~atoms cm$^{-3}$. The trap was $\simeq 0.4$~K deep and the atoms, once trapped, were cooled to less than a millikelvin and compacted into a region less than 0.5~mm in diameter.

abstract:
We report the viscous confinement and cooling of neutral sodium atoms in three dimensions via the radiation pressure of counterpropagating laser beams. These atoms have a density of about $\sim 10^{6}$~cm$^{-3}$ and a temperature of $\sim 240$~$\mu$K corresponding to a rms velocity of $\sim 60$~cm/sec. This temperature is approximately the quantum limit for this atomic transition. The decay time for half the atoms to escape a $\sim 0.2$~cm$^{3}$ confinement volume is $\sim 0.1$~sec.

abstract:
The development of a new and simplified system to create a Bose-Einstein condensate (BEC) is presented, as well as experimental studies of two spin-state condensed and non-condensed samples. The first part of this thesis describes in detail our apparatus and experiential procedure for creating a BEC. We designed a system to create a BEC that is simple and robust enough to be constructed by someone outside of the field of atom cooling and trapping. Our system includes several novel features that reduce the complexity of the apparatus, which include mechanical transfer of atoms and a hybrid magnetic trap. The second part of this thesis describes studies of two spin-state clouds. We studied the correlations and coherences of non-condensed clouds using Ramsey spectroscopy to gain an understanding of interatomic interactions. We were able measure precisely the mean-field interactions between coherent particles, as well as determine mechanisms that preserve and destroy coherence. These experiments led us to the study of spin waves, in which scattering of indistinguishable particles gives rise to coherent spin oscillations. Using sensitive Ramsey spectroscopy, we were able to fully examine the spatio-temporal spin-state oscillations. Following the studies of normal cloud coherence, we went on to explore coherence effects in condensates.

abstract:
We have Raman cooled sodium atoms below the photon recoil temperature in a novel type of blue-detuned optical dipole force trap. In this trap 4.5x10^5 atoms have been cooled to an effective three dimensional temperature of 1.0 microK at a final density of 4x10^11 cm^-3 . No atoms were lost during the cooling process. The phase space density increased by a factor of 320 over the uncooled sample. This is the highest phase space density achieved by an all-optical cooling method.

abstract:
We propose a technique for cooling optically trapped atoms to microkelvin temperatures, and lower, by using the dipole force of resonance-radiation pressure.

abstract:
We present a summary ofthe results of a simple two-level theory of Doppler cooling in optical molasses and contrast it with the recent theories of multilevel, polarization-gradient cooling. The effects of single-photon recoil and of trapping in microscopic optical potential wells are also considered. Experiments are described in which the temperature of sodium atoms released from optical molasses is measured and found to be well below the Dopplercooling limit. Measurements of the temperature dependence on many experimental parameters are found to be in good qualitative agreement with the new theories of polarization-gradient cooling.

abstract:
A theoretical analysis is given for laser cooling of a two-level atom with magnetic sublevels in the presence of polarization gradients. The optical Bloch equations for the multilevel system are solved numerically for four combinations of polarizations in one-dimensional optical molasses. The light-pressure force on the atom as given by a simple two-level theory is recovered in the absence of polarization gradients, whereas a spatial variation of the polarization is found to lead to a strong cooling force for slow atoms. The increased cooling force is responsible for the recent observations of atoms cooled in optical molasses to temperatures an order of magnitude below the Doppler limit.

abstract:
The cooling mechanisms for laser cooling of atoms in optical molasses have been investigated experimentally. A significant simplification over the usual three-dimensional geometry has been obtained by studying the optical molasses in one or two dimensions only. By proper choice of polarizations the behavior of a pure two-level system as well as the more complicated effects of polarization gradients on laser cooling of a multilevel atom were observed.

abstract:
We demonstrate a new cooling technique which is used to cool sodium atoms in one dimension to an effective temperature of 100~nK, less than 1/10 of the single photon recoil temperature $k_{B}T_{\mbox{rec}} = (\hbar k)^{2}/2M.

abstract:
Laser cooling and trapping of neutral atoms is a rapidly expanding area of physics research that has seen dramatic new developments over the last decade. These include the ability to cool atoms down to unprecedented kinetic temperatures (as low as one micro Kelvin) and to bold samples of a gas isolated in the middle of a vacuum system for many seconds. This unique new level of control of atomic motion is allowing researchers to probe the behavior of atoms in a whole new regime of matter where deBroglie wavelengths are much larger than the Bohr radius. Undoubtedly one of the distinct appeals of this research is the leisurely and highly visible motion of the laser cooled and trapped atoms. In this experiment you will operate a laser trap that is equal or superior in performance to what is used in many current research programs. This experiment uses the lasers and saturated absorption spectrometers used in the laser spectroscopy experiment' and thus you should have done that experiment before doing this one. A small fraction (- 10%) of the beams of each of the two lasers goes to their respective saturated absorption spectrometers. This allows for precise detection and control of the laser frequencies, which is essential for cooling and trapping. The remainder of the laser light goes into the trapping cell. Section 2 of this write-up provides a brief introduction to the relevant physics of the atom trap, section 3 discusses the laser stabilization, section 4 explains the optical layout for sending the laser beams into the cell to create the trap, section 5 explains the trapping cell construction and section 6 discusses the operation of the trap, measurement of the number of trapped atoms, and measurement of the time the atoms remain in the trap.

abstract:
This article presents a review of some of the principal techniques of laser cooling and trapping that have been developed during the past 20 years. Its approach is primarily experimental, but its quantitative descriptions are consistent in notation with most of the theoretical literature.

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We present two cooling mechanisms that lead to temperatures well below the Doppler limit. These mechanisms are based on laser polarization gradients and work at low laser power when the optical-pumping time between different ground-state sublevels becomes long. There is then a large time lag between the internal atomic response and the atomic motion, which leads to a large cooling force. In the simple case of one-dimensional molasses, we identify two types of polarization gradient that occur when the two counterpropagating waves have either orthogonal linear polarizations or orthogonal circular polarizations. In the first case, the light shifts of the ground-state Zeeman sublevels are spatially modulated, and optical pumping among them leads to dipole forces and to a Sisyphus effect analogous to the one that occurs in stimulated molasses. In the second case ($\sigma+ - \sigma-$ configuration), the cooling mechanism is radically different. Even at very low velocity, atomic motion produces a population difference among ground-state sublevels, which gives rise to unbalanced radiation pressures. From semiclassical optical Bloch equations, we derive for the two cases quantitative expressions for friction coefficients and velocity capture ranges. The friction coefficients are shown in both cases to be independent of the laser power, which produces an equilibrium temperature proportional to the laser power. The lowest achievable temperatures then approach the one-photon recoil energy. We briefly outline a full quantum treatment of such a limit.

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We present detailed instructions for the construction and operation of an inexpensive apparatus for laser cooling and trapping of rubidium atoms. This apparatus allows one to use the light from low power diode lasers to produce a magneto-optical trap in a low pressure vapor cell. We present a design which has reduced the cost to less than \$3000 and does not require any machining or glassblowing skills in the construction. It has the additional virtues that the alignment of the trapping laser beams is very easy, and the rubidium pressure is conveniently and rapidly controlled. These features make the trap simple and reliable to operate, and the trapped atoms can be easily seen and studied. With a few milliwatts of laser power we are able to trap $4 \times 10^{7}$ atoms for 3.5~s in this apparatus. A step-by-step procedure is given for construction of the cell, setup of the optical system, and operation of the trap. A list of parts with prices and vendors is given in the Appendix.

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Reviews the history of optical trapping and manipulation of small-neutral particles, from the time of its origin in 1970 up to the present. As we shall see, the unique characteristics of this technique are having a major impact on the many subfields of physics, chemistry, and biology where small particles play a role.

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We propose an atomic or molecular interferometer based on a series of femtosecond laser pulses, that allows the measurement and control of particle velocities. The probability for a particle to absorb a net photon momentum depends on the particle velocity and on relative phases of the laser pulses, but not on the absolute laser detuning from an optical transition. The scheme has prospects for the laser cooling of atoms and molecules.

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Laser cooled sodium atoms are trapped in an optical dipole force trap formed by the intersection of two 1.06~$\mu$m laser beams. Densities as high as $4 \times 10^{12}$~atoms/cm$^{3}$ at a temperature of $\sim 140$~$\mu$K have been obtained in a $\sim 900$~$\mu$K deep trap. By reducing the trap depth over a 2~s interval, we have evaporatively cooled the atoms to a final temperature of $\sim 4$~$\mu$K at a density of $6 \times 10^{11}$~atoms/cm$^{3}$. This corresponds to a factor of 28 increase in atomic phase-space density.

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Significant advances have been made in the ability to control the motion of neutral atoms. Cooling and trapping atoms present new possibilities for studies of ultracold atoms and atomic interactions. The techniques of laser cooling and deceleration of atomic beams, magnetic and laser trapping of neutral atoms, and a number of recent advances in the use of radiative forces to manipulate atoms are reviewed.

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Stimulated Raman transitions between the ground-level hyperfine states of atoms have been used to manipulate slowly moving atoms in an atomic fountain. An ensemble of sodium atoms with an inferred velocity spread along one dimension of 270~$\mu$m/sec has been prepared by this technique. We also show that this velocity-selection method is effective in measuring ultracold temperatures of laser-cooled atoms in a regime where traditional ballistic methods fail.

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It is possible to tune the scattering length for the collision of ultracold $^{1}$S$_{0}$ ground-state alkaline-earth-metal atoms using an optical Feshbach resonance. This is achieved with a laser far detuned from an excited molecular level near the frequency of the atomic intercombination $^{1}$S$_{0}$-$^{3}$P$_{1}$ transition. Simple resonant-scattering theory, illustrated by the example of $^{40}$Ca, allows an estimate of the magnitude of the effect. Unlike alkali metal species, large changes of the scattering length are possible while atom loss remains small, because of the very narrow linewidth of the molecular photoassociation transition. This raises prospects for control of atomic interactions for a system without magnetically tunable Feshbach resonance levels.

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For more than 100 years, optical atomic/molecular frequency references have played important roles in science and technology, and provide standards enabling precision measurements. Frequency-stable optical sources have been central to experimental tests of Einstein's relativity, and also serve to realize our base unit of length. The technology has evolved from atomic discharge lamps and interferometry, to narrow atomic resonances in laser-cooled atoms that are probed by frequency-stabilized cw lasers that in turn control optical frequency synthesizers (combs) based on ultra-fast mode-locked lasers. Recent technological advances have improved the performance of optical frequency references by almost four orders of magnitude in the last eight years. This has stimulated new enthusiasm for the development of optical atomic clocks, and allows new probes into nature, such as searches for time variation of fundamental constants and precision spectroscopy.

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The electron energy distribution function (EEDF) in laser-excited Rb vapour at the first resonance transition 5S-5P and 5p--$nl$ transitions is calculated numerically from the Boltzmann equation under the experimental conditions of Barbier and Cheret (1987 J. Phys. B: At. Mol. Phys. 20 1229). The calculations included all the interactions between electrons and ground or excited states of atomic rubidium. We have also studied the time evolution and the laser power dependences of the EEDF and the electron density which is created during the interactions. The energy spectra of the electrons emerging from the interaction contain a number of peaks corresponding to the low-energy electrons produced by the associative and Hornbeck-Molnar ionization in addition to the electrons created from the various excited states by the ionizing collisions (Penning ionization). The low-energy electrons are then heated by one or more superelastic collisions leading to further ionization. However the calculations indicated that, under the conditions of low power laser intensity and relatively low atom density, the dominant processes are collisional ionization and collisional excitation.

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We consider the storage and transmission of a Gaussian distributed set of coherent states of continuous variable systems. We prove a limit on the average fidelity achievable when the states are transmitted or stored by a classical channel, i.e., a measure and repreparation scheme which sends or stores classical information only. The obtained bound is tight and serves as a benchmark which has to be surpassed by quantum channels in order to outperform any classical strategy. The success in experimental demonstrations of quantum memories as well as quantum teleportation has to be judged on this footing.

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Optical trapping of dielectric particles by a single-beam gradient force trap was demonstrated for the first reported time. This confirms the concept of negative light pressure due to the gradient force. Trapping was observed over the entire range of particle size from 10~$\mu$m to $\sim 25$~nm in water. Use of the new trap extends the size range of macroscopic particles accessible to optical trapping and manipulation well into the Rayleigh size regime. Application of this trapping principle to atom trapping is considered.

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A new idea of controlling molecular processes by time-dependent external fields is proposed. Molecular processes in external fields are considered to be composed of a sequence of time-dependent nonadiabatic transitions in which the external fields play a role of adiabatic parameters. Unit final transition probability can be achieved with the use of the interference effects among various paths created by nonadiabatic transitions. The basic idea is to sweep the external field periodically at each avoided crossing and to control the transition there completely as we desire. This idea is quite general, and can hold whatever the external field is. Various control schemes can be proposed corresponding to the various types of time-dependent nonadiabatic transitions. The methods of $\pi$-pulse and chirped laser pulse with the adiabatic rapid passage may be considered as special cases of the present idea. As an example, a one-dimensional model of the laser-induced ring-puckering isomerization of trimethylenimine is considered, and comparative studies on the effectiveness and the stability of the various control schemes proposed in this paper are made together with presentation of numerical examples.

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A variety of powerful techniques to control the position and velocity of neutral particles has been developed. As examples of this new ability, lasers have been used to construct a variety of traps, to cool atoms to temperatures below $3 \times 10^{-6}$ kelvin, and to create atomic fountains that may give us a hundredfold increase in the accuracy of atomic clocks. Bacteria can be held with laser traps while they are being viewed in an optical microscope, and organelles within a cell can be manipulated without puncturing the cell wall. Single molecules of DNA can now be stretched out and pinned down in a water solution with optical traps. These new capabilities may soon be applied to a wide variety of scientific questions as diverse as precision measurements of fundamental symmetries in physics and the study of biochemistry on a single molecule basis.

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We show how the use of optimally shaped pulses to guide the time evolution of a system (``coherent control'') can be an effective approach towards quantum computation logic. We demonstrate this with selective control of decoherence for a multilevel system with a simple linearly chirped pulse. We use a multiphoton density-matrix approach to explore the effects of ultrafast shaped pulses for two-level systems that do not have a single photon resonance, and show that many multiphoton results are surprisingly similar to the single-photon results. Finally, we choose two specific chirped pulses: one that always generates inversion and the other that always generates self-induced transparency to demonstrate an ensemble CNOT gate.

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The mechanical effects of stimulated Raman transitions on atoms have been used to demonstrate a matter-wave interferometer with laser-cooled sodium atoms. Interference has been observed for wave packets that have been separated by as much as 2.4~mm. Using the interferometer as an inertial sensor, the acceleration of a sodium atom due to gravity has been measured with a resolution of $3 \times 10^{-6}$ after 1000 sec of integration time.

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A method of slowing, accelerating, cooling, and bunching molecules and neutral atoms using time-varying electric-field gradients is demonstrated with cesium atoms in a fountain. The effects are measured and found to be in agreement with our calculations. Time-varying electric-field-gradient slowing and cooling is applicable to atoms that have large dipole polarizabilities, including atoms that are not amenable to laser slowing and cooling, to Rydberg atoms, and to molecules, especially polar molecules with large electric dipole moments. The possible applications of this method include slowing and cooling thermal beams of atoms and molecules, launching cold atoms from a trap into a fountain, and measuring atomic dipole polarizabilities.

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We examine collective magnetoassociation of a Bose-Einstein condensate (BEC), focusing on rapid adiabatic passage from atoms to molecules induced by a sweep of the magnetic field across a wide ($\geq 10$~G) Feshbach resonance in $^{85}$Rb. This problem raises an interest because strong magnetoassociation is expected to favor the creation of molecular-dissociated atom pairs over the formation of molecular BEG [Javanainen and Mackie, Phys. Rev. Lett. \textbf{88}, 090403 (2002)]. Nevertheless, the conversion to atom pairs is found to depend on the direction of the sweep, so that a system initially above threshold (open dissociation channel) may in fact give efficient conversion to molecules.

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Inspired by the spectacular successes in the field of cold atoms, there is currently great interest in cold molecules. Cooling molecules is useful for various fundamental physics studies and gives access to an exotic regime in chemistry where the wave property of the molecules becomes important. Although cooling molecules has turned out to be considerably more difficult than cooling atoms, a number of methods to produce samples of cold molecules have been demonstrated over the last few years. This paper aims to review the application of cold molecules and the methods to produce them. Emphasis is put on the deceleration of polar molecules using time-varying electric fields. The operation principle of the array of electrodes that is used to decelerate polar molecules is described in analogy with, and using terminology from, charged-particle accelerators. It is shown that, by applying an appropriately timed high voltage burst, molecules can be decelerated while the phase-space density, i.e.~the number of molecules per position-velocity interval, remains constant. In this way the high density and low temperature in the moving frame of a pulsed molecular beam can be transferred to the laboratory frame. Experiments on metastable CO in states that are either repelled by or attracted to high electric fields are presented. Loading of slow molecules into traps and storage rings is discussed.

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This theoretical paper investigates the formation of ground state molecules from ultracold cesium atoms in a two-color scheme. Following previous work on photoassociation with chirped picosecond pulses [Luc-Koenig et al., Phys. Rev. A {\bf 70}, 033414 (2004)], we investigate stabilization by a second (dump) pulse. By appropriately choosing the dump pulse parameters and time delay with respect to the photoassociation pulse, we show that a large number of deeply bound molecules are created in the ground triplet state. We discuss (i) broad-bandwidth dump pulses which maximize the probability to form molecules while creating a broad vibrational distribution as well as (ii) narrow-bandwidth pulses populating a single vibrational ground state level, bound by 113~cm$^{-1}$. The use of chirped pulses makes the two-color scheme robust, simple and efficient.

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Certain molecules, it seems, may be laser cooled by methods technically similar to those applied with abundant success in atomic physics. We discuss the spectroscopic criteria molecules should meet to make methods of Doppler cooling technically feasible and identify diatomic candidates. Some candidates, such as the alkaline-earth monohydrides (e.g. BeH and CaH), are paramagnetic and amenable to magneto-optical trapping. Our experimental study concentrates on CaH, and we present our recent high-resolution, molecular-beam-based measurements of low-$J$ rotational lines within the $A-X(0,0)$ band of CaH. From these measurements we report hyperfine separations in the A-state, as important to laser-cooling spectroscopy, and centroidal transition frequencies for comparison with existing values. We conclude with an outline of a possible magneto-optical trap for CaH.

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A general scheme for reducing the center-of-mass entropy is proposed. It is based on the repetition of a cycle, composed of three concepts: velocity selection, deceleration and irreversible accumulation. Well-known laser techniques are used to represent these concepts: Raman $\pi$-pulse for velocity selection, STIRAP for deceleration, and a single spontaneous emission for irreversible accumulation. No closed pumping cycle nor repeated spontaneous emissions are required, so the scheme is applicable to cool a molecular gas. The quantum dynamics are analytically modelled using the density matrix. It is shown that during the coherent processes the gas is translationally cooled. The internal states serve as an entropy sink, in addition to spontaneous emission. This scheme provides new possibilities to translationally laser-cool molecules for high precision molecular spectroscopy and interferometry.

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Optimal control theory (OCT) is applied to laser cooling of molecules. The objective is to cool vibrations, using shaped pulses synchronized with the spontaneous emission. An instantaneous in time optimal approach is compared to solution based on OCT. In both cases the optimal mechanism is found to operate by a ``vibrationally selective coherent population trapping''. The trapping condition is that the instantaneous phase of the laser is locked to the phase of the transition dipole moment of v=0 with the excited population. The molecules that reach $v=0$ by spontaneous emission are then trapped, while the others are continually repumped. For vibrational cooling to $v=2$ and rotational cooling, a different mechanism operates. The field completely changes the transient eigenstates of the Hamiltonian creating a superposition composed of many states. Finally this superposition is transformed by the field to the target energy eigenstate.

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We point out a laser cooling method for atoms, molecules, or ions at low saturation and large detuning from the particles' resonances. The moving particle modifies the field inside a cavity with a time delay characteristic of the cavity linewidth, while the field acts on the particle via the light shift. The dissipative mechanism can be interpreted as Doppler cooling based on preferential scattering rather than preferential absorption. It depends on particle properties only through the coherent scattering rate, opening new possibilities for optically cooling molecules or interacting atoms.

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We report the cooling of nitric oxide molecules in a single collision between an argon atom and an NO molecule at collision energies of $5.65\pm 0.36$~kJ/mol and $14.7\pm 0.9$~kJ/mol in a crossed molecular beam apparatus. We have produced in significant numbers ( $\sim 10^{8}$ molecules cm$^{-3}$ per quantum state) translationally cold NO( $^{2}{\rm\Pi} _{1/2}$, v'=0, j'=7.5) molecules in a specific quantum state with an upper-limit laboratory-frame rms velocity of $14.8\pm 1.1$ m/s, corresponding to a temperature of $406\pm 28$~mK. The translational cooling results from the kinematic collapse of the velocity distribution of the NO molecules after collision. Increasing the collision energy by increasing the velocity of the argon atoms, as we do here, does shift the scattering angle at which the cold molecules appear, but does not result in an experimentally measurable change in the velocity spread of the cold NO. This is entirely consistent with our analysis of the kinematics of the scattering which predicts that the velocity spread will actually decrease with increasing argon atom velocity.

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This overview prefaces a collection of Insight review articles on the physics and applications of laser-cooled atoms. I will cast this work into a historical perspective in which laser cooling and trapping is seen as one of several research directions aimed at controlling the internal and external degrees of freedom of atoms and molecules.

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The translational motion of molecular ions can be effectively cooled sympathetically to temperatures below 100~mK in ion traps through Coulomb interactions with laser-cooled atomic ions. The distribution of internal rovibrational states, however, gets in thermal equilibrium with the typically much higher temperature of the environment within tens of seconds. We consider a concept for rotational cooling of such internally hot, but translationally cold heteronuclear diatomic molecular ions. The scheme relies on a combination of optical pumping from a few specific rotational levels into a ``dark state'' with redistribution of rotational populations mediated by blackbody radiation.

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We have constructed a compact extended-cavity diode laser (ECDL) that is based on a Littman configuration with a grating and a reflector. The whole structure is installed in a 2-inch kinematic mount. ECDLs operating at 852 nm (Cs D/sub 2/ line), 894 nm (Cs D/sub 1/ line), 780 nm (Rb D/sub 2/ line), and 794 nm (Rb D/sub 1/ line) were fabricated and tested. As a result of the performance test, up to 9 GHz continuous tuning without mode hopping could be obtained by tuning with a piezoelectric transducer only. The linewidth from the beat-note spectrum of two ECDLs was about 200 kHz.

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We propose a scheme of transient laser manipulation and cooling of two-level quantum systems with narrow natural linewidths by a sequence of counterpropagating laser pulses with special frequency chirping. Interaction with a large number of laser pulses within the decay time decreases drastically the cooling time of such systems.

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In this dissertation I describe the frequency stabilization of the diode laser pumped Non Planar Ring Oscillator (NPRO) and discuss its use in coherent optical communication systems. The requirements necessary to achieve stability at the sub-Hz level are emphasized and the experiments resulting in a relative stability of 330~mHz are included. Optical interferometers with extremely high finesse are used as frequency discriminators in the stabilization experiments discussed here. The development and characterization of one such interferometer with a finesse of 27,500 and an optical transmission bandwidth of 25~kHz is also presented. The spectral density of frequency noise associated with the NPRO was measured from 10~Hz to 100~kHz and an unambiguous pole in the spectrum was observed. This structure was assumed to be due to thermal filtering of the pump laser power fluctuations and theoretical modeling supported this hypothesis. This model also demonstrated that a significant reduction in the spectral density can be achieved when a shot noise limited laser is used as the optical pump in the NPRO. The magnitude of the spectral density of frequency noise was $\sim$115~Hz/$\sqrt{\mbox{Hz}}$ at 100~Hz and $\sim$3~Hz/$\sqrt{\mbox{(Hz)}}$ at 10~kHz. The diode laser pumped solid state laser, with its ability to be frequency stabilized to the sub-Hz level, is an ideal source for use in a coherent optical communications link. An optical phase locked loop (OPLL) at 1.06~$\mu$m for use in a coherent homodyne receiver was developed using two NPROs, and an extended theory describing its performance is pesented. This analysis emphasizes the advantages offered by the NPRO, as compared to. the diode laser, when used in a homodyne communications receiver.

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We have developed a wideband frequency-stabilized injection-locked Nd:YAG laser as a light source for the laser interferometric gravitational wave detector, in which short-term frequency stability of the laser improves the sensitivity of the interferometer and the long-term frequency stability aims for the stable long-time operation of the interferometer. The frequency of a 2-W injection-locked laser is locked to both a rigid Fabry-Perot cavity with ULE spacer and saturated absorption line of $^{127}$I$_{2}$ simultaneously with two nested servo loops, and the long-term as well as short-term frequency stability are obtained. The drift of the resonant frequency of the rigid Fabry-Perot cavity is measured and the stability of the Fabry-Perot cavity is estimated to be $20\times f^{-1}$~[Hz/$\surd$Hz]. The predicted frequency stabilities of the present dual-reference-locked laser are numerically simulated. Our wideband frequency-stabilized laser is also available for the high-resolution spectroscopy.

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In a previous paper we described details of the construction of stabilized 670-nm diode lasers for use in undergraduate physics laboratories. We report here a series of experiments that can be performed using the 670-nm diode laser, a homemade scanning Fabry-Perot cavity, a helium-neon laser, a simple photodiode, and a few pieces of electronics hardware. The experiments include: (1) an introduction to the scanning confocal Fabry-Perot cavity, and to its use as an optical spectrum analyzer; (2) laser frequency modulation and observation of FM sidebands using the optical spectrum analyzer; and (3) the Pound-Drever method for servo-locking a Fabry-Perot cavity to a laser. These experiments are relatively easy to set up and perform, yet they demonstrate a number of useful optical principles and experimental techniques.

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We present a simple modification of the traditional method of locking the laser frequency to the side of an atomic spectral line. We achieve first-order power insensitivity at arbitrary intensity and frequency and in this way eliminate one of the major drawbacks of the traditional method. A similar approach could also be used in locking a laser to a Fabry-Perot cavity.

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These are notes on the Pound-Drever-Hall technique for doing interferometry with a Fabry-Perot cavity. They are intended as introductory material to supplement the published literature for members of the Thermal Noise Interferometer group. Understanding the Pound-Drever-Hall technique is essential to understanding the workings of the Thermal Noise Interferometer (TNI), as well as LIGO. These notes include a review of the fundamentals of the technique as well as some quantitative predictions about sensitivity and noise.

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We report a diode laser system developed for narrow-line cooling and trapping on the $^{1}S_{0} - ^{3}P_{1}$ intercombination transition of neutral strontium atoms. Doppler cooling on this spin-forbidden transition with a line width of $\Gamma/2\pi = 7.1$~kHz enables us to achieve sub-micro Kelvin temperatures in a two-step cooling process. The required reduction of the laser line width to the kHz level was achieved by locking the laser to a tunable Fabry-Perot cavity. The long-term drift ($> 0.1$~s) of the reference cavity was compensated by employing the saturated absorption signal obtained from Sr vapor in a heat pipe of novel design. We demonstrate the potential of the system by performing spectroscopy of Sr atoms confined to the Lamb-Dicke regime in a one-dimensional opticallattice.

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Fast frequency fluctuations of a 780~nm grating-stabilized diode laser are directly observed by the measurement of the Allan variance. Time intervals as short as 10~ns and up to 1~s are realized by means of two independent time interval counters. Frequency fluctuations with the laser locked to a Doppler-free resonance transition of atomic rubidium are examined in detail.

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An improvement to the saturated absorption technique of long term stabilization of lasers is reported. This new method enables stabilization to the centre of saturated absorption features regardless of whether the peaks are symmetric or asymmetric. Another feature of the locking system is its ability to stabilize the laser at precisely known small detunings from the centre of the absorption line. The locking system has been applied to both dye laser and titanium sapphire laser systems. The long term drift for each was measured to be smaller than the laser jitter bandwidth of about 1~MHz for periods of hours, and no mode hops were recorded over several weeks of operation.

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Diode lasers with a power output superior to 100~mW are in widespread use in medical as well as research applications. However, for such diodes lasing oscillation generally occurs simultaneously in several longitudinal and transverse modes that are unsuitable for high-resolution spectroscopy. We spectrally narrow a 100-mW broad-area diode laser by first using an extended cavity and then an electrical feedback produced by a Pound-Drever-Hall stabilization on a low-finesse reference cavity. Reduction of the linewidth by more than 6 orders of magnitude is achieved (the output linewidth is narrowed from 1~THz to less than 500~kHz), making possible its use for high-resolution spectroscopy. The power and the spectral qualities of this diode laser allow us to induce quantum jumps toward the $D_{5/2}$ metastable level of single Ca$^{+}$ ions.

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The gas spectroscopy is a standard technique for frequency stabilization in semiconductor lasers. The Doppler effect is what exerts most influence on the linewidths of atomic spectrum increments. In the present work, the saturation spectroscopy, whose main characteristic is to cancel the undesirable consequences of this effect, is studied. Different techniques of dispersion-like signals generation are revised, and are used as error signals in the stabilization of semiconductor lasers. From a different point of view, these techniques are reviewed and being compared. Moreover, the experimental results of Cesium 133 spectroscopy made at Centro Nacional de Metrologia, CENAM, are exhibited herein.

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Sub-Doppler lines of Cs2 were investigated with a Nd:YAG laser. Modulation transfer spectroscopy and FM spectroscopy were applied to yield error signals that were used for absolute stabilization of the laser frequency. The frequency stability was characterized with an I2 -locked dual-wavelength Nd:YAG frequency reference. The root Allan variance of the beat frequency reached a minimum of 1.3x10^12 (beat frequency fluctuations of 3.65 kHz) for a measurement time of 20 s. Absolute frequencies of several Cs2 lines were determined with an accuracy of ~1 MHz. Further improvement in sensitivity was demonstrated by insertion of the absorption cell into a Fabry--Perot cavity. While the laser was locked to the cavity, the cavity length was modulated and the transmitted probe beam was detected at the third harmonic of the modulation frequency.

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A new type of wavelength-modulation laser spectroscopy is accomplished by utilizing an external phase modulator driven at radio frequencies large compared to the width of the spectral feature of interest. The spectral feature is probed by a single frequency-modulated (FM) sideband, and the associated absorption and dispersion are measured by monitoring the resulting radio-frequency beat signal. Experimental results are presented for the measurement of Fabry-Perot resonances, I$_{2}$ vapor absorption lines, and saturation holes in Na vapor.

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The emission frequency of a diode laser submitted to a frequency-dependent optoelectronic feedback is observed to have more than one stable operation point together with a stable power emission. This is, to our knowledge, the first observation of bistability exclusively in the frequency of an optical system. The experiment was carried out with a semiconductor laser coupled to the cesium $D_{2}$ line by an orthogonally polarized frequency-sensitive optical feedback.

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We demonstrate a novel method of stabilizing an external-cavity diode laser frequency to an atomic transition. This technique employs a sub-Doppler spectrum of Cs atoms in a thin 150~$\mu$m vapour cell. We obtain the result of the frequency fluctuation to be less than 0.8~MHz using the third derivative signal, with the root of Allan variance of the error signals reaching a minimum of $5.9 \times 10^{-11}$ for an averaging time of 200~s.

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Improvements in the frequency stability of the visible helium-neon laser are possible using saturated absorption in iodine vapour. Recent improvements in the design of these lasers are discussed, and the importance of third derivative locking methods in realizing the possible high reproducibility of the iodine reference frequencies is emphasized. Allan Variance measurements of the beat frequency between two stabilized lasers indicate a long term stability of better than 1 part in $10^{10}$.

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A novel and compact diode-pumped Er--Yb:glass laser, with single-frequency output power up to 20~mW in a wavelength range from 1531 to 1547~nm, was frequency stabilized against sub-Doppler rovibrational transitions of $^{13}$C$_{2}$H$_{2}$. The non-linear wavelength-modulation spectroscopy technique was used to lock the laser frequency to the molecular lines. The stability of the developed optical frequency standard was mainly dominated by a white frequency-noise contribution, which sets the Allan deviation at a level of $\sigma_{y}(\tau) = 2 \times 10^{-11}\tau^{-1/2}$ for integration times in the range of $0.01 < \tau < 100$~s.

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A simple frequency stabilization method for frequency-doubled output (532~nm) of an Nd:YAG laser is described. The frequency shift caused by an acousto-optic modulator is effectively applied to stabilize the laser frequency. For the frequency stabilization, there are two kinds of error signal: (1) the first-derivative of absorption of iodine molecules and (2) the first-derivative of dispersion were extracted to examine which signal is suitable for the actual stabilization of the laser frequency. Based on this examination, the stabilization was executed utilizing the error signal (1), resulting in a frequency instability of $2.4\times10^{-8}$

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The last part of the twentieth century has experienced a huge resurge of activity in the field of coherent light-matter interaction, more so in attempting to exert control over such interactions. Birth of coherent control was originally spurred by the theoretical understanding of the quantum interferences that lead to energy randomization and experimental developments in ultrafast laser spectroscopy. The theoretical predictions on control of reaction channels or energy randomization processes are still more dramatic than the experimental demonstrations, though this gap between the two is consistently reducing over the recent years with realistic theoretical models and technological developments. Experimental demonstrations of arbitrary optical pulse shaping have made some of the previously impracticable theoretical predictions possible to implement. Starting with the simple laser modulation schemes to provide proof-of-the-principle demonstrations, feedback loop pulse shaping systems have been developed that can actively manipulate some atomic and molecular processes. This tremendous experimental boost of optical pulse shaping developments has prospects and implications into many more new directions, such as quantum computing and terabit/sec data communications. This review captures certain aspects and impacts of optical pulse shaping into the fast developing areas of coherent control and other related fields. Currently available reviews focus on one or the other detailed aspects of coherent control, and the reader will be referred to such details as and when necessary for issues that are dealt in brief here. We will focus on the current issues including control of intramolecular dynamics and make connections to the future concepts, such as, quantum computation, biomedical applications, etc.

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We perform spectroscopy on the ground-state hyperfine splitting of $^{85}$Rb atoms trapped in far-off-resonance optical traps. The existence of a spatially dependent shift in the energy levels is shown to induce an inherent dephasing effect, which causes a broadening of the spectroscopic line and hence an inhomogeneous loss of atomic coherence at a much faster rate than the homogeneous one caused by spontaneous photon scattering. We present here a number of approaches for reducing this inhomogeneous broadening, based on trap geometry, additional laser fields, and novel microwave pulse sequences. We then show how hyperfine spectroscopy can be used to study the quantum dynamics of optically trapped atoms.

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We discuss the properties of a simple and robust scheme for preparing atoms and molecules in an arbitrary preselected coherent superposition of quantum states - fractional stimulated Raman adiabatic passage (fractional STIRAP) - first proposed by Marte et al (1991 Phys. Rev. A 44 R4118). As in STIRAP, the Stokes pulse arrives before the pump pulse, but unlike STIRAP, here the two pulses terminate simultaneously, while maintaining a constant ratio of amplitudes. We extend earlier research and suggest a realization of this scheme with two smoothly varying delayed laser pulses (which can be derived from a single laser), the parameters of the created superposition being controlled by the polarization of the delayed pulse. Furthermore, we provide simple analytic estimates of the robustness of this process against variations in the laser intensity, laser frequency and pulse delay. Finally, we discuss an extension to multistate systems which provides a possibility for creating coherent superpositions of more than two states.

abstract:
The optical scattering force, behind Doppler cooling and magneto-optical trapping, may be amplified without incurring additional spontaneous emission by the state-dependent coherent deflection produced by a pulsed or chirped laser field. At some cost in experimental complexity, amplified forces allow efficient cooling on narrow transitions and permit the compact deceleration of beams with reduced transverse heating, and will be of interest for molecules and atoms with open level schemes where losses following spontaneous emission would otherwise prevail. We present a general analysis of the amplification scheme, and propose an optimized, dynamic cooling scheme that allows the temperature of a sample to be reduced by around a factor of two per excited state lifetime.

abstract:
We describe frequency locking of a diode laser to a two-photon transition of rubidium using the Zeeman modulation technique. We locked and tuned the laser frequency by modulating and shifting the two-photon transition frequency with ac and dc magnetic fields. We achieved a linewidth of 500~kHz and continuous tunability over 280~MHz with no laser frequency modulation.

abstract:
In this experiment, a diode laser beam passes through a rubidium (Rb) vapor cell. The transmitted laser intensity is measured by a photodiode as the laser frequency is scanned across several resonance transitions in Rb. Laser absorption is observed and used to study Doppler and sub-Doppler broadening. A subDoppler technique called saturated absorption spectroscopy highlights this experiment and is used to observe and study the hyperfine energy levels in the two naturally occuring Rb isotopes.

abstract:
We demonstrate a new technique for saturated-absorption spectroscopy by use of copropagating beams that does not have the problem of crossover resonances. The pump beam is locked to a transition, and its absorption signal is monitored while the probe beam is scanned. As the probe comes into resonance with another transition, the pump absorption is reduced and