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arXiv:cond-mat/0406554 · Europhysics Letters (EPL)

Dressed atom vs. exciton polariton: From Rabi oscillations to the Fermi golden rule

François Dubin, Monique Combescot, Bernard Roulet

Published Feb 22, 2005• 8 citations
arXivHTMLPDF

Abstract

We rederive the dressed atom and the exciton polariton within the same framework to make clear that their major difference only comes from the number of electrons available for photoexcitations. This allows to analytically show how the decrease of the excited-state lifetime transforms the photon number time evolution from oscillations—at the stimulated Rabi frequency for an atom and the vacuum one for semiconductors—to an exponential decay identical for both. Although the matter ground state is in both cases coupled not to a continuum but to a discrete state , broadened by lifetime, this decay yet follows a Fermi golden rule, however having a different scale for the energy conservation. Our work actually provides a clean justification for using the golden rule for exciton absorption.

DOI
10.1209/epl/i2004-10447-3

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We studied the semiconductor response with respect to high intensity resonant excitation on short-time scale when the contribution of the Fermi statistics of the electrons and holes prevails. We studied both the single- and double-pulse excitations. For the latter case we considered the time evolution of the multiwave mixing exciton polarization. The main difference between the excitation by a single pulse or by two noncollinear pulses is that the Rabi oscillations of the multiwave mixing response are characterized by two harmonics. Analyzing the operator dynamics governed by the external excitation we found that there are three invariant spin classes, which do not mix with the evolution of the system. Two classes correspond to the bright exciton states and one contains all dark states. We found that the dynamics of the classes are described by six frequencies and the Rabi frequencies are only two of them (one for each bright class). We discuss the effect of the dispersion of the electrons and holes and the Coulomb interaction describing the semiconductor by the semiconductor Bloch equation (SBE). We show that if initially the system is in the ground state then the SBE preserves the invariant spin classes, thus proving the absence of the dark excitons in the framework of this description. We found that due to the mass difference between holes of different kind additional Rabi frequencies, two of those present in the operator dynamics should appear in the evolution of the exciton polarization.
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.
We reconsider the problem of the spontaneous emission of light by an excited atomic state. We scrutinize the survival probability of this excited state for very short times, in the so-called Zeno regime, for which we show that the dynamics is dictated by a coherent, in-phase, response of the on-shell and off-shell vacuum modes. We also develop a perturbative approach in order to interpolate between different temporal regimes: the Zeno, golden rule (linear), and Wigner-Weisskopf (exponential) regimes. We compare results obtained with the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{E}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{x}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{A}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$ interaction Hamiltonians, using successively the dipole approximation and the exact coupling.