Entropy-Driven Dynamics of Open Quantum Systems

The quantum computer promises to be superior to its classical counterpart for a vast amount of complex problems. The range of potential applications includes database search, prime number factorization and simulation of various physical processes. Moreover, recent advances in quantum machine learning raises expectations that it can provide a solution for big data analysis which is in such a big demand now. Unfortunately, our theoretical and experimental efforts have not resulted yet in working devices because of a single phenomenon, decoherence. The unavoidable interaction of a quantum system with its environment forbids us to see quantum miracles in the real world. Two very quantum properties, the superposition between quantum states and entanglement, which are in the heart of quantum algorithms, disappear almost instantaneously by interacting with classical objects. Quantum superposition, or coherence, usually decays exponentially, whereas entanglement exhibits the sudden death phenomenon, decaying in a finite time. However, recent advances in the theory of open quantum systems show that the system can be protected from the harmful influence of an environment or even exploit it. Discovery of the bath-induced coherence and entanglement, decoherence-free subspaces, quantum Zeno effect, creation and revival of quantum correlations, etc. show that there are many hidden secrets in open quantum systems and we need tools and methods to unveil them. Traditionally, quantum systems exposed to environments are studied by deriving master equations from the underlying microscopic picture. In the weak coupling regime, after the Born and Markov approximations, one obtains the Redfield master equation. Unfortunately, it violates the positivity of a system density matrix, therefore, an additional secular approximation is performed. It neglects fast oscillating terms resulting in an equation of the Lindblad form enjoying all properties of completely positive dynamics. However, there are situations where the restrictions imposed by the secular approximations lead to incorrect predictions. This raises an interest to alternative phenomenological equations which contain information about fast oscillating term and describe physically relevant dynamics. In this thesis, we propose an alternative view on quantum master equations. Motivated by analogy with classical nonequilibrium thermodynamics we assume that the quantum system is driven to equilibrium by the modular Hamiltonian which has a deep connections with the von-Neumann entropy. The resulting modular dynamical semigroup (MDS) preserves the positivity, Hermicity and trace of a density matrix. In addition, it produces non-negative entropy production rates ensuring the second law of thermodynamics. Remarkably, the secular master equation belongs to the class of MDS. Moreover, we provide the new nonlinear master equation which contains fast oscillating terms and does not violate positivity of the system state. Then we perform an extensive analysis to show that the thermodynamic master equation can be successfully applied in the Born-Markov regime. It correctly describes the decay and creation of coherence and quantum correlations in contrast to the secular master equation. The equation yields especially good results in the vicinity of the degenerate spectrum, where the fast oscillating terms play a crucial role. To linearize the equation, we propose a perturbative scheme, treating the nonsecular terms as the perturbation with respect to the secular dynamics and obtain results which match the exact solution in a wide range of parameters. The technique is especially well suited to study the steady states in the presence of multiple reservoirs. This led us to discover that the fast oscillating terms give rise to nonequilibrium thermal entanglement between weakly interacting qubits which was not observed before. Finally, we propose a new way to study quantum-classical systems by properly generalizing the thermodynamic master equation.