Making rare events typical in Markovian open quantum systems

Large dynamical fluctuations—atypical realizations of the dynamics sustained over long periods of time—can play a fundamental role in determining the properties of collective behavior of both classical and quantum nonequilibrium systems. Rare dynamical fluctuations, however, occur with a probability that often decays exponentially in their time extent, thus making them difficult to be directly observed and exploited in experiments. Here, using methods from dynamical large deviations, we explain how rare dynamics of a given (Markovian) open quantum system can always be obtained from the typical realizations of an alternative (also Markovian) system. The correspondence between these two sets of realizations can be used to engineer and control open quantum systems with a desired statistics “on demand.” We illustrate these ideas by studying the photon emission behavior of a three-qubit system which displays a sharp dynamical crossover between active and inactive dynamical phases.

Figure 1

Three-qubit system with dynamics described by Eqs. (4) and (5) with $\mathrm{\Omega }=V=\gamma =1$, and an initial state with all qubits in the excited state. A photon is emitted whenever the excited state $|\uparrow \rangle$ decays into the ground state $|\downarrow \rangle$. Photons are detected (blue vertical lines) and their statistics $P_t\left(K\right)$, with $K$ being the total amount of emitted photons, is sampled up to time $t=10$. Top: Typical records of emissions show that the average number of emitted photons is $K_a\sim 7$. Bottom: Modified quantum system for which the rare event $K_b=13$ of the original dynamics becomes typical, as shown by ${\tilde{P}}_t\left(K\right)$; green points are obtained by sampling quantum trajectories of the modified dynamics $\tilde{H},\tilde{L}$ discussed in the last section, while the red solid line corresponds to the biased ensemble of probabilities (2) for $s=-0.5$.

Figure 2

(a) Top: Scaled cumulant generating function (SCGF) of the original three-qubit system and its first two derivatives for $V=10$. The first-order derivative displays an abrupt crossover from an inactive phase to an active one. Bottom: Emission time record for $\mathrm{\Omega }=1$; each blue line corresponds to the detection of a photon. Trajectories are obtained performing quantum jump Monte Carlo for the modified dynamics. (b) SCGF of the original system (solid blue line) and the modified system (red dashed line) for $s_0=-2$ and $\mathrm{\Omega }=1$. (c) Top: Original LDF (solid blue line) with average activity ${\langle k\rangle }_0=2\times {10}^{-3}$ and the modified one (red dashed line) with average activity ${\langle k\rangle }_{s_0}=1.75$. Bottom: Corresponding probability distributions for $t=100$, whose maximum is at the average activity in each case.

Figure 3

Coefficients of the operator decomposition of the modified Hamiltonian and jump operators on the Pauli matrix basis as a function of $s$. (a) Real coefficients of ${\tilde{H}}_s$. (b) Absolute value of the complex coefficients of ${\tilde{L}}_k^s$. For $s_0=-11.5$ (active phase), ${\mathrm{\Omega }}^{\prime }\sim 26.5$ and ${\gamma }^{\prime }\sim 4.8$ give an emission rate of $\langle k\rangle \sim 55.4$. For $s_0=12$ (inactive phase), with $\alpha =0.95$, $\beta =0.4$, the typical emission rate is reduced to $\langle k\rangle \sim {10}^{-5}$.

Figure 4

(a) Biasing towards an active dynamics ($s=-0.5$) for $t=10$. Black circles correspond to the numerical probability distribution of the photon emission outcome for the original dynamics of three-qubit system. Red squares stand for the biased probability $P_t^s\left(K\right)$ obtained from the original distribution $P_t\left(K\right)$ while green triangles are the numerical results obtained with the time-dependent Doob dynamics. (b) Analogous results when biasing towards an inactive dynamics ($s=0.5$). (c) Same results biasing towards an active dynamics ($s=-0.5$) for $t=40$. The apparent discontinuous behavior of $P_t^s\left(K\right)$ stems from the undersampling of the far tails of the original distribution $P_t\left(K\right)$ due to the exponential suppression in time of atypical events. Green triangles are the numerical results obtained with the time-dependent Doob dynamics.