Metastability in an open quantum Ising model

We apply a recently developed theory for metastability in open quantum systems to a one-dimensional dissipative quantum Ising model. Earlier results suggest this model features either a nonequilibrium phase transition or a smooth but sharp crossover, where the stationary state changes from paramagnetic to ferromagnetic, accompanied by strongly intermittent emission dynamics characteristic of first-order coexistence between dynamical phases. We show that for a range of parameters close to this transition or crossover point the dynamics of the finite system displays pronounced metastability, i.e., the system relaxes first to long-lived metastable states before eventual relaxation to the true stationary state. From the spectral properties of the quantum master operator we characterize the low-dimensional manifold of metastable states, which are shown to be probability mixtures of two, paramagnetic and ferromagnetic, metastable phases. We also show that for long times the dynamics can be approximated by a classical stochastic dynamics between the metastable phases that is directly related to the intermittent dynamics observed in quantum trajectories and thus the dynamical phases.

Figure 1

(a) Real part of the spectrum of the master operator of the open quantum Ising model defined in Sec. 3, for size $N=7$, with spin interaction $V/\kappa =250$ and transverse field $\mathrm{\Omega }/\kappa =50$, where $\kappa$ is decay rate. Spectrum is plotted in units of $\kappa$. The two eigenvalues with largest real parts, ${\lambda }_{1,2}$, highlighted in gray, correspond to the only eigenmodes that contribute to the dynamics at long times. The large gap to the next eigenvalue, ${\lambda }_3$, causes metastability in the dynamics. (b) The metastable manifold (MM) induced by the spectral structure of (a). System states are represented by the average total magnetization per spin in the $x$ and $z$ directions. The MM (gray line) is bounded by two extreme metastable states ${\tilde{\rho }}_1$ (purple dot) and ${\tilde{\rho }}_2$ (green dot). The evolution from an initial state ${\rho }_{\uparrow }$ of all spins up (blue curve) undergoes a fast evolution for $t\ll {\tau }^{\prime }=-1/{\lambda }_3$ onto the MM. After a long period of apparent stationarity (${\tau }^{\prime }\ll t\ll \tau =-1/{\lambda }_2$), the states on the MM evolve towards the true stationary state ${\rho }_{\mathrm{ss}}$ (red circle) for $t\gg \tau$.

Figure 2

Illustration of the two-basin analogy of metastability when the metastable manifold is one dimensional. The figure represents a one-dimensional Landau free energy. The two eMSs are analogous to the two free-energy basins corresponding to two phases of the system, and their free-energy difference is analgous to the logarithm of their relative probabilities. In the thermodynamic case, the time scale for connecting the basins is the exponential of the free-energy barrier (divided by temperature—the usual Arrhenius law for thermal escape). In this sense, since in our dynamical case the time scale is controlled by the gap, the log of the gap is analogous to the barrier. See the main text for discussion.

Figure 3

[(a)–(c)] Flow induced on cross sections of the Bloch ball by the dynamics in Eq. (14) for parameters $V/\kappa =10, \mathrm{\Omega }/\kappa =4.4$ where the system is bistable. The color of the line indicates the magnitude of the component perpendicular to the plane. The green and purple dots correspond to the two stationary solutions of the mean-field approximation projected onto the planes through the Bloch sphere. [(d)–(f)] Basins of attraction of the two mean-field stationary solutions, with the indicated values of $〈M_z〉$, at three points in the bistable regime with parameters $V/\kappa =10$ and $\mathrm{\Omega }/\kappa =4.4,5.8,7.0$, respectively, marked by black crosses in panel (g). (g) Mean-field phase diagram: ${\overline{m}}_{\infty }$, as a function of model parameters. Spinodal lines separating one-solution from two-solution regions are indicated by white dashed lines.

Figure 4

(a) The ratio of the real parts of the second and third eigenvalues ${\lambda }_3/{\lambda }_2$, plotted over a region of the parameter space $0\le V\le 250$ and $0\le \mathrm{\Omega }\le 100$. (b) The inverse gap $-1/{\lambda }_2$ plotted over the same region. (c) A cross section of plots (a) and (b) for $V=200$. The dashed (purple) line is the ratio while the solid (blue) line is the inverse gap. (d) The scaling of the maximum value of the ratio (purple triangles) and inverse gap (blue circles) in this range of parameters with system size.

Figure 5

(a) Magnetization of the steady state $m_{\mathrm{ss}}$. The approximate boundary of the metastable region is indicated by the dashed white line. (b) The average $z$ magnetization $\overline{m}$ in the metastable regime. In this and the following panels we only show the area where metastability holds. [(c) and (d)] Expected $z$ magnetizations $m_i=\text{Tr}\left(M_z{\tilde{\rho }}_i\right)$ and [(e) and (f)] purities ${\gamma }_i=\text{Tr}\left({\tilde{\rho }}_i^2\right)$ of the two metastable phases.

Figure 6

Results calculated for parameters of $V/\kappa =250$ and $\mathrm{\Omega }/\kappa =50$. (a) Time evolution of the average $z$ magnetization, $〈M_z〉$, for the initial state ${\rho }_{\uparrow }$. The $z$ magnetizations of the metastable phases ${\tilde{\rho }}_{1,2}$ are represented by the dashed lines. (b) Same but for the initial state ${\rho }_{\to }$. [(c) and (d)] Time evolution of the average $x$ magnetization, $〈M_x〉$, for ${\rho }_{\uparrow }$ and ${\rho }_{\to }$. The average metastable values are given by dashed lines. (e) Histogram of the probability of finding ${\rho }_{\uparrow }, {\rho }_{\to }$, and ${\rho }_{\mathrm{ss}}$ in each eMS during the metastable regime, see Eq. (5) and (6). (f) Time-averaged autocorrelation function $C\left(t\right)$ of the projective measurement related to $M_z$ in ${\rho }_{\mathrm{ss}}$ (see the main text). In panels (a)–(d) and (f) the time is in log scale, with the two arrows indicating the time scales ${\tau }^{\prime }$ and $\tau$. The exact dynamics is plotted as solid curves, and the effective dynamics as dash-dotted lines.

Figure 7

(a) A 2D histogram of the average magnetization $m$ and activity $k$ of time bins of length $t_{\text{bin}}=6{\tau }^{\prime }$, for parameters $V/\kappa =250, \mathrm{\Omega }/\kappa =50$, and $N=7$. The arrows indicate the expectation values for ${\tilde{\rho }}_1$ and ${\tilde{\rho }}_2$, respectively, with the decay $e^{t_{\text{bin}}{\lambda }_2}$ of $R_2$ taken into account. The white dashed line indicates the relation $\langle k\rangle /N\kappa =\langle M_z\rangle +\frac{1}{2}$. [(b) and (c)] Sample trajectories for two different sets of parameters in the metastable region, $V/\kappa =250$ with $\mathrm{\Omega }/\kappa =20$, 50, for $N=7$ and time in linear scale. The first and seconds rows show the space- and time-resolved photon emissions and the corresponding magnetization of the state over time, respectively, for a sample QJMC trajectory. The third row shows a sample classical trajectory, jumping between the two metastable phases. The insets on the third row show sketches of the double-well potentials of free energy leading to analogous classical evolution between the metastable states of two wells.