We develop the algebraic method based on the Lie algebra of quadratic combinations of left and right superoperators associated with matrices to study the Lindblad dynamics of multimode bosonic systems coupled a thermal bath and described by the Liouvillian superoperator that takes into account both dynamical (coherent) and environment mediated (incoherent) interactions between the modes. Our algebraic technique is applied to transform the Liouvillian into the diagonalized form by eliminating jump superoperators and solve the spectral problem. The temperature independent effective non-Hermitian Hamiltonian,
H^eff, is found to govern both the diagonalized Liouvillian and the spectral properties. It is shown that the Liouvillian exceptional points are represented by the points in the parameter space where the matrix,
H, associated with
H^eff is non-diagonalizable. We use our method to derive the low-temperature approximation for the superpropagator and to study the special case of a two mode system representing the photonic polarization modes. For this system, we describe the geometry of exceptional points in the space of frequency and relaxation vectors parameterizing the intermode couplings and, for a single-photon state, evaluate the time dependence of the speed of evolution as a function of the angles characterizing the couplings and the initial state.
The Lindblad equation is commonly used to model dissipation and decoherence in open quantum systems. Yet many of the properties of Lindblad dynamics are still little understood. This thesis explores various aspects of the Lindblad equation and its connections with non-Hermitian quantum theories. The evolution speed of Lindblad dynamics is investigated and an explicit expression for the speed is derived. The radial component, connected to the rate at which the purity of the state changes, is shown to be determined by the modified skew information of the Lindblad operators. The semiclassical time evolution of Gaussian states under Lindblad dynamics is then considered and a new type of phase-space dynamics is derived for the centre of a Gaussian Wigner function. By viewing the phase-space Lindblad equation as a Schroedinger equation with a non-Hermitian Hamiltonian, a further set of semiclassical equations are derived. These can describe the interference terms in Wigner functions. Further, a PT-symmetric two-level system driven through two consecutive exceptional points at finite speed is considered and an analytical expression for the probability of a non-adiabatic transition is derived. It is shown that the transition through the exceptional points can be experimentally addressed in a PT-symmetric lattice using Bloch oscillations. Finally, a quantum implementation of this PT-symmetric system using ultracold atoms in an optical lattice with single-particle losses is investigated.
This study investigates the dynamics of two-qubit systems in open quantum environments, with a focus on the influence of exceptional points (EPs) and phase angle on entanglement dynamics. Two-qubit systems are pivotal in quantum information science, as they are the simplest system that reveals quantum entanglement, a crucial resource for quantum computing and communication. The research explores the effects of coupling, spontaneous emission, and environment on two-qubit states, specifically the
X-states, within the framework of the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) or Lindblad master equation. EPs are unique to non-Hermitian systems. They affect the system’s dynamics significantly when the system approaches EP. Analytical and numerical analyses reveal that third-order EPs occur at specific parameters for two-qubit system, leading to critical changes in eigenvalues and eigenvectors of the Liouvillian superoperator. The study demonstrates that EPs can enhance or suppress entanglement. The concurrence, a quantitative measure of entanglement, peaks around the EPs under certain initial conditions. Furthermore, the impact of phase angle,
θ, that parameterizes the relative phase of maximally entangled states is analyzed. For initial states labeled as
ρΨ(θ,p) in this thesis, the phase angle significantly influences concurrence evolution, enabling tunability of entanglement dynamics. In contrast, for other class of initial states labeled as
ρΦ(θ,p), concurrence exhibits phase invariance, ensuring stability across the variations in
θ. Besides this, the analysis shows that the maximum concurrence occurs around EP for some states, the phase angle can also be tuned to increase the concurrence for some initial states. These findings highlight the importance of both EPs and phase angle in optimizing entanglement generation and control. This research offers valuable insights into the interplay between non-Hermitian physics and quantum entanglement, paving the way for advancements in quantum technologies. The outcomes provide possible practical implications in the field of quantum computation.