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.
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.
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.
The previous configuration-space cluster integral treatment of the properties of the ground state of a many-boson system is modified by including those higher order diagrams consistent with the pair excitation approximation, which were previously omitted. The cluster expansions involve a parameter, analogous to the fugacity in classical expansions, whose definition automatically accounts for the depletion of the free particle ground state. The resulting expectation value for the Hamiltonian is in agreement with that obtained by field-theoretic methods for states of pair-excitation type.
We extend a standard stochastic theory to study open quantum systems coupled to a generic quantum environment. We exemplify the general framework by studying a two-level quantum system coupled bilinearly to the three fundamental classes of non-interacting particles: bosons, fermions, and spins. In this unified stochastic approach, the generalized stochastic Liouville equation (SLE) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied. Anharmonic effects of a non-Gaussian bath are precisely encoded in the bath multi-time correlation functions that noise variables have to satisfy. Starting from the SLE, we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi-time correlation functions in a complete basis of orthonormal functions. The general hierarchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics. For bosonic bath models, our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes. Similar efficiency and flexibility can be achieved for the fermionic bath models within our formalism. The spin bath models can be simulated with two complementary approaches in the present formalism. (I) They can be viewed as an example of non-Gaussian bath models and be directly handled with the general hierarchical equation approach given their multi-time correlation functions. (II) Alternatively, each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism.
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.
We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of noninteracting particles (bosons or fermions) on an arbitrary lattice of N sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the N=2 case, we show that RQME gives results which agree with exact analytical results for steady-state properties and with exact numerics for time-dependent properties over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in nonequilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed-form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED, and far-from-equilibrium quantum dot experiments.