---
title: Dynamical Squeezing Phase Transition
url: https://www.emergentmind.com/topics/dynamical-squeezing-phase-transition
type: topic
---

# Dynamical Squeezing Phase Transition

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 [1912.05150]. 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 [2503.11802]. 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 [2306.15227] [2505.18618] [2508.04126].

## 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 [1912.05150]. 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 [1912.05150].

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 [1912.05150]. 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 \(\xi_S^2(t)\), together with qualitative changes in squeezing directionality [2306.15227].

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 \(\xi_{\min}^2\sim 1/N\) or equivalently \(\mathrm{Var}[O^-]_{\min}\sim N^0\), whereas the partially collective phase has \(\mathrm{Var}[O^-]_{\min}\sim N^p\) with \(0<p<1\) [2503.11802] [2605.13969]. 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 [2508.04126]. 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 [1912.05150]. The effective Hamiltonian is
\[
\frac{H_1}{\hbar}=\sum_{i\neq j}^{N}\lambda_{ij}\bigl(\sigma_i^+\sigma_j^-+\sigma_i^-\sigma_j^+\bigr) + h^x \sum_{j=1}^{N}\sigma_j^x,
\]
with nearly uniform \(\lambda_{ij}\simeq \lambda\). In collective-spin notation, the dynamics is captured by
\[
H_{\mathrm{LMG}}= -\frac{J}{N}(S^z)^2 + \mu S^x,
\]
where \(J\equiv N\lambda\) and \(\mu=2h^x\). The quench starts from the fully polarized state \(|\psi(0)\rangle=|00\ldots0\rangle\) and suddenly changes the transverse field from \(h_i=0\) to \(h_f=h^x\) [1912.05150].

The dynamical critical point is
\[
\mu_c=\frac{|J|}{2}\quad \Rightarrow\quad h_c^x=\frac{N\lambda}{4},
\]
which for the measured coupling predicts \(h_c^x/2\pi\simeq 5.7\,\mathrm{MHz}\) [1912.05150]. Three independent observables converge near this value. First, \(\overline{m_z}\) crosses from nonzero to zero, distinguishing the dynamical ferromagnetic and dynamical paramagnetic phases. Second, the time-averaged, pair-averaged longitudinal correlator \(\overline{C_{zz}}\) develops a dip near the critical point, signaling enhanced quantum fluctuations and a change of dynamical phase. Third, the earliest Loschmidt minimum \(L_{\min}^{(1)}\) is large in the dynamical ferromagnetic phase and strongly suppressed in the dynamical paramagnetic phase, with a threshold \(L_{\min}^{(1)}\lesssim 10^{-2}\) marking the paramagnetic side for \(N=16\) [1912.05150].

The metrological aspect is central. Near the dynamical critical point the simulator achieved optimal squeezing \(\xi_{\mathrm{dB}}\approx -7.0\pm 0.8\,\mathrm{dB}\), corresponding to \(\xi^2\approx 0.20\pm 0.04\) and a metrological gain \(G\approx 5\) beyond the standard quantum limit [1912.05150]. The minimum of the time-optimized squeezing \(\xi^2_{\min}(h^x)\) occurs close to the same \(h_c^x/2\pi\simeq 5.7\,\mathrm{MHz}\). The interpretation given is that the collective interaction term \(-\frac{J}{N}(S^z)^2\) provides nonlinear twisting, while a quench to \(h_f\simeq h_c\) brings the dynamics close to an unstable fixed point where critical slowing-down and enhanced susceptibility amplify collective quantum fluctuations [1912.05150].

Finite size and noise remain essential qualifiers. Exact Loschmidt zeros are absent for \(N=16\), accessible evolution is limited to \(t_f\lesssim 600\,\mathrm{ns}\), and readout errors and residual crosstalk broaden minima in \(\xi^2\) and \(L_{\min}^{(1)}\) [1912.05150]. 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
\[
r(t)= -\frac{1}{N}\ln |G(t)|^2
\]
shows non-analytic cusps when Fisher zeros reach the real-time axis, at critical times
\[
t_c^p=\frac{\pi}{\epsilon_{k^*}^f}\left(p-\frac{1}{2}\right).
\]
The spin-squeezing parameter
\[
\xi_S^2(t)=\frac{4(\Delta J_{\hat n_\perp})^2}{N}
\]
is then found to exhibit pronounced extrema in the immediate vicinity of \(t_c\) when quenching across equilibrium phase boundaries [2306.15227]. For quenches between Ising phases, \(\xi_S^2(t)\) typically shows a local maximum just before the first DQPT; for quenches from the anisotropy boundary \(\delta_i=0\), \(\xi_S^2(t)\) instead attains a minimum near \(t_c\). Across the anisotropy boundary, two critical momenta can generate two critical times, and the squeezing vector reverses its rotation between them [2306.15227].

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: \(\sum_n G_n^{xx}(t)\) peaks near \(t_c\) for quenches to \(FM_x\), while \(\sum_n G_n^{yy}(t)\) plays the same role for quenches to \(FM_y\). Cross-correlations \(\sum_n(G_n^{xy}+G_n^{yx})\) vanish at the critical times [2306.15227]. The proposed criterion is therefore composite: a dynamical squeezing phase transition occurs when the Loschmidt rate has a non-analytic cusp and \(\xi_S^2(t)\) simultaneously exhibits an extremum together with a sharp change in squeezing directionality [2306.15227].

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 \(\phi=0\), the DQPT condition reduces to the unsqueezed one unless the squeezing strength reaches \(r=\pi/4\). At that special point, \(|A_k|^2=|B_k|^2=1/2\) for every \(k\), all Fisher zeros lie on the real-time axis, every \((k,-k)\) pair is maximally entangled, and the dynamical phase vanishes so that the evolution becomes purely geometric with \(\pi\)-jumps in the Pancharatnam geometric phase [2601.03494]. 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 \(\Delta_k=0\) across or away from the Brillouin zone [2601.03494].

## 4. Nonequilibrium universality in bilayer XXZ systems

A distinct line of work identifies a dynamical squeezing phase transition in power-law interacting spin-\(1/2\) bilayer XXZ models [2503.11802] [2605.13969]. The initial state has opposite layer polarizations, \(\langle S_A^z\rangle=-\langle S_B^z\rangle=N/2\), so the intralayer Heisenberg term does not drive dynamics from \(t=0\), 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,
\[
H_{\mathrm{TMS}}=\frac{N V_{\mathrm{av}}}{2}(\hat a^\dagger \hat b^\dagger+\hat a\hat b),
\]
with
\[
\mathrm{Var}[O^\pm](t)=\frac{N}{2}e^{\pm \gamma t},\qquad \gamma \approx \frac{N V_{\mathrm{av}}}{\hbar}.
\]
The fully collective phase is defined by \(\xi_{\min}^2\sim 1/N\), while the partially collective phase has \(\xi_{\min}^2\propto N^{p-1}\) and \(\mathrm{Var}[O^-]_{\min}\sim N^p\) with \(0<p<1\) [2503.11802] [2605.13969].

The transition is controlled by the instability of finite-momentum modes. In the Bogoliubov description,
\[
\omega(k)=\sqrt{E_k^2-|\Lambda_k|^2},
\]
and the phase boundary is reached when the smallest nonzero momentum mode becomes unstable:
\[
|\Lambda_{k_1}|=E_{k_1}.
\]
Two regimes follow. For \(\alpha<d+2\), the critical aspect ratio scales as \(a_Z^*\propto L\). For \(\alpha>d+2\), Bogoliubov theory yields the analytical scaling
\[
a_Z^* \propto L^{2/(\alpha-d)},
\]
while at the boundary case \(\alpha=d+2\), \(a_Z^*/L\sim 1/\log L\) [2605.13969]. This short-range regime was identified as previously unrecognized.

The nonequilibrium universality claim is supported by discrete truncated Wigner simulations. In 2D bilayers at \(\alpha=3\), data collapse gives exponents \(d_v\approx -1.2\), \(d_r\approx 1.05\)–\(1.12\), \(\delta\approx 0.11\)–\(0.16\), \(\nu\approx 0.68\)–\(0.82\), and \(p\approx 0.18\)–\(0.28\), consistent across square, triangular, and honeycomb geometries within uncertainties. In 1D ladders at \(\alpha=1\), exponents remain consistent for \(\lambda=0.8,1.0,1.4\), establishing robustness under symmetry-preserving rescaling of interlayer couplings [2605.13969]. The scaling ansatz for the variance near its minimum,
\[
\mathrm{Var}[O^-]\, a_Z^{d_v} N^\Phi = f\!\left((t-t_{\min}) a_Z^{d_r} N^\Psi\right),
\]
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 [2503.11802] [2605.13969].

## 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 [2505.18618]. In the squeezing-only limit, the classical spin Hamiltonian is
\[
H=\frac{\alpha}{2}S_x^2+\beta S_x S_y+\frac{\gamma}{2}S_y^2,
\]
with Hessian
\[
H_2=\begin{bmatrix}\alpha & \beta\\ \beta & \gamma\end{bmatrix},\qquad
\det(H_2)=\alpha\gamma-\beta^2.
\]
The sign of \(\det(H_2)\) determines whether the constant-energy surface is elliptic or hyperbolic; the boundary \(\alpha\gamma-\beta^2=0\) separates a positive-mass Euler top from an inverted top [2505.18618]. In tetragonal media, the effective Hamiltonian
\[
H_{\mathrm{tetra}}=-\frac{1}{2}\{\Delta\beta S_z + c_0 S^2 + c_z S_z^2 + c_x S_x^2 + 2c S_z S_x\}
\]
contains the unconventional cross-squeezing term \(2c S_z S_x\), and the associated bifurcation produces separatrices, divergent periods, and excited-state quantum phase transition signatures [2505.18618]. 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
\[
\Delta X^2=\frac{1}{2}e^{-2r},\qquad
\Delta P^2=\frac{1}{2}e^{+2r},
\]
and squeezing parameter \(r(t)=\zeta t\) [2508.04126]. 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 [2508.04126]. By increasing trap anisotropy to open an unstable window and quenching to \(\Omega=0.9\omega\) at \(\epsilon=0.3\), the simulations show early-time squeezing at rate \(\approx 1.7\zeta\) and a minimum near \(-6.7\,\mathrm{dB}\) at about \(14\,\mathrm{ms}\), faster and deeper than a quasi-adiabatic ramp protocol [2508.04126].

## 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 \(T_{\mathrm{sq}}\sim |\xi-\xi_c|^{-1/2}\), while the squeezing strength scales as \(\zeta_{s,\min}\sim |\xi-\xi_c|^{1/2}\) in the one-axis twisting case and generically as \((\zeta_{\min})^2=u+v|\xi-\xi_c|\) in the Dicke case, crossing over to \(\zeta_{\min}\sim |\xi-\xi_c|^{1/2}\) in the extreme-detuning limit [2006.04056]. 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 [2009.02630]. 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,
\[
H=\frac{V}{N}(J_x^2-J_y^2)-\frac{i\gamma}{2}J_z-\frac{i\gamma N}{4},
\]
the finite-\(N\) phase transition is an exceptional-point transition of the postselected steady state [1409.02630]. At the transition, the averaged quantum Fisher information saturates \((N^2+2N)/3\), implying full \(N\)-particle entanglement, while the optimal squeezing occurs at \(V^*=\gamma N/6\) with \(\xi^2(V^*)=27/(8N)\) in the bosonic approximation and numerically \(\xi_{\min}^2\approx 3/N\) [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 \(N\) [2305.12181]. That work explicitly states that it does not study dynamical quantum phase transitions such as non-analyticities in Loschmidt amplitudes [2305.12181]. 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 [1912.05150].

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.

Source: https://www.emergentmind.com/topics/dynamical-squeezing-phase-transition