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Physical Review A

Time dependence of photon correlations in a three-level atomic cascade

Harry J. Kimble, Anthony Mezzacappa, Peter W. Milonni

Published Jun 1, 1985• 22 citations
Publisher

Abstract

The question of the temporal correlation of photons arising from a three-level atomic cascade is considered by calculating the joint probability of photoelectric detection for two detectors placed in the far field of the atomic emission. The analysis is carried out in the Heisenberg picture and involves the use of dipole, adiabatic (Weisskopf-Wigner), and rotating-wave approximations. Except for a small interval about zero time delay, which is however too short to be detected, it is found that the temporal correlation between emitted photons is one of sequential but otherwise statistically independent events, even when Landau's generalized decay and frequency-shift terms are included. This result is compared with a well-known expression for the joint spectral density of the emitted photons.

DOI
10.1103/physreva.31.3686

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The question of the temporal correlation of photons arising from a three-level atomic cascade is considered by calculating the joint probability of photoelectric detection for two detectors placed in the far field of the atomic emission. The analysis is carried out in the Heisenberg picture and involves the use of dipole, adiabatic (Weisskopf-Wigner), and rotating-wave approximations. Except for a small interval about zero time delay, which is however too short to be detected, it is found that the temporal correlation between emitted photons is one of sequential but otherwise statistically independent events, even when Landau's generalized decay and frequency-shift terms are included. This result is compared with a well-known expression for the joint spectral density of the emitted photons.
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.
The photon correlations of resonance fluorescence emitted by a degenerate two-level atom near a metal surface are studied. An intense non-monochromatic optical laser drives a low-lying atomic transition, and the emitted fluorescence is detected by a photon counter. An expression for the spontaneous decay operator is derived, and it appears that the various substates of the upper level have a different lifetime due to the presence of the surface, or equivalently, due to the loss of spherical symmetry. The temporal photon correlation function is then evaluated in terms of a photon-emission operator and the time evolution operator for the atomic density matrix. The general theory is exemplified with a detailed elaboration of the j/sub g/ = 0 yields j/sub e/ = 1 transition, for which the two-photon correlation is obtained explicitly as a function of the lifetime A(h), depending on the atom surface distance h, the laser linewidth, the detuning from resonance, and the dipole-coupling strength with the external field.
Two uncoupled two-level atoms cannot be jointly excited by classical light under general circumstances, due to destructive interference of excitation pathways in two-photon absorption. However, with temporally entangled light, two-atom excitation is shown possible. Photons arising from three-level cascade decay are intrinsically ordered in time of emission. This field correlation induces a joint resonance in the two-atom excitation probability via suppression of one of the time-ordered excitation pathways. The relative gain in two-photon absorption increases with the time-frequency entanglement.
The photon correlations of resonance fluorescence emitted by a degenerate two-level atom near a metal surface are studied. An intense nonmonochromatic optical laser drives a low-lying atomic transition, and the emitted fluorescence is detected by a photon counter. An expression for the spontaneous-decay operator is derived, and it appears that the various substates of the upper level have a different lifetime due to the presence of the surface, or equivalently, due to the loss of spherical symmetry. The temporal photon correlation function is then evaluated in terms of a photon-emission operator and the time-evolution operator for the atomic density matrix. The general theory is exemplified with a detailed elaboration of the jg{j}_{g}jg​=0\ensuremath{\rightarrow}je{j}_{e}je​=1 transition, for which the two-photon correlation is obtained explicitly as a function of the inverse lifetime A(h), depending on the atom-surface distance h, the laser linewidth, the detuning from resonance, and the dipole-coupling strength with the external field.
We present a theory of cascade emission from an array of NNN fixed, ``three-level'' atoms. The total angular momentum of the ground state of each atom is J=0J=0J=0, that of the intermediate state is J=1J=1J=1, and that of the upper state is J=0J=0J=0. The atoms are prepared in a spatially phased superposition of their ground and upper states, collectively sharing a single excitation. We calculate the time-integrated, joint probability distribution for radiation to be emitted in a given direction with a given polarization on each transition and use this joint probability distribution to calculate the time-integrated intensity emitted on each transition. The dipole-dipole interaction between the atoms is taken into account. Analytic expressions are obtained for two atoms and the calculation is then extended formally to an ensemble of NNN atoms at arbitrary positions. As expected, the radiation emitted on the upper transition is unpolarized and isotropic. However, somewhat surprisingly, we find that for excitation by counterpropagating fields and for nearly equal ground to intermediate and intermediate to upper level transition frequencies, the radiation on the lower transition is also isotropic and unpolarized, provided the atoms are prepared in a fully symmetric state. We also establish conditions for which there can be a phase-matched enhancement of the joint probability density for specific directions of emissions on the upper and lower transitions.
New details of a well-known physical process—cascade transition—have been considered. The wave function of a system of two photons emitted in a cascade transition in the absence of a shift of the phase of the intermediate state was studied. The joint energy distribution density for two such photons is obtained by numerical calculation, and a correlation between their energies at certain probabilities of radiative transitions is ascertained. This phenomenon can be used to diagnose the plasma density.