We discuss Bose-Einstein condensation in harmonic traps where the confinement has undergone a splitting along one direction. We mostly consider three-dimensional potentials consisting of two cylindrical wells separated a distance $2a$ along the z axis. For ideal gases, the thermodynamics of the confined bosons has been investigated, performing exact numerical summations to describe the major details of the transition and comparing the results with the semiclassical density-of-states approximation. We find that for large particle number and increasing well separation, the condensation temperature evolves from the thermodynamic limit value $T_c^{\mathrm{}\left(0\right)\mathrm{}}\mathrm{}\left(N\right)\mathrm{}$ to $T_c^{\mathrm{}\left(0\right)\mathrm{}}\mathrm{}(N/2\mathrm{}).$ The effects of adding a repulsive interaction between atoms has been examined resorting to the Gross-Pitaevskii-Popov procedure, and it is found that the shift of the condensation temperature exhibits different signs according to the separation between wells. In particular, for sufficiently large splitting, the trend opposes the well-known results for harmonic traps, since the critical temperature appears to increase with growing repulsion strength.

We experimentally investigate Bragg reflection of atoms from a standing light wave. We focus on the influence of small angular deviations from the exact Bragg resonance onto the reflection probability. The Bragg resonance has a finite acceptance angle, which depends on the light intensity. Within this acceptance angle, an oscillatory behavior of the reflection probability as a function of the angle is observed.

Sympathetic cooling of an atomic Fermi gas by a Bose gas is studied by solution of the coupled quantum Boltzmann equations for the confined gas mixture. Results for equilibrium temperatures and relaxation dynamics are presented, and some simple models are developed. Our study illustrates that a combination of sympathetic and forced evaporative cooling enables the Fermi gas to be cooled to the degenerate regime where quantum statistics and mean-field effects are important. The influence of mean-field effects on the equilibrium spatial distributions is discussed qualitatively.

We consider a trapped multicomponent atomic Bose-Einstein condensate, concentrating specifically on condensates in the hyperfine ground state $F=1,$ where spin exchange collisions result in a transfer of population between $m=0\mathrm{}$ and $m=\pm 1$ internal states. Drawing an analogy with the optical situation, we show that this system can be regarded as a matter-wave analog of optical multiwave mixing. This opens up the way to realize matter-wave phase conjugation, whereby an incident atomic beam can be “time reversed.” In addition, matter-wave phase conjugation also offers novel diagnostic tools to study the coherence properties of condensates, as well as to measure the relative scattering lengths of hyperfine sublevels.

We report the measurement of the rate constant for dipolar relaxation from the $(F=2,$ $m_F=2)$ hyperfine state of ${}^7\mathrm{Li}.$ The atoms are confined in a permanent magnet trap at a field of ${10}^3\hspace{0.5em}\mathrm{G}.$ The measured value of $(1.05\mathrm{}\pm 0.10)\times {10}^{-14}\hspace{0.5em}{\mathrm{cm}}^3/\mathrm{s}$ agrees well with theory. Additionally, we determine an upper bound for the three-body molecular recombination rate constant of ${10}^{-27}\hspace{0.5em}{\mathrm{cm}}^6/\mathrm{s},$ which is also consistent with predictions.

The propagation of the electromagnetic field of a laser through a dense Bose gas is examined and nonlinear operator equations for the motion of the center of mass of the atoms are derived. The goal is to present a self-consistent set of coupled Maxwell-Bloch equations for atomic and electromagnetic fields generalized to include the atomic center-of-mass motion. Two effects are considered. The ultracold gas forms a medium for the Maxwell field which modifies its propagation properties. Combined herewith is the influence of the dipole-dipole interaction between atoms which leads to a density-dependent shift of the atomic transition frequency. It is expressed in a position-dependent detuning and is the reason for the nonlinearity. This results in a direct and physically transparent way from the quantum field-theoretical version of the local-field approach to electrodynamics in quantum media. The equations for the matter fields are general. Previously published nonlinear equations are obtained as limiting cases. As an atom-optical application the scattering of a dense beam of a Bose gas is studied in the Raman-Nath regime. The main conclusion is that for increasing density of the gas the dipole-dipole interaction suppresses or enhances the scattering depending on the sign of the detuning.

Three hyperfine states of Bose-condensed sodium atoms, recently optically trapped, can be described as a spin-1 Bose gas. We study the behavior of this system in a magnetic field, and construct the phase diagram, where the temperature of the Bose condensation $T_{\mathrm{BEC}}$ increases with magnetic field. In particular the system is ferromagnetic below $T_{\mathrm{BEC}}$ and the magnetization is proportional to the condensate fraction in a vanishing magnetic field. Second derivatives of the magnetization with regard to temperature or magnetic field are discontinuous along the phase boundary.

We consider the intrinsic stability of the vortex states of a pure Bose-Einstein condensate confined in a harmonic potential under the effects of a coherent atom-atom interaction. We find that stable vortices can be supported and vortex stability can be controlled by changing the interparticle interaction strength. At unstable regimes, a vortex will spontaneously disintegrate into states with different angular momenta even without external perturbations, with the lifetime determined by its imaginary excitation frequencies.

The mathematical methods that have been used to analyze the statistical properties of boson fields, and in particular the coherence of photons in quantum optics, have their counterparts for Fermi fields. The coherent states, the displacement operators, the P representation, and the other operator expansions all possess surprisingly close fermionic analogues. These methods for describing the statistical properties of fermions are based upon a practical calculus of anticommuting variables. They are used to calculate correlation functions and counting distributions for general systems of fermions.

We apply the Schwinger-Keldysh formalism to study the nonequilibrium dynamics of the BCS transition to the superfluid state in trapped atomic ${}^6\mathrm{Li}.\mathrm{}$ We find that the Fokker-Planck equation for the probability distribution of the order parameter is, sufficiently close to the critical temperature, identical to the equation that describes the switching on of a single-mode laser.

By including the contribution of the thermal cloud in the Lagrangian of the condensate of a Bose gas, we extend the time-dependent variational method at zero temperature to study temperature-dependent low collective excitation modes. A Gaussian trial wave function of the condensate and a static distribution density of the thermal cloud are used and analytical expressions for temperature-dependent excitation frequencies obtained. Theoretical results are compared with measurements in the JILA and MIT experiments.

The nonclassical measures of nonclassical properties of radiation fields are proposed in parallel with those of the nonclassical states. It is shown that the nonclassical measures of sub-Poissonian photon statistics and squeezing are bounded below $1/2$ and their physical implications are discussed in comparison with the nonclassical measures of states.

We study two-dimensional optical pattern formation in sodium vapor with a single-feedback mirror in a magnetic field. Complex transverse patterns sustained by resonant interactions arise under conditions when wave and static composite modes are excited simultaneously.

Here we report on what is to our knowledge, the first achievement of saturated output in a collisionally pumped, Ni-like Pd soft-x-ray laser at $14.7$ nm $(J=0\mathrm{}-1$ line). Using a single prepulse scheme and a 100-ps pulse duration we were able to observe the onset of saturated output and to measure a gain-length product of $15.8$ (gain=8.3 ${\mathrm{cm}}^{-1})$ on flat slab targets. A systematic investigation and discussion on beam divergence, deflection angle, and space-resolved measurements is also presented. All these aspects lead to the evidence of the onset of saturation. The measured drive-irradiance threshold for the x-ray laser emission was only $6.6$ ${\mathrm{T}\mathrm{W}/\mathrm{c}\mathrm{m}}^2$ (≈165 GW) while the maximum used drive irradiance was only $12$ ${\mathrm{T}\mathrm{W}/\mathrm{c}\mathrm{m}}^2$ (≈300 GW). These values correspond, respectively, to a pump energy of 16.5 J and 30 J in 100 ps and demonstrate the realization of a real table-top soft-x-ray laser. A key role in the achievement of these results was played by the reduction in the roughness of the target surface, which greatly improves the performances of the Pd x-ray laser.

The effects of intrinsic spontaneous-emission noise on the nonlinear dynamics of an optically injected semiconductor laser is quantitatively studied through a single-mode model. It is found that for periodic motions, the effect of this noise source is to kick phase points off the underlying deterministic limit cycle. When the noise is not too strong so that phase points can stay near the limit cycle, phase points execute Brownian-like motions. When the noise is very strong, or the convergent flow near the attractor is weak, noise instantly kicks phase points away from the attractor to a region where nonlinearity is very strong. Then the diffusional process is slower than a standard Brownian motion. For chaotic motions, the characteristic short-term predictability of chaos is destroyed with an increasing amount of noise. The chaotic motions near the periodic/chaotic boundary, as well as at some locations inside the chaotic region, are more susceptible to noise. This suggests that there may exist fine structures inside the chaotic region.

The operational approach to the measurement of phase studied by Noh, Fougerès, and Mandel [Phys. Rev. A 45, 424 (1992); 46, 2840 (1992); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 47, 4535 (1993)] is applied to the measurement of the state of polarization of fully polarized light. Operational counterparts of the quantum Stokes parameters are introduced and their fluctuations are examined. It is shown that if the polarized field is weak, the measured fluctuations are influenced not only by the quantum properties of the source field but also that of the measurement. This character is reflected on the measured probability distributions of the parameters of polarization, which are also investigated independently for the fully polarized coherent states and the Fock states as the initial field strength is varied. Finally, connection between the operational approach to the measurement of polarization and the su(2) interferometry is examined.

We investigate the modification of quantum noise in the phase quadratures of the resonance fluorescence of an atom dressed by an intense laser field and coupled to a frequency-tunable cavity mode with finite bandwidth. The in-phase quadrature of the atomic fluorescence emitted from the side of the cavity exhibits two-mode squeezing at the Rabi sideband frequencies. The squeezing may be greatly enhanced (or inhibited) by appropriately tuning the cavity resonant frequency.

Measurements of the Young interference pattern for down-converted light in the two-photon state are performed. Light in the two-photon state is generated at the output of a Hong-Ou-Mandel balanced interferometer. Two-photon interference patterns with visibilities up to 100% are obtained. Visibilities of the two-photon interference patterns as a function of the double-slit separation are obtained and show that the transverse coherence length of the two-photon light is much larger than the one-photon beam for the distances considered in the experiment.

We study the entanglement between the two beams exiting a Mach-Zehnder interferometer fed by a couple of squeezed-coherent states with arbitrary squeezing parameter. The quantum correlations at the output are functions of the internal phase shift of the interferometer, with the output state ranging from a totally disentangled state to a state whose degree of entanglement is an increasing function of the input squeezing parameter. A couple of squeezed vacuums at the input lead to maximum entangled state at the output. The fringe visibilities resulting from measuring the coincidence counting rate or the squared difference photocurrent are evaluated and compared to each other. Homodynelike detection turns out to be preferable in almost all situations, with the exception of the very-low-signal regime.

We consider a vectorial Kerr-medium model including transverse spatial effects. We analyze cases in which, immediately above the threshold of the spatial instability, the homogeneous pump wave gives rise to two tilted waves corresponding to a stripe pattern in the near field. We analyze both the self-focusing and the self-defocusing case and we point out the existence of anti-correlations between the quantum fluctuations of the intensity of the pump and the sum of the intensities of the two tilted waves creating the transverse pattern in the near field. We also evaluate the efficiency of this scheme as a quantum nondemolition (QND) scheme which uses the tilted waves as a “meter” to measure the intensity fluctuations of the pump. Our results show the posibility of a QND measurement in the self-defocusing case. In this case, and for a linearly polarized pump, the output pump beam (uniform in the transverse plane) and the pattern have orthogonal polarization and could be easily separated experimentally.