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Dynamical Mean-Field Theory for Markovian Open Quantum Many-Body Systems

Orazio Scarlatella, Aashish A. Clerk, Rosario Fazio, Marco Schirò

Published Jul 22, 2021• 27 citations
Publisher

Abstract

A number of experimental platforms relevant for quantum simulations, ranging from arrays of superconducting circuits hosting correlated states of light to ultracold atoms in optical lattices in the presence of controlled dissipative processes. Their theoretical understanding is hampered by the exponential scaling of their Hilbert space and by their intrinsic nonequilibrium nature, limiting the applicability of many traditional approaches. In this work, we extend the nonequilibrium bosonic dynamical mean-field theory (DMFT) to Markovian open quantum systems. Within DMFT, a Lindblad master equation describing a lattice of dissipative bosonic particles is mapped onto an impurity problem describing a single site embedded in its Markovian environment and coupled to a self-consistent field and to a non-Markovian bath, where the latter accounts for fluctuations beyond Gutzwiller mean-field theory due to the finite lattice connectivity. We develop a nonperturbative approach to solve this bosonic impurity problem, which dresses the impurity, featuring Markovian dissipative channels, with the non-Markovian bath, in a self-consistent scheme based on a resummation of noncrossing diagrams. As a first application of our approach, we address the steady state of a driven-dissipative Bose-Hubbard model with two-body losses and incoherent pump. We show that DMFT captures hopping-induced dissipative processes, completely missed in Gutzwiller mean-field theory, which crucially determine the properties of the normal phase, including the redistribution of steady-state populations, the suppression of local gain, and the emergence of a stationary quantum-Zeno regime. We argue that these processes compete with coherent hopping to determine the phase transition toward a nonequilibrium superfluid, leading to a strong renormalization of the phase boundary at finite connectivity. We show that this transition occurs as a finite-frequency instability, leading to an oscillating-in-time order parameter, that we connect with a quantum many-body synchronization transition of an array of quantum van der Pol oscillators.

DOI
10.6082/jvrs0-8xr60

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A number of experimental platforms relevant for quantum simulations, ranging from arrays of superconducting circuits hosting correlated states of light to ultracold atoms in optical lattices in the presence of controlled dissipative processes. Their theoretical understanding is hampered by the exponential scaling of their Hilbert space and by their intrinsic nonequilibrium nature, limiting the applicability of many traditional approaches. In this work, we extend the nonequilibrium bosonic dynamical mean-field theory (DMFT) to Markovian open quantum systems. Within DMFT, a Lindblad master equation describing a lattice of dissipative bosonic particles is mapped onto an impurity problem describing a single site embedded in its Markovian environment and coupled to a self-consistent field and to a non-Markovian bath, where the latter accounts for fluctuations beyond Gutzwiller mean-field theory due to the finite lattice connectivity. We develop a nonperturbative approach to solve this bosonic impurity problem, which dresses the impurity, featuring Markovian dissipative channels, with the non-Markovian bath, in a self-consistent scheme based on a resummation of noncrossing diagrams. As a first application of our approach, we address the steady state of a driven-dissipative Bose-Hubbard model with two-body losses and incoherent pump. We show that DMFT captures hopping-induced dissipative processes, completely missed in Gutzwiller mean-field theory, which crucially determine the properties of the normal phase, including the redistribution of steady-state populations, the suppression of local gain, and the emergence of a stationary quantum-Zeno regime. We argue that these processes compete with coherent hopping to determine the phase transition toward a nonequilibrium superfluid, leading to a strong renormalization of the phase boundary at finite connectivity. We show that this transition occurs as a finite-frequency instability, leading to an oscillating-in-time order parameter, that we connect with a quantum many-body synchronization transition of an array of quantum van der Pol oscillators.
Open quantum many body systems describe several quantum simulators, ranging from arrays of superconducting circuits hosting correlated states of light, to ultracold atoms in optical lattices in presence of controlled dissipative processes. Their theoretical understanding is hampered by the exponential scaling of their Hilbert space and by their intrinsic nonequilibrium nature, limiting the applicability of many traditional approaches. In this work we extend Dynamical Mean Field Theory (DMFT), a powerful nonperturbative technique developed for strongly correlated quantum many-body systems, to Markovian open quantum systems. Within Open-DMFT, a master equation describing a lattice of bosonic particles is mapped, at large but finite connectivity, onto an impurity problem describing a single site coupled to Markovian and non-Markovian environments, where the latter accounts for quantum fluctuations beyond Gutzwiller mean-field theory. We introduce a new non-perturbative approach to solve the impurity problem, which dresses the impurity, featuring Markovian dissipative channels, with the non-Markovian bath, in a self-consistent scheme based on a resummation of non-crossing diagrams. Finally, we apply our Open-DMFT to solve a driven-dissipative Bose-Hubbard model, which is relevant to current experiments with dissipative ultracold bosons. We show that this model undergoes a dissipative phase transition towards a superfluid phase, which can be equally interpreted as a quantum many-body synchronization transition of an array of quantum van der Pol oscillators. We show how Open-DMFT captures crucial effects due to finite lattice connectivity, such as hopping-induced dissipation and heating, which are completely missed by mean field theories and which lead to a drastic reduction of the broken symmetry phase that we interpret as a desynchronization driven by quantum fluctuations.
The study of nonequilibrium phenomena in correlated lattice systems has developed into one of the most active and exciting branches of condensed matter physics. This research field provides rich new insights that could not be obtained from the study of equilibrium situations, and the theoretical understanding of the physics often requires the development of new concepts and methods. On the experimental side, ultrafast pump-probe spectroscopies enable studies of excitation and relaxation phenomena in correlated electron systems, while ultracold atoms in optical lattices provide a new way to control and measure the time evolution of interacting lattice systems with a vastly different characteristic time scale compared to electron systems. A theoretical description of these phenomena is challenging because, first, the quantum-mechanical time evolution of many-body systems out of equilibrium must be computed and second, strong-correlation effects which can be of a nonperturbative nature must be addressed. This review discusses the nonequilibrium extension of the dynamical mean field theory (DMFT), which treats quantum fluctuations in the time domain and works directly in the thermodynamic limit. The method reduces the complexity of the calculation via a mapping to a self-consistent impurity problem, which becomes exact in infinite dimensions. Particular emphasis is placed on a detailed derivation of the formalism, and on a discussion of numerical techniques, which enable solutions of the effective nonequilibrium DMFT impurity problem. Insights gained into the properties of the infinite-dimensional Hubbard model under strong nonequilibrium conditions are summarized. These examples illustrate the current ability of the theoretical framework to reproduce and understand fundamental nonequilibrium phenomena, such as the dielectric breakdown of Mott insulators, photodoping, and collapse-and-revival oscillations in quenched systems. Furthermore, remarkable novel phenomena have been predicted by the nonequilibrium DMFT simulations of correlated lattice systems, including dynamical phase transitions and field-induced repulsion-to-attraction conversions.
We study a driven-dissipative Bose-Hubbard model in the presence of two-particle losses and an incoherent single-particle drive on each lattice site, leading to a finite-density stationary state. Using dynamical meanfield theory (DMFT) and an impurity solver based on exact diagonalization of the associated Lindbladian, we investigate the regime of strong two-particle losses. Here a stationary-state quantum Zeno effect emerges, as can be seen in the on-site occupation and spectral function. We show that DMFT captures this effect through its self-consistent bath. We show that, in the deep Zeno regime, the bath structure simplifies, with the occupation of all bath sites except one becoming exponentially suppressed. As a result, an effective dissipative hard-core Bose-Hubbard dimer model emerges, where the auxiliary bath site has single-particle dissipation controlled by the Zeno dissipative scale.
We introduce a versatile method to compute electronic steady-state properties of strongly correlated extended quantum systems out of equilibrium. The approach is based on dynamical mean-field theory (DMFT), in which the original system is mapped onto an auxiliary nonequilibrium impurity problem imbedded in a Markovian environment. The steady-state Green's function of the auxiliary system is solved by full diagonalization of the corresponding Lindblad equation. The approach can be regarded as the nontrivial extension of the exact-diagonalization-based DMFT to the nonequilibrium case. As a first application, we consider an interacting Hubbard layer attached to two metallic leads and present results for the steady-state current and the nonequilibrium density of states.
We develop the nonequilibrium extension of bosonic dynamical mean-field theory and a Nambureal-time strong-coupling perturbative impurity solver. In contrast to Gutzwiller mean-field theory andstrong-coupling perturbative approaches, nonequilibrium bosonic dynamical mean-field theory capturesnot only dynamical transitions but also damping and thermalization effects at finite temperature. We applythe formalism to quenches in the Bose-Hubbard model, starting from both the normal and the Bose-condensed phases. Depending on the parameter regime, one observes qualitatively different dynamicalproperties, such as rapid thermalization, trapping in metastable superfluid or normal states, as well aslong-lived or strongly damped amplitude oscillations. We summarize our results in nonequilibrium“phase diagrams” that map out the different dynamical regimes.
We study the nonequilibrium dynamics of a many-body bosonic system on a lattice, subject to driving and dissipation. The time evolution is described by a master equation, which we treat within a generalized Gutzwiller mean field approximation for density matrices. The dissipative processes are engineered such that the system, in the absence of interaction between the bosons, is driven into a homogeneous steady state with off-diagonal long-range order. We investigate how the coherent interaction affects the properties of the steady state of the system qualitatively and derive a nonequilibrium phase diagram featuring a phase transition into a steady state without long-range order. The phase diagram also exhibits an extended domain where an instability of the homogeneous steady state gives rise to a persistent density pattern with spontaneously broken translational symmetry. In the limit of low particle density, we provide a precise analytical description of the time evolution during the instability. Moreover, we investigate the transient following a quantum quench of the dissipative processes and we elucidate the prominent role played by collective topological variables in this regime.