Entanglement and dynamical phase transition in a spin-orbit-coupled Bose-Einstein condensate

Characterizing quantum phase transitions through quantum correlations has been deeply developed for a long time, while the connections between dynamical phase transitions (DPTs) and quantum entanglement is not yet well understood. In this work, we show that the time-averaged two-mode entanglement in the spin space reaches a maximal value when it undergoes a DPT induced by external perturbation in a spin-orbit-coupled Bose-Einstein condensate. We employ the von Neumann entropy and a correlation-based entanglement criterion as entanglement measures and find that both of them can infer the existence of DPT. While the von Neumann entropy works only for a pure state at zero temperature and requires state tomography to reconstruct, the experimentally more feasible correlation-based entanglement criterion acts as an excellent proxy for entropic entanglement and can determine the existence of entanglement for a mixed state at finite temperature, making itself an excellent indicator for DPT. Our work provides a deeper understanding about the connection between DPTs and quantum entanglement and may allow the detection of DPT via entanglement become accessible as the examined criterion is suitable for measuring entanglement.

Figure 1

The external perturbation $V_{\mathrm{ex}}$ described by Eq. (5) can induce resonant couplings between the two magnetized phases, marked as ${\psi }_R$ and ${\psi }_L$ at $k_m$ and $-k_m$, respectively.

Figure 2

The evolution of entropic entanglement $E$ is related to the evolution of pseudospin $\langle s_z\rangle$. From left to right, the external perturbation strength is (a, e) $V_0=0.6V_{0,\mathrm{crit}}$, (b, f) $0.999V_{0,\mathrm{crit}}$, (c, g) $1.001V_{0,\mathrm{crit}}$, and (d, h) $1.4V_{0,\mathrm{crit}}$. For each situation, we plot the entropic entanglement for various numbers of atoms $N=1$ (solid blue line), $N=10$ (dash-dotted green line), and $N=100$ (dotted red line). Other parameters are used as $g_{s}n=1.0k_0^2$ and $\mathrm{\Omega }=0.3k_0^2$.

Figure 3

The time-averaged entropic entanglement measure $\overline{E}$ calculated as a function of perturbation strength $V_0/V_{0,\mathrm{crit}}$ (left axis). Different lines represent different total numbers of atoms $N=1$ (solid blue), $N=10$ (dash-dotted green), and $N=100$ (dotted red). The entanglement displays a sharp peak at the critical value $V_{0,\mathrm{crit}}$ where the DPT occurs as characterized by the order parameter $\overline{M}$ (black dashed, right axis). This observation suggests that the critical behavior of entropic entanglement measure can be used to identify the DPT. Other parameters are used as those in Fig. 2.

Figure 4

A comparison between the evolution of HZ entanglement parameter $E_{\mathrm{HZ}}$ (solid blue) and the entropic entanglement $1-E$ with total number of atoms $N=10$ (dash-dotted green) and $N=100$ (dotted red). The strength of external perturbation is (a) $V_0=0.6V_{0,\mathrm{crit}}$, (b) $0.999V_{0,\mathrm{crit}}$, (c) $1.001V_{0,\mathrm{crit}}$, and (d) $1.4V_{0,\mathrm{crit}}$, respectively. The HZ entanglement parameters behaves similarly to the entropic entanglement for different $N$. Other parameters are used as those in Fig. 2.

Figure 5

The time-averaged HZ entanglement parameter ${\overline{E}}_{\mathrm{HZ}}$ (solid blue) calculated as a function of perturbation strength $V_0/V_{0,\mathrm{crit}}$, in comparison with the time-averaged entropic entanglement measure $1-\overline{E}$ with $N=10$ (dash-dotted green) and $N=100$ (dotted red). In the vicinity of the critical point $V_{0,\mathrm{crit}}$, ${\overline{E}}_{\mathrm{HZ}}$ shows a discontinuity in the first-order derivative, which signifies the occurrence of DPT. Other parameters are used as those in Fig. 2.

Figure 6

(a) The order parameter $\overline{M}$ and (b) the time-averaged HZ entanglement parameter ${\overline{E}}_{\mathrm{HZ}}$ at temperatures of $T=0\mathrm{nK}$ (solid blue), $30\mathrm{nK}$ (dash-dotted green), $50\mathrm{nK}$ (dotted red), and $70\mathrm{nK}$ (dashed cyan) by using Hartree-Fock-Bogoliubov-Popov approximation. The critical point of the DPT shifts towards a smaller value of perturbation strength with elevated temperature, where the HZ entanglement parameter features a sharp dip as an unambiguous signature. Other parameters are used as those in Fig. 2.