Canonical Quantum Mpemba Effect in a Dissipative Qubit

Abstract

The Mpemba effect, where a hotter system cools faster than a colder one under otherwise identical conditions, has been extensively studied in classical systems. In this work, we present the quantum analogue of the Mpemba effect using a dissipative qubit, which is referred to as the canonical quantum Mpemba effect. We demonstrate that, under the identical conditions, the relaxation dynamics of a qubit initialized in a thermal state with a higher temperature can be exponentially faster than those of a colder thermal state. Strikingly, this acceleration is determined solely by the initial temperature of the system, independent of other parameters. The relaxation is confirmed to be a genuine cooling process via the effective steady state temperature, mirroring its classical counterpart. Last, we propose a practical classical quantum hybrid algorithmic quantum circuit to realize this effect using superconducting qubits experimentally.

Figures (6)

• Figure 1: (a) Schematic of the canonical quantum Mpemba effect. Top panel: traditional quantum Mpemba effect focus on how fast the initial state relax to the steady state. Bottom panel: the canonical quantum Mpemba effect considered in this paper in addition requires both process to be cooling processes. (b) Schematic of the model: a qubit coupled to two independent baths, each inducing a distinct dissipation channel with decay rates $\gamma_{-}$ and $\gamma_{y}$. (c) Schematic of the canonical quantum Mpemba effect implemented on a quantum circuit, demonstrating our proposed classical-quantum hybrid algorithm with a thermal initial state.
• Figure 2: The overlaps $c_k$ between the eigenmodes $l_k$ and the initial thermal state as a function of its temperature $T$. The overlaps $c_2$ (red solid), $c_3$ (blue circles), and $c_4$ (green dashed) are plotted, with $\omega_z=2$ fixed throughout. (a) In the absence of off-diagonal term $\omega_{y}$, the inset shows that $c_2$ and $c_3$ are always zero. (b) For finite off-diagonal term $\omega_{y}=0.01$ and weak dissipation $\gamma=1$, $c_2$ and $c_3$ overlap and are smooth. (c) For finite off-diagonal term $\omega_{y}=0.01$ and strong dissipation $\gamma=5$, the sharp dip of $c_{2}$ at T=11.13 indicates the potential for the canonical quantum Mpemba effect.
• Figure 3: (a) Trace distance (orange solid line) and quantum relative entropy (green solid line) between the steady state and the thermal states as a function of temperature $T$. The dashed line indicates that when $T\approx 5.77$, the distance between $\varrho_{\text{SS}}$ and $\varrho_{\text{th}}[T]$ is minimized; i.e., the steady state is best described by a thermal state at $T\approx5.77$ (b) The inverse critical time $t_{c}^{-1}$ to reach the steady state as a function of the initial-state temperature $T$. The dashed line indicates that initial thermal state at $T\approx11.13$ reaches the steady state the fastest. (c) Trace distance between the time-evolved state $\varrho(t)$ and the steady state $\varrho_{\text{SS}}$ as a function of the dimensionless time $\gamma t$ for different initial temperatures. (d) The effective velocity $\upsilon_{\rm eff}$, i.e., the ratio of initial trace distance to the critical time, as a function of the initial-state temperature $T$. The gray zone with $T\in[5.77,11.13]$ indicates the region where the canonical quantum Mpemba effect occurs.
• Figure 4: The spectrum of Liouvillian $\pazocal{L}$. (a) real part and (b) imaginary part of the eigenvalues $\lambda_{k}$ as a function of the decay strength $\gamma$. The yellow squares, red solid, blue circles and green dashed line denote the eigenvalues $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, and $\lambda_{4}$, respectively. The other parameters are $\omega_{y}=0.01$ and $\omega_{z}=2$.
• Figure 5: Schematic of the quantum collision model to realize Eq. \ref{['Eq:ME']}. The system qubit $S$ interacts sequentially with two types of ancillas, $A_{1}$ and $A_{2}$, which are prepared in $|0\rangle$ state. Each interaction is followed by a partial trace over the ancillas, simulating the effect of the environment.