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Communications in Theoretical Physics

A study of the effective Hamiltonian method for decay dynamics*

Jing Chen, Xinyu Shan, Xiaoyun Wang, Yong-Gang Huang

Published Dec 30, 2022• 2 citations
Publisher

Abstract

Abstract The decay dynamic of an excited quantum emitter (QE) is one of the most important contents in quantum optics. It has been widely applied in the field of quantum computing and quantum state manipulation. When the electromagnetic environment is described by several pseudomodes, the effective Hamiltonian method based on the multi-mode Jaynes–Cummings model provides a clear physical picture and a simple and convenient way to solve the decay dynamics. However, in previous studies, only the resonant modes are taken into account, while the non-resonant contributions are ignored. In this work, we study the applicability and accuracy of the effective Hamiltonian method for the decay dynamics. We consider different coupling strengths between a two-level QE and a gold nanosphere. The results for dynamics by the resolvent operator technique are used as a reference. Numerical results show that the effective Hamiltonian method provides accurate results when the two-level QE is resonant with the plasmon. However, when the detuning is large, the effective Hamiltonian method is not accurate. In addition, the effective Hamiltonian method cannot be applied when there is a bound state between the QE and the plasmon. These results are of great significance to the study of the decay dynamics in micro-nano structures described by quasi-normal modes.

DOI
10.1088/1572-9494/acaf29

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Abstract The decay dynamic of an excited quantum emitter (QE) is one of the most important contents in quantum optics. It has been widely applied in the field of quantum computing and quantum state manipulation. When the electromagnetic environment is described by several pseudomodes, the effective Hamiltonian method based on the multi-mode Jaynes–Cummings model provides a clear physical picture and a simple and convenient way to solve the decay dynamics. However, in previous studies, only the resonant modes are taken into account, while the non-resonant contributions are ignored. In this work, we study the applicability and accuracy of the effective Hamiltonian method for the decay dynamics. We consider different coupling strengths between a two-level QE and a gold nanosphere. The results for dynamics by the resolvent operator technique are used as a reference. Numerical results show that the effective Hamiltonian method provides accurate results when the two-level QE is resonant with the plasmon. However, when the detuning is large, the effective Hamiltonian method is not accurate. In addition, the effective Hamiltonian method cannot be applied when there is a bound state between the QE and the plasmon. These results are of great significance to the study of the decay dynamics in micro-nano structures described by quasi-normal modes.
A bound state between a quantum emitter (QE) and surface plasmon polaritons (SPPs) can be formed, where the excited QE will not relax completely to its ground state and is partially stabilized in its excited state after a long time. We develop some theoretical methods for investigating this problem and show how to form such a bound state and its effect on the non-Markovian decay dynamics. We put forward an efficient numerical approach for calculating the analytical part of the self-energy for frequency below the lower energy threshold. We also propose an efficient formalism for obtaining the long-time value of the excited-state population without calculating the eigenfrequency of the bound state or performing a time evolution of the system, in which the probability amplitude for the excited state in the steady limit is equal to one minus the integral of the evolution spectrum over the positive frequency range. With the above two quantities obtained, we show that the non-Markovian decay dynamics of an initially excited QE can be efficiently obtained by the method based on the Green's function expression for the evolution operator when a bound state exists. A general criterion for identifying the existence of a bound state is presented. The performances of the above methods are numerically demonstrated for a QE located around a metal nanosphere and in a gap plasmonic nanocavity. Numerical results show that these methods work well and the QE becomes partially stabilized in its excited state at a long time for the transition dipole moment beyond its critical value. In addition, it is also found that this critical value is heavily dependent on the distance between the QE and the metal surface, but nearly independent on the size of the nanosphere or the rod. Our methods can be utilized to understand the suppressed decay dynamics for a QE in an open quantum system and provide a general picture on how to form such a bound state.
We put forward a general approach for calculating the energy level shift of a quantum emitter (QE) in arbitrary-shaped nanostructure. It is expressed by the sum of the real part and a simple integral of the imaginary part of the scattered photon Green's function (GF) in a rather narrow real frequency range without a principal value. Compared with the method of direct evaluation of the principal value integral along the positive frequency axis or the method of transferring the principal value into an integration along the imaginary frequency axis, our method avoids the principal value integral and the evaluation of the scattered GF with imaginary frequency. It is numerically demonstrated for a QE located around a nanosphere and in a gap plasmonic nanocavity. The decay dynamics of an excited QE is calculated using the time-domain method through solving the Schr\"odinger equation in the form of a Volterra integral of the second kind, and using the frequency-domain method based on the Green's function expression for the evolution operator. It is found that the frequency domain method requires much narrower band information of the scattered GF. These findings are instructive in the field of coherent light-matter interactions.
We propose an efficient numerical method to study the emission dynamics of a two-level quantum system placed in the vicinity of plasmonic nano-structures. All parameters of an inhomogeneous environment relevant for the evolution of the quantum system are appropriately included via the classical Green's function calculated with three-dimensional finite element method. Numerical solution for the emission dynamics of a two-level system near a gold nano-cylinder demonstrates excellent agreement with the available analytical solution in the limit of the Lorenzian-shaped local density of states. We show numerically that the strong coupling regime can be realized for realistic strength of the dipole moment for a quantum emitter in the gap of a gold bowtie nano-antenna.
Enabling the confinement of light to a scale far below the one of conventional optics, surface plasmon polaritons (SPPs) induced by an electromagnetic field in a metal-dielectric interface supply an ideal system to explore strong quantized light-matter coupling. The fast matter-SPP population exchange reported in previous works makes it a candidate for spin manipulation, but such reversible dynamics asymptotically vanishes accompanying the quantum matter relaxing completely to its ground state. Here, we study the exact dissipative dynamics of a quantum emitter (QE) coupled to SPPs. It is interesting to find that, qualitatively different from conventional findings, the QE can be partially stabilized in its excited state even in the presence of the lossy metal. Our analysis reveals that it is the formation of a QE-SPP bound state which results in such suppressed dissipation. Enriching the decoherence dynamics of the QE in the lossy medium, our result is helpful to understand QE-SPP interactions and apply plasmonic nanostructures in quantum devices.
We investigate theoretically the non-Markovian dynamics of a degenerate V-type quantum emitter in the vicinity of a metallic nanosphere, a system that exhibits quantum interference in spontaneous emission due to the anisotropic Purcell effect. We calculate numerically the electromagnetic Green's tensor and employ the effective modes differential equation method for calculating the quantum dynamics of the emitter population, with respect to the resonance frequency and the initial state of the emitter, as well as its distance from the nanosphere. We find that the emitter population evolution varies between a gradual total decay and a partial decay combined with oscillatory population dynamics, depending strongly on the specific values of the above three parameters. Under strong-coupling conditions, coherent population trapping can be observed in this system. We compare our exact results with results when the flat continuum approximation for the vacuum modified by the metallic nanosphere is applied. We conclude that the flat continuum approximation is an excellent approximation only when the spectral density of the system under study is characterized by nonoverlapping plasmonic resonances.
We report the results of numerical modelling of emission of a two-level atom near a metal nanoparticle under resonant interaction of light with plasmon modes of the particle. Calculations have been performed for different polarisations of light by a dipole approximation method and a complex multipole method. Depending on the distance between a particle and an atom, the contribution of the nonradiative process of electron tunnelling from a two-level atom into a particle, which is calculated using the quasi-classical approximation, has been taken into account and assessed. We have studied spherical gold and silver particles of different diameters . The rates of electron tunnelling and of spontaneous decay of the excited atomic state are found. The results can be used to develop nanoscale plasmonic emitters, lasers and photodetectors.