1310 Vol. 42, No. 6 / June 2025 / Journal of the Optical Society of America B
Research Article
Semiclassical calculation of the power saturation
of the Kerr effect in Rb vapor. II: the effect
of incomplete optical pumping
Zachary H. Levine
Quantum Measurement Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8410, USA (zlevine@nist.gov)
Received 20 December 2024; revised 7 April 2025; accepted 24 April 2025; posted 24 April 2025; published 21 May 2025
Rubidium vapor cells play a key role in contemporary quantum optics since they have strong nonlinear interactions
with a low-power incident laser light. The simplest nonlinear interaction is the Kerr effect. Although most calcu-
lations concentrate on the unsaturated Kerr effect, the saturated Kerr effect has been measured in many materials
including
85
Rb and
87
Rb vapor. Moreover, entanglement-generating vapor cells often operate in the saturated
regime. Here, the calculation of the saturated Kerr effect for rubidium vapor at the D2 line uses a density matrix
formulation that includes two hyperfine-split levels and all magnetic sublevels in each of the ground and excited
states to account for incomplete optical pumping: the transit time of the atoms through the beam is too short to
reach a final population distribution. The calculation shows a rapid qualitative change at low optical power, in
semiquantitative agreement with experimental data.
https://doi.org/10.1364/JOSAB.553632
1. INTRODUCTION
Nonlinear optical interactions are at the heart of quantum optics
[1,2], including phenomena such as entangled photon genera-
tion via parametric down-conversion [3], electromagnetically
induced transparency [4], stimulated Raman adiabatic passage
(STIRAP) [5], and ultrafast pulse generation [6]. Rubidium
vapor cells in particular play an important role in quantum
optics [7,8] including a compact, accurate clock [9].
The Kerr effect may be most fundamental of all nonlinear
processes since it involves one beam at a single frequency and
occurs in all media. In the usual formulation, the index of
refraction obeys
n(I) =n
0
+n
(u)
2
I, (1)
where n
0
is the linear index of refraction, n
(u)
2
is the unsatu-
rated Kerr coefficient, I is the optical intensity, and n(I ) is the
intensity-dependent index of refraction. Both n
0
and n
(u)
2
have
complex numerical values which are independent of the inten-
sity. While this works well in many cases, there are numerous
exceptions [10]. In particular, it is relatively easy to saturate
the Kerr effect in a rubidium vapor cell with a moderate laser
power [11].
In saturation, the Kerr coefficient depends on the optical
intensity I via the relation
n(I) =n
0
+n
(s )
2
(I)I , (2)
introducing the saturated Kerr coefficient n
(s )
2
(I). The term
“saturated” is used because typically |n
(s )
2
(I)| is monotonically
decreasing with I. As always, n
2
=1 +χ, where χ is the suscep-
tibility. If χ 1, which is typical of atomic vapor systems, then
to a good approximation
n
(s )
2
=
χ − χ
(1)
2I
, (3)
where χ
(1)
is the low-field susceptibility and χ is a function of
the optical intensity I.
The saturated susceptibility is not usually studied theo-
retically. The prevailing method has been to measure in the
saturated regime and extrapolate to the unsaturated regime for
a comparison to theory [11]. The formula for the unsaturated
Kerr coefficient used in Ref. [11] may be obtained from a Taylor
expansion of the susceptibility for a two-level system [12]. In
the first paper of this series, there is a direct comparison of the
saturated Kerr coefficient and the two-level system [13]. While
it showed reasonable agreement with the data for larger optical
power, the two-level theory did not capture the rapid variation
of the Kerr coefficient at low power. This might be counterintui-
tive given that the low-field limit corresponds most closely to the
unsaturated Kerr coefficient.
Our collective intuition is based on textbook arguments,
such as the derivation of Fermi’s Golden Rule, which rely on
the system being exposed to light for a very long time. Most
relevant here, the susceptibility formula for the two-level system
corresponds to the limit of long exposure to the laser field [12].
However, under the experimental conditions of McCormick
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