Dynamical Squeezing Phase Transition

• Dynamical squeezing phase transition is a nonequilibrium phenomenon where quench dynamics induce a qualitative reorganization of quantum fluctuations, spin squeezing, and symmetry-breaking behavior.
• It manifests in distinct regimes—such as fully collective (Heisenberg-limited squeezing) and partially collective phases—with scaling laws sensitive to interaction range and system geometry.
• Experimental and theoretical studies reveal critical signatures like optimal squeezing levels, non-analyticities in the Loschmidt echo, and universal fluctuation scaling across diverse quantum systems.

Dynamical squeezing phase transition denotes a family of nonequilibrium critical phenomena in which the generation, scaling, or orientation of squeezing changes qualitatively across a dynamical boundary. In quench dynamics of long-range interacting spin systems, it refers to a dynamical critical point at which symmetry-breaking dynamics, nonlocal correlations, and non-analyticities in the Loschmidt echo coalesce, while critical fluctuations catalyze metrologically useful spin squeezing (Xu et al., 2019). In power-law interacting bilayer XXZ models, the same term designates a transition between a fully collective phase with Heisenberg-limited squeezing and a partially collective phase with universal critical scaling (Duha et al., 14 Mar 2025). Closely related usages occur in integrable XY-chain quenches, squeezing-enhanced generalized Lipkin–Meshkov–Glick models, and rotating Bose–Einstein condensates, where squeezing extrema, Fisher-zero structure, bifurcations, or superfluid instabilities define the dynamical boundary (Wong et al., 2023, Kam, 24 May 2025, Chen et al., 6 Aug 2025).

1. Conceptual definitions and diagnostics

Two notions of dynamical criticality recur in the literature. The first is symmetry-breaking dynamics, where a nonequilibrium order parameter separates distinct long-time dynamical regimes. In the Lipkin–Meshkov–Glick setting, the order parameter is the time-averaged longitudinal magnetization

$$m_z(t)\equiv \frac{1}{N}\sum_{j=1}^N\langle \sigma_j^z(t)\rangle,\qquad \overline{m_z}\equiv \frac{1}{t_f}\int_0^{t_f} dt\, m_z(t),$$

with $\overline{m_z}\neq 0$ in a dynamical ferromagnetic phase and $\overline{m_z}=0$ in a dynamical paramagnetic phase (Xu et al., 2019). The second is the dynamical quantum phase transition, diagnosed by the Loschmidt echo

$$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$

whose non-analyticities are cusp singularities in the rate function $\lambda(t)$ in the thermodynamic limit (Xu et al., 2019).

Squeezing enters as both a fluctuation diagnostic and a metrological resource. In collective-spin language, the most widely used measure is the Kitagawa–Ueda parameter

$$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$

while metrological sensitivity is commonly expressed through a Wineland parameter and gain $G=1/\xi^2$ when $|\langle \mathbf{J}\rangle|$ remains close to maximal (Xu et al., 2019). In the XY chain, the spin-squeezing parameter is minimized over transverse directions in the $x$–$y$ plane, and a dynamical squeezing phase transition is proposed to occur when a cusp in the Loschmidt rate coincides with an extremum of $\overline{m_z}\neq 0$0, together with qualitative changes in squeezing directionality (Wong et al., 2023).

A different but related definition arises in long-range bilayer XXZ models. There, the transition is not formulated in terms of a Loschmidt singularity but in terms of the scaling of the minimal squeezed variance. The fully collective phase has $\overline{m_z}\neq 0$1 or equivalently $\overline{m_z}\neq 0$2, whereas the partially collective phase has $\overline{m_z}\neq 0$3 with $\overline{m_z}\neq 0$4 (Duha et al., 14 Mar 2025, Duha et al., 13 May 2026). In rotating Bose–Einstein condensates, the order parameter is the long-time exponential growth rate of the unstable collective mode; the transition separates oscillatory dynamics from exponential geometric squeezing (Chen et al., 6 Aug 2025). These usages indicate that the term is not restricted to a single microscopic mechanism, but consistently denotes a sharp dynamical reorganization of quantum fluctuations.

2. Quench criticality in the Lipkin–Meshkov–Glick model

The experimentally most direct realization was reported in a 16-qubit superconducting quantum simulator implementing the long-range LMG model with all-to-all connectivity (Xu et al., 2019). The effective Hamiltonian is

$\overline{m_z}\neq 0$5

with nearly uniform $\overline{m_z}\neq 0$6. In collective-spin notation, the dynamics is captured by

$\overline{m_z}\neq 0$7

where $\overline{m_z}\neq 0$8 and $\overline{m_z}\neq 0$9. The quench starts from the fully polarized state $\overline{m_z}=0$0 and suddenly changes the transverse field from $\overline{m_z}=0$1 to $\overline{m_z}=0$2 (Xu et al., 2019).

The dynamical critical point is

$\overline{m_z}=0$3

which for the measured coupling predicts $\overline{m_z}=0$4 (Xu et al., 2019). Three independent observables converge near this value. First, $\overline{m_z}=0$5 crosses from nonzero to zero, distinguishing the dynamical ferromagnetic and dynamical paramagnetic phases. Second, the time-averaged, pair-averaged longitudinal correlator $\overline{m_z}=0$6 develops a dip near the critical point, signaling enhanced quantum fluctuations and a change of dynamical phase. Third, the earliest Loschmidt minimum $\overline{m_z}=0$7 is large in the dynamical ferromagnetic phase and strongly suppressed in the dynamical paramagnetic phase, with a threshold $\overline{m_z}=0$8 marking the paramagnetic side for $\overline{m_z}=0$9 (Xu et al., 2019).

The metrological aspect is central. Near the dynamical critical point the simulator achieved optimal squeezing $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$0, corresponding to $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$1 and a metrological gain $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$2 beyond the standard quantum limit (Xu et al., 2019). The minimum of the time-optimized squeezing $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$3 occurs close to the same $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$4. The interpretation given is that the collective interaction term $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$5 provides nonlinear twisting, while a quench to $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$6 brings the dynamics close to an unstable fixed point where critical slowing-down and enhanced susceptibility amplify collective quantum fluctuations (Xu et al., 2019).

Finite size and noise remain essential qualifiers. Exact Loschmidt zeros are absent for $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$7, accessible evolution is limited to $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$8, and readout errors and residual crosstalk broaden minima in $$L(t)=|\langle \psi(0)|\psi(t)\rangle|^2,\qquad \lambda(t)=-\frac{1}{N}\ln L(t),$$9 and $\lambda(t)$0 (Xu et al., 2019). The experimental signatures are therefore finite-system precursors of the thermodynamic singularities.

3. Fisher zeros, squeezing extrema, and symmetry control

In the one-dimensional XY chain, the Loschmidt amplitude factorizes into momentum sectors after Jordan–Wigner, Fourier, and Bogoliubov transformations. The Loschmidt rate

$\lambda(t)$1

shows non-analytic cusps when Fisher zeros reach the real-time axis, at critical times

$\lambda(t)$2

The spin-squeezing parameter

$\lambda(t)$3

is then found to exhibit pronounced extrema in the immediate vicinity of $\lambda(t)$4 when quenching across equilibrium phase boundaries (Wong et al., 2023). For quenches between Ising phases, $\lambda(t)$5 typically shows a local maximum just before the first DQPT; for quenches from the anisotropy boundary $\lambda(t)$6, $\lambda(t)$7 instead attains a minimum near $\lambda(t)$8. Across the anisotropy boundary, two critical momenta can generate two critical times, and the squeezing vector reverses its rotation between them (Wong et al., 2023).

This correspondence is resolved at the level of correlations. Near DQPTs, the dominant parallel-spin correlations align with the preferred direction of the post-quench phase: $\lambda(t)$9 peaks near $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$0 for quenches to $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$1, while $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$2 plays the same role for quenches to $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$3. Cross-correlations $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$4 vanish at the critical times (Wong et al., 2023). The proposed criterion is therefore composite: a dynamical squeezing phase transition occurs when the Loschmidt rate has a non-analytic cusp and $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$5 simultaneously exhibits an extremum together with a sharp change in squeezing directionality (Wong et al., 2023).

A further refinement appears when the initial state is modified by double-mode squeezing in the XY chain. For a particle–hole-symmetry-preserving squeeze with $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$6, the DQPT condition reduces to the unsqueezed one unless the squeezing strength reaches $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$7. At that special point, $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$8 for every $$\xi^2 = \frac{4\,\min_{\vec{n}_\perp}\mathrm{Var}(S^{\vec{n}_\perp})}{N},$$9, all Fisher zeros lie on the real-time axis, every $G=1/\xi^2$0 pair is maximally entangled, and the dynamical phase vanishes so that the evolution becomes purely geometric with $G=1/\xi^2$1-jumps in the Pancharatnam geometric phase (Cao et al., 7 Jan 2026). When particle–hole symmetry is broken, squeezing becomes a control knob that can induce DQPTs within a single phase or suppress them by steering the condition $G=1/\xi^2$2 across or away from the Brillouin zone (Cao et al., 7 Jan 2026).

4. Nonequilibrium universality in bilayer XXZ systems

A distinct line of work identifies a dynamical squeezing phase transition in power-law interacting spin-$G=1/\xi^2$3 bilayer XXZ models (Duha et al., 14 Mar 2025, Duha et al., 13 May 2026). The initial state has opposite layer polarizations, $G=1/\xi^2$4, so the intralayer Heisenberg term does not drive dynamics from $G=1/\xi^2$5, while the interlayer XX exchange generates entanglement dynamically. At leading order in a collective Holstein–Primakoff mapping, the dynamics reduces to two-mode squeezing between layers,

$G=1/\xi^2$6

with

$G=1/\xi^2$7

The fully collective phase is defined by $G=1/\xi^2$8, while the partially collective phase has $G=1/\xi^2$9 and $|\langle \mathbf{J}\rangle|$0 with $|\langle \mathbf{J}\rangle|$1 (Duha et al., 14 Mar 2025, Duha et al., 13 May 2026).

The transition is controlled by the instability of finite-momentum modes. In the Bogoliubov description,

$|\langle \mathbf{J}\rangle|$2

and the phase boundary is reached when the smallest nonzero momentum mode becomes unstable: $|\langle \mathbf{J}\rangle|$3 Two regimes follow. For $|\langle \mathbf{J}\rangle|$4, the critical aspect ratio scales as $|\langle \mathbf{J}\rangle|$5. For $|\langle \mathbf{J}\rangle|$6, Bogoliubov theory yields the analytical scaling

$|\langle \mathbf{J}\rangle|$7

while at the boundary case $|\langle \mathbf{J}\rangle|$8, $|\langle \mathbf{J}\rangle|$9 (Duha et al., 13 May 2026). This short-range regime was identified as previously unrecognized.

The nonequilibrium universality claim is supported by discrete truncated Wigner simulations. In 2D bilayers at $x$0, data collapse gives exponents $x$1, $x$2–$x$3, $x$4–$x$5, $x$6–$x$7, and $x$8–$x$9, consistent across square, triangular, and honeycomb geometries within uncertainties. In 1D ladders at $y$0, exponents remain consistent for $y$1, establishing robustness under symmetry-preserving rescaling of interlayer couplings (Duha et al., 13 May 2026). The scaling ansatz for the variance near its minimum,

$y$2

identifies a divergent time scale and defines a non-equilibrium universality class within the symmetry class of intralayer SU(2) and interlayer U(1) couplings (Duha et al., 14 Mar 2025, Duha et al., 13 May 2026).

5. Optical and condensate analogues

In nonlinear optical analogues of a generalized LMG model, the transition is driven purely by quadratic squeezing structure rather than by a quench of a transverse field (Kam, 24 May 2025). In the squeezing-only limit, the classical spin Hamiltonian is

$y$3

with Hessian

$y$4

The sign of $y$5 determines whether the constant-energy surface is elliptic or hyperbolic; the boundary $y$6 separates a positive-mass Euler top from an inverted top (Kam, 24 May 2025). In tetragonal media, the effective Hamiltonian

$y$7

contains the unconventional cross-squeezing term $y$8, and the associated bifurcation produces separatrices, divergent periods, and excited-state quantum phase transition signatures (Kam, 24 May 2025). The paper does not compute Loschmidt echoes, so the result is formulated as a squeezing-driven bifurcation and ESQPT correspondence rather than as a DQPT.

A rotating interacting Bose–Einstein condensate provides a second analogue. In the non-interacting case, a sudden quench of the rotation frequency to the trapping frequency produces a single-mode geometrically squeezed state with

$y$9

and squeezing parameter $\overline{m_z}\neq 0$00 (Chen et al., 6 Aug 2025). For interacting condensates, however, the same quench can yield only periodic oscillations because the relevant quadrupole mode remains stable. The transition is therefore controlled by superfluid stability: the oscillatory phase has a real collective-mode frequency, whereas the squeezed phase has an imaginary frequency and exponential growth or decay of principal-axis fluctuations (Chen et al., 6 Aug 2025). By increasing trap anisotropy to open an unstable window and quenching to $\overline{m_z}\neq 0$01 at $\overline{m_z}\neq 0$02, the simulations show early-time squeezing at rate $\overline{m_z}\neq 0$03 and a minimum near $\overline{m_z}\neq 0$04 at about $\overline{m_z}\neq 0$05, faster and deeper than a quasi-adiabatic ramp protocol (Chen et al., 6 Aug 2025).

6. Relation to equilibrium criticality, non-Hermitian transitions, and conceptual boundaries

Several closely related results concern critical squeezing enhancement without introducing the same dynamical phase boundary. In the one-axis twisting model in a transverse field and in the Dicke model, the squeezing time diverges as $\overline{m_z}\neq 0$06, while the squeezing strength scales as $\overline{m_z}\neq 0$07 in the one-axis twisting case and generically as $\overline{m_z}\neq 0$08 in the Dicke case, crossing over to $\overline{m_z}\neq 0$09 in the extreme-detuning limit (Sharma et al., 2020). These results tie long-lived squeezing to equilibrium soft-mode physics near a quantum critical point, rather than to a separate nonequilibrium universality class.

The Dicke model also supports a stronger equilibrium statement: at the superradiant phase transition critical point, the ground state is a two-mode squeezed vacuum in the photon–matter basis, and the variance of a properly chosen two-mode quadrature vanishes while the conjugate variance diverges, saturating the Heisenberg bound (Hayashida et al., 2020). This is “perfect intrinsic squeezing” at a quantum critical point. It is equilibrium and ground-state based, but it provides a limiting case for any dynamical protocol that attempts to approach critical squeezing by ramps or quenches.

Non-Hermitian dynamics furnish another variant. In the non-Hermitian LMG model,

$\overline{m_z}\neq 0$10

the finite-$\overline{m_z}\neq 0$11 phase transition is an exceptional-point transition of the postselected steady state (1409.02630). At the transition, the averaged quantum Fisher information saturates $\overline{m_z}\neq 0$12, implying full $\overline{m_z}\neq 0$13-particle entanglement, while the optimal squeezing occurs at $\overline{m_z}\neq 0$14 with $\overline{m_z}\neq 0$15 in the bosonic approximation and numerically $\overline{m_z}\neq 0$16 (1409.02630). This is again a distinct mechanism: conditional non-Hermitian evolution rather than unitary quench criticality.

A common misconception is to identify any appearance of “squeezed” terminology with dynamical squeezing criticality. The “squeezed ensemble” developed for first-order phase transition points is a generalized statistical ensemble designed to realize phase coexistence in general quantum systems, and its main dynamical result is local stationarity on time scales diverging with $\overline{m_z}\neq 0$17 (Yoneta, 2023). That work explicitly states that it does not study dynamical quantum phase transitions such as non-analyticities in Loschmidt amplitudes (Yoneta, 2023). Another common misconception is to require exact Loschmidt zeros in finite systems; in finite-size LMG dynamics, true zeros are absent and sharply suppressed minima only track the DQPT (Xu et al., 2019).

Taken together, the literature supports a precise but plural usage. The term “dynamical squeezing phase transition” may denote quench-induced coincidence of DQPT and squeezing extrema, a universal transition in the scaling of optimal squeezed variance, a squeezing-driven bifurcation of classical polarization dynamics, or a superfluid-stability boundary between oscillatory and exponentially squeezed condensate motion. What remains common is the existence of a dynamical boundary across which squeezing ceases to be a perturbative fluctuation effect and becomes the defining signature of a reorganized nonequilibrium phase.