Coherence-Controlled Quantum Zeno Dynamics from Exact Reset Maps

Abstract

We develop an exact framework for quantum Zeno and anti-Zeno dynamics in a broad class of open systems, whose microscopic Hamiltonians are quadratic in bosonic or fermionic operators. We treat the environment through an exact stroboscopic resetting scheme acting at the level of the single-particle density matrix (SPDM). Within this framework, we consider two cases: a repeated-interaction (RI) protocol, in which the environment block is rethermalized and all system-environment coherences are erased after each step, and an evolving-correlation (EC) protocol, in which only the environment block is reset while system-environment coherences are preserved. For RI, we derive a general short-time Zeno law for the survival probability of a single-particle excitation and show that the corresponding decay rate scales linearly with the reset interval, implying Zeno freezing in the limit of infinitely frequent resets. Beyond the short-time regime, we formulate the RI dynamics directly in terms of the exact one-cycle propagator, which allows us to analyze finite-$τ$ anti-Zeno windows without additional approximations. For EC, we obtain a continuous-reset description in which the kept single-particle correlators obey a finite-dimensional linear differential equation. In this case the drift in the system block remains finite in the frequent-reset limit, so strict freezing is absent. We illustrate these results for a single fermionic level coupled to a semi-infinite tight-binding chain acting as a structured bath. Our results identify coherence erasure versus coherence retention as the key factor controlling the reset-induced Zeno physics.

Figures (3)

• Figure 1: Schematic comparison between the repeated-interaction (RI) and evolving-correlation (EC) protocols. In each stroboscopic step the global quadratic system $S+E$ evolves unitarily under $U(\tau)$, followed by a reset of the environment block $\rho_{EE}$ to a fixed reference state $\rho_{EE}^{(0)}$. In RI (panel a), system-environment coherences $\rho_{SE}$ and $\rho_{ES}$ are erased after each reset. In EC (panel b), $\rho_{SE}$ and $\rho_{ES}$ are retained and evolve continuously during the dynamics.
• Figure 2: (a) Exact repeated-interaction (RI) decay rate for a single fermionic level coupled to a semi-infinite tight-binding chain, illustrating strict Zeno freezing. The plotted quantity is the exact map-extracted rate $\Gamma_{\mathrm{eff}}(\tau)=-\tau^{-1}\ln |U_{00}(\tau)|^2,$ shown as a function of the reset interval $\tau J$ for three level positions: $\omega_0=0$, $\omega_0=0.8J$, and $\omega_0=3J$. In all cases the rate vanishes as $\tau\to0$, demonstrating strict Zeno suppression in the RI protocol. The inset shows the universal short-time collapse predicted by Sec. III: after normalization by $A_T J$, with $A_T=\sum_{x\in E}|M_{0x}|^2=t_{\rm c}^2$ for the present site-basis tight-binding model, the curves collapse onto the linear law $\Gamma_{\mathrm{eff}}(\tau)/(A_TJ)=\tau J$. Numerical parameters: $J=1$, $t_{\rm c}=0.2$, truncated bath size $N_{\rm b}=400$. (b) Exact finite-$\tau$ anti-Zeno behavior. The exact map-extracted decay rate $\Gamma_{\mathrm{eff}}(\tau)$ is shown for outside-band level positions $\omega_0=3J$, $4J$, $5J$, and $6J$, i.e. for levels lying above the tight-binding bath band $[-2J,2J]$. The dashed line is the universal small-$\tau$ Zeno asymptote, $\Gamma_{\mathrm{eff}}(\tau)\simeq t_{\rm c}^2 \tau,$ valid in the strict frequent-reset regime. All curves follow this Zeno law as $\tau\to0$, but at finite reset interval they develop nonmonotonic structure. The marked point identifies the clearest anti-Zeno peak.
• Figure 3: Dynamical design map for the repeated-interaction (RI) protocol. The plotted quantity is the exact map-extracted decay rate $\Gamma_{\mathrm{eff}}(\tau,\omega_0) = -\frac{1}{\tau}\ln |U_{00}(\tau)|^2,$ shown over the parameter plane $(\tau J,\omega_0/J)$. The dashed horizontal lines mark the tight-binding bath band edges, $\omega_0/J=\pm2$. The dark region near $\tau\to0$ corresponds to strict Zeno suppression, since $\Gamma_{\mathrm{eff}}(\tau)\to0$ in the frequent-reset limit. The white curve is a numerical ridge of finite-$\tau$ local maxima of $\Gamma_{\mathrm{eff}}$, indicating where anti-Zeno enhancement is strongest. Thus the same exact RI map displays both the strict Zeno regime and the finite-$\tau$ anti-Zeno sector within a single parameter-space representation. Numerical parameters: $J=1$, $t_{\rm c}=0.2$, $\tau J\in[0.01,2.0]$, $\omega_0/J\in[-6,6]$. The semi-infinite bath is represented numerically by a large truncated open chain.