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Quantum Information and Computation

Generalized Dicke states

HartmannStephan

Published Nov 1, 2016• 0 citations
Publisher

Abstract

Quantum master equations are an important tool in quantum optics and quantum information theory. For systems comprising a small to medium number of atoms (or qubits), the non-truncated equations ar...
DOI
10.5555/3179439.3179444

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Quantum master equations are an important tool in quantum optics and quantum information theory. For systems comprising a small to medium number of atoms (or qubits), the non-truncated equations are usually solved numerically. In this paper, we present a group-theoretical superoperator method that helps solving these equations. To do so, we exploit the SU(4)-symmetry of the respective Lindblad operator and construct basis states that generalize the well-known Dicke states. This allows us to solve various problems analytically and to considerably reduce the complexity of problems that can only be solved numerically. Finally, we present three examples that illustrate the proposed method.
Abstract Characterizing quantum entanglement in mixed states is a longstanding challenge. Among the various methods available, conditional entropies serve as a powerful tool. Notably, the AR q-conditional entropy introduced by Abe and Rajagopal in 2002 has demonstrated significant promise as it often surpasses other entropy-based criteria. The wide-ranging applications of conditional entropy in quantum information underscore the importance of studying and analyzing it for a deeper understanding of quantum correlations and their implications. In this paper, we investigate the non-separability of noisy Dicke states using the AR approach of conditional entropy. Our findings reveal that the entropic criterion is equally effective as the PPT criterion in identifying non-separability across a large subset of N -partite noisy Dicke states with even N and excitation number k=N/2k = N/2k=N/2 k = N / 2 . Additionally, for systems with N>30N > 30N>30 N > 30 and k=1k=1k=1 k = 1 , the separability thresholds derived from both criteria converge within 10−810^{-8}10−8 10 - 8 , highlighting their strong agreement in this parameter range. Furthermore, we established a condition based on AR q-conditional entropy for identifying genuine multipartite entanglement (GME) in noisy Dicke states and compared its effectiveness to previous methods. Notably, our condition identifies a broader range of GME, particularly when the number of excitations approaches half the number of qubits (i.e., N /2). In contrast, previous methods perform better when the number of excitations is significantly less than N /2. We believe these results will pave the way for further advancements in entanglement theory and the development of potential quantum-based applications for conditional entropy.
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.