A consistent quantum formalism based on the localized basis of the Wannier functions in the Heisenberg and Schrödinger pictures to describe propagation of an electromagnetic field in a three-dimensional media including diffraction is presented. In the Schrödinger picture the Fokker-Planck equation for the Glauber-Sudarshan quasi-probability and the corresponding Langevin equations are given. As a result the spacetime description is obtained by a simple change of variables in the temporal master equation of the field. Using this formalism it is shown that the existence of integrals of motion in the propagation of light in a medium under the condition of non-degenerated parametric and two-photon interactions results in amplification of modes when non-classical properties of the light are conserved. Quantum propagation of light in a linear medium taking into account the diffraction is considered and its solution is found.
A consistent quantum formalism based on the localized basis of the Wannier functions in the Heisenberg and Schrödinger pictures to describe propagation of an electromagnetic field in a three-dimensional media including diffraction is presented. In the Schrödinger picture the Fokker-Planck equation for the Glauber-Sudarshan quasi-probability and the corresponding Langevin equations are given. As a result the spacetime description is obtained by a simple change of variables in the temporal master equation of the field. Using this formalism it is shown that the existence of integrals of motion in the propagation of light in a medium under the condition of non-degenerated parametric and two-photon interactions results in amplification of modes when non-classical properties of the light are conserved. Quantum propagation of light in a linear medium taking into account the diffraction is considered and its solution is found.
A scheme for deriving a field kinetic equation for the problems of macroscopic quantum optics has been formulated. The general master equation obtained for the s-ordered quasi-probability, and Glauber-Sudarshan P-function in particular, has been concretized for the case of three-photon parametric processes in a multi-level medium. Thus the three-photon parametric phenomena have been examined without exploiting the traditional effective Hamiltonian. Discussed here is the problem of formation of field statistics depending on the type of excitation medium with two-level lasing levels that reduce to the Lamb-Scully or Haken models. The form of the kinetic equation obtained is shown to depend on the types of atomic transitions.
The basic equations for the statistical system of waves in a randomly fluctuating medium are presented in the same form as in quantum mechanics, assuming a Markov process for the temporal change of the medium. A simple model is chosen for the Fokker-Planck equation of the medium which gives rise to a fluctuation of the Gaussian process, and several results are given to illustrate how the methods used in quantum mechanics or quantum field theory can be applied almost without change. Thus, the physical variables are represented by linear operators, and their equations of motion are determined by an equation similar to the Heisenberg equation of motion. The system has two stationary states (corresponding to the vacuum states 〈0∣ and ∣0〉 in quantum field theory), and the rather standard methods in field theory can be used for the evaluation of the Green's functions. When the temporal changes of the medium are sufficiently large compared with those of the waves, an adiabatic approximation is possible, and the isomorphic transformation (corresponding to the unitary transformation) is employed to lead to the result that the statistical system of the waves for this model is in perfect correspondence with the waves of bosons which interact with each other only through a two-body potential.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
During these three years we focused on several topics in quantum otpics and quantum electrodynamics. A central theme in our investigations is that of the photon wave function. Can quantum optics and quantum electrodynamics experiments be described simply, in position space, with the help of a wave function describing the photon(s) featured in the experiment ? The answer to that question is not quite obvious: the usual description of photons takes place in the reciprocal space of wave vectors. But these experiments call for a wave mechanical description in the position representation, as is done in quantum mechanics textbooks in situations featuring massive particles. Moreover, in a recent experiment [1], single photon trajectories through a Young two-slit setup have been observed. In order to try and describe these trajectories formally, it is natural to build a wave mechanical formalism for photons. We therefore studied in detail the formal construction of the photon wave function, an object which was little studied until the 1990s. We also studied the properties of the photon wave function in the presence of sources.To do that, we considered several open (interacting) quantum systems. We saw that there exists in principle an infinite number of possibilities when defining the photon wave function. We emphasised several criteria on the basis of which it appears that only three choices for the wave function are interesting. One of them coincides with an object introduced and used by Glauber [2] to study light detection andthe correlations of the electromagnetic field in the quantum regime. We also saw that, in the absence of sources, the propagation equation for a single photon is formally equivalent to Maxwell’s equations. At low photon numbers, the wave function formalism can be very useful. We adapted it to interacting systems,first, to cavity quantum electrodynamics (QED) [3], in particular to the experiments carried out by Serge Haroche’s group [4]. We proposed a simple model to describe photons in QED cavities. With this model, and with the helpof the photon wave function, we studied the propagation of photons escaping a cavity. We also constructed the Lindblad master equation without introducing nonunitary quantum jumps (also see [5]). We finally investigated the spacetime evolution of a photon which is emitted during the decay of an atomic electron. After having carefully studied the dynamics of the electronic decay, especially at very short times [6, 7], we set out to describe the emitted electromagnetic field as rigorously as possible. This emitted field, surprisingly, does not evolve causally. Though this is not entirely unexpected in view of Hegerfeldt’s theorem, which states [8] that causality is impossible for quantum systems which are described by a Hamiltonian with a spectrum which is bounded by below, we identified [9] two other sources of non causality. One of them was predicted qualitatively by Shirokov [10], while the other one, which is completely new as far as we can tell, is still to be better understood
We have developed a quantum-mechanical formalism which permits the treatment of light propagation within the conceptual framework of quantum optics. The formalism rests on the calculation of the momentum operator for the radiation field, and yields directly a description for the spatial progression of the electromagnetic waves. In this paper we give a quantum-mechanical treatment for refraction and reflection by applying our formalism to propagation through a linear dielectric. The fidelity with which this formalism reproduces all results known from classical optics demonstrates its validity.
We again note that Langevin equations written for observables (– operators in quantum mechanics) conform to the Heisenberg view. On the other hand, an entity that embodies the wave function of the Schrödinger picture is the von Neumann density operator, usually denoted by t( )ρ , whose classical counterpart is the probability distribution function (pdf) discussed earlier. It is helpful to introduce a formalism for the pdf completely at par with the Langevin description. The resultant Fokker-Planck (FP) equations serve as the subjects of this chapter. The simplest FP equation can be derived from the master equation (2.25) for a stationary Markov process, as shown below (Agarwal 1983, Risken 1996).