We point out that the quantum dynamical map of an open quantum system can be generated by an effective Liouville operator. The effective Liouville shows the dynamical breaking of time reversibility. This breaking of reversibility is expressed by the effective Liouville's discrete spectrum having negative imaginary parts. This generated dynamics of open quantum systems is capable of memory effects described by a frequency dependence of the spectrum. When memory effects can be neglected or smoothed out, the effective Liouville generates the well known semi-group dynamics of open systems in the Markov approximation. The spectral analysis of the effective Liouville -with or without memory- allows to represent the quantum dynamical map and expectation values of physical quantities by metric expressions using complex eigenvalues and bi-orthonormal eigenmodes. The long time dynamics is dominated by the zero-mode, which forms the stationary state of the open system. The relaxation time scale $τ$ is set by the inverse of the smallest negative imaginary part of the non-vanishing effective eigenvalues of the effective Liouville. In generic cases of system-environment-coupling this time scale $τ$ is the relevant scale for both, the relaxation of diagonal elements (populations) and the decay of off-diagonal elements (coherences) of an initial density matrix in the stationary state's diagonal representation. It also sets the time scale after which the entropy tends to its stationary value. The effective Liouville and its spectral content thus give a framework to study the dynamic breaking of time reversibility, memory effects, decoherence, relaxation to a stationary state and the role of entropy in open quantum systems.
We point out that the quantum dynamical map of an open quantum system can be generated by an effective Liouville operator. The effective Liouville shows the dynamical breaking of time reversibility. This breaking of reversibility is expressed by the effective Liouville's discrete spectrum having negative imaginary parts. This generated dynamics of open quantum systems is capable of memory effects described by a frequency dependence of the spectrum. When memory effects can be neglected or smoothed out, the effective Liouville generates the well known semi-group dynamics of open systems in the Markov approximation. The spectral analysis of the effective Liouville -with or without memory- allows to represent the quantum dynamical map and expectation values of physical quantities by metric expressions using complex eigenvalues and bi-orthonormal eigenmodes. The long time dynamics is dominated by the zero-mode, which forms the stationary state of the open system. The relaxation time scale τ is set by the inverse of the smallest negative imaginary part of the non-vanishing effective eigenvalues of the effective Liouville. In generic cases of system-environment-coupling this time scale τ is the relevant scale for both, the relaxation of diagonal elements (populations) and the decay of off-diagonal elements (coherences) of an initial density matrix in the stationary state's diagonal representation. It also sets the time scale after which the entropy tends to its stationary value. The effective Liouville and its spectral content thus give a framework to study the dynamic breaking of time reversibility, memory effects, decoherence, relaxation to a stationary state and the role of entropy in open quantum systems.
An equation of motion for open quantum systems incorporating memory effects and initial correlations with the environment is presented in terms of an effective Liouville operator that solely acts on states of the system. The environment can induce memory effects via the frequency dependence of the effective Liouville and initial correlations can be mapped to a shifted frequency dependent initial state within the system. The equation of motion generalizes the well known semi-group dynamic equations. In generic systems the effective Liouville has a non-degenerate zero mode. By probability conservation one can demonstrate that a generic open system reaches, in the long time limit, a stationary state, which is independent of any initial condition.
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.
In this licenciate thesis, based on two articles, we discuss the properties of open quantum systems and investigate how their dynamics can be used to design a synthetic lattice framework and how their spectrabehave based on two. The first article presents an extension of the Fock state lattice (FSL) framework to open quantum systems. Using the Liouville space representation, density matrices can be vectorized and the dynamics are generated by a linear operator, the Liouvillian, allowing for the introduction of Liouville-Fock state lattices (LFSLs). This framework provides an intuitive and visual understanding of the dynamics of open quantum systems and shows how environmental effects can give rise to phenomena such as directional transport, sources and sinks, typical phenomena occurring in classical processes. As an example, we show how frustration in LFSL can be induced by the environment. The second article in the work focuses on non-normality of Liouvillians and spectral diagnostics of chaos. Non-normal operators are characterized by the fact that their eigenvectors are not orthogonal, which leads to the spectrum exhibiting strong sensitivity to perturbations. We show that strong non-normality can give rise to random matrix-like spectral statistics without the system dynamics exhibiting chaotic signatures. Therefore, we conclude that this violates established assumptions and points to the need for alternative methods to characterize chaotic dynamics in open quantum systems.
The mixed quantum-classical Liouville equation (QCLE) provides an approximate perturbative framework for describing the dynamics of systems with coupled quantum and classical degrees of freedom of disparate thermal wavelengths. The evolution governed by the Liouville operator preserves many properties of full quantum dynamics, including the conservation of total population, energy, and purity, and has shown quantitative agreement with exact quantum results for the expectation values of many observables where direct comparisons are feasible. However, since the QCLE density matrix operator is obtained from the partial Wigner transform of the full quantum density matrix, its matrix elements can have negative values, implying that the diagonal matrix elements behave as pseudo-densities rather than densities of classical phase space. Here, we compare phase-space distributions generated by exact quantum dynamics with those produced by QCLE evolution from pure quantum initial states. We show that resonance effects in the off-diagonal matrix elements differ qualitatively, particularly for low-energy states. Furthermore, numerical and analytical results for low-dimensional models reveal that the QCLE can violate the positivity of marginal phase-space densities, a property that should hold at all times for any physical system. A perturbative analysis of a model system confirms that such violations arise generically. We also show that the violations of positivity of the marginal densities vanish as the initial energy of the system increases relative to the energy gap between subsystem states. These findings suggest that a negativity index, quantifying deviations from positivity, may provide a useful metric for assessing the validity of mixed quantum\textendash{}classical descriptions.
Dissipation caused by nonhomogeneous bosonic finite chain environments (implementable in, e.g., segmented ion traps) is investigated through an exact diagonalization approach, avoiding common approximations used to describe open systems. Different spectral densities, including band gaps, can be engineered to separately assess different factors leading to memory effects, namely, the environment's size, temperature, proximity of the cutoff frequency, the spectral density's shape (sub-Ohmic, Ohmic, super-Ohmic), and the strength of its coupling to the system. Non-Markovianity is quantified with two recently introduced measures related to information backflow and nondivisibility of the system dynamical map. By sweeping the bath spectrum via tuning of the system frequency we show strongest memory effects at band-gap edges and provide an interpretation based on energy flow between system and environment. A system weakly coupled to a stiff chain ensures a Markovian dynamics, while the size of the environment as well as the local density of modes are not substantial factors. We show an opposite effect when increasing the temperature inside or outside the spectral band gap. Further, non-Markovianity arises for larger (negative and positive) powers of algebraic spectral densities, being the Ohmic case not always the most Markovian one.
The mixed quantum-classical Liouville equation (QCLE) provides an approximate perturbative framework for describing the dynamics of systems with coupled quantum and classical degrees of freedom of disparate thermal wavelengths. The evolution governed by the Liouville operator preserves many properties of full quantum dynamics, including the conservation of total population, energy, and purity, and has shown quantitative agreement with exact quantum results for the expectation values of many observables where direct comparisons are feasible. However, since the QCLE density matrix operator is obtained from the partial Wigner transform of the full quantum density matrix, its matrix elements can have negative values, implying that the diagonal matrix elements behave as pseudo-densities rather than densities of classical phase space. Here, we compare phase-space distributions generated by exact quantum dynamics with those produced by QCLE evolution from pure quantum initial states. We show that resonance effects in the off-diagonal matrix elements differ qualitatively, particularly for low-energy states. Furthermore, numerical and analytical results for low-dimensional models reveal that the QCLE can violate the positivity of marginal phase-space densities, a property that should hold at all times for any physical system. A perturbative analysis of a model system confirms that such violations arise generically. We also show that the violations of positivity of the marginal densities vanish as the initial energy of the system increases relative to the energy gap between subsystem states. These findings suggest that a negativity index, quantifying deviations from positivity, may provide a useful metric for assessing the validity of mixed quantum-classical descriptions.