Intermittency and dynamical Lee-Yang zeros of open quantum systems

We use high-order cumulants to investigate the Lee-Yang zeros of generating functions of dynamical observables in open quantum systems. At long times the generating functions take on a large-deviation form with singularities of the associated cumulant generating functions—or dynamical free energies—signifying phase transitions in the ensemble of dynamical trajectories. We consider a driven three-level system as well as the dissipative Ising model. Both systems exhibit dynamical intermittency in the statistics of quantum jumps. From the short-time behavior of the dynamical Lee-Yang zeros, we identify critical values of the counting field which we attribute to the observed intermittency and dynamical phase coexistence. Furthermore, for the dissipative Ising model we construct a trajectory phase diagram and estimate the value of the transverse field where the stationary state changes from being ferromagnetic (inactive) to paramagnetic (active).

Figure 1

Thermodynamics of trajectories and the $s$-ensemble formalism. (a) Time traces of quantum jumps. In the active phase, a large number of quantum jumps occur per unit time. When the system is intermittent, it switches randomly between the active phase and the inactive phase. (b) Large deviation function $(1/t)lnP(K_t,t)$ at long times. The tails of the distribution are determined by the fluctuations in each of the two phases. The intermediate region is governed by the random switching between the two phases. (c) Dynamical activity (number of quantum jumps) per unit time as a function of the counting field $s$. By tuning the counting field, the system may be driven through a phase transition (here close to $s=0)$, where it changes from being in an active phase with a large dynamical activity to an inactive phase with a low dynamical activity. Here we show a smeared first-order transition. (d) Dynamical Lee-Yang zeros in the complex plane of the counting field. With time, the dynamical Lee-Yang zeros move towards the complex values of the counting field, where the dynamical free energy (the CGF) displays singular features corresponding to a phase transition. The phase transition is smeared when it occurs for complex values of the counting field, and it only becomes sharp as the transition points move on to the real axis.

Figure 2

Driven three-level system. (a) Two lasers with frequencies ${\Omega }_1$ and ${\Omega }_2$ are on resonance with transitions between the ground state $|0\rangle$ and the excited states $|1\rangle$ and $|2\rangle$. The system decays from the state $|1\rangle$ under photon emission with rate $\kappa$. (b) Average dynamical activity $k\left(s\right)$ as a function of the counting field $s$. For ${\Omega }_1=1,\kappa =4$, and ${\Omega }_1\gg {\Omega }_2$, the dynamical activity exhibits a smooth first-order transition around $s=0$ between an active and an inactive phase. (c) Phase transition points in the complex plane of the counting field (in the long-time limit). As ${\Omega }_2$ is decreased, the phase transition points approach the origin. (d) We now fix ${\Omega }_2=0.15$ and consider the dynamical activity along the horizontal lines in the complex plane of the counting field. A phase transition point is marked with an asterisk. The real and imaginary parts of the dynamical activity calculated along the horizontal lines are shown in the following panels. (e) The real part of the dynamical $\text{R}\text{e}\left[k\right(s\left)\right]$ becomes sharper as the critical point is approached and eventually develops a discontinuity. (f) The imaginary part of the dynamical activity $\text{I}\text{m}\left[k\right(s\left)\right]$ diverges as the phase transition point is approached.

Figure 3

Lee-Yang zeros and high-order cumulants. (a) Dynamical Lee-Yang zeros extracted from the high-order cumulants of the dynamical activity. The leading pair of dynamical Lee-Yang zeros converge with time to the phase transition points (marked with asterisks) in the complex plane of the counting field. The parameters are ${\Omega }_1=1,\kappa =4$, and ${\Omega }_2=$ $0.1,0.2$, and $0.3$. (b, c, d) Numerically exact results for the cumulants of order $n=9$ (full lines) as functions of time together with the approximation given by Eq. (28) (dashed lines) based on the extracted pair of dynamical Lee-Yang zeros in panel (a).

Figure 4

Dissipative Ising model. (a) The system consists of $N$ spins in the transverse field $\lambda$ and nearest-neighbor interaction $V$. The spins decay with rate $\kappa$. (b) Motion of the leading pair of dynamical Lee-Yang zeros for different values of the transverse field $\lambda$. The other parameters are $\kappa =0.1,V=10$, and $N=12$. The Lee-Yang zeros converge to the complex values of the counting field, where the dynamical free energy displays singular features corresponding to a phase transition. (c) Phase diagram showing the real part of the trajectory transition points for different values of $\lambda$. The line separates the active and the inactive phases. We also show a density plot of $k\left(s\right)$ computed by exact diagonalization of $\mathbb{L}\left(s\right)$ for systems of size $N=4$. (d, e, f) Numerically exact results for the cumulants of order $n=9$ (full lines) together with the approximation given by Eq. (28) (dashed lines) based on the extracted pair of dynamical Lee-Yang zeros in panel (b).