Nonlinear magneto-optical rotation of elliptically polarized light
A. B. Matsko,
1
I. Novikova,
1,
*
M. S. Zubairy,
1,2
and G. R. Welch
1
1
Department of Physics and Institute for Quantum Studies, Texas A&M University, College Station, Texas 77843-4242
2
Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan
共Received 30 October 2002; published 11 April 2003兲
We predict theoretically and demonstrate experimentally an ellipticity-dependent nonlinear magneto-optic
rotation of elliptically polarized light propagating in a medium with atomic coherence. We show that this effect
results from hexadecapole and higher-order moments of the atomic coherence, and is associated with an
enhancement of Kerr and higher-order nonlinearities accompanied by suppression of the other linear and
nonlinear susceptibility terms of the medium. These nonlinearities might be useful for quantum signal process-
ing. In particular, we report an observation of enhancement of the polarization rotation of elliptically polarized
light resonant with the 5S
1/2
F⫽ 2→5P
1/2
F⫽ 1 transition of
87
Rb.
DOI: 10.1103/PhysRevA.67.043805 PACS number共s兲: 42.50.Gy, 03.67.⫺a, 42.65.An, 32.60.⫹i
I. INTRODUCTION
Nonlinear magneto-optic rotation 共NMOR兲 of the polar-
ization plane of light resonant with atomic transitions is at-
tracting increasing attention 关1,2兴. Ultranarrow 共up to 1 Hz
关3,4兴兲 spectral features accompanied by strong polarization
rotation observed in NMOR experiments are used 共or pro-
posed to be used兲 in sensitive magnetometry 关5–7兴, in time-
reversal-invariance violation experiments 关8–10兴, in mea-
surements of the electron dipole moment 关11,12兴, and in
measurements of various atomic constants 关13兴. Extremely
slow propagation of light has also been observed in NMOR
in hot rubidium vapor 关14兴.
The most accurate description of the properties of NMOR
signals is obtained from an analysis of density matrix equa-
tions for the atomic polarizations and populations along with
Maxwell equations describing propagation of the electro-
magnetic fields in the atomic medium. The exact solution of
this problem, however, is very complicated, and for most
cases may be obtained only numerically. The problem should
be somehow simplified to obtain analytical results.
The traditional approach to solution of the problem is
based on the approximation of weak electromagnetic fields
and low atomic vapor densities 关15–18兴, conditions found in
early experiments involving incoherent radiation from
atomic discharge lamps. In this case one can use perturbation
theory, and the atomic susceptibility may be decomposed
into a series of the electromagnetic fields involved.
Magnetic-field-dependent terms of the susceptibility decom-
position which are nonlinear in the electromagnetic fields are
responsible for NMOR. It can be demonstrated that only
two-photon processes are important in this approximation,
and therefore complicated multilevel systems may be re-
duced to systems with small level number 共such as ⌳,V,or
X schemes兲关17,19,20兴. In this approximation, NMOR is a
consequence of low-frequency ground-state coherence
formed by two-photon processes between Zeeman sublevels
with difference in magnetic quantum numbers equal to ⌬m
⫽⫾2.
In some cases it is convenient to describe the atom-light
interaction from the point of view of light-induced multipole
moments of the atomic electron distribution. Conventionally,
this is done in terms of an irreducible tensor representation
of the density matrix 关21–23兴. In this case, the ground-state
coherence is equivalent to the quadrupole moment, or align-
ment. It has been suggested that NMOR is a consequence of
the alignment to orientation conversion 关24兴, where the ori-
entation is equivalent to the population difference between
nearest Zeeman sublevels with ⌬m⫽⫾2.
The simplified theoretical approaches used for weak elec-
tromagnetic fields generally fail for strong ones. The ques-
tion that arises here is whether or not the interaction with
strong fields brings new physics, e.g., if the higher-order
atomic coherences influence NMOR. Alkali-metal atoms
have a level structure that allows for formation of a coherent
superposition of the magnetic sublevels with ⌬m⫽⫾4
共hexadecapole moment in the multipole decomposition of the
interaction process兲 and even higher. Such coherences should
be excited by multiphoton processes that include four or
more photons. Gawlik et al. 关25兴 observed strong narrow
features in a forward scattering experiment with free sodium
atoms, which were attributed to a hexadecapole moment.
However, subsequent work of Giraud-Cotton et al. 关15兴 and
other groups 关17,19,20兴 demonstrated that these features may
be explained using third-order perturbation theory, which in-
cludes only quadrupole moments.
There have been a number of publications where obser-
vation of hexadecapole and higher-order moments is re-
ported for the case where the magnetic field is perpendicular
to the light propagation direction 关26,27兴. At the same time,
the question of their influence on forward scattering and
NMOR signals in Faraday configuration is still open 关28兴.
Generally, the interpretation of the experimental results in
the case of strong laser fields and large multipole moments is
very complicated. The high-order coherence causes only
slight modifications of the rotation caused by the quadrupole
moment, which hinders a convincing demonstration of these
high-order effects.
We here solve both analytically and numerically the prob-
lem of the propagation of strong elliptically polarized
electro-magnetic fields through resonant atomic media in the
*
Electronic address: i.novikova@osa.org
PHYSICAL REVIEW A 67, 043805 共2003兲
1050-2947/2003/67共4兲/043805共18兲/$20.00 ©2003 The American Physical Society67 043805-1
presence of a magnetic field. We particularly investigate the
properties of the light that interacts with the magnetic sub-
levels in an M-like level configuration and, therefore, forms
coherences with ⌬m⫽ 4. We demonstrate that these coher-
ences are responsible for a modification of polarization rota-
tion that depends on both the light ellipticity and the applied
magnetic field. We observe this effect in a hot vapor of ru-
bidium atoms. Since such a rotation does not appear for an
isolated ⌳ scheme, our experiment may be treated as a clear
demonstration of the hexadecapole moment of atoms.
Another interesting and important feature of the system
under consideration is connected with the large Kerr nonlin-
earity that is associated with NMOR. We analyze Kerr non-
linearity in the M level configuration and show that the ratio
between the nonlinearity and the absorption may be large.
Moreover, we show that by increasing the number of Zee-
man sublevels 共e.g., by using another Rb isotope or different
alkali-metal atom with higher ground-state angular momen-
tum兲 it is possible to realize higher orders of nonlinearities.
Our method of creation of the highly nonlinear medium with
small absorption has prospects in fundamental as well as
applied physics. It can be used for construction of nonclas-
sical states of light as well as for coherent processing of
quantum information 关29兴.
To form a bridge between this and previous studies we
should note that NMOR may be attributed to coherent popu-
lation trapping 共CPT兲关30,31兴 and electromagnetically in-
duced transparency 共EIT兲关32兴. Both EIT and CPT are able to
suppress linear absorption of resonant multilevel media
while preserving a high level of nonlinear susceptibility 关33–
35兴. Previous theoretical studies of coherent media with large
optical Kerr nonlinearities have described nonlinearities re-
sulting from the effective self-action of an electromagnetic
field at a single photon energy level, such as a photon block-
ade 关36–39兴, or an effective interaction between two electro-
magnetic fields due to refractive 关34,35,40,41兴 and absorp-
tive 关42兴 Kerr nonlinearities. The absorptive
(3)
nonlinearities have been studied experimentally for quasi-
classical cases 关43,44兴. It was shown quite recently that a
similar approach may lead to achievement of even higher
orders of nonlinearity 关45兴.
A method of producing Kerr nonlinearity with vanishing
absorption is based on the coherent properties of a three-
level ⌳ configuration 关see Fig. 1共a兲兴. In such a scheme the
effect of EIT can be observed. Two optical fields
␣
1
and ⍀
1
,
resonant with the transitions of the ⌳ system, propagate
through the medium without absorption. However, because
an ideal EIT medium does not interact with the light, it also
cannot lead to any nonlinear effects at the point of exact
transparency 关31兴. To get a nonlinear interaction in the co-
herent medium one needs to ‘‘disturb’’ the EIT regime by
introducing, for example, additional off-resonant level共s兲
关level a
2
in Fig. 1共b兲兴. In the following we refer to the re-
sultant level configuration as an N-type scheme. Such a
scheme has been used in previous work 关34–39,41兴.Ifthe
disturbance of EIT is small, i.e., the detuning ⌬ is large, the
absorption does not increase significantly. At the same time,
the nonlinearity can be as strong as the nonlinearity in a
near-resonant two-level system.
This paper is based on the existence of CPT in multilevel
media. Unlike the early ideas of Kerr nonlinearity enhance-
ment, we propose to use, not a single ⌳ scheme, but several
coupled ⌳ schemes. In particular, we consider the M-type
configuration as shown in Fig. 1共c兲. Coherent population
trapping exists in such a scheme, as in a ⌳-type level system.
By introducing a small detuning
␦
we may disturb this
CPT and produce a strong nonlinear coupling among the
electromagnetic fields interacting with the atomic system,
while having small absorption of the fields 关46兴. The disper-
sion of the M level media and associated group velocity of
light propagating in the media are intensity dependent due to
the nonlinearity, as was theoretically predicted by Greentree
et al. 关47兴. Finally, in the case discussed below, the energy
levels of the M configuration correspond to Zeeman sublev-
els of alkali-metal atoms. The multiphoton detuning is intro-
duced by a magnetic field, resulting in the intensity-
dependent polarization rotation.
We show a simple way to reduce a five-level M configu-
ration to a four-level N configuration, and prove that these
completely different schemes demonstrate refractive nonlin-
earities of the same magnitude. This is a very interesting
result, because the nonlinearity of the M configuration is a
consequence of the hexadecapole part of atomic coherence,
while the nonlinearity in the N configuration results from
quadrupole atomic coherence.
Our paper is organized as follows. In Sec. II we analyze
the F⫽ 1→ F
⬘
⫽ 0 atomic transition, demonstrate that this
transition may be described by a ⌳ level configuration, and
show that the polarization rotation in the case of a ⌳ con-
figuration does not depend on the light ellipticity. In Sec. III
we study F⫽ 2→F
⬘
⫽ 1 atomic transitions, show that they
consist of ⌳ and M schemes, investigate the properties of the
M interaction scheme, and show that ellipticity-dependent
NMOR is possible. Using analytical calculations we show
that the hexadecapole moment plays an important role here.
In Sec. IV we expand our theory to the case of generalized M
energy level systems and discuss the possibilities of obser-
vations of
(5)
and higher-order nonlinearities. In Sec. V we
discuss applications of the nonlinearities for quantum-
information processing. The case of Doppler broadened ⌳,
M, and N systems is considered in Sec. VI for the particular
FIG. 1. Energy level schemes for 共a兲⌳sys-
tem; 共b兲 N system; 共c兲 M system.
MATSKO et al. PHYSICAL REVIEW A 67, 043805 共2003兲
043805-2
case of a weak probe field. We present experimental mea-
surements of the polarization dependent NMOR in hot
87
Rb
and
85
Rb atomic vapors in Sec. VII. Finally, in Sec. VIII, we
present our conclusions based on these results.
II. ANALYSIS OF NMOR FOR THE CASE
OF AN FÄ 1\ F
⬘
Ä 0 TRANSITION
A three-level ⌳ configuration is the simplest system that
results in NMOR. This system appears naturally in the con-
figuration of Zeeman sublevels of an F⫽1→F
⬘
⫽ 0 atomic
transition, where F and F
⬘
are the total angular momenta of
the ground and excited atomic states, respectively. This
scheme can be easily seen if the angular momentum quanti-
zation axis is chosen along the light propagation direction.
The effective interaction scheme for this case is shown in
Fig. 2共a兲. The ⌳ configuration consists of two circularly po-
larized components of the laser field, which create low-
frequency coherence between the magnetic sublevels m⫽
⫾ 1. Because of the selection rules, the electromagnetic
waves do not interact with the sublevel having m⫽ 0.
For zero magnetic field such a configuration demonstrates
coherent population trapping. A nonzero magnetic field col-
linear with the wave vector of the light leads to a Zeeman
shift of the magnetic sublevels m⫽⫾1, which disturbs CPT
and results in an interaction between the light and the atoms.
The nonlinear polarization rotation emerges as a conse-
quence of this interaction.
In the following, we briefly review the basic properties of
CPT in ⌳ systems and calculate the optical losses and polar-
ization rotation by solving the optical Bloch equations for the
density matrix elements. Finally, we note how the F⫽1
→F
⬘
⫽ 0 level configuration can be reduced to a ⌳ system
via proper renormalization of decay rates and density matrix.
A. Coherent population trapping in a ⌳ system
The Hamiltonian for the ⌳ system shown in Fig. 2共b兲 can
be written as
H
⌳
⫽ ប⌬
兩
a
典具
a
兩
⫺ ប
␦
兩
b
⫹
典具
b
⫹
兩
⫹ ប
␦
兩
b
⫺
典具
b
⫺
兩
⫹ ប
共
⍀
⫺
兩
a
典具
b
⫹
兩
⫹ ⍀
⫹
兩
a
典具
b
⫺
兩
⫹ H.c.
兲
, 共1兲
where E
⫹
and E
⫺
are the electric field amplitudes of two
opposite circularly polarized electromagnetic waves, ⍀
⫺
⫽ E
⫺
㜷
ab⫹
/ប, ⍀
⫹
⫽ E
⫹
㜷
ab⫺
/ប are the corresponding com-
plex Rabi frequencies, 㜷
ab⫹
and 㜷
ab⫺
are the atomic dipole
moments, ⌬ is the one-photon detuning of the laser fre-
quency from the exact atomic transition, and
␦
is the shift of
the ground-state sublevels resulting, for example, from inter-
action with a magnetic field.
The eigenvalues of this Hamiltonian
i
共where H
兩
典
⫽ ប
兩
典
) may be found from
冏
␦
⫺ ⍀
⫹
*
0
⍀
⫹
⌬⫺ ⍀
⫺
0 ⍀
⫺
*
⫺ ⫺
␦
冏
⫽ 0 共2兲
or
⫺
3
⫹
2
⌬⫹
共
␦
2
⫹
兩
⍀
⫹
兩
2
⫹
兩
⍀
⫺
兩
2
兲
⫺
␦
共
␦
⌬⫹
兩
⍀
⫺
兩
2
⫺
兩
⍀
⫹
兩
2
兲
⫽ 0. 共3兲
In the degenerate case (
␦
⫽ 0) the eigenvalues and corre-
sponding eigenstates are
D
⫽ 0,
兩
D
典
⫽
⍀
⫹
兩
b
⫹
典
⫺ ⍀
⫺
兩
b
⫺
典
冑
兩
⍀
⫹
兩
2
⫹
兩
⍀
⫺
兩
2
, 共4兲
B
1,2
⫽
⌬
2
⫾
冑
⌬
2
4
⫹
兩
⍀
⫹
兩
2
⫹
兩
⍀
⫺
兩
2
,
兩
B
1,2
典
⫽
冑
兩
B
1,2
兩
B
1
⫺
B
2
冉
兩
a
典
⫹
⍀
⫹
*
B
1,2
兩
b
⫺
典
⫹
⍀
⫺
*
B
1,2
兩
b
⫹
典
冊
.
共5兲
The state denoted as
兩
D
典
is called the ‘‘dark state’’ because
an atom in this state does not interact with the light fields
and, therefore, does not fluoresce. Atoms in the other two
states, called ‘‘bright states,’’ readily absorb light. Therefore,
atoms initially prepared in a bright state are optically
pumped into the dark state after some finite time comparable
with the lifetime of the excited level
兩
a
典
. Thus, in steady
state, the atomic ensemble does not interact with the electro-
magnetic fields, which is the essence of CPT. The dispersive
properties of the atomic system in the dark state are governed
by the coherence between the ground states of the ⌳ system.
The corresponding density matrix element may be found
from 共see 关Ref. 4兴兲
b⫹b⫺
⫽⫺
⍀
⫺
*
⍀
⫹
兩
⍀
⫺
兩
2
⫹
兩
⍀
⫹
兩
2
. 共6兲
The true dark state exists only for
␦
⫽ 0. As soon as the
exact resonant conditions are disturbed, the system starts in-
teracting with light. However, for small detunings
(
冑
兩
⍀
⫹
兩
2
⫹
兩
⍀
⫺
兩
2
Ⰷ
兩
␦
兩
,
冑
兩
⌬
␦
兩
) the disturbance of the dark
state is small, and most of the atomic population is concen-
FIG. 2. 共a兲 Schematic interaction of an electromagnetic wave
with an atomic transition
兩
b
典
, F⫽ 1→
兩
a
典
, F
⬘
⫽ 0. The electromag-
netic field is decomposed into two circularly polarized components
having Rabi frequencies ⍀
⫹
and ⍀
⫺
. 共b兲 Simplification of the
scheme 共a兲 for the case when there is a magnetic field applied
parallel to the wave vector of the electromagnetic wave.
NONLINEAR MAGNETO-OPTICAL ROTATION OF... PHYSICAL REVIEW A 67, 043805 共2003兲
043805-3
trated in the modified dark state
兩
D
˜
典
. In this case the eigen-
value
˜
D
corresponding to this state can be found by solving
Eq. 共3兲 and keeping only the terms linear in
␦
:
˜
D
⫽
␦
兩
⍀
⫺
兩
2
⫺
兩
⍀
⫹
兩
2
兩
⍀
⫹
兩
2
⫹
兩
⍀
⫺
兩
2
, 共7兲
兩
D
˜
典
⯝N
再
兩
D
典
⫹ 2
␦
⍀
⫹
⍀
⫺
(
兩
⍀
⫹
兩
2
⫹
兩
⍀
⫺
兩
2
)
3/2
兩
a
典
冎
, 共8兲
where N⯝1⫹ O(
␦
2
) is a normalization constant. From Eq.
共8兲 it is obvious that the population of the excited level
兩
a
典
is
proportional to
␦
2
.
B. Equations of motion
It is possible to obtain the equation of motion for the
electromagnetic fields, using the method reported in Refs.
关45,46兴. If we assume a small disturbance of CPT, almost all
atomic population remains in a dark state during the interac-
tion process, so that
兩
D
˜
典具
D
˜
兩
⯝I
ˆ
, where I
ˆ
is a unit operator,
and we can rewrite the interaction Hamiltonian as
H⯝ប
˜
D
I
ˆ
. 共9兲
In this case we can exclude the atomic degrees of freedom
from the interaction picture, and write the quasiclassical ana-
log of the interaction Hamiltonian with respect to the atomic
degrees of freedom: H⯝ប
˜
D
. This Hamiltonian may be fur-
ther rewritten in the Heisenberg picture, so that ⍀⬀a
ˆ
, where
a
ˆ
is the annihilation operator for the electromagnetic field
关45兴. The quantum mechanical equation for the electromag-
netic creation and annihilation operators may be presented in
the following form:
da
ˆ
dt
⫽⫺
i
ប
H
a
ˆ
†
. 共10兲
Strictly speaking, the right-hand side of this equation should
involve a functional derivative, rather than a partial one.
However, in this case the two give the same result. The
propagation equation for the electromagnetic field amplitude
E can be obtained from Eq. 共7兲 as a quasiclassical analog of
Eq. 共10兲关48兴:
E
z
⫽ 2
iN
c
H
E
*
, 共11兲
where N is the density of the atoms in the cell and
is the
carrier frequency of the electromagnetic wave. Using Eqs.
共11兲 and 共7兲共with H⯝ប
˜
D
) we arrive at the following
propagation equations for the Rabi frequencies ⍀
⫾
:
⍀
⫾
z
⫽⫿2i
␦
⍀
⫾
兩
⍀
⫿
兩
2
共
兩
⍀
⫹
兩
2
⫹
兩
⍀
⫺
兩
2
兲
2
, 共12兲
where
is a coupling constant given by
⫽
3
8
N
2
␥
r
共13兲
and is the wavelength of the light in vacuum. It is also
useful to rewrite the equation of motion for the field ampli-
tudes E
⫾
:
E
⫾
z
⫽⫿4i
ប
␦
N
c
E
⫾
兩
E
⫿
兩
2
共
兩
E
⫹
兩
2
⫹
兩
E
⫺
兩
2
兲
2
. 共14兲
Equation 共12兲 is suitable for describing the phase evolu-
tion of the electromagnetic fields. However, the decay pro-
cesses responsible for the optical losses cannot be correctly
included in this method and we need a density matrix ap-
proach. In the following section we explicitly calculate the
density matrix elements for the ⌳ system to verify Eq. 共12兲
and discuss the attenuation of the light.
C. Density matrix approach
In order to discuss a realistic model of the atom-field in-
teraction in an atomic cell we need to include atomic level
decay rates 关Fig. 2共b兲兴. We introduce the decay rate
␥
0
out-
side the system, which is inversely proportional to the finite
interaction time of the atoms and electromagnetic field. This
decay represents the atoms leaving the interaction region.
Another term that describes decay to outside levels,
␥
˜
r
, de-
scribes population pumping into states that do not interact
with the fields, for example, the state with zero magnetic
moment
关
m⫽ 0 in Fig. 2共a兲兴. The natural decay rate from
level
兩
a
典
to levels
兩
b
⫹
典
or
兩
b
⫺
典
is denoted as
␥
r
.
We also need to take into account the atoms entering the
laser beam. To do that, we include incoherent pumping to all
Zeeman sublevels from outside the system, which means that
atoms that enter the interaction region have equal popula-
tions of the ground-state sublevels and no coherence between
them. The value of the incoherent pumping rate is chosen to
be
␥
0
/2 to keep the sum of level populations equal to unity in
the case of
␥
˜
r
⫽ 0. When
␥
˜
r
⫽0, the sum of the populations
is less than unity because of the optical pumping, i.e.,
aa
⫹
b⫹ b⫹
⫹
b⫺ b⫺
⫽ 1⫺
␥
˜
r
␥
0
aa
. 共15兲
The time-evolution equations for the density matrix ele-
ments
ij
for the ⌳ system can be obtained from the Liou-
ville equation:
˙
⫽⫺
i
ប
关
H
⌳
,
兴
⫺
1
2
兵
⌫,
其
⫹ R, 共16兲
where
⫽ 兺
ij
兩
i
典具
j
兩
, H
⌳
is given by Eq. 共1兲, ⌫ is the matrix
describing the decays in the system, and R is the matrix of
incoherent pumping to the ground-state sublevels. Then the
equations for the atomic populations are
˙
b⫺ b⫺
⫽
␥
0
2
⫺
␥
0
b⫺ b⫺
⫹
␥
r
aa
⫹ i
共
⍀
⫹
*
ab⫺
⫺ c.c.
兲
,
共17兲
MATSKO et al. PHYSICAL REVIEW A 67, 043805 共2003兲
043805-4
˙
b⫹ b⫹
⫽
␥
0
2
⫺
␥
0
b⫹ b⫹
⫹
␥
r
aa
⫹ i
共
⍀
⫺
*
ab⫹
⫺ c.c.
兲
.
共18兲
Analogously, for the polarizations we have
˙
ab⫾
⫽⫺⌫
ab⫾
ab⫾
⫹ i⍀
⫿
共
b⫾ b⫾
⫺
aa
兲
⫹ i⍀
⫾
b⫿ b⫾
,
共19兲
˙
b⫺ b⫹
⫽⫺⌫
b⫺ b⫹
b⫺ b⫹
⫹ i⍀
⫹
*
ab
⫹
⫺ i⍀
⫺
b⫺ a
, 共20兲
where
⌫
ab⫾
⫽
␥
⫹ i
共
⌬⫾
␦
兲
, 共21兲
⌫
b⫺ b⫹
⫽
␥
0
⫹ 2i
␦
, 共22兲
with
␥
⫽
␥
r
⫹
␥
0
⫹
␥
˜
r
/2.
In the steady state case, we can solve Eqs. 共19兲 and 共20兲 in
terms of the atomic populations:
b⫺ b⫹
⫽⫺
⍀
⫹
*
⍀
⫺
共
n
b⫺ a
/⌫
b
⫺
a
⫹ n
b⫹ a
/⌫
ab
⫹
兲
⌫
b⫺ b⫹
⫹
兩
⍀
⫹
兩
2
/⌫
ab⫹
⫹
兩
⍀
⫺
兩
2
/⌫
b⫺ a
, 共23兲
ab⫾
⫽
i⍀
⫿
⌫
ab⫾
n
b⫾ a
共
⌫
b⫿ b⫾
⫹
兩
⍀
⫿
兩
2
/⌫
b⫿ a
兲
⫺ n
b⫿ a
兩
⍀
⫾
兩
2
/⌫
b⫿ a
⌫
b⫿ b⫾
⫹
兩
⍀
⫾
兩
2
/⌫
ab⫾
⫹
兩
⍀
⫿
兩
2
/⌫
b⫿ a
, 共24兲
where n
b⫾ a
⬅
b⫾ b⫾
⫺
aa
. Inserting these expressions into
Eqs. 共17兲 and 共18兲 and using the condition given in Eq. 共15兲,
we can derive linear equations for the atomic populations. In
the general case, however, their solution is very cumber-
some.
Let us consider the case of a strong electromagnetic field,
such that
兩
⍀
兩
2
/
␥
0
␥
Ⰷ 1. We also assume that
兩
␦
兩
,
␥
0
Ⰶ
␥
,
兩
⍀
兩
and ⌬⫽0. In the zeroth approximation the atomic
populations are determined by Eq. 共7兲:
b⫾ b⫾
(0)
⯝
兩
⍀
⫾
兩
2
兩
⍀
兩
2
, 共25兲
aa
(0)
⯝0, 共26兲
where
兩
⍀
兩
2
⫽
兩
⍀
⫹
兩
2
⫹
兩
⍀
⫺
兩
2
.
Now we can solve for the polarizations
ab⫾
keeping
only the terms linear in
␦
and
␥
0
,
ab⫾
⯝
i⍀
⫿
兩
⍀
兩
4
冉
␥
0
2
兩
⍀
兩
2
⫾ 2i
␦
兩
⍀
⫾
兩
2
冊
. 共27兲
It is important to note that this expression for the polariza-
tion, obtained for an open ⌳ system, coincides with the
analogous expression calculated by Fleischhauer et al. 关5兴
for a closed system, if the ground-state coherence decay rate
and the population exchange rate between ground states are
the same and equal to
␥
0
. This proves the equivalence of the
open and closed models for the description of ⌳ schemes,
which has been previously demonstrated by Lee et al. 关49兴
for the particular case of a weak probe field.
The stationary propagation of two circularly polarized
components of the laser field through the atomic medium is
described by the Maxwell-Bloch equations for the slowly
varying amplitudes and phases:
⍀
⫾
z
⯝⫺
⍀
⫾
兩
⍀
兩
4
冉
␥
0
2
兩
⍀
兩
2
⫾ 2i
␦
兩
⍀
⫿
兩
2
冊
. 共28兲
Note that Eq. 共12兲 can be obtained from Eq. 共28兲 in the limit
␥
0
⫽ 0.
Separating the real and imaginary parts of Eq. 共28兲 and
using ⍀
⫾
⫽
兩
⍀
⫾
兩
e
i
⫾
, one can find the propagation equa-
tions of the electromagnetic field intensity
兩
⍀
兩
2
and the ro-
tation angle of the polarization ellipse
⫽ (
⫹
⫺
⫺
)/2:
兩
⍀
兩
2
z
⫽⫺
␥
0
, 共29兲
z
⫽⫺
2
␦
兩
⍀
兩
2
. 共30兲
After integration, the following expressions for the light
transmission I
out
and the polarization rotation angle
are
obtained:
I
out
⫽ I
in
⫺
2
ប
c
␥
0
NL, 共31兲
⫽
2
␦
␥
0
ln
I
in
I
out
, 共32兲
where L is the interaction length. It is important to note that
the final expressions in Eqs. 共31兲 and 共32兲 include only the
total laser intensity, not the intensities of the individual cir-
cular components. This means that both transmission and
polarization rotation are independent of the initial polariza-
tion of light 关50兴.
NONLINEAR MAGNETO-OPTICAL ROTATION OF... PHYSICAL REVIEW A 67, 043805 共2003兲
043805-5
