In a typical compact atomic time standard a current modulated semiconductor laser is used to create the optical fields that interrogate the atomic hyperfine transition. A pair of optical sidebands created by modulating the diode laser become the coherent population trapping (CPT) fields. At the same time, other pairs of optical sidebands may contribute to other multiphoton resonances, such as three-photon N-resonance [Phys. Rev. A65, 043817 (2002)]. We analyze the resulting joint CPT and N-resonance (hereafter N+CPT) analytically and numerically. Analytically we solve a four-level quantum optics model for this joint resonance and perturbatively include the leading ac Stark effects from the five largest optical fields in the laser's modulation comb. Numerically we use a truncated Floquet solving routine that first symbolically develops the optical Bloch equations to a prescribed order of perturbation theory before evaluating. This numerical approach has, as input, the complete physical details of the first two excited-state manifolds of Rb87. We test these theoretical approaches with experiments by characterizing the optimal clock operating regimes.
We show that a characteristic two-photon lineshape asymmetry arises in coherent population trapping (CPT) and three-photon (N) resonances, because both resonances are simultaneously induced by modulation sidebands in the interrogating laser light. The N resonance is a three-photon resonance in which a two-photon Raman excitation is combined with a resonant optical pumping field. This joint CPT and N resonance can be the dominant source of lineshape distortion, with direct relevance for the operation of miniaturized atomic frequency standards. We present the results of both an experimental study and theoretical treatment of the asymmetry of the joint CPT and N resonance under conditions typical to the operation of an N resonance clock.
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
f 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,…). 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.
We have developed an effective mathematical model to calculate the coherent population trapping (CPT) resonance in periodically modulated light, when the modulation frequency
f varies near the fractional part of hyperfine splitting in the ground state
Δhfs/N (where
N=1,2,...). In such polychromatic field, only two frequency components that are most resonant with atomic optical transitions are taken into account accurately, while all other off-resonant components are taken into account using the second-order perturbation theory in the field. Within the presented concept, equation for atomic density matrix is obtained, in which the contribution of all off-resonant components is reduced to the appearance of two new operators (non-diagonal, in general case): the shift operator and relaxation operators. In the case of three-level
Λ-system, the adequacy of presented effective model was verified by numerical calculations of various dependencies, in which we did not find visual differences from the exact calculations. In addition to a significant mathematical simplification, our model provides a clear physical picture of various features of CPT spectroscopy in a periodically modulated laser field, including effects that have not been discussed in the scientific literature before. In particular, we show that the widespread viewpoint that the CPT resonance shift is determined by usual 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
Δhfs between different off-resonant frequency components can be comparable (or even dominate) with the standard ac Stark shift. Therefore, even if we have detailed information on the modulated field spectrum (e.g. using a spectrum analyzer), this is, in general, absolutely insufficient to determine the light shift of CPT resonance.
This letter presents for the first time a systematic study of an atomic frequency standard based on coherent population trapping (CPT) effect in87Rb vapour optically pumped with a multifrequency laser radiation formed with sidebands of a single-frequency laser whose injection current is modulated at a subharmonic frequency (1/2-1/6) relative to the frequency of hyperfine ground state splitting. Measured are the CPT resonance parameters and the atomic clock short-term stability at different subharmonic frequencies of microwave modulation. The possibility of CPT resonance light shift suppression is experimentally demonstrated at modulation frequencies between 1/3 and 1/5 of the resonant frequency for continuous wave optical pumping of the87Rb vapour D1 line. Analyzed are the capabilities and potential of reduced energy consumption of atomic clocks when CPT resonance in rubidium vapour is excited at subharmonic modulation frequencies.
Presented for the first time is our research into reduction of energy consumption of miniaturised Rb atomic clocks based on coherent population trapping (CPT) effect at lowered microwave modulation frequency. We studied properties of a CPT-based atomic frequency standard pumped with multi-frequency radiation from sidebands of a single-frequency semiconductor laser whose injection current was modulated at subharmonics (1/2–1/6) of the frequency of Rb hyperfine ground state splitting. Pumping by third-order sidebands is the optimum when the atomic clock stability drops only slightly (by 11%), whereas energy consumption of the microwave components is reduced by a factor of 1.5.
Atomic clocks based on CPT are interesting because of the narrow linewidths achievable at very low intensities. The linewidth of a CPT resonance in a sample of cold atoms mainly depends on the optical saturation by two lasers and the different relaxation terms responsible for destroying the hyperfine coherence. We propose a complete analytical solution for the excited population based on the classical Bloch formalism to determine the line shape of the resonance. Finally we present the future development of a cold atoms clock based on CPT linewidth.