Squeezing-Induced Symmetry Breaking

• Squeezing-induced symmetry breaking is the phenomenon where squeezing interactions or mechanical compression destabilize symmetric states, leading to lower-symmetry responses.
• It spans various domains, including non-Hermitian bosonic dynamics, cavity and spin systems, and elastic media, each with distinctive symmetry implications.
• These mechanisms enhance sensing, alter dynamical selection rules, and enable nonreciprocal transport by actively reshaping fluctuation sectors and effective dynamics.

Squeezing-induced symmetry breaking denotes a family of mechanisms in which squeezing interactions, squeezed fluctuations, or literal mechanical squeezing destabilize a symmetric description of a system and select, expose, or amplify a lower-symmetry response. In the current arXiv literature, the expression does not refer to a single canonical formalism. Instead, it covers Hermitian bosonic models whose Heisenberg dynamics acquire a non-Hermitian pseudo-anti-$\mathcal{PT}$ structure, cavity and spin protocols in which symmetry breaking produces squeezed dark or equilibrium states, quantum-light-driven violations of classical high-harmonic selection rules, reservoir-engineered transport in which asymmetric squeezing breaks exchange symmetry, and elastic media in which radial or axial compression induces non-radial responses (Luo et al., 2021, Zheng, 2012, Comparin et al., 2022, Stammer et al., 27 May 2026, Liu et al., 1 Jul 2026, Kumar et al., 2023, Conti et al., 2015). This suggests a unifying theme: squeezing reshapes either the fluctuation sector, the effective dynamical matrix, or the energetic competition strongly enough that the original symmetry no longer controls the observable physics.

1. Scope and taxonomy of the phenomenon

The phrase spans several distinct symmetry concepts. In non-Hermitian bosonic dynamics, the relevant structure is pseudo-$\mathcal{APT}$ symmetry and its exceptional point. In cavity QED and nonlinear optics, the central issue is whether a detuning or a continuous polarization degree of freedom lifts or preserves dark-state degeneracy. In spin models, the relevant symmetries are $U(1)$, $SU(2)$, or a weak $\mathbb Z_2$ Liouvillian symmetry. In high-harmonic generation, the operative object is a dynamical symmetry combining rotation and time translation. In transport problems, squeezing breaks exchange symmetry between channels. In elasticity, “squeezing” is literal compression, and the broken symmetry is radial or axial rather than quadrature-based.

Domain	Squeezed or compressed object	Symmetry consequence
Two-mode bosonic dynamics	Pair-creation coupling and two-mode squeezing factor $S$	pseudo-$\mathcal{APT}$ breaking at an exceptional point
Cavity and spin systems	Dark-state, polarization, or spin squeezing	Unique dark states, spontaneous polarization symmetry breaking, equilibrium SSB, or DSSB
Quantum-light HHG	Squeezed-field fluctuations	Breakdown of classical dynamical selection rules
Reservoir-engineered transport	Unequal cavity squeezing	Exchange-symmetry breaking and enhanced nonreciprocity
Elastic media	Radial or axial compression	Non-radial elastic response or non-radial minimizers

A recurring misconception is that the term always means that squeezing itself is the sole cause of symmetry breaking. Several papers use squeezing as the diagnostic of a broken-symmetry phase rather than its unique microscopic origin. Others use the phrase for compression-induced instability in purely elastic media, where the operative mechanism is mode coupling or variational energy reduction rather than bosonic quadrature squeezing (Debecker et al., 15 Apr 2025, Kumar et al., 2023).

2. Pseudo-$\mathcal{APT}$ symmetry, exceptional points, and squeezing dynamics

A particularly explicit realization appears in a two-mode bosonic model with Hermitian second-quantized Hamiltonian

$$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$

whose Heisenberg equations generate the non-Hermitian dynamical matrix

$$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$

This matrix obeys the anti-commutation relation $\mathcal{APT}$0, with $\mathcal{APT}$1 and $\mathcal{APT}$2, while the underlying Hamiltonian remains Hermitian and therefore preserves bosonic commutation relations without Langevin noise. The authors therefore describe the construction as quantum pseudo-$\mathcal{APT}$3 symmetry rather than ordinary $\mathcal{APT}$4 symmetry (Luo et al., 2021).

The symmetry-breaking criterion follows from the eigenvalues

$\mathcal{APT}$5

Here the pseudo-$\mathcal{APT}$6-symmetric region has purely imaginary eigenvalues, whereas the broken region has purely real eigenvalues, the opposite assignment from conventional $\mathcal{APT}$7 language. The exceptional point is at $\mathcal{APT}$8. Exact operator evolution takes the Bogoliubov form

$\mathcal{APT}$9

so the two-mode squeezing factor

$U(1)$0

directly tracks the symmetry sector. In the broken region $U(1)$1, $U(1)$2 and $U(1)$3 are trigonometric and the dynamics are periodic; $U(1)$4 oscillates with period $U(1)$5, returns to $U(1)$6 at $U(1)$7, and reaches

$U(1)$8

at half-periods. In the symmetric region $U(1)$9, $SU(2)$0 and $SU(2)$1 are hyperbolic and $SU(2)$2 grows exponentially. At the exceptional point, the paper states that the squeezing factor crosses over to linear long-time growth, $SU(2)$3 (Luo et al., 2021).

This system is also used for sensing. Near the exceptional point, small parameter changes cause large changes in $SU(2)$4, $SU(2)$5, and $SU(2)$6. For coherent input and quadrature readout by homodyne detection, the observable susceptibility scales as $SU(2)$7, while the quantum Fisher information behaves as $SU(2)$8 at the working points $SU(2)$9, where $\mathbb Z_2$0 again. An important clarification is that the stable-side sensing enhancement does not rely on large output squeezing at the operating point; the enhancement comes from proximity to the pseudo-$\mathbb Z_2$1 transition itself (Luo et al., 2021).

3. Symmetry-selected squeezed states in cavities and spin models

In cavity-QED adiabatic-passage protocols, controlled symmetry breaking is used to lift dark-state degeneracy and thereby select a unique squeezed target state. After adiabatic elimination of far-detuned excited states and a Holstein-Primakoff reduction, the atom-cavity system becomes a bilinear interaction between a cavity mode $\mathbb Z_2$2 and a bosonic collective atomic mode $\mathbb Z_2$3. If one chooses $\mathbb Z_2$4 and $\mathbb Z_2$5, a squeezing transformation of the atomic mode maps the problem to one whose dark state is the two-mode vacuum in the transformed frame, so the original dark state is $\mathbb Z_2$6. If instead $\mathbb Z_2$7 and $\mathbb Z_2$8, the unique dark state is $\mathbb Z_2$9. The paper emphasizes that when both detunings vanish, the dark state is degenerate; the nonzero detuning breaks the symmetry and makes the squeezed state unique. The squeezing parameter is $S$0, with phase $S$1, and the relevant gap grows as $S$2, so the time needed for state preparation decreases with system size (Zheng, 2012).

A distinct mechanism appears in polarization-isotropic nonlinear cavities. There the system has a continuous polarization symmetry, and above threshold the output is always linearly polarized but with arbitrary orientation. The selected polarization angle is the free polarization parameter, a Goldstone variable. Quantum diffusion of that angle leaves its canonically conjugate dark-mode quadrature maximally damped, producing complete noise suppression in one quadrature irrespective of parameter values. The central point is that the squeezing is noncritical: it does not require fine tuning to threshold because it follows from spontaneous polarization symmetry breaking and the bright/dark mode decomposition itself (Garcia-Ferrer et al., 2010).

In equilibrium spin systems with continuous symmetry, spontaneous symmetry breaking can itself generate metrologically useful squeezing. For XXZ models in $S$3, a small field $S$4 mixes the Anderson tower of states, preserves a finite transverse order parameter $S$5, and suppresses fluctuations of the symmetry generator $S$6. The relevant Wineland parameter is

$S$7

and in the symmetry-broken phase it scales as $S$8 down to $S$9, implying $\mathcal{APT}$0 at the smallest significant field for a finite system and a phase uncertainty $\mathcal{APT}$1. The adiabatic preparation time scales linearly, $\mathcal{APT}$2. The paper also stresses that these low-field states are nearly minimal-uncertainty states and that Ramsey interferometry about the $\mathcal{APT}$3-axis is optimal (Comparin et al., 2022).

A related open-system extension appears in the non-Markovian Lipkin-Meshkov-Glick model, where a single weak $\mathcal{APT}$4 symmetry can break in two distinct directions, yielding directional spontaneous symmetry breaking. Phase (I) has $\mathcal{APT}$5, phase (III) has $\mathcal{APT}$6, and the squeezing axis is orthogonal to the broken-symmetry direction. The paper explicitly cautions that this is not “squeezing alone causes symmetry breaking” in a generic sense; rather, non-Markovian dissipation reshapes the phase diagram, and orthogonal spin squeezing is the clearest observable signature of which broken-symmetry direction is selected (Debecker et al., 15 Apr 2025).

4. Quantum-light fluctuation-induced breaking of dynamical selection rules

In high-harmonic generation driven by quantum light, the central symmetry object is not the crystal point group alone but the dynamical symmetry of the full driven Hamiltonian. Classically, circular driving in a $\mathcal{APT}$7 crystal obeys the combined rotation-plus-time-translation symmetry

$\mathcal{APT}$8

which enforces the standard rule $\mathcal{APT}$9. For graphene under circularly polarized classical driving, only $\mathcal{APT}$0 harmonics are allowed; for MoS$\mathcal{APT}$1, the allowed orders are $\mathcal{APT}$2. Under bicircular classical driving, the familiar rule is $\mathcal{APT}$3, while $\mathcal{APT}$4 is forbidden (Stammer et al., 27 May 2026, Stammer et al., 25 Mar 2026).

Squeezed quantum light breaks these selection rules by a more subtle mechanism than a change in the mean field. The mean field can remain symmetry-respecting,

$\mathcal{APT}$5

while the fluctuation sector violates the classical invariance. In the solid-state HHG analysis, the full Hamiltonian obeys $\mathcal{APT}$6 because the variance contains an explicit $\mathcal{APT}$7 term that is not invariant under the same transformation. The paper identifies anomalous correlations $\mathcal{APT}$8 as a zero-helicity channel that breaks the dynamical symmetry while preserving the crystal symmetry. As a consequence, classically forbidden $\mathcal{APT}$9 harmonics, notably $$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$0 and $$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$1, appear in graphene and MoS$$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$2 under circular quantum light, and under bicircular quantum light the $$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$3 channels become allowed (Stammer et al., 27 May 2026).

The bicircular quantum-light treatment makes the same point in operator language. The mean field still respects the threefold bicircular symmetry, but the field variance

$$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$4

does not. The paper interprets the resulting harmonic state as a Husimi average over symmetry-violating field realizations generated by the squeezed driver. This produces otherwise forbidden $$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$5 harmonics and also modifies the helicity structure of classically allowed orders (Stammer et al., 25 Mar 2026).

A key diagnostic is photon statistics. The newly generated forbidden harmonics exhibit squeezing-like super-Poissonian correlations with $$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$6 in the strong-squeezing regime, whereas the corresponding thermal-fluctuation-induced channels give $$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$7. In the solid-state HHG study, the denser spectrum produced by quantum-allowed harmonics also shortens the reconstructed central pulse full width at half maximum from $$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$8 fs to $$\mathcal{H}=\delta(\hat a_1^\dagger \hat a_1+\hat a_2^\dagger \hat a_2)+i\kappa(\hat a_1^\dagger \hat a_2^\dagger-\hat a_1\hat a_2),$$9 fs in the MoS$$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$0 example. A common misconception is therefore corrected: in these works, symmetry breaking is not inferred from a distorted classical driver but from higher-order quantum correlations of a driver whose mean field can still look classically symmetric (Stammer et al., 25 Mar 2026, Stammer et al., 27 May 2026).

5. Transport, chirality, and nonreciprocity under squeezing asymmetry

In reservoir-engineered nonreciprocity, squeezing enhances directional transport only when it breaks exchange symmetry between the two cavity channels. The basic unsqueezed setup contains two degenerate modes $$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$1 and $$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$2, a coherent coupling

$$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$3

and a collective dissipator $$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$4. Under the directional condition $$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$5, $$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$6, and $$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$7, the system is nonreciprocal. After local cavity squeezing, the effective directional coupling becomes

$$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$8

with enhancement factor

$$H_{APT}=\delta \sigma_z+i\kappa \sigma_x = \begin{pmatrix} \delta & i\kappa\ i\kappa & -\delta \end{pmatrix}.$$9

If the two cavities are squeezed identically, $\mathcal{APT}$00 and $\mathcal{APT}$01, the system remains exchange symmetric and the squeezing-generated pathways cancel, leaving $\mathcal{APT}$02. By contrast, asymmetric squeezing prevents this cancellation. For symmetric total squeezing $\mathcal{APT}$03, the maximum occurs at $\mathcal{APT}$04, where $\mathcal{APT}$05. The paper further emphasizes that reservoir squeezing alone does not modify the first-order dynamics; it reshapes noise correlations but does not change the effective nonreciprocal coupling. This is why the term “squeezing-induced symmetry breaking” here refers specifically to asymmetric redistribution of squeezing resources across the two channels (Liu et al., 1 Jul 2026).

The same paper connects this mechanism to performance metrics. In the quantum-battery setting, the squeezing-enhanced contributions to the steady-state stored energy scale as $\mathcal{APT}$06, while in the optical-isolator setting the transmission coefficient contains $\mathcal{APT}$07 and $\mathcal{APT}$08, leading to the reported second-order exponential enhancement of the output signal. The important conceptual point is that the enhancement is not a trivial rescaling of a coupling constant; it depends on symmetry-breaking interference between squeezing-generated pathways (Liu et al., 1 Jul 2026).

A related but structurally different transport-oriented realization appears in optomechanical networks with time-modulated radiation pressure. Difference-frequency modulation produces beam-splitter couplings, whereas sum-frequency modulation generates squeezing interactions of the form

$\mathcal{APT}$09

Because these particle-nonconserving terms mix annihilation and creation operators, the appropriate Bogoliubov-de Gennes description is non-Hermitian even without physical dissipation, and the spectrum appears in quartets $\mathcal{APT}$10. In the squeezing dimer, the effective flux $\mathcal{APT}$11 controls a non-Hermitian Aharonov-Bohm effect with eigenvalues

$\mathcal{APT}$12

yielding $\mathcal{APT}$13-symmetric and broken phases, exceptional points, flux-tuned squeezing, and unidirectional phononic amplification. Here squeezing is the resource that changes the symmetry class of the effective dynamics from number-conserving Hermitian transport to particle-hole non-Hermitian chirality (Pino et al., 2021).

6. Elastic compression analogues, disorder-driven responses, and conceptual boundaries

In amorphous solids, squeezing-induced symmetry breaking refers to literal radial compression. A two-dimensional annulus filled with a disordered assembly of particles is inflated at the inner boundary by a tiny amount with $\mathcal{APT}$14, while the outer boundary is fixed. Linear elasticity would predict a purely radial displacement field, yet the observed response develops angular structure and breaks radial symmetry even though the forcing is strictly radial. The paper attributes this to disorder-induced nonlinear mode coupling in the elastic response, not to plasticity. The displacement field is decomposed into angular Fourier components, and the observed low-order harmonics are quantitatively captured by Michell solutions of the biharmonic elasticity equation. A further reconstruction shows that the non-radial pattern can be reproduced by replacing the pure radial inner-boundary shift with a finite superposition of Fourier perturbations. The effect weakens when the inner radius exceeds the disorder scale and disappears in a perfectly ordered hexagonal crystal, which sharpens the claim that the symmetry breaking originates in amorphous disorder (Kumar et al., 2023).

A second compression-driven example is the indented elastic cone. In the radially symmetric von Kármán reduction, the energy obeys the scaling law

$\mathcal{APT}$15

corresponding to a disclination-dominated regime, a linear-response regime near the outer boundary, and a localized central inversion regime. However, in the full admissible class a non-radial ridge-based construction achieves

$\mathcal{APT}$16

which beats every radial competitor when $\mathcal{APT}$17. The resulting minimizer is therefore non-radially symmetric for sufficiently strong squeezing relative to thickness. The symmetry-breaking state is a faceted, pyramid-like buckling pattern with localized ridges rather than a circular inversion (Conti et al., 2015).

These mechanical examples clarify that the term can refer to compression-induced loss of geometric symmetry even when no bosonic squeezing operator is present. They also delimit the concept relative to nearby phenomena. In photoexcited group-V semimetals, a pump pulse creates an anisotropic carrier distribution that exerts a symmetry-breaking force on the low-symmetry $\mathcal{APT}$18 mode; the force then decays exponentially, with Bi and Sb lifetimes on the order of $\mathcal{APT}$19 fs, because electron-phonon scattering restores symmetry. This is a symmetry-breaking-force relaxation problem rather than a squeezing-induced mechanism (O'Mahony et al., 2019). Likewise, in inflationary dark-matter production from curvature-induced $\mathcal{APT}$20 breaking, the authors explicitly work in a heavy-field regime where stochastic infrared growth and large superhorizon squeezing effects are not important; the broken phase is generated classically by the $\mathcal{APT}$21 term, not by fluctuation-driven squeezing (Laulumaa et al., 2020). This suggests that “squeezing-induced symmetry breaking” is best reserved for cases where squeezing, squeezed fluctuations, or literal compression plays an operational role in selecting or diagnosing the lower-symmetry response, rather than for symmetry breaking in general.