Squeezed Coherent States in Quantum Optics

• Squeezed coherent states are quantum Gaussian states that combine displacement and squeezing, leading to anisotropic quadrature noise and altered phase-space geometry.
• They can be generated via nonlinear photonic crystals or conditional linear-optical methods, offering practical routes for entanglement and enhanced metrological applications.
• The noncommutativity of displacement and squeezing operations necessitates specific ordering, affecting state parameterization, photon statistics, and measurement outcomes.

A squeezed coherent state is a single-mode bosonic state obtained by combining displacement and squeezing on the vacuum. In standard notation, the relevant unitaries are the displacement operator

$$\hat D(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a)$$

and the single-mode squeezing operator

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$

The two standard orderings,

$$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$

define Gaussian states with displaced first moments and anisotropic quadrature noise; they are distinct because $\hat D$ and $\hat S$ do not commute, but they are related by a parameter transformation rather than by a change of state family (Azuma, 2023, Munguia-Gonzalez et al., 2021).

1. Operator structure and canonical definitions

For a single bosonic mode with annihilation and creation operators $\hat a,\hat a^\dagger$, coherent states are generated from the vacuum by $\hat D(\alpha)$, while squeezed vacuum is generated by $\hat S(\zeta)$. A squeezed coherent state combines both operations and therefore interpolates between the classical-like phase-space displacement of a coherent state and the quadrature anisotropy of a squeezed state (Azuma, 2023).

Because ordering matters, the relation between the two conventional forms is central: $$\hat S(\zeta)\hat D(\alpha)\hat S^\dagger(\zeta) =\hat D(\alpha\cosh r-\alpha^*e^{i\varphi}\sinh r).$$ Hence

$$|\zeta,\alpha\rangle = ||\zeta,\ \alpha\cosh r-\alpha^*e^{i\varphi}\sinh r\rangle\!\rangle.$$

The noncommutativity is therefore kinematic rather than taxonomic: the two orderings have different mean fields and different phase-space parametrizations, but belong to the same Gaussian family (Azuma, 2023).

In quantum-optical quadrature notation,

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$0

a coherent state satisfies $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$1. A single-mode state is squeezed when one quadrature variance falls below $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$2 while the uncertainty product remains bounded by the canonical lower limit. A common convention for quoting squeezing is

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$3

relative to the vacuum variance $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$4 (Azuma, 2023).

2. Wavefunction, covariance, and phase-space geometry

Squeezed coherent states are Gaussian in both position and momentum representations, with a modified, generally complex width parameter. Their position- and momentum-space wavefunctions remain Gaussian under displacement and squeezing, which makes them the most general Gaussian wavepackets of the harmonic oscillator obtained from vacuum by those two unitaries (Munguia-Gonzalez et al., 2021).

For suitable squeeze phase alignment, notably $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$5 or $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$6, the variances reduce to the familiar minimum-uncertainty form

$$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$7

so that $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$8. More generally, the Gaussian width becomes complex and the uncertainty product need not be minimal at every instant, although the state remains Gaussian (Munguia-Gonzalez et al., 2021).

A basic structural fact is that displacement shifts only the first moments. The covariance ellipse, squeezing level, and uncertainty product depend on $$\hat S(\zeta)=\exp\!\left[\frac{1}{2}(\zeta^*\hat a^2-\zeta \hat a^{\dagger 2})\right],\qquad \zeta=r e^{i\varphi}.$$9, not on $$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$0. This is explicit in quadrature-variance calculations for both conventional squeezed coherent states and the Lie–Trotter “coherent squeezed like state”: displacement alters the centroid of the Wigner or Husimi distribution, but not the noise matrix (Azuma, 2023).

Geometrically, the state is represented in phase space by an ellipse rather than the circular Gaussian associated with vacuum or a coherent state. The squeeze parameter $$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$1 fixes the eccentricity, and the squeeze phase $$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$2 fixes the orientation. This phase-space picture underlies later uses in metrology, continuous measurement, coding, and variational many-body methods (Munguia-Gonzalez et al., 2021).

3. Equivalent formulations and nonstandard variants

One nonstandard construction appears in the Lie–Trotter formulation introduced as a “coherent squeezed like state”: $$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$3 Its closed form is

$$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$4

but the state is exactly equivalent, up to a global phase and a renormalized displacement parameter, to an ordinary squeezed coherent state: $$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$5 This result rules out interpreting the Lie–Trotter state as a genuinely new nonclassical species; it is a reordered parametrization of the standard family (Azuma, 2023).

Another extension is the $$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$6-photon squeezed coherent state,

$$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$7

The set $$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$8 forms an orthonormal basis of the single-mode Hilbert space, because it is obtained from the Fock basis by a unitary map. These states are useful when one wants excitations above a squeezed coherent background rather than above vacuum, and even a few added photons can strongly modify collapse-and-revival behavior in Jaynes–Cummings dynamics (Alexanian, 2022).

A distinct object is the mixed squeezed coherent state,

$$|\zeta,\alpha\rangle=\hat S(\zeta)\hat D(\alpha)|0\rangle,\qquad ||\zeta,\alpha\rangle\!\rangle=\hat D(\alpha)\hat S(\zeta)|0\rangle,$$9

which is a non-Gaussian convex mixture of a coherent state density operator and a squeezed-state density operator. It is not equivalent to the pure squeezed coherent state $\hat D$0, and the difference is operationally significant: photon-counting distributions, atomic inversion, and entanglement dynamics differ qualitatively between the pure and mixed constructions (Uzun, 2021).

4. Generation, conditioning, and monitored evolution

A concrete generation proposal employs a nonlinear photonic crystal made of LiNbO$\hat D$1 and air, placed inside a bow-tie ring resonator. The $\hat D$2 nonlinearity and reduced group velocity enhance the effective interaction, while repeated cavity circulation yields a finite Lie–Trotter alternation of displacement and squeezing. With realistic parameters, the scheme gives a real squeezing parameter $\hat D$3, corresponding to a practically achievable squeezing level of $\hat D$4, and the emitted field is a weak squeezed coherent beam with mean photon number of order $\hat D$5–$\hat D$6 (Azuma, 2023).

The same work analyzes entanglement generation from such states by sending the squeezed output into a balanced beam splitter together with vacuum. The output contains a two-mode squeezing component, and the Duan–Simon criterion is violated for any $\hat D$7, so the beams are entangled. In that sense, single-mode squeezed coherent light acts as a standard continuous-variable entanglement resource even when the source construction is nonstandard (Azuma, 2023).

Under continuous heterodyne observation, an initially squeezed coherent state remains within the squeezed coherent manifold in the a posteriori dynamics. The conditional state can be written as

$\hat D$8

with $\hat D$9 stochastic and the squeeze parameter deterministic. The mean quadratures depend on the measured noise, whereas the quadrature uncertainties evolve deterministically; the long-time limit approaches the vacuum variances as the squeezing decays under monitoring (Dabrowska et al., 2012).

In optomechanical weak-measurement schemes, a mechanical pointer initially prepared in a squeezed coherent state can exploit enhanced fluctuations and operator noncommutativity to produce amplification beyond the usual Gaussian-pointer limit. The amplified displacement can reach the anti-squeezed fluctuation scale $\hat S$0 or even $\hat S$1, a regime inaccessible to an unsqueezed coherent pointer in the same weak-coupling setting (Li et al., 2015).

A different preparation route uses only linear optics, a coherent input, a number-state input, and conditional photon detection. The heralded output can be written as a displaced qudit,

$\hat S$2

Because displacement plays no role in squeezing, the optimization reduces to a finite-dimensional problem in the coefficients $\hat S$3. For lower number-state inputs, the conditional scheme reaches the maximal squeezing allowed by the truncated Fock space (Esakkimuthu et al., 2024).

5. Roles in many-body, continuous-variable, and communication settings

In the quantum Rabi model, coherent-squeezed states serve as variational packets that capture both displacement and width deformation. A one-packet coherent-squeezed-state ansatz improves substantially on a pure coherent-state ansatz near the critical regime, while a two-packet superposition captures the two delocalized Gaussian wave packets that characterize the strongly correlated phase. Ground-state energy and average photon number agree well with exact numerics even in strong-correlation regimes (Chen et al., 2019).

In continuous-variable coding, squeezed comb states are finite superpositions of equidistant squeezed coherent states arranged along a line in phase space. They provide realistic approximations to ideal Gottesman–Kitaev–Preskill codewords. Their robustness depends strongly on the encoding parameters, and a notable conclusion is that finite squeezed combs are more suitable and less error-prone under amplitude damping than under Gaussian diffusion (Shukla et al., 2020).

In private quantum communication, squeezed coherent states enhance Gaussian private quantum channels. A GPQC with squeezed coherent states generalizes the displacement-only coherent-state construction and yields a smaller upper bound on accessible information than the coherent-only version. The same work also argues that squeezed states have an advantage over coherent states against a beam-splitting attack in continuous-variable QKD (Jeong et al., 2014).

The concept also admits broader analogues and extensions. Spin-squeezed coherent states generated by two-axis counter-twisting play the role of SU(2) analogues of bosonic squeezed coherent states in metrology; $\hat S$4-photon squeezed coherent states provide a natural basis adapted to squeezed-displaced modes; and hydrodynamic treatments reinterpret generic squeezed coherent states in terms of phase-space Gaussian kinetic theory and Bohmian conditional momentum fields (Yukawa et al., 2014, Alexanian, 2022, Uzun, 2021).

6. Conceptual clarifications, misconceptions, and analogies

A frequent misconception is that any reordered or device-specific construction defines a new class of state. The Lie–Trotter “coherent squeezed like state” is the clearest counterexample: although its formal definition differs from $\hat S$5 and $\hat S$6, it is exactly an ordinary squeezed coherent state up to a global phase and a transformed displacement parameter (Azuma, 2023).

A second misconception is that displacement changes the squeezing strength. It does not. Across standard Gaussian treatments, continuous-measurement analyses, and conditional linear-optical constructions, the displacement controls only the first moments, while the squeezing level is determined by the covariance structure. This is why displaced qudits can be optimized for squeezing by ignoring the displacement part altogether (Munguia-Gonzalez et al., 2021, Esakkimuthu et al., 2024).

A third point concerns purity. The pure squeezed coherent state is a Gaussian state produced by coherent application of displacement and squeezing. A convex mixture of coherent and squeezed states is not the same object and is generically non-Gaussian. The distinction matters operationally, because photon statistics and Jaynes–Cummings dynamics differ sharply between the pure and mixed cases (Uzun, 2021).

Finally, there are classical analogies. Structured-light constructions can reproduce the algebraic form of $\hat S$7 in a “novel Hilbert space” built from spatial modes, and can map phase-space squeezing to real-space beam shaping. But these analogies do not reproduce nonclassical photon statistics, genuine quantum noise reduction, entanglement, or measurement back-action; they reproduce geometric and algebraic aspects rather than the full quantum content (Wang et al., 2021).