The original References [
3,
4,
7] show how the polarizability measurements in
Table 1 compare to theoretical predictions [
13,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41]. In this article, we devote our attention to interpreting the atomic polarizability measurements in
Table 1 in a systematic and tutorial manner. In the rest of
Section 3, we show how to use these polarizability measurements to predict other atomic properties such as oscillator strengths, lifetimes, matrix elements, line strengths, and van der Waals
coefficients, following procedures described earlier by Derevianko and Porsev [
47], Amini and Gould [
42], and Mitroy, Safronova, and Clark [
13], among others. Then, in
Section 4, we use the polarizabilities in
Table 1 to provide experimental constraints on the residual polarizabilities,
, for each of the alkali atoms.
3.1. Reporting Oscillator Strengths, Lifetimes, Matrix Elements, and Line Strengths from Static Polarizabilities
The dynamic polarizability,
, of an atom in state
can be written as sum over electric-dipole transition matrix elements
, Einstein coefficients
, oscillator strengths
, or line strengths
as
where
e and
m are the charge and mass of an electron,
are resonant frequencies for excitation from state
to state
, and
is the degeneracy of state
. The squares of electric dipole transition matrix elements
, or equivalently
, are related to the reduced dipole matrix elements (denoted with double bars) by
using the Wigner–Eckart theorem. For ground state alkali atoms, line strength
.
The expressions for polarizability
in Equations (
1)–(
4) each have dimensions of
times volume, as expected from the definitions
and
, where
is the induced dipole moment and
U is the energy shift (Stark shift) of an atom in an electric field
. When polarizability is reported in units of volume (typically Å
or
cm
), it is implied that one can multiply by
to get polarizability in SI units. The atomic unit (au) of polarizability,
, is equivalent to
, where
is the Bohr radius, and
is a Hartree. Since
in au, polarizability is naturally expressed in atomic units of volume of
(and for reference
Å
).
Since the principal D1 and D2 transitions of alkali metal atoms (denoting the
-
and
-
transitions, respectively, where
n = 6 for Cs,
n = 5 for Rb,
n = 4 for K,
n = 3 for Na, and
n = 2 for Li), account for over 95% of those atoms’ static polarizabilities [
33], it is customary to decompose polarizability as
where
represents the contribution from the principal transitions and
is the residual polarizability due to all other excitations. The residual polarizability itself can be further decomposed as
where
is due to excitations of the valence electron to higher-energy valence states as well as continuum states,
is the polariability due to the core electrons, and
is due to correlations between core and valence electrons. Sometimes, the notation
is used to denote a subset of
with
[
48], or
[
32], or an even higher cutoff such as
[
40].
Using the decomposition in Equation (
5), we can rewrite Equations (
1)–(
4) for static (
) polarizabilities:
Equation (
8) is written in terms of lifetimes
, rather than Einstein
A coefficients because alkali metal atom
states decay with a branching ratio of 100% to their respective ground
states. To support our analysis of polarizabilites, here in
Section 3, we use theoretically calculated values of residual static polarizabilities
= 2.04(69) au for Li,
= 1.86(12) au for Na,
= 6.26(33) au for K,
= 10.54(60) au for Rb, all from Safronova et al. [
48], and
= 16.74(11) au for Cs from Derevianko et al. [
47].
Table A1 in
Appendix A lists these and several other published values for
,
,
and
. We do not need to consider the hyperfine structure of the
,
, and
levels because the transition frequencies
and
are defined with respect to the center of gravity of the hyperfine states associated with each fine structure level. Furthermore, hyperfine splitting results in multiple resonance frequencies that are each shifted by less than a few parts in
, which is insignificant compared to the experimental uncertainties in atomic property measurements discussed throughout this article.
Since
and
are well known [
49], we can further use Equations (
7)–(
10) to derive expressions for
,
, and
in terms of
,
, and a ratio of line strengths
R:
where
R is defined as
To support our analysis of polarizabilities, we will use
R = 2.0000 for Li inferred from [
18],
R = 1.9994(37) for Na [
50],
R = 1.9976(13) for K [
51],
R = 1.99219(3) for Rb [
12] , and
R = 1.9809(9) for Cs [
52]. It is noteworthy that References [
11,
12,
51] determined
R experimentally using atom interferometry measurements of tune-out wavelengths.
Table 2 shows principal transition matrix elements, lifetimes, line strengths, and oscillator strengths inferred from experimental polarizability measurements [
3,
4,
5] and theoretical
values [
47,
48] using Equations (
11)–(
17). Our inferred lifetimes for K, Rb, and Cs are based on
measurements with 0.11% uncertainty, yet our derived lifetimes have slightly larger uncertainty. In the case of Li, Na, K and Rb, this is because roughly half of the total uncertainty comes from uncertainty in
, whereas for Cs, the uncertainties in
τ are dominated by contributions from uncertainty in
. Because there have been many high precision measurements of alkali metal principal transition lifetimes, it is useful to compare our derived lifetimes to those measurements. Our derived K and Rb lifetimes agree well with and have comparable uncertainty to those measured by Volz et al. [
50], Wang et al. [
53,
54], and Simsarian et al. [
55]. Because the
measurements used to derive the Li and Na lifetimes in
Table 2 are less precise, our inferred Na lifetimes have about twice the uncertainty (about 0.4%) of measurements by Volz et al. [
50], and our inferred Li lifetimes have much greater uncertainty than measurements by Volz et al. [
50] and McAlexander et al. [
56].
For Cs, the lifetimes we report in
Table 2 for this work have an uncertainty of less than 0.15%, which is slightly smaller than the uncertainty of four previous high-precision determinations of the Cs
state lifetimes [
47,
57,
58,
59].
Table 3 and
Figure 4 show how our semi-empirical lifetime results are consistent with [
47,
58] but differ from lifetimes reported in [
57,
59]. Our results deviate by 1.5
σ from
found in [
57] and by 3
σ from
in [
59], where
σ for the deviations here refers to the combined uncertainty (added in quadrature) for the experiments. Comparing the sum of line strengths (
), a quantity that is mostly independent of
R, provides a similar conclusion: our results are consistent with [
47,
58] but differ by two and three
σ from [
57,
59].
Because the two recent measurements of
by Gregoire et al. [
3] and Amini and Gould [
42] were made using very different methods, we combine these measurements using a weighted average in order to report a value for
with even smaller (0.03 ns) uncertainty in
Table 3. We note that, due to the uncertainty in
R and
, the uncertainty in
would still be 0.01 ns even if the polarizability measurements had no uncertainty.
The Cs
value calculated ab initio by [
60] is also consistent with our results for
. Since our results come from independent measurements of
and
R, combined with theoretical values for
, the agreement between our result for
with that of Derevianko and Porsev [
47,
60] adds confidence to their analysis of atomic parity violation [
60,
61].
3.2. Deriving van der Waals Coefficients from Polarizabilities
Since polarizability determines the strengths of van der Waals (vdW) potentials, we can also use measurements of
to improve predictions for atom-atom interactions. Two ground-state atoms have a van der Waals interaction potential
where
r is the inter-nuclear distance and
,
, and
are dispersion coefficients. For long-range interactions in the absence of retardation (i.e. for
), the
term is most important. The
coefficient for homo-nuclear atom-atom vdW interactions depends on dynamic polarizability as
Even though
in au, we write
ℏ explicitly in Equation (
19) to emphasize that the dimensions of
are energy × length
.
The London result of
can be found from Equation (
19) by using Equation (
1) for
with a single term in the sum to represent an atom as a single oscillator of frequency
with static polarizability
. However, calculating
gets more difficult for atoms with multiple oscillator strengths. In light of this complexity, we instead use the decomposition in Equation (
5) to express
as
Because of the cross term, the integration over frequency, and the way
remains relatively constant until ultraviolet frequencies,
is significantly more important for
than for
. Contributions from
account for 15% of
, whereas
contributes only 4% to
for Cs, as pointed out by Derevianko et al. [
33].
The fact that
and
depend on
in different ways (compare Equations (
5) and (
20)) suggests that it is possible to determine
based on independent measurements of
and
. We will explore this in
Section 4. First, we want to demonstrate how to use experimental
measurements and theoretical
spectra to improve predictions of
coefficients. For this, we begin by factoring
out of the
term in the integrand of Equation (
20) to get
where the spectral shape function
uses
R defined in Equation (
17). We are now able to calculate
using our choice of
, which we can relate to static polarizability measurements via
. The formula for
can then be written as
To use Equation (
23) to infer values of
from our static polarizability measurements, one still needs to know
and
. Derevianko et al. calculated and tabulated values
in [
63] of polarizability for all the alkali atoms, where the principal component
was calculated using experimental lifetime measurements by Volz and Schmoranzer [
50] for Li, Na, K, and Rb and by Rafac et al. [
57] for Cs. Therefore, we know that the residual component
of Derevianko et al.’s tabulated values of
is
Figure 5 shows an example of how
for Cs tabulated by Derevianko et al. [
63] can be decomposed into principal and residual parts.
Figure 5 also shows the small adjustment to
that can be recommended based on measurements of
. In essence, this procedure makes the assumption that any deviation between the measured and the tabulated [
63] values of static polarizability are due to an error in the
part of the tabulated values, and that the
component of the tabulated values is correct. To assess the impact of this assumption, we next examine how uncertainty in
propagates to uncertainty in
.
Equation (
23) shows how
calculations depend on
and
with opposite signs. This helps explain why uncertainty in
propagates to uncertainty in
with a somewhat reduced impact. For example, if
accounts for 15% of
, and
itself has an uncertainty of 5%, one might naively expect that uncertainty in
due to uncertainty in
would be 0.75%. However, using Equation (
23), one can show that the uncertainty in
is smaller (only 0.48% due to
). To explain this, if a theoretical
is incorrect, say a bit too high, then when we subtract this from the measured
, we will deduce an
that is too small, and the error from this contribution to
has the opposite sign from the error caused by adding back
in Equation (
23).
We can also rewrite Equation (
23) by adding and subtracting the tabulated
so that
depends explicitly only on the measured and tabulated (total) polarizabilities:
where
and
refer to values tabulated by Derevianko et al. This way
does not explicitly depend on
.
Using Equation (
25), or equivalently Equations (
23) and (
24), our calculated
values for Rb and Cs agree with recent theoretical and experimental
values, as shown in
Figure 6. For K, our predicted
is different from that measured by D’Errico et al. using Feshbach resonances by roughly 3
σ. Of course, this discrepancy may be at least partly explained by statistical errors in the
and
measurements for K atoms. In the next section, however, we will explore how an error in the
used to construct
for K could partly explain this discrepancy.
To interpret the
values that we report in
Table 4, we compare these semi-empirical results to direct measurements and earlier predictions of
in
Figure 6. One sees that the uncertainty of
measurements that we report based on atom interferometry measurements of polarizability are comparable to direct measurements [
64,
65,
66] and slightly more precise than previous semi-empirical predictions [
63].