Defect production due to time-dependent coupling to environment in the Lindblad equation

Recently, defect production was investigated during nonunitary dynamics due to non-Hermitian Hamiltonian. By ramping up the non-Hermitian coupling linearly in time through an exceptional point, defects are produced in much the same way as approaching a Hermitian critical point. A generalized Kibble-Zurek scaling accounted for the ensuing scaling of the defect density in terms of the speed of the drive and the corresponding critical exponents. Here we extend this setting by adding the recycling term and considering the full Lindbladian time evolution of the problem with quantum jumps. We find that by linearly ramping up the environmental coupling in time, and going beyond the steady-state solution of the Liouvillian, the defect density scales linearly with the speed of the drive for all cases. This scaling is unaffected by the presence of exceptional points of the Liouvillian, which can show up in the transient states. By using a variant of the adiabatic perturbation theory, the scaling of the defect density is determined exactly from a set of algebraic equations. Our study indicates the distinct sensitivity of the Lindbladian time evolution to exceptional points corresponding to steady states and transient states.

Figure 1

The defect production of the gapped system $(\mathrm{\Delta }\ne 0)$ scaled with $\tau$ and ${\gamma }_0=\mathrm{\Delta }$. The solid line depicts the analytical solution from Eq. (26), the red dots and yellow crosses correspond to the numerical solution of Eq. (10) with $\mathrm{\Delta }\tau =100$ and 1000, respectively.

Figure 2

The ${\lambda }_{2,3}$ eigenvalues of the Liouvillian for $\mathrm{\Delta }\ne 0$ and $\gamma \left(t\right)={\gamma }_{0}t/\tau$, with ${\gamma }_0=2\mathrm{\Delta }$ as time evolves from $0\to \tau$ depicted by the arrows on the curves. The red curves denote the eigenvalues for $p=0$, as we can see at the end of the evolution they coalesce. The blue curves denote the eigenvalues whenever $p\ne 0$. If ${\gamma }_0=\mathrm{\Delta }$ the eigenvalues will look like the blue curves for every $p$, signing the absence of EPs.

Figure 3

Comparing $c_1$ from Eq. (30) to the numerical solution of Eq. (10). The solid line depicts the analytical solution, the red dots and yellow crosses correspond to the numerical solution with ${\gamma }_0\tau =100$ and 1000, respectively.

Figure 4

For different momenta, the $c_k$ coefficients increase exponentially with $k$. The figure shows the coefficients in the gapped system, the inset shows the coefficients in the gapless system. From the slope of the curves, we can conclude that in order for the series from Eqs. (24) and (29) to converge the rate has to have magnitude $\mathrm{\Delta }\tau \gg 10$ for the gapped case and ${\gamma }_0\tau \gg 10$ for the gapless case.

Figure 5

The figure shows the coefficients in the gapped system for dynamics that reaches the EP: $\epsilon =2$. For different momenta, the $c_k$ coefficients increase exponentially with $k$ the same way as in for dynamics that does not reach the EP. We can conclude that the presence of an EP in the transient states has no effect on the dynamics.

Figure 6

The momentum-dependent defect density obtained from the numerical solution of Eq. (A5). The main plot shows the dynamics reaching the EP, with scaling in agreement with Eq. (A8), for $\mathrm{\Delta }\tau ={10}^2$ and ${10}^3$. The inset shows dynamics that does not reach the EP and a scaling in agreement with the series expansion solution: solid blue, the numerical solution of Eq. (A2) for $\mathrm{\Delta }\tau ={10}^3$ and $x=0.95$; solid red, the analytical result based on the series expansion coefficients.