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Physical Review A

Quantum complementarity ad infinitum : Switching higher-order coherence from infinity to zero

Arash Azizi

Published Dec 6, 2025• 0 citations
Publisher

Abstract

We reveal a sharp form of complementarity in the \emph{Janus state}—a coherent superposition of two squeezed vacua. Using a compact generating-function formalism, we derive closed-form expressions for $g^{(k)}$ at all orders and introduce \emph{squeezing polynomials} $P_k(z)$ that factorize the moments and expose phase-dependent scaling. The Janus state operates as a phase-tunable quantum switch for multi-photon correlations: when which-path/which-state information is available (trivial interference), the higher-order coherences $g^{(k)}$ reproduce the single-squeezed limit and diverge; when path information is erased and genuine interference from coherent superposition is enabled, multi-photon events are extinguished, driving $g^{(k)}\!\to\!0$ for all $k\!\ge\!3$. We obtain a hierarchy of suppression with exponents set by the interference phase and the parity of $k$, and we visualize the mechanism via pronounced Wigner-function negativity. The framework is general and experimentally accessible, enabling engineered nonclassical light with tailored higher-order statistics.

DOI
10.1103/gxzt-164h

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We report a profound manifestation of quantum complementarity in the higher-order photon statistics of the ``Janus state,'' a coherent superposition of two squeezed vacua. We find that the state acts as a perfect quantum switch for multi-photon correlations, toggled by the availability of which-path information. Erasing this information activates quantum interference that can be tuned to be maximally destructive. This reveals a remarkable hierarchy of suppression: while two-photon correlations remain finite, we prove analytically and demonstrate numerically that it is possible to drive all higher-order correlations (g(k)g^{(k)}g(k) for k≥3k \ge 3k≥3) to zero. This transition from the extreme bunching of the constituent states (g(k)→∞g^{(k)} \to \inftyg(k)→∞) to a state of profound quantum order is visualized by the emergence of negativity in the state's Wigner function, an unambiguous signature of non-classicality. This work provides a foundational demonstration of quantum complementarity in multi-photon statistics and introduces a new paradigm for engineering highly ordered, non-classical light from Gaussian resources.
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Janus states, defined as coherent superpositions of two single-mode squeezed vacua, provide a simple but genuinely non-Gaussian setting for studying how interference reshapes quantum Fisher information (QFI) beyond the Gaussian squeezed-vacuum picture. Using an exact analytic treatment, we determine the QFI of Janus states and identify the benchmarks under which they can or cannot offer a metrological advantage over the single squeezed vacuum. We find that, under a fair comparison at fixed mean photon number, the single squeezed vacuum remains optimal for principal second-moment squeezing, so no genuine Janus advantage exists at that level. By contrast, within a fixed two-state span, a Janus superposition can simultaneously outperform its constituents in a laboratory quadrature variance and in number-generated phase QFI. We also introduce an operational benchmark based on fixed measured squeezing and show that, at the same observed squeezing level, Janus interference can substantially enhance the QFI for quadratic-generator sensing beyond the pure-Gaussian squeezed-vacuum reference. These results show that the metrological performance of Janus states is controlled not only by quadrature squeezing, but also by higher-order fluctuations and by the benchmark used for comparison.
Non-Gaussian states are essential for achieving a quantum advantage in continuous-variable (CV) information processing. Among these, coherent superpositions of squeezed states are a foundational resource. While exact higher-order statistics are available in the undisplaced case, a complete and analytically tractable treatment with a common displacement has been missing. We introduce and solve the displaced Janus state---a coherent superposition of two squeezed coherent states that share the same displacement---and develop an analytical framework, based on a family of generalized squeezing polynomials, that yields closed-form expressions for arbitrary-order factorial moments and coherence functions g(k)(0){g}^{(k)}(0)g(k)(0), the full Wigner function, and the quantum Fisher information. The analysis shows how interference at a fixed mean, driven by a mismatch of the component covariances rather than by mean separation, can be precisely engineered to transform the extreme photon bunching of the constituents into strong sub-Poissonian and even perfect multiphoton suppression. We further provide a rigorous quantum Fisher information analysis, proving that parameters encoded by linear generators (for example, the number operator) are bounded by the standard quantum limit, whereas parameters encoded by quadratic generators (for example, squeezing transformations) achieve Heisenberg-limited scaling. Together, these results furnish a complete analytical toolkit for a versatile class of non-Gaussian states, establishing the displaced Janus state as a key primitive for hybrid quantum protocols, quantum metrology, and fault-tolerant continuous-variable computation.
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 kkk-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)g^{(k)}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.
We define and study the properties of ``squeezed quantum multiplets''. Ordinary multiplets are sets of DDD-orthonormal quantum states formed by superpositions of states squeezed along DDD equally spaced directions in quadrature space. More generally, we also discuss superpositions of ``higher-order squeezed states'', including tri-squeezed and quad-squeezed states. All these states involve superpositions of multiples of ppp photons. We compare states in ordinary (p=2p=2p=2) multiplets and higher-order ones (p>2p>2p>2) in the most relevant cases, showing that ordinary squeezed multiplets and higher-order ones share some important similarities, as well as some differences. Finally, we present analytical expressions for phase-space distributions (Wigner and characteristic functions) representing ordinary squeezed multiplets. We use this to show that some squeezed multiplets are highly sensitive to perturbations in all phase-space directions, making them interesting for metrological applications.