Collisional resonances

As shown by Fig. 14.2, there are four collisional resonances for each ns state. We label them by me, mu), where me and mu are the m values of the lower and upper final p states. As shown by Fig. 14.2 in order of increasing field the... 

Normally one might expect that if the transition probability vanishes on resonance it also vanishes off resonance. However, such is not the case. When the transition probability is calculated off resonance, by numerically solving Eqs. (14.16) using a Taylor expansion method, it is nonzero for both v E and v 1E.14,16 In Fig. 14.6 we show the transition probabilities obtained using two different approximations for v E, and vlE for the 17s (0,0) collisional resonance.16 To allow direct comparison to the analytic form of Eq. (14.21) we show the transition probabilities calculated with EAA = VBB = 0. For these calculations the parameters ju2l = pLz, = 156.4 ea0, b = 104ao, and v = 1.6 x 10-4 au have been used. The resulting transition probability curves are shown by the broken lines of Fig. 14.6. As shown by Fig. 14.6 these curves are symmetric about the resonance position. The vlE curve of Fig. 14.6(b) has an approximately Lorentzian form, but the v E curve of Fig. 14.6(a), while it vanishes on resonance as predicted by Eq. (14.24), has an unusual double peaked structure. 

Fig. 14.8 The observed Na 17p ion signal after population of the 17s state vs dc electric field, showing the sharp collisional resonances. The resonances are labeled by the m values of the lower and upper p states (from ref. 12).

Verification that the collision time r increases as n2, or that the linewidth of the collisional resonances decreases as n-2, is simply a matter of measuring the widths of the collisional resonances. In Fig. 14.10 we show a plot of the width of the (0,0) resonance as a function of n. The observed widths exhibit magnitudes and an 1.95(20) dependence in agreement with Eq. (14.8). 

One of the more interesting aspects of the collision process is the calculated dependence on the orientation of v relative to E. In particular, do the lineshapes for collisions with v E and v 1E differ as dramatically as shown in Fig. 14.6 To answer this question unambiguously required two improvements upon the initial measurements. First, the Na atoms must be in a well defined beam, otherwise v is not well defined. This requirement is easily met by enclosing the interaction region with a liquid N2 cooled box, which ensures that the only Na atoms in the interaction region are those in the beam. Second, the field homogeneity must be adequate, 1 part in 104, to resolve the intrinsic lineshape of the collisional resonances. A pair of Cu field plates 1.592(2) cm apart with 1 mm diameter holes in the top plate to allow the ions to be extracted is adequate to meet this requirement. 

The observed cross sections for the 18s (0,0) collisional resonance with v E and v 1 E are shown in Fig. 14.12. The approximately Lorentzian shape for v 1 E and the double peaked shape for v E are quite evident. Given the existence of two experimental effects, field inhomogeneties and collision velocities not parallel to the field, both of which obscure the predicted zero in the v E cross section, the observation of a clear dip in the center of the observed v E cross section supports the theoretical description of intracollisional interference given earlier. It is also interesting to note that the observed v E cross section of Fig. 14.12(a) is clearly asymmetric, in agreement with the transition probability calculated with the permanent electric dipole moments taken into account, as shown by Fig. 14.6. 

One of the potentially most interesting aspects of the resonant collisions is that, in theory, the collision time increases and the linewidth narrows as the collision velocity is decreased. According to Eqs. (14.6) and (14.8) the collision time is proportional to l/v3/2. Collisions between thermal atoms with temperatures of 500 K lead to linewidths of the collisional resonances that are a few hundred MHz at n = 20. In principle, substantially smaller linewidths can be observed if the collision velocity is reduced. 

The first and most obvious question is whether or not a narrower velocity distribution leads to narrower collisional resonances. In Fig. 14.14 we show the Na 26s + Na 26s — Na 26p + Na 25p resonances obtained under three different experimental conditions.20 In Fig. 14.14(a) the atoms are in a thermal 670 K beam. In Figs. 14.14(b) and (c) the beam is velocity selected using the approach shown in Fig. 14.13 to collision velocities of 7.5 X 103 and 3.8 X 103 cm/s, respectively. The dramatic reduction in the linewidths of the collisional resonances is evident. The calculated linewidths are 400, 28, and 10 MHz, and the widths of the collisional resonances shown in Figs. 14.14(a)-(c) are 350,40, and 23 MHz respectively. The widths decrease approximately as l/v3/2 until Fig. 14.14(c), at which point the inhomogeneities of the electric field mask the intrinsic linewidth of the collisional resonance. 

In the Na collision process of Eq. (14.1) the inhomogeneities in the static tuning fields required preclude the observation of very narrow collisional resonances. In... 

Fig. 14.14 Na 26s + Na 26s —> Na 26p + Na 27p collisional resonances observed with (a) no velocity selection, collision velocity 4.6 x 1(T cm/s, (b) velocity selection to a collision velocity 7.5 x 103 cm/s, (c) velocity selection to a collision velocity 3.8 x 103 cm/s (from...

As shown by Fig. 14.15, the resonances occur near zero field, and it is easy to calculate the small Stark shifts with an accuracy greater than the linewidths of the collisional resonances. As a result it is straightforward to use the locations of the collisional resonances to determine the zero field energies of the p states relative to the energies of the s and d states. Since the energies of the ns and nd states have been measured by Doppler free, two photon spectroscopy,22 these resonant collision measurements for n = 27, 28, and 29 allow the same precision to be transferred to the np states. If we write the quantum defect dp of the K np states as... 

While collisional resonances 5 MHz wide are interesting for spectroscopic purposes, what makes them most interesting is that the 5 MHz line width implies that the collision lasts at least 200 ns, a time not much less than the 1 fxs period allowed for the collisions to occur. If the collision linewidths can be reduced to the inverse of the time allowed for the collisions to occur, the collisional resonances become transform limited, and we know when each collision begins and ends. 

It is hardly surprising that, as the microwave power is raised, higher order multiphoton processes are observed. On the other hand, it may be surprising that for each m 0 the cross sections first increase then decrease with microwave power. For example, the m = cross section is clearly zero in the lowest trace. Similarly, the m = 0 cross section vanishes in the trace one above the lowest but reappears in the lowest trace. Such behavior, typical of the strong field regime, is not predicted by perturbation theory. Close inspection of Fig. 15.5 reveals that the positions of the collisional resonances shift to lower static field as the microwave power is raised. Finally, in contrast to the usual observation of broadening with increased power, the (0,0)m resonances, which are well isolated from other resonances, develop from broad asymmetric resonances to narrow symmetric ones as the microwave power is raised. 

Fig. 15.5 Observed Na 18p ion signal after the population of the 18s level vs the static field with a 15.4 GHz microwave field. Trace (a) corresponds to no microwave power input to the cavity and shows the set of four zero-photon collisional resonances. Traces (b), (c), (d), and (e) correspond, respectively, to 13.5, 50, 105, and 165 V/cm microwave field amplitudes inside the cavity and show additional sets of four collisional resonances corresponding to one, two, and three-photon radiatively assisted collisions. The peaks labelled 0,1,2, and 3 correspond to the lowest field member of the set of four resonances corresponding to zero-, one-, two-, and three-photon assisted collisions, (0,0)°, (0,0), ...

To describe the shifts and intensities of the m-photon assisted collisional resonances with the microwave field Pillet et al. developed a picture based on dressed molecular states,3 and we follow that development here. As in the previous chapter, we break the Hamiltonian into an unperturbed Hamiltonian H(h and a perturbation V. The difference from our previous treatment of resonant collisions is that now H0 describes the isolated, noninteracting, atoms in both static and microwave fields. Each of the two atoms is described by a dressed atomic state, and we construct the dressed molecular state as a direct product of the two atomic states. The dipole-dipole interaction Vis still given by Eq. (14.12), and using it we can calculate the transition probabilities and cross sections for the radiatively assisted collisions. 

Eq. (15.39) tells us where the collisional resonances occur, but it does not tell us how strong they are. The strength is determined by the phase contributions of the two parts of the rf cycle, when WA < WB and WA > WB. When the phases accumulated in these two half cycles are both integral multiples of 2n the collisional resonances are strong. This requirement leads to an oscillation in the intensity of the N photon assisted collisional resonance proportional to cos(kAE[fIoj + y), where y is a small constant, which has the same period as the Bessel function expression of Eq. (15.29). 

A final aspect of these radiation collisions, shown experimentally by Thomson et a/.,17 is that the cross section integrated over all the collisional resonances increases with the rf field. Since the mth collisional resonance has a cross section of o = oR Jm KEmvJ/oj) and 2m J2m(x) = 1, in general, 2 om > crR.17... 

When the phase of a low frequency rf field is controlled the observed collisional resonances change dramatically. With the field of Eq. (15.32), when the atoms are allowed to collide during the interval -0.5 fis < t < 0.5 pts with 0 = 0 or tt, we observe the collisional resonances shown in Figs. 15.12(b) and (c) respectively, in agreement with Eq. (15.33). With no rf field the resonance occurs at Es = 6.44... 

Figure 1. Na energy levels showing the locations of the collisional resonances for the process Na(17s) + Na(17s) — Na(17p) + 7Va(16p).

An interesting aspect of the collisional resonances shown in Fig. 2 is that they are quite easy to observe. In fact, most of the Rydberg atoms undergo collisions, and we can estimate the cross section rather easily using... 

Eq. (8) is given in atomic units, and if we re-express it in laboratory units, for n = 20, we find a = 3.10-8 cm2 and r = 10-9 s. This value of the cross section is in accord with our earlier rough estimate, and the inverse of the collision time is consistent with the linewidths of the collisional resonances shown in Fig. 2. The n scaling of both the cross section and the resonance width has been verified, and in Fig. 3 we show the observed dependence of the width A of the (0, 0) resonance on n [Gallagher 1982],... 

One of the more interesting aspects of the collisional resonances is how sharp they are. Most atomic collision processes have durations of 10-12 s, and these collsions last for times in excess of a nanosecond. Furthermore, the collisional resonances should become even narrower as the collision velocity is reduced. To explore this issue we used the K system... 

The radiatively assisted collisional resonances shown in Fig. 8 are taken under the condition that the microwave frequency far exceeds the linewidth of the collisions. If the frequency is far less than the linewidth, or equivalently the duration of the collision is short compared to the rf period, then the rf field simply adds to the static field. If in a static field the collisional resonance occurs at the field Er, then if the collision occurs at time i - 0 in the combined field... 

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