Periodic quenches across the Berezinskii-Kosterlitz-Thouless phase transition

The quenched dynamics of an ultracold homogeneous atomic two-dimensional Bose gas subjected to periodic quenches across the Berezinskii-Kosterlitz-Thouless (BKT) phase transition are discussed. Specifically, we address the effect of periodic cycling of the effective atomic interaction strength between a thermal disordered state above and a highly ordered state below the critical BKT interaction strength, by means of numerical simulations of the stochastic projected Gross-Pitaevskii equation. Probing the emerging dynamics as a function of the frequency of sinusoidal driving from low to high frequencies reveals diverse dynamical features, including phase-lagged quasiadiabatic reversible condensate formation, resonant excitation consistent with an intrinsic system relaxation timescale, and the gradual establishment of dynamically recurring or time-averaged nonequilibrium states with enhanced coherence, which are neither condensed nor thermal. Our study paves the way for experimental observation of such driven nonequilibrium ultracold superfluid states.

Figure 1

Repeated quenches through the BKT phase transition, with time normalized to the quench period through the time scale $2\pi /{\omega }_D$. (a) Quench protocol [Eq. (3)]. Dotted line through $g=g_c$ separates the parameter space corresponding to incoherent (thermal) ($g>g_c$) and (partly) superfluid ($g<g_c$) states at equilibrium. (b) Evolution of the c-field density averaged over space and stochastic realisations, compared to its initial value $\langle n(t=0)\rangle \equiv \langle n_{\text{bg}}\rangle$ for various driving frequencies ${\omega }_D$, ranging from quasiadiabatic (${\omega }_D/2\pi =1/40$, red) to very rapid (${\omega }_D/2\pi =500$, pink). Also shown are the equilibrium densities for the initial thermal state (${\text{eq}}_{\text{th}}$, lower purple horizontal line) and the final-parameter part-superfluid system ${\text{eq}}_{\text{c}}$ (upper blue horizontal line) respectively, found by fixing $g\left(t\right)=g_{\text{max}}$ and $g\left(t\right)=g_{\text{min}}$, and the equilibrium density $n_0\left(t\right)=\mu /g\left(t\right)$. The slowest quench and condensate equilibrium data are reduced by a factor of 4 for visual aid. For a given driving frequency, the phase lag between the $i_{\text{th}}$ trough of $g\left(t\right)$ and the $i_{\text{th}}$ peak of the average density is denoted ${\varphi }_{\text{lag}}^{n_i}$. The vertical grey dashed line throughout all three panels highlights the time of the first trough in $g\left(t\right)$. (c) Evolution of the average number of vortices, $\langle N_v\rangle$, for the same quench frequencies and equilibria as above.

Figure 2

Quench dependent time shifts probed here in the physically interesting range $0.05\lesssim ({\omega }_D/{\omega }_0)\lesssim 10$. (a) The mean phase shift for the density (${\varphi }^n$) and vortex number (${\varphi }^V$) between the $i^{\text{th}}$ peaks of $g$ and the peak average density, as a function of ${\omega }_D$. Error bars are smaller than the marker size. Points corresponding to the quench frequencies considered previously are colored according to the legend of Fig. 1. (b) Average number of oscillation cycles, $n_{\text{cycle}}$, between the first quench and density (vortex number) reaching 90% of its maximum value (10% of its initial value). The relaxation frequency ${\omega }_0$ [Eq. (5)] is marked by a vertical dashed red line, with the grey-shaded area corresponding to the “quasiadiabatic” regime ${\omega }_D<{\omega }_0$. Horizontal black lines in (a) and (b) respectively denote the values of ${\varphi }_{\mathrm{lag}}/2\pi =1/4$, marking the transition from the observed phase lag occurring within a quarter of the driving cycle (i.e., while the system is within the “coherent” phase), and $n_{\mathrm{cycle}}=1$, marking the transition between vortex number counts that are independent and dependent on cycle number.

Figure 3

Dynamics of the momentum distribution during the periodic quench. (a) Evolution of the average proportion of atoms in mode $k$, ${\widehat{n}}_k\left(t\right)$, defined in Eq. (6). The value of ${\omega }_D/2\pi$ (in Hz) is stated at the top of each panel and a colored block appears above to highlight its relation to previous figures. The solid, dashed, and dotted lines in (i) and (iv) pertain to the first maximum, the tenth minimum and the tenth maximum of $g$, respectively. The occupation of the zero-momentum mode is shown along the bottom parts of these subplots. (b) Slices of momentum density occupation taken through the top panels, showing the profile of ${\widehat{n}}_k$ at different times, given as vertical lines in (a) (i) and (iv). Also shown is the initial $t=0$ distribution prior to the initiation of our periodic quenching (hollow purple circles), with error bars indicative of the magnitude of the error in all of the data shown, and the distribution at equilibrium after a half-quench cycle, followed by a constant $g=g_{\min }$ post-quench relaxation period (dot-dashed blue line). The expected behavior in the two limiting cases is shown by solid grey lines: exponential behavior for a thermal cloud $\sim e^{-\xi k}$ [with a fitted coherence length $\xi =0.91\mu \mathrm{m}\ll L_x,L_y$] (see also Sec. 3c [107]) and a $k^{-2}$ scaling for a superfluid system. (c)(i) Occupation of the ${\widehat{n}}_{k=0}\equiv N_0/N$ mode for the first density peak (circles) and a late density peak corresponding to the time $\gamma t_{*}=0.12\mathrm{s}$ (squares) [see later]. (c)(ii) Total atom number evaluated at the same times as (c)(i). A vertical red line depicting the relaxation frequency ${\omega }_0$ [Eq. (5)] appears in both panels of (c).

Figure 4

Growth of coherence through repeated quenches. (a) First order correlation function, $g^{\left(1\right)}\left(r\right)$, evaluated at $t=t_{p_1}$, the time of the first peak in the average density in Fig. 1. Each curve is fitted with an algebraic function, $g^{\left(1\right)}\left(r\right)\sim r^{-\alpha }$, plotted using a dashed line. (b) Correlation function evaluated at the density peak closest to $\gamma t_{*}=0.12\mathrm{s}$. (c) Value of the correlation function evaluated at $r_{★}=40\mu \mathrm{m}$, [indicated by the dashed vertical grey lines in (a)–(b)], as a function of time for times $g\left(t\right)=g_{\min }$. Error bars are located where $g^{\left(1\right)}$ is measured at a peak in the average density. Data for ${\omega }_D/2\pi =1/40$ (red) and $5/33$ (grey) Hz extend far beyond $\gamma t=0.12\mathrm{s}$, with corresponding red/grey bands denoting error bars over the total range covered. Plotted error bars indicate one standard deviation of $g^{\left(1\right)}\left(r\right)$.

Figure 5

Resonances of observables with the relaxation time. (a) Correlation function evaluated at $r_{★}=40\mu \mathrm{m}$, and at a late time $\gamma t_{*}\approx 0.12\mathrm{s}$ with $g\left(t_{*}\right)=g_{\min }$, see Fig 4. (b) Absolute change in $g^{\left(1\right)}(r_{★},t)$ from $t_1$ to $t_{*}$ (early to late times). (c) Exponent from algebraic fits of Fig. 4. (d) Average time delay in the average peak density.