In this chapter we have presented a new approach to the treatment of electronic correlations in a nanostructure coupled to continuous electronic states, phonons, and photons. For such a system, the direct solution of the vNL equation is only possible if the electronic Hilbert space is small, which is the case for a single or few emitters and a single cavity mode with a limited number of photons. On the other hand, an EoM approach, in connection with the CE method to truncate the infinite hierarchy of equations, has been used in the past as a valuable method to describe luminescence-related phenomena, laser emission and photon correlations for systems with many QDs or other active materials with a continuous density of states. We have addressed the situation in finite-sized systems, in which the small number of electronic degrees of freedom plays an important role. This is for example the case in QDs with few confined states. We have demonstrated that boundary conditions play an important role and lead to an enhancement of correlations. For this, we have devised a formalism that combines the exact representation of the electronic degrees of freedom of the vNL approach, with the truncation of the photonic hierarchy from the CE method, resulting in the FSH method.
The second major point addressed in this chapter is the inclusion of scatter-ing and dephasscatter-ing by usscatter-ing the Lindblad formalism in the EoM-based approaches.
This regards scattering and dephasing in a consistent manner and on equal footing with the Hamiltonian contributions to the time evolution of the system. A correct
treatment of dissipative processes is a key requirement for making quantitative pre-dictions in correlated systems. In earlier attempts, simpler models for the dephasing have been used to obtain an estimate for the impact of correlations, such as adding an estimated constant dephasing term to the EoM. Such an approach is subject to artifacts that are overcome by the presented theory.
The FSH method allows for a description of much larger systems than it is possible by means of the vNL equation, the emission into free space via a mode continuum.
For a single QD we have presented free-space emission spectra comprising multi-excitonic effects, as well as time-resolved photoluminescence for a QD coupled to a microcavity mode. The latter case allows for a comparison with the vNL equation to benchmark the theory. The outcome is that the FSH method provides an accurate description of the dynamics predicted by the vNL equation.
Expectation Value
Based Cluster Expansion
≈ + +
In this chapter a new method to formulate equations of motion (EoM) for open quantum many-particle systems is presented. Our approach allows for a numerically exact treatment as well as for approximations necessary in large systems and can be applied to systems involving both bosonic and fermionic particles. The method generalizes the cluster expansion (CE) technique by using expectation values (EVs) instead of correlation functions (CFs), which we will term expectation value based cluster expansion (EVCE). The use of EVs not only makes the equations more trans-parent, but also considerably reduces the amount of algebraic effort to derive the equations. The proposed formulation offers a unified view on various approxima-tion techniques presented recently in the literature. The convergence properties of the EVCE are studied for the Jaynes-Cummings model (JCM) explicitly, and three additional examples for the application of the EVCE are shown schematically.
Parts of this chapter are published in [Leymann et al., 2014, Leymann et al., 2013b]. The basic theoretical concept of the EVCE where developed by H.A.M. mann in collaboration with A. Foerster, the EoM where mainly derived H.A.M. Ley-mann, the numerical integration of the EoM was mainly done by A. Foerster, all authors of [Leymann et al., 2014, Leymann et al., 2013b] discussed the results and physical implications of the results.
4.1 Numerical approaches for interacting many-particle systems
The finite size hierarchy described in chapter 3 was developed with a specific ap-plication in mind, in this chapter we introduce the EVCE as a general technique to describe interacting many-particle systems. In order to do justice to this general approach we need to reintroduce some concepts already introduced in the former chapter in a slightly different language or form, and the reader is kindly asked to pardon for inevitable repetitions. Interacting many-particle systems can drive strong correlations between the interacting particles. A straight forward way to describe the dynamics of correlated many-particle systems is to directly derive the equations
of motion for the quantities of interest. Equation of motion (EoM) techniques have been used successfully to realize microscopic descriptions of quantum systems, and are a way to systematically incorporate many-particle correlations into the descrip-tion of exciton dynamics in quantum wells [Hoyer et al., 2003], ultracold Bose gases [Köhler and Burnett, 2002], spin dynamics [Kapetanakis and Perakis, 2008], pho-toluminescence [Kira et al., 1998], resonance fluorescence [Kira et al., 1999], cavity phonons [Kabuss et al., 2012,Kabuss et al., 2013], cavity-quantum-electrodynamics [Carmele et al., 2010b], and microcavity quantum dot (QD) lasers [Gies et al., 2007].
The basic idea of EoM approaches is to truncate the unfolding hierarchy of differ-ential equations at a certain level to allow for a numerical solution. The details of the truncation depend strongly on the used technique and the investigated system and are the subject of this chapter. Many different formulations and approximation techniques are known in the field of EoM approaches. However, we will distinguish between two basic types of formulations using correlation functions (CFs) [Wiersig et al., 2009,Kapetanakis and Perakis, 2008,Kira et al., 1998,Kira et al., 1999,Hoyer et al., 2003, Hohenester and Pötz, 1997] as in the CE [Fricke, 1996a, Hoyer et al., 2004] on the one hand and EVs (or density matrix elements) on the other hand [Gart-ner, 2011, Richter et al., 2009, Witthaut et al., 2011, Kabuss et al., 2012, Carmele et al., 2010b, Richter et al., 2015]. The formulation in CFs is algebraically demand-ing but has proven to be very effective in approximately describdemand-ing large systems.
Expectation value based formulations are algebraically less demanding and produce a linear and very clear system of EoM, but are usually limited to small systems.
The proposed approach combines the two formulations (in CFs and EVs) with their respective advantages and adds a new perspective on former techniques used in the literature.
The outline of this chapter is as follows. In Sec. 4.2, we will revisit the general concept of CFs and the factorization of EVs. The approximation techniques pre-sented in Sec. 4.2 are the basis for the truncation variants prepre-sented in Sec. 4.3.
Section 4.3 is devoted to the derivation of EoM and we show how the introduced formulation can be used to truncate the unfolding hierarchy of EoM. We focus on the truncation of EoM for systems involving bosons and fermions and provide de-tails on the various truncation possibilities. In Sec. 4.4, we will give an example for the EoM of a coupled quantum system and show how different truncation schemes result in known models.