Collective Excitations and Nonequilibrium Phase Transition in Dissipative Fermionic Superfluids

We predict a new mechanism to induce collective excitations and a nonequilibrium phase transition of fermionic superfluids via a sudden switch on of two-body loss, for which we extend the BCS theory to fully incorporate a change in particle number. We find that a sudden switch on of dissipation induces an amplitude oscillation of the superfluid order parameter accompanied by a chirped phase rotation as a consequence of particle loss. We demonstrate that when dissipation is introduced to one of the two superfluids coupled via a Josephson junction, it gives rise to a nonequilibrium dynamical phase transition characterized by the vanishing dc Josephson current. The dissipation-induced collective modes and nonequilibrium phase transition can be realized with ultracold fermionic atoms subject to inelastic collisions.

Figure 1

(a) Schematic illustration of the amplitude and phase modes in a Mexican-hat free-energy potential as a function of the complex order parameter $\mathrm{\Delta }$, when either the interaction $U_R$ or the dissipation $\gamma$ is suddenly switched on. A sudden quench of the interaction $U_R$ and that of the dissipation $\gamma$ kick $\mathrm{\Delta }$ in a direction parallel and perpendicular to the radial direction, respectively. Note that a finite change of $\gamma$ excites both the phase and amplitude modes. (b) Two superfluids coupled via a Josephson junction, where one superfluid (system 2) is subject to two-body loss.

Figure 2

Dynamics of a superfluid after the atom loss with $\gamma =2.81{\mathrm{\Delta }}_0$ is switched on for the initial state with $U_R=12.2{\mathrm{\Delta }}_0$ and bandwidth $W=46.8{\mathrm{\Delta }}_0$, where ${\mathrm{\Delta }}_0$ is the superfluid order parameter in the absence of the atom loss. (a) Real parts (light green), imaginary parts (blue), and the amplitude (violet) of the order parameter. (b) Angular velocity (pink) and particle number (yellow) plotted against time. The figures indicate a chirped phase rotation and an amplitude oscillation of $\mathrm{\Delta }$.

Figure 3

Dynamics of two fermionic superfluids after the switch on of the atom loss $\gamma$ and the tunnel coupling $V=0.02{\mathrm{\Delta }}_0$ with $U_R=3.06{\mathrm{\Delta }}_0$ and bandwidth $W=5.11{\mathrm{\Delta }}_0$, where $\gamma =0.03{\mathrm{\Delta }}_0$ for (a1)–(d1) and $\gamma =0.06{\mathrm{\Delta }}_0$ for (a2)–(d2). (a),(b) Real parts (light green), imaginary parts (blue), and amplitudes (violet) of the order parameter for systems 1 and 2. (c) Particle numbers of system 1 (red) and system 2 (yellow), and their difference [green, in (c1)]. (d) Josephson current (pink) and phase difference (light blue) between the two systems. The black curve in (d1) shows an oscillation at frequency ${\omega }_L$ for comparison.

Figure 4

(a) dc component of the Josephson oscillation defined by [${\max }_{0\le t\le t_f}\{\sin ({\theta }_2(t)-{\theta }_1(t)+\delta )\}+{\min }_{0\le t\le t_f}\{\sin ({\theta }_2(t)-{\theta }_1(t)+\delta )\}]/2$ with $t_f=97.9/{\mathrm{\Delta }}_0$. (b) Phase difference between the two systems (blue) and particle numbers of system 1 (red) and system 2 (yellow) after a sufficiently long time ($t_f=97.9/{\mathrm{\Delta }}_0$). The parameters used are $U_R=3.06{\mathrm{\Delta }}_0$, $V=0.02{\mathrm{\Delta }}_0$, and $W=5.11{\mathrm{\Delta }}_0$.