Given a quantum many-body system and the expectation-value dynamics of some operator, we study how this reference dynamics is altered due to a perturbation of the system's Hamiltonian. Based on projection operator techniques, we unveil that if the perturbation exhibits a random-matrix structure in the eigenbasis of the unperturbed Hamiltonian, then this perturbation effectively leads to an exponential damping of the original dynamics. Employing a combination of dynamical quantum typicality and numerical linked cluster expansions, we demonstrate that our theoretical findings for random matrices can, in some cases, be relevant for the dynamics of realistic quantum many-body models as well. Specifically, we study the decay of current autocorrelation functions in spin-1/2 ladder systems, where the rungs of the ladder are treated as a perturbation to the otherwise uncoupled legs. We find a convincing agreement between the exact dynamics and the lowest-order prediction over a wide range of interchain couplings.
Given a quantum many-body system and the expectation-value dynamics of some operator, we study how this reference dynamics is altered due to a perturbation of the system's Hamiltonian. Based on projection operator techniques, we unveil that if the perturbation exhibits a random-matrix structure in the eigenbasis of the unperturbed Hamiltonian, then this perturbation effectively leads to an exponential damping of the original dynamics. Employing a combination of dynamical quantum typicality and numerical linked cluster expansions, we demonstrate that our theoretical findings for random matrices can, in some cases, be relevant for the dynamics of realistic quantum many-body models as well. Specifically, we study the decay of current autocorrelation functions in spin-1/2 ladder systems, where the rungs of the ladder are treated as a perturbation to the otherwise uncoupled legs. We find a convincing agreement between the exact dynamics and the lowest-order prediction over a wide range of interchain couplings.
Understanding how the dynamics of a given quantum system with many degrees of freedom is altered by the presence of a generic perturbation is a notoriously difficult question. Recent works predict that, in the overwhelming majority of cases, the unperturbed dynamics is just damped by a simple function, e.g., exponentially as expected from Fermi's golden rule. While these predictions rely on random-matrix arguments and typicality, they can only be verified for a specific physical situation by comparing to the actual solution or measurement. Crucially, it also remains unclear how frequent and under which conditions counterexamples to the typical behavior occur. In this work, we discuss this question from the perspective of projection-operator techniques, where exponential damping of a density matrix occurs in the interaction picture but not necessarily in the Schrödinger picture. We show that a nontrivial damping in the Schrödinger picture can emerge if the dynamics in the unperturbed system possesses rich features, for instance due to the presence of strong interactions. This suggestion has consequences for the time dependence of correlation functions. We substantiate our theoretical arguments by large-scale numerical simulations of charge transport in the extended Fermi-Hubbard chain, where the nearest-neighbor interactions are treated as a perturbation to the integrable reference system.
Since perturbations are omnipresent in physics, understanding their impact on the dynamics of quantum many-body systems is a vitally important but notoriously difficult question. On the one hand, random-matrix and typicality arguments suggest a rather simple damping in the overwhelming majority of cases, e.g., exponential damping according to Fermi's golden rule. On the other hand, counterexamples are known to exist, and it remains unclear how frequent and under which conditions such counterexamples appear. In our work, we consider the spin-1/2 XXZ chain as a paradigmatic example of a quantum many-body system and study the dynamics of the magnetization current in the easy-axis regime. Using numerical simulations based on dynamical quantum typicality, we show that the standard autocorrelation function is damped in a nontrivial way and only a modified version of this function is damped in a simple manner. Employing projection-operator techniques in addition, we demonstrate that both the nontrivial and simple damping relation can be understood on perturbative grounds. Our results are in agreement with earlier findings for the particle current in the Hubbard chain.
Abstract Evidently, some relaxation dynamics, e.g. exponential decays, are much more common in nature than others. Recently there have been attempts to explain this observation on the basis of “typicality of perturbations” with respect to their impact on expectation value dynamics. These theories suggest that a majority of the very numerous, possible Hamiltonian perturbations entail more or less the same type of alteration of the decay dynamics. Thus, in this paper, we study how the approach towards equilibrium in closed quantum systems is altered due to weak perturbations. To this end, we perform numerical experiments on a particular, exemplary spin system. We compare our numerical data to predictions from three particular theories. We find satisfying agreement in the weak perturbation regime for one of these approaches.
Concepts like ``typicality'' and the ``eigenstate thermalization hypothesis'' aim at explaining the apparent equilibration of quantum systems, possibly after a very long time. However, these concepts are not concerned with the specific way in which this equilibrium is approached. Our point of departure is the (evident) observation that some forms of the approach to equilibrium, such as, e.g., exponential decay of observables, are much more common than others. We suggest to trace this dominance of certain decay dynamics back to a larger stability with respect to generic Hamiltonian perturbations. A numerical study of a number of examples in which both the unperturbed Hamiltonians and the perturbations are modeled by partially random matrices is presented. We furthermore develop a simple heuristic, mathematical scheme that describes the result of the numerical investigations remarkably well. According to those investigations, the exponential decay indeed appears to be most stable. Dynamics that are in a certain sense at odds with the arrow of time are found to be very unstable.
Understanding the dynamics of open quantum systems is intrinsically important to control and utilize quantum hardware in various applications. The generic behavior of purely dissipative open quantum many-body systems with k-local, normal dissipation processes can be investigated using random matrix theory, revealing different decay timescales of observables determined by their complexity, as shown by Wang et al. [K. Wang et al., Phys. Rev. Lett. 124, 100604 (2020)]. The time evolution of an open quantum system is entirely described by the eigenvalue spectrum of the Lindbladian, in which the hierarchy of decay timescales is reflected in the formation of well-separated eigenvalue clusters. Here, we extend the analysis of this spectrum to the presence of unitary dynamics using a random matrix model. In the case of strong dissipation, the unitary dynamics can be treated perturbatively and the k locality of the Hamiltonian determines how susceptible the spectrum is to such a perturbation. For the physically most relevant case of (dissipative) two-body interactions, the correction in the first order of the perturbation vanishes, leading to the relative robustness of the spectral features, which in turn implies the same robustness of the corresponding separation of decay timescales. We find the perturbative approach for the spectrum to yield an excellent agreement with exact-diagonalization results. For weak dissipation, the spectrum flows into clusters except for outlier eigenmodes, which we identify to be the local symmetries of the Hamiltonian. We support our analytic findings by numerical simulations and show that our generic model applies to a nonrandom Heisenberg chain.
In this work we discuss three problems connected to the damping of oscillations in quantum systems: dynamics and decay in a random matrix model with conserved quantities, energy redistribution and decay of the amplitude mode in a superconductor, and the decay of the gapped mode in a molecular condensate, a bosonic analog to the superconductor. These problems highlight different aspects of damping and energy redistribution in quantum dynamics, while being simple enough that analytic control is possible. The first system captures the idea of a "partially conserved" quantity: in ergodic quantum systems, physical observables have a non-relaxing component if they overlap with a conserved quantity, but how to isolate the non-relaxing component is in general unclear. We compute exact dynamical correlators governed by a Hamiltonian composed of two large interacting random matrices, H=A+B, and we analytically obtain the late-time value of ⟨A(t) A(0)⟩, which quantifies the non-relaxing part of the observable A. We show that the relaxation to this value is governed by a power-law determined by the spectrum of the Hamiltonian H, independent of the observable A, while the long--time value and the amplitude of the oscillations depend on the trace--overlap between the operator and the Hamiltonian. For Gaussian matrices, we further compute out-of-time-ordered-correlators (OTOCs) and find that the existence of a non-relaxing part of A leads to modifications of the late time values and exponents. Our results follow from exact resummation of a diagrammatic expansion and hyperoperator techniques. The above problem deals with energy redistribution in a system with a complex internal structure, but without any spatial dependence nor many--body effects. In the second part of this work we discuss energy relaxation in a system with both: a BCS superconductor. In particular, we study the excitation of the collective Higgs oscillations of the order parameter by incoherent short pulses of light with frequency much larger than the superconducting gap. We find that the excitation amplitude of the Higgs mode is controlled by a single parameter, determined by the total number of quasiparticles excited by the pulse, which we trace back to the universality of the shape of the light-induced quasiparticle cascade at energy below the Debye frequency and above the gap. Our analysis is primarily based on the Keldysh technique for non--equilibrium field theory and the Boltzmann kinetic equation. Finally, we study the damping of the gapped mode in a molecular Bose--Einstein condensate, the Bosonic analogue of a BCS superconductor. This system has the advantage of giving the experimentalist fine control over the interatomic interactions using Feshbach resonances, and is the object of renewed interest as the molecular superfluid phase has only very recently been realized in the lab. We discuss damping in the nontrivial thermodynamic phases: in the molecular superfluid phase the gapped excitation is protected by parity, and is damped only above a threshold momentum --as in the Cherenkov effect--, while in the atomic superfluid phase the gapped mode is damped at all momentum scales. We propose a class of experiments where our results are measurable: transmission (and reflection) of an atom beam through a molecular condensate cloud.