Single-Photon-Added Squeezed Vacuum

• Single-photon-added squeezed vacuum is a non-Gaussian state obtained by applying a photon creation operator to a squeezed vacuum, converting its even photon-number distribution into an odd one.
• A key finding shows that for pure squeezed vacua, normalized photon addition and subtraction yield the same state (up to a global phase), effectively rendering the state equivalent to a squeezed single-photon.
• This state serves as a versatile resource in applications such as optical Schrödinger-cat generation, quantum metrology, boson sampling, and multimode entanglement engineering.

Single-photon-added squeezed vacuum is a non-Gaussian optical state obtained by applying a single creation operator to a squeezed vacuum, typically written as $\hat a^\dagger \hat S(\zeta)\lvert 0\rangle$ (Chen et al., 2023). In the single-mode setting, the underlying squeezed vacuum contains only even photon-number components, while single-photon addition converts it into a state with only odd photon-number components (Olson et al., 2014). This parity change places the state at the intersection of non-Gaussian state engineering, optical Schrödinger-cat approximations, continuous-variable quantum information, multimode entanglement generation, and photonic complexity-theoretic sampling (Chen et al., 2023, Walschaers et al., 2017, Olson et al., 2014). A 2024 result sharpens its conceptual status further by proving that, for pure squeezed vacuum states, normalized single-photon addition and single-photon subtraction yield the same output state up to a global scalar (Steuernagel et al., 2024).

1. Definition and state structure

A single-mode squeezed vacuum is obtained by applying the squeezing operator to vacuum,

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$

with $\zeta=re^{i\phi}$ the squeezing parameter (Steuernagel et al., 2024, Chen et al., 2023). In the Fock basis, the squeezed vacuum has only even photon numbers (Steuernagel et al., 2024, Olson et al., 2014). For real squeezing and $\phi=0$, one representation is

$$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$

as given in the cited treatment (Steuernagel et al., 2024).

A single-photon-added squeezed vacuum is obtained by acting with $\hat a^\dagger$,

$$|\mathrm{PASV}_1(\xi)\rangle \propto \hat a^\dagger \hat S(\xi)|0\rangle=\hat a^\dagger|\xi\rangle,$$

or, in a normalization-explicit notation,

$$|\xi,1\rangle=\frac{\hat a^\dagger|\xi\rangle}{\sqrt{N_1(|\xi|)}}$$

(Olson et al., 2014, Chen et al., 2023). In the single-photon case, the Fock expansion becomes an odd-photon superposition,

$$\hat a^\dagger|\xi\rangle= \frac{1}{\sqrt{\cosh r}}\sum_{m=0}^{\infty} (-1)^m \frac{\sqrt{(2m+1)!}}{2^m m!} e^{i m\theta}\tanh^m r\,|2m+1\rangle$$

(Olson et al., 2014). Thus the state has only odd photon-number support (Olson et al., 2014, Chen et al., 2023).

Several equivalent normalizations appear in the literature covered here. For the single-mode, real-squeezing convention of one analysis, the normalized state is

$$|\psi_{\text{PASV}}\rangle=\frac{1}{\cosh\lambda}\,a^\dagger S_1(\lambda)|0\rangle =\frac{1}{\cosh\lambda}\,S_1(\lambda)|1\rangle$$

(Hu et al., 2011). This directly identifies the state with a squeezed single-photon. A related account likewise notes that, when the input is a squeezed vacuum, photon-added and squeezed-single-photon descriptions coincide up to normalization (Jezek et al., 2012).

In the weak-squeezing limit, the state reduces to a single photon: $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$0 (Olson et al., 2014). This makes PASV a continuous interpolation between a Fock-state resource and a strongly nonclassical odd-parity non-Gaussian state.

2. Addition, subtraction, and the squeezed-mode annihilation operator

A central 2024 result proves that for pure squeezed vacuum states, normalized single-photon addition and normalized single-photon subtraction produce identical output states up to a global phase (Steuernagel et al., 2024). The question is posed as whether

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$1

represent the same quantum state. In position representation, the condition reduces to a differential equation whose solutions are Gaussian wavefunctions corresponding to pure squeezed vacua (Steuernagel et al., 2024). For these states, one obtains

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$2

(Steuernagel et al., 2024). After normalization, the output states are therefore the same up to phase.

The same paper states the corresponding characterization result more strongly: among pure states, the identity-of-outcome holds precisely for possibly rotated pure squeezed vacuum states (Steuernagel et al., 2024). Displaced states, such as squeezed coherent states, do not satisfy the same condition (Steuernagel et al., 2024). Mixed Gaussian states with $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$3 also do not exhibit this property, although certain non-Gaussian mixed states formed as incoherent sums of pure squeezed vacua with the same squeezing do (Steuernagel et al., 2024).

This equivalence is tied to the Bogoliubov-transformed annihilation operator

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$4

for which the squeezed vacuum satisfies

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$5

(Steuernagel et al., 2024). The same analysis rewrites the identity-of-outcome condition in terms of an operator

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$6

that annihilates the squeezed-vacuum wavefunction and is proportional to $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$7 (Steuernagel et al., 2024). This gives a reinterpretation of the familiar statement that squeezed vacuum is the vacuum of a Bogoliubov mode: the linear combination of $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$8 and $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}\big(\zeta \hat a^2-\zeta^* \hat a^{\dagger 2}\big)\right],\qquad |0,\zeta\rangle=\hat S(\zeta)|0\rangle,$$9 that annihilates the squeezed vacuum is exactly the combination that makes single-photon addition and subtraction equivalent on that state (Steuernagel et al., 2024).

This does not imply that photon addition and subtraction are generally the same operation. The result is explicitly stated to be strictly single-photon and specific to pure squeezed vacuum input; after one addition or subtraction, the state is no longer a pure Gaussian squeezed vacuum, so repeated operations do not preserve the same symmetry (Steuernagel et al., 2024).

3. Phase-space properties, non-Gaussianity, and nonclassical signatures

Squeezed vacuum is Gaussian, whereas applying $\zeta=re^{i\phi}$0 or $\zeta=re^{i\phi}$1 is a non-Gaussian operation, so single-photon-added and single-photon-subtracted squeezed vacua are non-Gaussian states (Olson et al., 2014). Their Wigner functions are correspondingly non-Gaussian and typically exhibit negativity (Olson et al., 2014, Chen et al., 2023).

For pure squeezed-vacuum input, one explicit Wigner-space result gives

$\zeta=re^{i\phi}$2

with

$\zeta=re^{i\phi}$3

(Steuernagel et al., 2024). At the origin,

$\zeta=re^{i\phi}$4

which is the quoted maximal negativity for these parameters (Steuernagel et al., 2024).

A broader treatment of photon-added squeezed thermal states shows that, in the squeezed-vacuum limit, the single-photon-added state has

$\zeta=re^{i\phi}$5

independently of the squeezing parameter $\zeta=re^{i\phi}$6 (Hu et al., 2011). The same work emphasizes that the state is maximally nonclassical in this sense and that adding one photon to a squeezed vacuum produces a Wigner function with a negative dip at the origin for all $\zeta=re^{i\phi}$7 (Hu et al., 2011). For the more general squeezed thermal case, the single-photon-added Wigner function remains negative at the phase-space center (Hu et al., 2011).

Nonclassicality is also discussed beyond Wigner negativity. One analysis of noisy squeezed single-photon states, directly relevant because ideal photon-subtracted and photon-added squeezed vacua both reduce to a squeezed single-photon for pure squeezed-vacuum input, distinguishes nonclassicality from quantum non-Gaussianity (Jezek et al., 2012). There, a state is called quantum non-Gaussian if it cannot be expressed as a mixture of Gaussian states, and a witness based on the vacuum and one-photon probabilities is used to certify this even when the Wigner function is positive (Jezek et al., 2012). A plausible implication is that SPASV under realistic loss may remain certifiably quantum non-Gaussian even after Wigner negativity is experimentally washed out.

For multimode Gaussian backgrounds, a general phase-space theory shows that single-photon addition yields a Wigner function of the form

$\zeta=re^{i\phi}$8

that is, a quadratic polynomial times the original Gaussian Wigner function (Walschaers et al., 2017). The same work states that single-photon addition to any non-displaced Gaussian state always produces a negative Wigner function, even with impurity or mixing in the addition process (Walschaers et al., 2017). In that multimode setting, nonzero higher-order truncated quadrature correlations beyond second order serve as direct signatures of the non-Gaussianity created by photon addition (Walschaers et al., 2017).

4. Relation to Schrödinger-cat states and experimental generation

Because a squeezed vacuum has only even photon-number components and single-photon addition produces only odd ones, a single-photon-added squeezed vacuum naturally approximates an odd Schrödinger-cat state (Chen et al., 2023). The cat target is written as

$\zeta=re^{i\phi}$9

an odd superposition containing only odd Fock components (Chen et al., 2023). The same source states explicitly that, for one added photon, only a negative-cat superposition state can result (Chen et al., 2023).

An experimental realization reported in 2023 generated optical Schrödinger cats by adding a photon to a squeezed vacuum state (Chen et al., 2023). In that experiment, photon addition is modeled as the completely positive map

$\phi=0$0

up to normalization (Chen et al., 2023). The setup used a squeezed vacuum from an optical parametric oscillator, injected as a seed into a second cavity implementing spontaneous parametric down-conversion; a click in the idler channel heralded photon addition on the squeezed state (Chen et al., 2023). Balanced homodyne detection with trigger-gated tomography reconstructed the output density matrix (Chen et al., 2023).

The reported generation rate was

$\phi=0$1

at the highest OPO pump power and fixed SPDC pump (Chen et al., 2023). The paper states that this is “at least one order of magnitude higher than all previously reported realizations” based on photon subtraction (Chen et al., 2023). It further reports pronounced Wigner negativity down to $\phi=0$2 squeezing, even when the initial squeezed-vacuum input state had low purity (Chen et al., 2023). The reconstructed central Wigner values were listed as $\phi=0$3, $\phi=0$4, $\phi=0$5, and $\phi=0$6 for increasing pump powers (Chen et al., 2023).

The same study optimized fidelity with an ideal odd cat and found:

• $\phi=0$7 with $\phi=0$8 at $\phi=0$9,
• $$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$0 with $$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$1 at $$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$2,
• $$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$3 with $$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$4 at $$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$5,
• $$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$6 with $$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$7 at $$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$8 (Chen et al., 2023).

To benchmark the addition operation itself, that work introduced a relative fidelity against ideal photon addition applied to the experimentally reconstructed degraded squeezed input and reported

$$|0,z\rangle=\frac{1}{\sqrt{\cosh z}}\sum_{m=0}^{\infty}(-\tanh z)^m\frac{\sqrt{(2m)!}}{2^m m!}\,|2m\rangle$$9

at the highest squeezing and largest cat amplitude $\hat a^\dagger$0 (Chen et al., 2023). The same paper argues that photon addition offers experimental advantages over photon subtraction because it uses heralded signal photons as triggers and avoids the weak tap-off constraint intrinsic to subtraction protocols (Chen et al., 2023).

A later theoretical study on optical GKP-state generation used single-photon-added squeezed vacuum as the non-Gaussian input resource and reported a maximum fidelity of $\hat a^\dagger$1 at a squeezing level of $\hat a^\dagger$2 for the approximate finite-energy GKP target (Senjaya, 9 Jun 2026). That work compares favorably with squeezed odd-cat approaches and treats SPASV as a simpler non-Gaussian ingredient in this context (Senjaya, 9 Jun 2026).

5. Quantum metrology, sampling complexity, and other applications

In Mach–Zehnder interferometry with a coherent state in one port and a photon-added squeezed vacuum state in the other, a 2018 analysis studies parity detection and phase sensitivity (Wang et al., 2018). It defines the normalized $\hat a^\dagger$3-photon-added squeezed vacuum as

$\hat a^\dagger$4

so that for the single-photon case

$\hat a^\dagger$5

(Wang et al., 2018). The paper states that, when $\hat a^\dagger$6, the single-photon PASVS and single-photon PSSVS are the same non-Gaussian squeezed state (Wang et al., 2018), matching the equivalence later formalized for pure squeezed vacua (Steuernagel et al., 2024).

That interferometric study finds that, when the phase shift approaches zero, the squeezed vacuum state is the optimal state within a constraint on average photon number, but when the phase shift slightly deviates from zero, the optimal state is the photon-added squeezed vacuum state when the states carry many photons (Wang et al., 2018). It also states that the quantum Cramér–Rao bound can be reached by parity detection (Wang et al., 2018). This suggests that the non-Gaussian deformation introduced by photon addition changes the parity fringe structure in a way favorable away from the strict $\hat a^\dagger$7 limit.

In photonic sampling complexity, a 2014 result shows that sampling arbitrary photon-added or photon-subtracted squeezed vacuum states is in the same complexity class as boson sampling (Olson et al., 2014). The multimode input is

$\hat a^\dagger$8

with the first $\hat a^\dagger$9 modes carrying PASV states and the remaining modes carrying squeezed vacua, all with the same squeezing parameter $$|\mathrm{PASV}_1(\xi)\rangle \propto \hat a^\dagger \hat S(\xi)|0\rangle=\hat a^\dagger|\xi\rangle,$$0 (Olson et al., 2014). After a Haar-random orthogonal interferometer and parity measurements, the parity pattern recovers the same binary occupation configuration as in Aaronson–Arkhipov boson sampling (Olson et al., 2014). The paper emphasizes that the sampling amplitudes are completely independent of the squeezing (Olson et al., 2014).

The same source states that, under the standard assumptions $$|\mathrm{PASV}_1(\xi)\rangle \propto \hat a^\dagger \hat S(\xi)|0\rangle=\hat a^\dagger|\xi\rangle,$$1 or stronger, lossless interferometer, perfect parity detection, and indistinguishable photons, the PASV/PSSV sampling task is exactly in the same computational complexity class as standard boson sampling (Olson et al., 2014). This places single-photon-added squeezed vacuum among non-Gaussian optical resources supporting classically hard linear-optical sampling.

Theoretical and experimental discussions also connect photon-added squeezed vacuum to non-Gaussian resource generation for continuous-variable quantum information, entanglement distillation, and metrology (Chen et al., 2023, Olson et al., 2014). These are stated as application domains rather than as detailed protocol-level demonstrations in the cited material.

6. Multimode generalizations, entanglement, and mixed-state extensions

Multimode photon-added squeezed states are treated systematically in several sources. One multimode formalism defines single-photon-added states from an arbitrary multimode Gaussian state $$|\mathrm{PASV}_1(\xi)\rangle \propto \hat a^\dagger \hat S(\xi)|0\rangle=\hat a^\dagger|\xi\rangle,$$2 by

$$|\mathrm{PASV}_1(\xi)\rangle \propto \hat a^\dagger \hat S(\xi)|0\rangle=\hat a^\dagger|\xi\rangle,$$3

where the mode-selective creation operator acts in a chosen mode $$|\mathrm{PASV}_1(\xi)\rangle \propto \hat a^\dagger \hat S(\xi)|0\rangle=\hat a^\dagger|\xi\rangle,$$4 of the full phase space (Walschaers et al., 2017). For a pure multimode squeezed vacuum, this becomes the multimode SPASV state (Walschaers et al., 2017). The same work shows that if the photon is added in a mode that is not aligned with a single supermode, the resulting state is inherently entangled in the sense that no passive linear optics can render it separable (Walschaers et al., 2017).

In a separate four-mode study, photon addition and subtraction are applied to a four-mode squeezed vacuum (Das et al., 2015). The general photon-added state is written as

$$|\mathrm{PASV}_1(\xi)\rangle \propto \hat a^\dagger \hat S(\xi)|0\rangle=\hat a^\dagger|\xi\rangle,$$5

(Das et al., 2015). When exactly one mode is a player mode, the entanglement in the $$|\mathrm{PASV}_1(\xi)\rangle \propto \hat a^\dagger \hat S(\xi)|0\rangle=\hat a^\dagger|\xi\rangle,$$6 bipartition increases monotonically with the number of added photons, and for a single player mode the photon-added and photon-subtracted states have the same entanglement in the player:spectator cut (Das et al., 2015). However, in other bipartitions and mode allocations, photon-subtracted states can possess higher entanglement than photon-added states (Das et al., 2015). The same paper finds that photon addition always produces larger relative-entropy non-Gaussianity than photon subtraction, while that non-Gaussianity does not reliably predict entanglement (Das et al., 2015).

The 2024 equivalence result also extends beyond pure states in a restricted way. It states that any incoherent sum of pure squeezed vacuum states with the same squeezing, but arbitrary phase-space orientations, satisfies the identity-of-outcome condition for addition and subtraction (Steuernagel et al., 2024). A maximally angularly mixed state built by integrating rotated pure squeezed vacua over all angles is explicitly given as a rotationally symmetric non-Gaussian mixed state that still satisfies the condition (Steuernagel et al., 2024). By contrast, mixtures with different squeezing parameters do not satisfy it (Steuernagel et al., 2024).

These multimode and mixed-state analyses correct a possible misconception. Single-photon-added squeezed vacuum is not only a single-mode odd-parity state; in multimode settings it is a mode-selective non-Gaussian operation whose entanglement properties depend sharply on mode basis, player versus spectator assignments, and the purity or coherence structure of the underlying squeezed resource (Walschaers et al., 2017, Das et al., 2015, Steuernagel et al., 2024).

7. Conceptual status and current perspective

Single-photon-added squeezed vacuum occupies a distinctive position among non-Gaussian states of light. Formally, it is the odd-parity state obtained from a Gaussian squeezed vacuum by one creation operation (Olson et al., 2014, Chen et al., 2023). In the pure squeezed-vacuum setting, it is also proportional to a squeezed single-photon and, after normalization, identical to the photon-subtracted squeezed vacuum state (Hu et al., 2011, Jezek et al., 2012, Steuernagel et al., 2024). This identity is exact for possibly rotated pure squeezed vacua and fails for displaced states and generic mixed Gaussians (Steuernagel et al., 2024).

Operationally, SPASV is used as a non-Gaussian resource for odd-cat generation (Chen et al., 2023), parity-based interferometry (Wang et al., 2018), multimode entanglement engineering (Walschaers et al., 2017, Das et al., 2015), boson-sampling-class sampling models (Olson et al., 2014), and, in recent theory, finite-energy optical GKP-state generation (Senjaya, 9 Jun 2026). Across these contexts, three structural features recur: odd photon-number support, polynomial-times-Gaussian Wigner structure, and strong sensitivity to mode selectivity.

A recurring theme in the cited research is that the label “photon-added squeezed vacuum” can obscure a deeper equivalence class. In ideal single-mode theory it is the same state as a photon-subtracted squeezed vacuum and as a squeezed single-photon (Steuernagel et al., 2024, Jezek et al., 2012). In realistic or multimode settings, however, the preparation route still matters because impurity, heralding structure, loss, and mode mismatch determine whether the experimentally generated state retains negativity, cat overlap, or inherent entanglement (Chen et al., 2023, Walschaers et al., 2017, Jezek et al., 2012).

This suggests a current perspective in which SPASV is best understood not merely as one named state, but as a canonical one-photon non-Gaussian excitation of squeezed vacuum whose exact algebraic simplicity in the pure single-mode limit coexists with a rich and mode-dependent phenomenology in realistic continuous-variable platforms.