Photonic realization of an electromagnetic cavity field with uniform mirror motion

Abstract

We perform a detailed mathematical study of the effective Hamiltonian describing a non-stationary electromagnetic cavity field, introduced by C.~K.~Law, in the presence of uniform linear motion. We find that the corresponding dynamical equations admit a unitary diagonalization and that the normalized velocity of motion satisfies the two-sided bound $\lvert v/c \rvert \lesssim 10.1716$. Although this result is mathematically consistent, the interval obtained is an order of magnitude larger than the physically admissible range $\lvert v/c \rvert < 1$. To address this limitation, we propose a reinterpretation of bosonic coupling terms as generators of the symplectic algebra $\mathrm{Sp}(4,\mathbb{R})$, identifying them with transformations that can be implemented through linear optical elements. Within this photonic approach, the coupling parameters are not restricted to small values, allowing us to explore regimes of strong effective interactions. In this case, we demonstrate the generation of two-mode squeezed states and derive analytical expressions for the average photon number in each mode. Furthermore, we analyze statistical properties that characterize mixedness and entanglement, including intermode correlations, the Mandel parameter, and the linear entropy. The corresponding Wigner function provides a phase-space visualization of the system's behavior, illustrating the connection between cavity motion, mode coupling, and nonclassical field correlations via the underlying symplectic algebra.

The paper studies Law’s effective Hamiltonian for a cavity with one uniformly moving mirror, asking how its two-mode dynamics can be reinterpreted as a photonic model beyond the mechanical constraint $|v/c|<1$. For $q(t)=q_0+vt$, the Hamiltonian factorizes into a scalar time dependence and a fixed quadratic bosonic Hamiltonian containing mode conversion, single-mode squeezing, and two-mode squeezing. The analysis identifies these interactions with generators of $\mathrm{sp}(4,\mathbb{R})$ and replaces the normalized mirror velocity by an effective photonic coupling $g=v/c$. Three time-independent quadratic unitary transformations decouple the position–momentum Hamiltonian. Requiring real transformation parameters gives the reported bound $|g|\lesssim10.1716$; the explicit transformation becomes singular at the endpoints. The paper identifies a lower frequency-softening threshold near $|g|\approx3.75$, beyond which an oscillator coefficient product becomes negative and the corresponding normal-mode frequency becomes imaginary. It describes these intervals as complex-pseudoenergy or forbidden regions. Explicit Bogoliubov coefficients yield analytical photon-number dynamics for initial product Fock states, compared with QuantumOptics.jl simulations. Examples at $g=2.1,3.1,3.5$ examine photon generation, Mandel statistics, reduced-state linear entropy, and single-mode Wigner functions for vacuum and single-photon inputs, including reduced-state Wigner-negativity loss. The treatment is a two-mode, closed, quadratic theoretical model, not an experimental demonstration. Uniform mirror motion is distinguished from acceleration-driven radiation. The photonic reinterpretation does not describe a superluminal mechanical cavity; unstable regimes are stated to require nonlinearities or dissipation for physical steady states.