In the following, we study the dissipative time evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the uni- tary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad approach to open quantum systems, one can perform the rotating-wave approximation with respect to the interacting or noninteracting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different timescales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behavior of fidelity between them versus interoscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical interoscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger interoscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of interoscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.
In the following, we study the dissipative time evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the uni- tary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad approach to open quantum systems, one can perform the rotating-wave approximation with respect to the interacting or noninteracting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different timescales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behavior of fidelity between them versus interoscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical interoscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger interoscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of interoscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.
In the following, we study the dissipative time-evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the unitary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) approach to open quantum systems, one can perform the rotating wave approximation with respect to the interacting, or non-interacting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different time-scales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behaviour of fidelity between them versus inter-oscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical inter-oscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger inter-oscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of inter-oscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.
This dataset contains: 1-Fidelity between the local and global steady state versus inter-oscillator coupling, 2- The evolution of fidelity between the local staedy state and evolved local steady state under the evolution of exact approach as well as the evolution of fidelity between the global staedy state and evolved global steady state under the evolution of exact approach, 3- Critical inter-oscillator coupling versus the temperature of the right bath, for the paper: "An open harmonic chain: Exact vs global and local reduced dynamics" submitted to Physical Review A.
The Lindblad theory of open quantum systems has been successfully applied in various fields ranging from quantum optics, condensed matter physics, and quantum thermodynamics to quantum chemistry. Nevertheless, there are situations where we can compare its predictions to those of exact methods or numerically exact results and find discrepancies between them. One example is the case where a single harmonic oscillator is coupled to a thermal bath of independent harmonic oscillators. In this case, for certain parameters, the exact master equation describes equilibration of the system to a thermal state with a temperature different from the one predicted by the Lindblad master equation. The Lindblad theory, based on a weak coupling between the system and environment as well as the Born-Markov approximation, predicts that the system will evolve to a thermal state whose temperature is equal to the initial temperature of the bath. This result is a consequence of the Born approximation, which supposes that throughout the evolution, the bath only fluctuates around its initial equilibrium state. In this master thesis we study the validity of the Born approximation as well as its consistency with other approximations. By starting from a Hamiltonian for the total system we show that, if the the Born approximation is not invoked, the dynamics of the system is determined by a hierarchy of equations for matrices which can be used to reconstruct the reduced density matrix of the system. This hierarchy approach is fundamentally different from the standard approach, which describes the system dynamics with a weak-coupling master equation, because the hierarchy does not restrict the amount of entanglement possible between the system and the environment. Furthermore, we show how invoking the Born approximation reduces to the usual result and how a generalised ansatz, which we refer to as a generalised Born approximation, reduces the hierarchy to a master equation where the bath temperature is an additional degree of freedom. Finally, we analyse the conservation of the energy of the total system and its relationship to the Markov and secular approximations, and discuss whether imposing it as a condition can fix the time-dependent temperature.
Identifying which master equation is preferable for the description of a multipartite open quantum system is not trivial and has led in recent years to the local vs global debate in the context of Markovian dissipation. We treat here a paradigmatic scenario in which the system is composed of two interacting harmonic oscillators A and B, with only A interacting with a thermal bath---collection of other harmonic oscillators---and we study the equilibration process of the system initially in the ground state with the bath finite temperature. We show that the completely positive version of the Redfield equation obtained using coarse-grain and an appropriate time-dependent convex mixture of the local and global solutions give rise to the most accurate approximations of the whole exact system dynamics, i.e., both at short and at long timescales, outperforming the local and global approaches.
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.
A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature T>0 . We show that for fixed, small values of the coupling constant λ , the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is O(|λ|1/4) for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy O(|λ|1/4) for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as 1/t4 or faster, which is a significant improvement relative to the previously required exponential decay.