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Modern Physics Letters B

NEW NON-MARKOVIAN TREATMENT OF THE SPONTANEOUS DECAY OF AN UNSTABLE ATOMIC STATE IN RELATIVISTIC QED

JOSIP SEKE

Published Aug 30, 1996• 0 citations
Publisher

Abstract

It is shown that the application of a self-consistent projection-operator method, developed recently by the author, makes a complete treatment of the spontaneously decaying atomic state in relativistic QED possible. In a new physical real-transition interaction picture (in which the unobservable interaction of the free electron with the vacuum radiation field is eliminated) explicit finite analytic results for the non-Markovian time evolution and the radiative line shape in the case of relativistic Lyman-α spontaneous emission are obtained.

DOI
10.1142/s0217984996001103

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It is shown that the application of a self-consistent projection-operator method, developed recently by the author, makes a complete treatment of the spontaneously decaying atomic state in relativistic QED possible. In a new physical real-transition interaction picture (in which the unobservable interaction of the free electron with the vacuum radiation field is eliminated) explicit finite analytic results for the non-Markovian time evolution and the radiative line shape in the case of relativistic Lyman-α spontaneous emission are obtained.
It is shown that the application of a self-consistent projection-operator method, developed recently by the author, makes possible a complete and gauge-invariant treatment of spontaneously decaying atomic state. All the shortcomings of the Weisskopf-Wigner gauge-dependent method (Markov approximation and the neglect of virtual transition, retardation and electron-spin effects) are removed. After deriving a gauge-invariant equation of motion for the decaying state, a new physical interaction picture (in which the unobservable interaction of the free electron with the vacuum radiation field is eliminated) is introduced. In this new interaction picture, apart from a natural solution of the renormalization problem, explicit finite analytic results for the whole non-Markovian Lyman- alpha spontaneous decay are derived. Furthermore, a quasi-Lorentzian expression for the radiative line shape containing the correct (complete) finite non-relativistic frequency shift, stemming from the Lamb shifts of both levels 1S and 2P, is calculated.
For the first time to the author's knowledge, a rigorous mathematical treatment of the spontaneous 2P 1/2 to 1S 1/2 transition, going beyond the usual Markov approximation, is presented in the case of the Dirac hydrogen atom. The relevant transition matrix elements of the Dirac Hamiltonian are calculated without the dipole approximation. Further, a non-Markovian equation of motion for the non-decay probability of the unstable 2P 1/2 2 state is derived, by using a self-consistent projection-operator method recently developed by the present author. By a rigorous analytical treatment of the non-Markovian equation, deviations from exponential decay for asymptotic times are obtained. Moreover, by careful error estimations deviations from exponential decay for finite times with definite validity range are also calculated. Finally, for the radiative lineshape a new quasi-Lorentzian distribution is obtained.
Spontaneous emission of a quantum emitter, such as an excited atom, is a fundamental process in quantum electrodynamics (QED), typically associated with exponential decay to the ground state accompanied by irreversible photon emission. This simple Markovian picture, however, is profoundly modified in the presence of time-delayed feedback, structured continua, or cooperative emission, as occurs when an emitter radiates in front of a mirror, when several emitters radiate collectively, or in the case of a giant atom. In such regimes, strong non-Markovian dynamics arise from photon reabsorption and interference effects, leading to pronounced deviations from exponential decay. Here we demonstrate the emergence of exceptional points (EPs) in these highly non-Markovian waveguide-QED environments, i.e., non-Markovian EPs. These EPs appear directly in the relaxation dynamics as sharp transitions to oscillatory behavior, manifested by the appearance of real zeros in the excited-state amplitude. We analyze in detail the spontaneous emission of giant atoms with two or more coupling points, highlighting the mechanisms leading to non-Markovian EPs, and show that similar phenomena arise in other waveguide-QED settings, such as the collective spontaneous emission of spatially separated point-like emitters. Our results reveal waveguide-QED systems as experimentally accessible platforms for realizing and exploring non-Markovian EP physics.
It is pointed out that the usual Heisenberg equations-of-motion or Liouvillian superoperator perturbation-theory approaches to systems with arbitrary statistical ensembles can give spurious, nonconvergent results when attempts are made to calculate the expectation value for certain operators which commute with the zeroth-order Hamiltonian. The non-Markovian, long-time tail of the Weisskopf-Wigner spontaneous-emission decay of a two-level atom is presented as an example of this failure. The general reason for such failures is elucidated and a new method applicable to arbitrary equilibrium or nonequilibrium statistical ensembles is formulated. The new method is applied to the same example of spontaneous emission to derive the correct short- and long-time behavior. The physical model of a two-level atom is discussed and some mathematical techniques of general usefulness are presented. The present results are also briefly discussed in the context of previous work on nonexponential decay. A fully quantum-mechanical definition of time-dependent spectral line shape is given in terms of the response of a two-level---atom detector with variable frequency interacting with electromagnetic radiation. The non-Lorentzian line shape recorded by such a detector for the spontaneous emission of a source two-level atom is calculated as another example of a problem for which the usual methods give incorrect results. The extension of the present method to Liouvillian or master-equation approaches is discussed.