Resonant Energy Transfer, with Creation of Hyper-Excited Atoms, and Molecular Auto-Ionization in a Cold Rydberg Gas

A cold Rydberg gas, with its atoms prepared initially all in the excited state 0 n , with 0 1 n  , contains an excessive amount of energy, and presumably is to relax by the Penning-type molecular auto-ionization (MAI), in which a portion of excess energy of one atom is given to another near-by atom and ionizing it. Its complementary process, the resonant energy transfer (RET), is discussed, in which the excess energy of one atom is used on another to form a hyper-excited atomic state a n with 0 a n n  . This process is always present, provided certain resonance energy conditions are satisfied. In this report, the n 0 and density dependences of the RET rates are studied in detail, employing a simple model: 1) at low densities, the RET is mediated by the dipole-dipole coupling V dd and its rates are generally much smaller than that of MAI, especially for small n 0 . But 2) as the density increases, our model shows that the rates become of comparable magnitude or even larger than the MAI rates. The V dd is no longer adequate. We, then construct a semi-empirical potential to describe the RET process. 3) At high densities, we show that the atomic orbital of a n overlaps with that of neighboring atoms, and the electron-electron potential becomes prominent, resulting in much higher rates.


Introduction
The relaxation of a cold Rydberg gas produced initially by exciting the atoms to , the decay rates were observed [8] to be too large, by as much as a factor of two orders of magnitude or more of the MAI rates. Besides, a large number of low energy free electrons as well as readily field-ionizable excited state atoms were detected.
In this paper, we discuss the resonant energy transfer (RET), a Forster-type [9] [10] [11] process, in which a pair of RyA's shares its internal energies and creates new bound states, one higher and the other lower than the original state 0 n . This RET process is complementary to MAI and is always present if certain energy resonance conditions are satisfied. However, the RET process has been omitted in the past, because its rates are known to be very small compared to that of MAI, especially when n 0 is small and the density of the gas N A is low. By a simple model, we examine in detail the n 0 and density dependences of the RET rates as well as that of MAI. Especially for the RET, as the density increases, the dipole-dipole interaction V dd between the RyA's may no longer be valid and must be modified.
The density of a cold gas N A (cm −3 ) defines the average separation R N between a pair of atoms, given by g. N A = 10 11 cm −3 gives R N ≈ 4 × 10 4 a B , where a B is the Bohr radius which we set equal to one. Furthermore, the radius of the RyA is defined simply as 2 0 0 r n = and 2 a a r n = , neglecting the factor 3/2 and angular momentum dependence. In discussing the RET, the r aT = 2r a is the important parameter [1] [3], and we simply define (a) the low density as R N > 2r aT , (b) the moderate density as 2r aT > R N > r aT , and (c) the high density as R N < r aT .

The MAI at Low Density
For a pair of RyA's A and B, each in state 0 0 , n l , and neglecting the l 0 part, the where I, II denote the ions and 1, 2 denote the ionized electrons. While being attracted to each other by the van der Waal's potential (which is quadratic in V dd and adiabatic), the MAI proceeds as The MAI amplitude (superscript ma) is defined, in the "prior" form, as where ( ) which makes the Born amplitude integral separable in r 1 and r 2 .
2) The total MAI probabilities. The MAI transition probability per unit time is given by The R −3 dependence of V dd , causes the ma fi P to decrease as R −6 . The total MAI probability is obtained by summing the da fi P over the restricted set of final states [2], as In (2.9), the sum ∑ nb is over the range 1 . It has been established that the transition involv- , as the both dipole matrix elements associated with V dd decrease as n b decreases from x b n : the threshold dominance.  ( ) 6 , , We have A y = 1/(4π) for l = 0 and A y = 1/(5π) for l = 1, m = 0. Explicitly, C p0 = 1/(2π) ≈ 0.16 for l = 0, e.g. and C p1 = 8/(25π) ≈ 0.11 for l = 1. We obtain

The RET at Low Density
In RET, a pair of RyA's, A and B, can exchange some of their internal energy, such that both atoms remain in the bound states of different n's; the process must satisfy the strict resonance energy condition. This can be satisfied either by an accidental matching of relevant energies, by a Stark shift, or by imposing an external electric field. An estimate of the Stark shift for example shows that, at a given density, high enough n a can satisfy the resonance condition. In the following, we simply assume that such conditions are met, and focus on the rates. r n a r a ≈ ≈ = × ; in this case, R N must be larger than 8R 0 to have the V dd applicable. Incidentally, we note that the RET for near-by states transitions, such as (48, 52) and (49, 51), have often been considered [11], but not for maximum n a . (The RET rates for small n 0 and large n a are usually very small, but not for large n 0 < n a .) Generally, the maximum n a depends sensitively on the lowest b n′ , and varies widely, roughly between 150 to 300 and more. For the general discussion, we simply take the typical value n a = 200. Since at N aT b T aT R r r r ′ ≈ + ≈ , the "tails" of two orbital functions for a n and b n′ start to overlap each other [1] [3], and where the transition probabilities start to increase exponentially, the r aT is the very parameter that can be used to define the regions of different densities, as given in Section 1.
2) The total RET probability. The total probability that includes the on-shell part of the sum in  As expected, the quantum defect theory gives 3 a Q n − = , for n 0 < 10, but slowly increases with n 0 , and becomes large. n  . Our model calculation, with V dd and the parameters n 0 = 20, n b = 10, and n a = 50, shows that Q is very small for n 0 < n a /2. Evidently, the RET is negligible for low n 0 , and this may be the principal reason for neglecting it. More generally, because of the unusual dependence of I(n 0 , n a ) on n 0 for a fixed n a (Figure 2), Q behaves approximately as

Summary and Conclusion
We have presented the RET as the potentially important process that affects the relaxation of cold Rydberg gas. This process has been neglected in all the previous studies of the decay of the gas, presumably because of its extreme low rates at small n 0 . Our present study has shown the importance of the RET, especially because of the large physical size of the hyper-Rydberg state; it basically changes the effective density by many folds.
The MAI is shown to be the dominant process at low density, R N > 2r aT , where the V dd is effective, while the RET is always present, but at very low probabilities. The n 0 dependence of the P re and P da is studied in detail; for the initial excited state 0 n not too high, 1 ≤ n 0 < 3n a /4. The ratio Q = P re /P da is found to be very small, of the order of 3 a n − to 1 a n − . The lower limit of Q follows from the quantum defect theory, which is approximately valid for n 0 < 10, but starts to break down for higher n 0 . As the n 0 approaches n a , Q increases to one, and grows rapidly, to as much as 2 a n .
For the cold Rydberg gas at moderate density, r aT < R N < 2r aT , the wave function ϕ a starts to overlap with the neighboring RyA's, and the V dd is no longer applicable. It is replaced by the electron-electron interaction V ee . The modified RET probabilities, re M ji P − , are estimated in several approximations, all of which indicate the resulting Q M to be much larger than the Q, of the order to n a . As expected, the re h ji P − at high density is much larger, with the overlap of orbital wave functions.
The RET can create a hyper RyA of large size, which in turn immediately forms a giant auto-ionizing clusters, enveloping many near-by atoms, and contains a huge amount of excess internal energies, making it highly unstable. Multiple production of clusters in the gas leads to a cascade decay of the gas. This problem requires a careful analysis with rate equations.
The dominance of states near the ionization threshold both in the MAI and RET processes yields multitude of low energy free electrons, via MAI, and many weakly bound electrons, via RET. These results are consistent with the experimental observation [8].