Mixed Squeezed Coherent State

• Mixed squeezed coherent state is a quantum optics concept defined by distinct classical and effective mixtures of coherent and squeezed states.
• The topic is examined using varied methodologies such as photon-counting distribution, interference observables, and geometric phase analysis to clarify its operational significance.
• Experimental implementations demonstrate that loss, phase randomization, and imperfect conditioning yield mixed states with altered dynamical and statistical signatures compared to pure states.

A mixed squeezed coherent state is not a universally standardized object in quantum optics. In the literature, the phrase can denote several distinct constructions: a classical probabilistic mixture of a coherent state and a squeezed vacuum, an incoherent mixture of displaced-squeezed branches, a two-mode classical mixture built from pure squeezed-coherent configurations, or an effective mixed description that arises when only a restricted sector of observables is accessible. Closely related work also shows that some papers often associated with the term do not define such a state literally, but instead analyze squeezed-vacuum experiments driven by phase-randomized lasers, or experimentally reconstructed squeezed coherent-state superpositions whose mixedness is induced by loss and imperfect conditioning (Mandal et al., 2021, Lo et al., 2014, Almas et al., 2024, Müller et al., 2015, Calixto et al., 2020).

1. Terminological scope and main usages

The strongest obstacle to a single encyclopedia-style definition is terminological heterogeneity. Different subfields use “mixed,” “squeezed,” and “coherent” in operationally different ways, and several representative constructions are all legitimate within their local contexts.

Usage	Representative form	Representative source
Classical mixture of coherent and squeezed states	$$\hat{\rho}_{\text{mixed}}=q\ket{\alpha}\bra{\alpha}+(1-q)\ket{\zeta}\bra{\zeta}$$	(Mandal et al., 2021)
Incoherent mixture of displaced-squeezed branches	$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$	(Lo et al., 2014)
Two-mode mixed squeezed-coherent state	$$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$	(Almas et al., 2024)
Effective mixture over excitation manifolds	$$\hat\varrho_{\mathrm{pol}}=\bigoplus_S \hat\varrho^{(S)}$$	(Müller et al., 2015)

A further complication is that some nearby papers are about related but different objects. “Describing squeezed-light experiments without squeezed-light states” explicitly states that it does not introduce or define a state literally called a “mixed squeezed coherent state”; instead it studies single-mode squeezed-vacuum experiments when the laser is represented by a phase-randomized mixed state (Calixto et al., 2020). This suggests that the term should always be read together with its local definition rather than assumed to have a unique canonical meaning.

2. Single-mode mixed-state constructions

The most explicit single-mode definition appears in the Jaynes–Cummings study of the “mixed squeezed coherent state version” of the model. There the usual pure squeezed coherent state is

$$\ket{\alpha,\zeta}\equiv \hat{D}(\alpha)\hat{S}(\zeta)\ket{0},$$

with

$$\hat{D}(\alpha)=\exp(\alpha \hat a^\dagger-\alpha^*\hat a), \qquad \hat{S}(\zeta)=\exp\left(-\frac{1}{2}\zeta \hat a^{\dagger 2}+\frac{1}{2}\zeta^*\hat a^2\right),$$

whereas the mixed squeezed coherent state is defined as the classical mixture

$$\hat{\rho}_{\text{mixed}}=q\ket{\alpha}\bra{\alpha}+(1-q)\ket{\zeta}\bra{\zeta},$$

with $0\le q\le 1$ (Mandal et al., 2021). In that construction, “coherent photons act as the signal” and “squeezed photons act as noise,” but in a mixed-state sense rather than as a single pure Gaussian state.

The same paper emphasizes that the photon-counting distribution of this mixed state is simply

$$P(n)=q|\langle n|\alpha\rangle|^2+(1-q)|\langle n|\zeta\rangle|^2.$$

Because the squeezed-vacuum component contributes only to even photon numbers, the mixed state exhibits an even–odd oscillatory modulation rather than the localization characteristic of the pure squeezed coherent state. The paper further states that, in the studied regime, the Mandel parameter is always positive and the Wigner function remains positive as a sum of two Gaussian distributions at different positions in phase space (Mandal et al., 2021).

A second single-mode mixed construction appears in trapped-ion spin-motion dynamics. There the basic pure branch is the displaced-squeezed state

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

and loss of phase coherence between two opposite branches yields the explicit oscillator mixed state

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$0

This is not a mixture of a coherent state and a squeezed vacuum, but an incoherent mixture of two displaced-squeezed components (Lo et al., 2014). The same work gives the branch overlap

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$1

which makes explicit the anisotropic effect of squeezing orientation on distinguishability.

3. Effective mixedness, two-mode variants, and sector-restricted descriptions

In polarization optics, “mixedness” can be operational rather than statistical. “Parsing Squeezed Light into Polarization Manifolds” studies the pure two-mode state

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$2

but shows that polarization observables only access its block-diagonal part in fixed-photon-number sectors,

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$3

because

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$4

In that operational sense the polarization state behaves like an effective mixture over excitation manifolds, even though the global optical state is pure (Müller et al., 2015). The same paper reports that squeezing appears in all manifolds except vacuum and one-photon manifolds, and that full-Stokes/global covariance methods can miss squeezing visible inside individual manifolds.

A different two-mode meaning is adopted in the geometric-phase study of mixed squeezed-coherent states. There the single-mode building block is

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$5

and the mixedness is classical, introduced by convex combination,

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$6

The paper analyzes three distinct mixed states characterized by different configurations of the squeezed-coherent constituents and studies the geometric phase under unitary cyclic evolution (Almas et al., 2024). The results reported there state that increasing the squeezing parameters of individual modes compresses the geometric-phase contours in linearly, hyperbolically, and elliptically patterned ways, depending on the mixed-state configuration.

A related but conceptually distinct case arises when the laser itself is mixed. In the squeezed-light reinterpretation paper, the laser is represented by the phase-averaged mixed state

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$7

equivalently

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$8

That work explicitly argues that it is about mixed coherent laser states and squeezed-vacuum signatures, not about a formally named “mixed squeezed coherent state” (Calixto et al., 2020).

4. Dynamical consequences

The mixed squeezed coherent state of the Jaynes–Cummings paper is not merely a terminological variant; it produces qualitatively different dynamics from the pure squeezed coherent state. The resonant interaction Hamiltonian is

$$\hat\rho_{\rm mixed}=\frac12\left(\ket{\alpha,\xi}\bra{\alpha,\xi}+\ket{-\alpha,\xi}\bra{-\alpha,\xi}\right)$$9

and the atomic inversion is determined by the initial photon-number distribution through

$$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$0

Because the mixed state has a weighted-sum photon-counting distribution rather than the Hermite-structured distribution of the pure squeezed coherent state, the paper reports sharply different collapse–revival behavior and entanglement dynamics (Mandal et al., 2021). In particular, increasing squeezing in the pure squeezed coherent state enhances ringing revivals, whereas increasing squeezing in the mixed squeezed coherent state “very significantly alters” both the collapse–revival pattern and the negativity dynamics.

The geometric-phase analysis provides a second dynamical characterization. For the mixed two-mode squeezed-coherent states, the geometric phase is computed from

$$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$1

The paper’s main conclusion is that the geometric phase depends jointly on the squeezing parameters and the classical mixing weight $$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$2, with distinct contour geometries for the three mixed-state architectures (Almas et al., 2024).

A third operational consequence appears in the trapped-ion branch-mixture construction. There, number-state populations alone cannot distinguish the equal mixture

$$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$3

from either pure branch individually, because both branches have the same $$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$4. The paper therefore stresses that interference-sensitive observables such as spin revival are needed to certify coherence rather than mere population structure (Lo et al., 2014).

5. Experimental and near-experimental realizations

Several experiments and proposals approach mixed squeezed coherent states indirectly through imperfect state preparation. “Optical synthesis of large-amplitude squeezed coherent-state superpositions with minimal resources” targets pure squeezed even or odd coherent-state superpositions,

$$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$5

but emphasizes that the experimentally reconstructed traveling state is a density matrix affected by losses, finite homodyne efficiency, and phase errors. In the reported two-photon implementation, the generated state has fidelity

$$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$6

with an even CSS of size $$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$7 and squeezing of $$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$8 dB, at a heralding rate of $$\rho(0)=\lambda |\psi_1\rangle\langle\psi_1|+(1-\lambda)|\psi_2\rangle\langle\psi_2|$$9 (Huang et al., 2015). This is therefore an experimentally mixed approximation to a pure squeezed coherent-state superposition, rather than an intentionally prepared mixed squeezed coherent state.

A later free-space experiment similarly targets a pure odd squeezed coherent-state superposition

$$\hat\varrho_{\mathrm{pol}}=\bigoplus_S \hat\varrho^{(S)}$$0

with target parameters $$\hat\varrho_{\mathrm{pol}}=\bigoplus_S \hat\varrho^{(S)}$$1 and $$\hat\varrho_{\mathrm{pol}}=\bigoplus_S \hat\varrho^{(S)}$$2, but reconstructs a mixed state because of optical loss, storage loss, finite homodyne efficiency, imperfect heralded resources, and finite postselection window. The final corrected state has fidelity

$$\hat\varrho_{\mathrm{pol}}=\bigoplus_S \hat\varrho^{(S)}$$3

with the ideal target and displays “three well-resolved negative regions” in its Wigner function (Caron et al., 14 Jan 2026). This again belongs to the experimental neighborhood of mixed squeezed coherent-state resources.

A more explicit route from pure to mixed conditional states appears in optical GKP breeding with approximate squeezed coherent-state superpositions. That work notes that finite-resolution homodyne postselection does not produce a pure conditional ket but a mixed conditional state

$$\hat\varrho_{\mathrm{pol}}=\bigoplus_S \hat\varrho^{(S)}$$4

and states unambiguously that “strictly speaking, a finite resolution post-selected homodyne measurement yields a mixed state” (Pizzimenti et al., 2024). This provides a concrete mechanism by which approximate squeezed coherent-state resources become mixed in realistic conditioning protocols.

6. Conceptual issues and limitations

Two persistent misconceptions recur across the literature. The first is to assume that “mixed squeezed coherent state” has a single standard definition. The cited papers show instead that the phrase may refer to a classical mixture of $$\hat\varrho_{\mathrm{pol}}=\bigoplus_S \hat\varrho^{(S)}$$5 and $$\hat\varrho_{\mathrm{pol}}=\bigoplus_S \hat\varrho^{(S)}$$6, an incoherent mixture of displaced-squeezed branches, a two-mode convex mixture of pure squeezed-coherent configurations, or an effective manifold-resolved polarization description (Mandal et al., 2021, Lo et al., 2014, Almas et al., 2024, Müller et al., 2015).

The second misconception is to identify mixedness with loss of all squeezing signatures. The squeezed-light reinterpretation paper is particularly clear that a phase-randomized mixed laser state does not destroy the observed homodyne noise-reduction formulas. Its claim is instead interpretational: in the Fock-basis description the generated mode alone need not be a squeezed pure state, even though the measured phase-sensitive statistics are unchanged (Calixto et al., 2020). This suggests that operational signatures, reduced states, and ensemble decompositions should be distinguished carefully.

The available literature is also highly model-specific. Some analyses are restricted to single-mode squeezed vacuum and balanced homodyne detection (Calixto et al., 2020); others concern two-mode polarization manifolds (Müller et al., 2015), Jaynes–Cummings dynamics (Mandal et al., 2021), or cyclic geometric phase in specially chosen two-mode mixtures (Almas et al., 2024). A plausible implication is that any rigorous use of the term “mixed squeezed coherent state” should specify, at minimum, whether the mixedness is classical or effective, whether the state is single-mode or multimode, and whether the reference pure objects are coherent states, squeezed vacua, displaced-squeezed states, or squeezed coherent-state superpositions.