Coherent Squeezed-Like States

• Coherent squeezed-like states are bosonic constructions that combine displacement-induced coherence with squeezing-driven quadrature deformation, controlling both first and second moments.
• Canonical and extended formulations use operator orderings, Lie-Trotter products, and su(1,1) algebra to generate states that range from pure Gaussian to non-Gaussian conditioned types.
• Experimental protocols, including heralded optical synthesis and conditional measurements, demonstrate practical applications in quantum metrology and hybrid quantum information.

Searching arXiv for recent and foundational papers on coherent squeezed-like states and related formulations. A coherent squeezed-like state is a bosonic-state construction in which displacement-type coherence and squeezing-type quadrature deformation are combined, but the term is not used in a single universal sense across the literature. In the canonical single-mode setting it is the Gaussian state obtained from the vacuum by displacement and squeezing, while in other works it denotes Lie–Trotter-interleaved displacement and squeezing, heralded finite-Fock approximants to squeezed coherent-state superpositions, generalized $su(1,1)$ coherent states, or mixed and conditional variants. Across these formulations, the common structural feature is the simultaneous control of first moments and second moments, often supplemented by non-Gaussian conditioning or parity-sector engineering (Dabrowska et al., 2012, Azuma, 2023).

1. Canonical single-mode construction

In the standard harmonic-oscillator formulation, a squeezed coherent state is defined by

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

with

$$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$

where $\alpha\in\mathbb C$ fixes the phase-space center and $\xi=r e^{i\theta}$ fixes the squeezing amplitude and orientation (Dabrowska et al., 2012). Closely related conventions also write

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

or

$$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$

with the orderings related by

$$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$

(Chen et al., 2019, Trenggana et al., 28 Apr 2025).

The associated quadratures may be written as

$$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$

or, in another normalization,

$$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$

For the squeezed vacuum with $|\,\alpha,\xi\rangle=D(\alpha)S(\xi)|0\rangle,$0, one finds

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

while in the general displaced state the means shift but the variances remain controlled by the squeezing sector (Trenggana et al., 28 Apr 2025). In the more general covariance form,

$|\,\alpha,\xi\rangle=D(\alpha)S(\xi)|0\rangle,$2

with

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

and the Schrödinger–Robertson bound is saturated: $|\,\alpha,\xi\rangle=D(\alpha)S(\xi)|0\rangle,$4 (Sukhanov et al., 2012).

These formulas identify the canonical coherent squeezed state as a pure Gaussian state. The displacement $|\,\alpha,\xi\rangle=D(\alpha)S(\xi)|0\rangle,$5 governs the first moments, whereas the complex squeeze parameter governs the covariance matrix and the orientation of the squeezing ellipse. This decomposition remains the reference point even when later constructions depart from strict Gaussianity.

2. Nonstandard definitions and generalized state manifolds

A distinct “coherent squeezed like” construction was introduced by Azuma through the Lie–Trotter product formula: $|\,\alpha,\xi\rangle=D(\alpha)S(\xi)|0\rangle,$6 This state differs mathematically from the conventional ordered products $|\,\alpha,\xi\rangle=D(\alpha)S(\xi)|0\rangle,$7 and $|\,\alpha,\xi\rangle=D(\alpha)S(\xi)|0\rangle,$8, yet it can be rewritten as

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

with a modified displacement $$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$0 that depends on $$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$1, $$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$2, $$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$3, and $$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$4 (Azuma, 2023). In this formulation the quadrature variances are

$$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$5

and for real $$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$6 they reduce to the minimal-uncertainty form

$$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$7

A further generalization replaces the standard one-parameter squeezing operator by the $$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$8 group element

$$D(\alpha)=\exp\!\big(\alpha a^\dagger-\bar\alpha a\big),\qquad S(\xi)=\exp\!\Big(\tfrac12\bar\xi\,a^2-\tfrac12\xi\,(a^\dagger)^2\Big),$$9

acting not only on $\alpha\in\mathbb C$0 but on the most general normalized state annihilated by $\alpha\in\mathbb C$1,

$\alpha\in\mathbb C$2

The resulting state

$\alpha\in\mathbb C$3

contains two additional control parameters, $\alpha\in\mathbb C$4 and $\alpha\in\mathbb C$5, beyond ordinary squeezing. In this family, both quadrature variances and Mandel’s $\alpha\in\mathbb C$6 parameter can be tuned in regimes inaccessible to the standard $\alpha\in\mathbb C$7 sector (Raffa et al., 2019).

Other noncanonical extensions retain the coherent–squeezed logic while changing the underlying algebra or state space. For the Morse potential, squeezed coherent–like states are constructed from finite-spectrum ladder operators through the generalized eigenvalue equation

$\alpha\in\mathbb C$8

yielding finite-sum expansions over Morse eigenstates rather than the infinite Fock series of the harmonic oscillator (Angelova et al., 2011). In the $\alpha\in\mathbb C$9-deformed oscillator formalism, “Morse-like squeezed coherent states” are introduced either as approximate eigenstates of a linear combination of deformed ladder operators or as deformed photon-subtracted coherent states (Sánchez et al., 2018).

Taken together, these constructions show that the phrase “coherent squeezed-like state” functions as a family resemblance term. The state may remain Gaussian, become non-Gaussian, or even migrate to a different algebraic setting, provided coherence and squeezing-like deformation continue to coexist in the same object.

3. Heralded optical synthesis and conditional preparation

The most explicit optical state-engineering realization in the cited literature is the protocol of Huang et al., which uses a two-mode squeezed vacuum,

$\xi=r e^{i\theta}$0

followed by weak linear mixing and $\xi=r e^{i\theta}$1-photon heralding. Upon detecting exactly $\xi=r e^{i\theta}$2 photons in one arm, the other arm is projected onto the “core” state

$\xi=r e^{i\theta}$3

For $\xi=r e^{i\theta}$4, this becomes a $\xi=r e^{i\theta}$5–$\xi=r e^{i\theta}$6 superposition that approximates the non-Gaussian core of a squeezed even coherent-state superposition

$\xi=r e^{i\theta}$7

Experimentally, with $\xi=r e^{i\theta}$8 corresponding to half-wave plate angle $\xi=r e^{i\theta}$9, the freely propagating state exhibits $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$0 fidelity with a squeezed even coherent-state superposition of size $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$1 and $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$2 squeezing. The type-II OPO operated at $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$3 with escape efficiency $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$4, system detection efficiency $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$5, two-photon heralding rate $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$6, and balanced homodyne tomography over $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$7 quadrature samples per phase. The reconstructed Wigner function showed two negative lobes reaching $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$8 (Huang et al., 2015).

A separate conditional-measurement scheme starts from the beam-splitter input $$|\psi(\alpha,\zeta)\rangle=S(\zeta)D(\alpha)|0\rangle$$9 and postselects on one-photon detection in one output arm. The remaining state can always be written as

$$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$0

so that the displacement can be factored out and all non-Gaussian squeezing is carried by the finite-dimensional “undisplaced qudit” $$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$1. For the rotated quadrature

$$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$2

the minimal variance is

$$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$3

For $$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$4 detection the optimized values reported are

$$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$5

$$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$6

$$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$7

with detector inefficiency and source impurity degrading the squeezing approximately through convex mixing with vacuum noise (Esakkimuthu et al., 2024).

Non-Gaussian conditioning also appears in interferometric metrology through the photon-catalyzed squeezed vacuum state

$$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$8

which, when mixed with a coherent state at the input of a Mach–Zehnder interferometer, improves both parity-detection phase sensitivity and the quantum Fisher information relative to the coherent-plus-squeezed-vacuum benchmark. Multi-photon catalysis ($$|0;z\rangle=S(z)|0\rangle,\qquad |\,\alpha,z\rangle=S(z)D(\alpha)|0\rangle,$$9) yields the strongest improvement, and the performance persists under internal and external loss, with external loss degrading the sensitivity more severely (Zhao et al., 2024).

These protocols emphasize a recurring design principle: Gaussian operations and displacements are used to shape the accessible phase-space geometry, while a small non-Gaussian trigger supplies the decisive nonclassical component.

4. Dynamics under measurement and interaction

Under continuous nondemolition heterodyne observation, a squeezed coherent state remains Gaussian at all times. If the initial posterior state is

$$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$0

then the posterior evolves as

$$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$1

The posterior means of the quadratures depend explicitly on the measurement record because $$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$2 is noise-driven, but the quadrature uncertainties are deterministic functions of time. In the double-heterodyne case,

$$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$3

so the squeezing decays monotonically toward zero while the squeeze axis rotates at frequency $$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$4 (Dabrowska et al., 2012).

In the Jaynes–Cummings setting, the distinction between a pure squeezed coherent state and a mixed squeezed coherent state is dynamical rather than merely terminological. The pure state is

$$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$5

whereas the mixed squeezed coherent state is the convex mixture

$$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$6

The reported comparison shows that mild squeezing strongly localizes the photon-counting distribution for the pure state, while no such localization occurs for the mixed state; instead the mixed-state photon-counting distribution develops oscillations. Increasing the squeezing strengthens ringing revivals and irregularity in the atomic inversion and entanglement dynamics for the pure state, whereas in the mixed state it destroys the familiar collapse–revival structure and makes the entanglement oscillations more regular (Mandal et al., 2021).

Variational coherent-squeezed-state expansions for the quantum Rabi model use

$$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$7

as the basic wave packet. A single coherent-squeezed state already improves the coherent-state variational description near the critical regime, while a two-packet ansatz,

$$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$8

captures the double-peaked “delocalized wake packet” structure and yields ground-state energies and average photon numbers in good agreement with numerical results in strongly correlated regimes (Chen et al., 2019).

In a different control-theoretic direction, a single squeezed state can itself encode a qubit when the dynamics is governed by a quadratic dissipative equation. In the deconvolution picture, there exists a basis of squeezed states that evolves into another such basis, and a non-unitary impurity filter can restore the coherence lost to Gaussian noise at a privileged time $$S(z)D(\alpha)=D(\beta)S(z)e^{i\,\mathrm{Im}(\alpha\beta^*)}, \qquad \beta=\alpha\cosh r-\alpha^*e^{i\phi}\sinh r$$9 (Tameshtit, 2016).

5. Analogues, deformations, and non-optical realizations

The coherent squeezed-like paradigm extends beyond the single optical mode. In spin systems, an initial coherent spin state

$$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$0

evolves under the two-axis counter-twisting Hamiltonian

$$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$1

into highly entangled squeezed-like states. After suitable evolution times and a $$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$2 rotation, the state has high fidelity with the equally weighted superposition state, the twin-Fock state, and the state that maximizes $$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$3, yielding sensitivities that beat the standard quantum limit and approach the Heisenberg limit (Yukawa et al., 2014).

Classical optics also admits a direct analogue. By replacing photon-number states with structured spatial modes such as Hermite–Gaussian or Laguerre–Gaussian modes, one may define

$$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$4

and construct

$$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$5

In this classical structured-light analogue, the quadratures map directly to beam coordinates, with

$$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$6

and positive $$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$7 producing an elliptically deformed intensity profile whose widths scale as $$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$8 and $$X=\frac{a+a^\dagger}{\sqrt2},\qquad P=\frac{a-a^\dagger}{i\sqrt2},$$9 along orthogonal axes (Wang et al., 2021).

A gravitational-wave version has been proposed for a detector in a harmonic trap potential. There one uses

$$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$0

or its equivalent reordered form, interprets $$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$1 as a graviton-mode displacement and $$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$2 as squeezing, and studies detector transitions induced by

$$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$3

In that model, coherent, squeezed, and squeezed coherent graviton states generate transition pathways that are absent for a graviton number state $$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$4, including graviton annihilation and increasing the energy level of the detector by two levels in final state (Trenggana et al., 28 Apr 2025).

These realizations indicate that coherence-plus-squeezing is portable across bosonic, spin, spatial-mode, and field-theoretic settings. What changes is the algebra, the state basis, and the operational meaning of displacement and squeezing, not the basic strategy of combining center shifts with variance deformation.

6. Conceptual distinctions, limitations, and applications

Several cited works make clear that coherent squeezed-like states should not be conflated with every appearance of sub-shot-noise behavior. One analysis of squeezed-light experiments replaces the laser’s coherent-state description by the phase-averaged mixture

$$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$5

and shows that the usual homodyne variance reduction can be reproduced without assigning a pure squeezed state to a single optical mode. In that description, the decisive mechanism is the sharpening of the relative phase between signal and local oscillator, rather than single-mode squeezing as an ontological prerequisite (Calixto et al., 2020).

A second limitation concerns thermodynamic interpretation. Although squeezed coherent and squeezed vacuum states are Gaussian and may be called “thermal-like,” any pure squeezed state satisfies

$$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$6

whereas a true thermal harmonic-oscillator state has

$$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$7

with

$$X=\tfrac12(a+a^\dagger),\qquad Y=\tfrac1{2i}(a-a^\dagger).$$8

On this basis, squeezed coherent states cannot represent an oscillator in equilibrium with a heat bath at any nonzero temperature (Sukhanov et al., 2012).

These caveats do not reduce the operational importance of coherent squeezed-like states. In optical hybrid quantum information, squeezed coherent-state superpositions are identified as key resources for tests of quantum-to-classical transition, coherent-state qubit teleportation, coherent–discrete entanglement distribution, repeater architectures, linear-optical gate implementations, quantum metrology, non-Gaussian gate injection, entanglement distillation, and storage in atomic memories after suitable squeezing adaptation (Huang et al., 2015). In interferometric metrology, coherent input mixed with photon-catalyzed squeezed vacuum improves phase sensitivity and the quantum Fisher information beyond the coherent-plus-squeezed-vacuum baseline, with multi-photon catalysis providing the strongest enhancement (Zhao et al., 2024).

A coherent squeezed-like state is therefore best understood not as a single textbook state but as a technically specific class of coherence-and-squeezing constructions whose exact definition depends on the algebra, ordering, conditioning, and measurement context. The literature shows both the power of this class as a resource and the importance of distinguishing carefully among pure Gaussian states, non-Gaussian conditioned states, mixed states, and alternative descriptions based on relative-phase correlations.