Quasi-electrostatic trap for neutral atoms

15 February

1995

OPTICS COMMUNICATIONS Optics Communications

114 ( 1995

) 42 l-424

Quasi-electrostatic trap for neutral atoms T. Takekoshi, J.R. Yeh, R. J. Knize Department

of Physics and Astronomy, Received

26 August

University of Southern California.

Los Angeles, CA 90089-0484.

1994; revised version received 10 November

USA

I994

Abstract We show that it is possible to trap any neutral atom using a focused high power infrared laser. The trap could hold several different species of atoms simultaneously. Small photon scattering and heating rates allow the possibility of long trap lifetimes.

Neutral atom traps are valuable tools for precision spectroscopy, the study of cold collisions, and for quantum collective effects. There have been several neutral atom traps demonstrated using either optical, magnetic, or microwave fields. Magnetostatic fields have been used to trap low field-seeking states of hydrogen [ 11, sodium [ 21, and cesium [3]. An ac magnetic trap has also been demonstrated for cesium [ 41. Optical dipole force traps [ 51, and magneto-optical traps [6] have been demonstrated for various alkali and metastable inert gas atoms [ 71 using nearresonant laser fields. Recently, far off resonance optical dipole force traps have been demonstrated with sodium [ 81 and rubidium [9] atoms, with light detuned up to 65 nm below the first Dl resonance. A magnetic dipole force trap has been proposed for hydrogen [ lo], and has been recently demonstrated with cesium atoms [ 111 using microwave fields nearly resonant with the ground state hyperfine splitting. A dynamic electric trap using an oscillating electric field has also been proposed [ 121. In this paper, we discuss a non-resonant optical dipole force trap using intense infrared light. This trap can confine all ground state atoms of several atomic species simultaneously for long periods of time with small scattering and heating rates. An electrostatic Stark effect trap would be idea1 for 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI0030-4018(94)00638-5

several reasons. First, there would be no photon scattering. Second, since the static electric polarizability is positive for any atomic ground state, multiple atomic species and ions, could be trapped at a local field maximum. Unfortunately, Maxwell’s equations forbid a charge-free local electrostatic field maximum [ 131. It is well known that this problem can be circumvented by creating a local rms electric field maximum using an ac field, such as in an optical dipole force trap. In this paper, we consider an optical trap where the laser frequency w and Rabi frequency I Q 1 are much smaller than the frequency of the first allowed electric dipole resonance w,. The trap frequency is far below all the atomic electric dipole resonances (w < wi /2 ), that in many respects, this is essentially a quasi-electrostatic trap (QUEST). The proposed trap can be realized at the focus of a high intensity CO2 laser beam (1~ 10.6 pm ) Previous optical dipole force traps have been analyzed and demonstrated with laser detuning 6 small compared to the resonance frequency, 6= (w, - w ) < w, / 10. In this regime, a two-level atom in a spatially varying optical field experiences an optical dipole potential U [ 141

[!=-@ln I+ 2 [

lQIZ 2[ (r/2)2+@]

1

(1)

422

T Takekoshi et al. /Optics Communications 114 (1995) 421-424

and the photon scattering rate S is r15212/2 s= 2[(r/2)2+&2]+

ls212’

trapping potential including gravity can be approximated by (2)

Here S2=d,*E,/h, where 1Q) is the Rabi frequency, E= (E, e-‘“‘+EL eiw’)/2 is the electric field, & is the dipole matrix element between the excited and ground states, and ris the spontaneous decay rate of the excited state. Photon scattering causes trap losses through recoil heating, excited state collisions [ 15 1, and dipole force fluctuations. The heating rate in the trap can be estimated from the momentum diffusion constant. For a two-level atom, the excited and ground states will have equal but opposite Stark shifts for near resonance light, and the momentum diffusion constant D,is [14]

D P-- y

[k2+ (Vlnl91 )2+ (V0)2] ,

(3)

where L2= lr(21e”, and k is wavevector of the laser. The first optical trap [ 51 for neutral atoms used a focused 220 mW dye laser beam detuned about 130 GHz to the red of the sodium Dl resonance. The trap depth was about 10 mK, and the photon scattering rate was about 2600 s-‘. Recently, a far off resonance trap with detunings of up to 0.080,, was demonstrated for rubidium atoms [ 91. For a typical detuning of O.O23w, below the Dl resonance and 0.8 watts of laser power, the trap depth was about 6 mK and the photon scattering rate was about 400 s-‘. Since a QUEST operates with a laser frequency far below any electric dipole resonances, Eqs. ( 1 )-( 3 ) for an optical dipole force trap need to be modified. We estimate the saturation parameter [ 141 and obtain the result, that for practically attainable laser intensities, the excited state population is always extremely small. The trapping potential is therefore given by the lowest order perturbation theory expression for the Stark shift [ 161 of a ground state g, due to the excited states e,

&_!

lJ%ns12+wz,

where m is the mass of the atom, and w, is the frequency of the first Dl transition. For an alkali or hydrogen atom, the discrepancy between Eq. (4) and the Stark shift term in Eq. ( 5 ) is < 0.1 Oh,for w< o, / 2. At CO2 laser wavelengths, the Stark shift in Eq. (5) approaches the dc Stark shift AEg= - asE2/2. In addition to the electric dipole force, there will be light shifts due to the magnetic dipole interaction [lo] with the ground state hyperfine levels. This shift is a factor 2pLhr / (fia,o) smaller than the Stark shift. For cesium atoms, this magnetic dipole trap potential is a factor of 10’ smaller than the electric dipole trap potential. Since atoms are trapped without using a near-resonant laser, this trap may be able to confine several species simultaneously, provided the atoms can be cooled, and loaded into the trap. it may even be possible to confine atoms and ions simultaneously in the same QUEST. The total scattering rate S can be calculated using the Kramers-Heisenberg formula [ 171 S=

8na2m31 ri’l Im)(mlr~l c ul 6.&--o 3fic2 m,g ?

+
Ii)

(6)

where i, m, and f are the initial, intermediate, and final states, Q indicates the incident laser polarization, q is the scattered light polarization, I is the incident light intensity, and (Yis the fine structure constant. It is possible to write S=SRaylclgh+SRaman.In this expression, Rayleigh scattering leaves the atom in its original state (i=fl, whereas Raman scattering leaves the atom in a different hyperfine or Zeeman sublevel ( i #f) . SRayleigh can be described by the usual Rayleigh formula [ 18 ] S

where 0%~ (E,-E,)/k At QUEST frequencies this expression can be simplified, by using the ground state static scalar atomic polarizability, (Y,. The overall

ffs 2 [l-(dw1)21

(7)

wheref, is the oscillator strength between the excited and ground states, and r, is the classical electron radius. At CO2 laser frequencies, this expression is al-

T Takekoshi et al. 1 Optics Communications 114 (1995) 411-424

most exact. At o=co, 12, the discrepancy between Eq. ( 7 ) and the exact result from Eq. (6 ) is about 30%. For an alkali or hydrogen atom, where > 95% of the oscillator strength is in the first Dl and D2 transitions. the ratio between the Raman and Rayleigh scattering rates for o < wi /2 can be written as S Raman

SRayleIgh &w(2w

+A)

3w:+4d~,0:+(Af;-3w2)0~-2~~,~* ,g 9

[A,w*1’ wf

1 2

(8)

where An is the fine structure frequency splitting. Eq. (8) shows that the scattering is almost entirely due to Rayleigh scattering. The factor (ArJoi)* appears, because the atoms do not spend enough time in the upper state to have their spin perturbed by the fine structure interaction L-S [ 19 1. The factor ( w/w1 )* in Eq. (8) appears, because the Raman scattering matrix elements for the anti-rotating terms ( w, + w ) - ’ in Eq. ( 6 ) are equal but opposite in sign to the matrix elements for the corresponding rotating terms (w,-co-‘. In Eq. (S), we have neglected the excited state hyperfine splitting dhrs, which would introduce a very small amount of additional Raman scattering on the order of (d,,w/c$ )’ SRayleigh. Because the QUEST laser frequency is far below any electric dipole resonance frequency, the atom can no longer be treated as a simple two-level system. The Stark shift of the nearest excited state is no longer equal in magnitude and opposite in sign to the ground state’s. The second term in Eq. (3) for the momentum diffusion constant may need to be modified. The order of magnitude may be calculated using the scattering rate (7 ) in Eq. (3 ). The momentum diffusion constant is expected to be very small. Velocity dependent dipole forces are negligible [ 141. In order to operate a QUEST, it is desirable to have an intense infrared laser with the following characteristics: The laser frequency should be small compared to the first electric dipole transition frequency, but large compared to possible ground state hyperfine frequencies. The wavelength should be small, and the mode quality of the laser should be good enough so that the laser can be focused to a waist small enough

423

for a trapping potential to be obtained in the presence of gravity, Eq. ( 5 ). Commercial CO, lasers and associated optics satisfy these requirements. They are fairly efficient, high cw powers are available, and the strongest CO? transition is at 10.6 pm. There are several possible COz laser configurations for producing a QUEST. A high power laser could be focused in a manner similar to previous traveling wave optical traps. As an example, it should be possible to focus a 3 kW laser down to a waist radius of 15 urn [ 201. This trap would provide a potential well in excess of a mK for any neutral atom, including hydrogen and helium. It is also possible to operate a QUEST using a lower power laser with a buildup cavity, or in an intracavity configuration. In a typiallO0 W CO, laser, there is about 2.5 kW of circulating power that could be focused intracavity to form a trapping potential. The problem of large heating rates in intense standing wave optical traps [ 2 1 ] is negligible because the excited state population is extremely small. We have analyzed a CO1 laser QUEST for alkali and hydrogen atoms. The trap is formed by focusing a horizontal TEMoo laser beam to a waist of radius M’~.Fig. 1 shows the trap depth V. for cesium atoms evaluated from Eq. (7), as a function of power and waist radius. Fig. 2 shows the total trap volume Vfor cesium atoms, determined numerically from Eq. ( 5 ). If the waist radius is increased, eventually a point is reached where gravity prevents the Stark shift from forming a potential well. The figures show that for laser powers from 10 to 1O4W, the trap depth varies from 1O-5 to 1 IS, and the total trap volume varies from 1O-5 to 1 cm3. As a specific example for cesium atoms, we consider a 1 kW CO1 laser beam focused to a waist radius of 50 pm. In this case, the trap depth will be about 23 mK, and the trap volume will be about 3.5 mm3. At the center of the trap. where maximum scattering occurs, the Rayleigh scattering rate is 0.045 s-‘, and the Raman scattering rate is a factor of 1O4 smaller. Using Eq. (3 ) we find the maximum momentum diffusion constant Dy -2x 10P5’ erg gram/s. The lifetime of the atoms in the trap will be determined by D,,background collisions with residual gases, and three-body recombination. Multiphoton ionization is expected to be negligible [ 221. Since all ground state levels are trapped, there is no loss due to dipolar collisions such as in magnetic traps [ 1 1. The trap life-

424

T. Takekoshi et al. /Optics Communications 114 (1995) 421-424

1

E

10

-1 -2

10 -3

Do

10 -4 10 _A

w

References

,

-5

A”

-3

10

I

(4 [ I] H. Hess, G.P. Kochanski, J.M. Doyle, N. Masuhara, D. Kleppner and T.J. Greytak, Phys. Rev. Len. 59 ( 1987) 672.

-2

-1 10

Wolo(M~ 1

m

10

-1

(b)

-2 10 -3 10 -4 10 -5 Fz#zTiI 10

-1

-2

-3

10

10

10

w

(cm)

0 Fig. I. (a) Trap depth V,/k, (including gravity) versus waist radius w, for cesium atoms. (b) Trap volume Vversus waist radius IV,for cesium atoms. The curves (from top to bottom ) correspond to laser powersof 10 kW, 1kW, 100 W, IO W.

time due to momentum diffusion can be estimated This particular QUEST has a from ~~~~~~ rn&,ID~. r trap~4~ 10” s which shows that the heating rate is negligible. A QUEST with the same parameters, in the standing wave configuration, will have a similar lifetime. We have shown that it is possible to trap neutral atoms in a CO, laser optical trap. The laser frequency is far below any electric dipole resonances of atoms, so it is possible to simultaneously trap all the ground states of multiple atomic and ionic species, with low scattering and heating rates. Excluding background collisions, and three body recombination, the trap lifetime can be on the order of years. Two possible QUEST applications include a search for an atomic electric dipole moment (EDM), and an attempt to produce Bose-Einstein condensation of a cold atomic vapor. This work was supported by the US Army Research Office, under grant number 3020 I-PH, and by the National Science Foundation.

[2] Alan L. Migdall, John V. Prodan, William D. Phillips, Thomas H. Bergeman and Harold J. Metcalf, Phys. Rev. Lett. 54 (1985) 2596; K. Helmerson, A Martin and D.E. Pritchard, J. Opt. Sot. Am.B9(1992)483;9(1992)1988. [3] C.R. Monroe, E.A. Cornell, C.A. Sackett, C.J Myatt and C.E. Wieman, Phys. Rev. Lett. 70 ( 1993) 414. [4] Eric A. Cornell, Chris Monroe and Carl E. Wieman, Phys. Rev. J&t. 67 ( 1991) 2439. [ 51 S. Chu, J.E. Bjorkholm, A. Ashkin and A. Cable, Phys. Rev. Lett. 57 (1986) 314. [6] E.L. Raab, M. Prentiss, Alex Cable, Steven Chu and D.E. Pritchard, Phys. Rev. Lett. 59 ( 1987) 263 1. [ 7) Hidetori Katori and Fujio Shim& Phys. Rev. Lett. 70 (1993) 3545. [8] S. Rolston, C. Gerz, K. Helmerson, P. Jessen, P. Lett, W. Phillips, R. Spreeuw and C. Westbrook, Proc. 1992 Shanghai International Symposium on Quantum Optics, eds. Yuzhu Wang, Yiqui Wang and Zugeng Wang, Proc. SPIE 1726 ( 1992) 205. [9 ] J.D. Miller, R.A. Cline and D.J. Heinzen, Phys. Rev. A 47 ( 1993) R4567. [ IO] Charles C. Agosta, Issac F. Silvera, H.T.C. Stoof and B.J. Verhaar, Phys. Rev. Lett. 62 (1989) 2361. [ 111R.J.C. Spreeuw, C. Gerz, Lori S. Goldner, W.D. Phillips, S.L. Rolston, CL Westbrook, M.W. Reynolds and Isaac F. Silvera, Phys. Rev. Lett. 72 ( 1994) 3162. [ 121 F. Shimizu andM. Morinaga, Japn. J. Appl. Phys. 31 (1992) Ll721. [ 131 W. Ketterle and D.E. Pritchard, Appl. Phys. B 54 ( 1992) 403; W.H. Wing, Laser Cooled and Trapped Atoms, Proc. Workshop on Spectroscopic Applications of Slow Atomic Beams, ed. W.D. Phillips, NBS Special Publication No. 653 (U.S. GPO, Washington, DC, 1983). [ 141 J.P. Gordon and A. Ashkin, Phys. Rev. A 2 1 ( 1980) 1606. [ 151 M. Prentiss et al., Optics Lett. 13 (1988) 452. [ 161 Butcher and Cotten, Elements of Nonlinear Optics (Cambridge University Press, 1990) p. 183. [ 171 Rodney Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1993) p. 314. [ 181 C. Cohen-Tannoudji, J. DuPont-Rot and G. Grynberg, Atom-Photon Interactions (Wiley, 1992) p. 526. [ 19] R.A. Cline, J.D. Miller, M.R. Matthews and D.J. Heinzen. Optics Lett. 19 ( 1994) 207. [2O] Laser Power Optics Inc., San Diego CA USA, technical data 1992. [2l] J. Dalibard and C. Cohen-Tannoudji, J. Opt. Sot. Am. B I 1 (1989) 1707. [22] P. Lambropoulos, private communication.