Exact nonequilibrium quantum observable statistics: A large-deviation approach

The exact statistics of an arbitrary quantum observable is analytically obtained. Due to the probabilistic nature of a sequence of intermediate measurements and stochastic fluctuations induced by the interaction with the environment, the measurement outcomes at the end of the system's evolution are random variables. Here, we provide the exact large-deviation form of their probability distribution, which is given by an exponentially decaying profile in the number of measurements. The most probable distribution of the measurement outcomes in a single realization of the system transformation is then derived, thus achieving predictions beyond the expectation value. The theoretical results are confirmed by numerical simulations of an experimentally reproducible two-level system with stochastic Hamiltonian.

Figure 1

Statistics of $\mathrm{\Theta }$'s outcomes. By repeating several times the stochastic evolution of the system and measuring each of the outcomes ${\theta }_{m,j}$ at the final time instant $t_m$, an ensemble of conditional probabilities $p_{{\theta }_{m,j}}$ is obtained, where each of them is computed after the single realization of the system dynamics. Thus, just by counting the occurrence relative frequencies of the $p_{{\theta }_{m,j}}$'s, one can derive the corresponding probability distributions (blue dashed lines). If the number of realizations is relatively small, such distributions do not obey the Gaussian approximation. This means that one has to distinguish between two different statistics for $\theta$'s: one (green solid line) linking the ensemble averages $\langle p_{{\theta }_{m,j}}(\vec{o},\vec{\lambda })\rangle$ (red dots) of the conditional probabilities for each measurement outcome, and the other (orange dotted line)—also called most probable distribution—connecting all the realizations of $p_{{\theta }_{m,j}}$ in correspondence to the maximum value of the conditional probability distributions (blue dots).

Figure 2

Experimental proposal: Exact quantum observable statistics of a 2LS with stochastic Hamiltonian. (a) Theoretical most probable values $p_{{\theta }_1}^{★}$ and $p_{{\theta }_2}^{★}$ as a function of $m$ compared with different values of the probability $p_{{\theta }_1}$, which has been numerically obtained from single realizations of the system transformation. (b) Time behavior of system populations ${\rho }^{\left(11\right)}$ and ${\rho }^{\left(22\right)}$ by numerically solving the stochastic evolution of the 2LS for one realization of $\vec{o}$ and $\vec{\phi }$. (c) Plot of $ln\left(p_{{\theta }_1}\right)/m$ from data of inset (a). (d) Histogram of $p_{{\theta }_1}$, with $m=14\times {10}^3$, over $20\times {10}^3$ repetitions of the system dynamics.