Photoassociation of Sodium in a Bose-Einstein Condensate

arXiv:cond-mat/0111501v2 [cond-mat.mes-hall] 30 Nov 2001 Photoassociation of sodium in a Bose-Einstein condensate C. Mc Kenzie, J. Hecker Denschlag,∗ H. Häffner,∗ A. Browaeys, Luı́s E. E. de Araujo,† F. K. Fatemi,‡ K. M. Jones,§ J. E. Simsarian,¶ D. Cho,∗∗ A. Simoni,†† E. Tiesinga, P. S. Julienne, K. Helmerson, P. D. Lett, S. L. Rolston, and W. D. Phillips National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. (Dated: Preprint November 18, 2018) Abstract We report on the formation of ultra-cold Na2 molecules using single-photon photoassociation of a Bose-Einstein condensate. The photoassociation rate, linewidth and light shift of the J = 1, v = 135 vibrational level of the A1 Σ+ u molecular bound state have been measured. We find that the photoassociation rate constant increases linearly with intensity, even where it is predicted that many-body effects might limit the rate. Our observations are everywhere in good agreement with a two-body theory having no free parameters. PACS numbers: 03.75.Fi, 33.70.Jg, 34.20.Cf, 33.20.Kf 1 Bose-Einstein condensates (BECs) of atomic gases [1, 2] are versatile systems for the experimental study of quantum behavior. Of particular interest are the suggestions for the coherent coupling of a BEC of atoms with a BEC of molecules [3, 4, 5, 6] and the possibility of creating entangled pairs of atoms in the BEC via coupling with molecular levels [7]. Twophoton photoassociation processes using stimulated Raman transitions have formed ground state molecules from ground state atoms in a BEC [8, 9] but at very low rates. Here we explore the fundamental upper limits of molecule formation by making them at high rates using the elementary process of single-photon photoassociation. In single-photon photoassociation two atoms collide in the presence of a light field and form an excited state molecule. Photoassociative spectroscopy is used extensively to study collisions between laser cooled atoms [10]. Photoassociation in a BEC presents quite a different regime: the collision energies are orders of magnitude lower than in a laser cooled sample (the de Broglie wavelength is as big as the sample) and the densities are higher. This puts us in a regime where many-body effects may be important. We concentrate on a particular photoassociation transition and measure the photoassociation spectra for various intensities and durations of the light pulse. From these we determine the photoassociation rate, lineshape and the shift of the resonance. Finally we examine various limits on the photoassociation rate. Figure 1 shows the photoassociation process. The molecular level chosen for study is the J = 1, v = 135, rotational-vibrational level of the A1 Σ+ u Na2 molecular state, excited from free atoms by a laser frequency of 16913.37(2) cm−1 [22]. We chose this level because its detuning from the D1 resonance is far enough (43 cm−1 ) for atomic absorption to be negligible and because earlier experiments using laser cooled atoms indicated a high photoassociation rate. The lifetime of our excited molecules is about 8.6 ns. Decay of excited molecules into hot atoms or ground state molecules constitutes loss from the condensate. This loss is how we detect photoassociation. We prepare an almost pure condensate of N ≈ 4 × 106 sodium atoms in the |F = 1, mF = −1i ground state with a peak density of n0 ≈ 4 × 1014 cm−3 . The condensate is held in an √ √ anisotropic magnetic TOP [11] trap with oscillation frequencies of ωx / 2 = ωy = 2ωz = √ √ 2π × 198 Hz and corresponding Thomas-Fermi radii of 2rx = ry = rz / 2 = 15 µm. To induce photoassociation we illuminate the condensate with a Gaussian laser beam focused to 120 µm FWHM at the condensate. The peak intensity is varied from 50 to 1200 2 W·cm−2 . The polarization is linear and parallel with the rotation (z) axis of the TOP trap bias field. The light is applied as a square pulse for between 10 and 400 µs with rise and fall times of less than 0.5 µs. The condensate number is measured using phase contrast imaging [12], taking two images before and two images after the photoassociation pulse to determine loss. The imaging occurs at 40 ms intervals using a 100 µs probe pulse from a laser tuned 1.78 GHz to the red of the 3S1/2 , F = 1 → 3P3/2 , F = 2 transition. The imaging rate is limited by the readout time of the CCD camera. The photoassociation pulse begins halfway between the second and third images. We use multiple imaging pulses to improve statistics and to partially correct for small losses other than those due to the photoassociation pulse. These losses are typically 4% between images. Once the number of atoms is extracted from the images [23] we calculate f , the fraction of atoms remaining after the photoassociation pulse. Figure 2 shows a typical photoassociation spectrum. Each point represents a freshly prepared condensate. We use a Fabry-Perot etalon and a reference laser locked to an atomic Na line to measure differences in the photoassociation frequency with a precision of 5 MHz. The laser linewidth is < 3 MHz. All detunings quoted are relative to the center of the photoassociation line in the low intensity limit. For small trap loss we expect the line to be Lorentzian (in contrast to photoassociation lines in an uncondensed thermal sample where the kinetic energy distribution distorts the line shape [13]). For significant trap loss, as in figure 2, one must account for the change of the density profile during the photoassociation pulse. A two-body collisional loss process changes the local atomic density as: ṅ(t, r) = −K(I, ω)n2 (t, r), where K(I, ω) is the intensity and frequency dependent photoassociation rate constant. Because the characteristic time for the motion of the atoms, the trap oscillation period, is long compared to the photoassociation pulse, we assume that the local density only changes due to photoassociation. We then obtain: n(t, r) = n(0, r) . 1 + K(I, ω)n(0, r)t (1) The density distribution flattens with time. Spatially integrating equation (1), assuming an initial, parabolic (Thomas-Fermi), density distribution and a uniform intensity I, leads to 3 an expression for the fraction of atoms remaining in the condensate  15 −5/2 1/2 1 3/2 f (η) = η η + η 2 3 o p − (1 + η)1/2 tanh−1 ( η/(1 + η)) , (2) where η = K(I, ω)n(0, 0)t = Km (I)n(0, 0)t/(1 + (2(ω − ω0 (I))/γ(I))2 ). We use a three parameter fit of equation (2) to the spectra to extract the on-resonance rate constant Km (I), effective line width γ(I) and central frequency ω0 (I) (for example, the dotted line in figure 2). The fit is good. To further verify equation (2) we plot the measured 1 − f as a function of pulse length for I = 140 W·cm−2 and ω = ω0 , along with a one parameter (Km ) fit to the data (figure 3). The error bars are the fitting uncertainty. By fitting spectra obtained at various intensities we measure Km (I), γ(I), and ω0 (I). We calculate the unbroadened molecular linewidth of the chosen state to be γ0 /2π = 18.5 MHz (nearly twice the atomic linewidth). This is in good agreement with the measured linewidth of 19.5(25) MHz in the low intensity limit where it is independent of intensity. At higher intensities we observe broadening with a maximum linewidth of 60 MHz at 1 kW·cm−2 . Homogeneous power broadening is calculated to be three orders of magnitude too low to explain this width. It is, however, partially explained by differential light shifts of the unresolved molecular hyperfine states. These states are calculated to be split by less than 1 MHz at low intensities and about 30 MHz at our maximum intensity. Another possible contribution is the inhomogeneity of the photoassociation beam intensity combined with the large light shift (discussed below). Variations due to either local spatial inhomogeneity (e.g. interference fringes) or displacement of the cloud from the center of the Gaussian beam could account for the extra width. Assuming these inhomogeneous broadening mechanisms do not change the area of the line (verified by a simulation), we take the on-resonance photoassociation rate constant to be K0 (I) = Km (I)γ(I)/γ(I → 0). Figure 4 shows corrected and uncorrected K as a function of intensity (for various pulse lengths). The error bars are the fitting uncertainties. Once we correct for the inhomogeneous broadening we get a linear dependence on intensity with a slope of dK0 /dI = 3.5(2)(10) × 10−10 (cm3 · s−1 )/(kW · cm−2 ). The uncertainties are due, respectively, to fitting and to the combined uncertainties in the measurement of the intensity and density. For intensities above 1.2 kW·cm−2 , which we could only achieve by more tightly focusing the photoassociation laser, atomic dipole forces significantly perturb the 4 condensate, thwarting meaningful measurements. A coupled-channels, two-body scattering calculation with no adjustable parameters [14] yields a photoassociation rate constant of dK0 /dI = 4.1 × 10−10 (cm3 · s−1 )/(kW · cm−2 ) for our range of intensities. This includes a factor of 2 decrease relative to K for a non-condensed gas and agrees well with our experimental result. We study the photoassociation light shift, previously observed in a non-condensed gas [15], in a set of experiments where the total fluence (intensity × pulse length) of the pulse was kept constant, to maintain the depth of the photoassociation dip in an easily observable  kW regime. The results are shown in figure 5. The measured light shift is −164(35) MHz cm 2 , which leads at high intensity to a shift large compared to the linewidth. The principal contribution to the uncertainty is the intensity calibration. During preparation of this work we became aware of similar results in lithium [16]. While the strength of the photoassociation resonance is dominated by s-wave scattering, the dominant contribution to the light shift is due to a d-wave shape resonance. A theoretical calculation of the light shifts using equation (3.7) of reference [17], including the effect of the d-wave shape resonance embedded in the continuum, gives an average value of  kW  kW due to the hyperfine structure. −130 MHz cm 2 with a spread of ±13 MHz cm2 We now consider the upper limit to the photoassociation rate constant K [this implies a lower limit on the photoassociation time τ = (Kn)−1 ]. If one uses a semi-classical theory that is commonly applied to collisions of laser-cooled atoms [18] then K0 = σv = πRC2 P v, where RC is the Condon radius (see figure 1) and P is the probability of a photoassociative transition with a maximum value of 1. If we take the relative velocity v of the atoms to be h/(2mry ) = 0.6 mm·s−1 , where m is the atomic mass, then the maximum photoassociation rate constant is four orders of magnitude lower than our highest measured value. This reveals the inadequacy of a semi-classical approach, which fails to take into account threshold laws [10]. Quantum theories for the photoassociation rate constant can be compared by expressing K as K = (h/m)L, where L is a characteristic length. Two-body s-wave scattering theory for a BEC gives Ls = |S(k)|2 /k where h̄k is the relative collision momentum and S(k) is the S-matrix element for atom loss via photoassociation. References [17, 19] show that, on 5 resonance, Ls (I) = 4γ0 Γ(k, I)/k (γ0 + Γ(k, I))2 (3) where h̄Γ(k, I) = 2π|he|h̄Ω|ki|2 is the Fermi-golden-rule stimulated-decay width of the ex√ cited molecular state |ei due to the optical coupling h̄Ω ∝ I with the colliding atoms. Since Γ ∝ k as k → 0, and Γ/γ0 < 0.001 in our range of power and collision energy, Ls is independent of k. We calculate γ0 /2π = 18.5 MHz. Ls is linear in I for our experimental kW conditions and dLs /dI is calculated to be 24 nm/ cm 2 . This gives the above-quoted rate con- stant in good agreement with the experiment. In our power range, Ls can be significantly larger than the Condon point for the transition, 2.0 nm. Note that equation (3) shows that Ls (I) will saturate with increasing I and decrease for sufficiently large I. The upper limit to the two-body quantum K is the unitarity limit where |S|2 = 1 so Lu = 1/k = λ/2π, where λ is the de Broglie wavelength. Since λ is on the order of the BEC size Ls /Lu ≪ 1 so that our experiment is well below the unitarity constraint. Recent many-body theoretical work [3, 20, 21] has suggested an upper limit to K in a BEC of K ∼ h L , m J where LJ = n−1/3 2π and n−1/3 is the mean distance between particles. One might question if two-body scattering methods are applicable at densities where Ls becomes larger than LJ . At our maximum density LJ = 22 nm, so Ls /LJ ≈ 1 at our highest intensities. This is the regime where one might expect two-body theory to start to fail. Nevertheless the linearity of K(I) at our highest intensities (figure 4) suggests that two-body theory continues to be valid. Larger values of Ls /LJ might be accessible by a modification to our experimental design. We can use the atomic dipole force (which currently limits our ability to use high intensities) to our advantage by trapping the atoms with the photoassociation laser. Without changing the atomic dipole forces, the laser can be suddenly brought from far off molecular resonance to on molecular resonance to induce photoassociation. Difficulties due to the molecular light shift might be reduced by finding a transition with a smaller light shift. In conclusion, we have measured the single-photon photoassociation in a BEC, in good agreement with two-body theory. This agreement represents a confirmation of the factorof-two reduction for a two-body inelastic process in a BEC. The characteristic time for photoassociation is as short as 5 µs, much shorter than the 100 µs to traverse the mean distance between atoms, another demonstration of the extreme quantum nature of the col6 lisions. Our largest rate is still much smaller than the unitarity limit, but is about equal to a limit suggested on the basis of many-body effects; we do not see evidence that the rate saturates at this limit. We acknowledge funding support from the US Office of Naval Research and NASA. J.H.D. and H.H. acknowledge funding from the Alexander von Humboldt foundation. A.B. was partially supported by DGA (France). ∗ Institut für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria. † The Institute of Optics, University of Rochester, Rochester, NY 14627, USA. ‡ Naval Research Lab, Washington, DC 20375, USA. § Department of Physics, Williams College, Williamstown, MA 01267, USA. ¶ Bell Laboratories, Lucent Technologies, Holmdel, NJ 07733, USA. ∗∗ Dept. of Physics, Korea University, 5-1 Ka Anam-dong, Sungbuk-ku, Seoul 136-701, Korea. †† INFM and LENS, Università di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy. [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995). [2] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [3] M. Kos̆trun, M. Mackie, R. Côté, and J. Javanainen, Phys. Rev. A 62, 063616 (2000). [4] D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, Phys. Rev. Lett. 84, 5029 (2000). [5] E. Timmermans, P. Tommasini, R. Côté, M. Hussein, and A. Kerman, Phys. Rev. Lett. 83, 2691 (1999). [6] P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, Phys. Rev. A 58, R797 (1998). [7] K. Helmerson and L. You, Phys. Rev. Lett. 87, 170402 (2001). [8] R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, and D. J. Heinzen, Science 287, 1016 (2000). [9] J. M. Gerton, D. Strekalov, I. Prodan, and R. G. Hulet, Nature 408, 692 (2000). [10] J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod. Phys. 71, 1 (1999). [11] M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L. Rolston, and 7 W. D. Phillips, Phys. Rev. Lett. 82, 871 (1999). [12] M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 273, 84 (1996). [13] K. M. Jones, P. D. Lett, E. Tiesinga, and P. S. Julienne, Phys. Rev. A 61, 012501 (2000). [14] C. Samuelis, E. Tiesinga, T. Laue, M. Elbs, H. Knöckel, and E. Tiemann, Phys. Rev. A 63, 012710 (2001). [15] K. M. Jones, S. Maleki, L. P. Ratliff, and P. D. Lett, J. Phys. B 30, 289 (1997). [16] J. M. Gerton, B. J. Frew, and R. G. Hulet, Phys. Rev. A 64, 053410 (2001). [17] J. L. Bohn and P. S. Julienne, Phys. Rev. A 60, 414 (1999). [18] A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989). [19] J. L. Bohn and P. S. Julienne, Phys. Rev. A 54, R3647 (1996). [20] J. Javanainen and M. Mackie, cond-mat/0108349 (2001). [21] J. Javanainen, private communication. [22] Unless otherwise stated, all uncertainties reported here are one standard deviation combined systematic and statistical uncertainties. [23] Because we are in a regime where the phase shift of the light passing through the BEC is large (up to several radians) the extracted density is a multi-valued function of the phase contrast signal. We assume a Thomas-Fermi distribution for the atoms to resolve this ambiguity. 8 Figures 9 32S1/2 + 32P1/2 Energy A1Σ+u 591 nm v = 135 2 2 3 S1/2 + 3 S1/2 1 + X Σg Rc Internuclear separation R FIG. 1: The two-atom potentials for the ground state and the excited state used for photoassociation. The atoms are initially unbound and on the ground state asymptote and are excited into the J = 1, v = 135 level. From there they decay and are lost from the condensate. RC = 2.0 nm is the Condon radius, the internuclear separation where the energy of a resonant photon matches the difference between the potentials. 10 Remaining fraction f 1 0.8 0.6 0.4 0.2 0 −100 −50 0 50 Detuning (MHz) FIG. 2: A typical photoassociation loss spectrum. A 140 W·cm−2 pulse was applied for 100 µs. The dotted line is a fit to equation (2). The uncertainty in the frequency for each point is 5MHz. 11 Fractional atom loss 1 − f 1 0.8 0.6 0.4 0.2 0 0 100 200 300 400 Laser pulse length (µ s) FIG. 3: The dependence of the maximum condensate loss on the photoassociation pulse length for I = 140 W·cm−2 ; the curve is a fit of equation (2). 12 3 −1 cm . s ) −10 Photoassociation rate constant K (10 5 4 3 2 1 Corrected data Uncorrected data 0 0 0.2 0.4 0.6 0.8 1.0 1.2 Intensity ( kW . cm−2 ) FIG. 4: Photoassociation rate constant as a function of intensity. The corrected data has been adjusted to account for the inhomogeneous broadening. 13 0 Shift (MHz) −50 −100 −150 −200 0 0.2 0.4 0.6 0.8 1.0 1.2 Intensity ( kW . cm −2 ) FIG. 5: The light shift of the resonance as a function of laser intensity. 14