Experimental observation and application of the genuine Quantum Mpemba Effect

TL;DR

The paper tackles accelerating quantum relaxation by experimentally realizing the genuine Quantum Mpemba effect (QME) in a Markovian Davies-map setting. It combines a theoretical framework that decomposes the Lindbladian into population and coherence subspaces and uses a unitary pre-processing to minimize overlap with the slowest eigenmode, with a measurement of non-equilibrium free energy $\mathcal{F}_{neq}=\mathrm{Tr}(\hat{H}\hat{\rho}) + \frac{1}{\beta}\mathrm{Tr}(\hat{\rho}\ln\hat{\rho})$ to identify genuine speed-ups via crossing of relaxation curves. In the laboratory, a spin-1/2 system is realized on an NMR platform using $^1H$ as the working qubit and $^{13}C$ as a heat sink, implementing a Davies map through Kraus operators and comparing direct relaxation to a QME-prepared state $\hat{\rho}_{mb}(0)=\hat{U}\hat{\rho}(0)\hat{U}^{\dagger}$; the experiments confirm the predicted crossing and the role of coherence. The QME is then embedded into a quantum Otto refrigerator, where a measured cooling-power gain of up to ~10% is achieved at optimal timing, illustrating a practical advantage for quantum thermal tasks and highlighting the trade-off between speed, energy cost, and efficiency.

Abstract

Coherence is an inherently quantum property that deeply affects microscopic processes, including thermalization phenomena. A striking example is the quantum Mpemba effect (QME), in which a system can exhibit anomalous relaxation, thermalizing faster from a state initially farther from equilibrium than from one closer. Here, we experimentally investigate the genuine QME and observe how the dynamics of a spin-1/2 system interacting with a heat sink can be sped-up to equilibrium. Furthermore, we apply the QME in a quantum Otto refrigerator, thereby increasing its cooling power. This proof-of-concept experiment unveils new practical paths for improving quantum thermal tasks.

Figures (6)

• Figure 1: Schematics for the quantum Mpemba effect. Theoretical simulation of the non-equilibrium free energy dynamics, $\mathcal{F}_{\text{neq}}$, in the cooling protocol employed in our experiment (as described in Fig. \ref{['fig:fig2']} and in the supplementary material). The equilibrium state of this dynamics is the Gibbs state of the Hamiltonian $\hat{\mathcal{H}}_0^{\text{eq}} = -2\pi\hbar\nu_1\hat{\sigma}_z$ at effective temperature $k_{B} T/ h = 4.77 \; \text{kHz}$, where the energy gap is determined by the frequency $\nu_1 = 2\; \text{kHz}$. Here, the following set of initial states is considered, $\hat{\rho}_\theta(0) = \hat{R}_y(\theta)\hat{\rho} (0)\hat{R}_y(-\theta)$, where $\hat{\rho} (0) = 0.3 \dyad{x_{+}} + 0.7 \dyad{x_{-}}$, $|x_{\pm}\rangle$ are the $\hat{\sigma}_x$ Pauli operator eigenstates (with eigenvalues $\pm 1$) and $\hat{R}_y(\theta)$ is the usual SU2 rotation. The angle $\theta$, in the longitudinal axis, is directly related to the projection of the initial state onto the slowest eigenmode of the Lindbladian super-operator, $\langle\!\langle\hat{\xi}_2|\hat{\rho}_{\theta}(0)\rangle\!\rangle$, whereas the time duration of the cooling protocol, $\tau$, is represented in the horizontal axis. The initial states indicated by $\hat{\rho}_{\text{mb}}(0)$ and $\hat{\rho}(0)$ correspond to those in the experimental realization, and $\hat{\rho}_{\text{ps}}(0)$ is the passive state of the mentioned Hamiltonian. We note that, although $\mathcal{F}_{\text{neq}}(\hat{\rho}_{\text{mb}}(0)) > \mathcal{F}_{\text{neq}}(\hat{\rho}(0))$, the former goes to equilibrium faster than the latter, implying an intersection in the cooling curves signature of the genuine quantum Mpemba effect as observed in Fig. \ref{['fig:fig3']} .
• Figure 2: A schematic depiction of the experimental protocols. In the first implementation, the heat exchange protocol is realized directly following the initial state preparation. Conversely, the second implementation introduces an intermediate procedure that transitions the state to the QME state before heat exchange. Blue (red) circles denote rotations along the $x$ ($y$) direction, realized through rf-pulses. Yellow markings indicate periods of free evolution under scalar coupling with duration $\tau\in [0,(2J)^{-1}]$, as described by the interaction Hamiltonian $\hat{H}_{\text{int}}=2\pi\hbar J\hat{\sigma}_z\otimes\hat{\sigma}_z$, where $J=215.1$ Hz is the scalar coupling strength.
• Figure 3: Non-equilibrium free energy variation of the working medium along the heat exchange protocol. A comparison is shown between a system initially prepared in the state $\hat{\rho} (0)$ in red, and in the QME state, $\hat{\rho}_{\text{mb}}(0) = \hat{U} \hat{\rho} (0) \hat{U}^\dagger$, in blue. Circles showcase experimental measurements, whereas solid lines depict the theoretical dynamics of the experimental states. The quantum Mpemba effect is evidenced by the characteristically faster dynamics towards equilibrium presented for the transformed state (in red), despite its initially higher non-equilibrium free energy.
• Figure 4: Trace distance between the evolved state $\hat{\rho}(\tau_2)$ and the target thermal state $\hat{\rho}^{\text{eq,h}}$ during thermalization. The blue (red) curve represents the theoretical cooling of the system prepared with (without) the optional QME preparation step, whereas the markers correspond to experimental measurements. We observe that preparing the QME state leads to a characteristically faster approach to equilibrium. The threshold $\delta$ quantifies the distinguishability between the instantaneous state of the system, $\hat{\rho}(\tau_2)$, and the asymptotic state of the dynamics, $\hat{\rho}_0^{\text{eq,h}}$
• Figure 5: Experimental and theoretical power gain ratio $\mathcal{R}$ seen in Eq. \ref{['eq:pw']} as a function of the threshold $\delta$ depicted in Fig. \ref{['fig:fig4']}. Blue points represent experimental measurements, while the solid red curve shows the theoretical prediction. The cooling power advantage peaks at the point of maximum separation between the thermalization curves in Fig. \ref{['fig:fig4']}, which corresponds to $\tau_2\approx 1.99$ ms.