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Physical Review A

Photon correlations and successive emission

M. S. Kim, P. L. Knight

Published Jul 1, 1989• 22 citations
Publisher

Abstract

We describe the coherence properties of light emitted by driven atoms in terms of dressed-atom survival amplitudes. Such amplitudes can be used to construct analytic expressions for the function describing the delay between successive photon emissions and used to construct the appropriate radiation correlation functions. This approach is used to calculate the radiative dynamics of three-level atoms appropriate to recent quantum-jump experiments. We contrast our simple approach to that normally employed which relies on the numerical solutions of density-matrix equations of motion followed by the application of the quantum-regression theorem.

DOI
10.1103/physreva.40.215

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We describe the coherence properties of light emitted by driven atoms in terms of dressed-atom survival amplitudes. Such amplitudes can be used to construct analytic expressions for the function describing the delay between successive photon emissions and used to construct the appropriate radiation correlation functions. This approach is used to calculate the radiative dynamics of three-level atoms appropriate to recent quantum-jump experiments. We contrast our simple approach to that normally employed which relies on the numerical solutions of density-matrix equations of motion followed by the application of the quantum-regression theorem.
The statistics of spontaneous photon emission times in single atom resonance fluorescence are investigated through the radiative cascade of the dressed atom. The delay function which gives the distribution of the delays between two successive emissions is calculated for a coherent as well as an incoherent laser excitation. For a two-level atom, various signals that reflect the fluorescence intensity (average value, photon counting, fluctuation spectrum, etc.) are reviewed by this method. For a three-level atom, this approach is applied to the analysis of the recently observed phenomenon of intermittent fluorescence and quantum jumps.>
We present a microscopic model for the so-called quantum jumps in three-level atoms with one transition between two levels being strongly driven and the third level being metastable. Our model is based on calculating the elementary probability density P(0,t)(t1, … , t n ) which gives the probability density that exactly n photons are emitted at times t1, … , t n by the atom in the time interval (0, t) on the strong transition. This enables us to calculate the conditional probabilities c ˜ ( τ ) that given a photon has emitted at τ = 0 the next photon emission occurs at time τ. This quantity is shown to be the sum of two exponentials which for sufficient differetial in the time scales describe the periods of light and darkness. Using this conditional probability it is possible to numerically simulate the individual photon emissions.
Photon pairs generated from electronic transitions in atomic three-level systems are now considered as a highly promising quantum state for constructing qubit, because of easy on-demand generation and low photon-photon interaction [1]. While polarization-entangled biphotonic states have been successfully demonstrated by using diamond-shaped four level systems with degenerated intermediates states in atoms and quantum dots and verified through the violation of Bell inequality [2,3], quantifying spectrally entangled photon pairs in experiments is still hardly available even with state-of-art techniques. In this theoretical study, we investigated the possibility of the cross correlation function of two photons as an observable for verifying the entanglement. To calculate the cross photon density-density correlation function and the degree of entanglement, Heisenberg's quantum kinetic equations of motion for single electron interacting with photons and two coherent pump fields are derived within the framework of density matrix formalism. In figure 1 (a), the energy configuration of a three-level light emitter is depicted with specially focused observables such as photon number densities (Ck†Ck) and (Cq†Cq) and their normalized cross correlation function g12(2)(t) = (ck†cq†cqck)/(ck†ck)(cq†cq). We compared (2) the temporal behavior of g(2)12(t) and the entanglement criterion D(t) introduced by Duan et al. [4] and find peculiar similarities between them for coherently generated photon pairs, as presented in figure (b).
We theoretically investigate temporal dynamics of the second order cross correlation function at zero delay time ( G 12 ( 2 ) ( t ) ) and spectral entanglement of two photons emitted from an atomic three-level cascade. In Heisenberg’s picture, a closed set of quantum kinetic equations of motion for G 12 ( 2 ) ( t ) is derived within density matrix formalism with cluster expansion rule. G 12 ( 2 ) ( t ) shows qualitatively distinctive features depending on the spectral entanglement of two photons. Although incoherent photon pairs generated from spontaneous radiation of the excited electron are not entangled, their correlation and anti-correlation properties can be found in G 12 ( 2 ) ( t ) depending on the radiative decay rates. In the coherent excitation regime where the light emitter is located in a high Q-cavity, and its atomic polarizations are predominantly initialized, spectral entanglement between two coherent photons is established. We show that G 12 ( 2 ) ( t ) is well fitted by the entanglement criterion by Duan-Giedke-Cirac-Zoller and explain the close relationship between them by means of the optically forbidden transition in the three-level cascade.
A set of coupled non-Markovian equations of motion is derived for the Green's function of a driven two-level atom. These equations can be expressed in terms of the density-matrix elements only in the Markovian limit, in which case the set reduces to optical Bloch equations. The non-Markovian effect arises from correlations between successive photon emissions, an effect which is neglected in the existing equations of motion. The resulting correction is of the order of ${\ensuremath{\gamma}}_{0}$/${\ensuremath{\Omega}}_{0}$ in the strong-field region (${\ensuremath{\gamma}}_{0}$${\ensuremath{\Omega}}_{0}$) and it vanishes in the weak-field region, where ${\ensuremath{\gamma}}_{0}$ is the Einstein A coefficient and ${\ensuremath{\Omega}}_{0}$ is the Rabi frequency.