We study spectral and steady-state properties of generic Markovian dissipative systems described by quadratic fermionic Liouvillian operators of the Lindblad form. The Hamiltonian dynamics is modeled by a generic random quadratic operator, i.e., as a featureless superconductor of class D, whereas the Markovian dissipation is described by M M random linear jump operators. By varying the dissipation strength and the ratio of dissipative channels per fermion, m=M/(2N_F) m = M / ( 2 N F ) , we find two distinct phases where the support of the single-particle spectrum has one or two connected components. In the strongly dissipative regime, this transition occurs for m=1/2 m = 1 / 2 and is concomitant with a qualitative change in both the steady-state and the spectral gap that rules the large-time dynamics. Above this threshold, the spectral gap and the steady-state purity qualitatively agree with the fully generic (i.e., non-quadratic) case studied recently. Below m=1/2 m = 1 / 2 , the spectral gap closes in the thermodynamic limit and the steady-state decouples into an ergodic and a nonergodic sector yielding a non-monotonic steady-state purity as a function of the dissipation strength. Our results show that some of the universal features previously observed for fully random Liouvillians are generic for a sufficiently large number of jump operators. On the other hand, if the number of dissipation channels is decreased the system can exhibit nonergodic features, rendering it possible to suppress dissipation in protected subspaces even in the presence of strong system-environment coupling.
We study spectral and steady-state properties of generic Markovian dissipative systems described by quadratic fermionic Liouvillian operators of the Lindblad form. The Hamiltonian dynamics is modeled by a generic random quadratic operator, i.e., as a featureless superconductor of class D, whereas the Markovian dissipation is described by M M random linear jump operators. By varying the dissipation strength and the ratio of dissipative channels per fermion, m=M/(2N_F) m = M / ( 2 N F ) , we find two distinct phases where the support of the single-particle spectrum has one or two connected components. In the strongly dissipative regime, this transition occurs for m=1/2 m = 1 / 2 and is concomitant with a qualitative change in both the steady-state and the spectral gap that rules the large-time dynamics. Above this threshold, the spectral gap and the steady-state purity qualitatively agree with the fully generic (i.e., non-quadratic) case studied recently. Below m=1/2 m = 1 / 2 , the spectral gap closes in the thermodynamic limit and the steady-state decouples into an ergodic and a nonergodic sector yielding a non-monotonic steady-state purity as a function of the dissipation strength. Our results show that some of the universal features previously observed for fully random Liouvillians are generic for a sufficiently large number of jump operators. On the other hand, if the number of dissipation channels is decreased the system can exhibit nonergodic features, rendering it possible to suppress dissipation in protected subspaces even in the presence of strong system-environment coupling.
Abstract We study generic open quantum systems with Markovian dissipation, focusing on a class of stochastic Liouvillian operators of Lindblad form with independent random dissipation channels (jump operators) and a random Hamiltonian. We establish that the global spectral features, the spectral gap, and the steady-state properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular quantity. Within each regime, we determine the scaling exponents with the dissipation strength and system size. We find that, for two or more dissipation channels, the spectral gap increases with the system size. The spectral distribution of the steady state is Poissonian at low dissipation strength and conforms to that of a random matrix once the dissipation is sufficiently strong. Our results can help to understand the long-time dynamics and steady-state properties of generic dissipative systems.
We study generic open quantum systems with Markovian dissipation, focusing on a class of stochastic Liouvillian operators of Lindblad form with independent random dissipation channels (jump operators) and a random Hamiltonian. We perform a thorough numerical study, focusing on global spectral features, the spectral gap, and the steady-state purity and statistics. We establish that all properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular observable. Within each regime, we determine the scaling exponents with the dissipation strength and system size. On the analytical side, we compute the average spectral gap at arbitrary dissipation and provide simple closed-form expressions for the asymptotic values at strong and weak dissipation strength. We also consider spectral correlations in generic complex spectra, such as that of the Liouvillian. We introduce and study, both analytically and numerically, the complex ratio of the nearest-neighbor spacing by next-to-nearest-neighbor spacing. Besides the usual level repulsion, we find that complex spacing ratios have a nontrivial angular dependence which offers a distinct signature of quantum chaos. It distinguishes integrable systems, which support the flat ratio distribution characteristic of Poisson statistics, and chaotic dissipative quantum systems, which conform to Random Matrix Theory statistics. We apply this new signature of quantum chaos to random Liouvillians and boundary-driven dissipative spin-chains. The results of this thesis can help understand the long-time dynamics, steady-state properties, and spectral correlations of generic dissipative systems.
In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
The study of dissipation and decoherence in generic open quantum systems recently led to the investigation of spectral and steady-state properties of random Lindbladian dynamics. A natural question is then how realistic and universal those properties are. Here, we address these issues by considering a different description of dissipative quantum systems, namely, the discrete-time Kraus map representation of completely positive quantum dynamics. Through random matrix theory (RMT) techniques and numerical exact diagonalization, we study random Kraus maps, allowing for a varying dissipation strength, and their local circuit counterpart. We find the spectrum of the random Kraus map to be either an annulus or a disk inside the unit circle in the complex plane, with a transition between the two cases taking place at a critical value of dissipation strength. The eigenvalue distribution and the spectral transition are well described by a simplified RMT model that we can solve exactly in the thermodynamic limit, by means of non-Hermitian RMT and quaternionic free probability. The steady state, on the contrary, is not affected by the spectral transition. It has, however, a perturbative crossover regime at small dissipation, inside which the steady state is characterized by uncorrelated eigenvalues. At large dissipation (or for any dissipation for a large-enough system), the steady state is well described by a random Wishart matrix. The steady-state properties thus coincide with those already observed for random Lindbladian dynamics, indicating their universality. Quite remarkably, the statistical properties of the local Kraus circuit are qualitatively the same as those of the nonlocal Kraus map, indicating that the latter, which is more tractable, already captures the realistic and universal physical properties of generic open quantum systems.
We propose the Sachdev-Ye-Kitaev Lindbladian as a paradigmatic solvable model of dissipative many-body quantum chaos.It describes N strongly coupled Majorana fermions with random all-to-all interactions, with unitary evolution given by a quartic Hamiltonian and the coupling to the environment described by M quadratic jump operators, rendering the full Lindbladian quartic in the Majorana operators.Analytical progress is possible by developing a dynamical mean-field theory for the Liouvillian time evolution on the Keldysh contour.By disorder-averaging the interactions, we derive an (exact) effective action for two collective fields (Green's function and self-energy).In the large-N, large-M limit, we obtain the saddle-point equations satisfied by the collective fields, which determine the typical timescales of the dissipative evolution, particularly the spectral gap that rules the relaxation of the system to its steady state.We solve the saddle-point equations numerically and find that, for strong or intermediate dissipation, the system relaxes exponentially, with a spectral gap that can be computed analytically, while for weak dissipation, there are oscillatory corrections to the exponential relaxation.In this letter, we illustrate the feasibility of analytical calculations in strongly correlated dissipative quantum matter.
We study dissipative translationally-invariant free fermionic theories with quadratic Liouvillians. Using a Lie-algebraic approach we solve the Lindblad equation and find explicitly the density matrix at all times for arbitrary time dependence of the Liouvillian. We then investigate the Liouvillian spectral properties and derive a generic criterion for the closure of the dissipative gap, which is believed to be linked with nonequilibrium dissipative phase transitions. We illustrate our findings with a few exotic examples. Particularly, we show the presence of gapless modes with a linear spectrum for fermions with a long-range hopping, which might be related to non-unitary conformal field theories. The predicted effects can be probed in experiments with ultracold atomic and quantum-optical systems using currently available experimental facilities.