Microwave multiphoton transitions

Rydberg atoms and microwave fields constitute an ideal system for the study of atom-strong field effects, and they have been used to explore the entire range of one electron phenomena [5]. Here we focus on an illustrative example, which has a clear parallel in laser experiments, a series of experiments which show that apparently non-resonant microwave ionization of nonhydronic atoms proceeds via a sequence of resonant microwave multiphoton transitions and that this process can be understood quantitatively using a Floquet, or dressed state approach. 

In the following section the experimental approach is briefly described. The initial observations of microwave ionization and the completely non-resonant picture initially used to describe it are then presented. Then microwave multiphoton transitions in a two level system analogous to the rate limiting step of microwave ionization are described both experimentally and theoretically. Experiments on this two level system with well controlled pulses of microwaves to show the applicability of an adiabatic Floquet theory to pulses are then described. We finally return to microwave ionization to see evidence for the resonant nature of the process. 

The pair of levels 21s - (16,3) is exactly analogous to the extreme blue and red Na Stark states of n and + 1. The fact that only one has a permanent dipole moment is of no consequence it is only the difference in the permanent moments which is significant. Based on the single cycle Landau-Zener description of microwave ionization we expect that if atoms in the 18s state are exposed to a microwave field of amplitude equal to the crossing field, Eq = 753 V/cm, they would make transitions to the (16,3) state. On the other hand, if a static field is present as well as the microwave field it should be possible to see resonant microwave multiphoton transitions between these two bound states, and seeing the connection between these processes is part of our objective. 

Sweeping the static field, which alters the energy spacing between the levels, while keeping the microwave field fixed leads to the observation of resonant multiphoton transitions. An example, shown in Fig. 10.9, is the set of the 18s—>(16,k) transitions observed by sweeping the static field with different strengths of the 10.35 GHz microwave field.8 The sequence of 18s—>(16,3) transitions 25 V/cm apart is quite evident. At the top of Fig. 9 there is a scale in terms of the number of 10.35 GHz photons required to drive the 18s—> (16,3) transition. 

The data of Fig. 10.9 show clearly that the K (n + 2)s— (n,k) transitions are multiphoton transitions. On the other hand, most of the data obtained by sweeping the microwave field agree qualitatively with the Landau-Zener description. To reconcile these apparently different descriptions we consider the problem shown in Fig. 10.10, in which there are two states 1 and 2, with a linear Stark shift and no Stark shift respectively.10 States 1 and 2 are coupled by V, the core... 

Fig. 10.19 The microwave frequency dependence of the n changing signals at low microwave power, where n changes up or down only by 1. Resonant multiphoton transitions are observed near the expected static field Stark shifted frequencies indicated. These resonances involve the absorption of four or five microwave photons. The down n changing atom production curve was obtained with the state analyzer field EA set at 50.0 V/cm, while up n changing was studied as n = 60 atom loss with EA = 45.5 V/cm. The locations of resonances for larger direct (not stepwise) changes in n are indicated along with...

When the static field brought the two states into multiphoton resonance a sharp increase in the detected signal was observed, as shown in Fig. 7, which is a sequence of field scans showing the K 18s - (16, transitions with different microwave field amplitudes. In Fig. 7 the sequence of N photon 18s - (16,3) transitions, separated by 28 V/cm is quite apparent. At higher microwave fields transitions to other (16, fe) states are also present, cluttering the spectrum. From the data of Fig. 7 it is evident that the highest number of photons, N, which can be absorbed increases with the microwave field. In fact, it increases approximately linearly with the microwave field, and a linear fit of N to the microwave field yields... 

Laser-microwave spectroscopy based on nonlinear phenomena developed from the type of experiments on molecules already discussed in Section 3.2 which make use of optical pumping or double resonance. Occasionally, the laser and the rf power were high enough to create the nonlinear phenomena mentioned above, i.e., to saturate the transitions involved and/or to induce multiphoton transitions. The intermediate level in, e.g., two-photon transitions did not have to be a real state but could be virtual as well. Therefore, a drawback often encountered in earlier infared laser-microwave experiments could be avoided if the laser transition frequency did not exactly coincide with the molecular absorption line the Stark or Zeeman effect had to be used for tuning. This results in an undesired line splitting. With laser-microwave multiphoton processes, however, the laser can be operated at its inherent transition frequency. Exact resonance with molecular lines is then achieved by using a nonlinear effect, i.e., a radiofrequency quantum is added to or subtracted from the laser frequency (see Figure 28). 

Having considered the connection between the multiphoton resonances and the microwave threshold field for the K (n + 2)s —> (n,k) transitions, it is now interesting to return to the analogous n — n + 1 transitions which are responsible for microwave ionization and consider them from this point of view. We start with a two level description based on the extreme n and n + 1 m = 0 Stark states, a description which is the multiphoton resonance counterpart to the single cycle Landau-Zener model presented earlier. The problem is identical to the problem... 

Due to the n(n + 1) possible n— n + 1 transitions it is in general difficult to observe resonance effects in microwave ionization as obvious as those shown in Fig. 10.9. Nonetheless several experiments show clearly the importance of multiphoton resonance in microwave ionization. In Ba and in He the observed microwave ionization thresholds are structured by resonances3,6. An excellent example is the microwave ionization probability of the He 28 3S state shown in Fig. 10.14. In He the 3S states intersect the Stark manifold at fields approaching l/3n5, and as a result making transitions from the energetically isolated 3S state requires a field comparable to the field required to drive n — n + 1 transition. The structure in Fig. 10.14 is quite similar to the structure in Fig. 10.8, which is not... 

The ionization curve of Fig. 10.18 is obtained in the same way as the data shown in Fig. 10.14, by exciting atoms in zero field and then exposing them to a strong microwave field. When atoms are excited in the presence of a static field, to a single Stark state, and held in single Stark state by the continued application of the field, resonances became more apparent when a microwave field in the same direction is applied. Bayfield and Pinnaduwage have observed transitions from the extreme red H n = 60, m = 0 Stark state to other nearby extreme Stark states in static fields of 5-10 V/cm.29 As shown by Fig. 10.19 resonances corresponding to the four photon transition to the extreme red n = 61 Stark state and four and five photon transitions to the extreme red n = 59 Stark state are visible. These experiments are similar to the K and He multiphoton resonance experiments described earlier, but are inherently simpler because the extreme red n = 60 Stark state is only coupled to the extreme n = 59 Stark state. In contrast, the K (n + 2)s state is coupled to all the (n,k) Stark states. 

Using Rydberg atoms and microwave fields it has been possible to observe virtually all one electron strong field phenomena. The attraction of these experiments is that they can be more controlled than most laser experiments, with the result that more quantitative information can be extracted. The insights gained from these experiments can be profitably transferred to optical experiments. To demonstrate the latter point we demonstrate that apparently non-resonant microwave ionization, in fact, occurs by resonant transitions through intermediate states. These experiments demonstrated clearly the power of Floquet analysis of such processes, and the ideas were subsequently applied to the analogous problem of laser multiphoton ionization. 

A relatively unexplored extension of the Kramers theory is the escape of a Brownian particle out of a potential well in the presence of an external periodic force. Processes such as multiphoton dissociation and isomerization of molecules in high-pressure gas or in condensed phases/ laser-assisted desorption/ and transitions in current-driven Josephson junctions under the influence of microwaves " may be described with such a model, where the pieriodic force results from the radiation field. 

There is also a variety of experimental methods, including, for example, simple optical pumping with subsequent rf transitions, but also photon-echo microwave nuclear double resonance, or the generation of multiphoton Lamb dips with a laser and a microwave field. 

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