Odinger Picture Research Articles

An explicit schrödinger picture for aharonov’s modular-variable concept

We propose to address in a natural manner, the modular variable concept explicitly in a Schr\"odinger picture. The idea of Modular Variables was introduced in 1969 by Aharonov, Pendleton and Petersen to explain certain non-local properties of quantum mechanics. Our approach to this subject is based on Schwinger's finite quantum kinematics and it's continuous limit.

Model of a relativistic oscillator in a generalized Schrödinger picture

AbstractIn a generalized Schrödinger picture, we consider the motion of a relativistic particle in an external field (like in the case of a harmonic oscillator). In this picture the analogs of the Schrödinger operators of the particle are independent of both the time and the space coordinates. These operators induce operators which are related to Killing vectors of the Anti de Sitter (AdS) space. We also consider the nonrelativistic limit.

Implementation of the time-dependent configuration-interaction singles method for atomic strong-field processes

We present an implementation of the time-dependent configuration-interaction singles (TDCIS) method for treating atomic strong-field processes. In order to absorb the photoelectron wave packet when it reaches the end of the spatial grid, we add to the exact nonrelativistic many-electron Hamiltonian a radial complex absorbing potential (CAP). We determine the orbitals for the TDCIS calculation by diagonalizing the sum of the Fock operator and the CAP using a flexible pseudospectral grid for the radial degree of freedom and spherical harmonics for the angular degrees of freedom. The CAP is chosen such that the occupied orbitals in the Hartree-Fock ground state remain unaffected. Within TDCIS, the many-electron wave packet is expanded in terms of the Hartree-Fock ground state and its single excitations. The virtual orbitals satisfy nonstandard orthogonality relations, which must be taken into consideration in the calculation of the dipole and Coulomb matrix elements required for the TDCIS equations of motion. We employ a stable propagation scheme derived by second-order finite differencing of the TDCIS equations of motion in the interaction picture and subsequent transformation to the Schr\odinger picture. Using the TDCIS wave packet, we calculate the expectation value of the dipole acceleration and the reduced density matrix of the residual ion. The technique implemented will allow one to study electronic channel-coupling effects in strong-field processes.

Spontaneous emission of a moving atom in vacuum

Spontaneous emission of an excited two-level nonrelativistic atom is examined in the Schr\"odinger picture by treating the atom's external and internal degrees of freedom on the same quantum footing. After all atomic transitions that lead to the evolution of this atom-vacuum system are analyzed, it is found that the atom's spontaneous emission rate is reduced by its center-of-mass motion. It is also found that, as a result of the entanglement between atomic and photonic states, the spontaneous emission process cannot be explained as a process that takes place in one coordinate system moving with the atom but is observed in another stationary system.

Von Neumann Mutual Information for Anisotropic Coupled Oscillators Interacting with a Single Two-Level Atom

We consider the interaction between a two-level atom and two electromagnetic fields injected simultaneously within a cavity, with the interaction between the fields in parametric frequency-converter form. The wave function in Schr odinger picture is obtained under certain conditions and consequently the density matrix. By employing a generalization of the von Neumann mutual information (in the context of Tsallis' nonextensive statistics) we measure the degree of entanglement for the present system. An important change is observed in the generalized mutual information depending on the entropic index. We also measure the minimum degree of entanglement during the transition from collapse to revival and vice-versa. Successive revival peaks show a lowering of the local maximum point indicating a dissipative irreversible change in the atomic state.

Short distance and initial state effects in inflation: Stress tensor and decoherence

We present a consistent low energy effective field theory framework for parameterizing the effects of novel short distance physics in inflation, and their possible observational signatures in the Cosmic Microwave Background. We consider the class of general homogeneous, isotropic initial states for quantum scalar fields in Robertson-Walker (RW) spacetimes, subject to the requirement that their ultraviolet behavior be consistent with renormalizability of the covariantly conserved stress tensor which couples to gravity. In the functional Schr\"odinger picture such states are coherent, squeezed, mixed states characterized by a Gaussian density matrix. This Gaussian has parameters which approach those of the adiabatic vacuum at large wave number, and evolve in time according to an effective classical Hamiltonian. The one complex parameter family of $\alpha$ squeezed states in de Sitter spacetime does not fall into this UV allowed class, except for the special value of the parameter corresponding to the Bunch-Davies state. We determine the finite contributions to the inflationary power spectrum and stress tensor expectation value of general UV allowed adiabatic states, and obtain quantitative limits on the observability and backreaction effects of some recently proposed models of short distance modifications of the initial state of inflation. For all UV allowed states, the second order adiabatic basis provides a good description of particles created in the expanding RW universe. Due to the absence of particle creation for the massless, minimally coupled scalar field in de Sitter space, there is no phase decoherence in the simplest free field inflationary models. We apply adiabatic regularization to the renormalization of the decoherence functional in cosmology to corroborate this result.

Energy density of vortices in the Schrödinger picture

The one-loop energy density of an infinitely thin static magnetic vortex in $\mathrm{SU}(2)$ Yang-Mills theory is evaluated using the Schr\"odinger picture. Both the gluonic fluctuations as well as the quarks in the vortex background are included. The energy density of the magnetic vortex is discussed as a function of the magnetic flux. The center vortices correspond to local minima in the effective potential. These minima are degenerated with the perturbative vacuum if the fermions are ignored. Inclusion of fermions lifts this degeneracy, raising the vortex energy above the energy of the perturbative vacuum.

Fractional quantum Hall effect and quantum symmetry

Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group ${U}_{q}{(su}_{2})$ with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.

Wilson loops as precursors

There is substantial evidence that string theory on ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}_{5}$ is a holographic theory in which the number of degrees of freedom scales as the area of the boundary in Planck units. Precisely how the theory can describe bulk physics using only surface degrees of freedom is not well understood. A particularly paradoxical situation involves an event deep in the interior of the bulk space. The event must be recorded in the (Schr\"odinger picture) state vector of the boundary theory long before a signal, such as a gravitational wave, can propagate from the event to the boundary. In a previous paper with Polchinski, we argued that the ``precursor'' operators which carry information stored in the wave during the time when it vanishes in a neighborhood of the boundary are necessarily non-local. In this paper we argue that the precursors cannot be products of local gauge invariant operators such as the energy momentum tensor. In fact gauge theories have a class of intrinsically non-local operators which cannot be built from local gauge invariant objects. These are the Wilson loops. We show that the precursors can be identified with Wilson loops whose spatial size is dictated by the UV-IR connection.

Quantum fields at any time

The canonical quantum theory of a free field using arbitrary foliations of a flat two-dimensional spacetime is investigated. It is shown that dynamical evolution along arbitrary spacelike foliations is unitarily implemented on the same Fock space as that associated with inertial foliations. It follows that the Schr\"odinger picture exists for arbitrary foliations as a unitary image of the Heisenberg picture for the theory. An explicit construction of the Schr\"odinger picture image of the Heisenberg Fock space states is provided. The results presented here can be interpreted in terms of a Dirac constraint quantization of parametrized field theory. In particular, it is shown that the Schr\"odinger picture physical states satisfy a functional Schr\"odinger equation which includes a slice-dependent $c$-number quantum correction, in accord with a proposal of Kucha\ifmmode \check{r}\else \v{r}\fi{}. The spatial diffeomorphism invariance of the Schr\"odinger picture physical states is established. Fundamental difficulties arise when trying to generalize these results to higher-dimensional spacetimes.

The quantum-jump approach to dissipative dynamics in quantum optics

Dissipation, the irreversible loss of energy and coherence, from a microsystem is the result of coupling to a much larger macrosystem (or reservoir) that is so large that one has no chance of keeping track of all of its degrees of freedom. The microsystem evolution is then described by tracing over the reservoir states, which results in an irreversible decay as excitation leaks out of the initially excited microsystems into the outer reservoir environment. Earlier treatments of this dissipation used density matrices to describe an ensemble of microsystems, either in the Schr\"odinger picture with master equations, or in the Heisenberg picture with Langevin equations. The development of experimental techniques to study single quantum systems (for example, single trapped ions, or cavity-radiation-field modes) has stimulated the construction of theoretical methods to describe individual realizations conditioned on a particular observation record of the decay channel. These methods, variously described as quantum-jump, Monte Carlo wave function, and quantum-trajectory methods, are the subject of this review article. We discuss their derivation, apply them to a number of current problems in quantum optics, and relate them to ensemble descriptions.

Causality in the excitation exchange between identical atoms

The interaction between two identical atoms, separated by a distance r that is large compared to an optical wavelength, is considered using an amplitude approach in the Schr\"odinger picture. When atom 1 is excited by a radiation pulse whose duration is long compared to an optical period but short compared to the time it takes for light to travel between the atoms, the probability for atom 2 to be excited is found to be identically zero for times t/c, provided only r-dependent terms in the probability are retained. The r-dependent probability amplitude for atom 2 to be excited and atom 1 to be in its ground state is not identically equal to zero for t/c. Rather it oscillates at the natural frequency of the optical transition. The Schr\"odinger picture approach allows one to clearly identify the various terms that give rise to the excitation probabilities. A diagrammatic technique is developed that facilitates the calculation.

Phase-sensitive reservoir modeled by beam splitters.

The superposition of input fields in a lossless beam splitter is studied in the Schr\"odinger picture by using the convolution of the positive P representations, and the convolution law for these representations is extended to other quasiprobability functions such as the Wigner and Q functions. We show that the reservoir can be modeled by an infinite array of beam splitters, and we use the convolution law and this model to derive the Fokker-Planck equation for a system coupled with a phase-sensitive reservoir. Solving this equation shows that a phase-sensitive attenuation and amplification can be described by the superposition of two independent quantum fields, one of which is the initial signal field and the other the squeezed thermal noise field representing the reservoir.

Heisenberg-picture approach to the exact quantum motion of a time-dependent harmonic oscillator.

The generalized invariant and the exact quantum motions are found in the Heisenberg picture for a harmonic oscillator with time-dependent mass and frequency in terms of classical solutions. It is shown that the Heisenberg pictures gives a relatively simpler picture that the Schr\"odinger picture and also manifestly exhibits the time independency of the invariant. We apply this method to the system with a linear sweep of frequency and Paul trap and study the squeezing properties.

Finite temperature quantum field theory in the functional Schrödinger picture.

We calculate the finite temperature Gaussian effective potential of scalar ${\mathrm{\ensuremath{\varphi}}}^{4}$ theory in the functional Schr\odinger picture. Our method is the direct generalization of the variational method proposed by Eboli, Jackiw, and Pi for quantum-mechanical systems, and gives the same result as that of Amelino-Camelia and Pi who used the self-consistent composite operator method.

Exact dynamics of a quantum dissipative system in a constant external field.

We study the quantum dynamics of the simplest dissipative system, a particle moving in a constant external field and interacting with a bath of harmonic oscillators with Ohmic spectral density. Applying the main idea and methods developed in our recent work [L. H. Yu and C. P. Sun, Phys. Rev. A 49, 592 (1994)] to this system, we obtain the simple and exact solutions for the coordinate operator of the system in the Heisenberg picture and the wave function of the composite system of the system and the bath in the Schr\"odinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions. The meaning of the wave function described by this effective Hamiltonian is clarified by analyzing the effect of the Brownian motion. In particular, the general effective Hamiltonian for an arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting result that the dissipation suppresses the wave-packet spreading.

Gaussian approximation of the (2+1)-dimensional Thirring model in the functional Schrödinger picture.

The (2+1)-dimensional Thirring model is studied by using the Gaussian approximation method in the functional Schr\"odinger picture. Although the dynamical symmetry breaking does not occur in the large N limit, it does occur in the Gaussian approximation which includes the higher order contributions in 1/N.

Effective Hamiltonian for the radiation in a cavity with a moving mirror and a time-varying dielectric medium.

We study the quantized field in a one-dimensional electromagnetic resonant cavity. The cavity contains a linear and lossless dielectric medium with frequency-independent polarizability. The dielectric permittivity is an externally prescribed function of both the space and the time. We also allow one of the cavity's mirrors to move in a given trajectory. Unlike other previous studies on the same system, we formulate an effective Hamiltonian so that the dynamics of the cavity field can be described in the Schr\"odinger picture. The effective Hamiltonian is quadratic in structure, therefore two-photon generation from the vacuum state can occur. We also discuss the case of resonant behavior of the system.

BRST symmetry in the Schrödinger picture.

We show that the effective Lagrangian including the gauge-fixing and ghost terms of the non-Abelian gauge theories can be derived in the functional Schr\"odinger picture by using the residual symmetry of the gauge-fixed Lagrangian. This residual gauge symmetry is shown to be equivalent to the well-known Becchi-Rouet-Stora-Tyutin symmetry.

Theory of four-wave mixing using an amplitude approach.

A theory of four-wave mixing is developed using an amplitude approach in the Schr\"odinger picture. It is shown that the four-wave mixing signal produced when three fields are incident on an ensemble of two-level atoms can be viewed as arising from constructive interference of the signals emitted at the various atomic sites. Fourteen distinct amplitudes contribute to the four-wave mixing signal. Some of these amplitudes involve final states in which one or two atoms are excited, and some involve final states in which photons are emitted into modes other than a phase-matched mode of interest. The amplitude approach has the advantage of clearly showing the atom-field correlations. It helps one to understand the underlying physical mechanisms responsible for creation of the four-wave mixing signal, as well as features in the line shape resulting from collisions and effects relating to atomic recoil on the absorption or emission of radiation. These so-called collision- and recoil-induced resonances are calculated explicitly using the amplitude approach, and an interpretation of the results in terms of the interference between different amplitudes is given. Our results, calculated for a finite interaction time between the atoms and the fields, are compared with analogous calculations in which an effective ground-state decay time is assumed. It is shown that the results differ if the atom-field interaction time is not the longest relevant time parameter in the problem.