Download Polarization rotation of slow light with orbital angular momentum in

PHYSICAL REVIEW A 76, 053822 共2007兲
Polarization rotation of slow light with orbital angular momentum in ultracold atomic gases
1
Julius Ruseckas,1 Gediminas Juzeliūnas,1 Patrik Öhberg,2 and Stephen M. Barnett3
Institute of Theoretical Physics and Astronomy of Vilnius University, A. Goštauto 12, Vilnius 01108, Lithuania
SUPA, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
3
SUPA, Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom
共Received 5 June 2007; published 16 November 2007兲
2
We consider the propagation of slow light with an orbital angular momentum 共OAM兲 in a moving atomic
medium. We have derived a general equation of motion and applied it in analyzing propagation of slow light
with an OAM in a rotating medium, such as a vortex lattice. We have shown that the OAM of slow light
manifests itself in a rotation of the polarization plane of linearly polarized light. To extract a pure rotational
phase shift, we suggest to measure a difference in the angle of the polarization plane rotation by two consecutive light beams with opposite OAM. The differential angle ⌬␣l is proportional to the rotation frequency of the
medium ␻rot and the winding number l of light, and is inversely proportional to the group velocity of light. For
slow light the angle ⌬␣l should be large enough to be detectable. The effect can be used as a tool for measuring
the rotation frequency ␻rot of the medium.
DOI: 10.1103/PhysRevA.76.053822
PACS number共s兲: 42.50.Gy, 03.75.Kk, 32.70.Jz
I. INTRODUCTION
Light can be slowed down by seven orders of magnitude
using the electromagnetically induced transparency 共EIT兲
关1–3兴. The EIT makes a resonant, opaque medium transparent by means of quantum interference between the optical
transitions induced by the control and probe laser fields. As a
result, a weak probe pulse travels slowly and almost without
a dissipation in a resonant medium controlled by another
laser 关1–9兴. Electromagnetically induced transparency was
shown not only to slow down dramatically laser pulses 关1–3兴
but also to store them 关10,11兴 in atomic gases. Following the
proposal of Ref. 关12兴, storage and release of a probe pulse
has been demonstrated 关10,11,13兴 by dynamically changing
the intensity of the control laser. The possibility to coherently
control the propagation of quantum light pulses opens up
interesting applications such as generation of nonclassical
states in atomic ensembles and reversible quantum memories
for light waves 关14–18兴.
Associated with EIT is a dramatic modification of the
reflective properties of the medium. In this paper we analyze
the manifestation of the orbital angular momentum 共OAM兲
of light in such a highly dispersive medium. During the last
decade optical OAM has received a great deal of interest
关19,20兴. However, most studies on the topic deal with ordinary 共not slow兲 light. Here we consider the propagation of a
slow light with an OAM in a moving atomic medium, the
emphasis being placed on a rotating medium, such as a vortex lattice in an atomic Bose-Einstein condensate 共BEC兲. Unlike the previous studies on the slow light in moving media
关21–27兴, we allow the light to carry an OAM along the
propagation direction. We show that the OAM of slow light
manifests itself in a rotation of the polarization plane of a
linearly polarized light. To extract a pure rotational phase
shift, we suggest to measure a difference in the angles of the
rotation of the polarization plane for two light beams characterized by opposite OAM. The differential angle ⌬␣l is
proportional to the rotation frequency of the medium ␻rot, as
well as to the OAM of light. The effect can be used as a tool
1050-2947/2007/76共5兲/053822共5兲
for measuring the rotation frequency ␻rot of the medium
and/or the OAM of slow light. We should note that rotational
phase shifts have been shown to lead to rotation of an image
by propagation through a rotating medium 共Ref. 关28兴, and
references therein兲. This suggests that such effects might also
be observable in the configuration discussed here.
II. FORMULATION
Consider an ensemble of ⌳-type atoms characterized by
two hyperfine ground levels 1 and 2, as well as an electronic
excited level 3 共Fig. 1兲. The translational motion of atoms is
described by the three component or vector operator for the
ជ 共r兲 in which the three components
translational motion ⌿
⌿1 ⬅ ⌿1共r , t兲, ⌿2 ⬅ ⌿2共r , t兲, and ⌿3 ⬅ ⌿3共r , t兲 correspond to
the three internal atomic states. The quantum nature of the
atoms comprising the medium will, of course, determine
whether these field operators obey Bose-Einstein or FermiDirac commutation relations. The atoms interact with two
light fields: a strong classical control laser induces a transition 兩2典 → 兩3典 and a weaker quantum probe field, which
drives a transition 兩1典 → 兩3典, see Fig. 1.
A. Equation for the probe field
The electric field of the probe beam is
FIG. 1. 共Color online兲 Slow light propagating in a medium of
⌳-type atoms.
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©2007 The American Physical Society
PHYSICAL REVIEW A 76, 053822 共2007兲
RUSECKAS et al.
E共r,t兲 = ê
冑
ប␻
E共r,t兲e−i␻t + H.c.,
2␧0
⳵ 2E 1 ⳵ 2P
=
,
⳵ t2 ␧0 ⳵ t2
共2兲
where
P = ê␮⌿†1⌿3 + H.c.
共3兲
is the polarization field of atoms, ␮ being the dipole moment
for the atomic transition 兩1典 → 兩3典. Let us introduce the
slowly varying matter field operators
⌽ 1 = ⌿ 1e i␻1t ,
共4兲
⌽2 = ⌿2ei共␻1+␻−␻c兲t ,
共5兲
⌽3 = ⌿3ei共␻1+␻兲t ,
共6兲
where ប␻1 is the energy of the atomic ground state 1 and ␻c
is the frequency of the control field. The probe field is quasimonochromatic, so that its amplitude E ⬅ E共r , t兲 varies very
slowly in time during an optical cycle. In this case the following equation holds for the slowly varying amplitude of
the probe field:
冉
冊
⳵
␻
c2
− i ⵜ2 − i E = ig⌽†1⌽3 ,
2
⳵ t 2␻
共8兲
characterizes the strength of coupling of the probe field with
the atoms. Note that, unlike in the usual treatment of slow
light 关14–17,24–27兴, we have retained the second-order derivative ⵜ2 in the equation of motion 共7兲. This allows us to
account for the fast changes of E in a direction perpendicular
to the wave vector k, i.e., in the xy plane. Therefore, our
analysis can be applied to the twisted beams of light E共r , t兲
⬃ exp共il␸兲 carrying an OAM of បl per photon.
B. Equations for the matter field-operators
In the following we shall adapt a semiclassical picture in
which both electromagnetic and matter field operators are
replaced by c numbers. The equations for the matter fields
are then
ⴱ
K̂ = iប
K̂⌽1 = V1共r兲⌽1 − បgE ⌽3 ,
共9兲
K̂⌽3 = ប␻31⌽3 + V3共r兲⌽3 − ប⍀c⌽2 − បgE⌽1 ,
共10兲
⳵
ប2 2
+
ⵜ .
⳵ t 2m
共12兲
Here m is the atomic mass, V j共r兲 is the trapping potential for
an atom in the electronic state j 共j = 1 , 2 , 3兲, ⍀c is the Rabifrequency of the control laser driving the transition 兩2典
→ 兩3典; ␻21 = ␻2 − ␻1 + ␻c − ␻ and ␻31 = ␻3 − ␻1 − ␻ are, respectively, the frequencies of electronic detuning from the twoand one-photon resonances. In the limit of infinite mass we
can ignore the spatial derivatives in these equations which
then reduce to the familiar equations for the probability amplitudes describing the coherent excitation of three-level atoms in the ⌳ configuration 关8兴. For us, however, retention of
this center of mass motion is essential in order to describe
light dragging effects. Note also that the equation of motion
共9兲 for ⌽1 does not explicitly accommodate collisions between the ground-state atoms. In the case of a BEC where
the state 1 is mostly populated, the collisional effects can be
included replacing V1共r兲 by V1共r兲 + g11兩⌽1兩2 in Eq. 共9兲 to
yield a mean-field equation for the condensate wave function
⌽1, where g11 = 4␲ប2a11 / m and a11 is the scattering length
between the condensate atoms in state 1. In Eqs. 共9兲–共11兲 the
coupling with the probe and control fields has been written
using the rotating wave approximation. Therefore the last
term in Eq. 共9兲 has a positive frequency part of the probe
field 共i.e., Eⴱ兲, whereas the last term in Eq. 共10兲 has a negative frequency part 共E兲. For the same reason, Eq. 共10兲 contains a factor ⍀c, whereas Eq. 共11兲 contains a factor ⍀ⴱc .
Equation 共11兲 relates ⌽3 to ⌽2 as
⌽3 =
共7兲
where the parameter
g = ␮冑␻/2␧0ប
共11兲
with
where ␻ = ck is the central frequency for probe photons, k
= ẑk is the wave vector, and ê ⬜ ẑ is the unit polarization
vector. This field may be considered as either a classical
amplitude or a quantum operator. We have chosen the dimensions of the electric field amplitude E so that its squared
modulus represents the number density of probe photons.
The probe field E共r , t兲 obeys the following wave equation:
c 2ⵜ 2E −
K̂⌽2 = ប␻21⌽2 + V2共r兲⌽2 − ប⍀ⴱc ⌽3 ,
共1兲
1
ប⍀ⴱc
关− K̂ + V2共r兲 + ប␻21兴⌽2 .
共13兲
Initially the atoms are in the ground level 1 and the Rabi
frequency of the probe field is considered to be much smaller
than ⍀c. Consequently one can neglect the last term in Eq.
共9兲 that causes depletion of the ground level 1, giving K̂⌽1
= V1共r兲⌽1. In the case of a BEC the wave-function ⌽1
= 冑n exp共iS1兲 represents an incident variable determining the
atomic density n and velocity field ប ⵱ S1 / m.
III. PROPAGATION OF PROBE BEAM
A. Adiabatic approximation
Suppose the control and probe beams are close to the
two-photon resonance. Application of the two beams cause
electromagnetically induced transparency 共EIT兲 关1–9兴 in
which the transitions 兩2典 → 兩3典 and 兩1典 → 兩3典 interfere destructively preventing population of the excited state 3. The adiabatic approximation is obtained neglecting the excited state
population in Eq. 共10兲 to yield
⌽2 = − g
⌽1
E.
⍀c
共14兲
Equations 共7兲, 共13兲, and 共14兲 provide a closed set of the
equations for the electric field amplitude E:
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PHYSICAL REVIEW A 76, 053822 共2007兲
POLARIZATION ROTATION OF SLOW LIGHT WITH ...
冉
冊
⌽ 1E
g2⌽ⴱ1
⳵
␻
c2
.
− i ⵜ2 − i E = − i
ⴱ 关− K̂ + V2共r兲 + ប␻21兴
2
⳵ t 2␻
⍀c
ប⍀c
共15兲
This equation suffices to describe a wide variety of phenomena. In particular it can be used to model light storage by
introducing time dependence in 兩⍀c兩 关12兴 and light dragging
due to spatial variation of ⍀c and/or ⌽1. Here we shall be
especially interested in image rotation associated with spatial
variations of ⌽1, corresponding to a moving medium and
with light beams carrying orbital angular momentum so that
E ⬀ eil␾.
C. Rotating medium
E共r,t兲 = Ẽ共r,t兲eikz ,
共16兲
Consider a probe beam carrying an orbital angular moment 共OAM兲 បl per photon along the propagation axis
Ẽ共r , t兲 ⬃ eil␸, where ␸ is the azimuthal angle. Suppose the
medium is rotating as a rigid body with a frequency ␻rot in
the xy plane. For instance, an atomic vortex lattice exhibits
such rigid body rotation on a coarse grained scale 关29兴. In
such a situation the linear velocity is va = ␻rot␳e␸, where e␸ is
the unit vector along the azimuthal direction and ␳
= 冑x2 + y 2 is the cylindrical radius. Under these conditions the
dragging term featured in the equation of motion 共18兲 takes
the form
⍀c共r,t兲 = ⍀共r,t兲eikcz ,
共17兲
va · ⵱Ẽ共r,t兲 = il␻rotẼ共r,t兲.
B. Co-propagating control and probe beams
Suppose the probe and control beams co-propagate:
where kc = ␻c / c is the wave number of the control beam. For
paraxial beams the amplitudes Ẽ共r , t兲 and ⍀共r , t兲 depend
only weakly on the propagation direction z. Furthermore the
amplitude of the control field ⍀共r , t兲 is considered to change
weakly in the transverse direction as well. Neglecting the
spatial dependence of ⍀共r , t兲, Eq. 共15兲 for the probe field
takes the form
冋 冉 冊
册
1 2
⳵
⳵
1 1
Ẽ + vg
+
− 共va · ⵱ + i␦兲 − i ⵜ⬜
Ẽ = 0,
2k
⳵t
⳵z
vg c
2
where we have replaced ⵜ2 by its transverse part ⵜ⬜
= ⳵2 / ⳵x2 + ⳵2 / ⳵y 2 because of the paraxial approximation.
Here
vg =
c兩⍀c共r,t兲兩2
兩⍀c共r,t兲兩2 + g2n共r兲
iប 1
ប
iប 1
⵱n
⵱ ⌽1 = ⵱ S1 −
m ⌽1
m
2m n
共20兲
is a complex atomic velocity field, and
1
ប
␦ = ␻21 + 关V2共r兲 − V1共r兲兴
冕冉
z
␪l = 共l␻rot + ␦兲
共21兲
is the two-photon frequency mismatch. As the beams copropagate, the ordinary two-photon Doppler shift va · ẑ共k
− kc兲 and two photon recoil ប共k − kc兲2 / 2m are of little importance and hence have been omitted. Furthermore, the radiative velocity vg is much larger than the velocity of the atomic
recoil vrec ⬅ បk / m, so the latter vrec has been neglected in the
equation of motion 共18兲.
The term with the spatial derivative ⳵ / ⳵z in Eq. 共18兲 describes the radiative propagation along the z axis with the
group velocity vg. On the other hand, the term containing
冊
1
1
− dz⬘ ,
vg共z⬘兲 c
共23兲
where z0 is the entrance point into the medium. The first term
in 共l␻rot + ␦兲 represents the rotational frequency shift due to
the orbital angular momenta បl of the probe photons. If the
control and probe fields are circularly polarized, the second
term accommodates the rotational two-photon frequency
shift ±2␻rot, as well as the residual detuning ␦⬘
␦ = ± 2␻rot + ␦⬘ ,
共19兲
is the radiative group velocity,
共22兲
Suppose the atomic density 共and hence the group velocity兲 is
approximately constant in the transverse direction 关30兴, so
that the group velocity depends on the propagation coordinate only: vg = vg共z兲. Using Eqs. 共18兲 and 共22兲, the total phase
shift then acquired by the beam propagating along the z axis
is
z0
共18兲
va = −
va · ⵱ represents the dragging of light by a moving medium.
The amount of such a dragging is determined by the difference in the group velocity and the speed of light. For fast
light there is no dragging, but for slow light a substantial
dragging is possible.
共24兲
where the sign of ±2␻rot depends on the polarizations of the
control and probe field. The factor 2 appears because the
control and probe beams are considered to have opposite
circular polarizations leading a double frequency shift. In the
slow-light experiments involving a ⌳ scheme 关10,11兴 it is
quite common for the control and probe fields to have the
opposite circular polarizations which drive the atomic transitions with ⌬m = ± 1. The residual detuning ␦⬘ depends neither on the orbital angular momentum of a photon nor on its
polarization and is hence of little importance for the subsequent discussion.
D. Polarization rotation of slow light with OAM
A characteristic feature of an EIT medium is that only one
circularly polarized component of light 共say left polarized
light兲 is slowed down. The right circularly polarized light has
a forbidden resonant transition to the excited state and is
propagating with a group velocity close to the speed of light
in vacuum. If the medium rotates and the probe beam carries
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PHYSICAL REVIEW A 76, 053822 共2007兲
RUSECKAS et al.
an OAM, the left circularly polarized beam of light acquires
a large rotational frequency shift because of its small group
velocity. On the other hand, the right polarized light is a fast
light and hence experiences almost no rotational frequency
shift. If the incoming light beam is linearly polarized, this
leads to the rotation of the polarization plane by an angle
␣l = 共␪l + ␪fast兲 / 2, where ␪fast is the dynamical phase shift acquired by the fast 共right polarized兲 light, ␪l being the phase
shift acquired by the slow 共left polarized兲 light given by Eq.
共23兲. To extract a pure rotational phase shift, we suggest
measuring the difference in the angles of the polarization
plane rotation for two consecutive probe beams characterized
by opposite OAM
冕冉
z
⌬␣l = ␣l − ␣−l = l␻rot
z0
冊
1
1
− dz⬘ ,
共z
兲
c
vg ⬘
共25兲
where we have taken advantage of the fact that ␪fast and ␦ do
not depend on the winding number l.
The angle ⌬␣l is proportional to the rotation frequency of
the medium ␻rot and to the winding number l. Therefore the
effect is due to the orbital angular momentum of slow light.
Furthermore, the angle ⌬␣l is inversely proportional to the
group velocity of light. This suggests that for slow light the
angle ⌬␣l should be large enough to be detectable. Let us
make some numerical estimates for typical BEC and slow
light experimental parameters. In the experiment by Hau et
al. 关1兴 the radiative group velocity is vg = 17 m / s, whereas
the length of a cigar shaped BEC is ⌬z = z1 − z0 = 0.3 mm
= 3 ⫻ 10−4 m. Typically the transverse frequency of an elon-
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IV. CONCLUSIONS
In the present paper we have considered the propagation
of a slow light with an orbital angular momentum in a moving atomic medium. We have derived a general equation of
motion and applied it to propagation of a slow light with an
orbital angular momentum in a rotating medium. We have
shown that the orbital angular momentum of slow light
manifest itself in a rotation of the polarization plane of a
linear polarized light. To extract a pure rotational phase shift,
we suggest to measure the difference in the angles of the
polarization plane rotation for two consecutive beams of
probe light characterized by opposite OAM. The differential
angle ⌬␣l is proportional to the rotation frequency of the
medium ␻rot, as well as to the winding number l. The effect
can be used as a tool for measuring the rotation frequency
␻rot of the medium and/or the orbital angular momentum of
slow light.
ACKNOWLEDGMENTS
This work was supported by the Royal Society, the Engineering and Physical Sciences Research Council 共U.K.兲, the
Royal Society of Edinburgh, and by the Wolfson Foundation.
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关15兴
关16兴
关17兴
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