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Japanese Journal of Applied Physics

Theory of Optical-Microwave Double Resonance. III. Double Resonance with Off-Resonant Optical Radiation

Michio Takami

Published Jan 1, 1978• 11 citations
Publisher

Abstract

The theory of optical-microwave double resonance is extended to the case of the off-resonant optical field. Line shapes for fixed optical and variable microwave frequencies are calculated under various conditions by the density matrix method. Analytical expressions are given for small microwave or small optical saturation. When the off-resonance is much greater than the Doppler width of the optical transition, optical Stark effect and optical-microwave two-photon transition are the principal effects. When the signal is observed as a variation of microwave absorption, the former effect is much stronger than the latter. When the signal is detected optically, the two-photon and the stepwise transitions for small microwave saturation are the effects with the same order of magnitude, while the optical Stark effect is a secondary effect. The line shapes are also calculated numerically for arbitrary microwave and optical saturation.

DOI
10.1143/jjap.17.125

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jun 10, 1996•1 citation
The theory of optical-microwave double resonance is extended to the case of the off-resonant optical field. Line shapes for fixed optical and variable microwave frequencies are calculated under various conditions by the density matrix method. Analytical expressions are given for small microwave or small optical saturation. When the off-resonance is much greater than the Doppler width of the optical transition, optical Stark effect and optical-microwave two-photon transition are the principal effects. When the signal is observed as a variation of microwave absorption, the former effect is much stronger than the latter. When the signal is detected optically, the two-photon and the stepwise transitions for small microwave saturation are the effects with the same order of magnitude, while the optical Stark effect is a secondary effect. The line shapes are also calculated numerically for arbitrary microwave and optical saturation.
The effect of optical pumping on the lineshapes and the peak values of microwave absorption line is calculated by a density matrix method in the Doppler limit approximation. The calculated double resonance signal consists of two terms. One is proportional to the unperturbed population difference of the optical transition and has a Lorentzian lineshape at weak microwave saturation. In this term the two-photon processes cause broadening of the line. The other is proportional to the population difference of the microwave transition and gives the higher-order dynamic Stark effect. At moderate optical saturation the former term is much greater than the latter. At strong optical saturation the latter term becomes noticeable and lineshape anomaly occurs. The peak values of the double resonance signals are also calculated and compared with an experimental result.
From resonance equations using density matrix formalism, we obtain several features of rf or mw optical double resonance experiments at zero field. Resonance signal intensity formulas are derived. The question of the spatial and polarization dependence is then reduced to a minimum number of parameters through the use of multipole tensors; the effect of a resonance within the excited state is formulated as a redistribution of excited state multipole moments. In the case of transitions between hyperfine sublevels which are separated on the order of the natural linewidth, coherence is produced by interference between different rf-optical transition pathways. Depending on the method of detection, such interference may lead to a dispersion-shaped contribution to the ODR line shape, as observed in the experiments on the NO A state (Bergeman and Zare, accompanying paper). Second order radiation field corrections for multiplet transitions (generalizations of the Bloch-Siegert shift) are also derived.
Novel spectral line shapes of optical-microwave double resonance with an extremely narrow line width were observed in a rubidium atomic clock pumped by a semiconductor laser. It was confirmed that these novel spectral line shapes of double resonance were due to the modulation-transfer effect. These spectral line shapes were calculated by solving the equation of motion of the density matrix that describes the modulation-transfer process. The calculated results agree with the experimental results. By using the calculated results of the influence of the optical Stark effect on these spectral line shapes, characteristics of the inhomogeneous light shift were evaluated quantitatively for the frequency-modulated microwave. Furthermore, optimum values of modulation parameters were found by computer simulation in order to obtain the highest microwave frequency stability of the atomic clocks. An experiment was carried out employing these optimum parameters, and a short-term frequency stability as high as σy(τ) = 7.9 × 10−13τ−1/2 was obtained, where τ is an integration time of measurement.
A theoretical study on double-resonance optical pumping for the transitions $5{S}_{1/2}\text{\ensuremath{-}}5{P}_{3/2}\text{\ensuremath{-}}4{D}_{3/2,5/2}$ of 87Rb^{87}\text{R}\text{b}87Rb atoms is presented. The full-density-matrix equations in the presence the probe and the coupling laser beams for all relevant magnetic sublevels of the hyperfine levels were solved numerically, and the absorption of a probe beam was calculated as a function of the detuning of the coupling beam. The calculated results were compared with experimental data, and they showed an excellent agreement. The results were also compared with the approximate results calculated when the coherence between the ground and the highest excited levels was neglected. We found that the dominant operating mechanism for double-resonance signals was the optical pumping rather than the two-photon coherence, although the latter affected the signals significantly.
The interaction of a four-level atomic system with two electromagnetic radiations has been studied. The two intermediate levels are closely spaced such that a single radiation frequency can induce transitions from the ground state to both of them. In a cascade-type double resonance, the pump electric-field amplitude is much stronger than the amplitude of the signal field, which can induce transitions from both of the intermediate levels to the upper state. The 15 optical Bloch equations obtained from the Liouville equations are solved analytically to obtain the Doppler-free signal line shape. The computed signals, exhibiting two Lorentzian line profiles, are presented for different pump frequencies. In addition to these one-photon transitions, we obtain a weak two-photon resonance peak. The two-photon resonance becomes weaker when the detuning of the pump frequency from a resonance frequency is increased. When the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is intermediate between the pump resonance frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, the two-photon resonance is very weak and almost vanishes when ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. When the pump frequency is on resonance with one of the transition frequencies, the Rabi splitting of the signal shows asymmetry because of the presence of a close-lying transition which is pumped off resonance. The intensity-dependent shifts of the transitions are such that they repel each other more and more as the intensity of the pump field is increased. These shifts arise from the nonresonant interaction between the atom and the field.
The theory of the optical Stark effect is developed for quantum-size structures. The effect is due to the interaction of the electron system with intense light, whose frequency, (omega) , is in resonance between two subbands in the conduction band. The probe light of the frequency (Omega) falls in resonance with the adjacent transition between the ground state and the state with an electron-hole pair. In the quasi-steady-state mode, the singularities in the interband (Omega) -light absorption spectra are related to the critical points of the bands, rearranged by the (omega) - light. The shape of the higher conduction band can be evaluated from the spectral positions of the singularities. In the case of two-photon interband transitions, a sharp dependence of the absorption on the light intensity is predicted. In the non-steady-state mode, the probe beam consists of two femtosecond (Omega) -light pulses following each other with the delay time, (tau) d. The dependence of energy, absorbed from the second pulse, on the (tau) d value is calculated. Under the (omega) -pumping, one of the two phenomena should be observed, depending on the type of band structure, namely, the decay of the induced polarization with the characteristic time approximately 1/(omega) R or oscillations with the frequency approximately (omega) R, where (omega) R is the Rabi frequency. The principal effect, controlling the decay of the Rabi oscillations, is the spreading of the packet of states generated by the (Omega) -light pulse.