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arXiv:2605.27660

Photon-Conditioned Squeezed States for Directional Displacement Response in Continuous-Variable Photonics

Boris Kiefer, Olivier Pfister

quant-phSubmitted May 26, 2026• Announced May 28, 2026• 0 citations
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Abstract

Squeezed Fock states, photon-subtracted squeezed states, and optical cat states are established non-Gaussian resources in continuous-variable quantum optics. Here we compare these known state families from a task-oriented perspective: matched mean photon number, scalar Wigner negativity, and directional displacement-fidelity response. Starting from squeezed vacuum, single-photon subtraction prepares a state proportional to \(S(r,θ)\ket{1}\), while two-photon subtraction prepares an even-parity squeezed Fock superposition rather than a pure \(S(r,θ)\ket{2}\). We benchmark photon-conditioned squeezed states against Fock and coherent-cat references using the integrated Wigner negativity \(δ\), the energy-normalized metric \(δ/\langle n\rangle\), and fidelity-threshold displacement radii \(R_F(φ)\). Cat benchmarks remain strong scalar Wigner-negativity resources, whereas photon-conditioned squeezed states provide an origin-centered alternative with tunable anisotropic displacement response. In particular, the two-photon-subtracted squeezed state shows favorable displacement-fidelity radii over selected quadrature directions at matched \(\langle n\rangle\). These results identify a regime relevant to homodyne-aligned continuous-variable control and anisotropic displacement-noise mitigation, with directional sensing as a natural dual application.

Comments
Submitted to Physical Review A. 21 pages, includes 3 main-text figures and 1 appendix figure

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Squeezed Fock states, photon-subtracted squeezed states, and optical cat states are established non-Gaussian resources in continuous-variable quantum optics. Here we compare these known state families from a task-oriented perspective: matched mean photon number, scalar Wigner negativity, and directional displacement-fidelity response. Starting from squeezed vacuum, single-photon subtraction prepares a state proportional to (S(r,θ)\ket{1}), while two-photon subtraction prepares an even-parity squeezed Fock superposition rather than a pure (S(r,θ)\ket{2}). We benchmark photon-conditioned squeezed states against Fock and coherent-cat references using the integrated Wigner negativity (δ), the energy-normalized metric (δ/\langle n\rangle), and fidelity-threshold displacement radii (R_F(φ)). Cat benchmarks remain strong scalar Wigner-negativity resources, whereas photon-conditioned squeezed states provide an origin-centered alternative with tunable anisotropic displacement response. In particular, the two-photon-subtracted squeezed state shows favorable displacement-fidelity radii over selected quadrature directions at matched (\langle n\rangle). These results identify a regime relevant to homodyne-aligned continuous-variable control and anisotropic displacement-noise mitigation, with directional sensing as a natural dual application.
Schrödinger cat states are important non-Gaussian resources for quantum information processing and quantum-enhanced metrology. Although photon subtraction from squeezed vacuum states has proven useful for generating such states in the optical domain, scaling to large amplitudes while maintaining high fidelity and heralding probability remains challenging. To address these obstacles, we analyze alternating squeezing and photon-number-subtraction operations. Rather than optimizing for a single heralding outcome, we formulate cat-state generation as a fidelity-constrained multibranch problem, jointly optimizing the total probability of all photon-number outcomes satisfying a prescribed fidelity threshold at one common experimental setting. Already for plain photon subtraction, this provides a more operational measure of the usable generation probability and can favor parameter configurations that differ from single-branch optima. Considering experimental constraints on available squeezing and photon-number-resolving capability, we demonstrate improvements in fidelity and success probability compared with plain photon subtraction, without necessarily increasing the required squeezing resources. Without restrictions on photon-number resolution, we find that the optimal architecture of the alternating scheme reduces to one of two single-subtraction endpoint configurations, depending on the available squeezing resources. When photon-number resolution is additionally limited, genuinely alternating branches can become advantageous. We derive analytical expressions for the conditional states, cat-state fidelities, and heralding probabilities, enabling efficient multibranch optimization and a systematic analysis of resource trade-offs, sensitivity to experimental parameter deviations, and heralding loss. Together, these results identify the architecture best suited to a given resource budget.
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The generation of non-Gaussian quantum states is a key requirement for universal continuous-variable quantum information processing. We report the experimental generation of large-amplitude squeezed coherent-state superpositions (squeezed cat states) on free-space optical pulses, reaching an amplitude of α=2.47α= 2.47α=2.47, which, to our knowledge, exceeds all previously reported values. Our protocol relies on the controlled mixing of the Fock states ∣1⟩|1\rangle∣1⟩ and ∣2⟩|2\rangle∣2⟩ through a tunable beam splitter, followed by heralding via homodyne detection. The resulting state displays three well-resolved negative regions in its Wigner function and achieves a fidelity of 0.530.530.53 with the target state ∝S^(z)(∣α⟩−∣−α⟩)\propto \hat{S}(z)(|α\rangle - |-α\rangle)∝S^(z)(∣α⟩−∣−α⟩), with α=2.47α= 2.47α=2.47 and squeezing parameter z=0.56z = 0.56z=0.56. These results constitute a significant milestone for temporal breeding protocols and for the iterative generation of optical GKP states, opening new perspectives for scalable and fault-tolerant photonic quantum architectures.
A near-deterministic source of non-Gaussianity is the missing piece to reach universal and fault-tolerant quantum computing with continuous-variable optics. Due to a lack of strong non-linearities, the probabilistic photon-subtracted Gaussian states remain the most promising contender. In combination with Gaussian measurements and single-mode feed-forward operations, they have been shown to provide sufficient non-Gaussianity at high rates. However, these existing breeding protocols lack the necessary loss tolerance to be experimentally feasible. In this work, we introduce a mathematical formalism based on the application of a Gaussian blur and filter on the measured Fock Wigner function to describe general photon-subtracted Gaussian states. Within our representation, the initial squeezing, Gaussian measurements, and photon loss can all be understood as contributions to the total blurring and interchanged accordingly. We find that this intuitive approach can be used to find improvements to cat, cubic phase, and GKP state generation by compromising between success rates, feed-forward compatibility, and loss tolerance. In the case of multiple photon subtractions, this is achieved by introducing a hybrid setup variant bridging the gap between protocols based on breeding and post-selection. Finally, we establish a general lower bound on the resources needed to generate a given target state near-deterministically by introducing the expected Wigner logarithmic negativity as an additive monotone of non-Gaussianity.
Multiphoton subtraction from a traveling squeezed vacuum is a standard route to non-Gaussian optical-state engineering. High state-generation rates call for ultrashort optical pulses driven by high-repetition-rate lasers, but combining higher-order subtraction with ultrashort pulsed light and full state reconstruction remains challenging. Here we demonstrate up to four-photon subtraction from 10-ps squeezed-vacuum wave packets generated by a broadband waveguide optical parametric amplifier. Using a Ti-Au transition-edge sensor for photon-number-resolved heralding and homodyne tomography on the signal mode, we reconstruct the generated states up to the four-photon-subtracted case. The reconstructed states exhibit clear parity-dependent structure and visible Wigner-function negativity without loss correction. These results extend the experimentally accessible regime of multiphoton subtraction in broadband traveling optical modes and establish a platform for non-Gaussian optical resource-state engineering in continuous-variable quantum optics.
We extend our previous study of squeezed two-component Schrödinger cat states to a phase-referenced single-mode family with an arbitrary number NNN of coherent components and general relative phases. Within the established circular-state construction, we derive a closed complex overlap kernel for the ordered action of displacement, phase rotation, and squeezing, retaining exact finite-amplitude normalization. We distinguish finite-displacement orthogonality from local phase-estimation sensitivity quantified by quantum Fisher information (QFI). For equal-phase states, we compare N=2,…,9N=2,\ldots,9N=2,…,9 at a common mean photon number and include coherent-state and squeezed-vacuum benchmarks. A bound on the higher photon-number sectors establishes the fixed-energy asymptote FQ,N=4nˉ(N−nˉ)+o(1)F_{Q,N}=4\bar n(N-\bar n)+o(1)FQ,N​=4nˉ(N−nˉ)+o(1). Its leading term agrees with the known QFI of a vacuum--NNN-photon superposition. The resulting decrease of the ideal local quantum Cramér--Rao bound does not establish unlimited attainable precision: prior phase information, sample size, phase-reference resources, optical loss, state preparation, and measurement choice remain relevant.