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Identical emitters, collective effects and dissipation in quantum optics

Michael Gegg

Published Jan 1, 2017• 4 citations
Publisher

Abstract

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.

DOI
10.14279/depositonce-6526

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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 a recent publication we showed that permutation symmetry reduces the numerical complexity of Lindblad quantum master equations for identical multi-level systems from exponential to polynomial scaling. This is important for open system dynamics including realistic system bath interactions and dephasing in, for instance, the Dicke model, multi-Λ system setups etc. Here we present an object-oriented C++ library that allows to setup and solve arbitrary quantum optical Lindblad master equations, especially those that are permutationally symmetric in the multi-level systems. PsiQuaSP (Permutation symmetry for identical Quantum Systems Package) uses the PETSc package for sparse linear algebra methods and differential equations as basis. The aim of PsiQuaSP is to provide flexible, storage efficient and scalable code while being as user friendly as possible. It is easily applied to many quantum optical or quantum information systems with more than one multi-level system. We first review the basics of the permutation symmetry for multi-level systems in quantum master equations. The application of PsiQuaSP to quantum dynamical problems is illustrated with several typical, simple examples of open quantum optical systems.
Quantum cooperativity is evident in light-matter platforms where quantum emitter ensembles are interfaced with confined optical modes and are coupled via the ubiquitous electromagnetic quantum vacuum. Cooperative aspects such as dipole-dipole interactions and subradiance find applications in the design of nanoscale coherent light sources and highly-reflective quantum metasurfaces made up of hundreds of optically trapped atoms, in the implementation of topological quantum optics on subwavelength arrays of emitters, in quantum metrology and quantum information. The quick bursts of radiation from a collection of quasi-indistiguishable emitters provides an alternative approach to standard lasers by introducing superradiant lasers operating at extremely low intracavity power. This tutorial provides a set of theoretical tools to tackle the behavior responsible for the onset of cooperativity in light-matter systems by extending open quantum system dynamics methods, such as the master equation and quantum Langevin equations, to electron-photon interactions in strongly coupled and correlated quantum emitters ensembles. These analytical approaches are then also extended to frequency disordered or vibronically coupled quantum emitter ensembles with wide relevance ranging from atoms in optical lattices, quantum dots in solid state environments or molecular quantum systems.
Multi-photon lasing has been realized in systems with strong nonlinear interactions between emitters and cavity modes, where single-photon processes are suppressed. Coherence between the internal states of a quantum emitter, or among multiple emitters, plays a key role. Such continuous nonclassical sources of light can find applications in quantum computation, quantum sensing, quantum metrology, and quantum communication. This thesis explores the multi-photon lasing phenomena in various quantum dot-photonic crystal cavity quantum electrodynamic (QED) setups. Exciton-phonon interactions are inevitable in such systems and are incorporated using the polaron-transformed master equation. The Born-Markov approximation is employed to obtain the reduced density matrix rate equation. Using quantum laser theory, we derived the Scully-Lamb laser rate equations and evaluated the single- and multi-photon excess emission rates defined as the difference between emission and absorption rates into the cavity mode without mean-field approximations. We investigated cooperative two-photon lasing, correlated emission lasing, hyperradiant lasing, non-degenerate two-mode two-photon lasing, and continuous variable entanglement in open quantum systems with single or multiple semiconductor quantum dots (two-level, three-level, and four-level) driven coherently/incoherently and coupled to single/ bimodal cavities.
Reduced abstract. This Thesis explores emergent cooperative phenomena in collective light-matter systems. We study ensembles of interacting quantum emitters coherently driven by a laser field and coupled to photonic structures, focusing on the hybrid description of emitters dressed by light. The interplay among quantum emitters, coherent driving, and photonic environments reveals a rich landscape of cooperative effects. While single-emitter dressing has been widely studied, we address collective phenomena in two non-identical emitters modelled as two-level systems. Strong interaction forms a dimer exhibiting superradiant and subradiant states, with two-photon resonances directly connecting ground and doubly excited states. This nonlinear process, central to the Thesis, enables new regimes of cooperative light-matter physics. Analytical studies show how interactions reshape observables such as emission intensity, photon statistics, and spectra, offering implications for quantum metrology. The sensitivity of two-photon processes to emitter distance and driving strength enables high-precision sensing and sub-wavelength imaging. Unexpectedly, we find that off-resonant virtual states may gain population through dissipation, redefining their role in open systems. To capture this, we develop a hierarchical adiabatic elimination method for metastable dynamics. We also analyze entanglement in emitters coupled to a lossy cavity, identifying five mechanisms, including the frequency-resolved Purcell effect introduced here. This selective enhancement stabilizes cooperative states and enables scalable entanglement. Our models, tailored to solid-state platforms such as quantum dots and molecular aggregates, address challenges like inhomogeneous broadening and decoherence, demonstrating the feasibility of harnessing cooperative light-matter effects for quantum technologies.
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.
Quantum cooperativity is evident in light-matter platforms where quantum-emitter ensembles are interfaced with confined optical modes and are coupled via the ubiquitous electromagnetic quantum vacuum.Cooperative effects can find applications, among other areas, in topological quantum optics, in quantum metrology, or in quantum information.This tutorial provides a set of theoretical tools to tackle the behavior responsible for the onset of cooperativity by extending open quantum system dynamics methods, such as the master equation and quantum Langevin equations, to electron-photon interactions in strongly coupled and correlated quantum-emitter ensembles.The methods are illustrated on a wide range of current research topics such as the design of nanoscale coherent-light sources, highly reflective quantum metasurfaces, or low intracavity power superradiant lasers.