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Physica status solidi. C, Conferences and critical reviews/Physica status solidi. C, Current topics in solid state physics

Expectation value based cluster expansion

H. A. M. Leymann, A. Foerster, Jan Wiersig

Published Jun 7, 2013• 12 citations
Publisher

Abstract

Abstract We outline a new formulation of the cluster expansion to derive equations of motion for open quantum systems. Our method uses expectation values instead of correlation functions. The algebraic effort to derive the equations is reduced and the approach is more intuitive. We demonstrate the efficiency of our method by applying it to the damped Jaynes‐Cummings model. This model is a good starting point to investigate quantum dot based laser systems. Further we will examine the convergence properties of the cluster expansion for various orders. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssc.201200711

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Cluster-expansion representation of the quantum dynamics

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In this book, we encounter the hierarchy problem when we apply the equation-of-motion technique to analyze the quantum dynamics of the coupled light–matter system. To obtain systematic approximations, we use the so-called cluster-expansion method where many-body quantities are systematically grouped into cluster classes based on how important they are to the overall quantum dynamics. With increasing complexity, the clusters contain independent single particles (singlets), interacting pairs (doublets), three-particle terms (triplets), coupled four-particle contributions (quadruplets), as well as higher-order correlations. In this context, the N -particle concept is somewhat formal because it refers to generic N -particle expectation values 〈 N 〉, which may consist of an arbitrary mixture of carrier, photon, and phonon operators, as discussed in Section 14.2. In order to truncate the hierarchy problem at a given level, 〈 N 〉 is approximated through a functional structure that includes all clusters up to the predetermined level while all remaining clusters with a higher rank are omitted. It is natural that the corresponding approximations can be systematically improved by increasing the number of clusters included. To the best of our knowledge, the idea of coupled-clusters approaches was first formulated by Fritz Coester and Hermann Kümmel in the 1950s to describe nuclear many-body phenomena. The approach was then modified for the needs of quantum chemistry by Jiri Cizek 1966 to deal with many-body phenomena in atoms and molecules. Currently, it is one of the most accurate methods to compute molecular eigenstates.

Cambridge University Press eBooks•Nov 17, 2011•0 citations
Abstract We outline a new formulation of the cluster expansion to derive equations of motion for open quantum systems. Our method uses expectation values instead of correlation functions. The algebraic effort to derive the equations is reduced and the approach is more intuitive. We demonstrate the efficiency of our method by applying it to the damped Jaynes‐Cummings model. This model is a good starting point to investigate quantum dot based laser systems. Further we will examine the convergence properties of the cluster expansion for various orders. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We present a new method to formulate equations of motion for open quantum many-particle systems. 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 technique by using expectation values instead of correlation functions. The use of expectation values not only makes the equations more transparent but also considerably reduces the amount of algebraic effort to derive the equations. The proposed formulation offers a unified view on various approximation techniques presented recently in the literature. The microscopic semiconductor model for quantum-dot-based microcavity lasers is extended to higher-order photon-autocorrelation functions and the validity of the cluster expansion is shown for this system. We study photon-autocorrelation functions up to fifth order and monitor the onset of lasing in quantum-dot-based microcavity lasers. We observe a successive vanishing of photon bunching in the higher-order photon-autocorrelation functions with increasing pump rates. Our results reveal that the laser threshold is not only softened in microcavity laser systems but is centered around different pump rates with respect to the photon-autocorrelation functions.
Abstract Maxwell‐Bloch and luminescence equations are presented which describe vacuum Rabi splitting and the quantum rungs on the Jaynes‐Cummings ladder for strongly‐coupled dot‐cavity systems. Resonance fluorescence conditions are considered where an optical pump is exciting the dot‐cavity system while the re‐emitted light is detected. An analytical formula for the vacuum Rabi splitting is derived and a pumping mechanism for the direct generation of the second rung is presented and analyzed. An optimum pumping frequency and optimum pumping intensity are identified for the generation of the second rung. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The Jaynes-Cummings model provides a well established theoretical framework for single electron two level systems in a radiation field. Similar exactly solvable models for semiconductor light emitters such as quantum dots dominated by many particle interactions are not known. We access these systems by a generalized cluster expansion, the photon-probability cluster expansion: a reliable approach for few-photon dynamics in many body electron systems. As a first application, we discuss vacuum Rabi oscillations and show that their amplitude determines the number of electrons in the quantum dot.
Abstract The subjects of the paper are the likelihood method (LM) and the expected Fisher information (FI) considered from the point od view of the construction of the physical models which originate in the statistical description of phenomena. The master equation case and structural information principle are derived. Then, the phenomenological description of the information transfer is presented. The extreme physical information (EPI) method is reviewed. As if marginal, the statistical interpretation of the amplitude of the system is given. The formalism developed in this paper would be also applied in quantum information processing and quantum game theory (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
In this book, we encounter the hierarchy problem when we apply the equation-of-motion technique to analyze the quantum dynamics of the coupled light–matter system. To obtain systematic approximations, we use the so-called cluster-expansion method where many-body quantities are systematically grouped into cluster classes based on how important they are to the overall quantum dynamics. With increasing complexity, the clusters contain independent single particles (singlets), interacting pairs (doublets), three-particle terms (triplets), coupled four-particle contributions (quadruplets), as well as higher-order correlations. In this context, the N -particle concept is somewhat formal because it refers to generic N -particle expectation values 〈 N 〉, which may consist of an arbitrary mixture of carrier, photon, and phonon operators, as discussed in Section 14.2. In order to truncate the hierarchy problem at a given level, 〈 N 〉 is approximated through a functional structure that includes all clusters up to the predetermined level while all remaining clusters with a higher rank are omitted. It is natural that the corresponding approximations can be systematically improved by increasing the number of clusters included. To the best of our knowledge, the idea of coupled-clusters approaches was first formulated by Fritz Coester and Hermann Kümmel in the 1950s to describe nuclear many-body phenomena. The approach was then modified for the needs of quantum chemistry by Jiri Cizek 1966 to deal with many-body phenomena in atoms and molecules. Currently, it is one of the most accurate methods to compute molecular eigenstates.