Nonexponential decoherence and momentum subdiffusion in a quantum lévy kicked rotator

We investigate decoherence in the quantum kicked rotator (modeling cold atoms in a pulsed optical field) subjected to noise with power-law tail waiting-time distributions of variable exponent (Lévy noise). We demonstrate the existence of a regime of nonexponential decoherence where the notion of a decoherence rate is ill defined. In this regime, dynamical localization is never fully destroyed, indicating that the dynamics of the quantum system never reaches the classical limit. We show that this leads to quantum subdiffusion of the momentum, which should be observable in an experiment.     

Quantum reflection times for attractive potential tails

When a wave packet with a narrow momentum distribution is quantum reflected in a purely attractive potential proportional to -1/r(alpha), alpha>2, it generally experiences a time gain compared to a free particle reflected at r=0; for alpha=3 and very low energies there are large time delays. In quantum reflection of an atomic beam by a surface, such a time gain (delay) represents an apparent plane of reflection which is shifted in front of (behind) the surface. The quantum reflected wave is always delayed with respect to the classical particle accelerated in the attractive potential.     

Heating map in classical and quantum mechanics

Heating map of the classical probability-distribution function (in the phase space) and of density matrix (in the position representation) in quantum mechanics is introduced and its positivity is proved. The relation of the heating map to scaling transform and unitary squeezing transform of the momentum variable in the Wigner function is used to prove that noncanonical scaling transform of the position and momentum provides positive (but not completely positive!) map of density operator. The connection of momentum scaling transform with time scaling transform and Plancks constant scaling transform is discussed.     

A Remark on Nonequivalent Star Products via Reduction for CPn

In this Letter, we construct nonequivalent star products on CPⁿ by phase space reduction. It turns out that the nonequivalent star products occur very natural in the context of phase space reduction by deforming the momentum map of the U(1)-action on Cⁿ⁺¹\{0}; into a quantum momentum map and the corresponding momentum value into a quantum momentum value such that the level set, i.e. the constraint surface, of the quantum momentum map coincides with the classical one. All equivalence classes of star products on CPⁿ are obtained by this construction.     

Limitations of the strong field approximation in ionization of the hydrogen atom by ultrashort pulses

We present a theoretical study of the ionization of hydrogen atoms as a result of the interaction with an ultrashort external electric field. Doubly-differential momentum distributions and angular momentum distributions of ejected electrons calculated in the framework of the Coulomb-Volkov and strong field approximations, as well as classical calculations are compared with the exact solution of the time dependent Schr ödinger equation. We show that in the impulsive limit, the Coulomb-Volkov distorted wave theory reproduces the exact solution. The validity of the strong field approximation is probed both classically and quantum mechanically. We found that classical mechanics describes the proper quantum momentum distributions of the ejected electrons right after a sudden momentum transfer, however pronounced the differences at latter stages that arise during the subsequent electron-nucleus interaction. Although the classical calculations reproduce the quantum momentum distributions, it fails to describe properly the angular momentum distributions, even in the limit of strong fields. The origin of this failure can be attributed to the difference between quantum and classical initial spatial distributions.     

受线性阻尼和含时外力作用粒子的量子行为

运用广义线性量子变换理论求解了采用两种不同正则化变换给出的受线性阻尼和含时外力作用粒子的哈密顿量;给出了演化算符的严格解,以及粒子坐标和动量的期望值、 量子涨落.结果表明: 1)两种正则化变换是等价的; 2)线性阻尼对粒子的动量存在压缩效应, 动量的偏差随时间t按负指数规律衰减,阻尼系数越大,衰减越快; 3)粒子坐标和动量的期望值与经典值相同.     

Approximate Hidden Variables

The usual definition of (non-contextual) hidden variables is found to be too restrictive, in the sense that, according to it, even some classical systems do not admit hidden variables. A more general concept is introduced and the term approximate hidden variables is used for it. This new concept avoids the aforementioned problems, since all classical systems admit approximate hidden variables. Standard quantum systems do not admit approximate hidden variables, unless the corresponding Hilbert space is 2-dimensional. However, an appropriate non-standard quantum system, which arises by focussing on momentum and position and neglecting the remaining observables, admits approximate hidden variables. Systems associated with JBW-algebras (resp. von Neumann algebras) and satisfying some mild conditions admit approximate hidden variables iff they are classical, that is, iff the corresponding JBW-algebra (resp. von Neumann algebra) is associative (resp. commutative).     

Quantum theory of single events: Localized De Broglie wavelets,Schrödinger waves,and classical trajectories

For an arbitrary potential V with classical trajectoriesx=g(t), we construct localized oscillating three-dimensional wave lumps (x, t,g) representing a single quantum particle. The crest of the envelope of the ripple follows the classical orbitg(t), slightly modified due to the potential V, and (x, t,g) satisfies the Schrödinger equation. The field energy, momentum, and angular momentum calculated as integrals over all space are equal to the particle energy, momentum, and angular momentum. The relation to coherent states and to Schrödinger waves is also discussed.     

Noncommutativity effects in FRW scalar field cosmology

We study effects of noncommutativity on the phase space generated by a non-minimal scalar field which is conformally coupled to the background curvature in an isotropic and homogeneous FRW cosmology. These effects are considered in two cases, when the potential of scalar field has zero and nonzero constant values. The investigation is carried out by means of a comparative detailed analysis of mathematical features of the evolution of universe and the most probable universe wave functions in classically commutative and noncommutative frames and quantum counterparts. The influence of noncommutativity is explored by the two noncommutative parameters of space and momentum sectors with a relative focus on the role of the noncommutative parameter of momentum sector. The solutions are presented with some of their numerical diagrams, in the commutative and noncommutative scenarios, and their properties are compared. We find that impose of noncommutativity in the momentum sector causes more ability in tuning time solutions of variables in classical level, and has more probable states of universe in quantum level. We also demonstrate that special solutions in classical and allowed wave functions in quantum models impose bounds on the values of noncommutative parameters.     

Stability and equilibrium states of infinite classical systems

We prove that any stationary state describing an infinite classical system which is stable under local perturbations (and possesses some strong time clustering properties) must satisfy the classical KMS condition. (This in turn implies, quite generally, that it is a Gibbs state.) Similar results have been proven previously for quantum systems by Haag et al. and for finite classical systems by Lebowitz et al.Supported by N.S.F. Grant MPS 71-03375 A03. Part of this work carried out at the Courant Institute where it was supported by N.S.F. Grant GP-37069X.Supported in part by AFOSR Grant #73-2430 and N.S.F. Grant MP S75-20638.Supported by N.S.F. Grant # GP33136X-2. Part of this work was carried out at the Institute for Advanced Study.     

On the spin of Solitons in 2+1 dimensional models

It is believed that the charged vortex in the Chern-Simons-Higgs model can have fractional spin, since the extra angular momentum of the static vortex calculated from the classical energy-momentum tensor is nonzero. We re-examine the spin of the charged vortex by use of quantum mechanical method, by which the baby skyrmion in theO(3) nonlinear -model with the Hopf term is pointed out to have fractional spin. It is shown that the spin of the charged vortex obtained from the quantum mechanical argument does not necessarily coincide with the value of classical extra angular momentum. Moreover, it is found that its value is not unique, since it is one of quantities which depend on the gauge condition of the Chern-Simons gauge field.     

Quantum particles and classical particles

The relation between wave mechanics and classical mechanics is reviewed, and it is stressed that the latter cannot be regarded as the limit of the former as 0. The motion of a classical particle (or ensemble of particles) is described by means of a Schrödinger-like equation that was found previously. A system of a quantum particle and a classical particle is investigated (1) for an interaction that leads to stationary states with discrete energies and (2) for an interaction that enables the classical particle to act as a measuring instrument for determining a physical variable of the quantum particle.     

A Relativistic Schrödinger-like Equation for a Photon and Its Second Quantization

Maxwell's equations are formulated as a relativistic Schrödinger-like equation for a single photon of a given helicity. The probability density of the photon satisfies an equation of continuity. The energy eigenvalue problem gives both positive and negative energies. The Feynman concept of antiparticles is applied here to show that the negative-energy states going backward in time (t –t) give antiphoton states, which are photon states with the opposite helicity. For a given mode, properties of a photon, such as energy, linear momentum, total angular momentum, orbital angular momentum, and spin are equivalent in both classical electromagnetic field theory and quantum theory. The single-photon Schrödinger equation is field (or second) quantized in a gauge-invariant way to obtain the quantum field theory of many photons. This approach treats the quantization of electromagnetic radiation in a way that parallels the quantization of material particles and, thus, provides a unified treatment.     

Velocity Operators and Time-Energy Relations in Relativistic Quantum Mechanics

According to both Dirac's and Kemmer's relativistic quantum theories, the eigenvalues of the velocity operator are +c and –c. This false result is avoided if certain alternative particle coordinates are adopted. Another advantage is that the new coordinates occur in additional constants of the motion. These are sui generis angular momenta obtained by taking the vector product of the nonstandard coordinates with the linear momentum. An additional virtue of the new velocity operator is that, like in classical mechanics, it is proportional to the linear momentum. Besides, the zeroth component of the new set of coordinates does not commute with the hamiltonian, which results in a genuine indeterminacy relation between time and energy.     

Chaotic filtering of moving atoms in pulsed optical lattices by control of dynamical localization

We propose a mechanism for a velocity-selective device which would allow packets of cold atoms traveling in one direction through a pulsed optical lattice to pass undisturbed, while dispersing atoms traveling in the opposite direction. The mechanism is generic and straightforward: for a simple quantum kicked rotor pulsed with unequal periods, the quantum suppression of momentum diffusion (dynamical localization) yields momentum localization lengths L which are no longer isotropic, as in the standard case, but vary smoothly and controllably with initial momentum.     

Quantum Theory of Self‐Phase Modulation of Ultrashort Light Pulses in a Medium with Inertial Electronic Nonlinearity

We develop a systematic quantum theory of formation of ultrashort light pulses in a squeezed state at selfphase modulation. The response time of the electronic Kerr nonlinearity of the medium is accounted for and the dispersion of linear properties of the medium is described in first approximation. The theory uses the approach based on the momentum operator for the electric field. The response function of the nonlinearity is contained in the interaction operator. The results obtained are valid when the pulse duration is far greater than the nonlinear response time and the carrier pulse frequency is offresonance. It is established that the instantaneous spectral distribution depends quasistatically on the pulse form. In the general case, the spectral distribution of the quadraturesqueezed component is defined by both the response time of the nonlinearity and the magnitude of the nonlinear phase addition. It is also shown that the frequency at which suppression of quantum fluctuations is greatest can be controlled by adjusting the phase of the initial coherent light pulse. It is found that, by registering the quadraturesqueezed pulse using a balanced homodyne detector, the squeezing effect can be efficiently measured if the timedependent phase of the heterodyne pulse obeys a specific dependence defined by the pulse form. The spectral photonnumber distribution of the quadraturesqueezed pulse is studied, and the photon antibunching effect is found for the photon number in a limited spectral band.     

BRST Cohomology and Phase Space Reduction in Deformation Quantization

In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a "quantum" Chevalley-Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is "sufficiently nice", e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counterexamples are discussed.     

Observation of quantum particles on a large space-time scale

A quantum particle observed on a sufficiently large space-time scale can be described by means of classical particle trajectories. The joint distribution for large-scale multiple-time position and momentum measurements on a nonrelativistic quantum particle moving freely inR ^v is given by straight-line trajectories with probabilities determined by the initial momentum-space wavefunction. For large-scale toroidal and rectangular regions the trajectories are geodesics. In a uniform gravitational field the trajectories are parabolas. A quantum counting process on free particles is also considered and shown to converge in the large-space-time limit to a classical counting process for particles with straight-line trajectories. If the quantum particle interacts weakly with its environment, the classical particle trajectories may undergo random jumps. In the random potential model considered here, the quantum particle evolves according to a reversible unitary one-parameter group describing elastic scattering off static randomly distributed impurities (a quantum Lorentz gas). In the large-space-time weak-coupling limit a classical stochastic process is obtained with probability one and describes a classical particle moving with constant speed in straight lines between random jumps in direction. The process depends only on the ensemble value of the covariance of the random field and not on the sample field. The probability density in phase space associated with the classical stochastic process satisfies the linear Boltzmann equation for the classical Lorentz gas, which, in the limith0, goes over to the linear Landau equation. Our study of the quantum Lorentz gas is based on a perturbative expansion and, as in other studies of this system, the series can be controlled only for small values of the rescaled time and for Gaussian random fields. The discussion of classical particle trajectories for nonrelativistic particles on a macroscopic spacetime scale applies also to relativistic particles. The problem of the spatial localization of a relativistic particle is avoided by observing the particle on a sufficiently large space-time scale.