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Physical Review Letters

Multicolored Coherent Population Trapping in a Λ System Using Phase Modulated Fields

Harshawardhan Wanare

Published May 11, 2006• 19 citations
Publisher

Abstract

We demonstrate the simultaneous occurrence of coherent population trapping at a series of frequencies separated by modulation frequency of phase-modulated fields. The two arms of the system are coupled to two phase-modulated fields and the response of the atomic system to such fields is calculated nonperturbatively. A judicious choice of modulation characteristics allows one to selectively switch on or off the occurrence of coherent population trapping at specific frequencies. A new technique is developed to compute two-dimensional tridiagonal matrix equations. This generalized technique provides the vital methodology needed to calculate the response of such systems in the strong modulation regime and for arbitrary field strengths.

DOI
10.1103/physrevlett.96.183601

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We demonstrate the simultaneous occurrence of coherent population trapping at a series of frequencies separated by modulation frequency of phase-modulated fields. The two arms of the system are coupled to two phase-modulated fields and the response of the atomic system to such fields is calculated nonperturbatively. A judicious choice of modulation characteristics allows one to selectively switch on or off the occurrence of coherent population trapping at specific frequencies. A new technique is developed to compute two-dimensional tridiagonal matrix equations. This generalized technique provides the vital methodology needed to calculate the response of such systems in the strong modulation regime and for arbitrary field strengths.
We demonstrate new subharmonic features that arise in the strong modulation regime when a three-level Λ system is driven by two independent phase-modulated fields. In the strong modulation regime the system exhibits subharmonic spectral features even under weak-field excitation conditions, and the peaks occur precisely at integer combinations of the modulation frequencies. The spectral technique we use, apart from being nonperturbative, systematically accounts for the hierarchy of Fourier components underlying the wave-mixing phenomena. We also identify the dominant Fourier components responsible for individual peaks of the subharmonic signal. Under symmetric excitation one obtains multicoloured coherent population trapping with the associated atomic coherence. The magnitude of the multicoloured atomic coherence is invariably lower than the usual monochromatic case, and it is shared across a series of detunings as well as across (response) 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.
Coherent population trapping (CPT) resonance formation is modeled numerically in a three-level Λ -system with one of the near-resonance fields being frequency-modulated. The model is based on density matrix equations in RW approximation with atomic relaxation properly taken into account. Slow modulation is shown to be equivalent to CW excitation with the frequency changed point by point. As the modulation period approaches the relaxation times, the delayed response of the system is shown to cause the CPT resonance shift and reshaping, as well as the appearance of transient oscillations of the energy level populations.
We theoretically investigate the dynamic regime of coherent population trapping (CPT) in the presence of frequency modulation (FM). We have formulated the criteria for quasi-stationary (adiabatic) and dynamic (non-adiabatic) responses of atomic system driven by this FM. Using the density matrix formalism for Λ system, the error signal is exactly calculated and optimized. It is shown that the optimal FM parameters correspond to the dynamic regime of atomic-field interaction, which significantly differs from conventional description of CPT resonances in the frame of quasi-stationary approach (under small modulation frequency). Obtained theoretical results are in good qualitative agreement with different experiments. Also we have found CPT-analogue of Pound-Driver-Hall regime of frequency stabilization.
We have developed and tested an efficient mathematical model for calculating a coherent population trapping (CPT) resonance in a periodically modulated laser field when the modulation frequency fff is close to the fractional part of the hyperfine splitting frequency (HFS) in the ground state ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{HFS}}/N$ (where N=1,2,3,…N=1,2,3,\dots{}N=1,2,3,…). In such a polychromatic light, only two frequency components that are most resonant with the working optical transitions in the atom are taken into account exactly, while all other off-resonant frequency components are taken into account by using the second-order perturbation theory in the field magnitude. Within the framework of our concept, an equation for the atomic density matrix is obtained, in which all off-resonant frequency components are described by two operators (which are nondiagonal in the general case): a shift operator and a relaxation operator. This model has been verified by numerical calculations for various dependencies, in which we have not found visual differences from the results of exact calculations. Thus, in our bichromatic model, the infinite variety of periodically modulated fields in CPT spectroscopy can be uniformly described by only five parameters. In addition to a significant mathematical simplification, our model provides a clear physical picture of the various features of CPT spectroscopy in a periodically modulated laser field, including some effects that have not yet been discussed in the scientific literature. In particular, we show a widespread viewpoint that the CPT resonance shift is determined by standard ac Stark shifts of the lower levels is, in general, fundamentally incorrect, since the contribution to the light shift due to beats at the frequency ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{HFS}}$ between different off-resonant frequency components can be comparable to the value of standard ac Stark shift (or even dominate it). Therefore, even if we have detailed information on the spectral composition of the modulated field (e.g., using a spectrum analyzer), this is, in general, absolutely insufficient for determining the light shift of the CPT resonance.