Two-Mode Janus States: non-Gaussian generalizations of thermofield double

TL;DR

The paper develops the Two-Mode Janus State (TMJS), a non-Gaussian generalization of the thermofield double, defined as a coherent superposition of two TMSSs with a controllable Janus phase δ that steer higher-order coherences. It provides a complete analytic framework for arbitrary k-th order photon statistics via squeezing polynomials Pk(x) and cross-state kernels, revealing a phase-driven suppression or enhancement of g^(k) and a Wigner-negativity signature. The TMJS is physically realizable through coherently superposed Dynamical Casimir Effect trajectories, tying state-generation dynamics to relativistic quantum information and distinguishing it from observer-dependent Unruh physics. This interference-based control offers a versatile platform for engineering Unruh-DeWitt detector responses and probing non-Gaussian physics in relativistic settings. The work opens avenues for phase-controlled non-Gaussian resources in quantum information processing under relativistic or dynamical boundary conditions.

Abstract

We introduce the Two-Mode Janus State (TMJS), a non-Gaussian quantum state defined as a coherent superposition of two distinct Two-Mode Squeezed States (TMSS). This construction serves as a direct, non-Gaussian generalization of the canonical thermofield double (TFD) state, which is itself a single, Gaussian TMSS. We develop a complete analytical framework for the TMJS's arbitrary $k$-th order photon statistics, identifying a new family of "squeezing polynomials" that govern all diagonal and off-diagonal moments. Our central result is that the state's non-Gaussianity is dynamically steerable via an external "Janus phase." This phase acts as a switch, allowing the higher-order coherences ($g^{(k)}$) to be tuned from perfectly thermal (matching the TFD marginal) to deeply sub-Poissonian, a regime marked by strong Wigner negativity. We further establish a physical realization for the TMJS, proposing its generation via coherently superposed Dynamical Casimir Effect (DCE) trajectories, distinguishing it from the static, observer-dependent Unruh effect. The TMJS provides a versatile, interference-enhanced platform for engineering Unruh-DeWitt detector responses and probing non-Gaussian physics in relativistic settings.

Figures (8)

• Figure 1: Baseline (no interference): Diagonal cross–mode coherences for a single TMSS, $g_{ab}^{(k)}(\xi)=P_k(x)/x^{2k}$, where $P_k$ are the squeezing polynomials. The four curves (from bottom to top) show $k=1,2,3,4$ on a log scale as a function of $r$ ($x=\tanh^2 r$). As $r\to0$ one has $P_k(x)\sim (k!)^2 x^k$ so $g_{ab}^{(k)}\sim (k!)^2 x^{-k}\propto r^{-2k}$, explaining the steep small–$r$ divergence that sets the reference scale for all Janus effects.
• Figure 2: Interference turns the knob:Left—the same single–TMSS divergences from Fig. \ref{['fig:fig_pk_polynomials']} on a log–log axis. Right—the antisymmetric TMJS ($\Delta=\pi,\ \delta=\pi$) where destructive interference suppresses cross–mode coherences by many orders of magnitude at small $r$. The opposite slopes of the two panels emphasize that the Janus superposition replaces the $r^{-2k}$ blow–up by deep cancellations that can drive $g_{ab}^{(k)}(\Psi)\ll1$.
• Figure 3: Mean brightness landscape$\log_{10}\!(\langle a^\dagger a\rangle_\Psi)$ over $(r,\delta)$ for $\Delta\in\{0,\pi/2,\pi,3\pi/2\}$ with $r=s$. This quantity is the denominator seed that ultimately enters all normalized single–mode coherences. The dark trench at $\delta=\pi$ for $\Delta=\pi$ pinpoints maximal destructive interference in the marginal, while the bright lobes for $\Delta=\pi/2$ reflect quadrature–skewed constructive terms. Reading this figure first helps interpret where single–mode $g_a^{(k)}(\Psi)$ will be most strongly enhanced or suppressed.
• Figure 4: Single–mode coherence $g_a^{(2)}(\Psi)$:$\log_{10}$ heat maps vs. $(r,\delta)$ at fixed $\Delta$. When $\Delta=0$ the landscape is essentially flat: the two constituents are in phase and the normalized second–order coherence reduces to its single–state value with negligible $\delta$–dependence. For $\Delta=\pi$ a deep valley opens along $\delta=\pi$, mirroring the dark trench in Fig. \ref{['fig:brightness_landscape']}; moving away from $\delta=\pi$ restores super–Poissonian structure. Quarter–period shifts ($\Delta=\pi/2,3\pi/2$) rotate the bright–dark lobes as expected from the $z=e^{i\Delta}\tanh r\tanh s$ kernel.
• Figure 5: Single–mode coherence $g_a^{(3)}(\Psi)$: Compared to $k=2$, phase selectivity is sharper: the nodal trench at $(\Delta,\delta)=(\pi,\pi)$ is narrower and the surrounding ridges are higher. This is consistent with the higher–order cross kernels $P_k(z)/(1-z)^{2k+1}$: larger $k$ amplifies both interference gains and cancellations, so the dynamic range of $\log_{10}g_a^{(3)}$ increases while its extrema track those in Fig. \ref{['fig:brightness_landscape']}.

Theorems & Definitions (1)

• Remark 1