Fig. 1: Comparison for classic and quantum Mpemba effect.
a The ME can be understood in an intuitive way: the amplitude of the overlap of the initial state with the slowest decaying mode (SDM) depends on the initial temperature in a nonmonotonic way. The sME appears when the overlap with the SDM vanishes. b Weak ME: If an initial high temperature state has a smaller SDM amplitude than that of the lower temperature state, it can reach the thermal equilibrium faster. Strong ME: the system reaches equilibrium at an exponentially faster rate. No ME: the initial high temperature state has a larger overlap with the SDM and thus reaches the equilibrium slower. c By applying a unitary operation, one can realize an initial sME state and approaches the stationary state with a faster rate. d The energy levels for observing the sME (with κ2 ≪ κ1). e The overlap ∣c1∣ of a rotated initial random state with the SDM as a function of the rotation angle s. f The distance between the time-relaxed state ρ(t) and the stationary state ρss for different initial states: \(\left\vert 0\right\rangle\) (blue), \(\left\vert 2\right\rangle\) (green) and \(\left\vert {{{\rm{sME}}}}\right\rangle\) (red), respectively. The initial sME state starts with a longer distance from ρss than the initial state \(\left\vert 0(2)\right\rangle\) but reaches ρss faster. g The logarithmic scale of the distance evolves with time for different initial states. An exponential speed-up of relaxation is clearly observed for the sME initial state. The experimental parameters for (e–g) are Ω1 = 2π × 20 kHz, Ω2 = 0.06Ω1, κ1 = 2Ω1, κ2 = 0.0015Ω1, and the solid lines here are the theoretical predictions based on the experimental parameters.
