Universal and nonuniversal probability laws in Markovian open quantum dynamics subject to generalized reset processes

We consider quantum-jump trajectories of Markovian open quantum systems subject to stochastic in time resets of their state to an initial configuration. The reset events provide a partitioning of quantum trajectories into consecutive time intervals, defining sequences of random variables from the values of a trajectory observable within each of the intervals. For observables related to functions of the quantum state, we show that the probability of certain orderings in the sequences obeys a universal law. This law does not depend on the chosen observable and, in the case of Poissonian reset processes, not even on the details of the dynamics. When considering (discrete) observables associated with the counting of quantum jumps, the probabilities in general lose their universal character. Universality is only recovered in cases when the probability of observing equal outcomes in the same sequence is vanishingly small, which we can achieve in a weak-reset-rate limit. Our results extend previous findings on classical stochastic processes [N. R. Smith et al., Europhys. Lett. 142, 51002 (2023)] to the quantum domain and to state-dependent reset processes, shedding light on relevant aspects for the emergence of universal probability laws.

Figure 1

Open quantum reset dynamics and trajectory observables. (a) A quantum trajectory of an open quantum system subject to stochastic resetting is naturally partitioned into different time windows, with extension ${\tau }_i=[t_{i-1},t_i]$, by the times $t_i$ in which reset events occur. The last time window is special since it does not end with a reset event. Given a function of the state $f\left(\nu \right)$, it is possible to construct random variables within the different time windows, as $F_i={\int }_{t_{i-1}}^{t_i}\mathrm{d}uf\left({\nu }_u\right)$. As we show, the probability of observing certain sequences for the observables $F_i$ is universal, i.e., it does not depend on the open quantum dynamics and not even on the chosen function $f\left(\nu \right)$. (b) When considering instead of a state function the number of jump events (or dynamical activity) observed within the different time windows $A_i$, the probability of the same sequences is not universal in general. This is due to the discrete nature of the random variable $A_i$.

Figure 2

Open quantum dynamics with Poissonian resets. (a) Probability $P_1$ for the one-qubit system. The functions of the state $f_{1,2,3}\left(\nu \right)$ (see Sec. 4c) are a measure of coherence in the qubit ($f_1$), the average qubit excitation ($f_2$), and a combination of the two ($f_3=f_1+f_2$). We run 50 000 trajectories for two different parameter regimes, i.e., $\mathrm{\Omega }={\gamma }_1=\mathrm{\Gamma }$ and ${\gamma }_2=2\mathrm{\Gamma }$, and $\mathrm{\Omega }=2\mathrm{\Gamma }, {\gamma }_1=\mathrm{\Gamma }$, and ${\gamma }_2=\mathrm{\Gamma }/2$, with $\mathrm{\Gamma }$ the Poissonian reset rate, ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=\mathrm{\Gamma }$. (b) Probability $P_1$ for the two-qubit system. The functions $f_{4,5,6}\left(\nu \right)$ (see Sec. 4c) are the bipartite entanglement entropy of the system ($f_4$), the average number of excited qubits ($f_5$), and a combination of the two ($f_6=f_4-f_5$). We run 50 000 trajectories for two different parameter regimes, i.e., $\mathrm{\Omega }={\gamma }_1={\gamma }_2=\mathrm{\Gamma }$ and $V=2\mathrm{\Gamma }$, and $\mathrm{\Omega }=\mathrm{\Gamma }, V=\mathrm{\Gamma }/2$, and ${\gamma }_1={\gamma }_2/2=\mathrm{\Gamma }$, with $\mathrm{\Gamma }$ the Poissonian reset rate ${\mathrm{\Gamma }}_i=\mathrm{\Gamma }$ for $i=1,2,3,4$. In both panels, numerical simulations are compared with the prediction shown after Eq. (33), first derived in Ref. [51].

Figure 3

Open quantum dynamics with non-Poissonian resets. (a) Probability $P_1^{\prime }$ for the one-qubit system. The functions $f_{1,2,3}\left(\nu \right)$ are as in Fig. 2. We run 50 000 trajectories for two different parameter regimes, i.e., $\mathrm{\Omega }={\gamma }_1=\mathrm{\Gamma }$ and ${\gamma }_2=2\mathrm{\Gamma }$ (bottom markers) and $\mathrm{\Omega }=\mathrm{\Gamma }/10$ and ${\gamma }_1={\gamma }_2/2=\mathrm{\Gamma }$ (top markers), both with reset rates ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2/2=\mathrm{\Gamma }$. (b) Probability $P_1^{\prime }$ for the two-qubit system. The functions $f_{4,5,6}\left(\nu \right)$ are as in Fig. 2. We run 50 000 trajectories for two different parameter regimes, i.e., $\mathrm{\Omega }=V/2=\mathrm{\Gamma }{\gamma }_1={\gamma }_2=\mathrm{\Gamma }$ (top markers) and $\mathrm{\Omega }=2\mathrm{\Gamma }, V=\mathrm{\Gamma }/4$, and ${\gamma }_1={\gamma }_2=\mathrm{\Gamma }$ (bottom markers). The state-dependent reset rates are, in both cases, ${\mathrm{\Gamma }}_1=\mathrm{\Gamma }, {\mathrm{\Gamma }}_2={\mathrm{\Gamma }}_3=2\mathrm{\Gamma }$, and ${\mathrm{\Gamma }}_4=4\mathrm{\Gamma }$. In both panels, as a reference, we provide the prediction in Eq. (36) for a Poissonian reset with rate $\mathrm{\Gamma }$.

Figure 4

Quantum-jump observables in the one-qubit system. (a) Probability $P_1^{\prime }$ for the one-qubit system for a Poissonian reset. Symbols refer to different functions of the quantum-jump activities. We run 50 000 trajectories for two different parameter regimes, i.e., $\mathrm{\Omega }={\gamma }_1={\gamma }_2/2=\mathrm{\Gamma }$ and $\mathrm{\Omega }={\gamma }_1/2={\gamma }_2/2=\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is the Poissonian reset rate. (b) Probability $P_1^{\prime }$ for the one-qubit system for a non-Poissonian reset. Symbols refer to the different functions of the quantum-jump activities. We run 50 000 trajectories for two different parameter regimes, i.e., $\mathrm{\Omega }={\gamma }_1={\gamma }_2/2=\mathrm{\Gamma }$ and $\mathrm{\Omega }={\gamma }_1/2={\gamma }_2/2=\mathrm{\Gamma }$, with reset rates ${\mathrm{\Gamma }}_1=\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}_2=2\mathrm{\Gamma }$. In both panels, as a reference, we provide the prediction in Eq. (36) for a Poissonian reset with rate $\mathrm{\Gamma }$.

Figure 5

Emergent universality for quantum-jump observables in the weak-reset-rate limit. (a) Probability $P_1^{\prime }$ for the one-qubit system for a Poissonian reset process. Symbols refer to different functions of the quantum-jump activities. We run 5000 trajectories, for which fluctuations are evident, for a parameter regime with $\mathrm{\Omega }={\gamma }_1={\gamma }_2/2=100\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is the (weak) Poissonian reset rate. (b) Same as in (a) but for a non-Poissonian reset process specified by the rates ${\mathrm{\Gamma }}_1=2{\mathrm{\Gamma }}_2=\mathrm{\Omega }/100$. In both panels, as a reference, we provide the prediction in Eq. (36) for a Poissonian reset with rate $\mathrm{\Gamma }$.