Engineering quantum Mpemba effect by Liouvillian skin effect

Abstract

We propose a new approach to engineer the quantum Mpemba effect (QME) -- wherein an initial state farther from system relaxes faster than a close one -- by the Liouvillian skin effect (LSE) in open quantum systems. Moreover, the LSE serves as an ideal platform for realizing the QME and the spatial profile of the LSE provides a straightforward pathway for the initial state preparation, thereby enabling readily accessible experimental preparation. Focusing on the quadratic Lindbladians, we consider two concrete cases to design the initial states, thereby realizing the QME. Interestingly, we uncover a new kind of QME (QME-III) that is distinct from the two typical scenarios, manifested as two reversals in the Hilbert-Schmidt distance at two different times. In particular, the LSE provides a physically more intuitive understanding of the QME.

Figures (2)

• Figure 1: Schematic illustration of the quantum Mpemba effect (QME) engineered by the Liouvillian skin effect (LSE). (a) Sketch of the model and the selection of the initial states $C_1$ and $C_2$ according to the spatial profile of density of particle numbers, corresponding to the more and less particles. The purple arrows represent the linear gain and loss operators $L^{g/l}_j$ at site $j$. (b) and (c) are the two typical QME (QME-I and QME-II). For (b), QME-I is realized when one unitary transforms the initial state $\rho_2$ to another state $\rho_1$ which makes the coefficient of the slowest mode disappear so that $\rho_1(t)-\rho_{ss}\propto {\rm exp}(t{\rm Re}\epsilon_3)$. For (c), QME-II can occur at early times even though asymptotic decay rates remain unchanged if $|a_2^c|>|a_2^f|$, where the coefficients $a_2^{c(f)}$ denote the overlaps between the initial state and the slowest modes for close (farther) state. (d) and (e) present the main results in our work. When the eigenstates of the system exhibit LSE, labeled by the solid line in (d), we can realize the QME-II (solid lines in (e)), given that the two initial states $C_1$ and $C_2$ are selected according to the LSE. For comparison, QME-II can not appear (dashed lines in (e)) when the eigenstates are extended (dashed line in (d)). In the main text, we only consider the fermionic system. The bosonic system is discussed in SM.
• Figure 2: Quantum Mpemba effects in quadratic fermionic Lindbladians for Case-I ((b)-(c)) and Case-II ((e)-(g)), corresponding to the QME-II and QME-III. (a) The eigenvalues (Eq. (\ref{['eigen']})) of the effective Hamiltonian $\mathcal{H}_{{\rm eff}}$ for Type-I (yellow) and Type-II (blue) gain and loss jump operators. Parameters are $J=1$, $\gamma_g=\gamma_l=0.2$ and $L=40$. The corresponding eigenstates of Type-I and Type-II jump operators are shown in the inset. (b) The reduced distance $\mathcal{D}_{R}(n(t),n_{ss})=\sqrt{\sum_j^L(n^j(t)-n_{ss}^j)^2}$ varies as a function of time for two initial states $C_1$ and $C_2$. For Type-I jump operators, the initial state $C_1$ farther from the steady state relaxes faster than a close one $C_2$ at early times, the so-called QME. The inset is for Type-II jump operators. (c) The deviation of density of particle numbers $\delta n$ vary as a function lattice site at different times for the two initial states $C_1$ and $C_2$. (e) The two initial states $C_1$ and $C_2$. The diagonal elements of both states are $1$. In addition, initial state $C_1$ also has off-diagonal elements at the right edge (the opposite direction of LSE). Parameters are $J=1$, $\gamma_g=\gamma_l=0.2$ and $L=20$. (f) The Hilbert-Schmidt distance $\mathcal{D}_{HS}\left(C(t),C_{ss}\right)=\sqrt{{\rm Tr}[(C(t)-C_{ss})^2]}$ changes as a function of time. For Type-I jump operators, the two typical QME in Figs. \ref{['model']}(b) and (c) can occur. The inset is for Type-II jump operators. (g) The off-diagonal elements of initial state $C_1$ reside in the left edge (the direction of LSE) so that there is no QME for Type-I jump operators.