We investigate a non-Hermitian model featuring nonreciprocal gradient hoppings. Through an in-depth analysis of the Liouvillian spectrum and dynamics, we confirm the emergence of the Liouvillian skin effect resulting from the nonreciprocal nature of hoppings in this model. Furthermore, we observe that the presence of gradient hopping strength leads to an accelerated relaxation time for the system. Through numerical investigations of the Liouvillian gap, relaxation time, and steady-state localization length, we discover that the relaxation time in this model cannot be explained by the currently established relationship associated with the Liouvillian skin effect. This discrepancy highlights the need for further exploration and theoretical advancements to fully comprehend the intricate mechanisms underlying quantum relaxation processes. Motivated by these findings, we propose a theoretical approach to realize this non-Hermitian model in an atomic system with a sideband structure by employing an adiabatic elimination technique. These results contribute to our deeper comprehension of quantum relaxation dynamics and provide theoretical backing for the development of techniques aimed at controlling quantum relaxation processes.
We investigate a non-Hermitian model featuring nonreciprocal gradient hoppings. Through an in-depth analysis of the Liouvillian spectrum and dynamics, we confirm the emergence of the Liouvillian skin effect resulting from the nonreciprocal nature of hoppings in this model. Furthermore, we observe that the presence of gradient hopping strength leads to an accelerated relaxation time for the system. Through numerical investigations of the Liouvillian gap, relaxation time, and steady-state localization length, we discover that the relaxation time in this model cannot be explained by the currently established relationship associated with the Liouvillian skin effect. This discrepancy highlights the need for further exploration and theoretical advancements to fully comprehend the intricate mechanisms underlying quantum relaxation processes. Motivated by these findings, we propose a theoretical approach to realize this non-Hermitian model in an atomic system with a sideband structure by employing an adiabatic elimination technique. These results contribute to our deeper comprehension of quantum relaxation dynamics and provide theoretical backing for the development of techniques aimed at controlling quantum relaxation processes.
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 L being the system size. This implies that the longest relaxation time can diverge for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ without gap closing.
We unveil a quantum Pontus-Mpemba effect enabled by the Liouvillian skin effect in a dissipative tight-binding chain with asymmetric incoherent hopping and coherent boundary coupling. The skin effect, induced by non-reciprocal dissipation, localizes relaxation modes near the system boundaries and gives rise to non-orthogonal spectral geometry. While such non-normality is often linked to slow relaxation, we show that it can instead accelerate relaxation through a two-step protocol - realizing a quantum Pontus-Mpemba effect. Specifically, we consider a one-dimensional open chain with coherent hopping J, asymmetric incoherent hoppings JR=JL, and a controllable end-to-end coupling ε. For ε=0, the system exhibits the Liouvillian skin effect, with left and right eigenmodes localized at opposite edges. We compare two relaxation protocols toward the same stationary state: (i) a direct relaxation with ε=0, and (ii) a two-step (Pontus) protocol where a brief coherent evolution transfers the excitation across the lattice before relaxation. Although both share the same asymptotic decay rate, the two-step protocol relaxes significantly faster due to its reduced overlap with the slow boundary-localized Liouvillian mode. The effect disappears when JR=JL, i.e., when the skin effect vanishes. Our results reveal a clear connection between boundary-induced non-normality and protocol-dependent relaxation acceleration, suggesting new routes for controlling dissipation and transient dynamics in open quantum systems.
Open quantum systems far from thermal equilibrium can exhibit remarkable physical phenomena including topological properties without a direct equilibrium counterpart. Along these lines, in periodically driven-dissipative systems within the effective non-Hermitian (NH) Hamiltonian approximation spectral winding numbers have been linked to intriguing nonreciprocal transport properties. Here, going beyond an NH Hamiltonian description, we introduce and study a microscopic lattice model of a driven open quantum system described by a Markovian quantum master equation, which exhibits the mentioned spectral winding within an NH approximation. By encompassing quantum jump processes in the topological analysis, we uncover a distinct \emph{jump-induced} topological phase, which qualitatively corresponds to the richer nonreciprocal transport properties of the fully quantum model. In addition, we find that the NH skin effect, i.e., the accumulation of a macroscopic number of eigenstates at one end of the system, is already visible in the transient dynamics even for systems with periodic boundary conditions. Our results exemplify the subtle correspondence between NH topological properties and physical manifestations of Liouvillian topological properties in open quantum systems, thus providing a theoretical framework towards understanding unidirectional transport in quantum dissipative Floquet dynamics.
The non-Hermitian skin effect, in which eigenstates of non-Hermitian Hamiltonians are localized at one boundary in the open boundary condition, has attracted great interest recently. In this paper, we investigate the skin effect in one-dimensional dissipative quantum many-body systems, which we call the Liouvillian skin effect (LSE). We rigorously identify the existence of the LSE for generalized boundary conditions by solving the Liouvillian superoperator of an exactly solvable model with the advantage of the Bethe ansatz. The LSE is sensitive to boundary conditions where the signature is reflected in eigenfunctions of the system. We confirm that the LSE is fragile to a tiny coflow boundary hopping with non-Hermitian current but can survive a counterflow boundary hopping in the thermodynamic limit. Our work provides a prototypical example of exactly solvable dissipative quantum many-body lattice systems exhibiting the LSE for generalized boundary conditions. It can be further extended to other integrable open quantum many-body models.
Non-Hermitian systems with globally reciprocal couplings -- such as the Hatano-Nelson model with stochastic imaginary gauge fields -- avoid the conventional non-Hermitian skin effect, displaying erratic bulk localization while retaining ballistic transport. An open question is whether similar behavior arises when non-reciprocity originates at the Liouvillian level rather than from an effective non-Hermitian Hamiltonian obtained via post-selection. Here, a lattice model with globally reciprocal Liouvillian dynamics and locally asymmetric incoherent hopping is investigated, a disordered setting in which Liouvillian-specific effects have remained largely unexplored. While the steady state again shows disorder-dependent, erratic localization without boundary accumulation, {\color{black}excitations in the incoherent-hopping regime spread via {\em Sinai-type subdiffusion}, dramatically slower than ordinary diffusion in symmetric stochastic lattices.} {\color{black}This highlights that the genuinely distinct Liouvillian signature is the coexistence of global reciprocity with ultra-slow, disorder-induced subdiffusive transport, rather than the erratic localization itself.} {\color{black}These results reveal a fundamental distinction between globally reciprocal Hamiltonian and Liouvillian systems: in both cases the skin effect is suppressed, but only in Liouvillian dynamics erratic skin localization can coexist with subdiffusive transport
Strongly interacting nonequilibrium systems are of great fundamental interest, yet their inherent complexity make them notoriously hard to analyze. We demonstrate that the dynamics of the Bose-Hubbard model, which by itself evades solvability, can be solved exactly at any interaction strength in the presence of loss tuned to a rate matching the hopping amplitude. Remarkably, the full solvability of the corresponding Liouvillian, and the integrability of the pertinent effective non-Hermitian Hamiltonian, survives the addition of disorder and generic boundary conditions. By analyzing the Bethe ansatz solutions we find that even weak interactions change the qualitative features of the system, leading to an intricate dynamical phase diagram featuring non-Hermitian Mott-skin effects, disorder induced localization, highly degenerate exceptional points, and a Bose glasslike phase of fragmented condensates. We discuss realistic implementations of this model with cold atoms. Published by the American Physical Society 2024