Symmetries in an open quantum system lead to degenerated Liouvillians that physically imply the existence of multiple steady states. In such cases, obtaining the initial condition independent steady states is highly nontrivial since any linear combination of the true asymptotic states, which may not necessarily be a density matrix, is also a valid asymptote for the Liouvillian. Thus, in this work, we consider different approaches to obtain the true steady states of a degenerated Liouvillian. In the ideal scenario, when the open system symmetry operators are known, we show how these can be used to obtain the invariant subspaces of the Liouvillian and hence the steady states. We then discuss two other approaches that do not require any knowledge of the symmetry operators. These could be powerful numerical tools to deal with quantum many-body complex open systems. The first approach that is based on Gram-Schmidt orthonormalization of density matrices allows us to obtain all the steady states, whereas the second one based on large deviations allows us to obtain the non-degenerated maximum and minimum current carrying states. We discuss the symmetry-decomposition and the orthonormalization methods with the help of an open para-benzene ring and examine interesting scenarios such as the dynamical restoration of Hamiltonian symmetries in the long-time limit and apply the method to study the eigenspacing statistics of the nonequilibrium steady state.
Symmetries in an open quantum system lead to degenerated Liouvillians that physically imply the existence of multiple steady states. In such cases, obtaining the initial condition independent steady states is highly nontrivial since any linear combination of the true asymptotic states, which may not necessarily be a density matrix, is also a valid asymptote for the Liouvillian. Thus, in this work, we consider different approaches to obtain the true steady states of a degenerated Liouvillian. In the ideal scenario, when the open system symmetry operators are known, we show how these can be used to obtain the invariant subspaces of the Liouvillian and hence the steady states. We then discuss two other approaches that do not require any knowledge of the symmetry operators. These could be powerful numerical tools to deal with quantum many-body complex open systems. The first approach that is based on Gram-Schmidt orthonormalization of density matrices allows us to obtain all the steady states, whereas the second one based on large deviations allows us to obtain the non-degenerated maximum and minimum current carrying states. We discuss the symmetry-decomposition and the orthonormalization methods with the help of an open para-benzene ring and examine interesting scenarios such as the dynamical restoration of Hamiltonian symmetries in the long-time limit and apply the method to study the eigenspacing statistics of the nonequilibrium steady state.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
Abstract We study the null space degeneracy of open quantum systems with multiple non-abelian, strong symmetries. By decomposing the Hilbert space representation of these symmetries into an irreducible representation involving the direct sum of multiple, commuting, invariant subspaces we derive a tight lower bound for the stationary state degeneracy. We apply these results within the context of open quantum many-body systems, presenting three illustrative examples: a fully-connected quantum network, the XXX Heisenberg model and the Hubbard model. We find that the derived bound, which scales at least cubically in the system size the SU(2) symmetric cases, is often saturated. Moreover, our work provides a theory for the systematic block-decomposition of a Liouvillian with non-abelian symmetries, reducing the computational difficulty involved in diagonalising these objects and exposing a natural, physical structure to the steady states—which we observe in our examples.
Markovian open quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate toward the steady state, but it does not necessarily give a correct estimate of the relaxation time because the crossover time to the asymptotic regime may be too long. We here give a rigorous upper bound on the transient decay of autocorrelation functions in the steady state by introducing the symmetrized Liouvillian gap. The standard Liouvillian gap and the symmetrized one are identical in an equilibrium situation but differ from each other in the absence of the detailed balance condition. It is numerically shown that the symmetrized Liouvillian gap always gives a correct upper bound on the decay of the autocorrelation function, but the standard Liouvillian gap does not.
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 characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent , and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbladian evolution (spin, fermions, bosons, …), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other diagonalization techniques and retrieves the Liouvillian low-lying spectrum even for system sizes for which it would be impossible to perform exact diagonalization.
We discuss the numerical solution methods available when solving for the steady-state density matrix of a time-independent open quantum optical system, where the system operators are expressed in a suitable basis representation as sparse matrices. In particular, we focus on the difficulties posed by the non-Hermitian structure of the Lindblad super operator, and the numerical techniques designed to mitigate these pitfalls. In addition, we introduce a doubly iterative inverse-power method that can give reduced memory and runtime requirements in situations where other iterative methods are limited due to poor bandwidth and profile reduction. The relevant methods are demonstrated on several prototypical quantum optical systems where it is found that iterative methods based on iLU factorization using reverse Cuthill-Mckee ordering tend to outperform other solution techniques in terms of both memory consumption and runtime as the size of the underlying Hilbert space increases. For eigenvalue solving, Krylov iterations using the stabilized bi-conjugate gradient method outperform generalized minimal residual methods. In contrast, minimal residual methods work best for solvers based on direct LU decomposition. This work serves as a guide for solving the steady-state density matrix of an arbitrary quantum optical system, and points to several avenues of future research that will extend the applicability of these classical algorithms in absence of a quantum computer.