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arXiv:1401.2924 · Mathematical Methods in the Applied Sciences

Time‐dependent photon statistics in variable media

Sergey Kryuchkov, Erwin Suazo, Sergeĭ K. Suslov

Published Sep 27, 2018• 2 citations
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Abstract

We find explicit solutions of the Heisenberg equations of motion for a quadratic Hamiltonian, which describes a generic model of variable media in the case of multiparameter squeezed input photon configuration. The corresponding probability amplitudes and photon statistics are also derived in the Schrödinger picture in an abstract operator setting of the quantum electrodynamics; a comparison discussion is made in Heisenberg's picture as well. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. The time‐dependent photon probability amplitudes with respect to the Fock basis are indeed derived in an operator form. Further, explicit expressions for the matrix elements of the displacement and squeeze operators are derived in terms of hypergeometric functions and solutions of a certain Ermakov‐type system. In the Supporting Information , we provide a computer algebra verification of the derivation of the Ermakov‐system and of the solutions of the Heisenberg equations.

DOI
10.1002/mma.5285

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We find explicit solutions of the Heisenberg equations of motion for a quadratic Hamiltonian, which describes a generic model of variable media in the case of multiparameter squeezed input photon configuration. The corresponding probability amplitudes and photon statistics are also derived in the Schrödinger picture in an abstract operator setting of the quantum electrodynamics; a comparison discussion is made in Heisenberg's picture as well. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. The time‐dependent photon probability amplitudes with respect to the Fock basis are indeed derived in an operator form. Further, explicit expressions for the matrix elements of the displacement and squeeze operators are derived in terms of hypergeometric functions and solutions of a certain Ermakov‐type system. In the Supporting Information , we provide a computer algebra verification of the derivation of the Ermakov‐system and of the solutions of the Heisenberg equations.
We find explicit solutions of the Heisenberg equations of motion for a general quadratic Hamiltonian in a variable medium in the case of multi-parameter squeezed input photons. The corresponding photon statistics are also derived in the Schrodinger picture in an abstract Hilbert space operator settings. Their time evolution is given in terms of solutions of certain Ermakov- type system. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. 1. An Introduction
We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a + , which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.
This chapter explores the Schrödinger and Heisenberg pictures in quantum mechanics, emphasizing their mathematical equivalence and physical consistency. The two formulations are shown to be connected by a unitary transformation. The Heisenberg equation of motion is derived, and the time evolution of operators—particularly those not explicitly dependent on time—is expressed in terms of their initial values. It is demonstrated that scalar products, expectation values, matrix elements, and commutation relations remain invariant under the change of picture. Drawing a close analogy with classical mechanics, Dirac's insight is presented: the quantum mechanical counterpart of the classical Poisson bracket is the operator commutator divided by iℏ . The chapter concludes by deriving the Lorentz force law within the quantum framework using commutator algebra. Metadata/Keywords: Schrödinger picture, Heisenberg picture, unitary transformation, Heisenberg equation of motion, Poisson bracket, Lorentz force, time evolution, expectation values
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
In this paper, we treat the problem of the multi-photon Tavis–Cummings classically, where the effect of time-dependence takes place. Under certain integrability conditions the time-dependent wave function in the Schrodinger picture is obtained. Some statistical properties of the system are discussed based on the solution for the equations of motion in the Heisenberg picture. Using the atomic coherent state as a basis, the phenomenon of squeezing is observed in the quadrature variances Fx and Fy; however, it was pronounced in the quadrature Fy. The phenomenon of squeezing is also reported for the usual atomic state case where squeezing is only observed in the first quadrature Fx. An examination of the correlation function using the atomic state showed us that the system is sensitive to variation in the integrability parameter β. The nonclassical effect is also found to be pronounced for small values of the parameter β.