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Communications in Theoretical Physics

Green Function and Perturbation Method for Dissipative Systems Based on Biorthogonal Basis

Zhang Li, Gao Yi-Bo, Cheng Hai Wang

Published Jun 1, 2009• 5 citations
Publisher

Abstract

Based on the approach of biorthogonal basis, we carry out the quasinormal modes (QNMs) expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Hermitian system, the eigenfunction perturbation expansions and Green function method, which are based on the orthogonal eigenvectors of the Hermitian Hamiltonian for the closed quantum system, can be generalized in terms of the biorthogonal basis, the two sets of eigenfunctions of H and its adjointness H † . The time-independent perturbation theory for the complex frequencies can be also developed.

DOI
10.1088/0253-6102/51/6/11

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Based on the approach of biorthogonal basis, we carry out the quasinormal modes (QNMs) expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Hermitian system, the eigenfunction perturbation expansions and Green function method, which are based on the orthogonal eigenvectors of the Hermitian Hamiltonian for the closed quantum system, can be generalized in terms of the biorthogonal basis, the two sets of eigenfunctions of H and its adjointness H † . The time-independent perturbation theory for the complex frequencies can be also developed.
Dissipative quantum systems are sometimes phenomenologically described in terms of a non-Hermitian Hamiltonian HHH, with different left and right eigenvectors forming a biorthogonal basis. It is shown that the dynamics of waves in open systems can be cast exactly into this form, thus providing a well-founded realization of the phenomenological description and at the same time placing these open systems into a well-known framework. The formalism leads to a generalization of norms and inner products for open systems, which in contrast to earlier works is finite without the need for regularization. The inner product allows transcription of much of the formalism for conservative systems, including perturbation theory and second quantization.
Dissipation can sometimes be described by a non-Hermitian Hamiltonian H , whose left and right eigenvectors { f j , f j } form a bi-orthogonal basis (BB). For waves in a class of open systems, this is known to lead to exact, complete BB expansions if ⟨ f j | f j ⟩≠0 for all j . If not, normalization seems impossible and many familiar formulae fail; examples are given. The problem is related to the merging of eigenmodes, so that H can only be diagonalized to Jordan blocks. The resolution involves a generalized BB containing extra vectors, whose dynamics are modified by polynomials in the time t . The splitting of merged modes under a perturbation is also treated. One thus obtains a non-trivial extension of the BB formalism for open systems.
The concept of eigenfunction expansions for the wave equation is generalized to open systems, in which waves escape to the outside. These non-conservative systems are non-Hermitian in the usual sense. It is shown that the natural framework is an eigenfunction expansion within a two-component formalism that treats the wavefunction and its conjugate momentum together. Provided the system approaches spatial infinity rapidly `without tails', and possesses spatial discontinuities, the expansion in terms of the eigenfunctions (which are now quasinormal modes) is shown to be valid.
Dissipative quantum systems are sometimes phenomenologically described in terms of a non-hermitian hamiltonian HHH, with different left and right eigenvectors forming a bi-orthogonal basis. It is shown that the dynamics of waves in open systems can be cast exactly into this form, thus providing a well-founded realization of the phenomenological description and at the same time placing these open systems into a well-known framework. The formalism leads to a generalization of norms and inner products for open systems, which in contrast to earlier works is finite without the need for regularization. The inner product allows transcription of much of the formalism for conservative systems, including perturbation theory and second-quantization.
The non-Hermitian skin effect, namely, that the eigenvalues and eigenstates of a non-Hermitian tight-binding Hamiltonian have significant differences under open or periodic boundary conditions, is a remarkable phenomenon of non-Hermitian systems. Inspired by the presence of the non-Hermitian skin effect, we study the evolution of wave packets in non-Hermitian systems, which can be determined using the single-particle Green's function. Surprisingly, we find that in the thermodynamic limit, the Green's function does not depend on boundary conditions, despite the presence of skin effect. We provide a general proof for this statement in arbitrary dimension with finite hopping range, with an explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model. We also explore its applications in noninteracting open quantum systems described by the master equation. We demonstrate that the evolution of the density matrix is independent of the boundary condition.
Devices involving bound or propagating photons are among the basic building blocks for modern quantum technologies. For example, the bound modes of open optical resonators, coupled to quantum emitters such as atoms, molecules, or quantum dots, are used for an enhanced emission in lasing applications, for increased sensitivity in quantum sensing, and as qubits in quantum computing. Meanwhile, propagating photons are crucial for the transmission of quantum states between spatially separated systems. In this thesis, we develop a rigorous theory of multiple, spatially separated, and individually addressable cavities (or plasmonic resonators) for applications in modern quantum devices. Losses are modeled via the use of quasinormal modes (QNMs), which are solutions to the source-free Helmholtz equation under open boundary conditions, making them the natural modes of open resonators. Quantum emitters are modeled as two-level systems (TLSs) which couple to the electric field via dipole coupling. The multi-cavity theory is developed in three steps: 1. We present a classical scattering theory for a system of multiple QNM cavities, where the multi-cavity Green's function is formulated in terms of the single-cavity modes. The inter-cavity scattering is frequency-dependent to account for retardation delays, and is efficiently calculated numerically, using common (commercial) solvers. 2. We formulate a quantization scheme for QNMs of spatially separated open resonators. We introduce the \textit{cavity separation parameter} as a simple measure for when QNM cavities can be quantized separately and when they have to be treated as a single system with a joint set of operators. We derive a general system-bath Hamiltonian that contains coupling between bound modes, quantum emitters, and a bath of propagating photons, expressed via rigorous, numerically calculable coupling parameters. 3. We discuss the time-dependent theory of QNMs and TLSs coupled to propagating photons. The system-bath interaction is formulated in terms of correlation functions that are again rigorously defined in terms of numerically calculable QNM parameters. The correlation functions are immediately applicable to common methods for calculating the dynamics of open quantum systems. Steps 1-3 are all formulated for general cavity shapes and materials, as well as (in principle) arbitrary background structures. Hence, the theory is applicable to a wide range of devices, including cavities or plasmonic resonators in free space, on a photonic crystal, or on a substrate. For each step, we show example calculations for one-dimensional or three-dimensional cavities.