IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 225503 (8pp) doi:10.1088/0953-4075/41/22/225503
Coherent control of magneto-optic
rotation
Kanhaiya Pandey, Ajay Wasan
1
and Vasant Natarajan
Department of Physics, Indian Institute of Science, Bangalore 560 012, India
E-mail: vasant@physics.iisc.ernet.in
Received 8 August 2008, in final form 29 September 2008
Published 10 November 2008
Online at stacks.iop.org/JPhysB/41/225503
Abstract
We experimentally study the rotation of the plane of polarization of a laser beam passing
through room-temperature Rb vapour. The rotation occurs because the medium behaves
differently for t he two orthogonally-polarized components, displaying what is known as
circular birefringence or linear dichroism. The difference is induced either by a control laser
applied to an auxiliary transition of a ladder-type system, or by an applied axial magnetic field.
In the presence of both control laser and magnetic field, the line shape shows an interesting
interplay between the two effects with regions of suppressed and enhanced rotation. The line
shapes can be understood qualitatively based on a density-matrix analysis of the system.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The rotation of the plane of polarization of light by a
medium in the presence of a magnetic field, generally called
magneto-optical rotation (MOR), has been studied for a long
time. The phenomenon arises due to the birefringence or
dichroism of the medium induced by the magnetic field,
and is known as the Faraday effect [1–4] when the applied
magnetic field is longitudinal (causing circular birefringence)
and the Voigt effect [5–7] when the field is transverse (causing
linear dichroism). MOR has important applications [8]in
diverse areas such as polarization control, Faraday optical
isolators, magnetometry [9] and laser-frequency stabilization
[10, 11]. Magneto-optical effects have also played an
important role in measuring parity-violating optical activity
in atoms caused by the weak interaction [12]. In recent
times, another technique of optical rotation has been studied,
which uses the fact that the properties of a multilevel atomic
system can be ‘coherently controlled’ by using suitable control
lasers. In such systems, the strong control laser applied
on one transition induces birefringence or dichroism for a
probe laser on a second transition, similar to the phenomenon
of electromagnetically induced transparency (EIT) where the
control laser modifies the absorption properties of the probe
[13]. Coherent control of optical rotation [14–18]again
1
Present address: Department of Physics, Indian Institute of Technology,
Roorkee 247 667, India.
has important applications, e.g. in precision measurements of
parity violation in atoms [19], and polarization control in the
deep UV region where standard optical elements are currently
unavailable. The chirality of the optical gain induced by a
control laser has also been used to measure parity violation
in atoms [20]. Finally, Faraday rotation in the presence of a
coupling laser has been studied recently with cold atoms in a
magneto-optic trap [21].
It has been theoretically predicted that the combination
of coherent control and magnetic fields for optical rotation
[22–24], termed coherent control of magneto-optical rotation
(CCMOR), can result in enhanced rotation and useful
modifications to the line shape. In this work, we
experimentally study the phenomenon of CCMOR in a ladder-
type three-level system in room-temperature Rb vapour. The
changes in the line shape in the presence of both the control
field and the magnetic field can be explained qualitatively
by a density-matrix analysis of the system. In a previous
experiment demonstrating coherent control of optical rotation
(i.e. without a magnetic field) in a ladder system in Rb [16],
the detection method had contributions from both i nduced
ellipticity and rotation. By contrast, we use balanced
detection of both polarization components, which is a standard
technique to eliminate any effects of induced ellipticity of the
probe beam [18]. The measured signal is then purely due
to optical rotation and the resultant line shapes are easier to
understand.
0953-4075/08/225503+08$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 225503 K Pandey et al
5S
1/2
5P
3/2
Control, Ω
c
Probe, Ω
p
(780 nm)
Δ
c
Γ
2
Γ
3
7S
1/2
Δ
p
(741 nm)
3
2
282
F=3
2
121
4
3
63
Figure 1. Three-level ladder system in
85
Rb. The relevant hyperfine
levels in each state are shown, with intervals given in MHz.
2. Theoretical analysis
We first consider the theoretical prediction for optical rotation
in a ladder system, as shown in figure 1. For specificity, we
consider the 5S
1/2
→ 5P
3/2
→ 7S
1/2
transition in
85
Rb. The
strong control laser drives the upper |2↔|3 transition (at
741 nm) with the Rabi frequency
c
and detuning
c
, while
the weak probe is scanned across the lower |1↔|2 transition
(at 780 nm) with the Rabi frequency
p
. The decoherence rate
from the 5P
3/2
state is
2
(= 6 MHz) while the corresponding
rate from the 7S
1/2
state is
3
(= 11 MHz).
2.1. Circularly-polarized control
It is well known that a control laser which is circularly
polarized induces circular birefringence for the probe laser,
i.e., it changes the refractive index for the right- and left-
circularly polarized components. This can be understood from
the simplified energy-level diagram (neglecting fine-structure
and hyperfine-structure effects) shown in figure 2(a). There
are three sublevels for the intermediate state (m
F
= 0, ±1),
but only one of these levels forms a ladder system with the
circularly-polarized components of the probe laser. It is this
differential coupling that causes the circular birefringence.
The angle of rotation for the plane of polarization of the probe
laser i s therefore given by
θ =
αl
2
[Re(s
+
) − Re(s
−
)], (1)
where α is the absorption coefficient, l is the interaction
length and s
+
(s
−
) is the susceptibility for right- (left-)
circularly polarized light. From the steady-state solution of
the density matrix equations for this three-level system, the
two susceptibilities are (to all orders in
c
and to first order
in
p
)[25]
s
+
=
i
2
/2
[
2
/2+i
p
]+
2
c
/4
3
/2−i(
p
+
c
)
,
(2)
s
−
=
i
2
/2
[
2
/2+i
p
]
.
Note that the control laser also induces circular dichroism for
the probe beam, i.e., different degrees of absorption for right-
and left-circularly polarized components. This arises from
differences in the imaginary part of the two susceptibilities
and results in ellipticity of the probe beam, but does not cause
rotation.
The above analysis is correct for a stationary atom. In a
gas of atoms at room temperature, we have to account for the
thermal velocity of moving atoms. The effect of the velocity is
to change the frequency of the beams by ±v/λ, with the sign
depending on whether the atom is moving along or opposite
to the light-propagation direction. The calculated rotation
after accounting for the Maxwell–Boltzmann distribution of
velocities is shown in figure 2(b), for αl = 25%, and
c
=
20 MHz. The inset shows the real part of the susceptibility for
the two circular polarizations of the probe.
Now let us consider what happens when we add an axial
magnetic field B. Since the direction of the magnetic field and
the direction of propagation of the control laser are the same,
both cause circular birefringence. The effect of the field, as
shown in figure 2(a), is to shift each magnetic sublevel by an
amount gμ
B
mB, where g is the Land
´
e g factor, μ
B
is the
Bohr magneton and m is the magnetic quantum number of the
sublevel. Therefore, the two susceptibilities now become
s
+
=
i
2
/2
[
2
/2+i(
p
+ gμ
B
B)]+
2
c
/4
3
/2−i(
p
+
c
)
,
(3)
s
−
=
i
2
/2
[
2
/2+i(
p
− gμ
B
B)]
.
The calculated line shape for rotation in the presence of
both the control laser and magnetic field is shown in figure 2(c).
The value of the field is 30 G, and again the calculation takes
into account thermal averaging in room-temperature vapour.
The magnitude of optical rotation is in the range of
milliradians. Since the rotation angle is directly proportional
to the optical path length αl, it can be increased by
several orders of magnitude by increasing the cell length
and heating the cell to increase the atomic density. In Rb,
increasing the cell temperature from room temperature to 90
◦
C
increases the atomic density by more than two orders of
magnitude. The magneto-optic rotation can also be increased
by increasing the magnetic field strength. The coherent control
rotation can be increased by increasing the Rabi frequency
of the control laser. For example, a previous experiment
on coherent control of optical rotation with the same ladder
system in Rb used a Rabi frequency of 870 MHz and a heated
vapour cell (αl = 28) to observe a rotation of about 1 rad [16].
Theoretical calculations of CCMOR have assumed αl = 300
to predict rotation of few radians [24]. However, even with our
modest values of Rabi frequency (20 MHz) and optical path
length (αl = 0.25), we see that the control laser can enhance
or suppress the magneto-optic rotation by 60% on either side
of the optical resonance. Note that, for MOR applications such
as magnetometry and laser-frequency stabilization, it is not the
angle of rotation but the signal-to-noise ratio in measuring the
angle t hat is important.
As mentioned earlier, the above simplified picture
ignores the effects of fine-structure and hyperfine-structure
2
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 225503 K Pandey et al
(a)
Control
σ
+
gμ
B
B
Probe
σ
−
σ
−
m = −1
0+1
lin
(d)
σ
−
+2+10−1−2−3 +3
σ
+
σ
−
+2+10−1−2−3 +3
+2+10−1−2−3 +3−4 +4
1/84 1/31/45/285/421/141/28
(b)
-0.4
-0.2
0.0
0.2
0.4
Rotation (mrad)
-60 -40 -20
0 20 40 60
Δ
p
(MHz)
Ω
c
= 20 MHz
B = 0
(e)
-0.4
-0.2
0.0
0.2
0.4
Rotation (mrad)
-60 -40 -20
0 20 40 60
Δ
p
(MHz)
Ω
c
= 60 MHz
B = 0
(c)
1.2
0.9
0.6
0.3
0.0
Rotation (mrad)
-400 -200
0 200 400
Δ
p
(MHz)
Ω
c
= 20 MHz
B = 30 G
(f)
3.0
2.0
1.0
0.0
Rotation (mrad)
-400 -200
0 200 400
Δ
p
(MHz)
Ω
c
= 60 MHz
B = 30 G
Control
Probe
Figure 2. Optical rotation in the presence of circularly-polarized control. In (a), we show a simplified energy-level diagram without
fine-structure or hyperfine-structure effects. Only one of the magnetic sublevels of the intermediate level is coupled by the two lasers. The
dotted levels represent the shift in the presence of a magnetic field. In (b) and (c), we show coherent control of rotation with and without a
magnetic field and αl = 0.25. The insets (with the same x-axis) show the real part of the susceptibility for opposite circular polarizations of
the probe. The energy-level diagram in (d) shows the complete sublevel structure taking into account the F value for each level, and gives
the Clebsch–Gordan coefficients for the control-laser transitions. The corresponding rotation curves are shown in (e) and (f). The dotted
curve in (f) is the rotation without including the additional F = 2, 3 hyperfine levels in the intermediate state.
interactions. This is also the procedure followed in previous
work on this system [16]. A more accurate picture would
incorporate the total angular momentum F and all the m
F
sublevels for each level. The complete energy-level diagram
for the main F = 3 → F = 4 → F = 3 transition is
shown in figure 2(d). As before, the control l aser causes
circular birefringence because some of the sublevels of the
intermediate level do not couple to form a ladder system. We
can see that this condition would be satisfied as long as the
value of F in the intermediate level is greater than or equal
to that of the upper level. Thus, from the hyperfine levels
shown in figure 1, there will be an additional contribution only
from the F = 3 level of the intermediate state. The calculated
rotation with the f ull sublevel structure is shown in figure 2(e).
Since all the transitions coupled by the control laser do
not contribute equally (due to the different Clebsch–Gordan
coefficients), the same amount of rotation requires an increase
in the Rabi frequency of the control laser to 60 MHz. Apart
from this difference, the line shape is almost identical to that
shown in (b), which is calculated from the simplified picture.
In addition, the contribution of the F = 3 hyperfine level of the
intermediate state is negligible, and the rotation calculated with
and without this level shows no difference. This is because
the F = 3 level is 121 MHz lower [26], and the ac Stark
shift of the detuned level is ten times smaller. Indeed, EIT
experiments on this ladder system show a single transparency
dip from the resonant F = 4 level, and no additional dip due
to the off-resonant F = 3level[27].
3
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 225503 K Pandey et al
(a)
Control
Probe
⊥
m = −1
0+1
⊥
||
||
(b)
0.8
0.6
0.4
0.2
0.0
Rotation (mrad)
-60 -40 -20
0 20 40 60
Δ
p
(MHz)
Ω
c
= 20 MHz
B = 0
Figure 3. Optical rotation in the presence of linearly-polarized control. In (a), we show the magnetic sublevels coupled by the two lasers for
β = 45
◦
, inducing linear dichroism. In (b), we show coherent control of rotation for the probe polarization. The inset (with the same x-axis)
shows the imaginary part of the susceptibility for parallel and perpendicular linear polarizations.
In the presence of a magnetic field, the s hift of a hyperfine
level is given by g
F
μ
B
m
F
B, with different Land
´
e g
F
’s for
each level. The calculated line shape in the presence of both
the control laser and a magnetic field is shown in figure 2(f).
The other hyperfine levels of the intermediate state are also
included since they contribute to the Faraday rotation, though
not to the coherent-control part. The effect of the additional
hyperfine levels is to cause an asymmetry in the line shape.
When these levels are not included, the line shape is similar to
the simplified calculation shown in (c).
Therefore, in the following, we use the simplified picture
for our calculations, with the understanding that the presence
of the other hyperfine levels results in some asymmetry of the
line shape.
2.2. Linearly-polarized control
In the presence of a control laser which is linearly polarized,
it is the linear dichroism induced in the medium that is
responsible for the rotation of the plane of polarization of
the probe beam, similar to what happens in the Voigt effect.
In other words, the differential absorption of parallel and
perpendicular polarization components of the probe results
in the rotation. The rotation angle is therefore related to the
imaginary part of the susceptibility and is given by
θ =
αl
2
sin(2β)[Im(s
) − Im(s
⊥
)], (4)
where β is the angle between the planes of polarization of the
control laser and the incoming probe beam, and and ⊥ are
defined with respect to the control. The magnetic sublevels
coupled by the probe and control lasers for β = 45
◦
are shown
in figure 3(a). Note how only the m = 0 s ublevel of the
intermediate s tate forms a ladder system.
As before, the density-matrix analysis yields
s
=
i
2
/2
(
2
/2+i
p
) +
2
c
/4
3
/2−i(
p
+
c
)
,
(5)
s
⊥
=
i
2
/2
(
2
/2+i
p
)
.
The calculated line shape in room-temperature vapour for
β = 45
◦
and
c
= 20 MHz is shown in figure 3(b). Note again
that the linear birefringence induced in the medium (arising
from the real part of the susceptibilities) causes ellipticity but
no rotation.
The addition of an axial magnetic field complicates
matters because the rotation is an interplay between the linear
dichroism induced by the control and the circular birefringence
induced by the B field. In other words, the linear-dichroism
axis of the control laser will precess around the magnetic-field
axis, which will become significant when the field is strong
enough. However, it is possible to calculate the rotation when
the control and probe are polarized parallel to each other, i.e.
β = 0
◦
. Under these conditions, equation (4) shows that there
is no rotation in the presence of the control laser alone. But
there will be rotation when an axial B field is present due to
the circular birefringence induced by the field, now modified
by the presence of the linearly polarized control laser. As
seen from the energy levels in figure 4(a), both the m =−1
and m = +1 sublevels form ladder systems, but the magnetic
field shifts them in opposite directions. The rotation angle is
therefore
θ =
αl
2
[Re(s
+
) − Re(s
−
)], (6)
with the susceptibilities given by
s
±
=
i
2
/2
[
2
/2+i(
p
± gμ
B
B)]+
2
c
/4
3
/2−i(
p
+
c
)
. (7)
The calculated line shape in room-temperature vapour
with a 30 G field is shown in figure 4(b). The real
part of the susceptibility for the two circular components
(shown in the inset) has the same line shape but is pulled
in different directions by the magnetic field. The rotation
is therefore strongly suppressed by the control laser on
resonance.
3. Results and discussion
We now turn to the experimental confirmation of these
predicted line shapes. The experiments were done in a room-
temperature Rb vapour cell, as shown schematically in figure 5.
4
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 225503 K Pandey et al
(a)
Control
σ
+
gμ
B
B
Probe
σ
−
σ
+
m = −1
0+1
σ
−
(b)
1.2
0.9
0.6
0.3
0.0
Rotation (mrad)
-400 -200
0 200 400
Δ
p
(MHz)
Ω
c
= 20 MHz
B = 30 G
Figure 4. CCMOR with linearly-polarized control and a magnetic field. In (a), we show the magnetic sublevels coupled by the two lasers
for β = 0
◦
, and the circular birefringence in the presence of a magnetic field due to the opposite energy shifts of the sublevels. In (b) we
show control of magneto-optic rotation for B = 30 G. The inset (with the same x-axis) shows the real part of the susceptibility for opposite
circular polarizations.
The probe beam was derived from a feedback-stabilized diode
laser system operating on the D
2
line at 780 nm [28]. The
linewidth of the laser after stabilization was around 500 kHz.
It was scanned across the 5S
1/2
(F = 3) → 5P
3/2
(F = 2, 3, 4)
hyperfine transitions of
85
Rb. Note that the F = 2 hyperfine
level of the ground state is 3 GHz away, and does not play
a role in the experiment. The size of the probe beam was
about 2.5 mm and its power was 50 μW. It was linearly
polarized and the angle of rotation after passing through the
cell was determined by first splitting it into its two orthogonal
components (using a polarizing beam splitter cube with an
extinction ratio of 1000:1) and then measuring the power in
each component. This kind of balanced detection [18]gives
the rotation angle independent of any induced ellipticity in
the beam. The counter-propagating control beam came from
a ring-cavity Ti:sapphire laser (Coherent 899-21) tuned to
741 nm. The laser was stabilized to an ovenized reference
cavity that gave it a linewidth of 500 kHz. It was kept on the
5P
3/2
(F = 4) → 7S
1/2
(F = 3) hyperfine transition in
85
Rb.
The control beam had a diameter of 4 mm and its power was
varied from 100 to 350 mW. The on-resonance absorption of
the probe beam (in the absence of the control beam) through
Probe
(diode)
780 nm
Digital Storage
Oscilloscope
Rb cell
with B field coil
M
Control
(Ti:sapphire)
741 nm
M
PD
M
PD
PBS
Figure 5. Schematic of the experiment. The angle between the
beams i n the vapour cell has been exaggerated for clarity, in reality
it is less than 10 mrad. Figure key: M, mirror; PBS, polarizing beam
splitter; PD, photodiode.
the 5 cm long vapour cell was 25%. It had a 30 cm long
solenoid coil wound around it so as to produce a uniform
magnetic field of up t o 30 G.
3.1. Circularly-polarized control
For the first set of experiments, we used a control beam which
was circularly polarized and had power of 210 mW. We first
measured probe rotation in the absence of a magnetic field.
The results are shown in figure 6(a). The primary peak is a
dispersive peak showing a line shape similar to the calculated
one in figure 2(a), with a maximum rotation of 1.5 mrad.
However, there is a second smaller peak of 282 MHz to the
left that has a different (non-dispersive) line shape. This is due
to the F = 2 hyperfine level of the upper state, from which the
control laser is detuned by +282 MHz. Thus the two-photon
resonance condition will be satisfied only when the probe is
detuned by −282 MHz. The effect of this additional hyperfine
level is also seen in EIT experiments, and can be used for high-
resolution hyperfine spectroscopy of excited states, as shown
by us in earlier work [29]. The detuning has a significant effect
on the line shape of rotation, as seen in figure 6(b). Note the
progressive increase in the lower lobe compared to the upper
one as the control is increasingly detuned to the blue.
The combined effect of the control laser and the magnetic
field (CCMOR) is shown in figure 7, for control power of
210 mW and three values of the magnetic field. The line
shape agrees qualitatively with our theoretical prediction with
the appearance of a dispersive region near the peak of the
Doppler-broadened curve. The change in rotation due to the
control laser is ±75% at the line centre for a field of 10 G. But
the effect becomes progressively smaller as the field strength
is increased, and reduces to ±10% at 30 G. The asymmetry
in the line shape is due to the presence of the additional
hyperfine levels of the intermediate state, which is also seen
in the calculated line shape (shown in figure 2(f)) when these
levels are included. However, there is a difference between
the widths of the upper and lower lobes, which is not seen in
the calculation.
5
