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 |n₀> , with n ₀ »1, 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 | n _a > with n _a » n ₀. 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 | n _a > overlaps with that of neighboring atoms, and the electron-electron potential becomes prominent, resulting in much higher rates.

• Resonant Energy Transfer (RET)
• Molecular Auto-Ionization (MAI)
• Cold Rydberg Gas

1. Introduction

The relaxation of a cold Rydberg gas produced initially by exciting the atoms to Rydberg states | n 0 〉 , with the principal quantum number n 0 ≫ 1 , has been the subject of much recent studies, both theoretically [1] - [7] and experimentally [8]. At low densities, the Rydberg atoms (RyA’s) in the gas interact with each other by the dipole-dipole coupling V_dd. One of the prime breakdown mechanisms of the gas, mediated by V_dd, is the Penning-type molecular auto-ionization (MAI), in which the de-excitation energy of one RyA is transferred to the second one and ionizing it. However, at the density of N A e x p ≈ 10 9     cm − 3 , 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 | n 0 〉 . 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 n₀ is small and the density of the gas N_A is low. By a simple model, we examine in detail the n₀ 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 R N ( a B ) ≈ 1.9 × 10 8 [ N A ( cm − 3 ) ] − 1 / 3 . E.g. N_A = 10¹¹ cm⁻³ gives R_N ≈ 4 × 10⁴a_B, where a_B is the Bohr radius which we set equal to one. Furthermore, the radius of the RyA is defined simply as r 0 = n 0 2 and r a = n a 2 , 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.

2. The MAI at Low Density

For a pair of RyA’s A and B, each in state | n 0 , l 0 〉 , and neglecting the l₀ part, the two-body MAI involves the transition {i → f}, where [ i ] = [ ( n 0 ) A , ( n 0 ) B ] and [ f ] = [ ( n b ) A , ( c ) B ] , and where n b ≪ n 0 and (c)_B denote the ionized electron from B. At low densities and low temperature of interest here, the motion of the pair and van der Waal’s attraction may be neglected. States are specified by [., .], while transitions are specified by {[..] → [..]}.

1) The MAI amplitude. The system Hamiltonian is defined as

H = ( K I , I I + K 1 , I + K 2 , I I + V 1 , I + V 2 , I I ) + ( V 1 , I I + V 2 , 1 + V 12 + V I , I I ) ≡ H 0 + V (2.1)

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

{ i → f } = { ( n 0 → n b < n 0 ) A ( n 0 → c b = c o n t i n u u m ) B } (MAI) (2.2)

where simply [i] = [n₀, n₀] and [f] = [n_b, c_b]; the transition (2.2) produces a free electron, an ion II = B⁺, and a bound atom A in state | n b 〉 with n b ≪ n 0 . The energy conservation for (2.2) is

E i ( = 2 e 0 ) = E f ( = e n b + e c b ) (2.3)

where e 0 = − 1 / n 0 2   Ry , e n b = − 1 / n b 2   Ry , etc, and e_cb ≥ 0 for ionization to take place. The highest allowed value n b x of n_b is obtained by setting e_cb = 0 in (2.3), as

n b x = n 0 / 2 , (or the closest integer below it) (2.4)

For n₀ = 50, we have n b ≤ 35 ≡ n b x .

The MAI amplitude (superscript ma) is defined, in the “prior” form, as

M f i m a = ( Ψ f , V i Φ i ) (2.5)

where ( H − E ) Ψ f = 0 , and ( H i − E ) Φ i = 0 with H_i = H₀ and V_i = V. In the Born approximation and at low densities, we let Ψ f → Φ f with ( H 0 − E ) Φ f = 0 , while V i → V d d , as

M f i m a ≈ ( Φ f , V d d Φ i ) (2.6)

where

V d d = 2 [ ( r 1 ⋅ r 2 ) − 3 ( r 1 ⋅ ε ) ( r 2 ⋅ ε ) ] / R 3 , ε = R / R (2.7)

which makes the Born amplitude integral separable in r₁ and r₂.

2) The total MAI probabilities. The MAI transition probability per unit time is given by

P f i m a = 2 π | M f i m a | 2 (2.8)

The R⁻³ dependence of V_dd, causes the P f i m a to decrease as R⁻⁶.

The total MAI probability is obtained by summing the P f i d a over the restricted set of final states [2], as

Γ m a = ∑ n b = 1 n b x P f i m a , f = ( n b , c b ) (2.9)

In (2.9), the sum å_nb is over the range 1 ≤ n b ≤ n b x . (For type-setting, we used n b x ≡ n b x , c b ≡ c n b , n b ≡ n b ). It has been established that the transition involving n b = n b x is the most dominant [1], as the both dipole matrix elements associated with V_dd decrease as n_b decreases from n b x : the threshold dominance. Figure 1 illustrates two cases, n b = n b x and n b = n b x − 1 .

It is important to point out that the states with n ′ b > n b x are omitted in the sum (Equation (2.9)). They represent the RET, and thus the n b x is the dividing point of the spectrum between the MAI and RET.

3) A Model Calculation. For the actual calculation of the P f i m a , we define the model parameters, as n_b = 10, n₀ = 20, and n_a = 50 = 1/k for the continuum function. (These parameters are not quite the actual MAI in specific cases, but represent the physics reasonably well, both for MAI and RET.) We let the orbital functions φ = ( u / r ) ⋅ Y l m . Since the V_dd is separable and proportional to R⁻³, the amplitude P f i m a can be written in the form

P m a ( n l , n l → n b l ′ , c b l ′ ) = C p I J ⋅ J / R N 6 (2.10)

where C p = 8 π A y 2 , with A y = | ∫ d Ω Y l m Y l m + m ′ Y l ′ m ′ * | 2 for l ′ = l ± 1 . 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

I J ( n l , n b l ′ ) = ∫ r 1 u n l u n b l ′ d r 1 = 3.19 (2.11a)

J ( n l , c b l ′ ) = ∫ r 2 u n l u c b l ′ d r 2 = 7.47 × 10 4 (2.11b)

Thus, with R N = r a T = 2 r a = 5.0 × 10 3 and R N 6 = 1.6 × 10 22 ,

P m a ≈ C p 0 × 1.5 × 10 − 17 (2.12)

3. 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.

1) The RET probabilities (superscript re). The RET transition involves [ i ] = [ ( n 0 ) A ( n 0 ) B ] ≡ [ n 0 , n 0 ] and [ j ] = [ ( n ′ b ) A , ( n a ) B ] , with n ′ b ≪ n 0 and n a ≫ n 0 . Thus, we have

{ i → j } = { ( n 0 → n ′ b = n b x + m ′ ) A ( n 0 → n a ≫ n 0 ) B } (3.1)

with m ′ = 1 , 2 , ⋯ , n 0 − n b x , and the resonance energy conditions

E i = 2 e n 0 = e n b ′ + e n a = E j ,     n ′ b > n b x (3.2)

The RET probability P j i r e for {i ®j} is given in the Born approximation as

P j i r e = 2 π | M j i r e | 2 , with M j i r e ≈ ( Φ j , V d d Φ i ) (3.3)

A complete list of allowed n_a for n₀ = 50 and n b x = 35 is obtained from (Equation (3.2)) for the final state of RET [ j ] = ( n ′ b , n a ) : (36, 188), (37, 120), (38, 96), (39, 84), ⋯ , (48, 52), (49, 51), provided (Equation (3.2)) is satisfied. For the maximum n_a = 188, we have r a ≈ n a 2 a B ≈ 16 r 0 = 3.5 × 10 4 a B ; in this case, R_N must be larger than 8R₀ 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 n₀ and large n_a are usually very small, but not for large n₀ < n_a.)

Generally, the maximum n_a depends sensitively on the lowest n ′ b , 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 R N ≈ r a T + r b ′ T ≈ r a T , the “tails” of two orbital functions for | n a 〉 and | n ′ b 〉 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 j = ( n ′ b , n a ) is given by

Γ r e = ∑ n b ′ = n b x + 1 n 0 − 1 P j i r e ,     j = ( n ′ b , n a ) , (3.4)

where the sum over n ′ b is n b x + 1 ≤ n ′ b ≤ n 0 − 1 , provided the resonance condition is satisfied. (For typesetting we used n b ′ = n ′ b , n b x = n b x , and n 0 − 1 = n 0 − 1 .) This is to be compared with the total MAI probability given by (2.4), Γ d a = ∑ n b = 1 n b x P f ′ i d a , where f ′ = ( n b , c b ) . Evidently, Γ^re complements Γ^da as the combined set [ n b ] f + [ n ′ b ] j is complete; the sum includes all values up to n₀ and the contributions from values above n₀ are already included as the states for A are exchanged to that of B.

For the special case with hyper RyA, we can estimate the RET probability P j i r e for a two-body system by comparing with the P f i m a of MAI and adopting the quantum defect theory [12] [13]. To have a smooth analytic extrapolation over the threshold e_c = 0, the continuum function in the P f i m a is assumed energy-normalized [14] [15] [16]. For densities R_N > 2r_aT, define

Q ≡ P j i r e ( n a , n 0 , R N ) / P f i m a ( k n b = 1 n a , n 0 , R N ) . (3.5)

As expected, the quantum defect theory gives Q = n a − 3 , for n₀ < 10, but slowly increases with n₀, and becomes large.

3) Our model calculation. Consider again the model with the parameters n_b’ = 10, n₀ = 20, and n_a = 50. This corresponds roughly to one-half of the third set [ j ] = ( n ′ b , n a ) = ( 38 , 96 ) . (Here, or simplicity, we take the same n_b' as n_b of MAI.)

We write P f i r e as

P j i r e ( n 0 l , n 0 l → n ′ b l ′ , n a l ′ ) = C p I I ⋅ I / R N 6 , (3.6)

where C_p is defined earlier, and l ′ = l ± 1 . Taking l = 0, we obtain

I I ( n 0 l , n ′ b l ′ ) = ∫ r 1 u n 0 l u n b ′ l ′ d r 1 = 3.19 . (3.7a)

I ( n 0 l , n a l ′ ) = ∫ r 2 u n 0 l u n a l ′ d r 2 = 1.67 . (3.7b)

Thus, with R N = r a t = 5.0 × 10 3 we have

P j i r e ≈ C p 0 × 3.3 × 10 − 22 . (3.8)

This is to be compared with Equation (2.12)

P f i m a ( n 0 l , n 0 l → n b l ′ , k = n a − 1 l ′ ) = C p 0 × 1.5 × 10 − 17 .

In general, I_I > I_J and are nearly of same magnitudes for n b = n b x and n ′ b = n b x + 1 .

The main difference between the RET and MAI comes from I and J. They are conveniently compared by

Q = P r e / P m a = ( I I ⋅ I ) / ( I J ⋅ J ) ≈ I / J , (3.9)

especially when n b ≤ n b x and n ′ b ≥ n b x + 1 are close to each other and n b x ≫ 1 .

Our model calculation, with V_dd and the parameters n₀ = 20, n_b = 10, and n_a = 50, shows that Q is very small for n₀ < n_a/2. Evidently, the RET is negligible for low n₀, and this may be the principal reason for neglecting it. More generally, because of the unusual dependence of I(n₀, n_a) on n₀ for a fixed n_a (Figure 2), Q behaves approximately as

n a − 3 ≤ Q < n a − 2 , for 1 ≤ n 0 < n a / 4 (QDT), (3.10a)

n a − 2 < Q < 1 , for n a / 4 < n 0 < 3 n a / 4 , (3.10b)

1 < Q < n a 2 , for 3 n a / 4 < n 0 < n a . (3.10c)

For applications, (3.10b) is the most relevant. The above trend continues to hold for larger n_a.

4. The RET at High Densities

The RET probabilities discussed in the previous section are for low density, with R > 2r_aT, where the dipole-dipole coupling V_dd is effective. However, as the density increases and R approaches r_aT, the V_dd is no longer adequate in so far as treating the RET is concerned. By contrast, for the MAI, the V_dd should be valid for R N ≥ r a T / 4 ≥ R 0 = 4 r 0 . Thus, the result of Section 2 is unchanged; in fact, the MAI does not involve hyper-Rydberg states.

1) The RET amplitude (superscript h for high density) via V₁₂. We write the RET amplitude in the “post” form as

M j i r e − h = ( Φ j M , V j Ψ i ) , (4.1)

where Φ j M now satisfies ( H j − E ) Φ j M = 0 , with H j = H − V 12 and thus V j = V e e = V 12 . Note that the Φ j M is a two-center molecular orbitals. In the Born approximation for the Ψ i → Φ i , we have (superscript h for high density) for the post form

M j i r e − h = ( Φ j M , V 12 Φ i ) , (4.2)

and the RET probability is

P j i r e − h = 2 π | M j i r e M | 2 . (4.3)

2) Evaluation of the amplitude integrals—approximations. The integrals involved in the M j i r e are rather complex, and in the following, we make some simple approximations to obtain an estimate of the amplitude. Firstly, the molecular Φ j M is replaced by its LCAO (Linear Combination of Atomic Orbitals) form, Φ_j which is a solution of H₀; this will give only a crude estimate of the amplitude. Secondly, the variables involved in the V₁₂ = 1/r₁₂ are complicated, as r 12 = | r 2 I − r 1 | with r 2 I = | R + r 2 | ≡ s 2 , or r 12 = | r 2 − r 1 , I I | with r 1 , I I = | r 1 − R | ≡ s 1 . We first set the dipole case, as V 12 → Y 1 m ( 1 ) Y 1 m ∗ ( 2 ) r ′ < / r ′ > 2 , where r' = r₂ or s₁. Second, we let all the orbital functions be in the form φ = ( u / r ) ⋅ Y l m . To further simplify the task, e.g., let s₁ to be strictly a function of r₁ and R, but not its angular dependence. Then, the angular parts become trivial, and we have

P j i r e − h = C p × ( 1 / 4 ) × | T r e | 2 ,

where the C_p was defined earlier, and the amplitude integral is given by

T r e = ∫ 0 ∞ d r 2 u n l u n a l ′ { ( 1 / r 2 2 ) ∫ 0 r 2 d r 1 s 1 u n l u n b ′ l ′ + r 2 ∫ r 2 ∞ d r 2 u n l u n b ′ l ′ / s 1 2 } ≡ T 1 + T 2

Various simplifying approximations are considered: (si) s 1 = r 1 , (sii) s 1 = 2 n 0 2 , (siii) s 1 = 2 n a 2 . All three cases make the angular integrals simple.

We also define for convenience, W 1 = ∫ 0 r 2 d r 1 s 1 u n l u n a l ′ and W 2 = ∫ r 2 ∞ d r 1 u n l u n b ′ l ′ / s 1 2 Then, e.g. for (si) W 1 ~ ∫ 0 r 2 d r 1 r 1 u n l u n a l ′ and W 2 ~ ∫ r 2 ∞ d r 1 u n l u n b ′ l ′ / r 1 2 , etc.

The results for the model are:

si: T₁ = 7.11 × 10⁻⁷, T₂ = -1.68 × 10⁻⁷; T² = 2.9 × 10⁻¹³, Molecular

sii: T₁ = 2.54 × 10⁻⁵, T₂ = 5.73 × 10⁻⁹; T² = 6.45 × 10⁻¹⁰, n₀ Overlapping

siii: T₁ = 1.59 × 10⁻⁴, T₂ = 1.47 × 10⁻¹⁰; T² = 2.53 × 10⁻⁸, n_a touching.

5. 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 n₀. 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 n₀ dependence of the P^re and P^da is studied in detail; for the initial excited state | n 0 〉 not too high, 1 ≤ n₀ < 3n_a/4. The ratio Q = P^re/P^da is found to be very small, of the order of n a − 3 to n a − 1 . The lower limit of Q follows from the quantum defect theory, which is approximately valid for n₀ < 10, but starts to break down for higher n₀. As the n₀ approaches n_a, Q increases to one, and grows rapidly, to as much as n a 2 .

For the cold Rydberg gas at moderate density, r_aT < R_N < 2r_aT, the wave function j_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, P j i r e − M , 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 P j i r e − h 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].

Acknowledgements

This research work was supported by the National Science Foundation (NSF). The award number is 1901397.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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