Photon recoil

Let us regard an atom with rest mass M in the energy level which moves with the velocity v. If this atom absorbs a photon of energy Ek i and momentum ftk it is excited into the level Ej. Its momentum changes 

Expanding (14.2) into a power series of (1/c) (n = 0,1,2...) we obtain for the resonant absorption frequency 

The first term represents the absorption frequency Wq = (Ek-Ej) of an atom at rest if the recoil of the absorbing atom is neglected. The second term describes the linear Doppler shift (Ist-order Doppler effect) caused by the motion of the atom at the time of absorption. The third term expresses the quadratic Doppler effect (2nd-order Doppler effect). Note that this term is independent of the direction of the velocity v. It is therefore not eliminated by the Doppler-free techniques described in Chaps.7-10, which only overcome the linear Doppler effect. 

A parallel beam of Ne ions accelerated by 10 KeV move with the velocity Vj, = 3-10 m/s. When the beam is crossed perpendicularly by a single-mode laser beam tuned to a transition with A = 500 nm, even ions with v = Vy = 0 show a quadratic, relative Doppler shift of Ay/v = 5-10 . It yields an absolute shift of Au = 250 MHz. This should be compared with the linear Doppler shift of 600 GHz, which appears when the laser beam is parallel to the ion beam (Example 9.8). 

The last term in (14.3) represents the recoil energy of the atom due to momentum conservation. The energy hw of the absorbed photon has to be larger than that for recoil-free absorption by the amount 

When we extract (Mqc E ) from the first root and (Mqc + Ei) from the second, we obtain by a Taylor expansion the power series for the resonant absorption frequency 

In the previous discussion the third field was used to detect the Ramsay fringes through resonances in the absorbed power. It is also possible to omit the third field. If two standing waves at x = 0 and x = L resonantly interact with the molecules, continuous coherent radiation may be observed at x = 2L, which is due to polarization transfer. The radiation intensity has a sharp peak at the center frequency u) 2 basic principle of this phenomenon is similar to that of photon echoes (see Sect.11.4.2). Due to a phase jump at the nonlinear interaction with the second field the Doppler phase, caused by the transverse velocity v, is exactly cancelled if the transfer times T 2 = v /(x2 x ) and T23 = v (x - X2) are equal [13.9]. 

Assume an atom or molecule with mass M and energy levels E, Ej, which moves at a velocity y, absorbs a photon of energy hu) and momentum h] resulting in a transition E Conservation of momentum demands that 

Expanding (13.13) in powers of 1/c yields the absorption frequency for the resonance case 

The first term wq = (Eg -Ej )/h represents the eigen frequency of the atom in rest if recoil is neglected. The second term is the linear (first-order) Doppler effect, describing the well-known Doppler shift Aw =J in the absorption frequency of a moving atom. The third term represents the second-order Doppler effect. Note that this term is independent of the direction of V and cannot be eliminated by the methods, discussed in Chap.10, which only overcome the first-order Doppler effect. The last term in (13.14) describes the photon recoil effect, where has been approximated by wq. 

A similar consideration for the emission of a photon by a molecule in level with momentum yields with P = Pg + Hji the emission frequency 

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Another metrological application of simple atoms is the determination of values of the fundamental physical constants. In particular, the use of the new frequency chain for the hydrogen and deuterium lines [6] provided an improvement of a value of the Rydberg constant (Roc)- But that is not the only the constant determined with help of simple atoms. A recent experiment on g factor of a bound electron [27,11] has given a value of the proton-to-electron mass ratio. This value now becomes very important because of the use of photon-recoil spectroscopy for the determination of the fine structure constant [41] (see also [8])-... 

In the near future a small improvement in 1 can be expected from ongoing determinations of fr/me in measurements of the photon recoil in Cs atom spectroscopy and a Cs atomic mass measurement [29], The present limitation for accuracy of aj1 arises mainly from the muon mass uncertainty. Therefore any better determination of the muon mass, e.g. through a precise measurement of the reduced mass shift in Z zvls2s, will result in an improvement of 1. 

In addition to the Rydberg constant a number of different quantities, all based on intrinsically accurate frequency measurements, are needed. Experiments are under way in Stanford in S. Chu s group to measure the photon recoil shift free = fmh/2mcs( of the cesium Di line [48]. Together with the proton-electron mass ratio mp/me, that is known to 2 x 10-9 [49] and even more precise measurements of the cesium to proton mass ratio mcs/mp in Penning traps, that have been reported recently [50], our measurement has already yielded a new value of a [45]. 

A study of the electron mass is now of interest also because of determination of the fine structure constant a from the photon-recoil-spectroscopy [22,23], A measurement of the recoil frequency shift... 

An additional line ( two-transverse-photon-recoil line) appears to be ------------- ... 

An additional line ( two-transverse-photon-recoil line) appears to be... 

An interesting aspect of collision-aided radiative excitation is its potential for optical cooling of vapors. Since the change in kinetic energy of the collision partners per absorbed photon can be much larger than that transferred by photon recoil (Sect. 9.1), only a few collisions are necessary for cooling to low temperatures compared with a few thousand for recoil cooling [1093]. 

The corresponding peaks in the population distribution NkiVz) of molecules in the upper level A ) are shifted due to photon recoil (Fig. 9.2a). They show up, according to (9.5), at the velocity components... 

Fig. 9.6 Simplified experimental realization for the deceleration of atoms in a collimated beam by photon recoil...

An interesting application of atomic deflection by photon recoil is the collimation and focusing of atomic beams with lasers [1138]. Assume atoms with the velocity u = 0, pass through a laser resonator, where an intense standing op-... 

Fig. 9.13 Cooling, deflection and compression of atoms by photon recoil. The electro-optic modulators (EOM) and the acousto-optic modulator (AOM) serve for sideband generation and frequency tuning of the cooling laser sideband [1136]...

Another application is the deflection of atoms by photon recoil. For sufficiently good beam collimation, the deflection from single photons can be detected. The distribution of the transverse-velocity components contains information about the statistics of photon absorption [1207]. Such experiments have successfully demonstrated the antibunching characteristics of photon absorption [1208]. The photon statistic is directly manifest in the momentum distribution of the deflected atoms [1209]. Optical collimation by radial recoil can considerably decrease the divergence of atomic beams and thus the beam intensity. This allows experiments in crossed beams that could not be performed before because of a lack of intensity. 

Optical sideband cooling is quite analogous to the Doppler cooling by photon recoil discussed in Sect. 9.1. The only difference is that the confinement of the ion within the trap leads to discrete energy levels of the oscillating ion, whereas the translational energy of a free atom corresponds to a continuous absorption spectrum within the Doppler width. 

One example of an application is the measurement of the gravitational acceleration g on earth with an accuracy of 3 x 10 g with a light-pulse atomic interferometer [1291]. Laser-cooled wave packets of sodium atoms in an atomic fountain (Sect. 9.1.9) are irradiated by a sequence of three light pulses with properly chosen intensities. The first pulse is chosen as 7r/2-pulse, which creates a superposition of two atomic states 1) and 2) and results in a splitting of the atomic fountain beam at position 1 in Fig. 9.71 into two beams because of photon recoil. The second pulse is a tt-pulse, which deflects the two partial beams into opposite directions the third pulse finally is again a tt/2-pulse, which recombines the two partial beams and causes the wave packets to interfere. This interference can, for example, be detected by the fluorescence of atoms in the upper state 2). 

The force on an atom with mass M excerted by photon recoil is... 

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping. J. Opt. Soc. Am. B 6, 2112 (1989)... 

The photon recoil can be used not only for the deceleration of collimated atomic beams but also for the deflection of the atoms, if the laser beam... 

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