Nonequilibrium Critical Scaling of a Squeezing Phase Transition

Abstract

We investigate phase transitions in the nonequilibrium dynamics of power-law interacting spin-1/2 bilayer XXZ models, which have recently been shown to allow generation of entanglement in the form of two-mode squeezing. We find a transition between a collective phase characterized by Heisenberg limited squeezing and a partially collective phase with scalable squeezing. We identify universal scaling of the squeezing dynamics in terms of system parameters and a divergent time-scale, establishing these as distinct dynamical phases within the framework of non-equilibrium critical phenomena. Our work demonstrates a novel dynamical phase transition with potential applications in quantum sensing and quantum simulation in cold-atomic, molecular or Rydberg platforms.

Figures (4)

• Figure 1: Dynamical phases of squeezing. (a) Time evolution of the variance of the squeezed quadrature ${\rm Var}[\hat{\mathcal{O}}^-]$ in the fully collective regime (blue, $a_Z/L=0.17$) and partially collective regime (red, $a_Z/L=0.04$) for $\alpha = 3$ in $2d$. The opacity increases with system size $N \in \{9\times 10^2,10^4\}$. Inset: System size scaling of the minimum variance ${\rm Var}[\hat{\mathcal{O}}^-]_{\rm min}$ defining the fully, ${\rm Var}[\hat{\mathcal{O}}^-]_{\rm min}\sim N^0$, and partially collective regime $\sim N^{0.28}$. (b, c) Dynamical phase diagram as a function of power law exponent $\alpha$ and aspect ratio $a_Z/L$ in (b) $2d$ and (c) $1d$. The fully collective regime spans all $a_Z/L$ for $\alpha \leq d/2$, whereas a transition occurs at a critical $a_Z/L$ value (yellow marker) when $\alpha>d/2$.
• Figure 2: Comparison of phase-boundary within Bogoliubov analysis and dTWA. Critical aspect ratio $(a_Z/L)_*$ (Bogoliubov, solid line) as a function of powerlaw exponent $\alpha$ separating the region, in which only the $k=0$ mode is unstable (shaded blue), and the region with multiple unstable modes. Compared with dTWA results (markers, dashed line) (Fig 1(b,c)).
• Figure 3: Universal scaling collapse of minimal squeezing. Inset: Dependence of minimal variance $\mathrm{Var}[\mathcal{O}^-]_{\rm min}$ on layer spacing $a_Z$, for different system sizes ($L = 30$ to 100) with $\alpha =3$ in $2d$. The opacity increases with system size. Main panel: Rescaled minimal variance $\mathrm{Var}[\mathcal{O}^-]_{\rm min}/L^{0.55}$ versus aspect ratio $a_Z/L$ for the same system sizes, demonstrating a universal collapse onto a single curve. Red dashed line power-law $(a_Z/L)^{-1.29}$ obtained from the scaling ansatz, Eq. \ref{['eq:minscale']}.
• Figure 4: Universality of the partially collective phase. (a) ${\rm Var}[\hat{\mathcal{O}}^{-}] a_Z^{-d_{V}}$ vs rescaled time $t a_Z^{-d_{\tau}}$ for different $a_Z$ values in the partially collective regime (colorbar). Inset: raw data ${\rm Var}[\hat{\mathcal{O}}^{-}]$ vs time $t$ for a 1d system with $\alpha = 1.5$. (b) Same as in (a) but for a range of system sizes (legend) and with additional system size rescaling, giving variance ${\rm Var}[\hat{\mathcal{O}}^{-}]N^{-\nu}a_{Z,\delta}^{-d_{V}}$ vs rescaled time $(t-t_{\rm min}) a_{Z,\delta}^{-d_{\tau}}$. For each $N$, a range of $a_Z$ values are plotted with increased fading for smaller values. Inset: data without $N$ rescaling. The exponents are $d_{\tau} = 0.63$, $d_V = -0.69$, $\nu = 0.81$ and $\delta=0.25.$