Accelerated relaxation and Mpemba-like effect for operators in open quantum systems

Abstract

Quantum Mpemba effect occurs when a quantum system, residing far away from the steady state, relaxes faster than a relatively nearer state. We look for the presence of this highly counterintuitive effect in the relaxation dynamics of the operators within the open quantum system setting. Since the operators evolve under a non-trace preserving map, the trace distance of an operator is not a monotonically decaying function of time, unlike its quantum state counterpart. Consequently, the trace distance can not serve as a reliable measure for detecting the Mpemba effect in operator dynamics. We circumvent this problem by defining a \textit{dressed} distance between operators that decays monotonically with time, enabling a generalized framework to explore the Mpemba-like effect for operators. Applying the formalism to various open quantum system settings, we find that, interestingly, in the single qubit case, only accelerated relaxation of operators is possible, while genuine Mpemba-like effects emerge in higher-dimensional systems such as qutrits and beyond. Furthermore, we demonstrate the existence of Mpemba-like effects in nonlocal, non-equilibrium operators, such as current, in a double-quantum-dot setup. Our results, besides offering fundamental insight about the occurrence of the Mpemba-like effect under non-trace preserving dynamics, open avenues for new experimental studies where quicker relaxation of observables could be of significant interest.

Figures (5)

• Figure 1: (a-c) Monotonic decay of the dressed distance, defined in Eq. \ref{['dressed_norm']}, for single-qubit Pauli operators before (dashed line) and after (solid line) applying a unitary transformation. For (a) and (b) we choose the dissipator in Eq. \ref{['eqn_adlind']} as $\sigma_{+}$ and $\sigma_{-}$ with rates satisfying the detailed balance condition. For (c) the dissipator is chosen as $\sigma_x$. In all cases, only accelerated relaxation is observed, as the initial dressed distance after unitary transformation is always less or equal (such as for unital map) than before applying unitary transformation, as shown clearly in the inset (a)-(c). (d-f) The signature of the respective operator relaxation is reflected in the time evolutions of their expectation values computed for a random initial state $\rho_0$. However, this signature is also reflected for an arbitrary chosen initial state. For (d)-(e), the operators before and after unitary protocol, relaxes to different steady-states, whereas for the unital map with $\sigma_x$ dissipator in (f), the operators before and after unitary relaxes to the same steady state. The simulations are done using parameters $\gamma = 1$, $\omega_0 = 1$, $k_BT = 2$ for dissipators $\sigma_{\pm}$ and $\gamma=1$, $\omega_0=1$ for dissipator $\sigma_{x}$.
• Figure 2: Mpemba-like effect in the time dynamics of dressed distance for Hamiltonian operator for a qutrit setup, where the unitary transformed Hamiltonian i.e., $\widetilde{H}=U^{\dagger}HU$ relaxes much faster to $\widetilde{H}_{\text{ss}}$ than the corresponding relaxation of $H$ to $H_{\text{ss}}$, even though initially $\mathcal{D}_{\rm dd}(\widetilde{H}(0),\widetilde{H}_{\text{ss}})>\mathcal{D}_{\rm dd}(H(0), H_{\text{ss}})$. The corresponding time evolution of the expectation values (inset) of the respective operators also carries the signature of the Mpemba-like effect. Here, the average is computed using a diagonal initial state $\rho_0={\mathrm{diag}} (0.5,0.3,0.2)$. The parameters for the setup are chosen as $\omega_0=1$, $\gamma=0.2$, $k_B T=2.5$.
• Figure 3: (a) Occurrence of the Mpemba-like effect in the relaxation dynamics of the current operator $I$, and its unitarily transformed counterpart $\widetilde{I}$ in the DQD model. Inset: enlarged view of the short-time dynamics highlighting the initial difference between the dressed distance. (b) Corresponding time evolution of the expectation values of $\langle I \rangle$ and $\langle \widetilde{I} \rangle$, computed for a random initial state $\rho_0$, exhibiting distinct slow and fast relaxation dynamics. The simulations are done with parameter values $\epsilon_{\mathrm{d}1}=0.2$, $\epsilon_{\mathrm{d}2}=0.1$, $g=0.05$, $k_BT_1=15$, $\mu_1=0$, $k_BT_2=1$, $\mu_2=0$, $\gamma_1=0.1$, $\gamma_2=0.5$
• Figure 4: (a) Accelerated relaxation of the transformed operator $\widetilde{I}$ following a unitary transformation on the original current operator ${I}$, with both converging to the same steady-state operator at long times. (b) Corresponding operator dynamics reflected in the time evolutions of the respective expectation values, computed for a random initial state $\rho_0$. All simulations are done with parameter values $\epsilon_{\mathrm{d}1}=0.2$, $\epsilon_{\mathrm{d}2}=0.1$, $g=0.05$, $k_BT_1=1.5$, $\mu_1=0$, $k_BT_2=1$, $\mu_2=0$, $\gamma_1=0.1$, $\gamma_2=0.5$.
• Figure S1: Non-monotonic decay of the trace distance $\mathcal{D}_{\mathrm {tr}}$ for unitarily transformed energy current operator $\tilde{I}$ for the double-quantum dot (DQD) setup, as introduced in example (3) of the main text. The inset shows the plot for dressed distance $\mathcal{D}_{\mathrm {dd}}$ involving the same operator, which decays monotonically. Simulations are done with parameter values $\epsilon_{\rm{d1}}=2$, $\epsilon_{\rm{d2}}=1$, $g=0.05$, $T_1=1$, $\mu_1=0$, $T_2=0.1$, $\mu_2=0$, $\gamma_1=0.1$, $\gamma_2=0.5$.