In a two-level system, the radiative lifetimeτradof the excited state∣n′l′⟩is determined by the Einstein A-coefficient
1 τrad(n,l)
=Anln′l′=
(ωnln′l′)3
3πε0ħc3∣ ⟨n′l′∣er∣nl⟩ ∣2 (1.14) whereωnln′l′= (En′l′−Enl)/ħis the transition frequency and⟨n′l′∣er∣nl⟩ = dnln′l′is the dipole matrix element of the transition.
Treating Rydberg atoms as two-level systems, however, is an oversimplifica-tion since atoms in the level∣nl⟩can not only decay to one level∣n′l′⟩but to a variety of lower lying levels. To account for all of these decay channels, the de-cay rates for each possible channel∣nl⟩ → ∣n′l′⟩- the Einstein A-coefficients - have to be summed up. The radiative lifetime then read
τrad(n,l) =
⎡⎢
⎢⎢
⎢⎣
∑
En′l′<Enl
Anln′l′
⎤⎥
⎥⎥
⎥⎦
−1
, (1.15)
where the summation is only over final states∣n′l′⟩with lower energy than the initial state∣nl⟩. From (1.14) it can be seen that the Einstein coefficient Anln′l′ accounts for both, the transition dipole matrix element dnln′l′ and the transition frequencyωnln′l′. Since the level spacing in the region aroundn=40 is on the order of 100 GHz and the transition energy to the 5p-states is on the order of 600 THz, the energy term dominates the Einstein coefficient over the dipole matrix element. Rydberg atoms innsorndlevels thus will most dominantly decay into the 5p1/2and 5p3/2levels.
The transition frequenciesωnln′l′can be determined from quantum defect the-ory and the transition dipole matrix elementsdnln′l′from a numerical calcu-lation of the wave functions of Rydberg states, and thus the radiative lifetime of Rydberg atom can be calculated. An overview of the calculated radiative lifetimes for rubidium is given in Figure1.3. Since the theoretical predic-tion depend on the parameters used in the theoretical models employed in the calculations, experimental measurements of the radiative lifetimes serve as valuable benchmarks. Radiative lifetimes of rubidium have already been measured in the 1970ies for nsandnd42, np43and n f states44. Recently, the effects of collisions and superradiance in hot gases has been overcome by using cold samples. In these experiments, the radiative lifetimes of Rb(np)
28
1 Rydberg atoms
state Gounand38 Beterov39 Oliveira40 Branden41
τ′ γ τ′ γ τ′ γ τ′ γ
s1/2 1.43 2.94 1.368 3.0008 1.45(3) 3.02(2) 1.4(1) 2.99(3) p1/2
2.76 3.02 2.4360 2.9989 2.80(3) 3.01(3) 3.5(4) 2.90(3)
p3/2 2.22135 3.00256
d3/2
2.09 2.85 1.0761 2.9898 2.10(2) 2.89(2) 1.8(3) 2.84(4)
d5/2 1.0687 2.9897
Table 1.3 Lifetime parametersTheoretical and experimental parameterτ(inns) andγfor thendependence of the radiative lifetime of85Rb as defined in (1.16).
states and their dependency on the principal quantum numbernin cold sam-ples in the range of 31≤n≤4540, those of Rb(ns) and Rb(nd) states even for 26 ≤ n ≤ 4545,46 could be determined with enhanced accuracy. The most recent measurement of the radiative lifetime has been reported by Branden et al.41 and is shown in Figure1.3together with the lifetimes calculated by Beterov et al. To compare the different results of theories and experiments, it is usual to describe the radiative lifetime using the analytic expression
τrad(nl) =τ′(n−δnl j)γ, (1.16)
where the parameterτ′andγare fitted to experimentally obtained lifetimes.
Table1.3summarizes these parameters for the experiments done by Oliveira et al.40and Branden et al.41along with with theoretical predictions given by Gounand38and Beterov et al.39.
From Figure1.3it can be seen that the experimentally obtained lifetimes are not explained by the calculated radiative lifetimes. The reason is that apart from the radiative lifetime, there is a second decay process that can con-siderably shorten the lifetime of Rydberg atom in experimental investiga-tions. Although the decay via low-energy transitions to neighboring states is suppressed in spontaneous emission, these transitions can be induced by blackbody radiation as soon as the temperature of the environmentT be-comes comparable to or larger than the spacing of the Rydberg levelkBT ≈ Enl−En′l′36.
In this regime, the modes of blackbody radiation that are resonant with atomic transitions have high photon occupation numbersN≫1 and the stimulated transfer of population of the initial Rydberg state∣nl⟩to neighboring states
Part I. Theoretical Foundations
lifetime/ µsτ
principal quantum numbern
20 30 40 50 60 70 80
1 5 10 50 100 500
1000 ns
np nd
Figure 1.3 Lifetimes of Rydberg states. The straight curves show the radiative lifetimes τrad for l = s1/2,p3/2,d5/2 Rydberg states and the dashed curves the calculated effective lifetimes τeff including blackbody induced decay at temperatureT=300Kas calculated by Beterov et al.39The circled data points are the measured lifetimes from Branden et al.41
∣n′l′⟩by emission or absorption of photons from the thermal radiation field thus can no longer be neglected. The probability of transitions originated from state∣nl⟩are now dominated by their transition dipole matrix elements, which are largest for transitions to the neighboring states∣(n±1)(l±1)⟩ be-cause their wave functions have large spacial overlap36.
The most widely used model to calculate the effective lifetime of Rydberg states incorporating both, radiative decay and blackbody induced transitions, was developed by Gallagher and Cooke47. Based on the fact that the Einstein coefficientWnnl′l′for the stimulated emission process∣nl⟩ → ∣n′l′⟩is
Wnnl′l′=N¯(ωnln′l′,T) ×Anln′l′, (1.17) the product of Einstein A-coefficient for spontaneous emission of the same transition and the number of photons in the blackbody radiation field at the
30
1 Rydberg atoms
transition frequencyωnln′l′, they give the simple expression Wnnl′l′= Anln′l′
exp(ħωnln′l′/kBT) −1, (1.18) employing the Planck distribution at temperatureTto obtain the number of photons per mode, ¯N(ωnln′l′,T). For the transitionsns→ (n−1)pornd→ (n+1)p, which are at microwave frequencies for Rydberg atom, the transition dipole moment dnnl′l′ is typically large and on the order of some 1000ea0. Inspecting (1.17) reveals that a sufficiently large number of photonsN can further increase the total transition probability. This can partly compensate for the penalty due to the low transition frequency for transitions to adjacent levels in the spontaneous decay (1.14), making the blackbody induced decay comparable to the spontaneous decay. At room temperature, the transition to the closest pstate is enhanced by the number of photons in the thermal field byN=70 for Rb(35s)states and byN=200 for Rb(45d).
In analogy to the case of spontaneous decay (1.15), the blackbody limited life-time τbb is defined as the inverse of the sum over all possible blackbody-induced transitions
τbb(nl) = [ ∑
n′l′Wnnl′l′]
−1
=
⎡⎢
⎢⎢
⎣
∑
n′l′
Anln′l′ exp(ħωnln′l′/kBT) −1
⎤⎥
⎥⎥
⎦
−1
. (1.19)
Consequently, the effective lifetime is given by the sum of the depopulation rates due to spontaneous and stimulated emission as
τeff(nl) = [ 1 τrad(nl)
+ 1
τbb(nl) ]
−1
. (1.20)
The calculation of the blackbody induced lifetimes can also be reduced to the calculation of the transition matrix elements of neighboring Rydberg states
⟨n′l′∣er∣nl⟩since the photon number in selected thermal modes is deter-mined only by temperature. Beterov et al. deterdeter-mined and compared the blackbody induced decay rates of rubidium39with experimental data40,45,46. Their results for the effective lifetimeτeff fors,panddstates as well as the radiative lifetimesτradof Branden et al.41are shown in Figure1.3.
Part I. Theoretical Foundations