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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. 053822-1 ©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−it + H.c., 20 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 i1t , 共4兲 ⌽2 = ⌿2ei共1+−c兲t , 共5兲 ⌽3 = ⌿3ei共1+兲t , 共6兲 where ប1 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 = ប31⌽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 = 4ប2a11 / 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/20ប 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 = ប21⌽2 + V2共r兲⌽2 − ប⍀ⴱc ⌽3 , 共1兲 1 ប⍀ⴱc 关− K̂ + V2共r兲 + ប21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: 053822-2 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 + ប21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 = rote, 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 = ilrotẼ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lrot + ␦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lrot + ␦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 ±2rot, as well as the residual detuning ␦⬘ ␦ = ± 2rot + ␦⬘ , 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 ±2rot 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 053822-3 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 = lrot 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. 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