Detecting nonclassicality in randomly-displaced copies of a squeezed state

Abstract: We address a fundamental question: Can one determine whether a received signal is squeezed when each copy arrives with a different displacement/amplitude? We introduce an interaction Hamiltonian that converts quadrature squeezing into number squeezing. Using this conversion, we test whether the copies satisfy $g^{(2)}(0)<1$. The Hamiltonian itself does not create nonclassicality; it only transfers it from quadrature squeezing to number squeezing. This allows us to identify squeezing even when individual copies have random displacements.

• The paper proposes a protocol that converts quadrature squeezing into number squeezing using a beam splitter–like Hamiltonian, preserving nonclassical features despite random displacements.
• It employs counter-displacement and post-selection techniques to amplify the signature of sub-Poissonian statistics (g²(0) < 1) in challenging noisy environments.
• Analytical proofs and numerical simulations confirm the protocol's robustness, with significant implications for quantum communication, sensing, and error-correction.

Detecting Nonclassicality in Randomly-Displaced Copies of a Squeezed State

Problem Formulation and Motivation

The determination of squeezing in photonic quantum states forms a cornerstone of continuous-variable quantum information processing, quantum communications, and quantum-enhanced sensing. In realistic environments, especially in open or dynamically fluctuating channels such as biological imaging or free-space quantum communication, successive received copies of an initially identically prepared squeezed state are subject to unknown, random displacements and attenuation. This randomization precludes direct use of standard quadrature measurement-based squeezing detection, as averaging over noisy displacements obliterates evidence of squeezing.

The paper presents a rigorous protocol to witness nonclassical signatures, specifically quadrature squeezing, under these practical constraints. The protocol's central approach is to convert inaccessible quadrature squeezing in randomly displaced states into measurable number squeezing and subsequently detect nonclassicality via photon-number statistics, particularly using the second-order correlation function $g^{(2)}(0)$.

Theoretical Construction: Squeezing Conversion Hamiltonian

The core technical insight is the introduction of an interaction Hamiltonian $\hat{B}_n(\theta)$ which implements a transformation analogous to a beam splitter, but between the standard optical annihilation operator $\hat{a}$ and a specially constructed number-phase operator $$\hat{b}_n = (\hat{n} + i \gamma_0 \hat{\Phi})/\sqrt{2\gamma_0}$$. The conjugate pair $(\hat{n}, \hat{\Phi})$ satisfy $[\hat{n}, \hat{\Phi}] = i$ in the large-occupation regime, ensuring a well-defined phase operator and enabling construction of coherent states in the number-phase plane.

The Hamiltonian $\hat{B}_n(\theta)$ does not generate nonclassicality de novo: its action merely transfers nonclassical features from the quadrature domain of the $\hat{a}$ mode (the input state) into the number domain of the $\hat{b}_n$ mode. This property is proven by showing that acting on products of coherent states (classical states) yields only classical states at output, with no induced squeezing or sub-Poissonian statistics, echoing the behavior of standard linear optics (2605.18708).

Mathematically, the action of $\hat{B}_n(\theta)$ mediates a conversion:

• Quadrature squeezing in $\hat{B}_n(\theta)$0 $\hat{B}_n(\theta)$1 Number squeezing in $\hat{B}_n(\theta)$2.

Rigorous analysis includes transformation properties of the operators and explicit calculation of their effect on field moments and uncertainties. Analytical calculations are corroborated by high-fidelity numerical simulations in finite-dimensional Hilbert spaces, leveraging the Pegg-Barnett formalism for the phase operator.

Experimental Protocol and Statistical Detection

Given that direct photon-number variance measurements are challenging at high occupations, the protocol instead measures $\hat{B}_n(\theta)$3, which is sensitive to sub-Poissonian statistics (number squeezing). However, for large mean photon numbers, $\hat{B}_n(\theta)$4 approaches unity regardless of the underlying squeezing, potentially masking the nonclassical signature.

To resolve this, the protocol applies a counter-displacement operation $\hat{B}_n(\theta)$5 to the $\hat{B}_n(\theta)$6-mode output, reducing the mean photon number while preserving the normalized nonclassicality, thus amplifying the visibility of $\hat{B}_n(\theta)$7. This operational step never creates nonclassicality in a classical state, as rigorously shown. Strategies further include attenuation (with proof of invariance of $\hat{B}_n(\theta)$8 under loss) and post-selection to suppress copy-to-copy number fluctuations and improve statistical resolution.

Experimental feasibility is maintained by designing protocol variants suited for both number-resolving detectors and standard single-photon detectors, accommodating both mesoscopic and single-photon-number regimes.

Numerical Results and Strong Claims

The work provides robust numerical evidence that quadrature squeezing can be efficiently converted into measurable number squeezing with $\hat{B}_n(\theta)$9, even for states subjected to random displacements and photon loss. For example, the protocol demonstrates that initial $\hat{a}$0 values as high as 0.995 can be transformed, via counter-displacement, into significantly more pronounced sub-Poissonian statistics, making nonclassicality statistically discernible with practical detector resolution (2605.18708). It is emphatically proven that:

• $\hat{a}$1 does not generate nonclassicality from classical input states.
• Observation of $\hat{a}$2 within this protocol directly certifies that the incoming state was originally nonclassical, i.e., squeezed.

Implementation Considerations

Potential realizations of the Hamiltonian $\hat{a}$3 include both solid-state (circuit QED, Josephson systems) and atomic ensemble platforms. In the optical domain, interactions with large-spin atomic ensembles coupled via Raman transitions allow experimental construction of the required number-phase coupling, with tunable access to number squeezing in the atomic modes.

Attenuation via optical elements or absorptive media is shown analytically and experimentally to leave $\hat{a}$4 invariant, supporting the practicality of the proposed detection in low-photon-number statistical regimes.

Implications and Future Directions

Practically, the protocol addresses a critical bottleneck in verification of nonclassicality in dynamically fluctuating or poorly characterized quantum channels and is applicable to diverse settings including quantum key distribution, long-distance quantum transmission in free space, and quantum imaging through scattering materials.

Theoretically, the work emphasizes the locality and structure-preserving nature of $\hat{a}$5, suggesting future investigations into generalized squeezing conversion protocols, extension to multimode or entangled nonclassicality detection, and experimental realization in near-term quantum technologies.

Prospective developments include integration into quantum process benchmarking, channel characterization for CV quantum cryptography under adversarial noise, and advanced protocols for squeezing-preserving quantum error correction in bosonic codes using number-phase media.

Conclusion

The paper delivers a mathematically rigorous and experimentally feasible protocol for detecting squeezing in randomly displaced copies of a squeezed state using a conversion Hamiltonian tailored to migrate nonclassicality into the accessible number domain. The work substantiates with analytical proofs and numerical simulations that nonclassicality, as evidenced by $\hat{a}$6, can be robustly detected in challenging environments where conventional homodyne or quadrature measurements fail, provided that the initial state was indeed squeezed. This lays groundwork for future quantum communication and metrology protocols robust to uncontrolled displacement and channel-induced decoherence (2605.18708).