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Trapping of Bose-Einstein condensates in a three-dimensional dark focus generated by conical refraction

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 laser light exploiting the phenomenon of conical refraction occurring in biaxial crystals. We present experiments on confining a Rb87 Bose-Einstein condensate in this potential and derive the trapping frequencies and potential barriers under the harmonic approximation and the conical refraction theory.


I. INTRODUCTION
The intensity gradient of a light field can be used to efficiently trap ultra-cold atoms and Bose-Einstein condensates (BECs) using either red-or blue-detuned light. Taking advantage of this light-matter interaction, atoms are attracted to regions of high light intensity, when using light whose frequency is lower than the atomic twolevel transition (red-detuned case). These attractive optical traps are the most widely used optical traps due to their simplicity, since only a tightly focused laser beam is needed, producing a strong intensity gradient. Despite their simplicity, attractive dipole potentials introduce energy shifts and enhanced light scattering onto the trapped atoms, depending on the state and intensity of the trap. In this case, the fidelity of high precision measurements based on dipole traps also suffers from coherence loss caused by inhomogeneous differential light shifts [1]. In contrast, blue-detuned optical traps, allowing for a confinement of atoms in an local intensity minimum, have substantively decreased scattering rate and decoherence for atoms cooled close to the minimum of the trap potential. Therefore, they are ideal candidates in highly sensitive experiments. Blue-detuned optical potentials 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 trapped has null intensity and is surrounded by av steep intensity gradients, which creates an confining potential. Such light beams are refered to as 'optical bottle' [6]. Different methods have been proposed to generate optical bottle beams, such as interfering Laguerre-Gauss beams [7], scanning blue-detuned laser beams for time averaged potentials [8], crossing two or more vortex beams [9] or using optical c-cut uniaxial and biaxial crystals [10]. However, most of these methods have associated limitations such as the extreme precise control on the optical elements needed to generate and align the complex beam geometry or the fact that the intensity minimum is not exactly equal to zero [11]. Recently, the generation of an optical bottle beam with a point of exact null intensity, i.e. with a three-dimensional (3D) dark focus, by using a biaxial crystal exploiting the phenomenon of conical refraction (CR) [12] was reported. In this article, we will analyze in detail both experimentally and theoretically the implementation of such a 3D dark focus beam for trapping of ultra-cold atoms and BECs. In Section II we present the main characteristics of the CR phenomenon, its theoretical basis and the characteristics of the 3D dark focus beam. Utilizing such a beam, we show in Section IV, that optical trapping of a 87 Rb BEC can be achieved. In Section III we apply the harmonic approximation around the dark focus and derive expressions for trapping frequencies and maxima of the potential barriers as a function of the parameters of the trapping system. Finally, we sum up the main conclusions that can be drawn from this work in Section V.

II. CONICAL REFRACTION
The CR phenomenon transforms a focused input light beam with waist radius w 0 into a bright ring of radius R 0 , at the focal plane, when the input light beam passes along one of the optical axes of a biaxial crystal [13][14][15][16]. The CR ring radius R 0 = lα is the product of the crystal length l, and the CR semi-angle α [16]. The CR semi-angle α depends on the principal refractive indices of the crystal as α = (n 2 − n 1 )(n 3 − n 2 )/n 2 , where it is assumed that n 1 < n 2 < n 3 . At any point of the CR rings, the electric field is linearly polarized with the polarization plane rotating so that every pair of diagonally opposite points has orthogonal polarizations. This polarization distribution only depends on the orientation of the plane of optical axes [16,17].
The theoretical model describing the beam propagation in CR is based on the Belsky-Khapalyuk-Berry (BKB) integrals [13,15]. For an input light beam with electric field E and cylindrically symmetric 2D Fourier arXiv:1705.02425v1 [cond-mat.quant-gas] 6 May 2017 transform a (κ) = 2π ∞ 0 rE in (r) J 0 (κr) dr, the normalized BKB integrals in cylindrical coordinates can be written as follows [24]: where η = κw 0 , ρ = r/w 0 , Z = z/z R and J q is the q th -order Bessel function of the first type. The annular CR intensity distribution for an circular polarized input beam is given by This can be derived from a fundamental Gaussian input beam with power P and normalized transverse profile of the electric field amplitude E(ρ) = 2P/πw 2 0 exp(−ρ 2 ) and a 2D Fourier transform a (η) = 2P π/w 2 0 exp −η 2 /4. Depending on the parameter ρ 0 ≡ R 0 /w 0 the shape of the annular CR intensity pattern can vary drastically, and therefore it will be used as a control parameter. If the condition ρ 0 1 is satisfied,the intensity pattern at the focal plane will form by two bright rings separated by a dark (Poggendorff) ring. [18][19][20][21][22][23]. This pattern of concentric rings can, while illuminated with blue-detuned light, be used to create an repulsive ring shaped guiding potential for cold atoms and BECs [24]. However, for other values of the ρ 0 parameter, the CR beam changes substantially giving rise to different optical ring potentials [12,25,26].
Generation of a 3D dark focus using CR For the value ρ 0 = ρ DF 0 = 0.924, it has been shown that the CR beam possesses a 3D dark focus at the focal plane, i.e. it is a perfect bottle beam [6,12], as shown in Fig. 1 In this case, the transverse pattern at the focal plane is formed by a doughnut-like intensity distribution ( Fig. 1 (a)). Along the axis of propagation, the on-axis intensity increases with distance from the focal plane, as shown in Fig. 1 (b). At the focal plane, the maximum intensity region forms a ring with a radius ρ max = 1.1 and relative peak intensity I(ρ = 1.1, Z = 0) = 0.2. Along the axial direction, there are two maximum intensity points at Z max = ±1.388 and relative intensity I(ρ = 0, Z = ±1.388) = 0.14. Therefore, under the condition ρ 0 = 0.924, a circularly polarized input beam is transformed into a perfect optical bottle beam with a point of exact null intensity at ρ, z = 0. This feature makes this CR beam an ideal candidate for atom trapping experiments with blue-detuned light. To proof that there is a 3D trapping potential without additional light sheet, we measured the lowest and highest light intensity for various planes along the z direction. The region of highest intensity for a given plane is located at the ring radius, the lowest intensity is expected to be at the center of the annular CR pattern (see Fig. 1(b)). Experimental data is depicted in Fig. 2 and additionally, the calculated intensities are shown. A difference in intensity for a given plane results in a radial confining potential. This is given for every plane up to the point where the 3D dark trap closes off axially (|z| ≤ 7.7 mm). From that we conclude for every point inside the 3D dark trapping region, there is a light field gradient present to confine atoms.

III. THEORETICAL FORMULAE FOR THE 3D DARK FOCUS TO ATOM TRAPPING
In this section we will study the behavior of the CR beam close to the trap center, i.e. for r = (ρ ≈ 0, Z ≈ 0), for the value ρ 0 ≡ ρ DF 0 = 0.924 and we will deduce the trapping frequencies and potential depths for its application in ultra-cold atom trapping using the harmonic approximation. The strength of the dipolar potential will be considered as: InŨ 0 we have applied the rotating-wave approximation and we consider the case of Alkali atoms. In this equation, c is the speed of light in vacuum, Γ Di and ω Di (i = 1, 2) are, respectively, the natural line width and frequency of the D i line of the atomic species used, and ω L is the frequency of the input light. The spatial intensity 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 where is the gamma function. Under this expansion and for an input beam with transversally Gaussian profile, Eqs. (1) and (2) can be rewritten as: Eqs. (7) and (8) can be analytically solved, obtaining the following expressions: where 1 F 1 (a; b; z) is the Kummer confluent hypergeometric function. This formulation is valid for all values of ρ 0 as long as the point of intensity minimum remains at ρ = 0. Note that the minimum intensity point ρ min will depend on the value of ρ 0 . For the 3D dark focus beam (ρ 0 = 0.924) the minimum intensity in the focal plane is zero. We have found that close to ρ = 0, the expression for k = 0 is a good approximation to the full CR intensity profile. Therefore it is enough to keep the k = 0 terms of the series in Eqs. (9) and (10). In this case, the intensity of the CR beam reads The first term in Eq. (11) is an offset to the potential that appears for z = 0, as shown in the inset of Fig. 1(b). As a consequence, trapping atoms outside of the focal plane can increase the atom-photon interactions. To obtain the trapping frequencies of the potential in radial direction, one applies the harmonic approximation to the second term of Eq. (11), which yields: where m is the atomic mass. Note that, in the axial direction this approximation is valid only in the region where the optical bottle is formed, i.e for Z ∈ [−1.388, 1.388]. Additionally note that in Eq. (12), we have undone the normalization of the radial coordinate, i.e. we have changed ρ by r/w 0 . The potential barrier along the radial direction, i.e. at the point r = (ρ = 1.1, Z = 0), is not well described by the harmonic approximation. To give a full description of the radial maximum, the expressions up to at least k = 4 must be considered. in Eqs. (9) and (10). Its value is:

Axial direction
We also study the trapping confinement along the axial direction. In this case, a compact expression for any value of ρ 0 cannot be obtained since the minimum radial intensity point is a function of ρ 0 . For the case of ρ DF 0 = 0.924, the point of minimum intensity is at ρ = 0. Here, the approximation from Eq. (6) used before is not needed since J 1 (0) = 0 and J 0 (0) = 1 and, as a consequence B S (ρ = 0, Z) = 0. Therefore, the light intensity is solely described by B c as follows where D(x) is the Dawson function. The second order of the Taylor series of this analytical solution leads to the following expression for the trapping frequency (ω z ) along the axial direction: The height of the potential barriers along the axial direction, i.e at r = (ρ = 0, Z = ±1.388) can be obtained directly from Eq. (15). Their values are:

IV. RADIAL CONFINEMENT OF A 87 RB BEC IN THE DARK FOCUS
In order to demonstrate the experimental feasibility of atom confinement in the CR dark focus beam, we trapped a BEC of 87 Rb in a combination of the bottle beam potential and an additional light sheet as support against gravity. We measured the axial trap frequency and potential depth and show that an extrapolation to optimized experimental parameters should allow for a full 3D confinement in the bottle beam without additional support against gravity. The experimental setup is presented in Fig. 3(a) and a sketch of the trapping potential is shown in Fig. 3(b). A pair of intersecting focused laser beams generated by a 1070 nm fiber laser, forming a crossed optical dipole trap is used to create a BEC of 25000 87 Rb atoms [27]. The plane formed by the two beams is oriented perpendicular to gravity. The light for the CR potential is obtained from a tunable Ti:Sapphire laser at λ CR = 793.5 nm, providing a power of P CR = 24 mW inside the vacuum chamber. To generate the 3D dark focus potential, we align the focused input beam with waist radius of 42.7 µm and Rayleigh length z R = 7.23 mm along one of the optical axes of a KGd(WO 4 ) 2 biaxial crystal by using a lens f 1 of 150 mm focal length. A combination of a λ/2 and a λ/4 waveplate is used to ensure that the impingin light beam has a circular polarization state. A pinhole in front the focusing lens f 1 is arranged in order to increase the spatial coherence of the input beam and ensure its symmetrical transverse profile. The KGd(WO 4 ) 2 crystal has a length of l = 2.1 mm, a conicity of α(λ = 793.5 nm) = 19 mrad, and therefore results in radius R 0 = 40) µm. This value, together with the measured w 0 gives a value of the control parameter ρ 0 = 0.937, which is close to the theoretical value ρ DF 0 = 0.924 for the 3D dark focus. For an easier alignment of the biaxial crystal, the polished input facet is cut perpendicular to one of its optical axes within less than 1.5 mrad of misalignment angle. The facet has a cross-section of 6 × 4 mm 2 and is cut with a parallelism under 10 arc sec of misalignment. The transformed CR pattern appears at the focal plane of the system, which is shifted longitudinally from the source plane of the focused input beam in the absence of the crystal by the ∆L = l(1 − 1/n BC ) introduced by the biaxial crystal, where n BC = 2.06 is the mean value of the refractive indices of the biaxial crystal [16]. Lenses f 2 , f 3 , f 4 and f 5 are used in a telescope configuration to re-image the focal plane into the vacuum chamber on top of the crossed dipole trap. This re-imaging system demagnifies the CR potential by a factor of 0.75, so that the final radius of the CR ring and of the focused beam are R 0 = 30 µm and w 0 = 32.0 µm, respectively.
As discussed, the radial direction of the 3D dark focus is in a plane perpendicular to gravity. Since the ax-ial confinement provided by the 3D dark focus depends on z R , a smaller input beam waist w 0 or a higher laser power would be required to trap atoms against gravity along the axial direction in our configuration. For this reason, we use an additional sheet of light generated by a cylindrical lens that focuses a Gaussian beam to hold atoms against gravity. This additional attractive dipole potential is created with light from an tapered amplifier with a wavelength of λ LS = 783.55 nm and power of P LS = 137 mW at the experiment. The waist radius of the focused beam in the axial direction of the system is w Z = 26.2 µm. These parameters yield a measured trapping frequency of ω Z = (169 ± 2) Hz. Therefore, in the setup studied here, the full 3D optical trapping potential is formed by the dark focus beam with blue-detuned light, allowing for confinement in the radial direction, and by a red-detuned light sheet for axial confinement. The loading of the BEC into this potential is performed adiabatically, i.e. the crossed dipole trap is switched off slowly while the CR potential and the light sheet are switched on. The total duration of the loading process is 40 ms. Fig. 4 shows an experimental absorption image of cold 87 Rb atoms in this trapping configuration, which is used to spatially align the crossed dipole trap with the confining 3D dark focus trap. After turning off the crossed beam dipole trap, atoms formely confined in each of the two legs of the crossed beam dipole trap away from the crossing region are still visible. Approaching the center, there is a circular region without atoms as a result of the repulsive ring potential. Strong confinement within the dark-focus potential results in an amplified number of atoms right at the center of the bottle beam.

Measurement of the potential barriers and trapping frequencies of the dark focus
In order to measure the radial potential height of the blue-detuned dark focus trap, we use an optical Bragg lattice that accelerates the trapped BEC in the direction of one of trapping beams of the crossed dipole trap, see Fig. 3(a). Experimentally, both legs of the optical Bragg lattice can be controlled independently in intensity and frequency, resulting in a moving optical Bragg lattice for the manipuilation of BECs. More details about the optical Bragg lattice can be found in Ref. [28]. The trapped BEC is accelerated with a fixed momentum of 2 k and, depending on the potential height, the ratio of escaped to remaining atoms can be measured. The resulting potential height of the 3D trapping potential in the radial direction was termined to U r = 20E R , where E R is the recoil energy (E R / = 2π · 3.771 kHz). Using the experimental parameters and Eq. 13, we compute a potential barrier of U (ρ = 1.1, Z = 0) = 80.7 E R which differs by a significant factor from the corresponding measurement. We attribute this deviation to the variation of the maximum along the azimuthal direction in the experimental potential (see inset of Fig. 3(a)) since an atom loss already occurs at a potential heigth which corresponds to the minimum around the ring and not to the average of the barrier height as calculated. FIG. 5. Observed radial oscillation of a trapped BEC after acceleration with a Bragg lattice. After a variable trapping, time the confining dark-focus potential is removed and the resulting mean momentum of the atom cloud is measured. The oscillating mean momentum derictly reflects the oscillatory motion to the atoms in the radial dark-focus potential and the vibrational frequency of the trapped atoms can be extracted.
For a measurement of the trapping frequency in the 3D dark focus potential we used the optical Bragg lattice to impose a momentum of 2 k to the trapped atoms and measure their velocity after different oscillation times, as shown in Fig. 5. We found a trapping frequency of ω R = 2π · (283 ± 16) Hz. The experimental error is due to the asymmetry of the potential height of the ring potential. Inserting the experimental parameters into Eq. (12), the corresponding calculated trapping fre-quency is ω r (ρ 0 = 0.937(0.88), Z = 0) = 2π ·310(284) Hz, which is in good agreement with the measured trapping frequency. This confirms that our harmonic model for the radial trapping potential gives a good approximation for the real potential at the trap minimum. It also supports our assumption that the discrepancy in the measured and calcultated height of the potential barrier is caused by systematic uncertainties in the measurement technique.
Bare CR potential for 3D dark focus confinement Encouraged by these results, we conclude this presentation by a discussion of the possibility of creating a 3D dark trapping potential using blue-detuned light without the support of an additional light sheet. Considering the force created by gravity, we model a CR trapping potential that is sufficiently strong in the z direction to trap the atoms in the dark trap. By varying the input power P and the size of the radial trap R 0 (and by that the size of the input beam w 0 ), it is indeed possible to create a 3D trapping potential for ultra cold atoms and BECs. In Tab. I we summarize the experimental and the corresponding theoretical values for the experiments presented in this paper. Additionally we show parameters that are sufficient for trapping atoms in the CR potential without the support of an light sheet. The parameter ρ 0 = 0.937 and the wavelength λ = 793.5 nm are kept unaltered although decreasing the detuning of the trapping light would allow to increase the trap strenght in addition.

V. CONCLUSIONS
We have demonstrated the experimental implementation of a blue-detuned 3D atomic trap obtained from a single focused Gaussian beam through the conical refraction phenomenon (CR) in biaxial crystals. We have deduced simple formulas for the trapping frequencies and potential barriers in three dimensions as a function of typical experimental parameters. In this work, the 3D dark focus beam was arranged with its propagation direction parallel to gravity. Since the axial confinement offered by such optical potential is not enough to compensate gravity, an additional trapping potential confining in this direction was needed. However, the 3D dark focus beam can be used as a blue-detuned optical trap with a single focused Gaussian beam by arranging the light potential in a plane orthogonal to gravity or optimizing the trap parameters as discussed. One of the advantages of this technique is, that CR provides the full conversion of the input power into the 3D dark focus beam and avoids diffraction losses in contrast to other methods based in spatial light modulators (SLMs), for instance, which introduce losses due to diffraction in the generation of LG beams. Moreover, biaxial crystals can be transparent to an extremely wide spectral range [29] (e.g. wavelengths of 350 nm -5.5 µm in KGd(WO 4 ) 2 ), in contrast to SLMs, which only work in a short spectral range, usually few hundreds of nm. These features make the 3D dark focus beam, produced by Conical Refraction, a very promising candidate for particle manipulation [23,30] and atom trapping [4,31].
Further applications can be expected: If instead of a Gaussian input beam, an elliptical beam is used, the 3D dark focus will lead to a pair of elliptical beams divided by a thin dark region. This configuration could be used as a dark light sheet potential, where cold atoms are trapped in between the bright regions. Also, the combination of the 3D dark focus with an array of micro lenses [32,33] would lead to the generation of a 2D array of 3D dark traps of significant interest in atom trapping for quantum computing experiments. Finally, the interference of the 3D dark focus beam with a plane wave along the beam propagation direction would generate a 1D stack of doughnut-like beams, ideal for quantum many-body experiments [34].
The authors gratefully acknowledge financial support through the Spanish MICINN contract, and FIS2011-23719, the Catalan Government contract SGR2009-00347, and the DAAD contracts 50024895 and 57059126. A. Turpin acknowledges financial support through grant AP2010-2310 from the MICINN and the DAAD grant 91526836.