Dynamical Phase Transitions and Instabilities in Open Atomic Many-Body Systems

We discuss an open driven-dissipative many-body system, in which the competition of unitary Hamiltonian and dissipative Liouvillian dynamics leads to a nonequilibrium phase transition. It shares features of a quantum phase transition in that it is interaction driven, and of a classical phase transition, in that the ordered phase is continuously connected to a thermal state. We characterize the phase diagram and the critical behavior at the phase transition approached as a function of time. We find a novel fluctuation induced dynamical instability, which occurs at long wavelength as a consequence of a subtle dissipative renormalization effect on the speed of sound.

Figure 1

Nonequilibrium phase diagram for the model in Eq. (3). The solid lines indicate the border of the dynamical quantum phase transition from a condensed to a homogeneous thermal steady state. The dashed lines delimit the region where the condensed state is stable. The black (blue) lines are the numerical results corresponding to average density $n=1.0$ ($n=0.1$). The red line corresponds to the analytical results for $n=0.1$.

Figure 2

Stroboscopic plot of the time evolution of the condensate fraction as a function of the interaction strength $U$, for $J=1.5\kappa$ and $n=1$. For large times it converges to the lower thick solid line. The critical point is $U_c\simeq 4.5\kappa z$. Inset: Near critical evolution for $J=0$, $n=1$, and $U\lesssim U_c$. The early exponential decay ($\times$) is followed by a scaling regime (○) with exponent $\alpha \simeq 0.5$. The final exponential runaway ($+$) is due to a small deviation from the critical point.

Figure 3

Real (dissipative) part of the spectrum ${\gamma }_{\mathbf{q}}$ from the analytical low-density limit for $J=0$, $n=0.1$, and $U=1.0\kappa$. The inset magnifies the parameter region with unstable modes (red solid line). The black solid line is the bare dissipative spectrum ${\kappa }_{\mathbf{q}}$.