Photon-Subtracted Squeezed States

• Photon-subtracted squeezed states are non-Gaussian states produced by applying photon annihilation operations to squeezed vacuums, resulting in parity-modified states that approximate coherent state superpositions.
• They are prepared via heralded photon subtraction using beam splitters and high-efficiency detectors, which improves squeezing gain and facilitates state distillation in quantum experiments.
• Experimental implementations show enhanced Wigner negativity and interferometric sensitivity, validating these states as valuable resources in quantum metrology and optical quantum information processing.

Photon-subtracted squeezed states are non-Gaussian states produced by applying photon annihilation operations to squeezed states, most commonly to a single-mode squeezed vacuum. In the idealized form, an $n$-photon-subtracted squeezed vacuum is written as $|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$, where $S(r)$ is the squeezing operator and $n$ is the number of subtracted photons. Because a squeezed vacuum contains only even photon numbers, photon subtraction changes the parity structure of the state and yields outputs that approximate even or odd coherent state superpositions, depending on $n$. Across theory and experiment, photon-subtracted squeezed states have become a central resource for non-Gaussian state engineering, cat-state preparation, squeezing distillation, metrology, continuous-variable sampling, and logical-state generation for optical quantum information processing (Gerrits et al., 2010).

1. Definition and parity structure

A coherent state superposition (CSS), often called a Schrödinger cat state, is a superposition of two coherent states with opposite phases,

$$|\mathrm{CSS}_{\pm}\rangle \propto |\alpha\rangle \pm |-\alpha\rangle,$$

where the $+$ and $-$ combinations correspond to even and odd photon-number sectors, respectively. The squeezed vacuum is

$|\xi\rangle = S(r)|0\rangle,$

and contains only even photon numbers. Photon subtraction acts as a non-Gaussian operation on this Gaussian resource and produces

$|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle.$

In the single-mode setting, one-photon subtraction yields an approximate odd CSS, two-photon subtraction yields an approximate even CSS, and three-photon subtraction yields a higher-amplitude odd CSS. Increasing the number of subtracted photons increases the mean photon number and the amplitude $|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$0 of the CSS approximation (Gerrits et al., 2010).

This parity structure persists in more general constructions. In generalized photon-subtracted squeezed vacuum states, even and odd sectors are explicitly separated into even and odd photon-subtracted families, with Fock expansions supported only on even or odd number states, respectively. In multimode encodings, orthogonality of the computational basis is tied to the parity contrast between photon-subtracted squeezed modes and unmodified squeezed modes. In two-mode settings, parity of the total number of subtracted photons controls the structure of the joint photon-number distribution: even subtraction permits diagonal and off-diagonal components, whereas odd subtraction yields a node along the diagonal and only asymmetric terms (Dey et al., 2020, Arzani et al., 2018, Datta et al., 2024).

A recurrent physical interpretation is that squeezing supplies an even-parity resource state, while subtraction breaks or reshapes parity in a controlled way. This is the basic reason photon-subtracted squeezed states are effective approximations to cat states and, more generally, a flexible source of non-Gaussianity.

2. Preparation methods and experimental realizations

The standard preparation protocol sends squeezed vacuum to a beam splitter, taps a small fraction into a heralding arm, and conditions the transmitted state on photon detection in the reflected arm. In the pulsed experiment “Generation of Optical Coherent State Superpositions by Number-Resolved Photon Subtraction from Squeezed Vacuum” (Gerrits et al., 2010), squeezed vacuum was generated with an optical parametric amplifier, photons were subtracted at a weakly reflecting beam splitter, and the heralded state was reconstructed by homodyne detection with maximum-likelihood quantum state tomography. A high-efficiency transition-edge sensor (TES) with 85% detection efficiency and photon-number resolution up to 10 photons enabled resolved subtraction events. The three-photon-subtracted CSS had mean photon number $|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$1 and fidelity $|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$2 with an ideal CSS, establishing experimentally that subtracting more photons results in higher-amplitude CSSs (Gerrits et al., 2010).

Scaling multi-photon subtraction was a long-standing limitation. “High-Rate Four Photon Subtraction from Squeezed Vacuum” (Endo et al., 13 Feb 2025) used a waveguide periodically poled lithium niobate crystal as a broadband optical parametric amplifier, pumped by a picosecond-pulsed laser at 5 MHz repetition rate, together with a Ti-Au superconducting TES cooled to 280 mK. The detector resolved photon numbers up to at least four with $|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$3 and system efficiency $|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$4 at $|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$5. The experiment reported an event rate of $|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$6 cps for three-photon subtraction and $|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$7 cps for four-photon subtraction, with Wigner negativity observed without loss correction and quantum coherence verified through off-diagonal density-matrix elements in the continuous-variable representation (Endo et al., 13 Feb 2025).

Several variants extend the basic subtraction mechanism. Generalized photon subtraction injects an additional squeezed vacuum in the orthogonal quadrature into the unused port of the beam splitter, rather than vacuum, to improve heralding rate while maintaining state quality. In experiment, this produced more than one order of magnitude generation-rate improvement for 2-dB- and 4-dB-squeezed single-photon states relative to standard photon subtraction, particularly for the 2 dB case (Tomoda et al., 2024). In another direction, beam-splitter and controlled-$|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$8 implementations of photon subtraction from a two-mode entangled Gaussian state were shown to generate arbitrary-order squeezed Fock states under a “universal solution regime,” with maximum generation probability

$|\psi_n\rangle \propto \hat{a}^n S(r)|0\rangle$9

and $S(r)$0 for the first squeezed Fock state (Korolev et al., 2023).

3. Phase-space structure, nonclassicality, and state characterization

The most direct signatures of photon-subtracted squeezed states are non-Gaussianity, parity-selective photon statistics, and Wigner-function negativity. In the lossless ideal squeezed-vacuum case, the photon-number distribution preserves parity: even $S(r)$1-photon subtraction yields only even-photon components and odd $S(r)$2-photon subtraction yields only odd-photon components. The Wigner function becomes non-Gaussian and develops negative regions, in contrast to the always-positive Gaussian Wigner function of the squeezed vacuum (Xu et al., 2019).

For squeezed thermal inputs, the state

$S(r)$3

admits analytic normalization and quasiprobability expressions. A notable result is that the normalization factor is a Legendre polynomial,

$S(r)$4

with $S(r)$5 and $S(r)$6 determined by the squeezing parameter $S(r)$7 and thermal occupation $S(r)$8. For single-photon subtraction, the Wigner function at the phase-space center is negative if

$S(r)$9

showing explicitly that squeezing can overcome thermal noise, while thermal photons destroy nonclassicality (Hu et al., 2010). Thermal-field-dynamics methods lead to the same Legendre-polynomial structure for normalization and photon-number distributions of photon-subtracted squeezed thermal states, and simplify the pure squeezed-vacuum limit as well (Hu et al., 2011).

More elaborate phase-space features appear in superpositions built from photon-subtracted squeezed-vacuum states. When a significant number of photons is subtracted from superpositions of squeezed-vacuum states, the Wigner function develops sub-Planck structures whose area becomes substantially smaller than the Planck scale, and the overlap with displaced versions of the state exhibits high displacement sensitivity. In that analysis, the size of the sub-Planck structures and the sensitivity are strongly influenced by the average photon number, with photon subtraction offering sensitivity much higher than the standard quantum limit, although photon addition yields even smaller tiles for the same subtraction order (Akhtar et al., 2022).

These results collectively establish that photon-subtracted squeezed states are not merely weak perturbations of Gaussian states. They are phase-space resources with parity-selective structure, analytically tractable non-Gaussianity in several models, and experimentally measurable negativity and coherence.

4. Loss, impurity, and decoherence

Loss and mode impurity are the principal mechanisms that degrade photon-subtracted squeezed states. A complete operator and characteristic-function description of lossy multiphoton subtraction considers losses before subtraction, beam-splitter mixing, losses in the heralding arm, and conditional projection onto a photon-number state. In this framework, success probability decreases rapidly with increasing loss, increasing subtraction order, or decreasing beam-splitter transmission. Loss also breaks the parity selection rule of ideal multiphoton-subtracted squeezed vacuum states, causing all photon-number components to appear, and it reduces Wigner negativity until nonclassicality is erased (Xu et al., 2019).

In continuous-wave implementations, impurity can arise even in the ideal case of no optical losses. “Purification of photon subtraction from continuous squeezed light by filtering” (Yoshikawa et al., 2017) identifies the source as mode-mismatch of squeezing produced by the non-flat frequency spectrum of an optical parametric oscillator. The mode-matching rate is

$n$0

and for a typical Lorentzian optical parametric oscillator with a wavepacket matched to the optical-parametric-oscillator bandwidth, the resulting impurity corresponds to a mode-matching rate of about $n$1, analogous to about $n$2 optical loss. Narrowband filtering in the subtraction path can make $n$3, and in principle enables pure photon-subtracted squeezed states from continuous-wave squeezed light (Yoshikawa et al., 2017).

Decoherence in a thermal environment is analytically tractable for photon-subtracted squeezed thermal states. The Wigner function remains negative at the center for single-photon subtraction as long as

$n$4

and becomes strictly positive after that threshold, while in the long-time limit the state relaxes to a thermal Gaussian state and loses memory of initial subtraction or squeezing (Hu et al., 2010).

This body of work indicates that the effective quality of a photon-subtracted squeezed state is controlled jointly by optical loss, detector inefficiency, spectral mode matching, and environmental decoherence. A plausible implication is that improvements in detector efficiency alone are insufficient if multimode impurity and spectral mismatch remain unaddressed.

5. Squeezing gain, distillation, and metrological performance

Photon subtraction can increase quadrature squeezing as well as generate non-Gaussianity. “Gain of squeezing via photon subtractions” (Podoshvedov et al., 5 Feb 2025) studies subtraction of $n$5, $n$6, and $n$7 photons from a single-mode squeezed vacuum. More photons lead to more gain of the squeezing, with more than $n$8 dB observed for higher-order even subtraction, but the two-photon strategy is practically preferred because it has higher success probability, wider squeezing gain width of about $n$9 dB, and the least quadrature variance in the corresponding range of initial squeezing. Odd-photon subtraction does not provide additional squeezing in that analysis. The same work also shows that mixing a single photon with the squeezed vacuum at the beam splitter input and conditioning on an odd detection event can herald an even continuous-variable state that is several times brighter than the initial state and has lower quadrature noise (Podoshvedov et al., 5 Feb 2025).

Two-photon subtraction also appears as a practical distillation primitive. “Multi-step two-copy distillation of squeezed states via two photon subtraction” (Grebien et al., 2022) experimentally demonstrated distillation of Gaussian squeezed states that suffered from Gaussian photon loss. The first step improved the squeeze factor from $n$0 dB to $n$1 dB by subtraction of two photons, and the second step improved it from $n$2 dB to $n$3 dB by a Gaussification protocol based on two copies and post-selection. This was implemented using an 8-port balanced homodyne detector and data post-processing, with reconstructed Wigner functions showing the non-Gaussian intermediate state and re-Gaussification after the second step (Grebien et al., 2022).

In interferometry, photon-subtracted squeezed states can outperform ordinary squeezed vacuum at accessible squeezing levels. For a Mach-Zehnder interferometer fed with a coherent state and an $n$4-photon-subtracted single-mode squeezed vacuum, the quantum Cramér-Rao bound is determined by a modified quantum Fisher information, and sensitivity gains exceeding $n$5 dB were reported for weakly squeezed inputs with squeezing below $n$6 dB when $n$7. The improvement persists under intensity-difference measurement and remains visible even for large coherent amplitude $n$8 when the squeezing is below $n$9 dB (Podoshvedov et al., 4 Feb 2025).

These results place photon subtraction at the intersection of non-Gaussian resource generation and practical squeezing enhancement. This suggests that, in regimes where very high Gaussian squeezing is technically difficult, subtraction from weakly squeezed light can be the more efficient strategy.

6. Quantum information, sampling complexity, and state engineering beyond the single mode

Photon-subtracted squeezed states have broad quantum-information roles. Large-amplitude cat states are described as crucial resources for Gottesman-Kitaev-Preskill logical qubits, and the high-rate four-photon-subtraction experiment was positioned as a critical foundation for generating resources essential for fault-tolerant quantum computing and ultrafast optical quantum processors (Endo et al., 13 Feb 2025). In multimode encodings, coherent mode-dependent single-photon subtraction from multimode squeezed states supports arbitrary $$|\mathrm{CSS}_{\pm}\rangle \propto |\alpha\rangle \pm |-\alpha\rangle,$$0-level state generation by shaping a classical gate field, with readout by parity or homodyne-based Wigner-at-the-origin measurements. The same framework was proposed for efficient vector-distance and inner-product estimation in quantum machine learning primitives (Arzani et al., 2018).

In computational complexity, arbitrary photon-added or photon-subtracted squeezed vacuum states processed by linear optics and measured by parity form a sampling model in the same complexity class as boson sampling, independently of the squeezing parameter (Olson et al., 2014). A related continuous-variable sampling model uses photon-subtracted squeezed states, passive linear optics, and eight-port homodyne detection, and proves worst-case and average-case hardness for exact sampling, with the zero-squeezing limit reducing to boson sampling with eight-port homodyne detection (Chabaud et al., 2017). In both formulations, the non-Gaussianity of the input states is the essential ingredient that precludes efficient classical simulation under the stated complexity assumptions.

Two-mode and multimode generalizations reveal further structure. For photon-subtracted two-mode squeezed vacuum states generated in a waveguide trimer, the parity of the number of subtracted photons controls whether correlations are classical or nonclassical at small squeeze parameter and whether the joint photon-number distribution has diagonal support (Datta et al., 2024). In four-mode squeezed vacuum states, photon subtraction can yield higher entanglement than photon addition in some bipartitions, unlike the two-mode case, showing that multimode photon subtraction can alter entanglement patterns in ways not captured by non-Gaussianity alone (Das et al., 2015). In two-mode squeezed coherent states used for continuous-variable teleportation, symmetric photon subtraction enhances teleportation fidelity over an extensive squeezing range, while asymmetric subtraction yields only marginal low-squeezing improvement and must be weighed against the associated success probabilities (Arora et al., 2024).

Generalized constructions extend beyond the harmonic oscillator. A model-independent framework based on generalized ladder operators $$|\mathrm{CSS}_{\pm}\rangle \propto |\alpha\rangle \pm |-\alpha\rangle,$$1 and $$|\mathrm{CSS}_{\pm}\rangle \propto |\alpha\rangle \pm |-\alpha\rangle,$$2 defines generalized photon-subtracted squeezed vacuum states for nonlinear and deformed systems, including the trigonometric Pöschl-Teller potential, and finds that nonclassicality increases almost proportionally with the number of subtracted photons in some quantification schemes (Dey et al., 2020).

Photon-subtracted squeezed states therefore occupy a distinctive position in quantum optics: they are experimentally heralded non-Gaussian states with parity-controlled structure, analytically rich behavior under loss and decoherence, and direct utility in cat-state generation, squeezing distillation, quantum metrology, complexity-theoretic sampling models, and multimode continuous-variable information processing.