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This paper discusses the number correlations and subsequent observables of Rindler photon in a Minkowski vacuum .

Historically, correspondence rules and quantum quasi-distributions were motivated by classical mechanics as a guide for obtaining quantum operators and quantum corrections to classical results. In this paper, we start with quantum mechanics and show how to derive the infinite number of quantum quasi-distributions and corresponding c-functions. An interesting aspect of our approach is that it shows how the c-numbers of position and momentum arise from the quantum operator.

A ground-state atom uniformly accelerated through the Minkowski vacuum can become excited by emitting an Unruh-Minkowski photon. We show that from the perspective of an accelerated atom, the sign of the frequency of the Unruh-Minkowski photons can be positive or negative depending on the acceleration direction. The accelerated atom becomes excited by emitting an Unruh-Minkowski photon which has negative frequency in the atom's frame, and decays by emitting a positive-frequency photon. This leads to interesting effects. For example, the photon emitted by accelerated ground-state atom cannot be absorbed by another ground-state atom accelerating in the same direction, but it can be absorbed by an excited atom or a ground-state atom accelerated in the opposite direction. We also show that similar effects take place for Cherenkov radiation. Namely, a Cherenkov photon emitted by an atom cannot be absorbed by another ground-state atom moving with the same velocity, but can be absorbed by an excited atom or a ground-state atom moving in the opposite direction.

We consider an arbitrary square integrable function $F$ on the phase space and look for the Wigner function closest to it with respect to the $L^2$ norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert-Schmidt operator with Weyl symbol $F$. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a {\it bona fide} Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics., 50 pages, to appear in SIAM J. Math. Anal

The Weyl procedure associates a function of two ordinary variables, called the c-function or symbol, with an operator, called the Weyl operator of the symbol. One generally formulates this association by defining the operator corresponding to a given symbol. In this paper we consider the reverse problem: Given the Weyl operator, what is the matching symbol? We give a number of explicit formulas for obtaining the symbol that would generate an arbitrary Weyl operator, and we illustrate each form with an example.

In 1932, Fermi presented a two-atom model for determining whether quantum mechanics is consistent with causality, and concluded that indeed it is. In the late 1960's, Shirokov and others found that Fermi's approximations may not have been sound, and when corrected, Fermi's model shows non-causal behavior. We show that if instead of time-dependent perturbation theory, the Magnus expansion is used to approximate the time-evolution operator, causality does follow., Comment: 7 pages, 1 figure

We present a new approach for obtaining quantum quasi-probability distributions, $P(\alpha,\beta)$, for two arbitrary operators, $\mathbf{a}$ and $\mathbf{b}$, where $\alpha$ and $\beta$ are the corresponding c-variables. We show that the quantum expectation value of an arbitrary operator can always be expressed as a phase space integral over $\alpha$ and $\beta$, where the integrand is a product of two terms: One dependent only on the quantum state, and the other only on the operator. In this formulation, the concepts of quasi-probability and correspondence rule arise naturally in that simultaneously with the derivation of the quasi-distribution, one obtains the generalization of the concept of correspondence rule for arbitrary operators., Comment: 14 pages

Quantum entanglement is a key resource that can be exploited for a range of applications such as quantum teleportation, quantum computation, and quantum cryptography. However, efforts to exploit entanglement in imaging systems have so far led to solutions such as ghost imaging, that have since found classical implementations. Here, we demonstrate an optical imaging protocol that relies uniquely on entanglement: Two polarizing patterns imprinted and superimposed on a metasurface are separately imaged only when using entangled photons. Unentangled light is not able to distinguish between the two patterns. Entangled single-photon imaging of functional metasurfaces promises advances towards the use of nanostructured subwavelength thin devices in quantum information protocols and a route to efficient quantum state tomography. MOE (Min. of Education, S’pore) Published version

We find that uniformly accelerated motion of a mirror yields excitation of a static two-level atom with simultaneous emission of a real photon. This occurs because of virtual transitions with probability governed by the Planck factor involving the photon frequency ν and the Unruh temperature. The result is different from the Unruh radiation of an accelerated atom, which is governed by the frequency of the atom, ω, rather than frequency of the emitted photon. We also find that the excitation probability oscillates as a function of the atomic position because of interference between contributions from the waves incident on and reflected from the mirror.

We consider the motion of a “particle” in a medium of variable index of refraction whose motion is governed by Snell’s law. We give a simple derivation of the Newtonian forces that dictate the motion and show that a position-dependent variable mass is necessary. We apply the equations of motion and find the conditions for a SOFAR channel, a phenomenon where rays are trapped in a channel.

A phase space approximation method for linear dispersive wave propagation with arbitrary initial conditions is developed. The results expand on a previous approximation in terms of the Wigner distribution of a single mode. In contrast to this previously considered single-mode case, the approximation presented here is for the full wave and is obtained by a different approach. This solution requires one to obtain (i) the initial modal functions from the given initial wave, and (ii) the initial cross-Wigner distribution between different modal functions. The full wave is the sum of modal functions. The approximation is obtained for general linear wave equations by transforming the equations to phase space, and then solving in the new domain. It is shown that each modal function of the wave satisfies a Schrödinger-type equation where the equivalent "Hamiltonian" operator is the dispersion relation corresponding to the mode and where the wavenumber is replaced by the wavenumber operator. Application to the beam equation is considered to illustrate the approach.

We present a method for calculating expectation values of operators in terms of a corresponding c-function formalism which is not the Wigner--Weyl position-momentum phase-space, but another space. Here, the quantity representing the quantum system is the expectation value of the displacement operator, parametrized by the position and momentum displacements, and expectation values are evaluated as classical integrals over these parameters. The displacement operator is found to offer a complete orthogonal basis for operators, and some of its other properties are investigated. Connection to the Wigner distribution and Weyl procedure are discussed and examples are given., 6 pages + conclusion, references, and appendices

We apply quasi-distribution methods developed for quantum mechanics to the propagation of pulses in dispersive media with attenuation. We show that a Schrodinger type equation follows for propagation of the pulse for each mode. One then transforms the equation to obtain an equation of evolution in the phase space of position and wavenumber. In this paper we emphasize windowed wave functions and their corresponding phase space quasi-distributions. We obtain the time evolution equation, discuss possible approximations, and compare to the Wigner distribution approximation previously derived by Loughlin and Cohen by different methods.

The equations of motion for a ray in a Snell's law medium with a varying index of refraction are derived. A stratified medium is considered. Explicit expressions are given for the velocity and acceleration components of the ray. These are derived directly from Snell's law. It is further shown that the propagation of a ray can be modeled in terms of Newtonian-like equations of motion and that momentum is conserved along the interface. It is shown that Snell's law follows from this conservation law. Properties of the motion are studied and an example is given.

We show that phase space methods developed for quantum mechanics, such as the Wigner distribution, can be effectively used to study the evolution of nonstationary noise in dispersive media. We formulate the issue in terms of modes and show how modes evolve and how they are effected by sources.We show that each mode satisfies a Schrödinger type equation where the “Hamiltonian” may not be Hermitian. The Hamiltonian operator corresponds to dispersion relationwhere thewavenumber is replaced by the wavenumber operator. A complex dispersion relation corresponds to a non Hermitian operator and indicates that we have attenuation. A number of examples are given.

We discuss a number of topics relevant to noise propagation in dispersive media. We formulate the problem of pulse propagation with a source term in phase space and show that a four dimensional Wigner distribution is required. The four dimensional Wigner distribution is that of space and time and also wavenumber and frequency. The four dimensional Wigner spectrum is equivalent to the space-time autocorrelation function. We also apply the quantum path method to improve the phase space approximation previously obtained. In addition we discuss motion in a Snell’s law medium.

We describe how one can obtain the phase space differential equation for joint position-wavenumber distributions for pulse propagation with dispersion and attenuation. We show that there are many advantages to the phase space equation both from the point of view of insight and practical calculation. We use the method to obtain new approximations for pulse propagation. The phase space distributions we use are the Wigner distribution and spectrogram.

Quantum phase-space distributions offer a royal road into the fascinating quantum–classical interface; the Wigner function being the first and best example. However, the subject is frequent with subtleties and textbook-level misinformation; e.g. “The Wigner distribution can give wrong answers for some operator expectation values” [1]. Since the Wigner distribution is just another representation of the density matrix, it must yield correct answers. To that end, Marlan Scully has asked at several international conferences (the 2015 Prague conference being one of them) the following question: “Starting with the density matrix (not the Moyal characteristic function), could you give me a simple direct derivation of the Wigner distribution?” Section 'A jump start' contains his answer. In Appendix D, we give a related treatment and make contact with other approaches. We hope that as a result of our studies, the Wigner distribution will become more deeply appreciated.

We have developed a simple Newtonian dynamics for the motion of rays as particles that are governed by Snell's law. In Newton's original formulation, the particle moves faster in a relatively higher index of refraction medium. We show that it is the constant mass assumption that leads to this conclusion. We derive an explicit expression for the mass as a function of position and show that the formulation leads to the conclusion that indeed the particle moves slower in a relatively higher index of refraction medium. Our approach leads to a simple Newtonian particle picture where the equations of motion may be simply written. We obtain explicit expressions for the velocity, acceleration, and forces which allow one to write the equations of motion. We also formulate the dynamics in terms of the Lagrangian and Hamiltonian formulations, taking variable mass into account. The solutions to the dynamics are such that the particle always follows Snell's law in a variable index of refraction medium. Exactly solvable analytic examples are given. We also we show that the SOFAR channel phenomenon, where a wave is trapped between two regions, is easily explained in the particle picture.

A phase space approximation method that extends a previous single-mode approximation [Loughlin, Cohen, J. Acoust. Soc. Am. 118, 1268 (2005)] for linear dispersive wave propagation is developed. We show that each mode is governed by a Schrodinger-type equation, where the corresponding Hamiltonian operator is non-Hermitian if the dispersion relation is complex, for which case there is absorption. The propagated wave is obtained by evolving each mode according to its respective Schrodinger equation. We show how to obtain the initial modes from the initial conditions on the wave. We then formulate the propagation problem in phase space and obtain the exact equation of motion for the phase space function. We also obtain an approximate solution for the phase space evolution of the wave, which involves a simple substitution into the initial phase space function. Examples are given for a parallel plate wave guide and the beam equation. [Work supported by ONR, code 321US.]