The two-component Bose-Einstein condensate is known to break its symmetry spontaneously. The stability of the symmetric solution has been investigated by solving the ground-state density profiles for different parameter regimes. We emphasize the importance of the trap geometry for the formation of the condensate.

We show that the quantum topological effect predicted by Aharonov and Casher (AC effect) [Phys. Rev. Lett. 53, 319 (1984)] may be used to create circulating states of magnetically trapped atomic Bose-Einstein condensates (BECs). A simple experimental setup is suggested based on a multiply connected geometry such as a toroidal trap or a magnetic trap pinched by a blue-detuned laser. We give numerical estimates of such effects within the current atomic BEC experiments, and point out some interesting properties of the associated quantized circulating states.

A theoretical formulation is outlined that allows us to extend the semiclassical Van Vleck–Gutzwiller formulation to the description of nonadiabatic quantum dynamics on coupled potential-energy surfaces. In this formulation the problem of a classical treatment of discrete quantum degrees of freedom (DoF) such as electronic states is bypassed by transforming the discrete quantum variables to continuous variables. The mapping approach thus consists of two steps: an exact quantum-mechanical transformation of discrete onto continuous DoF (the “mapping”) and a standard semiclassical treatment of the resulting dynamical problem. Extending previous work [G. Stock and M. Thoss, Phys. Rev. Lett. 78, 578 (1997)], various possibilities for obtaining a mapping from discrete to continuous DoF are investigated, in particular the Holstein-Primakoff transformation, Schwinger’s theory of angular momentum [in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. V. Dam (Academic, New York, 1965)], and the spin-coherent-state representation. Although all these representations are exact on the quantum-mechanical level, the accuracy of their semiclassical evaluation is shown to differ considerably. In particular, it is shown that a generalization of Schwinger’s theory appears to be the only transformation that provides an exact description of a general N-level system within a standard semiclassical evaluation. Exploiting the connection between spin-coherent states and Schwinger’s representation for a two-level system, furthermore, a semiclassical initial-value representation of the corresponding spin-coherent-state propagator is derived. Although this propagator represents an approximation, its appealing numerical features make it a promising candidate for the semiclassical description of large molecular systems with many DoF. Adopting various spin-boson-type models (i.e., a two-level system coupled to a single or many DoF), computational studies are presented for Schwinger’s and the spin-coherent-state representation, respectively. The performance of the semiclassical approximation in the case of regular and chaotic classical dynamics as well as for multimode electronic relaxation dynamics is discussed in some detail.

We discuss a method to launch gap solitonlike structures in atomic Bose-Einstein condensates confined in optical traps. Bright vector solitons consisting of a superposition of two hyperfine Zeeman sublevels can be created for both attractive and repulsive interactions between the atoms. Their formation relies on the dynamics of the atomic internal ground states in two far-off-resonance counterpropagating $({\sigma }^{+}-{\sigma }^{-})$-polarized laser beams that form the optical trap. Numerical simulations show that these solitons can be prepared from a one-component state provided with an initial velocity.

We study propagation of low-intensity light in a medium within a one-dimensional (1D) model electrodynamics. It is shown that the coupled theory for light and matter fields may be solved, in principle exactly, by means of stochastic simulations that account for both collective linewidths and line shifts, and for quantum statistical position correlations of the atoms. Such simulations require that one synthesize atomic positions that have correlation functions appropriate for the given type of atomic sample. We demonstrate how one may simulate both a Bose-Einstein condensate (BEC) and a zero-temperature noninteracting Fermi-Dirac gas. Results of simulations of light propagation in such quantum degenerate gases are compared with analytical density expansions obtained by adapting the approach of Morice, Castin, and Dalibard [Phys. Rev. A 51, 3896 (1995)] to the 1D electrodynamics. A BEC exhibits an optical resonance that narrows and stays somewhat below the atomic resonance frequency as collective effects set in with increasing atom density. The first two terms in the analytical density expansion are in excellent agreement with numerical results for a condensate. While fermions display a similar narrowing and shift of the resonance with increasing density, already in the limit of very dilute gas the linewidth is only half of the resonance linewidth of an isolated atom. We attribute this to the regular spacing between the atoms, which is enforced by the Pauli exclusion principle. The analytical density expansion successfully predicts the narrowing, and also gives the next term in the density expansion of the optical response in semiquantitative agreement with numerical simulations.

We discuss the use of the Born and Markov approximations in describing the dynamics of an atom laser. In particular, we investigate the applicability of the quantum optical Born-Markov master equation for describing output coupling. We derive conditions based on the atomic reservoir and atom dispersion relations for when the Born-Markov approximations are valid and discuss parameter regimes where these approximations fail in our atom laser model. Differences between the standard optical laser model and the atom laser are due to a combination of factors, including the parameter regimes in which a typical atom laser would operate, the different reservoir state that is appropriate for atoms, and the different dispersion relations between atoms and photons. We present results based on an exact method in the regimes in which the Born-Markov approximation fails. The exact solutions in some experimentally relevant parameter regimes give a nonexponential loss of atoms from a cavity.

Two-photon excitation of a three-level atom in a ladder configuration $\left(\overrightarrow{1}\overrightarrow{2}3\right)$ by simultaneous illumination with fields in squeezed vacuum and coherent states results in quantum interference for the excitation process. The particular configuration considered here is one for which the signal and idler output fields of a subthreshold nondegenerate optical parametric oscillator are in resonance with the two-stepwise dipole atomic transitions $(\overrightarrow{1}2,\overrightarrow{2}3),$ while a “reference oscillator” field is in two-photon resonance with the quadrupole transition $\left(\overrightarrow{1}3\right).$ In an extension of the work of Ficek and Drummond [Phys. Rev. A 43, 6247 (1991)], a theoretical formulation based on the full quantum master equation for the problem is presented. The combined effects of quantum interference and the nonclassical character of the squeezed state are investigated, and offer the potential for a new detection strategy for quantum fluctuations of the electromagnetic field with ultrahigh frequencies (10’s–100’s THz). Based on the theory developed, we analyze quantum interference in excitation in several special cases relevant to experimental realizations, including the effects of a small focusing angle of the squeezing onto the atoms, and unusual population inversions. Special emphasis is given to identifying intrinsically quantum optical field effects versus classical field effects. Procedures that could distinguish between the two (i.e., classical and nonclassical) are suggested.

A quantum theory of dispersion for an inhomogeneous solid is obtained, from a starting point of multipolar coupled atoms interacting with an electromagnetic field. The dispersion relations obtained are equivalent to the standard classical Sellmeir equations obtained from the Drude-Lorentz model. In the homogeneous (plane-wave) case, we obtain the detailed quantum mode structure of the coupled polariton fields, and show that the mode expansion in all branches of the dispersion relation is completely defined by the refractive index and the group velocity for the polaritons. We demonstrate a straightforward procedure for exactly diagonalizing the Hamiltonian in one-, two-, or three-dimensional environments, even in the presence of longitudinal phonon-exciton dispersion, and an arbitrary number of resonant transitions with different frequencies. This is essential, since it is necessary to include at least one phonon (IR) and one exciton (UV) mode, in order to represent dispersion in transparent solid media accurately. Our method of diagonalization does not require an explicit solution of the dispersion relation, but relies instead on the analytic properties of Cauchy contour integrals over all possible mode frequencies. When there is longitudinal-phonon dispersion, the relevant group-velocity term is modified so that it only includes the purely electromagnetic part of the group velocity.

The influence of quantum interference between different decay channels of three-level atoms on various characteristics of matter and radiation is considered for the case of thermodynamic equilibrium of atoms in a photon gas. It is shown that for rare gases when collisions between atoms are not important, the direct consequence of such an interference is a spontaneously induced atomic coherence between two upper levels. Therefore, for an accurate description of such atoms and radiation in thermodynamic equilibrium with each other it is not sufficient to specify only Boltzmann exponents, which correspond to the diagonal elements of the density matrix. The reason for this is that the conventional Boltzmann description is applied to infinitely sharp eigenstates of the unperturbed Hamiltonian. As we show, it becomes essential to take into account nonzero width of the levels if the levels are relatively close to each other. This produces nonzero nondiagonal elements of the density matrix that modify the emissivity of the equilibrium medium and lead to a zero point at a certain frequency as well as to an enhanced intensity in the red wing at the corresponding temperature. It is shown that the occupation number of the photons is not changed and obeys the equilibrium Planck distribution.

We propose a scheme which can effectively restore fixed points in the quantum dynamics of repeated Jaynes-Cummings interactions followed by atomic state measurements, when the interaction times fluctuate randomly. It is based on selection of superposed atomic states whose phase correlations tend to suppress the phase fluctuations of each separate state. One suggested realization involves the convergence of the cavity field distribution to a single Fock state by conditional measurements performed on two-level atoms with fluctuating velocities after they cross the cavity. Another realization involves a trapped ion whose internal-motional state coupling fluctuates randomly. Its motional state is made to converge to a Fock state by conditional measurements of the internal state of the ion.

We propose to measure the quantum state of a single mode of the radiation field in a cavity—the signal field—by coupling it via a quantum-non-demolition Hamiltonian to a meter field in a highly squeezed state. We show that quantum-state tomography on the meter field using balanced homodyne detection provides full information about the signal state. We discuss the influence of measurement of the meter on the signal field.

When laser light propagates through a resonant medium, the transmitted beam can exhibit excess intensity noise [amplitude modulation (AM)]. In a semiclassical description of the phenomenon, laser phase noise (PM) induces fluctuations in the medium’s electric susceptibility, which in turn cause fluctuations in the transmitted intensity. The process provides an efficient means for PM-to-AM conversion, and intuition suggests that large linewidth lasers should exhibit much greater PM-to-AM conversion than narrow linewidth lasers. Here we measure the relative intensity noise (RIN) for two diode lasers whose linewidths $\Delta {\nu }_L$ differ by more than ${10}^2,$ after the lasers have propagated through a resonant rubidium vapor. Though the RIN of the narrow linewidth laser is only reduced by a factor of about 6 compared to the broad linewidth laser, our results are nonetheless consistent with numerical simulations of the PM-to-AM conversion process. In particular, both computation and analytical theory indicate that RIN is a nonlinear function of $\Delta {\nu }_L.$ For single-mode laser linewidths less than the atomic dephasing rate, RIN increases like $\sqrt{\Delta {\nu }_L},$ while for linewidths greater than the atomic dephasing rate RIN is a decreasing function of $\Delta {\nu }_L.$

We consider a system of laser-cooled ions in a linear harmonic trap and study the phenomenon of squeezing exchange between their internal and motional degrees of freedom. An interesting relation between the quantum noise reduction (squeezing) and the dynamical evolution is found when the internal and motional subsystems are prepared in properly squeezed (intelligent) states. Specifically, the evolution of the system is fully governed by the relative strengths of spectroscopic and motional squeezing, including the phenomenon of total cancellation of the interaction when the initial squeezing parameters are equal.

We analyze the probe absorption spectra of a strongly driven $\Lambda$ system with arbitrary spacing between the two lower levels. Under the condition that the spacing between the two lower levels is of the order of a strong-pump-field Rabi frequency, important interference effects are observed due to same-field coupling with both the transitions. We report the possibility of a significant gain from the cross talk among optical transitions, and a strong dependence of gain and absorption on the relative polarizations of the pump and probe beams. We present an analysis based on dressed states to explain our numerical results.

A formalism is developed that allows one to calculate the propagation of quantum noise, technical noise, and signals through a linearized quantum optical bus, i.e., a complex composite multibeam optical quantum system. General expressions for the output squeezing, the correlation coefficient, and quantum correlation coefficient between output observables and the transfer coefficients are derived and discussed in detail. The total correlation is introduced as a measure of the overall correlation of a set of output quadratures. From its geometric interpretation a multidimensional uncertainty relation is derived. Direct quantum-state preparation (QSP) is introduced as a method complementary to the conventional, indirect QSP. The minimal variance achievable by direct QSP is shown to provide superior noise suppression, including quantum noise suppression, than the conditional variance provided by indirect QSP. As an application we calculate and discuss the properties of a balanced bus of beam splitters with squeezed vacua at the usually unused ports. An important experimental effect is imperfect beam mode match, which is shown to lead to dramatic effects. A theory is derived which quantitatively matches measurements. Analytic expressions for the imperfectly modematched squeezed-light beam splitter including detection losses are given. Finally, a direct method is used to calculate the input-output matrix for the multiport subthreshold degenerate optical parametric amplifier for arbitrary parametric and mirror coupling strengths and detuning.

By taking carrier momentum into account, we present an extension of the commonly used model that describes the polarization properties of vertical-cavity surface-emitting lasers (VCSEL’s). We study how this modifies the polarization dependence of the gain saturation, and introduce the role of the crystalline symmetry of the semiconductor material. On a macroscopic level we show that a limited number of parameters is sufficient to describe fully the polarization properties of the VCSEL. On a microscopic level we show how the magnitudes of these parameters are determined by the dependence of the interband-transition dipole matrix element on the relative orientation of the carrier momentum and optical polarization. It turns out to be essential that the component of the carrier momentum in the plane of the quantum well is non-negligible. Furthermore, the relaxation rates of the carrier momentum and carrier spin are crucial in determining the magnitude of the polarization effects. Inclusion of the carrier momentum changes the interpretation of experimental results on the polarization of VCSEL’s, and makes it possible to understand experimental results that were until now unexplained.

Many areas of science require locking a probe signal to the center of a Lorentzian line. In this paper we experimentally explore the limits to the resolution for locking a sine-wave modulated probe signal to a Lorentzian line. Surprisingly we find that in some cases relatively high-frequency phase modulation (PM) noise on the probe signal (even 100 times the linewidth of the resonance) can significantly degrade the stability of the lock. The 6.835-GHz hyperfine transition in ${}^{87}\mathrm{Rb}$ was used as a test bed for these studies because it affords very high resolution for investigating this effect. The details presented here should prove useful for testing a complete theory of this effect and give some insight into the requirements of PM noise of local oscillators to be locked to a Lorentzian line.

The field in the micromaser cavity, the atoms which both pump and probe the micromaser, and the detectors that determine the outgoing states of the atoms are treated as a single system in order to derive a nonlinear master equation for the field in the micromaser cavity which takes the effect of undetected atoms into account. Analytic solutions for this master equation are found for both phase-insensitive and phase-sensitive detection. These solutions can be used to explicitly show the back-action effects that measurement has on the spectrum of the micromaser. In addition, these solutions are used to find analytic expressions for the mean number of successive detector clicks of the same kind as well as atom-atom and photon-photon correlation functions. In this way, the analytic solutions are linked to experimentally accessible data.

We identify a parameter that controls the localization length in a driven quantum system. This parameter results from an additional quantum degree of freedom. The center-of-mass motion of a two-level ion stored in a Paul trap and interacting with a standing-wave laser field exhibits this phenomenon. We also discuss the influence of spontaneous emission.

A finite relativistic model for free particles, which describes the collapse of the state vector, is presented. The interaction of particles with a classical fluctuating field $w\left(x\right)\mathrm{}$ causes collapse, as in the (so far) physically satisfactory nonrelativistic continuous spontaneous localization (CSL) model. In previous relativistic models, where $w\left(x\right)\mathrm{}$ has a white-noise spectrum, collapse is accompanied by spontaneous creation of particles with all energies out of the vacuum [amounting to infinite energy/(sec-volume)]. In this paper we explore one way of eliminating such vacuum excitation. We present a formalism which allows an arbitrary spectrum for $w\left(x\right).$ We point out, in lowest order of perturbation theory, that it is timelike components of this spectrum which are responsible for vacuum excitation but (appropriate, it seems, to a fundamentally nonlocal phenomenon) it is spacelike components which are responsible for collapse. We restrict the spectrum of $w\left(x\right)\mathrm{}$ to that of a tachyon of mass $\mu =ħ/ac\approx 1\hspace{0.5em}\mathrm{eV},$ where $a\approx {10}^{-5}\hspace{0.5em}\mathrm{cm}\mathrm{}$ is the Ghirardi-Rimini-Weber (GRW) collapse model scale parameter used in CSL. However, in higher orders than the first, there is still vacuum excitation due to the energy-momentum supplied by particle propagators. We opt to explore a simple means of eliminating this vacuum excitation: by removing the time-ordering operation from the state vector evolution so that particle propagators are on the mass shell. The result is a finite theory: there is no vacuum excitation (and no need for renormalization) as each vertex describes a finite physical process, the spontaneous emission or absorption of a real tachyon by a particle (pair creation or annihilation is not permitted by energy-momentum conservation). This process may be thought of as analogous to the GRW “hitting” description of collapse. The cost of no time ordering is that at each vertex a mass M particle’s wave function can spread by as much as $\approx cT(\mu /M)$ in time T. This has the result that, beginning at perturbation order $2M/\mu ,$ there is a nonvanishing probability for the particle to be found outside its light cone: for electrons and nucleons, the probability of this spacelike transport is negligibly small. Apart from this effect, the density matrix of particles in one region evolves independently of particles in another spacelike separated region. We thus obtain a finite, reasonably sensible, relativistic collapse model for free particles.