Role reversal in quantum Mpemba effect

TL;DR

The paper addresses whether the quantum Mpemba effect (QME) can occur in open quantum systems and how its relaxation ordering can reverse under parameter changes. It derives a sufficient criterion for QME using the $l_1$-norm differential coherence in a dissipative Dicke model and introduces the concept of role reversal, where altering Hamiltonian parameters inverts which initial state relaxes faster. The authors validate these phenomena across multiple diagnostics—differential coherence, differential entanglement (via logarithmic negativity), and trace-distance to the steady state—often employing adiabatic elimination to obtain an effective spin model. The results demonstrate tunable relaxation pathways, with parameter choices (e.g., coupling $g$, frequencies, and coherence phase $eta$) enabling role reversals, offering insights for quantum control in dissipative settings.

Abstract

We investigate the quantum Mpemba effect in a dissipative Dicke model, which consists of a spin-1/2 ensemble coupled to a bosonic mode, which in turn is coupled to a bosonic bath. We derive a sufficient criterion for occurrence of the quantum Mpemba effect, characterized by quantum coherence, in this model. We introduce the phenomenon of role reversal in the Mpemba effect, wherein changes in the system parameters invert the relaxation ordering of a given pair of initial states that exhibit the Mpemba effect, causing the faster-relaxing state to become slower and vice versa. We find the existence of role reversal in Mpemba effect for this Dicke model using different relaxation measures, including differential quantum coherence and entanglement, and trace distance, between the time-evolved and steady states.

Figures (4)

• Figure 1: Quantum Mpemba effect characterized by differential coherence. We plot a heat map of $l_{1}(\rho'(t))-l_{1}(\rho(t))$ as a function of $t$ and $\beta$. We consider a qubit state represented in Bloch sphere representation, with $r_x = r_y = 0.4$ and the model parameters $\Omega= \omega = k =1$, $g=3$. We find that $l'_{1}(\rho(t))-l_{1}(\rho(t))>0$, for all finite $t \in (0,\infty)$ where $\beta \in (0, \frac{\pi}{4}] \cup (\frac{\pi}{2},\frac{3\pi}{4}] \cup (\pi,\frac{5\pi}{4}] \cup (\frac{3\pi}{4},\frac{7\pi}{4}]$, manifesting that the initial state obtained after the action of the unitary operator $U(\beta)$ decoheres more slowly towards the zero coherence state. Here, the time is described in the unit of $\frac{\Omega}{\hbar}$ whereas $\beta$ is dimensionless. At sufficiently large time, i.e., $t \rightarrow \infty$, $l'_{1}(\rho(t))-l_{1}(\rho(t)) = 0$. In the inset, we plot $l_1$-norm measure of coherence with time $(t)$, for $\beta=0.65 \pi$, and we observe that the coherence of $\rho'(t)$ reaches zero slower in comparison to that of $\rho(t)$, exhibiting QME.
• Figure 2: Schematic representation of the quantum Mpemba effect and role reversal. Let us consider a situation when Alice and Bob have two cups of tea, each with the same amount. The initial (at $t=0$) temperature of the tea in both cups is same, i.e., $80^{\circ} C$ and both the cups are in the same environment with temperature $25^{\circ} C$. Alice and Bob choose their cups with two different shapes with different opening areas. The tea in the flat cup reaches the environment temperature $25^{\circ} C$ faster than the other. We define this situation as QME in our context. Let us consider two situations now. Panel (a) depicts a situation when Alice chooses a flat cup while Bob does not. In this situation, if the tea Alice has spends $t_{Alice}$ amount of time while the tea Bob has spends $t_{Bob}$ amount of time before reaching the environment temperature $25^{\circ} C$, then $t_{Bob} > t_{Alice}$, exhibiting the QME according to the definition. In contrast, Panel (b) shows another situation when Bob chooses a flat cup while Alice does not, unlike the previous situation. In this situation, if the tea Alice has spends $t'_{Alice}$ amount of time while the tea Bob has spends $t'_{Bob}$ amount of time before reaching the environment temperature $25^{\circ} C$, then $t'_{Bob} < t'_{Alice}$. This also manifests the QME, but the roles of the two dynamics have been reversed. Here, the temperature, choices of cups, and the tea Alice and Bob possess, are analogous to the distance or quantity $\mathcal{M}$, the tuning parameters $\{\alpha_i\}$ and time-evolved pair of the initial states $\{\rho_0,\rho'_0\}$, respectively. Thus, Panel (a) and Panel (b) exhibit two situations of the QME with reversed roles w.r.t. each other.
• Figure 3: Mpemba effect and its role reversal, characterized by differential quantum coherence and entanglement. Panels (a) and (b) correspond to the dynamics of differential quantum coherence, keeping $N=1$, for the pair of initial states $\{\rho_0,\rho'_0\}$ for different parameter values of the Hamiltonian, i.e., $\{g=1$, $\omega = 0.1\}$ and $\{g=4.5, \omega = 1\}$ respectively. The other parameters of the Hamiltonian, the parameters of the initial state $\rho_0$ and the coherence-preserving unitary operation $U(\beta)$ are chosen as $r_{x}=r_{y}=0.4$, $\Omega = 1$, $k = 1$, $\beta = 0.33 \pi$. The smooth red and blue lines, in both sub-figures (a) and (b), correspond to the dynamics of coherence of $\rho(t)$ and $\rho'(t)$ corresponding to the initial states $\rho_0$ and $\rho'_0$, respectively. It is worth noting that both of the panels (a) and (b) manifest the QME characterized by differential quantum coherence, with reversed roles w.r.t. each other. Similarly, the panels (c) and (d) correspond to the dynamics of entanglement for the pair of evolved states $\{\rho(t),\rho'(t)\}$ corresponding to the initial states $\{\rho_0,\rho'_0\}$ respectively, for different parameter values of $\omega_{B}$, i.e., $8.88$ and $2.79$ respectively. The other parameters of the Hamiltonian are $N_{A}=N_{B}=3$, $\Omega_{A} = 3$, $g_{A} = 1$, $k_{A}=1$, $\omega_{A}=1$, $\Omega_{B} = 2.5$, $g_{B} = 3.5$ and $k_{B}=3$. The smooth red and blue lines, in both sub-figures (c) and (d), correspond to the dynamics of entanglement corresponds to $\rho_0$ and $\rho'_0$ respectively. Note that both of the panels (c) and (d) illustrate the QME characterized by differential entanglement, with reversed roles w.r.t. each other. In all the sub-figures, the time is described in the unit $\frac{\Omega}{\hbar}$.
• Figure 4: Mpemba effect and role reversal characterized by trace-distance -based measure. We plot the trace distance of time-evolved states $\{\rho(t),\rho'(t)\}$ at any time $t$ from the steady state $\rho_{ss}$. We consider $N=25$, $\Omega = 3, k = 1,$ and $g=1$. In panel (a), when we choose $\omega = 1,$$\rho'_0$ evolves faster than $\rho_0$ towards the steady state $\rho_{ss}$. Furthermore, we see that the two curves of the trace distance between time-evolved states and steady state corresponding to $\{\rho(t),\rho'(t)\}$, obtained in panel (a), interchange their roles for different value of $\omega$, i.e., $\omega = 0.1$ for the same pair of initial states $\{\rho_{0},\rho'_{0}\}$. Here the smooth red and blue lines correspond to the dynamics of trace distance of $\rho_0$ and $\rho'_0$ from the steady state in both the sub-figures (a) and (b). Therefore, panel (b) manifests the QME and role reversal by panel (a), characterized by a trace-distance -based measure. In all the sub-figures, the time is described in the unit $\frac{\Omega}{\hbar}$.

Theorems & Definitions (1)

• Theorem 1