Two-photon Ramsey resonances

In Sect. 7.4 we saw that the first-order Doppler effect can be exactly canceled for a two-photon transition if the two photons ha ] = ha 2 have opposite wave vectors, i.e., ki = — 2. A combination of Doppler-free two-photon absorp- 

The first two terms describe the conventional two-photon transitions in the first and second zones, while the third term represents the interference leading to the Ramsey resonance. Due to the longitudinal thermal velocity distribution f Vz), only the central maximum of the Ramsey resonance is observed with a theoretical halfwidth (for negligible natural width) of 

With a field separation of L = 2.5 mm and a mean velocity v = 270 m/s at 400 K, we obtain a halfwidth (FWHM) of Av = 1/37 = 36kHz for the central Ramsey fringe if the other contributions to the linewidth are negligible. 

The achievable spectral resolution is demonstrated by Fig. 14.39, which shows a two-photon Ramsey resonance for a hyperfine component of the two-photon Rydberg transition 32 of rubidium atoms Rb [14.106, 

measured with the arrangment of Fig. 14.38. The length of the Ramsey resonator must always be kept in resonance with the laser frequency. This is achieved by piezo-tuning elements and a feedback control system (Sect. 5.4.5). The halfwidth of the central Ramsey maximum was measured as Av = 37 kHz for a field separation of 2.5 mm and comes close to the theoretical limit. With L = 4.5 mm an even narrower signal with Av = 18 kHz could be achieved, whereas the two-photon resonance width in a single zone with a beam waist of coo = 150 fim was limited by transit-time broadening to about 600 kHz [14.107]. 

In Sect. 2.4 we saw that the first-order Doppler effect can be exactly canceled for a two-photon transition if the two photons hcoi = fuo2 have opposite wave vectors, i.e., k = —k2. A combination of Doppler-free two-photon absorption and the Ramsey method therefore avoids the phase dependence (p Vx) on the transverse velocity component. In the first interaction zone the molecular dipoles are excited with the transition amplitude a and precess with their eigenfrequency (jl 2 = ( 2 — E )/h. If the two photons come from oppositely traveling waves with frequency u), the detuning 

The quantitative description of the two-photon Ramsey resonance [1262] starts from the transition amplitude per second 

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Fortunately, several methods have been developed that overcome these difficulties and that allow ultranarrow Ramsey resonances to be obtained. One of these methods is based on Doppler-free two-photon spectroscopy, while another technique uses saturation spectroscopy but introduces a third interaction zone at the distance z = 2L downstream from the first zone to recover the Ramsey fringes [1257-1259]. We briefly discuss both methods. 

It is also possible to observe the Ramsey resonances at z = 2L without the third laser beam. If two standing waves at z = 0 and z = L resonantly interact with the molecules, we have a situation similar to that for photon echoes. The molecules that are coherently excited during the transit time t through the first field suffer a phase jump at r = T in the second zone, because of their nonlinear interaction with the second laser beam, which reverses the time development of the phases of the oscillating dipoles. At r = 2T the dipoles are again in phase and emit coherent radiation with increased intensity for co = a>ik (photon echo) [1258,1263]. 

This modification of the Ramsey technique was used by Hansch and coworkers for precision measurements on the hydrogen atom. The fundamental radiation of a cw dye laser at A. = 486 nm was frequency-doubled and through an acousto-optical modulator formed into an equidistant sequence of pulses. These pulses were sent into a high finesse resonator where they formed a standing wave with a corresponding pattern of spatially separated amplitude peaks [1270]. The achieved line width of the two-photon transition at X = 121 nm was below 5 kHz (Fig. 9.65). 

M.M. Salour, C. Cohen-Tannoudji Observation of Ramsey s interference fringes in the profile of Doppler-free two-photon resonances. Phys. Rev. Lett. 38, 757 (1977)... 

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