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Physical Review A

Quantum theory of time refraction

José Tito Mendonça, Ariel Guerreiro, Ana María Guerra Martins

Published Aug 10, 2000• 89 citations
Publisher

Abstract

The concept of time refraction is introduced to describe the effects of a sudden change of the optical properties of a dielectric medium. This can be seen as the most elementary process associated with photon acceleration and frequency upshifting. The quantum theory of such a process shows that the initial wave splits into time-transmitted and time-reflected waves propagating in opposite directions after the occurrence of a time discontinuity of the refractive index. The time refraction laws, analogous to the well known Fresnel formulas and Snell's law, are also derived. It is shown that, in quantum terms, time refraction is equivalent to a squeezing transformation.

DOI
10.1103/physreva.62.033805

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The concept of time refraction is introduced to describe the effects of a sudden change of the optical properties of a dielectric medium. This can be seen as the most elementary process associated with photon acceleration and frequency upshifting. The quantum theory of such a process shows that the initial wave splits into time-transmitted and time-reflected waves propagating in opposite directions after the occurrence of a time discontinuity of the refractive index. The time refraction laws, analogous to the well known Fresnel formulas and Snell's law, are also derived. It is shown that, in quantum terms, time refraction is equivalent to a squeezing transformation.
The effect of photon beam splitting in a time-varying medium is described by classical and quantum theoretical models. It generalizes the concept of time refraction, introduced recently by the authors as a natural extrapolation of the usual concepts of refraction and reflection into the time domain. Total time reflection is shown to exist. A sequence of time refraction processes is shown to lead to temporal interference effects. The concept of temporal beam splitter is introduced. Bogoliubov transformations for the temporal beam splitter are derived. Resonant amplification of light by change in time in the optical medium is shown to exist.
If a boundary between two static media is moving with a constant superluminal velocity, or there is a sudden change of the refractive index with time, this yields generation of entangled pairs of photons out of vacuum propagating in the opposite directions. Here we show that during this process, entanglement of Minkowski vacuum is transferred to the entanglement of the generated photon pairs. If initially an electromagnetic pulse is present in the medium the photon generation is stimulated into the pulse mode, and since photons are created as entangled pairs the counter-propagating photon partners produce a pulse moving in the opposite direction, which is known as time reflection. Thus, time reflection occurs due to stimulated generation of the entangled photon pairs out of entangled vacuum and no photons in the original pulse are in fact being reflected. This is different from the mechanism of light reflection from spatial inhomogeneities for which no photons are generated.
Abstract Three linear equations are proposed as quantum-mechanical equations for the free propagating photons. The solution of these equations is the vector potential of the electromagnetic field and it is a product of the Gaussian functions for the transverse coordinates and the eigenfunction of the harmonic oscillator for the longitudinal coordinate. This solutions is used to describe the reflection and refraction of photons on the boundary between two dielectrics. The amplitude of the reflected field coincides with the Fresnel formulae for the plane waves, but the amplitude of the refracted field is different. These amplitudes are the probability’s amplitudes of reflection and refraction of photons—like in the quantum mechanics. The photons propagated from the first into the second medium increase (decrease) their transverse size in the plane of incidence, if the refraction index of the second medium is greater (smaller) than in the first medium. The increasing or decreasing grow when increasing the angle of incidence. The difference between the quantum mechanics of the particles with a mass and the photons, as well as the interpretation of the wave function of photons, is discussed.
The foundations of quantum mechanics have traditionally relied upon the probabilistic interpretation where wave function moduli determine spatial probabilities, treating measurement transitions through sudden collapse or decoherence. This paper introduces the theoretical framework of Quantum Probability Refraction (QPR), proposing that quantum probability currents bend when crossing distinct measurement environments in a manner analogous to light refraction in optics. We define the probability refractive index as n_P equal to v_{P,0} divided by v_P and derive a generalized Snell's law for probability flows expressed as n_{P1} multiplied by the sine of theta_1 equals n_{P2} multiplied by the sine of theta_2. By establishing the mathematical formulation, continuity equations, conservation laws, and the optical-like phenomena of total internal reflection and evanescent probability waves, this work presents a geometric theory of probability flows. This paradigm offers a novel continuous perspective on quantum measurement boundaries and opens new pathways for controlling quantum information transport.
The generalization of the quasi mode theory of macroscopic quantization in quantum optics and cavity QED presented in the previous paper, is applied to provide a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes. The derivation of the laws of reflection and refraction is achieved through the dual application of the quasi mode theory and a quantum scattering theory based on the Heisenberg picture. Formal expressions from scattering theory are given for the reflection and transmission coefficients. The behaviour of the intensity for a localized one photon wave packet coming in at time minus infinity from the incident direction is examined and it is shown that at time plus infinity, the light intensity is only significant where the classical laws of reflection and refraction predict. The occurrence of both refraction and reflection is dependent upon the quasi mode theory coupling constants between incident and transmitted region quasi modes being non-zero, and it is seen that the contributions to such coupling constants come from the overlap of the mode functions in the boundary layer region, as might be expected from a microscopic theory.
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.