Detection of Mpemba effect through good observables in open quantum systems

Abstract

The Mpemba effect refers to the anomalous relaxation of a quantum state that, despite being initially farther from equilibrium, relaxes faster than a closer counterpart. Detecting such a quantum Mpemba effect typically requires full knowledge of the quantum state during its time evolution, which is an experimentally challenging task since state tomography becomes exponentially difficult as system size increases. This poses a significant obstacle in studying Mpemba effect in complex many-body systems. In this work, we demonstrate that this limitation can be overcome by identifying suitable observables that signal rapid relaxation. Moreover, as long as the system equilibrates to a known unique steady-state, it is possible to fully detect the occurrence of quantum Mpemba effect just by measuring the observable for known state preparations. Our approach thus significantly reduces experimental complexity and offers a practical route for observing the quantum Mpemba effect in complex and extended multi-qubit setups.

Figures (3)

• Figure 1: Plot for the detection of the QME through observables for a one-dimensional lattice subjected to bulk dephasing. (a) QME captured through the distance measure $\mathcal{D}_{\rho_t}(t)=S(\rho_{\mathrm{ss}})-S(\rho_t)$, defined as the von Neumann entropy difference between the steady state and the time-evolved state, following exact GKSL dynamics and the dynamics using effective equation involving populations as given in Eq. \ref{['pauli_rate_eqn']}. Here ${\rho}_0 = \frac{1}{L-1}\sum_{i=1}^{L-1}|i\rangle\langle i|$ represents a delocalized initial state and $\widetilde{\rho}_0 = |n_{\rm{mid}}\rangle\langle n_{\rm{mid}}|$, represents a localized initial state in the middle site of the lattice. Even though $\widetilde{\rho}_0$ is far from the steady-state it relaxes faster compared to delocalized state $\rho_0$. (b) and (c): Inferring the existence of QME by tracking the dynamics of good operators, which in this case are local density (populations) for any site except the middle site. The expectation values of $n_1=a_1^{\dagger}a_1$, $n_3=a_3^{\dagger}a_3$, and their absolute deviation from the steady state $|\delta n_1 (t)|$ (inset),$|\delta n_3 (t)|$ (inset) both capture this quicker acceleration. (d) The expectation value of $n_5=a_5^{\dagger}a_5$ and $|\delta n_5|$ does not capture the quick acceleration due to the presence of a node for the SDM at the middle site of the lattice. Here we choose the parameters as $L=9$, $J=1$, and $\gamma=5$.
• Figure 2: Plot for detection of the QME through observable for an $N$-level system evolving under the Davies map. For numerics, we consider here $N=3$. For (a) and (b), we choose diagonal initial states for both $\rho_0$ and $\widetilde{\rho}_0$. $\rho_0$ overlaps with the SDM of the population sector, whereas $\widetilde{\rho}_0$ does not. In addition, $\widetilde{\rho_0}$ is far from the thermal state $\rho_{\mathrm{ss}} = {e^{-\beta H}}/{\mathcal{Z}}$. As a result, QME emerges and crossing in trace distance measure [Eq. \ref{['eq: Trace-distance']}] is observed. To detect this effect through operators, the inset of (a) shows that purely off-diagonal operators, such as $\mathcal{O}_{ij}=1$ for $i \neq j$, do not carry any accelerated relaxation signature as they always remain orthogonal to the population subspace. Whereas (b) shows that the diagonal operators, such as the population of the ground state $\langle n_1(t)\rangle$, shows a clear acceleration for the transformed state $\widetilde{\rho_0}$ (see inset for late time relaxation in a semi-log scale). For (c) and (d), we choose initial states with coherences for both $\rho_0$ and $\widetilde{\rho}_0$. In this case, $\rho_0$ overlaps with the SDM of the Liouvillian, given as complex conjugate pairs, whereas $\widetilde{\rho}_0$ does not. In this case as the insets show, diagonal operators, such as $\langle n_1(t)\rangle$ do not detect accelerated relaxation, whereas purely off-diagonal operators, such as $\mathcal{O}$, defined above, can clearly detect the acceleration. We choose the parameters $\omega_{1 (2) (3)}=0.3 \, (0.7) \, (1)$, $\gamma=0.4$, $T=3$ for plots (a) and (b) and for plots (c) and (d) we choose $\omega_{1 (2) (3)}=1 \, (1.1)\, (1.2)$, $\gamma=1$, and $T=3$.
• Figure 3: Plot for detection of the QME in a boundary-driven $N$-qubit setup with $N=4$. The QME occurring at the quantum state level, as shown in (a) by plotting the trace distance [Eq. \ref{['eq: Trace-distance']}] (inset (i) shows the zoomed version at early times) is now being detected through tracking the dynamics of a local operator $\mathcal{O}=\mathbb{I}\otimes\sigma_x\otimes \mathbb{I}\otimes\mathbb{I}$, as shown in the inset (ii). For the numerics, we choose a random state $\rho_0$ as the initial state and obtain the transformed state $\widetilde{\rho}_0$ by applying a unitary operation using the Metropolis protocol, as discussed in Ref. PhysRevLett.133.140404. The transformed state $\widetilde{\rho}_0$ has a higher distance from the steady state in the trace distance measure [see inset (i)], but as the overlap with the SDM is reduced via the unitary protocol, QME emerges. In (b) we plot the dynamics of the absolute deviation of the operator from the steady state $|\delta \mathcal{O}(t)|$ and a clear accelerated relaxation for the case of transformed initial state $\widetilde{\rho_0}$ is observed. We choose the parameters $g=0.05, \gamma_1=0.2,\gamma_N=0.5,~\epsilon_{1\,(2)\,(3)\,(4)}=1\,(0.8)\,(0.6)\,(0.5),~T_1=5,T_N=2,\mu_1=\mu_N=0$ for the simulation.