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A generating-function approach to the interference of squeezed states with partial distinguishability

Matheus Eiji Ohno Bezerra, Valery Shchesnovich

TL;DR

This work develops a phase-space generating-function formalism to describe interference of single-mode squeezed states under partial photon distinguishability. Distinguishability enters through the overlap matrix $V$ in a compact matrix $H$ and a Gaussian-generated state, yielding a general expression $G_{f r}({\\bm\\eta}) = c_{f r} \det(I_{2M}-\\mathcal{M}_{\bf r})^{-1/2}$ that governs all output statistics, including threshold-detection probabilities via $P_{f r}(l) = G_{f r}({\\bm\\eta})|_{\\eta_l=0} - G_{f r}({\\bm\\eta})|_{\\eta_l=1}$. In the indistinguishable limit, the framework reduces to hafnian-based Gaussian-boson-sampling probabilities, while a homogeneous-overlap model yields a clean separation of classical noise and quantum interference, with a low-noise interpretation as an average over displaced states. A Gaussian internal-state model reveals how collective phases of internal overlaps influence zero-probability events, demonstrating phase sensitivity beyond simple visibility. Altogether, the approach provides a unified, physically transparent tool to quantify and interpret partial distinguishability in squeezed-state interferometry, with practical implications for assessing imperfections in GBS experiments and for probing internal-state phase information.

Abstract

Photon distinguishability is a fundamental property manifested in multiphoton interference and one of the main sources of noise in any photonic quantum information processing. In this work, rather than relying on first-quantization methods, we build on a generating-function framework based on the phase-space formalism to characterize the effects of partial distinguishability on the interference of single-mode squeezed states. Our approach goes beyond commonly used models that represent distinguishability via additional noninterfering modes and captures genuine multiphoton interference effects induced by the overlap of the internal state of the photons. This description provides a clear physical account of how distinguishability gives rise to effective noise in Gaussian boson sampling protocols while enabling a systematic investigation of phase effects arising from the overlap of the internal states.

A generating-function approach to the interference of squeezed states with partial distinguishability

TL;DR

This work develops a phase-space generating-function formalism to describe interference of single-mode squeezed states under partial photon distinguishability. Distinguishability enters through the overlap matrix in a compact matrix and a Gaussian-generated state, yielding a general expression that governs all output statistics, including threshold-detection probabilities via . In the indistinguishable limit, the framework reduces to hafnian-based Gaussian-boson-sampling probabilities, while a homogeneous-overlap model yields a clean separation of classical noise and quantum interference, with a low-noise interpretation as an average over displaced states. A Gaussian internal-state model reveals how collective phases of internal overlaps influence zero-probability events, demonstrating phase sensitivity beyond simple visibility. Altogether, the approach provides a unified, physically transparent tool to quantify and interpret partial distinguishability in squeezed-state interferometry, with practical implications for assessing imperfections in GBS experiments and for probing internal-state phase information.

Abstract

Photon distinguishability is a fundamental property manifested in multiphoton interference and one of the main sources of noise in any photonic quantum information processing. In this work, rather than relying on first-quantization methods, we build on a generating-function framework based on the phase-space formalism to characterize the effects of partial distinguishability on the interference of single-mode squeezed states. Our approach goes beyond commonly used models that represent distinguishability via additional noninterfering modes and captures genuine multiphoton interference effects induced by the overlap of the internal state of the photons. This description provides a clear physical account of how distinguishability gives rise to effective noise in Gaussian boson sampling protocols while enabling a systematic investigation of phase effects arising from the overlap of the internal states.
Paper Structure (10 sections, 63 equations, 2 figures)

This paper contains 10 sections, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Probability events and generating function
  3. Output probability for indistinguishable photons and the relation with the hafnian
  4. Threshold detectors
  5. Internal states with homogeneous overlap
  6. Distinguishability contribution as an average over displaced states
  7. Gaussian model for the internal states and zero-probability events
  8. Conclusion
  9. Derivation of the squeezed-state generating function
  10. Detailed calculations for the distinguishability as an average over displacements

Figures (2)

  • Figure 1: We consider an $M$-mode multiport described by a unitary matrix $U$. Each input mode $k$ is prepared in a single-mode squeezed vacuum state $|r_k\rangle$, as defined in Eq. (\ref{['smss_input']}). These states are injected into the multiport, which implements a linear transformation on the spatial modes according to Eq. (\ref{['spatial_transf']}). At the output, photon-number-resolving detectors with efficiencies $\eta_l$ measure the number of photons $n_l$ in each output mode.
  • Figure 2: Output probabilities for the interference of single-mode squeezed states with squeezing parameters $r_k=1$ in (a) a balanced beamsplitter and (b) a balanced tritter. The probabilities are shown as a function of the normalized relative delay $\Delta T/\sigma_t$ for two central frequencies, $\Omega_0=0$ (solid) and $\Omega_0=\pi/4$ (dashed).