We show that open quantum systems of two-level atoms symmetrically coupled to a single-mode photon field can be efficiently simulated by applying a SU(4) group theory to quantum master equations. This is important since many foundational examples in quantum optics fall into this class. We demonstrate the method by finding exact solutions for many-atom open quantum systems such as lasing and steady-state superradiance.
We show that open quantum systems of two-level atoms symmetrically coupled to a single-mode photon field can be efficiently simulated by applying a SU(4) group theory to quantum master equations. This is important since many foundational examples in quantum optics fall into this class. We demonstrate the method by finding exact solutions for many-atom open quantum systems such as lasing and steady-state superradiance.
We present a detailed numerical study of the one-atom laser, that is, a single two-level atom interacting with one lasing mode, whereby both the atom and the photon field are coupled to reservoirs. The stationary as well as the dynamical properties of the model are calculated directly from the quantum master equation with the help of two numerical methods. These numerical methods do not need any quasi-probability representation and they do not require approximations. We find that the one-atom laser exhibits most of the typical features of a normal laser. In the region far below threshold some aspects, among them the linewidth, are changed due to eigenvalues of the master equation with imaginary parts. In this regime the complexity of the eigenvalues prominently enters the dynamical behavior.
In this thesis a formalism for indistinguishable multi-level quantum emitters in quantum master equations is developed. The complexity of the approach scales only polynomially in the number of quantum emitters. Complexity here means the number of coupled equations or rather the dimension of the Liouville space. This approach opens new possibilities for calculating open quantum systems. The method is implemented in the PsiQuaSP library, which allows to setup and solve arbitrary master equations, in particular master equations with the reduced polyno-mial scaling. The introduced tools are utilized to study subradiance in the open Dicke model and various cQED lasers. One of the main current research goals in quantum optics and quantum information science is the generation and stabilization of quantum coherence between distinct quantum systems. These systems can for instance be photons or quantum emitters, qubits. From a theoretical stance the difficulty with these systems is that their complexity scales in general exponentially with the system size, like the number of quantum emitters. The reduced scaling of the formalism introduced in this thesis stems from the permutation symmetry of the indistinguishable emitters. The method is derived in the context of Lindblad quantum master equations. Lindblad quantum master equations for many identical multi-level systems have been used to study funda- mental quantum optical systems, such as lasing and laser like action, various phase transitions, optical bistability, cooperative resonance fluorescence, entanglement, quantum light generation, quantum to classical transitions, super- and subradiance etc. In most of these cases the single emitter, the many emitter and the weak correlation limit can be satisfactorily treated with existing techniques, such as direct integration, phase space methods or cluster expansion. However these techniques are not well suited for the few emitter case with strong correlations. The theory developed in this thesis allows to study all these systems for moderate numbers of quantum emitters and arbitrary correlation strengths, thus filling the gap left open by conventional methods. Even though the theory is developed around Lindblad quantum master equations it is not limited to this case – the requirement of indistinguishability is used to construct symmetrized Liouville space basis states and operators which can also be used to construct other master equations. In this thesis the general methodology for the polynomial scaling of the many emitter master equation is derived by two different approaches. Generalized permutation symmetric Liouville space states and operators are introduced that allow for the construction of arbitrary master equations and observables. These Liouville space states and operators are translated into a graphical representation that greatly facilitates their usage. This representation is found to have a close connection to Lie algebras as well as graph theory. The whole method is implemented in a general and modular way in a C++ library called PsiQuaSP, which allows to simulate arbitrary master equations based on a number state representation. The design of the library heavily relies on the introduced graphical representation, which greatly facilitates the setup of a simulation. The treatment of non-identical systems with PsiQuaSP is illustrated and further techniques for complexity reduction and computational speedup are discussed. These tools are applied to the two main cases of cQED lasers/spasers and super-, subradiance. In the latter case a new type of phase transition in the open Dicke model is predicted that leads to a deterministic generation of subradiant steady state coherences, with applications in quantum information science. Furthermore the interaction of quantum dots with the dissipative modes at metal interfaces is investigated in order to explain experimental findings.
In the framework of a general theory of open quantum systems, we obtain a quantum master equation for the two-level atom interacting with a single mode of the electromagnetic field. On this basis, we obtain new optical equations. In comparison with the conventional Bloch-Feynman optical equations which contain three phenomenological parameters, the new equations contain nine phenomenological parameters. New physical effects , due to the opening of the system, are predicted: 1) a resonance frequency shift, 2) an asymmetry of the spectral line, 3) a modification of the dependence of the absorbtion coefficient and of the nonlinear dephasing on the atomic detuning, and 4) an additional nonlinear dependence of the absorbtion coefficient on the electric field. For a rather small atom density when the fourth effect can be neglected,we found an analytical expression of the transmittivity characteristic of an optical bistable Fabry-Perot resonator and found a good agreement with the experiment.
Abstract Chapter 22 investigates the dynamics of open quantum systems that interact with their environment. We discuss the time-evolution of the joint system in terms of a time-ordered exponential of an integral over the Hamiltonian and determine a similar formulation for a Liouvillian super-operator. Based on these ideas we develop the notion of dynamical maps and apply the Born-Markov approximation to arrive at a dynamical semi-group and general Markovian master equations. We then turn to a specific example of such an equation of motion for the density operator: the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation which captures both dissipative dynamics and decoherence using Lindblad operators. We employ the GKLS equation to model the absorption, and spontaneous as well as stimulated emission of radiation by a two-level atom. We then discuss the Jaynes-Cummings model that describes the interaction of such an atom with a single mode of the electromagnetic field in a cavity
We propose the use of quantum optical systems to perform universal simulation of quantum dynamics. Two specific implementations that require present technology are put forward for illustrative purposes. The first scheme consists of neutral atoms stored in optical lattices, while the second scheme consists of ions stored in an array of micro--traps. Each atom (ion) supports a two--level system, on which local unitary operations can be performed through a laser beam. A raw interaction between neighboring two--level systems is achieved by conditionally displacing the corresponding atoms (ions). Then, average Hamiltonian techniques are used to achieve evolutions in time according to a large class of Hamiltonians.
A spectrally pure coherent light source is an important component in precision measurement applications, such as an atomic clock. The more spectrally pure the coherent light source, or the narrower the linewidth of its power spectrum, the better for atomic clock experiments. A coherent light light source, such as a laser, is intrinsically an open quantum system, meaning that it gains and loses energy from an external environment.\nThe aim of this thesis is to study various open quantum systems in an attempt to discover a scheme in which an extremely spectrally pure coherent light source might be realized. Therefore, this thesis begins by introducing the two main approaches to treating open quantum systems, the quantum master equation approach, and the quantum Langevin equation approach. In addition to deriving these from first principles, many of the solution methods to these approaches are given and then demonstrated using computer simulations. These include the quantum jump algorithm, the quantum state diffusion algorithm, the cumulant expansion method, and the method of c-number Langevin equations.\nUsing these methods, the theory of the crossover between lasing and steady state superradiance is presented. It is shown that lasing and steady state superradiance might be demonstrated in the same physical system, but in different parameter regimes. The parameter space between these two extreme limits is explored, and the benefits and drawbacks of operating a system at a given set of parameters, i.e. to achieve the most spectrally pure light source, are discussed.\nWe also consider the phase stability of a laser that is locked to a cavity QED system comprised of atoms with an ultra-narrow optical transition. Although the atomic motion introduces Doppler broadening, the standing wave nature of the cavity causes saturated absorption, which can be used to achieve an extremely high degree of phase stabilization. The inhomogeneity introduced by finite atomic velocities can also cause optical bistability to disappear, resulting in no regions of dynamic instability that would otherwise restrict operational parameters in the experiment to be tuned outside of the optimum region where the minimum linewidth occurs.