The optical manipulation of matter-wave vortices: An analogue of circular dichroism
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t The optical manipulation of matter-wave vortices: An analogue of circular dichroism
Pradip Kumar Mondal, Bimalendu Deb, and Sonjoy Majumder ∗ Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India. Department of Materials Science, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India.
The transfer of orbital angular momentum from an optical vortex to an atomic Bose-Einsteincondensate changes the vorticity of the condensate. The spatial mismatch between initial andfinal center-of-mass wavefunctions of the condensate influences significantly the two-photon opticaldipole transition between corresponding states. We show that the transition rate depends on thehandedness of the optical orbital angular momentum leading to optical manipulation of matter-wavevortices and circular dichroism-like effect. Based on this effect, we propose a method to detect thepresence and sign of matter-wave vortex of atomic superfluids. Only a portion of the condensate isused in the proposed detection method leaving the rest in its initial state.
I. INTRODUCTION
The interaction of chiral molecules with light is sen-sitive to the circular polarization or helicity of the pho-tons. One manifestation of this optical activity is circulardichroism (CD), i.e., light absorption in such materialis sensitive to the handedness of the circular polariza-tion. Such interaction is enantiometrically specific anddepends on the structure of the chiral matter [1]. Po-larization is associated with the spin angular momentum(SAM)of light. Therefore, CD effect is usually associ-ated with spin of light. The work of Allen et al. [2]has shown that in addition to SAM, Laguerre-Gaussian(LG) beams carry well defined orbital angular momen-tum (OAM) associated with its spatial mode. This hastriggered new research on interaction of matter with laserbeams having certain spatial profiles, like, LG or Besselprofile [3, 4]. From extensive theoretical [5–7] and exper-imental works [8, 9] it has been believed that the OAMof LG beam does not play a role in CD. On the con-trary, here we prove that the matter-wave vortex of aBose-Einstein condensate (BEC) interacting with an LGbeam can lead to CD-like effects, i.e., the light absorptionbecomes sensitive to the handedness of the field OAM.Numerous theoretical and experimental studies relatedto creation of matter-wave vortex states, persistent cur-rent in BEC and coherent control of the OAM of atomsusing interaction of ultra-cold atoms with optical vortex[10–25] have been reported over past two decades. Quan-tized vortex states play an essential role in macroscopicquantum phenomena like superfluidity and superconduc-tivity. There have been studies to detect sign of a vortexstate using different techniques, like, interference betweenvortex states [26, 27], exciting the quadrupole mode of aBEC using an auxiliary laser beam as stirrer [28], vortexcore precession [29, 30], method of twisted densities [31],and vortex gyroscope imaging (VGI) [32]. The primarydifficulty of in-situ observation of vortices in trapped con- densate is that the radius of a vortex core is on the orderof the condensate healing length, which is generally sev-eral times smaller than the wavelength of the laser usedfor imaging. Methods applying free expansion after ex-tinguishing the trap or interferometric steps disrupt thetime evolution irreversibly. In most of the experiments,reproducing precisely the initial condition of vortex gen-eration is very challenging. In some methods, like, rapidquench across the BEC transition during evaporation viaKibble-Zurek mechanism (KZM) [33, 34], the vortex gen-eration is even stochastic [35]. So, any detection methodusing a fraction of the condensate [30] or in-situ nonde-structive phase-contrast imaging [29] is always preferableto any destructive method.Here we show how the CD-like effect that arises ininteraction of a BEC with an LG beam can be useful todetect a single vortex of BEC and its handedness keepinga portion of initial BEC undisturbed. In this method,two-photon stimulated Raman transitions are used onlyin a portion of BEC that is spatially separated out fromthe rest of the condensate which remains trapped in itsinitial state. One can detect the presence and sign ofthe undisturbed vortex by doing measurements on theseparated part. In recent experiments of [11, 14, 15],two-photon transitions are used to transfer the OAM tothe atomic BEC employing an LG and a Gaussian beam.
II. THEORY
The mechanism of angular momentum transfer from anLG beam to ultracold atoms is studied in paraxial limitin our previous work [36]. Here we recall some of theessential features of this OAM transfer mechanism. Weconsider an LG beam without any off-axis node propagat-ing along z axis of the laboratory frame interacting with Na BEC whose de Broglie wavelength is large enoughto feel the intensity variation of LG beam but smallerthan the waist of the beam. Under this condition, thedipole interaction Hamiltonian can be written as ∗ [email protected] H I = s π | l | ! e (cid:18) w (cid:19) | l | r X σ =0 , ± ǫ σ Y σ ( ˆr ) R | l | c.m. ⊥ e il Φ c.m. e ikZ c.m. + H.c. (1)where l and w are the winding number and the waistof the LG beam. r is coordinate of the valence elec-tron with respect to center-of-mass (c.m.) of the atomand R is the coordinate of the c.m. in laboratoryframe of reference. Rabi frequency is given by Ω I = ~ h Υ f | H I | Υ i i , where Υ denotes an unperturbed atomicstate, Υ( R c.m. , r ) = Ψ c.m. ( R c.m. ) ψ e ( r ) with Ψ c.m. ( R c.m. )being the c.m. wavefunction that depends on the exter-nal trapping potential and ψ e ( r ) is the internal electronicwavefunction. Ψ c.m. ( R c.m. ) is calculated by solving theGross-Pitaevskii equation [36] and ψ e ( r ) may be taken asa correlated orbital obtained from many-body calculation[37]. Detail calculations are given in our previous work[36]. There we have proved that electric dipole interac-tion of LG beam with BEC allows the transfer of thetotal field OAM to the c.m. motion of BEC and thus,changes the vorticity of the matter-wave state. However,the SAM corresponding to polarization is transferred tothe internal electronic motion and controls the selectionrule.We consider sodium BEC initially prepared in state | κ ; F, m F i where κ is vorticity of c.m. state and F , m F correspond to hyperfine spin of ground-state Na atoms.Figure 1 (a) shows a stimulated Raman scheme usingtwo sets of counter propagating LG and G pulses. Anatom of mass M , initially at rest, absorbs an LG pho-ton and stimulatedly emits a G photon, acquiring 2 ~ k of linear momentum (LM) ( k = 2 π/λ with λ the pho-ton wavelength) in the direction of propagation the LGphoton. Here we reason out how the two-photon Rabifrequency will depend on the handedness of the OAM ofthe LG beam. Two-photon detuning ∆ is much largerthan the spontaneous emission linewidth of the excitedstate. After adiabatically eliminating the intermediateexcited state, the effective Hamiltonian of the system oftwo-coupled vortices can be written as H ± = H ± + H ′± where H ± = ~ κω i | κ ; η ih κ ; η | + ( ~ ω f + ǫ q ) | κ ± l ; η ′ ih κ ± l ; η ′ | (2)and H ′± = (cid:2) Ω ± e − iδ ± t | κ ± l ; η ′ ih κ ; η | + C . c . (cid:3) . (3)where η = { F, m F } and η ′ = { F ′ , m ′ F } are spin states ofun-scattered and Raman-scattered atoms, respectively.The vorticity of the scattered atoms is κ ± l . ω i ( ω f ) isfrequency corresponding to the total energy (c.m. + in-ternal) of initial (final) state. Ω ± is the two-photon Rabifrequency for LG ± l /G pulse and δ ± = ω LG − ω G is the de-tuning between the two beams. The states | κ ± l ; F ′ , m ′ F i receive the momentum transfer q = k LG − k G dueto two-photon stimulated light scattering. As a re-sult, the atoms in | κ ± l ; F ′ , m ′ F i gain kinetic energy ǫ q = ( ~ q ) / M where M is mass of an atom. The im-portant point here is that the two-photon Rabi frequencyΩ + = Ω G Ω +I / ∆ for LG + l /G pulse is different from Ω − for LG − l /G pulse. The radial portions of c.m. wavefunc-tions of BEC (Ψ c.m. ( R c.m. )) corresponding to vorticities κ + l and κ − l are different (see Figure 2 of [36]). Thismakes the Rabi frequencies Ω + and Ω − correspondingto the two transitions different. Thus, in principle, aBEC vortex state is expected to show CD-like behaviorin interaction with LG beams having OAM of oppositehandedness. However, if initially the BEC was in a non-vortex state ( κ = 0) then after interaction with LG + l /Gand LG − l /G pulses for an appropriate pulse durationthe final states of BEC will have vorticities + l and − l ,respectively. The radial portions of c.m. wavefunctionsof these two states are identical. This makes the c.m.matrix elements and Rabi frequencies corresponding tothese transitions identical. Hence, non-vortex state inBEC does not show CD-like behavior in interaction withLG beam.The Hamiltonian H ± given by Eqs. (2) and (3) de-scribes coherent dynamics of spin-vortex coupled states.Let the solutions of this Hamiltonian be | ψ ± ( t ) i = a ± ( t ) | κ ; F, m F i + b ± ( t ) | κ ± l ; F ′ , m ′ F i . (4)Under phase matching condition ǫ q = ~ δ ± and assuming( ω f − ω i ) << Ω ± we have | a ± ( t ) | = cos (Ω ± t ) (5)and | b ± ( t ) | = sin (Ω ± t ) . (6)This shows that for pulse duration t p given by Ω ± t p = π/
2, the vortex state | κ i will be coherently transferredinto the vortex state | κ ± l ; F ′ , m ′ F i . For Ω ± t p = π/ ψ + and ψ − are different onlydue to CD-like effect that arises when κ = 0 i.e. , if theBEC is initially in a vortex state. On the other hand, thetemporal evolution of the two states will be the same for κ = 0 due to the absence of CD-like effect. Hence, thiseffect will play an important role in quantum evolutionof spin-vortex coupled BEC in a spinor condensate. Thestate given by Eq. (4) is an entangled state of spin andmatter-wave vortices. III. PROPOSED APPLICATION
Now, we discuss how this CD-like effect can be usefulto detect vortex state in BEC and determine the handed-ness of vorticity using two LG beams having OAM of op-posite handedness. We consider sodium BEC is initiallyprepared in electronic state | S / , F = 1 , m F = − i [11]. Our proposed experimental scheme is shown infigure 1. LG l and G pulses propagate along + Z axisand LG − l and G pulses propagate along − Z . We applyLG l /G and LG − l /G pulses simultaneously. The two-photon Raman transition by LG l /G pulse is called type-1 transition and the other two-photon Raman transitionby LG − l /G pulse is called type-2 transition in this pa-per. The atoms taking part in the Raman transitions fora pulse duration of the order Ω − ± will be in final spin state | S / , F = 1 , m F = +1 i which is high field seeking state.Therefore, they will no longer be trapped. In addition,these atoms will gain 2 ~ k LM from the Raman processand propagate ballistically. After a few millisecond timeof flight (TOF), these atoms will be spatially separatedfrom the atoms which are still at rest and trapped. Let uscall the atom cloud which has gone through type-1 tran-sition as cloud-1 and the other one as cloud-2. Then onecan use focused pump beam spatially localized along z and selectively image atom clouds in different LM statesand different spatial positions. In accordance with theCD-like effect described in the previous paragraph, thenumber of atoms in cloud-1 will be different from that ofcloud-2 depending on the handedness of the initial BECvortex state. IV. NUMERICAL RESULTS
We now proceed to numerically evaluate how the num-ber of atoms in cloud-1 will differ from that of cloud-2depending on the sign of vorticity of the initial BEC.As mentioned in the previous paragraph, we considersodium BEC containing 10 number of atoms initiallyprepared in electronic state | S / , F = 1 , m F = − i [11]. The atoms are trapped in an anisotropic trap. Theasymmetry parameter ω Z /ω ⊥ = 2 with axial frequency ω Z / π = 40 Hz. The corresponding characteristic lengthis a ⊥ = 4 . µ m. s - wave scattering length is a = 2 . δν = 4 E r /h , where E r = ( ~ k ) / M is therecoil energy [11] (see figure 2). In addition to LM, theatoms pick up the OAM difference between the two pho-tons and this causes change in rotational motion of theatoms. Corresponding rotational kinetic energy differ-ence between final and initial vortex states is small com-pared to δν [11] and gives rise to rotational Doppler shift[39]. The laser beams are detuned from the D line( λ = 589 . − . ≈
150 linewidths,enough to prevent any significant spontaneous photonscattering). The frequency of G , are further decreased F=1F’=2m F = -2 -1 0 +1 +2LG l G Δ (a) LG G zLG l G LG G LG l G Trapped BEC vortex (b)
Cloud-2
Cloud-1 zPROBE 1 PROBE 2 P U M P P U M P (c) FIG. 1. Proposed experimental scheme illustrating the pro-cess to detect handedness of matter-wave vortex of an atomicBose-Einstein condensate using the CD-like effect. (a) Energylevel scheme of the two-photon transitions. The atomic statesshown correspond to the Na hyperfine structure. Atoms inthe BEC are initially in | S / , F = 1 , m F = − i electronicstate. Afte 2-photon transitions the final electronic statesbecome | S / , F = 1 , m F = 1 i . ∆ denotes two-photon de-tuning (b) LG l /G and LG − l /G pulses are simultaneouslyapplied to the BEC. (c) The atoms that have undergone thetype-1 Raman transitions have come out as cloud-1 and thosewhich have undergone the type-2 Raman transitions have sep-arated as cloud-2 after a few millisecond TOF. Two pumpbeams along with two probe beams enable independent imag-ing of the two clouds (for details, see the text). E n e r g y Momentum ( (cid:1) k)4E r (cid:0)(cid:2) FIG. 2. Diagram illustrating energy and LM conservation ofthe 2-photon Raman process for LG/G pulse.TABLE I. This illustrates cicular dichroism-like effect in BECvortex states undergoing type-1 and type-2 two-photon Ra-man transitions. κ i and κ f are vorticity of initial and finalstates of the condensate, respectively. l denotes the windingnumber of the LG beam associated with the transition. Ω + and Ω − are dipole Rabi frequencies (s − ) corresponding totype-1 and type-2 transitions, respectively. κ i l κ f Ω + l κ f Ω − × -1 0 6.53 × +2 3 2.42 × -2 -1 2.03 × -1 +1 0 6.53 × -1 -2 7.49 × +2 +1 2.03 × -2 -3 2.42 × from LG ± l beams by δν (may be using acousto-opticmodulators). The intensity of the laser beams are setto be I = 10 Wcm − . The waists of the LG beams areset to be w = 10 − m. The radius of the G , beamsis generally a order of magnitude larger than that of theBEC and hence, during interaction with the condensatewe have considered the G , beams as plane waves. Aftertype-1 (type-2) two-photon transition, the vorticity of fi-nal c.m. state is going to be changed by + l ( − l ). Thetwo-photon Rabi frequencies corresponding to type-1 andtype-2 transitions are defined as Ω + and Ω − , respectively,shown in Table I. Our calculation shows that if κ i > + > Ω − and if κ i < + < Ω − .Andersen et. al. [11] used similar two-photon Ramantransition technique to generate and image BEC vortex.They worked with BEC of 10 Na atoms and achievedmaximum efficiency of the two-photon transition as highas 53% for a 130 µ s LG/G pulse. From their experimen-tal observation we can estimate roughly how the numberof atoms in cloud-1 will vary from the number of atomsin cloud-2. Let us start with 10 number of atoms with initial vorticity κ i = +1 and we consider that Ω + andΩ − are experimentally calibrated so that they are equalfor a non-vortex initial state. We assume that the exper-imental parameters are set such that 30% of atoms takepart in type-2 transition. According to Table I, if | l | = 1,then almost 40% of atoms will take part in type-1 transi-tion and that means cloud-1 will have almost 10 numberof atoms more than cloud-2. If we use LG beams with | l | = 2 then this difference in number of atoms presentin cloud-1 and cloud-2 is almost 1 . × (13% of initialnumber of atoms). This difference of 10% (for | l | = 1)number of atoms is easily detectable by absorption imag-ing of cloud-1 and cloud-2 [40, 41] and thereby one candetermine the handedness of rotation of the initial BEC.It will be same as the rest of portion of the BEC whichhas not taken part in the Raman processes. If the num-ber of atoms in cloud-1 and cloud-2 are almost same thenthe BEC had no vortex state. V. CONCLUSION
In conclusion, we have demonstrated that CD-like ef-fects can arise in interaction of a matter-wave vortex withLG beams of opposite OAM. This effect will have wideapplications including detection of handedness of thematter-wave vortex as theoretically demonstrated in thispaper. While creating a vortex state from non-rotatingBEC using two-photon Raman transitions by applyingsquare pulses of a particular duration, the transfer ef-ficiency is limited by the spatial mismatch between therotating state and the initial BEC [11]. In this work, wehave shown that it is possible to take advantage of thislimitation that lies at the heart of the predicted CD-likeeffect. As an application of this effect we have proposed amethod to detect BEC vortex states and its handednessusing a portion of the condensate. The main advantagesof this method are as follows. Firstly, only single imag-ing of cloud-1 and cloud-2 is needed. Secondly, we donot need to know the exact number of atoms present inthe clouds. All we need to know is which of the cloudscontains more number of atoms. Thirdly, the rest of thecondensate which does not take part in the two-photontransitions will remain trapped in its initial state. Whilethis method is suitable for steadily rotating condensatecontaining single vortex, the situation is different for mul-tiply quantized vortices in a BEC. It needs further studiesto extend the method for many-vortex configurations.
ACKNOWLEDGMENT
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