Ergotropic Mpemba crossings in finite-dimensional quantum batteries

TL;DR

This work introduces and analyzes ergotropic Mpemba crossings (EMC) in finite‑dimensional quantum batteries, focusing on qubits and extending to qutrits. The authors derive precise EMC conditions under Markovian noise: for generalized amplitude damping EMC is controlled by the initial coherence ordering, while under anisotropic Pauli noise both coherence and energy matter. They decompose ergotropy into incoherent and coherent parts, showing distinct roles for each contribution in qubits, and reveal that qutrits can exhibit EMC even with purely incoherent ergotropy due to multiple relaxation channels. Extending to non‑Markovian dynamics, they prove EMCs occur in odd numbers and can exhibit quasiergotropic behavior with memory effects. The results connect EMC to the conventional state Mpemba effect, uncovering dimension‑dependent relationships and providing criteria to tailor initial states for faster energy extraction in quantum batteries.

Abstract

The quantum Mpemba effect is a counterintuitive phenomenon in which a state initially farther from equilibrium relaxes more rapidly than one that starts nearer to equilibrium. In the context of finite-dimensional quantum batteries interacting with an environment, we introduce the notion of an ergotropic Mpemba crossing (EMC), defined by the intersection of ergotropy trajectories during the dynamics. For qubit batteries subjected to amplitude damping noise, we derive a condition for the occurrence of EMC in terms of the relative coherence of the initial states and fully characterize the region of state space that exhibits EMC with respect to a fixed reference state. Interestingly, our analysis reveals that under anisotropic Pauli noise, the emergence of EMC is jointly governed by the coherence and the energy of the initial states. To elucidate the physical origin of EMC, we decompose ergotropy into coherent and incoherent contributions and show that, in qubit systems, the coherent component plays a crucial role for EMC, an observation that strikingly does not extend to three-level batteries. Further, by extending our analysis to non-Markovian environments, we demonstrate that, unlike the Markovian case, non-Markovian dynamics can give rise to multiple Mpemba crossings, with the total number of crossings always being odd. Moreover, analyzing the connection between the EMC and the conventional state Mpemba effect reveals that, for qubits, an EMC necessarily entails a state Mpemba crossing while this correspondence breaks down for qutrits, where EMCs may arise without any state Mpemba crossing.

Figures (11)

• Figure 1: Ergotropic Mpemba crossing in a qubit system under ADC. (a) Isoergotropic surfaces for different values of ergotropy are plotted on the Bloch sphere using Eq. (\ref{['eq:iso_ergo']}). This surface never intersects and forms a paraboloid surface in the Bloch sphere (see cf. malavazi2025). (b) Ergotropy (ordinate) against time (abscissa) for different initial ergotropic states. There are pairs of states that show ergotropic Mpemba crossings at $t=t_*$, which depend upon the state and the noise strengths. Also, there exist pairs of states that do not show crossings at any finite time. (c) Decay of ergotropy for isoergotropic states. There is no EMC if the initial state belongs to the isoergotropic surface and the state with highest $m_z$ relaxes faster than the others. Other parameters of the systems are $T=0$ and $\gamma_-=0.1$.
• Figure 2: No EMC region for ADC channel. (a) The non-EMC region is plotted in the $xz$-plane for a corresponding fixed state $\rho_1$. The no-EMC region is denoted by the colored space bounded by the red lines and $\mathcal{E}_o$ denotes the isoergotropic line in the $xz$-plane. (b) and (c) Side and down views of the EMC region where the dots in the sphere are the states that show EMC with a fixed state $\rho_1$ with ergotropy $\mathcal{E}_o$. Other parameters of the systems are $h_z=1$, $T=0$ and $\gamma_-=0.01$.
• Figure 3: Incoherent and coherent ergotropies against time, $t$ for ADC. (a) Incoherent ergotropy, $\mathcal{E}_{inc} (\rho)$ (ordinate) vs time, $t$. The incoherent ergotropy decreases exponentially, which does not show any crossings. (b) Coherent ergotropy, $\mathcal{E}_{c} (\rho)$ against $t$. It increases upto a certain time, $t_{c}$, and then decreases sharply. This increasing behavior makes the delay in total ergotropy, leading to the ergotropic Mempba crossing. Other parameters are same as in Fig. \ref{['fig:xz_plane']}.
• Figure 4: EMC for a qutrit battery under ADC channel. (a) Ergotropy dynamics against time $t$ for two arbitrary pure states to highlight that EMC can be observed in higher-dimensional system. (b) $\mathcal{E}(\rho)$ vs $t$ for two pairs of diagonal initial states, $\rho^{d=3}(0)=p_1\ket{2}\bra{2}+p_2\ket{1}\bra{1}+(1-p_1-p_2)\ket{0}\bra{0}$. Note that unlike two-level quantum battery, the incoherent ergotropy alone can show the ergotropic Mpemba crossing. The set of state parameters chosen are $\{p_1=0.481, p_2=0.103\}$ (light solid line) and $\{p_1=0.485,p_2=0.382\}$ (dark solid line). (c) The EMC region of all the diagonal states are plotted for a fixed diagonal reference state (marked as solid circle). The (blue) dots are the states which show EMC with respect to the fixed state and the (green) solid line represents the isoergotropic surface.
• Figure 5: Dynamics of Ergotropy under non-Markovian noise. For a qubit battery, the EMC can be observed under non-Markovian noisy channel. Interestingly, an odd number of crossings is observed in the transient regime, and these curves never intersect after a certain time, referred to as quasiergotropic Mpemba effect Li2025. All other system parameters are $\lambda=0.03$, $\Delta=0.13$ and $\gamma=0.3$.

Theorems & Definitions (8)

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