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Journal of Physics B Atomic and Molecular Physics

The effects of laser field fluctuations on coherent population trapping

Bryan J. Dalton, P. L. Knight

Published Nov 14, 1982• 152 citations
Publisher

Abstract

A three-level system driven by two coherent fields on two-photon resonance leads to the trapping of population in a superposition of initial and final states which is immune to further photoexcitation, manifesting itself in a narrow coherence minimum in the absorption spectrum. The authors analyse the effects of laser field bandwidths and cross-correlations on these phenomena. They find that the inclusion of laser bandwidths dephases atomic coherences which leads to the destruction of population trapping and of the narrow coherence minimum in the absorption spectrum. When the driving fields are not cross-correlated, the atomic coherences are similarly dephased in both lambda and ladder systems. If, however, the driving fields are critically cross-correlated, two-photon coherences in the lambda system are unaffected by laser fluctuations; trapping and the coherence minimum persist. For the ladder system, on the other hand, the cross-correlation does not restore the atomic coherences involved.

DOI
10.1088/0022-3700/15/21/019

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Journal of Physics B Atomic and Molecular Physics•Oct 14, 1982•62 citations
A three-level system driven by two coherent fields on two-photon resonance leads to the trapping of population in a superposition of initial and final states which is immune to further photoexcitation, manifesting itself in a narrow coherence minimum in the absorption spectrum. The authors analyse the effects of laser field bandwidths and cross-correlations on these phenomena. They find that the inclusion of laser bandwidths dephases atomic coherences which leads to the destruction of population trapping and of the narrow coherence minimum in the absorption spectrum. When the driving fields are not cross-correlated, the atomic coherences are similarly dephased in both lambda and ladder systems. If, however, the driving fields are critically cross-correlated, two-photon coherences in the lambda system are unaffected by laser fluctuations; trapping and the coherence minimum persist. For the ladder system, on the other hand, the cross-correlation does not restore the atomic coherences involved.
Coherent trapping in three level lambda and ladder systems is examined for the case of two fluctuating laser fields with unequal bandwidths, obtained by frequency multiplying methods. The inclusion of uncorrelated laser fields destroys trapping and washes out the narrow coherence hole in the fluorescent intensity. However, if the fields are positively critically cross-correlated, the relevant atomic coherences in lambda systems are much less affected by the laser fluctuations. A significant degree of trapping re-occurs and the narrow coherence hole is largely restored. An enhancement of the destruction of coherent trapping occurs in ladder systems. An application of the theory to molecular photodissociation is discussed.
The resonance fluorescence spectrum of a Λ-system excited by two resonant light fields is calculated using a Markov analysis. Analytical formulas are derived in the strong-field limit within and beyond the rotating wave approximation. It is shown that the resonance fluorescence of the system does not vanish during coherent population trapping. Its spectrum consists of two multiplets which are similar to a triplet in the resonance fluorescence spectrum of a two-level atom and lie at the electronic transition frequencies, together with two triplets located at the frequencies of four-photon processes involving the optical excitation fields. The latter are fundamental in character and impose limits on the lower bound of the dephasing rate for the Raman resonance owing to the effect of radiative decay of the dipole transitions on the dynamics of the ground state. The effect of four-photon dephasing on the absorption spectrum of a Λ-system is analyzed and found to lead to a substantial reduction in the depth of a dip in the absorption spectrum which vanishes as the laser field strength is increased.
Master equation methods are used to investigate the effects of a broad-band squeezed vacuum on a three-level atom of the lambda configuration. The two-mode squeezed vacuum is treated as a Markovian reservoir in a non-stationary phase-dependent state. In addition to the squeezed vacuum the atom is driven by two coherent laser fields each of which, depending on the polarization, can couple to one or both of the atomic transitions. We show that in general the optical Bloch equations for the atomic density matrix elements have oscillatory coefficients, thereby necessitating the use of Floquet methods. For the case of equal laser frequencies, which are also equal to the carrier frequency of the squeezed vacuum, the coefficients of the Bloch equations become time independent and stationary solutions for the populations and coherences are obtained by standard matrix methods. For the ordinary vacuum the usual coherent population trapping effect at two-photon resonance is obtained, with the upper state population being zero. An unsqueezed thermal field partially destroys the trapping effect as the upper state population is no longer zero at two-photon resonance. The squeezed vacuum has the effect of improving the trapping in that the coherence hole becomes more pronounced for some values of the relative phase between the squeezed vacuum and the driving fields. The additional effects of a coherence transfer rate between the two optical coherences, which occurs for special choices of angular momentum quantum numbers are also studied. For the case of equal laser frequencies, the inclusion of this coherence transfer process destroys population trapping and reduces the lambda system to a two-level system. However, for the case of unequal laser frequencies, the coherence transfer process in combination with the squeezed vacuum can restore to some extent the population trapping. We show that other features that do not occur for two-level atoms, such as stationary population inversions between pairs of the atomic levels, also depend on the relative phase and can be enhanced in the squeezed vacuum. In the case of unequal frequencies of the driving fields the population in the upper state depends on the relative phase only when the carrier frequency of the squeezed vacuum is equal to one of the two frequencies of the driving fields. When the carrier frequency of the squeezed vacuum is slightly detuned from both frequencies of the driving fields, the population in the upper state is insensitive to the relative phase but is dependent on the degree of squeezing. For large detunings, the population does not show any dependence on the degree of squeezing and its distribution in function of the two-photon detuning is similar to that in the thermal vacuum field.
We study the correlation between level populations, absorption and two-photon coherence in closed Λ- and V-shaped three-level systems interacting with two lasers of arbitrary intensity. Whereas it is well known that maximum two-photon coherence leads to population trapping in Λ-shaped three-level systems interacting with two equally detuned lasers, we demonstrate that minimum two-photon coherence, under saturation conditions, can lead to equal populations in all the levels of a Λ- or V-shaped system when the lasers are symmetrically detuned. Moreover we show that steady-state population inversion between one of the upper levels of a V-shaped system and the ground state can occur when either the detunings are asymmetrical or the upper states decay at different rates. It is shown that this population inversion does not lead to stimulated emission into the lasing modes. These effects are also correlated with two-photon coherence and are explained physically by a cross three-photon scattering process involving two laser photons of different frequencies.
We examine the effects of cross correlated phase fluctuations on the coherent population trapping (CPT) induced by a pair of phase-modulated fields with equal modulation frequencies in a three-level Λ system. The maximal coherence of -0.5, which appears when CPT occurs for equal modulation indices, is preserved in the presence of the critically cross-correlated fluctuations. Unexpectedly, the non-maximal coherence, which is established when CPT is obtained for different modulation indices, is significantly enhanced due to the critically cross-correlated fluctuations.
The authors consider population trapping in a three-level system which is driven by two applied fields of arbitrary intensity, frequency and bandwidth. An irreversible loss of population is allowed to take place from the common level. The conditions are investigated under which population trapping can occur when spontaneous emission and finite radiation bandwidths are allowed. For the lambda configuration with monochromatic lasers the authors find that increasing the spontaneous emission rate can increase the amount of population trapped in the system, whereas spontaneous emission destroys trapping in the ladder configuration. For the case of uncorrelated lasers, finite laser bandwidths destroy trapping for either configuration.