Universal relaxation speedup in open quantum systems through transient conditional and unconditional resetting

TL;DR

The paper demonstrates that transient stochastic resetting can universally accelerate relaxation in open quantum systems, including many-body setups with slow metastable dynamics and first-order phase transitions. By applying resets during a finite transient time $t_r$, either unconditionally to a fixed state or conditionally based on measurement outcomes, the authors show acceleration is achievable without fine-tuning of initial states, relying only on macroscopic properties of the target stationary state. The results reveal both weak and strong Mpemba effects in simple qubits and metastable qutrits, and exponential speedups across a Kerr oscillator phase transition and the Dicke model. Conditional resetting further guarantees initial-state-independent acceleration, offering a robust route to rapid state preparation and relaxation in realistic noisy quantum devices. The framework is supported by extensive analytic and numerical analyses and is experimentally accessible across multiple platforms such as cavities, superconducting circuits, and trapped ions.

Abstract

Speeding up the relaxation dynamics of many-body quantum systems is important in a variety of contexts, including quantum computation and state preparation. We demonstrate that such acceleration can be universally achieved via transient stochastic resetting. This means that during an initial time interval of finite duration, the dynamics is interrupted by resets that take the system to a designated state at randomly selected times. We illustrate this idea for few-body open systems and also for a challenging many-body case, where a first-order phase transition leads to a divergence of relaxation time. In all scenarios, a significant and sometimes even exponential acceleration in reaching the stationary state is observed, similar to the so-called Mpemba effect. The universal nature of this speedup lies in the fact that the design of the resetting protocol only requires knowledge of a few macroscopic properties of the target state, such as the order parameter of the phase transition, while it does not necessitate any fine-tuned manipulation of the initial state.

Figures (8)

• Figure 1: Illustration of stochastic resetting protocols and relaxation speedup. (a) For simplicity, we showcase a single qubit system, initialized in the state $\rho(0)$ and evolving under a reset-free Liouvillian $\mathcal{L}_0$. Times $\tau_j$ elapsing between consecutive reset projections, sketched with $R$ boxes, are random. Stochastic resets take place up to an initial transient time $t_r$: $0< t \leq t_r$. After this time, $t>t_r$, the system relaxes to the steady state $\rho_\text{ss}$ of $\mathcal{L}_0$. We consider two resetting protocols: $(b)$ unconditional, where the system resets to a fixed state $|{\psi_r}\rangle$ regardless of the instantaneous state $\rho(t)$, and $(c)$ conditional, where a state-specific operation (a rotation $\mathcal{R}_\theta^x$ gate in the qubit case) is applied based on measurement outcomes (measurement symbol). $(d)$ Both protocols lead to universal acceleration of relaxation to the stationary state. This is quantified by the faster decay in time of the distance measure $\bar{F}(\rho(t),\rho_\text{ss})$ to zero.
• Figure 2: Weak and strong Mpemba effect. Solid lines indicate dynamics without resets, while dashed and dashed-dotted lines represent dynamics with resets to different states $|{\psi_r}\rangle$ (see labels) up to time $t_r$ (indicated by arrows). For $t>t_r$, the system follows the reset-free dynamics. Initial states are reported at the top. (a) Modified overlap $|a_2^r(t)|$ for a qubit coupled to a thermal bath (sketched in the inset). (b) Corresponding distance measure $\bar{F}(\rho(t),\rho_{\text{ss}})$, illustrating WME where $a_2^r(t)$ reaches a minimum under resetting. (c) Modified overlap $|a_2^r(t)|$ for the qutrit (see inset). (d) Distance measure $\bar{F}(\rho(t),\rho_{\text{ss}})$, depicting SME where $a_2^r(t_r)=0$ for multiple reset states, yielding exponential speedup toward the steady state. Parameters for qubit (a,b) are $\omega/\kappa=1,n_{th}/\kappa=1$, and for qutrit (c,d) are $\Omega_1/\kappa=1,\Omega_2/\kappa=0.1,n_{th}/\kappa=2$, where $\gamma_\downarrow=\kappa (1+n_{th})$ and $\gamma_\uparrow=\kappa n_{th}$.
• Figure 3: Mpemba effects across 1st-order transition. (a) Sketch of driven dissipative Kerr oscillator, characterized by a drive strength $\Omega$, Kerr nonlinearity $U$, and a dissipation rate $\kappa$. (b) This model exhibits a 1st-order dissipative phase transition in the thermodynamic limit $N\rightarrow \infty$ of an infinite number of cavity modes. In panel (c), we analyze the case $\bar{\Omega}/\gamma=1$ with $\bar{\Omega}=\Omega/\sqrt{N}$, where the system relaxes to a stationary state devoid of cavity modes. One finds SME for the reset state $|{\psi_r}\rangle=|{0}\rangle$, while a WME emerges for other reset states. The initial state is taken as $\rho(0)=|{10}\rangle\langle{10}|$. In panel (d), we consider the case of $\bar{\Omega}/\gamma=3$, where a nonvanishing stationary occupation of the cavity is obtained. A WME is observed for any reset state with a nonzero Fock occupation. The initial state is $\rho(0)=|{0}\rangle\langle{0}|$. Lines in (c) and (d) follow the same convention as in Fig. \ref{['fig:fig2']}. Parameters are fixed as $N=10$ and $\Delta/\gamma=\bar{U}/\gamma=10$ where $\bar{U}=UN$. Arrows indicate the time $t_r$ at which resetting is switched off.
• Figure 4: Mpemba effect via conditional resetting. In panels (a) and (b), we show the emergence of SME and WME, respectively, in a qubit. Panels (c) and (d) show the SME and WME in a qutrit, respectively. Different initial states are reported, each labeled, while arrows mark the time $t_r$ up to which resets are performed. Lines follow the same convention as in Figs. \ref{['fig:fig2']} and \ref{['fig:fig3']}.
• Figure S1: Mpemba effect in a qutrit system with unconditional resetting to a mixed state infinite-temperature state $\mu_r=\mathbb{I}/3$. In panel (a) we plot the coefficient $|a_2^r(t_r)|$\ref{['eq:akr']} as a function of the rescaled time $\kappa t$, with $\kappa$ the dissipation rate of $|{0}\rangle-|{1}\rangle$ line. We take various initial states $|{\psi(0)}\rangle$ (dashed lines). For $|{\psi(0)}\rangle=|{2}\rangle$, the overlap $|a_2^r(t_r)|=0$ and strong Mpemba effect is found. For the other initial states $|{0}\rangle$ and $|{1}\rangle$, the overlap $|a_2^r(t_r)|$ is minimized, but it remains nonzero. This reflects into the occurrence of the weak Mpemba effect. In panel (b), we plot the distance measure $\bar{F}(\rho(t),\rho_{\text{ss}})$\ref{['eq:bar_fidelity']} and \ref{['eq:fidelity']} for initial states considered in (b). For all the initial states, relaxation is significantly accelerated compared to the reset-free dynamics (black solid line), which shows metastable, very slow, convergence to stationarity. The parameter values are as follows: $\Omega_1/\kappa=1,\Omega_2/\kappa=0.1,n_{th}/\kappa=2$ and $\Gamma=1/2$. Arrows indicate the time $t_r$ when stochastic resets are turned off.