Universal Rephasing Dynamics after a Quantum Quench via Sudden Coupling of Two Initially Independent Condensates

We consider a quantum quench in which two initially independent condensates are suddenly coupled and study the subsequent “rephasing” dynamics. For weak tunneling couplings, the time evolution of physical observables is predicted to follow universal scaling laws, connecting the short-time dynamics to the long-time nonperturbative regime. We first present a two-mode model valid in two and three dimensions and then move to one dimension, where the problem is described by a gapped sine-Gordon theory. Combining analytical and numerical methods, we compute universal time-dependent expectation values, allowing a quantitative comparison with future experiments.

Figure 1

(a) Dispersion relation before (dashed line) and after (solid line) a quantum quench opening a gap $\Delta$ and involving only the low-frequency modes. (b) Physical realization: sudden coupling of two independent condensates via a uniform tunneling $j_{\perp }$. (c) Lattice model used for the numerical calculations.

Figure 2

Time evolution of the interference contrast $C_{12}(t)=⟨{\psi }_1^{†}{\psi }_2+\mathrm{H}.\mathrm{c}.⟩/2N=⟨\cos (\sqrt{2}\varphi )⟩$, according to the two-mode quantum Hamiltonian (2) with $N=1000$. (a) Exact diagonalization for different values of the ratio $j_{\perp }/\mu =0.10$ (red, lowest), 0.060 (green), 0.037 (blue), 0.022 (cyan), 0.0135 (magenta, highest). (b) Same plots on rescaled axes, showing the universal behavior (1) with $\eta =0$, superimposed to the analytic solution of the simple pendulum (thick solid line).

Figure 3

Series expansion (9). Numerical results obtained by keeping the first $N_{\max }$ terms of the series (solid curve and error bars) and analytic results (Supplemental Material [22]) for the first-order contribution $N=1$ (dashed curve). (a) $K=1.2$, $N_{\max }=11$; (b) $K=5$, $N_{\max }=23$. In both cases, we used ${10}^6$ random evaluation points of $\{(x_i,t_i){\}}_{i=1}^N$.

Figure 4

Truncated Wigner approach (11). (a) Numerical results for $j_{\perp }/\mu =0.27$ (red, highest), 0.16 (green), 0.10 (blue), 0.060 (cyan), 0.036 (magenta, lowest) with $K=5$, system size $L=400$, number of random paths $M=400$. (b) Data collapse according to Eq. (10), superimposed on the result from the series expansion [black curve, identical to Fig. 3b].

Figure 5

Hard-core bosons model (12). (a) Numerical results for $j_{\perp }/j_{\parallel }=0.27$ (red, highest), 0.16 (green), 0.10 (blue), 0.060 (red), 0.036 (cyan), 0.022 (magenta, lowest), with $V/j_{\parallel }=-1/2$, or $K=\pi /[2\pi -2\arccos (V/2j_{\parallel })]\approx 1.2$. The calculations were performed using the time-evolving block decimation algorithm of Refs. 48, 49 for system size $L=100$, number of states $\chi =100$, and took around 3 days/curve on a single core. (b) Data collapse according to Eq. (10), superimposed to the result from the series expansion [black curve, identical to Fig. 3a after the appropriate rescaling].