Nonequilibrium phase transition of dissipative fermionic superfluids: Case study of multi-terminal Josephson junctions

Abstract

We investigate nonequilibrium dynamics of a triad of fermionic superfluids connected via Josephson junctions, following sudden switch-on of two-body loss in one of the three superfluids. By formulating the dissipative BCS theory for the Lindblad equation, we find that the superfluid order parameter exhibits a phase rotation, thereby giving rise to three types of dc Josephson currents corresponding to different junctions. We demonstrate that, when the tunneling amplitude $V_{31}$ between superfluids without two-body loss is weak, two-step nonequilibrium dynamical phase transition (NDPT) characterized by the vanishing dc Josephson currents occurs: dissipation first induces the NDPT by making one dc Josephson current finite, while further increasing dissipation makes this remaining dc Josephson current vanish. By contrast, when $V_{31}$ is strong, dissipation induces the NDPT in which all dc Josephson currents simultaneously vanish. An analytical study based on a simplified model further supports this observation.

Figures (10)

• Figure 1: Schematic image of a triad of fermionic superfluids coupled via Josephson junctions. The amplitude of the Cooper-pair tunneling between system $\nu\:(=1,2,3)$ and system $\mu\:(=2,3,1)$ is given by $V_{\nu\mu}$. The two-body loss is introduced in system 2 with rate $\gamma$.
• Figure 2: (a), (b), (c) Nonequilibrium dynamics of order parameters $\Delta_{1},\Delta_{2}$ and $\Delta_{3}$ following a sudden switch-on of two-body loss. (d), (e), (f) Phase differences $\Delta\theta_{12},\Delta\theta_{23},\Delta\theta_{31}$ and corresponding Josephson currents $J_{12},J_{23},$ and $J_{31}$ are plotted. In (a), (b), and (c), the real part (blue), the imaginary part (red), and the amplitude (black) of order parameters are shown. The case of weak loss $\gamma/W=0.016$ is displayed. All Josephson currents have finite dc components. The parameters are set to $\:V_{31}/W=0.001,\:V_{12}/W=0.004,\:V_{23}/W=0.008,\:U_{R}/W=0.6,$ and $\Delta_{0}/W=0.2$.
• Figure 3: (a), (b), (c) Nonequilibrium dynamics of order parameters $\Delta_{1},\Delta_{2}$ and $\Delta_{3}$ following a sudden switch-on of two-body loss. (d), (e), (f) Phase differences $\Delta\theta_{12},\Delta\theta_{23},\Delta\theta_{31}$ and corresponding Josephson currents $J_{12},J_{23},$ and $J_{31}$ are plotted. In (a), (b), and (c), the real part (blue), the imaginary part (red), and the amplitude (black) of order parameters are shown. The case of medium loss $\gamma/W=0.025$ is displayed. Only $J_{23}$ has a finite dc component. The other parameters are the same as in Fig. \ref{['V310.05g0.8_image']}.
• Figure 4: (a), (b), (c) Nonequilibrium dynamics of order parameters $\Delta_{1},\Delta_{2}$ and $\Delta_{3}$ following a sudden switch-on of two-body loss. (d), (e), (f) Phase differences $\Delta\theta_{12},\Delta\theta_{23},\Delta\theta_{31}$ and corresponding Josephson currents $J_{12},J_{23},$ and $J_{31}$ are plotted. In (a), (b), and (c), the real part (blue), the imaginary part (red), and the amplitude (black) of order parameters are shown. All Josephson currents does not have finite dc components. The case of strong loss $\gamma/W=0.03$ is displayed. The other parameters are the same as in Fig. \ref{['V310.05g0.8_image']}.
• Figure 5: (a), (b), (c) Nonequilibrium dynamics of order parameters $\Delta_{1},\Delta_{2}$ and $\Delta_{3}$ following a sudden switch-on of two-body loss. (d), (e), (f) Phase differences $\Delta\theta_{12},\Delta\theta_{23},\Delta\theta_{31}$ and corresponding Josephson currents $J_{12},J_{23},$ and $J_{31}$ are plotted. In (a), (b), and (c), the real part (blue), the imaginary part (red), and the amplitude (black) of order parameters are shown. All Josephson currents have finite dc components. The case of medium loss $\gamma/W=0.025$ is displayed and the tunneling amplitude is set to $V_{31}/W=0.005$. The other parameters are the same as in Fig. \ref{['V310.05g0.8_image']}.