Displaced Squeezed Vacuum State

• Displaced squeezed vacuum state is a Gaussian quantum state obtained by sequentially applying displacement and squeezing operators to the vacuum, which shifts its phase-space mean and redistributes noise anisotropically.
• Its phase-space representation features a Gaussian Wigner function that transforms a circular vacuum contour into an ellipse while preserving the phase-space area.
• It underpins quantum communication and metrology by enabling advanced signal encoding and optimal phase estimation through tailored displacement and squeezing parameters.

A displaced squeezed vacuum state is a bosonic quantum state obtained by acting on the vacuum with a squeezing operator and a displacement operator, most commonly in the form

$|\psi\rangle = \hat D(\alpha)\hat S(\xi)|0\rangle,$

with

$$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$

It is a single-mode Gaussian state whose mean field is shifted in phase space while its quadrature noise is anisotropically redistributed. In the literature it appears both as a basic state-preparation primitive in quantum optics and as an effective state family in communication, metrology, microwave photonics, high-harmonic generation, and Gaussian-kernel constructions (Sukumar, 2012, Bai et al., 14 Jan 2026, Theidel et al., 2024).

1. Definition and operator structure

In the standard single-mode construction, the displaced squeezed vacuum state combines the two elementary unitary operations generated by the bosonic ladder operators. The displacement operator translates the state in phase space, while the squeezing operator changes the quadrature fluctuations. Several works use the real-parameter specialization

$$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$

so that the state is written as $|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$ (Bai et al., 14 Jan 2026, Bai et al., 27 Jan 2026).

For binary phase-shift keyed encodings, the same state family is used in the pair

$$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$

which the communication literature denotes as S-BPSK (Bai et al., 14 Jan 2026). In this setting the total mean photon number is

$N=|\alpha|^2+\sinh^2(r),$

so the displacement and the squeezing both contribute to the energy budget (Bai et al., 14 Jan 2026).

The displaced squeezed vacuum is also the $n=0$ member of the broader displaced squeezed Fock-state family

$|z,\xi,n\rangle=\hat D(z)\hat S(\xi)|n\rangle,$

used in quantized lossy left-handed transmission lines; setting $n=0$ yields the displaced squeezed vacuum as a special case (Zhao et al., 2016).

At the operator-theoretic level, the underlying order-1 and order-2 generators associated with displacement and squeezing are distinguished from naive higher-order generalizations. In the framework based on

$$A^{(k)}:= i\left(e^{i\theta}a_+^k-e^{-i\theta}a_-^k\right),$$

the cases $$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$0 and $$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$1 are essentially selfadjoint and generate unitary one-parameter groups, whereas $$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$2 is not essentially selfadjoint in the same sense (Gorska et al., 2014). This places the displaced squeezed vacuum within the mathematically well-defined low-order sector.

2. Phase-space representation and statistical structure

For real displacement $$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$3 and real squeezing $$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$4, the Wigner function of a displaced squeezed vacuum state is Gaussian: $$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$5 In this representation, displacement shifts the center along the $$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$6 axis and squeezing deforms the circular vacuum contour into an ellipse while preserving the phase-space area, with $$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$7 (Bai et al., 14 Jan 2026).

The squeezed-vacuum component determines the nonclassical noise structure. For a squeezed vacuum state

$$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$8

the mean photon number in the original Fock basis is

$$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a),\qquad \hat S(\xi)=\exp\!\left[\frac{1}{2}\left(\xi^*\hat a^2-\xi \hat a^{\dagger 2}\right)\right].$$9

and only even Fock states occur in its Fock decomposition (Gietka, 2023). This even-parity structure becomes operationally relevant in receiver architectures and in mismatch analyses, where residual squeezed-vacuum statistics can generate parity-dependent photon-counting effects (Bai et al., 27 Jan 2026).

The relation between operator orderings is nontrivial. One explicit identity is

$$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$0

for $$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$1 and $$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$2 (Tao et al., 2021). This equivalence shows that a displaced squeezed state can be re-expressed with the operators reversed, but with a transformed displacement parameter. The state family is therefore closed under the interchange of the two operations, though the physically transparent parameters differ.

In multimode settings the same structure is applied modewise. One formulation used for high-harmonic generation is

$$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$3

with displacement and squeezing distributed over spectral modes (Theidel et al., 2024). A related product-state encoding for kernel methods is

$$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$4

with single-mode factors $$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$5 (Mehta et al., 2024).

3. Ordering, extreme limits, and symplectic evolution

The order in which squeezing and displacement are applied becomes physically consequential in extreme limits. Two definitions analyzed explicitly are the Yuen form

$$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$6

and the Caves form

$$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$7

When one attempts to recover position eigenstates by taking $$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$8, the Yuen construction loses the $$\hat S(r)=\exp\!\left[\frac{1}{2}r(\hat a^2-\hat a^{\dagger 2})\right],\qquad r\in\mathbb{R}_+,$$9-dependence, whereas the Caves construction yields the correct position eigenstate (Moya-Cessa et al., 2013).

In that limit, the position eigenstate can be written as

$|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$0

and also as

$|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$1

(Moya-Cessa et al., 2013). This makes the displaced squeezed vacuum the finite-$|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$2 precursor of sharply localized Gaussian wavepackets, while ideal position eigenstates remain non-normalizable mathematical limits.

A closely related limit appears in the quantum Rabi model. In the relativistic regime, position eigenstates are identified as limits of infinitely squeezed coherent states,

$|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$3

and the bosonic sector becomes a superposition of two infinitely squeezed, oppositely displaced vacua in phase space (Maldonado-Villamizar et al., 2019). This connects displaced squeezed vacua to spectral-collapse physics and continuous-spectrum regimes.

The time evolution of squeezed and displaced states under harmonic-oscillator dynamics is naturally described by symplectic transformations. For a squeezed parameter $|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$4, the harmonic-oscillator evolution gives

$|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$5

and for a general symplectic map

$|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$6

the squeezing parameter transforms as

$|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$7

This formalizes the connection between squeezing and linear canonical transformations (Sukumar, 2012).

4. Generation routes and experimental realizations

Several mechanisms generate highly displaced squeezed states, but the available brightness, squeezing, and uncertainty are not independent. A beam-splitter method mixes a squeezed vacuum with a strong coherent state, while parametric routes include optical parametric oscillators, optical parametric amplifiers, and dissipative optomechanical squeezers seeded with coherent states. The comparative analysis of these mechanisms finds significant tradeoffs between brightness, squeezing, and overall uncertainty, with the tradeoff attributed, for parametric amplifiers, to unavoidable pump-noise admixture once a nonzero seed amplitude is present (Young et al., 2023).

In the beam-splitter scheme, the displaced output quadrature variances are

$|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$8

and the uncertainty product becomes

$|\alpha,r\rangle=\hat D(\alpha)\hat S(r)|0\rangle$9

so increasing the output brightness increases the overall uncertainty product (Young et al., 2023). This does not imply that displacement always degrades squeezing in all platforms; it is specific to the state-generation mechanism under study.

A contrasting experimental result was obtained for propagating microwave states. There, a flux-driven Josephson parametric amplifier produced squeezed vacuum, and a cryogenic directional coupler superimposed a strong coherent microwave tone to implement the displacement operator. Even for strong displacement amplitudes, there was no degradation of the squeezing level in the reconstructed quantum states, and the path entanglement produced by splitting displaced squeezed states remained constant over a wide range of displacement power (Fedorov et al., 2016). The covariance matrix of the Gaussian path-entangled state in that experiment was explicitly independent of the displacement parameter $$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$0 (Fedorov et al., 2016).

Multimode displaced squeezed states have also been reported in semiconductor high-harmonic generation. By simultaneously measuring second- and third-order intensity correlation functions on a tripartite harmonic set and performing an effective Schmidt decomposition, the state was modeled as a displaced squeezed state with an almost single-mode structure for each harmonic. The estimated effective Schmidt numbers were $$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$1 and $$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$2, and the maximum squeezing values were 8.5 dB for H4 and 7.5 dB for H5 (Theidel et al., 2024). The same work reported a significant violation of a Cauchy-Schwarz-type inequality for three biseparable partitions by multiple standard deviations (Theidel et al., 2024).

5. Communication and metrology

In quantum-state discrimination, displaced squeezed vacuum states are used as the signal alphabet itself. For S-BPSK, the codewords are $$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$3 and $$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$4, and an important receiver strategy is to apply operations that convert the problem into one of distinguishing vacuum from a large-amplitude coherent state (Bai et al., 14 Jan 2026).

One such design is the displacement-squeeze receiver. It applies a displacement followed by a squeezing operation with the squeezing axis rotated by $$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$5, giving the effective mapping

$$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$6

for real $$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$7 (Bai et al., 14 Jan 2026). With the optimal squeezing fraction

$$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$8

the ideal error probability becomes

$$|\psi_0\rangle=|-\alpha,r\rangle,\qquad |\psi_1\rangle=|+\alpha,r\rangle,$$9

The same analysis gives

$N=|\alpha|^2+\sinh^2(r),$0

for all signal energies under equal priors and ideal conditions; in the low-energy regime the receiver beats the S-BPSK SQL at $N=|\alpha|^2+\sinh^2(r),$1, goes below the coherent-state BPSK Helstrom bound at $N=|\alpha|^2+\sinh^2(r),$2, and reaches $N=|\alpha|^2+\sinh^2(r),$3 near $N=|\alpha|^2+\sinh^2(r),$4 (Bai et al., 14 Jan 2026).

A related architecture is the inverse-squeezing Kennedy receiver. After the usual Kennedy displacement, it applies $N=|\alpha|^2+\sinh^2(r),$5, so that

$N=|\alpha|^2+\sinh^2(r),$6

again converting the signal pair into vacuum versus a coherent state with amplified effective displacement (Bai et al., 27 Jan 2026). Under ideal conditions its error probability approaches the Helstrom bound across the entire energy spectrum, remaining within a constant factor of 3 dB, and in the low-photon-number regime $N=|\alpha|^2+\sinh^2(r),$7 it also achieves an error rate below $N=|\alpha|^2+\sinh^2(r),$8 (Bai et al., 27 Jan 2026). Under squeezing-parameter mismatch, the analysis predicts a characteristic “parity photon-number step” saturation effect caused by the even-photon structure of residual squeezed-vacuum components (Bai et al., 27 Jan 2026).

In quantum metrology, the relative phase between displacement and squeezing is decisive. For

$N=|\alpha|^2+\sinh^2(r),$9

the phase-sensitive parameter

$n=0$0

controls the quantum Fisher information for phase-difference estimation, and the accuracy limit is periodic in $n=0$1 with period $n=0$2 (Tao et al., 2021). The explicit QFI is

$n=0$3

When $n=0$4 lies in the favorable interval stated in the analysis, large displacement and large squeezing can bring the estimation accuracy close to the ultimate quantum limit; in the unfavorable interval, optimal performance is recovered only when the state degenerates to squeezed vacuum (Tao et al., 2021).

For full-period optical phase estimation, displaced squeezed probes have also been assigned a specialized role in Stage I of a two-stage protocol. There, heterodyne measurements on displaced squeezed states perform coarse localization of the phase over the full circle, while Stage II uses squeezed-vacuum probes and adaptive homodyne measurements for local high-precision estimation. For $n=0$5 photons and squeezing limited to 12 dB, using displaced squeezed states in Stage I reduces the optimized two-stage bound relative to coherent-state Stage I and keeps the protocol within a factor of 3 to 30 of the idealized local squeezed-vacuum quantum Cramér-Rao bound (Rodríguez-García et al., 3 Jul 2026).

6. Multimode extensions, field-theoretic contexts, and broader uses

The multimode displaced squeezed vacuum state is a natural generalization in which displacement and squeezing are distributed over orthogonal spectral or spatial modes. In quantum kernel methods this structure is used to realize a Gaussian kernel with tunable hyperparameter. For single-mode states $n=0$6, the overlap for real data reduces to

$n=0$7

with

$n=0$8

For $n=0$9-mode product states this yields

$|z,\xi,n\rangle=\hat D(z)\hat S(\xi)|n\rangle,$0

so the squeezing parameters control the kernel bandwidth (Mehta et al., 2024).

In dissipative quantum-field settings, explicit calculations have focused on squeezed vacuum initial states rather than displaced squeezed ones. For continuum single-mode squeezed states characterized by $|z,\xi,n\rangle=\hat D(z)\hat S(\xi)|n\rangle,$1, the field correlators obey

$|z,\xi,n\rangle=\hat D(z)\hat S(\xi)|n\rangle,$2

and the long-time energy-momentum tensor and Casimir force acquire a multiplicative statistical factor $|z,\xi,n\rangle=\hat D(z)\hat S(\xi)|n\rangle,$3 (Lopez, 2016). The same formalism notes that displaced squeezed states are not treated explicitly but could be incorporated by supplying the corresponding nonzero-mean correlators. This suggests that displacement would enter such problems through additional mean-field contributions rather than through a change of the Green-function machinery itself (Lopez, 2016).

Field-theoretic production mechanisms also motivate displaced squeezed states. In linearized quantum gravity, time-dependent quadratic couplings generate squeezed states from the vacuum, while the simultaneous presence of linear terms produces displaced squeezed vacuum states (Das et al., 23 Dec 2025). This establishes a structural distinction: linear couplings generate coherent displacement, quadratic couplings generate squeezing, and their coexistence produces the combined state (Das et al., 23 Dec 2025).

In more specialized settings, displaced squeezed states appear as governing states of matter-coupled photonic sectors and metamaterial observables. In the quantum Rabi model they interpolate, through regime changes, between displaced-number-state superpositions and infinitely squeezed coherent-state limits (Maldonado-Villamizar et al., 2019). In quantum lossy left-handed transmission lines, the displaced squeezed vacuum is the $|z,\xi,n\rangle=\hat D(z)\hat S(\xi)|n\rangle,$4 limit of the displaced squeezed Fock state used to analyze quantum effects in the negative refraction index under dissipation (Zhao et al., 2016).

Taken together, these results position the displaced squeezed vacuum state as a unifying Gaussian-state family whose significance lies not only in its formal definition $|z,\xi,n\rangle=\hat D(z)\hat S(\xi)|n\rangle,$5, but also in the way displacement, squeezing strength, squeezing phase, and mode structure jointly determine distinguishability, Fisher information, bandwidth tunability, and dynamical behavior across a broad range of quantum-optical and field-theoretic problems.