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Molecular Physics

Spontaneous emission from nonhermitian perspective: complex scaling of the photon coordinates

Milan Šindelka, Daniel Šimsa

Published Jan 24, 2019• 4 citations
Publisher

Abstract

Spontaneous emission (SE) is usually studied within the framework of quantum dissipation theory. In this paper we pursue a different view of SE. Namely, we take advantage of Nonhermitian Quantum Mechanics (NHQM) and interpret SE as the decay of a Feshbach type resonance. Correspondingly, we aim at calculating the SE decay rate Γ (resonance width) and the associated resonance wavefunction via the complex scaling (CS) method. Standard application of CS in NHQM is based on scaling the dissociative coordinates of the system, x↦xeiθ. In the case of SE, the decay consists in emitting photons. Therefore the CS must be applied on (suitably defined) position coordinates of the photons. Feasibility of such a programme is demonstrated explicitly through choosing the adequate photon coordinates and performing CS of the photon wavefunctions in the position representation. It is anticipated in this way that CS makes the full resonance wavefunction square integrable, and enables thus the calculation of Γ by proceeding in complete analogy with the well established NHQM approaches. As an illustration, the standard Golden Rule formula for Γ is rederived from the CS nonhermitian perturbation theory.

DOI
10.1080/00268976.2019.1570368

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Spontaneous emission (SE) is usually studied within the framework of quantum dissipation theory. In this paper we pursue a different view of SE. Namely, we take advantage of Nonhermitian Quantum Mechanics (NHQM) and interpret SE as the decay of a Feshbach type resonance. Correspondingly, we aim at calculating the SE decay rate Γ (resonance width) and the associated resonance wavefunction via the complex scaling (CS) method. Standard application of CS in NHQM is based on scaling the dissociative coordinates of the system, x↦xeiθ. In the case of SE, the decay consists in emitting photons. Therefore the CS must be applied on (suitably defined) position coordinates of the photons. Feasibility of such a programme is demonstrated explicitly through choosing the adequate photon coordinates and performing CS of the photon wavefunctions in the position representation. It is anticipated in this way that CS makes the full resonance wavefunction square integrable, and enables thus the calculation of Γ by proceeding in complete analogy with the well established NHQM approaches. As an illustration, the standard Golden Rule formula for Γ is rederived from the CS nonhermitian perturbation theory.
Defining a quantum mechanical wavefunction for photons is one of the remaining open problems in quantum physics. Thus quantum states of light are usually treated within the realm of second quantization. Consequently, spontaneous emission (SE) in arbitrary photonic media is often described by Fock space Hamiltonians. Here, we present a real space formulation of the SE process that can capture the physics of the problem accurately under different coupling conditions. Starting from first principles, we map the unitary evolution of a dressed two-level quantum emitter onto the problem of electromagnetic radiation from a self-interacting complex harmonic oscillator. Our formalism naturally leads to an efficient computational scheme of SE dynamics using finite difference time domain method without the need for calculating the photonic eigenmodes of the surrounding environment. In contrast to earlier investigations, our computational framework provides a unified numerical treatment for both weak and strong coupling regimes alike. We illustrate the versatility of our scheme by considering several different examples.
The time-dependent spectral properties and nonlinear dynamics of a spontaneous photon emitted by two level atom, trapped in damped nanocavity and coupled to a single resonance mode, have been investigated. The results have been obtained with the help of new nonperturbative approach by solving exactly the Schrödinger equation. The theory accounts exactly and simultaneously processes of emission, reabsorptions and leakage of photon. The solution was found with the Green functions formalism, supplemented with the novel algorithm in operating causal singular functions and fundamentals of the theory of quasi-stationary systems. The obtained distributions and emission dynamics are presented with plots as functions of photon frequency detuning and time for various value Γ/4g. For Γ/4g <1 the spectrum is a triplet inside the cavity and doublet outside it. In this case the total emission probability is described by decaying oscillations. For Γ/4g ≥ 1 and Γ/4g >> 1 the spectrum consists of a single central line decaying exponentially with profile depending on value Γ/4g.
We demonstrate a fundamental breakdown of the photonic spontaneous emission (SE) formula derived from Fermi's golden rule, in absorptive and amplifying media, where one assumes the SE rate scales with the local photon density of states, an approach often used in more complex, semiclassical nanophotonics simulations. Using a rigorous quantization of the macroscopic Maxwell equations in the presence of arbitrary linear media, we derive a corrected Fermi's golden rule and master equation for a quantum two-level system (TLS) that yields a quantum pumping term and a modified decay rate that is net positive. We show rigorous numerical results of the temporal dynamics of the TLS for an example of two coupled microdisk resonators, forming a gain-loss medium, and demonstrate the clear failure of the commonly adopted formulas based solely on the local density of states.
Abstract “Spontaneous emission of radiation” calculates the rate of spontaneous electric-dipole emission of a photon by an excited atom or molecule. The calculation proceeds by using basic quantum mechanics (i.e. not using the short cuts of Chapter 19); it uses quantum electrodynamics but is not, on that account, particularly difficult. A 2p–1s transition in hydrogen is used as exemplar; the radiation is elliptically polarized. The spectral line profile (lineshape function) is approximately Lorentzian, but has a high-frequency cut-off, needed to prevent the power radiated from diverging. A radiation-induced frequency shift is negligible. The width of the line profile agrees with the Einstein A-coefficient. A high-frequency cut-off is shown to apply similarly in the derivation of Golden Rule Number Two.
The formulas of the quantum electrodynamics have been applied to calculate the spontaneous emission rate of excited atom in dielectric microcavity.The results exhibit damping oscillating patterns which depend sensitively on the scaling parameter and geometrical structure.Compared with the case that the emitting atom is immersed in dielectric,the spontaneous emission rate is depressed obviously and the center or the mean value of the oscillations is intimately related to the real refractive index of the local position where the atom is.In order to explain this phenomenon,we utilize the closed-orbit theory to deal with the classical trajectories of the emitted photon,and extract the corresponding frequencies of the oscillations by Fourier transform.It is found that the oscillations can be represented in terms of the closed-orbits of the photon motion constrained in dielectric microcavity,thus providing another perspective on the spontaneous emission of atom sandwiched by dielectric slabs.更多还原
t o Ph o t on i cs" , Z ak opa n e 200 1 Sp on t an eou s Em ission b y A tom s in Sim p le En v iron m en ts W . `ak o w ic z I nst i t ut e of P h ysi cs, Po li sh Academ y of Sciences al .Lo tni k §w 32/ 46, 02-668 Warszawa , Po l andSpontan eous ra diation by ato ms in t he presence of the planar dielectri c{ v acuum interf ace, planar dielectri c w aveguides and cylindri cal dielectri c w aveguides are discussed in the frame of cavity quantum electro dynami cs in full analogy with that in free space.H owever, quanti zation of the electromagnetic Ùeld should be based on the mo des appropriate to the selected space structure.T hese quantizations are usuall y based on incoming w aves.H ow ever, the discussion of the angular intensity pattern of spontaneous emission can be simpli Ùed if the quantization is based on outgoing mo des.U sing these outgoing photons the angular emission radiatio n pattern has b een obtained from a straightf orw ard appli catio n of the p erturbativ e metho d of the quantum radiation theory .A dding a contribution of the w aveguiding photons attached to the w aveguides (w hen they are present ) the total emission of the spontaneous radiation and excitation decay rates of atoms radiatin g in these systems ha ve been derived.