Abstract Quantum dynamics of many-body systems is a fascinating and significant subject for both theory and experiment. The question of how an isolated many-body system evolves to its steady state after a sudden perturbation or quench still remains challenging. In this paper, using the Bethe ansatz wave function, we study the quantum dynamics of an inhomogeneous Gaudin magnet. We derive explicit analytical expressions for various local dynamic quantities with an arbitrary number of flipped bath spins, such as: the spin distribution function, the spin-spin correlation function, and the Loschmidt echo. We also numerically study the relaxation behavior of these dynamic properties, gaining considerable insight into coherence and entanglement between the central spin and the bath. In particular, we find that the spin-spin correlations relax to their steady value via a nearly logarithmic scaling, whereas the Loschmidt echo shows an exponential relaxation to its steady value. Our results advance the understanding of relaxation dynamics and quantum correlations of long-range interacting models of Gaudin type.Abstract Quantum dynamics of many-body systems is a fascinating and significant subject for both theory and experiment. The question of how an isolated many-body system evolves to its steady state after a sudden perturbation or quench still remains challenging. In this paper, using the Bethe ansatz wave function, we study the quantum dynamics of an inhomogeneous Gaudin magnet. We derive explicit analytical expressions for various local dynamic quantities with an arbitrary number of flipped bath spins, such as: the spin distribution function, the spin-spin correlation function, and the Loschmidt echo. We also numerically study the relaxation behavior of these dynamic properties, gaining considerable insight into coherence and entanglement between the central spin and the bath. In particular, we find that the spin-spin correlations relax to their steady value via a nearly logarithmic scaling, whereas the Loschmidt echo shows an exponential relaxation to its steady value. Our results advance the understanding of relaxation dynamics and quantum correlations of long-range interacting models of Gaudin type.

Communications in Theoretical PhysicsFeb 10, 20220 citations

Dynamics of isolated quantum many-body systems far from equilibrium

Markus Schmitt

This thesis addresses different aspects of the dynamics of isolated quantum many-body systems far from equilibrium, ranging from transient phenomena at short times to questions about relaxation and equilibration at late times. The first part deals with nonequilibrium critical phenomena. Dynamical quantum phase transitions are characterized in a two-dimensional spin system by studying the distribution of Fisher zeros in the complex time plane. Moreover, a new topologically driven transition in the nonequilibrium steady state of quenched Chern insulators is investigated, resulting in the identification of universal nonanalytic behavior of the Hall conductance as a function of the quench parameter. The second part of the thesis comprises a study of transient dynamics of the transverse-field Ising model in two and three dimensions based on a perturbative derivation of classical networks to encode the time-evolved wave function; this construction is also applicable to represent the state as an artificial neural network. In the last part of the thesis a possible definition of the irreversibility of quantum many-body dynamics is introduced, which is based on the echo dynamics of observables under imperfect effective time reversal. Echo dynamics is studied in generic and integrable quantum systems as well as in the semiclassical limit. The relation to out-of-time-order correlators is explored, which were recently introduced to probe the quantum butterfly effect.This thesis addresses different aspects of the dynamics of isolated quantum many-body systems far from equilibrium, ranging from transient phenomena at short times to questions about relaxation and equilibration at late times. The first part deals with nonequilibrium critical phenomena. Dynamical quantum phase transitions are characterized in a two-dimensional spin system by studying the distribution of Fisher zeros in the complex time plane. Moreover, a new topologically driven transition in the nonequilibrium steady state of quenched Chern insulators is investigated, resulting in the identification of universal nonanalytic behavior of the Hall conductance as a function of the quench parameter. The second part of the thesis comprises a study of transient dynamics of the transverse-field Ising model in two and three dimensions based on a perturbative derivation of classical networks to encode the time-evolved wave function; this construction is also applicable to represent the state as an artificial neural network. In the last part of the thesis a possible definition of the irreversibility of quantum many-body dynamics is introduced, which is based on the echo dynamics of observables under imperfect effective time reversal. Echo dynamics is studied in generic and integrable quantum systems as well as in the semiclassical limit. The relation to out-of-time-order correlators is explored, which were recently introduced to probe the quantum butterfly effect.

Jan 1, 20181 citation

Steady-states of out-of-equlibrium inhomogeneous Richardson–Gaudin quantum integrable models in quantum optics

Hugo Tschirhart, Thierry Platini, Alexandre Faribault

Abstract In this work we present numerical results for physical quantities in the steady-state obtained after a variety of product-states initial conditions are evolved unitarily, driven by the dynamics of quantum integrable models of the rational (XXX) Richardson–Gaudin family, which includes notably Tavis–Cummings models. The problem of interest here is one where a completely inhomogeneous ensemble of two-level systems (spins-1/2) are coupled to a single bosonic mode. The long-time averaged magnetisation along the z -axis as well as the bosonic occupation are evaluated in the diagonal ensemble by performing the complete sum over the full Hilbert space for small system sizes. These numerically exact results are independent of any particular choice of Hamiltonian and therefore describe general results valid for any member of this class of quantum integrable models built out of the same underlying conserved quantities. The collection of numerical results obtained can be qualitatively understood by a relaxation process for which, at infinitely strong coupling, every initial state will relax to a common state where each spin is in a maximally coherent superposition of its and states, i.e. they are in-plane polarised, and consequently the bosonic mode is also in a maximally coherent superposition of different occupation number states. This bosonic coherence being a feature of a superradiant state, we shall loosely use the term superradiant steady-state to describe it. A finite value of the coupling between the spins and the bosonic mode then leads to a long-time limit steady-state whose properties are qualitatively captured by a simple ‘dynamical’ vision in which the coupling strength plays the role of a time at which this ‘relaxation process’ towards the common strong coupling superradiant steady-state is interrupted.Abstract In this work we present numerical results for physical quantities in the steady-state obtained after a variety of product-states initial conditions are evolved unitarily, driven by the dynamics of quantum integrable models of the rational (XXX) Richardson–Gaudin family, which includes notably Tavis–Cummings models. The problem of interest here is one where a completely inhomogeneous ensemble of two-level systems (spins-1/2) are coupled to a single bosonic mode. The long-time averaged magnetisation along the z -axis as well as the bosonic occupation are evaluated in the diagonal ensemble by performing the complete sum over the full Hilbert space for small system sizes. These numerically exact results are independent of any particular choice of Hamiltonian and therefore describe general results valid for any member of this class of quantum integrable models built out of the same underlying conserved quantities. The collection of numerical results obtained can be qualitatively understood by a relaxation process for which, at infinitely strong coupling, every initial state will relax to a common state where each spin is in a maximally coherent superposition of its and states, i.e. they are in-plane polarised, and consequently the bosonic mode is also in a maximally coherent superposition of different occupation number states. This bosonic coherence being a feature of a superradiant state, we shall loosely use the term superradiant steady-state to describe it. A finite value of the coupling between the spins and the bosonic mode then leads to a long-time limit steady-state whose properties are qualitatively captured by a simple ‘dynamical’ vision in which the coupling strength plays the role of a time at which this ‘relaxation process’ towards the common strong coupling superradiant steady-state is interrupted.

arXiv:1805.03479Journal of Statistical Mechanics Theory and ExperimentAug 1, 20184 citations

Bath-induced interactions and transient dynamics in open quantum systems at strong coupling: Effective Hamiltonian approach

Marlon Brenes, Brett Min, Nicholas Anto-Sztrikacs +2 more

Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.

arXiv:2403.03386The Journal of Chemical PhysicsJun 25, 202412 citations

Loschmidt echo in many-body localized phases

Maksym Serbyn, Dmitry A. Abanin

The Loschmidt echo, defined as the overlap between quantum wave function evolved with different Hamiltonians, quantifies the sensitivity of quantum dynamics to perturbations and is often used as a probe of quantum chaos. In this work we consider the behavior of the Loschmidt echo in the many-body localized phase, which is characterized by emergent local integrals of motion and provides a generic example of nonergodic dynamics. We demonstrate that the fluctuations of the Loschmidt echo decay as a power law in time in the many-body localized phase, in contrast to the exponential decay in few-body ergodic systems. We consider the spin-echo generalization of the Loschmidt echo and argue that the corresponding correlation function saturates to a finite value in localized systems. Slow, power-law decay of fluctuations of such spin-echo-type overlap is related to the operator spreading and is present only in the many-body localized phase, but not in a noninteracting Anderson insulator. While most of the previously considered probes of dephasing dynamics could be understood by approximating physical spin operators with local integrals of motion, the Loschmidt echo and its generalizations crucially depend on the full expansion of the physical operators via local integrals of motion operators, as well as operators which flip local integrals of motion. Hence these probes allow one to get insights into the relation between physical operators and local integrals of motion and access the operator spreading in the many-body localized phase.The Loschmidt echo, defined as the overlap between quantum wave function evolved with different Hamiltonians, quantifies the sensitivity of quantum dynamics to perturbations and is often used as a probe of quantum chaos. In this work we consider the behavior of the Loschmidt echo in the many-body localized phase, which is characterized by emergent local integrals of motion and provides a generic example of nonergodic dynamics. We demonstrate that the fluctuations of the Loschmidt echo decay as a power law in time in the many-body localized phase, in contrast to the exponential decay in few-body ergodic systems. We consider the spin-echo generalization of the Loschmidt echo and argue that the corresponding correlation function saturates to a finite value in localized systems. Slow, power-law decay of fluctuations of such spin-echo-type overlap is related to the operator spreading and is present only in the many-body localized phase, but not in a noninteracting Anderson insulator. While most of the previously considered probes of dephasing dynamics could be understood by approximating physical spin operators with local integrals of motion, the Loschmidt echo and its generalizations crucially depend on the full expansion of the physical operators via local integrals of motion operators, as well as operators which flip local integrals of motion. Hence these probes allow one to get insights into the relation between physical operators and local integrals of motion and access the operator spreading in the many-body localized phase.

arXiv:1701.07772Physical review. B./Physical review. BJul 12, 201737 citations

Exact dynamics in the inhomogeneous central-spin model

Michael Bortz, Joachim Stolze

We study the dynamics of a single spin $1∕2$ coupled to a bath of spins $1∕2$ by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. For initially completely polarized bath spins and small magnetic field, we find persistent oscillations of the central spin about a nonzero mean value. For a large number of bath spins ${N}_{b}$ , the oscillation frequency is proportional to ${N}_{b}$ , whereas the amplitude behaves as $1∕{N}_{b}$ , to leading order. No asymptotic decay of the oscillations due to the nonuniform couplings is observed, in contrast to some recent studies.We study the dynamics of a single spin $1∕2$ coupled to a bath of spins $1∕2$ by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. For initially completely polarized bath spins and small magnetic field, we find persistent oscillations of the central spin about a nonzero mean value. For a large number of bath spins ${N}_{b}$, the oscillation frequency is proportional to ${N}_{b}$, whereas the amplitude behaves as $1∕{N}_{b}$, to leading order. No asymptotic decay of the oscillations due to the nonuniform couplings is observed, in contrast to some recent studies.

arXiv:cond-mat/0612382Physical Review BJul 26, 2007112 citations

We study the dynamics of a single spin

1∕2

coupled to a bath of spins

1∕2

by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. For initially completely polarized bath spins and small magnetic field, we find persistent oscillations of the central spin about a nonzero mean value. For a large number of bath spins

{N}_{b}

, the oscillation frequency is proportional to

{N}_{b}

, whereas the amplitude behaves as

1∕{N}_{b}

, to leading order. No asymptotic decay of the oscillations due to the nonuniform couplings is observed, in contrast to some recent studies.