Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction
A. Turpin, J. Polo, Yu. V. Loiko, J. Küber, F. Schmaltz, T. K. Kalkandjiev, V. Ahufinger, G. Birkl, J. Mompart
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Blue-detuned optical ring trap for Bose-Einstein condensates based onconical refraction
A. Turpin, ∗ J. Polo, Yu. V. Loiko, J. K¨uber, F. Schmaltz, T. K. Kalkandjiev,
1, 4
V. Ahufinger, G. Birkl, and J. Mompart Departament de F´ısica, Universitat Aut`onoma de Barcelona, Bellaterra, E-080193, Spain Aston Institute of Photonic Technologies, School of Engineering& Applied Science Aston University, Birmingham, B4 7ET, UK Institut f¨ur Angewandte Physik, Technische Universit¨at Darmstadt,Schlossgartenstrasse 7, D-64289 Darmstadt, Germany Conerefringent Optics SL, Avda Cubelles 28, Vilanova i la Geltr´u, E-08800, Spain (Dated: September 13, 2018)We present a novel approach for the optical manipulation of neutral atoms in annular light struc-tures produced by the phenomenon of conical refraction occurring in biaxial optical crystals. Fora beam focused to a plane behind the crystal, the focal plane exhibits two concentric bright ringsenclosing a ring of null intensity called the Poggendorff ring. We demonstrate both theoretically andexperimentally that the Poggendorff dark ring of conical refraction is confined in three dimensionsby regions of higher intensity. We derive the positions of the confining intensity maxima and minimaand discuss the application of the Poggendorff ring for trapping ultra-cold atoms using the repulsivedipole force of blue-detuned light. We give analytical expressions for the trapping frequencies andpotential depths along both the radial and the axial directions. Finally, we present realistic numeri-cal simulations of the dynamics of a Rb Bose-Einstein condensate trapped inside the Poggendorffring which are in good agreement with corresponding experimental results.
PACS numbers: 37.10.Gh,42.25.Lc,42.62.Be
I. INTRODUCTION
Optical ring potentials (ORPs) with axial symmetryare considered as basic building blocks and the simplestnontrivial closed-loop circuits in atomtronics [1–5] andatom interferometry [6]. Atoms can be trapped by meansof the optical dipolar force in high or low intensity regionswith red-detuned [7, 8] or blue-detuned [9] light, in whatfollows called bright and dark potentials, respectively.On the one hand, bright ORPs have been proposedand demonstrated with high-azimuthal-order Laguerre–Gaussian (LG) beams [10] and also with annular mi-crolenses [11]. Azimuthal lattices within ORPs have beendemonstrated with time orbiting of light beams [12, 13]and by interference of LG beams of different azimuthalorders [14]. A one-dimensional stack of ORPs in a linehas been proposed in an optical cavity [15] and demon-strated with axicon beams [16]. Experimental storageand propagation of ultra-cold atoms and Bose-Einsteincondensates (BECs) in bright ORPs have been reportedrecently [17, 18]. Dark ORPs on the other hand are op-tical fields with an annular region of minimum intensity[19], such as closed-loop optical singularities [20, 21], forwhich the region of minimum intensity is exactly zero.For ultra-cold atoms, dark ORPs have the advantageof substantially reducing atom heating and decoherencerates [9] because of the low rate of spontaneous photonscattering as well as producing intrinsically flat potential ∗ Corresponding author: [email protected] minima. Blue-detuned ORPs have been experimentallyreported by means of LG beams generated with spatiallight modulators (SLMs) [22] and by amplitude masks[23–25]. These two techniques might experience the fol-lowing limitations: (i) a significant fraction of the inputpower is lost and, therefore, it does not contribute to cre-ate the optical trap, (ii) the smoothness and, therefore,the quality of the trapping potential is limited by thesize and number of pixels for the SLMs and the resolu-tion of the printing system for the amplitude masks, and(iii) an accurate control on the position and alignmentof the optical elements being used is required. As a con-sequence, these two techniques yield typically not nullintensity minima. Producing ORPs with zero-intensityannular regions both along the radial and axial direc-tions is a challenging task. In this case, the dark poten-tial forms toroidal dark focus, i.e., a region of minimumintensity confined by higher intensities (light walls) bothin the axial and radial directions. A toroidal dark focushas only been demonstrated using a superposition of twoLG beams [19].In this article, we present a new method to gener-ate a dark ORP by means of the phenomenon of coni-cal refraction (CR) [26–33], occurring in biaxial crystals.CR leads to a set of two concentric bright rings enclos-ing a dark ring of null intensity, known as Poggendorffdark ring (PDR). We theoretically investigate the three-dimensional (3D) field distribution around the CR PDRand show both theoretically and experimentally that it isa toroidal dark focus. We also discuss the use of the PDRas a blue-detuned ORP for ultra-cold atoms and demon-strate this with a Rb BEC. This technique has the ad-vantage of the easy generation of the toroidal dark trap,which only needs a focused Gaussian beam and a biaxialcrystal. In addition, CR provides the full conversion ofthe input power into the toroidal dark trap, in contrast tothe previously reported methods which introduce lossesdue to diffraction in the generation of LG beams. Thesefeatures make the CR toroidal dark-focus beam very at-tractive for particle [34] and atom [10, 11, 19] trapping,in particular with blue-detuned light beams [9].The article is organized as follows: In Section II wegive an introduction to the CR phenomenon, present-ing its main fundamental characteristics: Section II Apresents the exact paraxial solution of the light patternafter propagation along one of the optic axes of a biaxialcrystal, while its asymptotic approximation is presentedin Section II B. In Section III, we investigate the use ofthe toroidal dark trap provided by CR for the trappingof ultra-cold atoms with blue-detuned light. In SectionIII A we apply the harmonic approximation around thePDR and present expressions for the trapping frequenciesand height of the potential barriers as a function of thephysical parameters of the trapping system. Then, bothnumerical simulations and experimental data for a RbBEC trapped in the PDR are shown in Section III B. Fi-nally, the main conclusions are summarized in SectionIV.
II. CONICAL REFRACTION
Conical refraction [28–33] is observed when a focusedlight beam passing along an optic axis of a biaxial crystal(BC) is transformed into a light ring at the focal plane.Each pair of diagonally opposite points of the CR ringare orthogonally linearly polarized, as shown by doublearrows in Fig. 1. Therefore, a complete ring with uni-form azimuthal intensity distribution is observed only forrandomly (RP) or circularly (CP) polarized input beams(Fig. 1(a)), while for linearly polarized (LP) input beamsthe intensity pattern is azimuthally crescent with a zero-intensity point, where the polarization is orthogonal tothe one of the input beam. This polarization distribution,which constitutes an essential signature of the CR phe-nomenon, significantly differs from the usually known ra-dial and azimuthal polarization modes and only dependson the orientation of the plane of the crystal optic axes[30].The CR geometric optical approximation of the ringradius, R , is the product of the crystal length, l , andthe CR semi-angle α , i.e., R = lα [33]. The CR semi-angle α depends on the principal refractive indices of thecrystal as α = p ( n − n )( n − n ) /n , where we haveassumed n < n < n . Moreover, under conditionsof ρ ≡ R w ≫ w is the waistof the focused input beam, defined as the radius of the e − relative intensity, i.e. I ( r = w ) = e − I ( r = 0). Fi- (a) (b) FIG. 1. Intensity and polarization distribution (depicted withyellow double arrows) of conical refraction with input beamsof circular (a) and linear vertical (b) polarization. The darkring between the two bright ones is known as the Poggendorffdark ring (PDR). nally, as far as CR beam evolution is concerned, the focalplane is a symmetry plane along the beam propagationdirection [33]. The CR rings are observed at the focalplane, and more involved structures including secondaryrings are found as one moves along the beam propaga-tion direction. At points given by z Raman = ± p / ρ z R from the focal plane a bright spot known as the Ramanspot [34] appears on the beam axis, where z R denotesthe Rayleigh range of the focused input beam. In thissection we describe the properties of the optical field atand close to the dark region of the Poggendorff ring. A. Paraxial solution of the intensity distributionfor CR
The paraxial solution describing CR was derived byBelsky and Khapalyuk [29] and later reformulated byBerry [28]. For a uniformly polarized and cylindricallysymmetric input beam it gives an electric field vector: ~E ( ~ρ, Z ) = (cid:18) B C + C SS B C − C (cid:19) ~e , (1)where C = B S cos ( ϕ + ϕ ) and S = B S sin ( ϕ + ϕ ). ϕ is the azimuthal component in cylindrical coordinatesand ϕ is the orientation of the plane of the optic axesof the crystal. E and ~e = ( e x , e y ) are the amplitudeand unit vector of the electric field ~E = E ( ρ, z ) ~e of a focused input beam with waist w and Rayleighrange z R . Z ≡ z/z R and ~ρ ≡ (cos ϕ, sin ϕ ) r/w with ρ = | ~ρ | ≡ r/w define, respectively, normalized axial andradial components in cylindrical coordinates with originat the ring center ( ρ = 0) at the focal plane ( Z = 0). B C and B S are the main integrals of the Belsky–Khapalyuk–Berry (BKB) solution, which describes the general prop-erties of the CR beam. These integrals read: B C ( ρ, Z ) = 12 π Z ∞ ηa ( η ) e − i Z η cos ( ηρ ) J ( ηρ ) dη, (2) B S ( ρ, Z ) = 12 π Z ∞ ηa ( η ) e − i Z η sin ( ηρ ) J ( ηρ ) dη, (3)where η ≡ κw , κ being the spatial wave-vector and J q is the q th -order Bessel function of the first type and a ( η ) = 2 π R ∞ rE ( r ) J ( ηr ) dr is the radial part of the2D transverse Fourier transform of the input beam. ForCP and LP inputs the intensity distribution behind thecrystal becomes, respectively I CP = | B C | + | B S | , (4) I LP = I CP + 2 Re [ B C B ∗ S ] cos (2Φ − ( ϕ + ϕ )) , (5)where Φ is the polarization azimuth of the linearly po-larized input light with ~e = (cos Φ , sin Φ). B. Asymptotic solution close to the Poggendorffdark ring
The asymptotic solution for the Poggendorff dark ring,i.e., for ρ = R w ≫
1, is obtained by using the asymp-totic expansion of Bessel functions cos ( ηρ ) J ( ηρ ) ≈ sin ( ηρ ) J ( ηρ ) ≈ cos ( ηξ − π/ / √ πηρ . Here we havecentered the normalized radial component in cylindricalcoordinates at ρ by using ξ ≡ ρ − ρ = r/w − R /w . Inthis case B C ≈ B S and the electric field can be writtenas [29, 30]: ~E ( ξ, Z, ϕ )= f ( ξ, Z ) E ( ~e CR · ~e ) ~e CR , (6)where f ( ξ, Z ) = r π ρ Z ∞ dη √ ηa ( η ) e − i Z η cos (cid:16) ηξ − π (cid:17) , (7)and ~e CR = (cid:18) cos ϕ + ϕ sin ϕ + ϕ (cid:19) . (8)Therefore, the asymptotic intensity distributions I aCP and I aLP for CP and LP input beams are, respectively, I aCP ( ξ, Z ) = | f ( ξ, Z ) | , (9) I a LP ( ξ, Z, ϕ ) = I aCP cos (cid:18) Φ − ϕ + ϕ (cid:19) . (10)For LP input beams (see Eq. (10)), the output inten-sity distribution lacks azimuthal symmetry. In this casethe CR ring has a maximum and a zero intensity at az-imuthal angles ϕ max = 2Φ − ϕ and ϕ min = ϕ max + π ,respectively. These points possess, correspondingly, thesame and the orthogonal polarization relative to that ofthe input beam, respectively (Fig. 1(b)).In the following we will analyze the case of a CPinput beam, for which the CR output intensity is az-imuthally symmetric and its spatial distribution is de-scribed by Eq. (9). For a Gaussian input beam withnormalized transverse profile of the electric field ampli-tude E ( ρ ) = p P/πw exp( − ρ ), its Fourier transformis a ( η ) = q πPw exp (cid:0) − η / (cid:1) . P is the power of the inputbeam. For this case, Eq. (7) can be analytically evaluated through the Kummer confluent hyper-geometric function F ( a ; b ; z ) [32]: f ( ξ, Z ) = √ P ( w Z ) / p π w ρ (cid:20) Γ (cid:18) (cid:19) F (cid:18)
34 ; 12 ; − ξ w Z (cid:19) + 2 ξ √ w Z Γ (cid:18) (cid:19) F (cid:18)
54 ; 32 ; − ξ w Z (cid:19)(cid:21) , (11)where w Z = 1 + iZ . I ( ξ , Z=0) -4 -2 0 2 4 = (r-R )/w ξ = ρ - ρ ( a ) ( b ) -4 -2 0 2 41.432.864.295.717.00 I ( ξ=ξ , Z) x 10 Z = z/z R FIG. 2. Normalized CR intensity for a CP Gaussian inputbeam as given by Eq. (11) along the radial direction (a) atthe focal plane and (b) along the axial direction at the radialposition of the PDR ( ξ = ξ ). Blue solid circles representexperimental data with an experimental uncertainty of 5 %along both axis. The solid line in Fig. 2(a) shows the square modulus ofEq. (11) at the focal plane ( Z = 0). f ( ξ ,
0) = 0, givesthe radial position of the Poggendorff dark ring at the fo-cal plane, being ξ = − . R , by approximately half the waist of theinput beam. Note that ξ = ρ − ρ , with ρ ≡ R /w . Inthe radial direction the PDR is confined by two maximaat ξ + = 0 .
390 and ξ − = − . ξ = ξ as shown in Fig. 2(b). At this radial point thepositions of the intensity maxima along Z obtained fromEq. (9) and (11) are Z ± = ± . ξ ± ) and axial ( Z ± )directions. As a visualization of the toroidal dark trap TABLE I. Positions of the Poggendorff dark ring and of themaxima in the radial ( ξ ± ) and axial ( Z ± ) directions.Point name ξ Z Dark Ring: ξ -0.541 0Bright Rings: ξ + ξ − -1.235 0Maxima along Z: Z ± -0.541 ± provided by the PDR of CR, Fig. 3(a) shows the three-dimensional distribution of light intensity of the asymp-totic approximation of the BKB solution near the focalplane. Fig. 3(b) is a contour plot near the PDR, confirm-ing that it is a region of low intensity surrounded in alldirections by regions of higher intensity. Note that thePDR is an exact null intensity region only for input Gaus-sian beams under the asymptotic approximation, i.e. for ρ ≫
1, while non-zero intensity radial minimum pointsare found out of the paraxial approximation, as reportedin Refs. [38, 39]. For input beams with different trans-verse profile the CR pattern may change [40, 41].
FIG. 3. (a) Normalized light intensity in three dimensionsnear the PDR. (b) 2D contour density plot near the PDR ofthe normalized light intensity calculated from Eqs. (9) and(11) and for ρ = R /w = 20. Color map: black = nullintensity, white = high intensity. We have experimentally checked that near the PDRthe light intensity increases in all directions, see blue solidcircles in Fig. 2(a) and Fig. 2(b). These experimentson the CR PDR were carried out using a CP focusedinput Gaussian beam ( w = 40 µ m, z R = 7 . λ =640 nm and a KGd(WO ) biaxial crystal (cross-section6 × , l = 28 mm, α = 17 mrad) cut perpendicular toone of the optic axes (entrance surface parallelism betterthan 10 arc seconds) yielding a CR ring of R = 476 µ m( ρ ≈ III. APPLICATION TO ATOM TRAPPINGWITH BLUE-DETUNED LIGHTA. Harmonic potential approximation
In the previous sections we have described the intensitydistribution near the PDR in the radial and axial direc-tions, showing that this region is a dark ORP. This makesthe PDR a good candidate for atom trapping applicationswith blue-detuned light. For a given light intensity I ( ~r ),the trapping dipole potential reads U ( ~r ) = ˜ U I ( ~r ), wherefor alkali atoms for sufficiently large detuning and linear polarization˜ U = − πc (cid:20) Γ D ω D (cid:18) ω D − ω L (cid:19) + Γ D ω D (cid:18) ω D − ω L (cid:19)(cid:21) , (12)as given e.g. in [42]. Note that in ˜ U we have appliedthe rotating-wave approximation. Here, c is the speedof light in vacuum, Γ D i and ω D i ( i = 1 ,
2) are, respec-tively, the natural line width and frequency of the D i line of the atomic species, and ω L is the frequency ofthe input beam. In our case, I ( ~r ) is given by Eq. (9)and Eq. (11) and ˜ U > ω r ) and axial ( ω z ) trapping frequencies of the PDR( ξ = ξ , Z = 0) ω r,z = s A r,z ˜ U Pπ mw ρ , (13)with m being the atomic mass and the numerical con-stants A r ( Z = 0) = 4 .
63 and A z = 0 .
34. Eq. (13) isobtained by expanding Eq. (11) in Taylor series, intro-ducing it into Eq. (9) and considering the ξ coefficient.Note that from Eqs. (9), (11)–(13) for a given biaxialcrystal, i.e. for a fixed R and w , the trapping fre-quencies and the maxima of the potential barriers can betuned by modifying the power P and the frequency ω L of the input beam. We have obtained that at the focalplane the maxima of the potential barriers are describedby U ( ξ ± ,
0) = C ± ˜ U P π w ρ , (14)where C + = 2 .
54 (outer bright ring) and C − = 0 . Z = 0) and theplane Z = 4. In these cases, Eq. (13) can be utilized tocalculate the trapping frequency of the potential at anyaxial position Z by just replacing A r ( Z = 0) by A r ( Z ) = − .
051 + 8 . .
873 + 2 . Z . (15)Figure 4(b) presents the dependence of A r ( Z ) on Z . Notethat outside the focal plane an offset to the potential isoccurring, since the minimum intensity point is no longerof null intensity as plotted for Z = 4 as solid line inFig. 4(a). We have found that this non-zero minimumintensity point can be taken into account by means ofthe optical potential along the axial direction, U ( ξ , Z ) = ˜ U P π w ρ Z . (16)The confining maxima along Z are not well describedby the harmonic approximation and must be evaluatedusing Eqs. (9) and (11). They read U ( ξ , Z ± ) = 0 .
17 ˜ U P π w ρ . (17) I ( ξ ) ( a . u . ) ξ -8 -6 0 6 80.000.140.290.430.570.710.861.00 ( a ) ( b) A ( Z ) r Z-5 -4 -3 -2 -1 0 1 2 3 4 50.03.02.52.01.00.55.04.54.03.51.5 I( ξ , Z=4)I( ξ , Z=0) FIG. 4. (a) Profile of the trapping potential at Z = 0, i.e.at the focal plane (dashed curve), and at Z = 4 (solid curve)where the inner and the outer bright rings of CR have equalmaximum intensity. (b) Coefficient A r as a function of Z .The analytical expression for the A r ( Z ) is given by Eq. (15). B. Numerical simulations of a BEC of Rb atoms
To demonstrate the applicability of the PDR for ultra-cold gases, now we discuss the two-dimensional (2D) evo-lution of a BEC of Rb atoms confined in an annulargeometry within the focal plane by using the PDR of CRand a strong additional confinement along the axial di-rection so that ω axial ≫ ω r [44]. Such confinement canbe achieved by using an additional red-detuned sheet oflight (e.g. generated by focusing a Gaussian beam witha cylindrical lens) to compensate for the weak axial con-finement as well as, in case of a horizontal ring plane,the effect of gravity as shown in [44]. We use the 2DGross–Pitaevskii equation (GPE) in order to study thedynamics of the BEC along the ring potential: i ~ ∂∂t Ψ( ~r, t ) = (cid:18) − ~ m ~ ∇ + V ext ( ~r ) + g D | Ψ( ~r, t ) | (cid:19) Ψ( ~r, t ) , (18)where V ext ( ~r ) is the external potential, g D = 2 ~ a s N q π ~ ω z m , a s is the scattering length, ω z is the frequency of the confining potential in the axialdirection, m is the mass of the Rb atoms and N is thenumber of atoms.In our simulations, we consider trapping close tothe D and D lines of Rb. These lines pos-sess natural line widths of Γ D = 2 π × .
07 MHz andΓ D = 2 π × .
75 MHz and frequencies of ω D = 2 π × .
23 THz and ω D = 2 π × .
11 THz, respectively.Thus, to calculate the trapping frequencies and the max-ima of the potential barriers is straightforward by using Eqs. (12), (13), (14), and (17). Based on the experimen-tal parameters of [44], for a biaxial crystal yielding a CRring of R = 170 µ m, an input beam waist w = 18 µ m, alight frequency of ω L = 2 π × .
40 THz and a laser power P = 27 mW, at the focal plane, the maxima of the po-tential barriers and trapping frequencies are, respectively, U ( ξ − , Z = 0) /k B = 280 nK, U ( ξ + , Z = 0) /k B = 1314 nKand ω r = 2 π ×
265 Hz, where k B is the Boltzmann con-stant.Figure 5(a) shows the numerical simulation for a RbBEC, with scattering length a s = 5 .
45 nm , of N = 12000atoms trapped in a blue-detuned harmonic annular po-tential V r = mω r ( r − ( R − . w )) with radial fre-quency ω r = 2 π ×
265 Hz calculated using Eq. (13). Ournumerical simulations are based in the following loadingprocess: the BEC is created in a cross-dipole trap, seee.g. [45], and loaded into the red-detuned sheet of light.We consider that both the cross-dipole trap and the red-detuned sheet of light are orthogonal to the gravity field.The PDR potential, which also lies orthogonal to thegravity field, is placed tangent to one of the beams ofthe cross-dipole trap, see Fig. 5(c). The beam from thecross-dipole trap that is tangent to the PDR is switchedoff as the CR PDR potential is switched on, in an adi-abatic process. Finally, the remaining beam from thecross-dipole trap is switched off and the BEC expandsin the CR PDR potential. We plot the atomic densityof the BEC after 30 ms of expansion in the annular po-tential. In order to reduce the transverse excitations,the loading of the BEC into the CR ring potential hasbeen performed adiabatically (in our case during 20 ms)as reported in [44]. Fig. 5(b) shows the correspondingexperimental density distribution after 30 ms expansionof a Rb BEC trapped in the real CR PDR. The CRPDR was placed perpendicular to gravity and a sheetof light generated by focusing a Gaussian beam with acylindrical lens was used to hold atoms against grav-ity. The corresponding measured trapping frequenciesare ω exp r = 2 π × (300 ±
20) Hz and ω exp z = 2 π × (169 ±
2) Hz.The major discrepancy between experimental and nu-merical density plots is found in the radial width of theBEC. In the ideal case (Fig. 5(a)), the effects of broaden-ing due to finite optical resolution and photon scatteringof the detection light have not been considered to obtainthe image, which shows a BEC with a width of 3 µ m. Incontrast, the experimental image from Fig. 5(b), whichshows a BEC with a width of 25 µ m, was obtained byusing red-detuned light ( λ ill = 780 nm, P ill = 0 .
25 mW)to illuminate the BEC during a time of t ill = 200 µ s. Forthis illuminating light we have calculated a scatteringrate Γ sc = 3 . × s − that, together with the recoilvelocity of v rec = 5 .
89 mm / s, increases the width of theatomic cloud in the radial direction by 21 . µ m duringthe illumination time. Figure 5(c) shows the same nu-merical simulation as Fig. 5(a) where we have taken intoaccount now the increase of the width produced by thedetection process included. Now, numerical simulationand experimental result agree well. (a) (b) (c) Gravity
FIG. 5. (a) Plot of the atomic density from the numerical simulation of a trapped Rb BEC after 30 ms of expansion in the ring V r = mω r ( r − ( R − . w )) , with the frequency ω r = 2 π ×
265 Hz being calculated using the harmonic approximation.Parameter values used for the simulation: R = 170 µ m, w = 18 µ m, P = 27 mW, w z = 2 π ×
500 Hz, a s = 5 .
45 nm and N = 12000 atoms. (b) Experimental density distribution of a trapped Rb BEC in the CR ring potential using the sameexperimental parameters as for the numerical simulation, with the exception of the axial confinement, that was made using ared-detuned Gaussian beam focused with a cylindrical lens, providing a measured trapping frequency of w exp z = 2 π × (169 ±
2) Hz.The measured radial trapping frequency provided by the CR PDR was ω exp r = 2 π × (300 ±
20) Hz. (c) Numerical simulationunder the same conditions as in (a) but including the scattering induced by the position spreading during detection. Each Fig.is 600 µ m × µ m. Color map: dark blue (red) corresponds to null (high) intensity. White dashed lines in (c) indicate theposition of the cross-dipole trap with respect to the PDR, being both of them orthogonal to the gravity field. The waist radiusof each beam from the cross-dipole trap is 25 µ m. In order to further confirm the validity of the har-monic approximation, we also studied the ground state ofthe BEC trapped in the toroidal dark-focus (see Fig. 6).The physical system considered has the following pa-rameters: R = 170 µ m, w = 18 µ m, P = 27 mW, w r = 2 π ×
265 Hz, a s = 5 .
45 nm and N = 12000 atoms.The toroidal dark trap is placed orthogonal to gravityand, therefore, to provide confinement along the axial di-rection we have considered a sheet of light analogous tothe one discussed in [44] with a trapping frequency w z =2 π ×
500 Hz. The plots represent a section of the wave-function in the radial direction at the peak value of thedensity. The red-solid line in Fig. 6(a) shows the wave-function ground state of the BEC trapped in the PDRpotential (represented by the red-dashed line), while theblack-solid line is the ground state of the BEC trappedin the harmonically approximated potential (representedby black-dashed line) equivalent to the PDR. To provideconfinement in the azimuthal direction, an extra beamyielding a trapping frequency of w azi = 2 π ×
265 Hz isincluded. We have found a 0 .
7% of relative difference be-tween the energies of the two ground states. Figure 6(b)presents the BEC wave-function after 30 ms of expan-sion within the harmonically approximated ring poten-tial (black-solid line) and within the real PDR potential(red-solid line). Black- and red-dashed lines representthe harmonic ring potential and the PDR potential, re-spectively. In this case, the relative difference betweenboth wave-functions is negligible. These results confirmthe good agreement between the harmonic approxima-tion derived in Section III A and the original PDR.
IV. CONCLUSIONS
In summary, we have presented a novel approach forgenerating toroidal optical traps for ultra-cold neutralatoms by applying the PDR as a blue-detuned toroidaltrap for BECs We have studied the normalized intensitydistribution around the annular ring structure of the CRphenomenon in biaxial crystals. For a well developed CRring, i.e. when R ≫ w , experimental results of the in-tensity distribution are compared with the exact paraxialsolution and with its asymptotic approximation. We havefound the positions of the bright and dark rings of CR andthe position of the two points with maximum intensityalong the beam propagation direction, both experimen-tally and analytically. We have shown that the radius ofthe PDR is smaller than the optical geometric approxi-mation of the CR ring radius R , by approximately halfthe waist radius of the input beam ( − . w in Table I).All previous related works [28–32] were performed consid-ering that the radius of the PDR exactly coincided with R . The reported results show that the PDR is enclosedby higher intensity walls both in the radial as in the ax-ial directions, i.e. it is a toroidal dark-focus in all threedimensions, at variance with other light beams possess-ing only radial confinement, such as Laguerre–Gaussianmodes. We have applied the harmonic approximationaround the PDR and we have derived the expression forthe radial and axial trapping frequencies and the max-ima of the potential barriers for blue-detuned light asa function of common experimental parameters such asbeam power, beam waist, detuning and the parametersof the crystal. The reported results show the suitabil-ity of the PDR for trapping ultra-cold atoms with blue- | Ψ r | ( µ m − ) V ( r (cid:21) R + . w , ) ( µ K / k B ) r − R + 0 . w ( µ m)(b) × − × − × − × − | Ψ r | ( µ m − ) V ( r (cid:21) R + . w , ) ( µ K / k B ) r − R + 0 . w ( µ m)(a) × − × − × − × − FIG. 6. Radial sections of the atomic density of theBEC (a) before and (b) after 30 ms of azimuthal expan-sion of the BEC trapped in the harmonic potential V r = mω r ( r − ( R − . w )) (black-dashed line) and in thePoggendroff dark ring of CR (red-solid line). Black-dashedand red-dotted lines are the corresponding trapping poten-tials. Parameter values: R = 170 µ m, w = 18 µ m, P =27 mW, w r = 2 π ×
265 Hz, w z = 2 π ×
500 Hz, a s = 5 .
45 nmand N = 12000 atoms. The ground state (a) is obtainedby adding an extra confinement ( w azi = 2 π ×
265 Hz) in theazimuthal direction in order to reproduce the loading of theBEC in the CR ring trap. detuned light, making this technique ideal for experi-ments where well-defined potentials and high intensitybeams are required [9–11, 19, 44]. Therefore, as a proofof the usefulness of the derived theory we have performednumerical simulations of the dynamics of a trapped RbBEC with N = 12000 atoms in the dark ring potentialusing the harmonic approximation and have comparedthe obtained results with the solution of the original CRlight field. We have also compared the ground states inboth cases and we have found 0 .
7% relative difference inenergy between them. The numerical simulations agreewell with the experimental results on the dynamics of atrapped Rb BEC in the PDR of CR.The main advantages of the presented technique arethe simple generation and high quality of the CR toroidaldark trap, since the only requirements are a biaxial crys-tal and a focused input Gaussian beam, at variance withthe techniques using LG beams that need the interfer-ence of at least two beams [19]. Also, the minimum (andpractically null) intensity circle offered by the toroidaldark trap avoids photon scattering and presents no cor- rugation of the potential minimum at the focal plane.Additionally, and at variance with techniques basedon LG beams [22] or amplitude masks [23–25], the useof a biaxial crystal allows for the full input power tobe converted into the CR dark toroidal trap. which in-creases the efficiency in ultra-cold atom trapping experi-ments. Moreover, biaxial crystals can be transparent toan extremely wide spectral range[46] (0.35 µ m-5.5 µ m inKGd(WO ) , for instance), in contrast with spatial lightmodulators used in the generation of LG beams, whichonly work in a small spectral range of few hundreds ofnm, typically.A range of applications of this technique can be envi-sioned: for optimized beam geometries, i.e. small w , R ,and z R , the toroidal dark focus of the PDR generated byCR could be used to built an all-optical trap for BECsusing a single beam. Under such conditions, this poten-tial could be used as a basic element in atomic SQUIDexperiments [47, 48], as well as to study the dynamicsof matter waves with periodic boundary conditions andthe generation of persistent currents [18, 49]. For large R , the PDR can be used as a dark 2D ring potential byusing a 1D light sheet, along the axial direction, as anadditional confining potential. This configuration wouldallow to study wave-packet interference in a mesoscopicring simulating a quasi-one-dimensional system [44]. Bymodifying this 1D light sheet to a blue-detuned doublelayer also accessible via CR [38, 50] again a fully blue-detuned dark trap geometry with added flexibility is gen-erated.As an encouragement for future investigations, othercylindrically symmetric light structures of interest inatomic trapping experiments such as flat intensity re-gions or doughnut-like beams are also accessible via CR[38, 39]. Additionally, a radial optical lattice could begenerated by means of a cascade of biaxial crystals, gen-erating 2 N − dark rings for a cascade of N biaxial crystals[35, 36]. This could be also combined with the techniqueshown in [37] to generate an azimuthal optical latticewith controllable number of nodes and separation be-tween them, applicable in quantum-many body systemsexperiments [51, 52]. Also interesting is the possibility ofusing the PDR to coherently injecting, extracting, andvelocity filtering of particles, ultra-cold atoms and BECsas reported in [34, 53] by tuning the polarization of theinput beam and opening/closing the ring potential. Fi-nally, we would also like to note that by switching to red-detuned light, the inner and outer bright rings aroundthe PDR generate an intrinsically concentric system of adouble-ring potential which can be used for the genera-tion of coherent double wave packets for the investigationof wave packet tunneling and coupled persistent currentsof ultra cold atoms [54, 55]. ACKNOWLEDGMENTS
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