Trapping of Bose-Einstein condensates in a three-dimensional dark focus generated by conical refraction
F. Schmaltz, J. Küber, A. Turpin, V. Ahufinger, J. Mompart, G. Birkl
TTrapping of Bose-Einstein condensates in a three-dimensional dark focus generated byconical refraction
F. Schmaltz, J. K¨uber, A. Turpin, V. Ahufinger, J. Mompart, and G. Birkl Institut f¨ur Angewandte Physik, Technische Universit¨at Darmstadt,Schlossgartenstraße 7, D-64289 Darmstadt, Deutschland Departament de F´ısica, Universitat Aut`onoma de Barcelona, Bellaterra, E-080193, Spain (Dated: April 16, 2018)We present a novel type of three-dimensional dark focus optical trapping potential for ultra-cold atoms and Bose-Einstein condensates. This ‘optical bottle’ is created with blue-detuned laserlight exploiting the phenomenon of conical refraction occurring in biaxial crystals. We presentexperiments on confining a Rb Bose-Einstein condensate in this potential and derive the trappingfrequencies and potential barriers under the harmonic approximation and the conical refractiontheory.
PACS numbers: 67.85.Hj, 37.10.Gh
I. INTRODUCTION
The intensity gradient of a light field can be used toefficiently trap ultra-cold atoms and Bose-Einstein con-densates (BECs) using either red- or blue-detuned light.Taking advantage of this light-matter interaction, atomsare attracted to regions of high light intensity, when us-ing light whose frequency is lower than the atomic two-level transition (red-detuned case). These attractive op-tical traps are the most widely used optical traps due totheir simplicity, since only a tightly focused laser beamis needed, producing a strong intensity gradient. De-spite their simplicity, attractive dipole potentials intro-duce energy shifts and enhanced light scattering onto thetrapped atoms, depending on the state and intensity ofthe trap. In this case, the fidelity of high precision mea-surements based on dipole traps also suffers from coher-ence loss caused by inhomogeneous differential light shifts[1]. In contrast, blue-detuned optical traps, allowing fora confinement of atoms in an local intensity minimum,have substantively decreased scattering rate and deco-herence for atoms cooled close to the minimum of thetrap potential. Therefore, they are ideal candidates inhighly sensitive experiments. Blue-detuned optical po-tentials are used in the manipulation of Rydberg states[2], atomic clocks [3], quantum information processing[4], or Bose-Einstein condensation in uniform potentials[5]. Ideally the local minimum where atoms are trappedhas null intensity and is surrounded by av steep inten-sity gradients, which creates an confining potential. Suchlight beams are refered to as ‘optical bottle’ [6]. Differ-ent methods have been proposed to generate optical bot-tle beams, such as interfering Laguerre-Gauss beams [7],scanning blue-detuned laser beams for time averaged po-tentials [8], crossing two or more vortex beams [9] or usingoptical c-cut uniaxial and biaxial crystals [10]. However,most of these methods have associated limitations suchas the extreme precise control on the optical elementsneeded to generate and align the complex beam geome-try or the fact that the intensity minimum is not exactly equal to zero [11]. Recently, the generation of an opticalbottle beam with a point of exact null intensity, i.e. witha three-dimensional (3D) dark focus, by using a biaxialcrystal exploiting the phenomenon of conical refraction(CR) [12] was reported. In this article, we will analyzein detail both experimentally and theoretically the im-plementation of such a 3D dark focus beam for trappingof ultra-cold atoms and BECs. In Section II we presentthe main characteristics of the CR phenomenon, its the-oretical basis and the characteristics of the 3D dark focusbeam. Utilizing such a beam, we show in Section IV, thatoptical trapping of a Rb BEC can be achieved. In Sec-tion III we apply the harmonic approximation around thedark focus and derive expressions for trapping frequen-cies and maxima of the potential barriers as a functionof the parameters of the trapping system. Finally, wesum up the main conclusions that can be drawn fromthis work in Section V.
II. CONICAL REFRACTION
The CR phenomenon transforms a focused input lightbeam with waist radius w into a bright ring of radius R , at the focal plane, when the input light beam passesalong one of the optical axes of a biaxial crystal [13–16]. The CR ring radius R = lα is the product of thecrystal length l , and the CR semi-angle α [16]. The CRsemi-angle α depends on the principal refractive indicesof the crystal as α = (cid:112) ( n − n )( n − n ) /n , where itis assumed that n < n < n . At any point of the CRrings, the electric field is linearly polarized with the po-larization plane rotating so that every pair of diagonallyopposite points has orthogonal polarizations. This po-larization distribution only depends on the orientation ofthe plane of optical axes [16, 17].The theoretical model describing the beam propaga-tion in CR is based on the Belsky–Khapalyuk–Berry(BKB) integrals [13, 15]. For an input light beam withelectric field E and cylindrically symmetric 2D Fourier a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y transform a ( κ ) = 2 π (cid:82) ∞ rE in ( r ) J ( κr ) dr , the normal-ized BKB integrals in cylindrical coordinates can be writ-ten as follows [24]: B C ( ρ, Z ) = 12 π (cid:90) ∞ ηa ( η ) e − i Z η cos ( ηρ ) J ( ηρ ) dη (1) B S ( ρ, Z ) = 12 π (cid:90) ∞ ηa ( η ) e − i Z η sin ( ηρ ) J ( ηρ ) dη (2)where η = κw , ρ = r/w , Z = z/z R and J q is the q th -order Bessel function of the first type. The annularCR intensity distribution for an circular polarized inputbeam is given by I CP = | B C | + | B S | . (3)This can be derived from a fundamental Gaus-sian input beam with power P and normalizedtransverse profile of the electric field amplitude E ( ρ ) = (cid:112) P/πw exp( − ρ ) and a 2D Fourier transform a ( η ) = (cid:112) P π/w exp − η /
4. Depending on the param-eter ρ ≡ R /w the shape of the annular CR intensitypattern can vary drastically, and therefore it will be usedas a control parameter. If the condition ρ (cid:29) ρ pa-rameter, the CR beam changes substantially giving riseto different optical ring potentials [12, 25, 26]. Generation of a 3D dark focus using CR
For the value ρ = ρ DF0 = 0 . ρ max = 1 . I ( ρ = 1 . , Z = 0) = 0 . Z max = ± .
388 and relative intensity I ( ρ = 0 , Z = ± . .
14. Therefore, under the con-dition ρ = 0 . ρ, z = 0. This featuremakes this CR beam an ideal candidate for atom trap-ping experiments with blue-detuned light. To proof thatthere is a 3D trapping potential without additional lightsheet, we measured the lowest and highest light intensityfor various planes along the z direction. The region ofhighest intensity for a given plane is located at the ringradius, the lowest intensity is expected to be at the center (b)(a) z/z x / w R −5 −2.5 0 2.5 15-3 -33 -3 3 y / w x/w FIG. 1. (color online) (a) 2D density plot of the transversepattern of CR at the focal plane for ρ ≡ ρ DF0 = 0 . z - x plane showing thatfor ρ DF0 the CR beam forms a 3D dark focus. Top insets plotthe radial and axial transverse cross-sections at z = 0 and x = 0, respectively. of the annular CR pattern (see Fig. 1(b)). Experimentaldata is depicted in Fig. 2 and additionally, the calculatedintensities are shown. A difference in intensity for a givenplane results in a radial confining potential. This is givenfor every plane up to the point where the 3D dark trapcloses off axially ( | z | ≤ . −10 −5 0 5 10050100150200 z Position [mm] I n t e n s i t y [ a . u .] ρ = 0 ρ = γ ( z ) FIG. 2. Highest (red) and lowest (blue) values of intensity ofthe CR pattern at different planes in axial z direction. For theaxial region of | z | ≤ . III. THEORETICAL FORMULAE FOR THE 3DDARK FOCUS TO ATOM TRAPPING
In this section we will study the behavior of the CRbeam close to the trap center, i.e. for r = ( ρ ≈ , Z ≈ ρ ≡ ρ DF0 = 0 .
924 and we will deduce thetrapping frequencies and potential depths for its applica-tion in ultra-cold atom trapping using the harmonic ap-proximation. The strength of the dipolar potential willbe considered as: U ( r ) = − I ( r ) ˜ U (4)˜ 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) . (5)In ˜ U we have applied the rotating-wave approximationand we consider the case of Alkali atoms. In this equa-tion, c is the speed of light in vacuum, Γ D i and ω D i ( i = 1 ,
2) are, respectively, the natural line width andfrequency of the D i line of the atomic species used, and ω L is the frequency of the input light. The spatial inten-sity distribution I ( r ) is given by Eqs. (1)-(3). Radial direction
The Taylor series of the Bessel functions of order α , J α ( x ), around x = 0 can be written as J α ( x ) = ∞ (cid:88) k =0 ( − k k !Γ( k + α + 1) (cid:16) x (cid:17) k + α , (6)where Γ( t ) = (cid:82) ∞ x t − e − x dx is the gamma function. Un-der this expansion and for an input beam with transver-sally Gaussian profile, Eqs. (1) and (2) can be rewrittenas: B C ( ρ, Z ) = (cid:115) P πw ( Z ) (cid:90) ∞ ηe − η iZ )4 cos ( ηρ ) × (7) × ∞ (cid:88) k =0 ( − k k !Γ( k + 1) (cid:16) ηρ (cid:17) k dη,B S ( ρ, Z ) = (cid:115) P πw ( Z ) (cid:90) ∞ ηe − η iZ )4 sin ( ηρ ) × (8) × ∞ (cid:88) k =0 ( − k k !Γ( k + 2) (cid:16) ηρ (cid:17) k +1 dη. Eqs. (7) and (8) can be analytically solved, obtaining thefollowing expressions: B C ( ρ, Z ) = (cid:115) Pπw ( Z ) ∞ (cid:88) k =0 ( − k ρ k k !(1 + iZ ) m × (9) × F (cid:18) k + 1; 12 ; − ρ iZ (cid:19) ,B S ( ρ, Z ) = (cid:115) P ρ πw ( Z ) ∞ (cid:88) k =0 ( − k ρ k +1 k !(1 + iZ ) m +1 × (10) × F (cid:18) k + 2; 32 ; − ρ iZ (cid:19) , where F ( a ; b ; z ) is the Kummer confluent hyper-geometric function. This formulation is valid for all val-ues of ρ as long as the point of intensity minimum re-mains at ρ = 0. Note that the minimum intensity point ρ min will depend on the value of ρ . For the 3D darkfocus beam ( ρ = 0 . ρ = 0,the expression for k = 0 is a good approximation to thefull CR intensity profile. Therefore it is enough to keepthe k = 0 terms of the series in Eqs. (9) and (10). In thiscase, the intensity of the CR beam reads I ( ρ ≈ , Z ) = 2 Pπw ( Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:16) ; − ρ iZ (cid:17) iZ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ 4 ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ F (cid:16) ; − ρ iZ (cid:17) (1 + iZ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (11)The first term in Eq. (11) is an offset to the potential thatappears for z (cid:54) = 0, as shown in the inset of Fig. 1(b). Asa consequence, trapping atoms outside of the focal planecan increase the atom-photon interactions. To obtain thetrapping frequencies of the potential in radial direction,one applies the harmonic approximation to the secondterm of Eq. (11), which yields: ω r ( Z ) = (cid:115) ρ ˜ U Pπmw (1 + Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:16) ; − ρ iZ (cid:17) (1 + iZ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (12)where m is the atomic mass. Note that, in the axial direc-tion this approximation is valid only in the region wherethe optical bottle is formed, i.e for Z ∈ [ − . , . ρ by r/w . The potential barrier along the radialdirection, i.e. at the point r = ( ρ = 1 . , Z = 0), is notwell described by the harmonic approximation. To givea full description of the radial maximum, the expressionsup to at least k = 4 must be considered. in Eqs. (9) and(10). Its value is: U ( ρ = 1 . , Z = 0) = 0 . × ˜ U P πw . (13) Axial direction
We also study the trapping confinement along the axialdirection. In this case, a compact expression for anyvalue of ρ cannot be obtained since the minimum radialintensity point is a function of ρ . For the case of ρ DF0 =0 . ρ = 0. Here,the approximation from Eq. (6) used before is not neededsince J (0) = 0 and J (0) = 1 and, as a consequence B S ( ρ = 0 , Z ) = 0. Therefore, the light intensity is solelydescribed by B c as follows I ( ρ = 0 , Z ) = | B C ( ρ = 0 , Z ) | = (14)= P πw ( Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ ηe − η iZ )4 cos ( ηρ ) dη (cid:12)(cid:12)(cid:12)(cid:12) == 2 Pπw ( Z ) | iZ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ D (cid:16) ρ √ iZ (cid:17)(cid:112) (1 + iZ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where D ( x ) is the Dawson function. The second orderof the Taylor series of this analytical solution leads tothe following expression for the trapping frequency ( ω z )along the axial direction: ω z = (cid:115) ˜ U Pπmw z R . (15)The height of the potential barriers along the axial di-rection, i.e at r = ( ρ = 0 , Z = ± . U ( ρ = 0 , Z = ± . . × ˜ U P πw , (16) IV. RADIAL CONFINEMENT OF A RB BECIN THE DARK FOCUS
In order to demonstrate the experimental feasibilityof atom confinement in the CR dark focus beam, wetrapped a BEC of Rb in a combination of the bottlebeam potential and an additional light sheet as supportagainst gravity. We measured the axial trap frequencyand potential depth and show that an extrapolation tooptimized experimental parameters should allow for afull 3D confinement in the bottle beam without addi-tional support against gravity. The experimental setupis presented in Fig. 3(a) and a sketch of the trappingpotential is shown in Fig. 3(b). A pair of intersectingfocused laser beams generated by a 1070 nm fiber laser,forming a crossed optical dipole trap is used to createa BEC of 25000 Rb atoms [27]. The plane formed bythe two beams is oriented perpendicular to gravity. Thelight for the CR potential is obtained from a tunableTi:Sapphire laser at λ CR = 793 . P CR = 24 mW inside the vacuum chamber. To gen-erate the 3D dark focus potential, we align the focusedinput beam with waist radius of 42 . µ m and Rayleighlength z R = 7 .
23 mm along one of the optical axes of aKGd(WO ) biaxial crystal by using a lens f of 150 mmfocal length. A combination of a λ/ λ/ f is arranged in order to increase the spatialcoherence of the input beam and ensure its symmetricaltransverse profile. The KGd(WO ) crystal has a lengthof l = 2 . α ( λ = 793 . R = 40) µ m. This value, (a)(b) gravity FIG. 3. (a) Experimental setup for the creation of a darkfocus potential based on conical refraction. The CR axis isoriented along the axis of gravity and gives confinement in thehorizontal plane. Vertical confinement is achieved by a hor-izontally oriented red-detuned light sheet. The inset showsthe transverse intensity distributions in the focal plane. (b)Sketch of the 3D trapping potentials. Due to the weak con-finement of the dark focus trap in the axial direction (i.e alonggravity), an additional light sheet potential is used to holdatoms against gravity. The dark focus is positioned on top ofthe crossed optical dipole trap and the BEC is addibaticallyloaded into the trapping geometry. together with the measured w gives a value of the controlparameter ρ = 0 . ρ DF0 = 0 .
924 for the 3D dark focus. For an easieralignment of the biaxial crystal, the polished input facetis cut perpendicular to one of its optical axes within lessthan 1 . × and is cut with a parallelismunder 10 arc sec of misalignment. The transformed CRpattern appears at the focal plane of the system, whichis shifted longitudinally from the source plane of the fo-cused input beam in the absence of the crystal by the∆ L = l (1 − /n BC ) introduced by the biaxial crystal,where n BC = 2 .
06 is the mean value of the refractive in-dices of the biaxial crystal [16]. Lenses f , f , f and f are used in a telescope configuration to re-image the fo-cal plane into the vacuum chamber on top of the crosseddipole trap. This re-imaging system demagnifies the CRpotential by a factor of 0.75, so that the final radius ofthe CR ring and of the focused beam are R (cid:48) = 30 µ mand w (cid:48) = 32 . µ m, respectively.As discussed, the radial direction of the 3D dark fo-cus is in a plane perpendicular to gravity. Since the ax-ial confinement provided by the 3D dark focus dependson z R , a smaller input beam waist w or a higher laserpower would be required to trap atoms against gravityalong the axial direction in our configuration. For thisreason, we use an additional sheet of light generated bya cylindrical lens that focuses a Gaussian beam to holdatoms against gravity. This additional attractive dipolepotential is created with light from an tapered ampli-fier with a wavelength of λ LS = 783 .
55 nm and power of P LS = 137 mW at the experiment. The waist radius ofthe focused beam in the axial direction of the system is w Z = 26 . µ m. These parameters yield a measured trap-ping frequency of ω Z = (169 ±
2) Hz. Therefore, in thesetup studied here, the full 3D optical trapping potentialis formed by the dark focus beam with blue-detuned light,allowing for confinement in the radial direction, and by ared-detuned light sheet for axial confinement. The load-ing of the BEC into this potential is performed adiabat-ically, i.e. the crossed dipole trap is switched off slowlywhile the CR potential and the light sheet are switchedon. The total duration of the loading process is 40 ms.Fig. 4 shows an experimental absorption image of cold Rb atoms in this trapping configuration, which is usedto spatially align the crossed dipole trap with the con-fining 3D dark focus trap. After turning off the crossedbeam dipole trap, atoms formely confined in each of thetwo legs of the crossed beam dipole trap away from thecrossing region are still visible. Approaching the center,there is a circular region without atoms as a result ofthe repulsive ring potential. Strong confinement withinthe dark-focus potential results in an amplified numberof atoms right at the center of the bottle beam. x Position [mm] y P o s i t i o n [ mm ] FIG. 4. Experimental absorptive image of a cold atomic cloudpartially trapped in the 3D optical trap (center). The atomswere released from the crossed dipole potential and partiallyrecaptured by the dark focus trap. Atoms from the legs ofthe indvidual dipole traps are not recaptered but still visibleduring expansion.
Measurement of the potential barriers and trappingfrequencies of the dark focus
In order to measure the radial potential height of theblue-detuned dark focus trap, we use an optical Bragglattice that accelerates the trapped BEC in the directionof one of trapping beams of the crossed dipole trap, seeFig. 3(a). Experimentally, both legs of the optical Bragglattice can be controlled independently in intensity andfrequency, resulting in a moving optical Bragg lattice forthe manipuilation of BECs. More details about the opti-cal Bragg lattice can be found in Ref. [28]. The trappedBEC is accelerated with a fixed momentum of 2 (cid:126) k and,depending on the potential height, the ratio of escapedto remaining atoms can be measured. The resulting po-tential height of the 3D trapping potential in the radialdirection was termined to U r = 20 E R , where E R is therecoil energy ( E R / (cid:126) = 2 π · .
771 kHz). Using the exper-imental parameters and Eq. 13, we compute a potentialbarrier of U ( ρ = 1 . , Z = 0) = 80 . E R which differs by asignificant factor from the corresponding measurement.We attribute this deviation to the variation of the max-imum along the azimuthal direction in the experimentalpotential (see inset of Fig. 3(a)) since an atom loss al-ready occurs at a potential heigth which corresponds tothe minimum around the ring and not to the average ofthe barrier height as calculated. Trapping Time t trap [ms] M e a n M o m e m n t u m [ ¯ h k ] FIG. 5. Observed radial oscillation of a trapped BEC afteracceleration with a Bragg lattice. After a variable trapping,time the confining dark-focus potential is removed and theresulting mean momentum of the atom cloud is measured.The oscillating mean momentum derictly reflects the oscilla-tory motion to the atoms in the radial dark-focus potentialand the vibrational frequency of the trapped atoms can beextracted.
For a measurement of the trapping frequency in the3D dark focus potential we used the optical Bragg latticeto impose a momentum of 2 (cid:126) k to the trapped atoms andmeasure their velocity after different oscillation times,as shown in Fig. 5. We found a trapping frequencyof ω R = 2 π · (283 ±
16) Hz. The experimental erroris due to the asymmetry of the potential height of thering potential. Inserting the experimental parametersinto Eq. (12), the corresponding calculated trapping fre-quency is ω r ( ρ = 0 . . , Z = 0) = 2 π · Bare CR potential for 3D dark focus confinement
Encouraged by these results, we conclude this presen-tation by a discussion of the possibility of creating a 3Ddark trapping potential using blue-detuned light with-out the support of an additional light sheet. Consideringthe force created by gravity, we model a CR trappingpotential that is sufficiently strong in the z direction totrap the atoms in the dark trap. By varying the inputpower P and the size of the radial trap R (and by thatthe size of the input beam w ), it is indeed possible tocreate a 3D trapping potential for ultra cold atoms andBECs. In Tab. I we summarize the experimental andthe corresponding theoretical values for the experimentspresented in this paper. Additionally we show param-eters that are sufficient for trapping atoms in the CRpotential without the support of an light sheet. The pa-rameter ρ = 0 .
937 and the wavelength λ = 793 . Configuration in this Paper Theoretical Valuesfor R = 18 . µ mTheoretical Experimental w /µ m - 32 20 P/ mW - 24 30 ω r / π · Hz 310.4 283 888.5 ω z / π · Hz 2.3 - 10.6 U r /E R U z /E R V. CONCLUSIONS
We have demonstrated the experimental implementa-tion of a blue-detuned 3D atomic trap obtained from a single focused Gaussian beam through the conical refrac-tion phenomenon (CR) in biaxial crystals. We have de-duced simple formulas for the trapping frequencies andpotential barriers in three dimensions as a function oftypical experimental parameters. In this work, the 3Ddark focus beam was arranged with its propagation di-rection parallel to gravity. Since the axial confinementoffered by such optical potential is not enough to compen-sate gravity, an additional trapping potential confining inthis direction was needed. However, the 3D dark focusbeam can be used as a blue-detuned optical trap with asingle focused Gaussian beam by arranging the light po-tential in a plane orthogonal to gravity or optimizing thetrap parameters as discussed. One of the advantages ofthis technique is, that CR provides the full conversion ofthe input power into the 3D dark focus beam and avoidsdiffraction losses in contrast to other methods based inspatial light modulators (SLMs), for instance, which in-troduce losses due to diffraction in the generation of LGbeams. Moreover, biaxial crystals can be transparent toan extremely wide spectral range[29] (e.g. wavelengths of350 nm - 5.5 µ m in KGd(WO ) ), in contrast to SLMs,which only work in a short spectral range, usually fewhundreds of nm. These features make the 3D dark focusbeam, produced by Conical Refraction, a very promisingcandidate for particle manipulation [23, 30] and atomtrapping [4, 31].Further applications can be expected: If instead of aGaussian input beam, an elliptical beam is used, the 3Ddark focus will lead to a pair of elliptical beams dividedby a thin dark region. 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