Run and Tumble Dynamics of Biased Quantum Trajectories in a Monitored Qubit

Abstract

We investigate the active stochastic dynamics of a qubit subjected to continuous measurement and conditional feedback. The stochastic equation governing the state vector trajectory of the qubit can be mapped, in the high-diffusion limit, to the dynamics of a classical persistent Run-and-Tumble Particle (p-RTP) in a bounded one-dimensional domain. The mapping enables us to use analytical results from classical active matter to derive an approximate non-equilibrium steady-state (NESS) distribution for the monitored quantum system. The competition between the coherent Rabi drive and the measurement-induced feedback leads to a rich NESS phase displaying Zeno--anti-Zeno transition--which is statistically equivalent to the propulsion-induced trapping observed in confined active particles.

Figures (2)

• Figure 1: Classical-Quantum connection (a) Single trajectory of the effective coordinate $x(t)$ of p-RTP model from Eq.\ref{['eq:sde_p-RTP']}. The red-blue markings reflect the current internal state. (b) ) Phase diagram showing the transition from Zeno to bimodal Anti-Zeno stationary distributions obtained from stochastic averaged- Eq. \ref{['eq:BBeqn']} (Avg-QM). The effect of the feedback bias $\gamma'$ acts as a symmetry-breaking drift field. (d) The Mean First Passage Time (MFPT) to the boundary. The maroon lines represent the prediction from the p-RTP model, while black lines represent numerical simulations of the quantum Belavkin SME: Eq.\ref{['eq:BBeqn']} (Bel-QM). The agreement in the high-noise limit validates the effective p-RTP description.
• Figure 2: Analysis of the steady-state statistics for biased continuous quantum monitoring of a qubit. (a) Phase diagrams displaying the mean position $\langle X \rangle$ as a function of the effective diffusion rate $\gamma_{\text{eff}}$ and bias parameter $\gamma'$. computed from Bel-QM (b). Shows the averaged mean position from the effective p-RTP SDE approximation. The stars mark two specific parameter regimes: Point A (magenta, low diffusion $\gamma_{\text{eff}}=D= 0.05$) and Point B (cyan, high diffusion $\gamma_{\text{eff}}=D=10.0$). (c) Steady-state probability density distributions $P(X)$ for the parameters corresponding to points A and B for the p-RTP model and the quantum simulation (Bel-QM). The plots compare the p-RTP simulation (blue histogram), the quantum simulation (green line), the exact solution (red line), and the small-diffusion asymptotic approximation (dashed teal line) as formulas given in \ref{['sec:asym']}. The statistics of the phase becomes accurate in the higher diffusion limit where the quantum effects has been washed out as also evidence from the first passage times Fig. \ref{['fig:maxwell_demon_trajectory']}(c) (d) One-dimensional slices of the phase diagram at fixed diffusion levels ($\gamma_{\text{eff}} = 0.02, 0.7, 2.0$). The curves compare the mean position $\langle X \rangle$ obtained from the tilted Liouvillian evolution (Avg-QM) (purple), the stochastic quantum simulation (Bel-QM) (green), and the hyperbolic transformed p-RTP model Eq.\ref{['eq:RTP']} with $\ell = \pi$ (blue).