Spikes in Poissonian quantum trajectories

TL;DR

This work analyzes spikes decorating quantum trajectories governed by Poisson-noise stochastic master equations in the strong measurement limit for a monitored qubit. By reducing the dynamics to a one-dimensional piecewise deterministic Markov process with resetting, the authors derive that the counts of spikes between jumps follow a Poisson distribution with explicit intensities for three non-QND setups: Collapse-Unitary, Collapse-Thermal, and Collapse-Measurement. They provide exact generating functions via Laplace transforms, confirm the results numerically, and place the findings within a general 1D PDMP framework with a decorated spike process. The results advance understanding of spike phenomena beyond Gaussian noise, highlighting a universal spike structure tied to resetting dynamics and offering potential routes for experimental detection in high-rate quantum monitoring.

Abstract

We consider the dynamics of a continuously monitored qubit in the limit of strong measurement rate where the quantum trajectory is described by a stochastic master equation with Poisson noise. Such limits are expected to give rise to quantum jumps between the pointer states associated with the non-demolition measurement. A surprising discovery in earlier work [Tilloy et al., Phys. Rev. A 92, 052111 (2015)] on quantum trajectories with Brownian noise was the phenomena of spikes observed in between the quantum jumps. Here, we show that spikes are observed also for Poisson noise. We consider three cases where the non-demolition is broken by adding, to the basic strong measurement dynamics, either unitary evolution or thermal noise or additional measurements. We present a complete analysis of the spike and jump statistics for all three cases using the fact that the dynamics effectively corresponds to that of stochastic resetting. We provide numerical results to support our analytic results.

Figures (12)

• Figure 1: Collapse-Unitary setup: Typical trajectories generated by Eq. \ref{['eq:sde-angle-1-1-1-1']} for increasing values of $\gamma_1=7,25,10000$ for (a),(b),(c) respectively, with the Rabi frequency tuned as $k_{\gamma_1} = \sqrt{\omega \gamma_1}$. In (a) we see time segments with a downward deterministic flow from $\pi$ and then an upward vertical instantaneous reset to $\pi$ --- this constitutes a pre-spike. With increasing $\gamma_1$, these pre-spikes either form a jump to $\theta \approx 0$ or, more often, develop into spikes. Thus we finally see in (c) a structure consisting of jumps (flows from $\pi$ to $\theta \approx 0$ or instantaneous resets from $\theta \approx 0$ to $\pi$), interspersed with spikes emerging from the state $\theta =\pi$. After a jump to $\theta \approx 0$, the quantum trajectory remains here for a time $\sim 1/(4 \omega)$ during which no spikes occur. In the simulations we took $\omega=1$ and a time step $dt=10^{-5}$ for iterating the Poisson process.
• Figure 2: Collapse-Thermal setup: Typical trajectories generated by Eq. \ref{['eq:EMS-3-1-2-1-1-1-2']} for different values of $\gamma_1=7,25,10000$ for (a),(b),(c) respectively. Here we see that in (a), the trajectory has time segments with pre-spikes which consist of a upward deterministic flow from $q=0$, and a downward reset to the state $q=0$. In the large $\gamma_1$ limit in (c), the pre-spikes develop into sharp spikes and we again see an effective Poisson jump process between the pointer states at $q=0$ and $q=1$, decorated by spikes emerging from the lower branch. The parameters here are $W_\mp = W_\pm = 0.3$, $\eta_1 =1$, $dt = 10^{-5}$
• Figure 3: Collapse-Measurement setup: Typical trajectories from Eq. \ref{['eq:sde-2-2-1-1']} for increasing values of $\gamma_1=7,25,10000$ for (a),(b),(c) respectively. In this case the pre-spikes have the same form as in Fig. \ref{['fig:traj-thermal']} and again we see spikes emerging in the limit of large $\gamma_1$. However, in this case, the spikes are transient, since one has a vanishing transition rate from $q=1$ to $q=0$. The parameters are $\gamma_2 = 1.0, \eta_1 = 1.0,\eta_2 = 0.7, dt = 10^{-5}$.
• Figure 4: Pictorial depiction of the construction of $P_c(n:t,a,b)$: the probability of observing $n$ pre-spikes, given that there are no jumps in between. Quantum trajectories are characterized by quantum jumps (shown in violet) and are accompanied by spikes (shown in brown). The spike statistics occurring between consecutive jumps within a black box of width $|b-a|$ and length $t$ is the focus of this article. It is anticipated that at large measurement strength, these spike statistics follows a Poisson distribution.
• Figure 5: Collapse-Unitary case: (a) Mean (b) Variance of the distribution (conditioned on no-jumps) of the number of spikes (from $\theta_t=\pi$) per unit time in a space-time box $(0,b)\times(0,t)$ plotted against the box edge $b$ for $\gamma=10^6$. The data-points were obtained for $\omega=1$ by averaging over $10^4$ realizations. $\mu_\mathrm{u}$ and $\sigma^2_\mathrm{u}$ are the mean and variance, respectively, of the theoretically-predicted Poisson process in Eq. \ref{['eq:spikes-stat']}. The data in the insets, plotted for a fixed time $t=0.1$, converge to $\mu_\mathrm{u}/t$ and $\sigma^2_\mathrm{u}/t$ at large $\gamma$.