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arXiv:2406.04183 · Physical review. B./Physical review. B

Manipulating the relaxation time of boundary-dissipative systems through bond dissipation

Yi Peng, Chao Yang, Yucheng Wang

Published Sep 12, 2024• 7 citations
arXivHTMLPDF

Abstract

Relaxation time plays a crucial role in describing the relaxation processes of quantum systems. We study the effect of a type of bond dissipation on the relaxation time of boundary dissipative systems and find that it can change the scaling of the relaxation time ${T}_{c}\ensuremath{\sim}{L}^{z}$ from $z=3$ to a value significantly less than 3. We further reveal that the reason such bond dissipation can significantly reduce the relaxation time is that it can selectively target specific states. For Anderson-localized systems, the scaling behavior of the relaxation time changes from an exponential form to a power-law form as the system size varies. This is because the bond dissipation we consider can not only select specific states but can also disrupt the localization properties. Our work reveals that in open systems one type of dissipation can be used to regulate the effects produced by another type of dissipation.

DOI
10.1103/physrevb.110.104305

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Relaxation time plays a crucial role in describing the relaxation processes of quantum systems. We study the effect of a type of bond dissipation on the relaxation time of boundary dissipative systems and find that it can change the scaling of the relaxation time Tc\ensuremath∼Lz{T}_{c}\ensuremath{\sim}{L}^{z}Tc​\ensuremath∼Lz from z=3z=3z=3 to a value significantly less than 3. We further reveal that the reason such bond dissipation can significantly reduce the relaxation time is that it can selectively target specific states. For Anderson-localized systems, the scaling behavior of the relaxation time changes from an exponential form to a power-law form as the system size varies. This is because the bond dissipation we consider can not only select specific states but can also disrupt the localization properties. Our work reveals that in open systems one type of dissipation can be used to regulate the effects produced by another type of dissipation.
It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from the spectral gap of the Liouvillian that governs the time evolution of the density matrix. We investigate the relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect, which means that the eigenmodes of the Liouvillian are localized exponentially close to the boundary of the system, and find that the timescale for the system to reach a steady state depends not only on the Liouvillian gap $\mathrm{\ensuremath{\Delta}}$, but also on the localization length $\ensuremath{\xi}$ of the eigenmodes. In particular, we show that the longest relaxation time $\ensuremath{\tau}$ that is maximized over initial states and local observables is given by $\ensuremath{\tau}\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}1}(1+L/\ensuremath{\xi})$ with LLL being the system size. This implies that the longest relaxation time can diverge for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ without gap closing.
The Mpemba effect, where a state farther from equilibrium relaxes faster than one closer to it, is a striking phenomenon in both classical and quantum systems. In open quantum systems, however, the quantum Mpemba effect (QME) typically occurs only for specifically chosen initial states, which limits its universality. Here we present a general and experimentally feasible strategy to realize both QME and anti-QME. By applying a temporary bond-dissipation quench, we selectively suppresses or enhances slow relaxation modes, thereby reshaping relaxation pathways independently of both the system and the initial state. We demonstrate this mechanism in systems with dephasing and boundary dissipation, and outline feasible cold-atom implementations. Our results establish controllable dissipation as a versatile tool for quantum control, accelerated relaxation, and efficient nonequilibrium protocols.
We study relaxation times, also called mixing times, of quantum many-body systems described by a Lindblad master equation. We in particular study the scaling of the spectral gap with the system length, the so-called dynamical exponent, identifying a number of transitions in the scaling. For systems with bulk dissipation we generically observe different scaling for small and for strong dissipation strength, with a critical transition strength going to zero in the thermodynamic limit. We also study a related phase transition in the largest decay mode. For systems with only boundary dissipation we show a generic bound that the gap cannot be larger than ∼1/L. In integrable systems with boundary dissipation one typically observes scaling of ∼1/L(3), while in chaotic ones one can have faster relaxation with the gap scaling as ∼1/L and thus saturating the generic bound. We also observe transition from exponential to algebraic gap in systems with localized modes.
We investigate the effects of dissipation on the quantum dynamics of many-body systems at quantum transitions, especially considering those of the first order. This issue is studied within the paradigmatic one-dimensional quantum Ising model. We analyze the out-of-equilibrium dynamics arising from quenches of the Hamiltonian parameters and dissipative mechanisms modeled by a Lindblad master equation, with either local or global spin operators acting as dissipative operators. Analogously to what happens at continuous quantum transitions, we observe a regime where the system develops a nontrivial dynamic scaling behavior, which is realized when the dissipation parameter uuu (globally controlling the decay rate of the dissipation within the Lindblad framework) scales as the energy difference $\mathrm{\ensuremath{\Delta}}$ of the lowest levels of the Hamiltonian, i.e., $u\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$. However, unlike continuous quantum transitions where $\mathrm{\ensuremath{\Delta}}$ is power-law suppressed, at first-order quantum transitions $\mathrm{\ensuremath{\Delta}}$ is exponentially suppressed with increasing the system size (provided the boundary conditions do not favor any particular phase).
Conformal interfaces provide an important setting for studying universal transmission phenomena in one-dimensional quantum critical systems. While energy and information transmission across such interfaces are characterized by universal quantities in closed systems, the corresponding role of conformal interfaces in dissipative dynamics is less understood. In this work, we study relaxation in locally dissipative quantum critical chains with a conformal interface and show that a universal characterization emerges at the level of individual relaxation modes. We first show that the relaxation coefficient defined in the recent work [1] from the Liouvillian gap can become non-universal for certain boundary conditions because the mode determining the smallest decay rate can change as the interface transmission is varied. To resolve this ambiguity, we introduce a mode-resolved relaxation coefficient crelaxc_{\rm relax}crelax​ by continuously tracking the same Liouvillian rapidity mode as a function of the interface transmission. Using analytical and numerical calculations for a critical harmonic chain and a critical free-fermion chain, we find that, in the weak-dissipation regime, crelaxc_{\rm relax}crelax​ follows the same universal dependence on the interface transmission in all cases considered, independent of microscopic details such as the boundary conditions, dissipation strength, and location of the local dissipation. For boundary dissipation, this universal behavior persists even at finite dissipation strength. Our results establish a universal mode-resolved characterization of relaxation across conformal interfaces in open quantum critical systems.
We carry out a systematical study of the size scaling of the Liouvillian gap in boundary-dissipated one-dimensional quasiperiodic and disorder systems. By treating the boundary-dissipation operators as a perturbation, we derive an analytical expression of the Liouvillian gap, which indicates clearly the Liouvillian gap being proportional to the minimum of boundary densities of eigenstates of the underlying Hamiltonian, and thus give a theoretical explanation why the Liouvillian gap has different size scaling relation in the extended and localized phase. While the Liouvillian gap displays a power-law size scaling ${\mathrm{\ensuremath{\Delta}}}_{g}\ensuremath{\propto}{L}^{\ensuremath{-}3}$ in the extended phase, our analytical result unveils that the Liouvillian gap fulfills an exponential scaling relation ${\mathrm{\ensuremath{\Delta}}}_{g}\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\kappa}L}$ in the localized phase, where $\ensuremath{\kappa}$ takes the largest Lyapunov exponent of localized eigenstates of the underlying Hamiltonian. By scrutinizing the extended Aubry-Andr\'e-Harper model, we numerically confirm that the Liouvillian gap fulfills the exponential scaling relation and the fitting exponent $\ensuremath{\kappa}$ coincides pretty well with the analytical result of the Lyapunov exponent. The exponential scaling relation is further verified numerically in other one-dimensional quasiperiodic and random disorder models. We also study the relaxation dynamics and show the inverse of the Liouvillian gap giving a reasonable timescale of asymptotic convergence to the steady state.