Creation of vortices in a Bose-Einstein condensate by a Raman technique
Eric L. Bolda and Dan F. Walls
Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand. (February 1, 2008)
arXiv:cond-mat/9708189v1 25 Aug 1997
We propose a method for taking a Bose-Einstein condensate in the ground trap state simultaneously to a di?erent atomic hyper?ne state and to a vortex trap state. This can be accomplished through a Raman scheme in which one of the two copropagating laser beams has a higher-order LaguerreGaussian mode pro?le. Coe?cients relating the beam waist, pulse area, and trap potentials for a complete transfer to the m = 1 vortex are calculated for a condensate in the noninteracting and strongly interacting regimes. 03.75.-b, 03.75.Fi, 32.80.Lg, 32.80.Qk
the vortex can already be described in two dimensions, we will treat the condensate as though it simply has a thickness L in the z -direction. The light beams both enter from the top (or bottom) and are copropagating. An important assumption is that the condensate be optically thin in this dimension so that the light beams will a?ect all of the atoms. For the time being we neglect the collisions between atoms; their e?ect will be explained below. It is convenient to de?ne a⊥ =
1 ? h (ω1 ω2 )? 4 m 1 a1 = a⊥ α 4 1 ?4
(1) (2) (3) (4)
Based on the rapid experimental progress in the ?eld of atomic Bose-Einstein condensation [1–3], the prospects look good that experimenters will soon be generating and observing vortices in these sytems. Vortex solutions to the Gross-Pitaevski equation for the condensate wavefunction have already been found numerically [4,5]. It has been pointed out that unlike homogeneous Bose ?uids such as 4 He, there exist two distinct concepts of stability in a trapped Bose-Einstein condensate (BEC) . Although they are thermodynamically stable while the system is being rotated, this does not necessarily imply dynamical stability once the rotation ceases. Thus, one might be interested in the dynamics of the vortex. A necessary step is to start with a well-de?ned state; here we propose an optical method for coherently creating a vortex state from the ground state. It is well known that the higher order LaguerreGaussian modes of a laser resonator possess orbital angular momentum . (Note that this is possibly in addition to the angular momentum contributed by circular polarization of the light). If this angular momentum can be coherently transferred to a BEC, the circular motion of the atoms should result in a vortex state. Since this involves removing photons from the Laguerre-Gaussian mode, we need to avoid subsequent spontaneous emission and heating of the condensate . One way to accomplish this is to use a Raman scheme in which the second beam is an ordinary Gaussian mode, and the atoms make a transition between hyper?ne states along with the transfer of angular momentum. To transfer all of the atoms from the trap ground state to the vortex state, we need to ?nd the equivalent of a two-photon π pulse. We denote the condensate wavefunctions for hyper?ne states 1 and 2 by ψ1 and ψ2 . The trapping in current experiments is done magnetically, so we allow for di?erent trap frequencies ω1 , ω2 for the two atomic states. Since
a2 = a⊥ α ω2 α= ω1 and use the scaling r ?=
r a √⊥ ? = ω1 ω2 t t
(5) (6) (7)
?j (? ψ r ) = a⊥ ψj (r).
Then, after adiabatically eliminating the third level from the Raman transition , the modi?ed Gross-Pitaevski equations are given in polar coordinates by i ?1 ?ψ 1 ?1 + ? ?2 ?2 + 1r ?ψ ?? = ?2 ψ ? ?t 2 α ?2 ?ψ 1 ?1 . ?2 + ? ? ?ψ ? 2 + αr i ?? ?2 ψ = ? ?t 2 (8) (9)
Note that we use unit normalization, d3 x |ψ1 |2 + |ψ2 |2 = 1. The two-photon Rabi frequency is ?= √? ? ω1 ω2 and in the unscaled variables ?(x, t) = (d1 · E1 (x, t))? d2 · E2 (x, t) h2 ? ? (12) (11) (10)
where d1 , d2 are the dipole matrix elements between states 1 and 3, 2 and 3 respectively, and E1 , E2 are the near-resonant ?elds detuned by ? from those transitions. 1
The ?elds have oppositely circular polarizations as appropriate to a mF to mF +2 transition. We will choose the ?eld E1 to be an ordinary Gaussian mode (TEM00 ), while E2 will be the TEM11 Laguerre-Gaussian mode. (In fact it makes no di?erence which of the two ?elds has which mode structure.) We approximate these modes by their values at z = 0,
2 E1 (x, t) = A1 (t) exp ?r2 /w1 r 2 eiφ exp ?r2 /w2 E2 (x, t) = A2 (t) w2
always be unoccupied. Second, when collisions between the atoms are included (see below), the di?erences in chemical potentials for higher states will no longer be on resonance so the transitions are weakly driven in any case. Substituting into the Gross-Pitaevski equations (9) (m) and using the orthogonality of the ψj , we ?nd the equations of motion
(?1) (0) dc1 (?1) = g (t)c2 e?i(?2 ??1 )t dt (?1) (?1) (0) dc2 (0) = g ? (t)c1 ei(?2 ??1 )t i dt
Substituting into Eq. (12) yields ?(x, t) = ?0 (t) where 1 1 1 = 2+ 2 w2 w1 w2 and ?0 (t) = (d1 · A1 (t))? d2 · A2 (t) h2 ? ? (17) (16) r r2 exp ? 2 eiφ w2 w (15) where
? g (t) = L
r dr ψ1
(x)?(x, t) (25)
These are exactly equivalent to the Bloch equations for a two-level system. We note that an additional detuning (?1) (0) ?2 ? ?1 is required from the untrapped atom resonanant Raman transition; the energy this represents is imparted to the gas to produce the rotation. The e?ective Rabi frequency is given by ? 0 (t)Lα ? 1 4 g (t) = ? √ α a2 1 ⊥ + + √ w2 2 2 α
We begin with all the atoms in hyper?ne state 1 and the associated trap ground state. The ground trap states of each hyper?ne state are (we drop the bars on the scaled variables) 1 1 exp ? √ r2 π2 α 2 α √ 1 α 2 α 4 (0) r exp ? ψ2 (r) = π2 2 ψ1 (r) =
As usual , one may obtain a complete transfer to the upper level at time tf by choosing an on-resonance π pulse, g (t) = gr (t)ei(?2 gr (t) = gr (t)?
(27) (28) (29)
and the two vortex states rotating clockwise and counterclockwise are 1 2 1 = r exp ? √ r2 e±iφ πα 2 α √ 1 α α 2 (±1) ψ2 (r, φ) = r2 e±iφ exp ? π 2
(±1) ψ1 (r, φ)
gr (t′ ) dt′ = π
We can expect that the technique will remain applicable when interactions between the atoms are included. We replace the ground and vortex wavefunctions in Eq. (22) by their Thomas-Fermi approximations in the strongly interacting regime, ψ1 = ψ2
We expand the time-dependent condensate wavefunctions in this basis. ψj (x, t) = cj (t)ψj (x)e
(+1) (+1) cj (t)ψj (x)e (0) (0)
(0) ?i?j t
2?1 ? u11 2?2
+ + for j = 1, 2 (22) where =
(+1) ?i?j t (?1)
? αr2 ? u22
Here each ?j is the chemical potential when all atoms are in the hyper?ne state j and the condensate has azimuthal angular momentum m. We have neglected the higher terms of the series for two reasons. The Raman transition couples these higher trap states which are initially empty only to each other and therefore they will 2
8πN ajj a⊥
is the scaled interaction potential between two atoms in hyper?ne state j . These functions then appear in the integral for the e?ective Rabi frequency Eq. (25). (Note
that for the vortex, the Thomas-Fermi approximation includes the azimuthal kinetic energy but neglects the radial part.) The chemical potential of the ground state is found from normalization to be ?1 =
a simple way to do this is to detune the Gaussian laser mode above the single-photon resonance and leave that ?eld on after the Laguerre-Gaussian pulse has ended.
u11 ? 2παL
To ?nd the chemical potential of the vortex state, we must solve numerically the normalization equation u22 cosh ζ ζ ? = α sinh2 ζ ? ln coth 2 2π L and then set ?2 = coth ζ.
This research was funded by the Marsden Fund of the Royal Society of New Zealand, and University of Auckland Research Fund. We would like to thank M. Andrews, D. Rokhsar, M. Levenson, T. Wong and M. Olsen for helpful discussions.
It is desirable to make the e?ective Rabi frequency large, so that short pulses can be used and the dynamics of the condensate can be ignored while the vortex is being made. To this end, we numerically evaluate the integral Eq. (25) and determine its dependence on the waist w1 and w2 of each beam. We choose parameters similar to those of the JILA experiment with overlapping condensates in two hyper?ne states . For a total of N = 2 × 106 condensed 87 Rb atoms, mean of trap √ ω1 ω2 = 2π × 400Hz, thickness L = 10?6 m, frequencies √ α = 2, and scattering lengths ajj = 10?8 m, we obtain the ground and vortex states shown in Fig. 1. (The mean trap width is a⊥ = 540nm.) The required detuning (0) (?1) (?2 ? ?1 )/? h = 39.4kHz is mostly due to the di?erence in trap potentials in this example. In Fig. 2 we show the dependence of g/?0 on the beam waists. We note that there is a maximum as a function of w2 when w1 is held ?xed, for a value of w2 on the order several a⊥ . It is advantageous to choose w1 as large as possible; for the case of a plane wave A1 , the optimal waist size for the Laguerre-Gaussian beam is w2 = 19.3a⊥. The resulting vortex state has a zero condensate density at its center. However, current techniques are probably not capable of imaging this small core in the atomic density directly. In liquid 4 He, a negative ion can be trapped at the vortex core, and after accelerating the ion out of the ?uid it can be imaged on a phosphor screen . It might be possible to adapt this technique to the atomic BEC. An alternative way of observing the vortex is to allow it to interfere with another (non-vortex) condensate, and image the fringes as in . A similar technique has been used in nonlinear optics . The 2π phase shift around the vortex will show up as two interference fringes merging into one. We have shown how a vortex state can be created in a trapped Bose-Einstein condensate. The transfer to the new state is completely coherent, as it involves only stimulated optical transitions. It could also be applied to create higher vortex states with |m| > 1. To investigate persistent currents, one may transfer the atoms to the vortex while a repulsive potential is placed at the center; 3
 M. H. Anderson et al., Science 269, 198 (1995); K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995); C. C. Bradley et al., ibid, 1687 (1995); M. R. Andrews et al., Science 273, 84 (1996); M.-O. Mewes et al., Phys. Rev. Lett. 77, 416 (1996); M.-O. Mewes et al., ibid, 988 (1996); M.-O. Mewes et al., Phys. Rev. Lett. 78, 582 (1997).  M. R. Andrews et al.,Science 275, 637 (1997).  C. J. Myatt et al.,Phys. Rev. Lett. 78, 586 (1997).  M. Edwards et. al., Phys. Rev. A, 53, R1950, (1996).  F. Dalfovo and S. Stringari, Phys. Rev. A53, 2477, (1996).  D. Rokhsar, Vortex stability and persistent currents in trapped Bose gases to be published in Phys. Rev. Lett. 79, (1997).  A. E. Siegman, Lasers, pp. 647-648, University Science, Mill Valley, California (1986).  Light-induced torques on cooled atoms, including spontaneous emission, are discussed in M. Babiker, W. L. Power, and L. Allen, Phys. Rev. Lett. 73, 1239 (1994); W. L. Power et al, Phys. Rev. A 52, 479 (1995); and J. Twamley, Quantised motion of an atom in a GaussianLaguerre beam, preprint.  J. Javanainen and J. Ruostekoski, Phys. Rev. A 52, 3033 (1995).  L. Allen and J. H. Eberly, Optical resonance and two-level atoms, Dover, New York (1975).  G. A. Williams and R. E. Packard, J. Low Temp. Phys. 39, 553 (1980).  G. A. Swartzlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992). FIG. 1. Thomas-Fermi wavefunctions of the initial and ?nal condensate, appropriate for strongly interacting atoms, for the parameters given in text. Solid curve: Ground trap state, hyper?ne state 1 Dashed curve: Vortex trap state, hyper?ne state 2. Note this function has a small core at which the density goes to zero.
FIG. 2. Overlap integral g/?0 as a function of Laguerre-Gaussian beam waist w2 for w1 = ∞ (solid curve), w1 = 50a⊥ (long dash) and w1 = 20a⊥ (short dash).
r [a ⊥ ]
0.35 0.3 0.25
0.2 0.15 0.1 0.05
r [a ⊥ ]