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Physical Review A

Conservation law for electromagnetic fields in a space-time-varying medium and its implications

Junchi Zhang, William R. Donaldson, Govind Prasad Agrawal

Published Oct 30, 2024• 5 citations
Publisher

Abstract

We discuss the propagation of electromagnetic waves in a space-time modulated medium, for which neither energy nor momentum remains conserved. We have found a conservation law when modulation moves at a constant speed $v$ in a traveling-wave fashion. We use this law to study how light is scattered inside such a medium. We consider both the subluminal ($v<c$) and superluminal ($v>c$) cases and find that they differ considerably in terms of what remains conserved. We show that the total number of photons is conserved in the subluminal case. In contrast, it is the difference between the forward- and backward-moving photons that remains conserved in the superluminal case. We also study the reciprocity issue for a space-time modulated medium and find that reciprocity holds in all situations. We develop a quantum formulation of our scattering problem and show that the subluminal case is similar to the action of a beam splitter. In the superluminal case, photon pairs can be generated from vacuum in a fashion analogous to the two-mode frequency down-conversion process.

DOI
10.1103/physreva.110.043526

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We discuss the propagation of electromagnetic waves in a space-time modulated medium, for which neither energy nor momentum remains conserved. We have found a conservation law when modulation moves at a constant speed vvv in a traveling-wave fashion. We use this law to study how light is scattered inside such a medium. We consider both the subluminal (v<cv<cv<c) and superluminal (v>cv>cv>c) cases and find that they differ considerably in terms of what remains conserved. We show that the total number of photons is conserved in the subluminal case. In contrast, it is the difference between the forward- and backward-moving photons that remains conserved in the superluminal case. We also study the reciprocity issue for a space-time modulated medium and find that reciprocity holds in all situations. We develop a quantum formulation of our scattering problem and show that the subluminal case is similar to the action of a beam splitter. In the superluminal case, photon pairs can be generated from vacuum in a fashion analogous to the two-mode frequency down-conversion process.
It is shown that there are no dispersing media with «subluminal only» or «superluminal only» real group velocity – if in some medium at some frequencies the real group velocity is subluminal, then in the same medium there are necessarily frequencies at which the real group velocity is superluminal. Conversely, if a medium has frequencies with a superluminal group velocity, then it necessarily has frequencies with a subluminal group velocity. In other words, there is a necessary connection between subluminal and superluminal group velocity – if any medium is capable of generating subluminal group velocity, then the same medium is capable of generating superluminal group velocity. Similarly, there are no media with «only positive» or «only negative» imaginary part of the complex group velocity – if at some frequencies the imaginary part of the group velocity is positive, then at other frequencies in the same medium it is negative and vice versa. This means that if there is a carrier frequency in the medium at which some linearly frequency-modulated signal propagates at a speed lower than the group speed, then in the same medium there is necessarily a carrier frequency at which the same signal will propagate at a speed higher than the group speed and vice versa.
We consider rectilinear free-space propagation of electromagnetic wavepackets using electromagnetic field theory, scalar wavepacket propagation, and quantum-mechanical formalism. We demonstrate that spatially localized wavepackets are inherently characterized by a subluminal group velocity and a superluminal phase velocity, whose product equals c2c^2c2. These velocities are also known as the 'energy' and 'momentum' velocities, introduced by K. Milton and J. Schwinger. We illustrate general conclusions by explicit calculations for Gaussian and higher-order beams and wavepackets, and also highlight subtleties of the quantum-mechanical description based on the 'photon wavefunction'.
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.
The laws of reflection and refraction are typically derived using a dispersion relation that assumes that the angular frequency stays the same across a vacuum--dielectric interface and that the angular wave number increases. At the single photon level, this means that the total energy of the photon is conserved but the linear momentum is not. Here I derive the laws of reflection and refraction for a single photon using a dispersion relation that is consistent with the conservation of both total energy and linear momentum. To ensure that both conservation laws are upheld, the optical path length must be considered to be more fundamental than the geometrical path length when reckoning optical distances. As long as the medium is transparent, the optical path length traveled by a photon in a given duration of time is constant. By contrast, the geometrical path length traveled in a given duration of time is inversely proportional to the index of refraction. A photon propagating across a vacuum--dielectric interface is considered to be an indivisible package whose total energy (and angular frequency) and linear momentum (and angular wave number) remain constant, but whose velocity decreases with refractive index as a result of unspecified electromagnetic interactions.
The reflection of test radiation from a smooth inhomogeneity of medium characteristics propagating with a subluminal or superluminal velocity is analyzed. The equations describing the propagation of the forward- and counter-propagating waves in such an inhomogeneous medium are derived. Quasi-phase conjugation is demonstrated in the case of superluminal inhomogeneities. The Bragg resonance conditions are formulated and the conditions for increasing the reflection coefficient of radiation from an inhomogeneity are discussed.
Starting with the canonical quantization procedure for the electromagnetic field inside an effective (linear or nonlinear) medium, we present a direct-space formulation of the theory of quantum optics. This approach does not use the conventional modal decomposition of the field, but relies on the electromagnetic momentum operator defined in terms of local electric- and magnetic-field operators. The momentum operator contains all the information on the spatial characteristics of the field, and can describe the translation of a short light pulse in a nonlinear medium, without a modal analysis of the pulse. Propagation is described through an operatorial wave equation that relates the temporal evolution of an electromagnetic pulse to its spatial progression. Through this equation, the direct-space approach to quantum optics can treat traveling-wave nonlinear-optical phenomena and, at the same time, account for their quantum statistics. The theory is applied to squeezed-light generation by the parametric down-conversion of a short laser pulse, as an illustration.