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Physical Review A

Non-Markovian correlation spectra and quantum stochastic trajectory analysis of spontaneous emission of an excited two-level atom

Chang-qi Cao, Wei Long, Jikun Wei, Hui W. Cao

Published Sep 18, 2001• 5 citations
Publisher

Abstract

The non-Markovian correlation spectra of spontaneous emission of an excited two-level atom are derived including the effect of finite size of the atom and all the possible contribution of allowed multipole radiations. The emission process is then analyzed by the quantum stochastic trajectory approach. The non-Markovian effect is counted in by expanding the original system to an enlarged system with Markovian reservoirs. In the case of a hydrogenlike atom with large atomic number Z, the deviation from the Weisskopf-Wigner result is quite evident.

DOI
10.1103/physreva.64.043810

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The non-Markovian correlation spectra of spontaneous emission of an excited two-level atom are derived including the effect of finite size of the atom and all the possible contribution of allowed multipole radiations. The emission process is then analyzed by the quantum stochastic trajectory approach. The non-Markovian effect is counted in by expanding the original system to an enlarged system with Markovian reservoirs. In the case of a hydrogenlike atom with large atomic number Z, the deviation from the Weisskopf-Wigner result is quite evident.
We present a formula to evaluate the spontaneous emission spectra of an atom in contact with a radiation field with non-Markovian effects. This formula is written in terms of a two-time correlation of system observables and the environmental correlation function, and depends on the distance between the emitting atom and the detector. As an example, we use it to analyze the fluorescence spectra of a two level atom placed as an impurity in a photonic crystal. The radiation field within those materials has a gap or discontinuity where electromagnetic modes cannot propagate in the stationary limit. In that situation, the atomic emission occurs in the form of evanescent waves which are detected with less efficiency the farther we place the detector. The methodology presented in this paper may be useful to study the non-Markovian dynamics of any quantum open system in linear interaction with a harmonic oscillator reservoir and within the weak-coupling approximation.
The non-Markovian corrections to electric-dipole emission of hydrogen-like atoms of large atomic number Z are studied by multipole photon field formulation. The correlation functions are derived by the corresponding spectra, which show how the δ-function type correlation is approached. The decay of upper-level population is calculated by the relevant integro-differential equation. The numerical results show that the relativistic and finite atom-size corrections are evident; while the non-Markovian corrections are negligible in the main period of decay, they can manifest themselves in the far remote tail.
A relativistic and non-Markovian study is carried out on the magnetic-dipole emission of a hydrogen-like atom of large atomic number Z without perturbation approximation. The correlation spectra are derived analytically and the corresponding correlation functions are calculated numerically. The Markovian approximation is seen to be satisfied with more accuracy as compared with the case of electric-dipole emission while the relativistic corrections are quite larger. In transition 2S1/2 to 1S1/2, the relativistic values even become several times of the non-relativistic values. As a comparison with the magnetic-dipole emission, the electric-quadrupole emission in the case of the 2D3/2 to 1S1/2 transition is also studied.
Quantum feedback can stabilize a two-level atom against decoherence (spontaneous emission), putting it into an arbitrary (specified) pure state. This requires perfect homodyne detection of the atomic emission, and instantaneous feedback. Inefficient detection was considered previously by two of us. Here we allow for a non-zero delay time tau\\tautau in the feedback circuit. Because a two-level atom is a nonlinear optical system, an analytical solution is not possible. However, quantum trajectories allow a simple numerical simulation of the resulting non-Markovian process. We find the effect of the time delay to be qualitatively similar to that of inefficient detection. The solution of the non-Markovian quantum trajectory will not remain fixed, so that the time-averaged state will be mixed, not pure. In the case where one tries to stabilize the atom in the excited state, an approximate analytical solution to the quantum trajectory is possible. The result, that the purity (P=2rmTr[rho2]−1P=2{\\rm Tr}[\\rho^{2}]-1P=2rmTr[rho2]−1) of the average state is given by P=1−4gammatauP=1-4\\gamma\\tauP=1−4gammatau (where gamma\\gammagamma is the spontaneous emission rate) is found to agree very well with the numerical results.
We present a formulation of non-Markovian quantum trajectories for open systems from a measurement theory perspective. In our treatment there are three distinct ways in which non-Markovian behavior can arise: a mode-dependent coupling between the bath (reservoir) and system, a dispersive bath, and spectral detection of the output into the bath. In the first two cases the non-Markovian behavior is intrinsic to the interaction; in the third case the non-Markovian behavior arises from the method of detection. We focus in detail on the trajectories that simulate real-time spectral detection of the light emitted from a localized system. In this case the non-Markovian behavior arises from the uncertainty in the time of emission of particles that are later detected. The results of computer simulations of the spectral detection of the spontaneous emission from a strongly driven two-level atom are presented.
A non-Markovian microscopic theory for the spontaneous emission spectrum of a two-level atom driven by a strong laser field and undergoing dephasing via the interaction with a thermal bath is developed. The present theory is valid for an arbitrary correlation time scale of the bath relative to the linewidth and interpolates from the Markovian to the static limits. Model calculations using the stochastic line-shape function of Kubo show how the characteristic triplet spectrum is modified by the interaction with the bath. The present results generalize our previous studies of resonance Raman scattering in a three-level system in a strong radiation field.