Photoassociation of a Quantum Degenerate Gas

Abstract

We have measured the intensity dependent rate and frequency shift of a photoassociation transition in a quantum degenerate gas of 7Li. The rate increases linearly with photoassociation laser intensity for low intensities, whereas saturation is observed at higher intensities. The measured rates and shifts agree reasonably well with theory within the estimated systematic uncertainties. Several theoretically predicted saturation mechanisms are discussed, but a theory in which saturation arises because of quantum mechanical unitarity agrees well with the data.

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arXiv:cond-mat/0307447v1 [cond-mat.soft] 17 Jul 2003
Photoassociation of a quantum degenerate gas
Ionut D. Prodan, Marin Pichler, Mark Junker, and Randall G. Hulet
Department of Physics and Astronomy and Rice Quantum Institute,
Rice University, Houston, Texas 77251
John L. Bohn
JILA, National Institute of Standards and Technology and
University of Colorado, Boulder, Colorado 80309-0440
(Dated: July 13, 2003)
Abstract
We have measured the intensity dependent rate and frequency shift of a photoassociation tran-
sition in a quantum degenerate gas of7Li. The rate increases linearly with photoassociation laser
intensity for low intensities, whereas saturation is observed at higher intensities. The measured
rates and shifts agree reasonably well with theory within the estimated systematic uncertainties.
Several theoretically predicted saturation mechanisms are discussed, but a theory in which satura-
tion arises because of quantum mechanical unitarity agrees well with the data.
PACS numbers: 03.75.Fi, 33.20.Kf, 33.70-w, 34.20.Cf
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Photoassociation (PA) of ultracold atoms has been a remarkably useful tool for deter-
mining scattering lengths characterizing ultracold atom collisions, for producing ultracold
molecules, and for providing extremely precise measurements of atomic radiative lifetimes
(see Refs. [1, 2] for reviews). This utility is largely a consequence of the spectroscopic
precision afforded by the small thermal broadening in a laser or evaporatively cooled gas.
Quantum degenerate gases are especially interesting because the coherence of the atomic field
may enable the formation of a molecular Bose-Einstein condensate (BEC) from an atomic
one by coherent Raman transitions [3, 4], stimulated Raman adiabatic passage (STIRAP)
[5], or by other coherent adiabatic population transfer schemes [6, 7].
There have been extensive theoretical studies on the rate of PA [8, 9, 10, 11, 12]. The rate
is predicted to increase linearly with intensity at low intensities, while various mechanisms
have been proposed that cause saturation of the rate at higher intensities. Among these
mechanisms are the quantum mechanical unitarity limit on the rate of atomic collisions [11],
a break-down of the 2-mode approximation [4, 13] for PA of Bose condensates caused by cou-
pling to non-condensed atomic modes [12, 14, 15, 16], and the depletion of the atomic pair
correlation function [15]. Photoassociation resonances are also predicted to exhibit a spectral
shift proportional to the light intensity caused by coupling to the continuum of free-atom
states [11, 17, 18, 19]. In contrast to theory, there are relatively few experimental measure-
ments, and only two that could be considered “precise” (which we define as measurements
with uncertainties of less than a factor of 2). We previously measured the spectral light
shift using quantum degenerate7Li and obtained good agreement with theory [20]. Both
the spectral shift and the PA rate constant were recently measured in a Na condensate, and
good agreement with two-body theory was found [21]. Saturation was not observed in this
experiment. Saturation was observed in two other lower precision experiments [22, 23], but
these experiments were performed in a magneto-optical trap, where the temperatures were
greater than 100 µK and the corresponding unitarity-limited rates were quite small, of the
order of 1 s−1or less.
We report precise measurements of both the rate of PA and the spectral shift in a quantum
degenerate gas of7Li atoms as a function of laser intensity. The rate is observed to saturate
at the highest intensities, where the corresponding rate constants are nearly two orders of
magnitude larger than any previously measured. The large PA rate is a consequence of
quantum degeneracy, where the densities are relatively high and the unitarity limited rates
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are large, as well as favorable free-bound state overlap in lithium.
The apparatus will be discussed only briefly as it has been described in detail previously
[20, 24]. Atoms are confined by a magnetic trap with a bias field of ∼1000 G. Atoms in the
F = 2, mF= 2 hyperfine sublevel of7Li are cooled by evaporation to quantum degeneracy.
Attractive interactions between atoms restrict the number of atoms in the Bose-Einstein
condensate, N0, to a relatively small number before the condensate collapses [25]. For the
experimental conditions here, N0<
∼1250, a small fraction of the total number of atoms. This
fact, however, facilitates the achievement of repeatable temperatures, total atom numbers,
and densities because quenching the gas below the BEC transition temperature Tccauses
it to attain thermal equilibrium with a temperature T ≃ Tc [26]. Therefore, following
evaporation we allow the gas to equilibrate for approximately 12 s, at which point the total
number of atoms N ≃ 8×105, the gas is very near the BEC transition at T ≃ 600 nK, and
the peak density is ∼4×1012cm3.
Following equilibration, the atoms are exposed to a pulse of photoassociation light. Up to
85 mW of PA light, coupled out of a single-mode fiber is focused on to the atoms. The laser
frequency is tuned to near resonance with the v′= 83 vibrational level of the 13Σ+
gstate,
which has a binding energy of 60 GHz relative to the 2P1/2asymptote [27]. The relative
frequency of the PA laser is monitored by comparing it with a reference laser locked to an
atomic transition using a scanning Fabry-Perot etalon. The v′= 83 level was chosen for its
large ratio of the photoassociation rate to the rate of off-resonant scattering from the atomic
resonance.
Photoassociation can be detected because it causes a reduction in atom number when
excited molecules spontaneously decay into pairs of hot atoms that escape the trap [1, 2].
The number of atoms remaining following the PA pulse is determined by polarization phase-
contrast imaging [24], and compared with a “background” measurement of the number
of atoms without the PA pulse. Since imaging destroys the sample, the trap must be
reloaded and the atoms evaporatively cooled for each image. Both N and T are extracted
by fitting the images to the distribution function for a Bose gas in an external potential. The
photoassociation rate is determined for a given laser intensity I by adjusting the laser pulse
length τ to cause a 20% - 30% loss of the initial number of atoms. For the data presented
here, τ is between 12 µs at the highest intensity and 4 ms at the lowest. Other phenomena
besides PA, such as spontaneous scattering from the atomic resonance and dipole forces, can
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cause light-induced trap losses. Dipole forces, arising from gradients of the laser field, are
minimized by using a large laser beam waist (1/e2radius of between 220 and 320 µm). The
background loss rate is measured by tuning the PA laser several GHz from the molecular
resonance, and the data is adjusted accordingly, by up to 2% at the highest intensities. Fig.
1 shows a typical photoassociation resonance curve for relatively low intensity. The curve
is well-described by a Lorentzian with a width only slightly larger than the expected value
of twice the atomic natural linewidth. Although we did not perform a systematic study of
lineshapes, the resonances broaden as expected as I is increased. The resonance position
shifts with laser intensity, as shown in Fig. 2. A linear fit to the data gives a slope of
(−1.7 ± 0.2) MHz/(W-cm−2).
The measurement of the rate is expressed in terms of the on-resonance rate coefficient
Kp, which is defined in terms of the time derivative of the density ˙ n(r,t) = −Kp(I)n2(r,t).
We neglect the effect of elastic collisions on the density distribution since τ is much less
than the characteristic time for elastic scattering of ∼300 ms. The solution for the evolution
of the density, n(r,t) = n(r,0)/{1 + Kp(I)n(r,0)t}, shows that the distribution becomes
flatter during the PA pulse since the rate is largest at the highest densities [21].The
initial density n(r,0) prior to the PA pulse is described by an equilibrium Bose-Einstein
density distribution. The fractional trap loss, for a given value of Kpand τ, is calculated
by taking the ratio between the spatial integral of n(r,τ) and the initial number of atoms.
The best value for Kpis found numerically by matching the computed fractional loss with
the measured one. Fig. 3 shows the dependence of the on-resonance rate coefficient Kp
on the laser intensity I. In the low-intensity limit Kpis found to depend linearly on the
laser intensity, with a slope of (7.9 ± 1.6) × 10−10(cm3-s−1)/(W-cm−2). The rate constant
saturates at high intensities, with a maximum value of (2.2 ± 0.2) × 10−8cm3-s−1. The
stated uncertainties are dominated by systematic effects.
Several factors contribute to the uncertainties of the measured quantities. There is no
direct way to know N before exposure to the PA pulse and therefore the ability to measure
PA-induced loss depends on the reproducibility of N and T. We find that the statistical
variations in N and T for the background measurements have standard deviations of only
3% and 2%, respectively. Furthermore, the phase-space density, proportional to NT−3, is
found to be within 5% of the critical value for all background measurements. Background
measurements are typically made every fourth shot to monitor and make adjustments for
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long-term drifts. Of more concern are systematic uncertainties. N and T are subject to
systematic uncertainties of 5% and 3%, respectively, due to uncertainties in the image mag-
nification and probe laser polarization [24]. Finally, the uncertainty in intensity is dominated
by a systematic uncertainty in power measurement, which we estimate to be ∼10%. These
effects combine to give a 20% uncertainty to the slope of Kp, and a 10% uncertainty to both
the rate constant maximum and the slope of the frequency shift.
The data are compared with the theory of Ref. [11]. The transition dipole, dm, for
the free-bound transition to this high-lying vibrational level is simply related to the atomic
dipole da. Since the binding energy of v′= 83 is large compared to the fine-structure splitting
of the 2P atomic state and any hyperfine interactions, the electronic and nuclear spins of
the atoms decouple from the PA light field. Therefore, for this Σ → Σ transition, dmlies
along the internuclear axis. Averaging over the unknown orientation of the molecular axis
relative to the polarization axis of the linearly polarized laser field gives a factor of
?
1/3, so
that dm=
?
2/3da[28]. The free-bound Franck-Condon factors are numerically evaluated
using molecular potentials determined previously [29, 30]. Finally, because of quantum
degeneracy, to calculate Kpwe average the scattering matrix element over a Bose-Einstein
energy distribution.
The light-induced spectral shift is calculated to be −2.1 MHz/(W-cm−2), in reasonably
good, but not perfect agreement with the measured value. The solid line in Fig. 3 shows
the theoretical results for Kp. The dashed line is a low-intensity extrapolation of the theory
which has a slope of 5.8×10−10(cm3-s−1)/(W-cm−2), which again is in reasonable agreement.
Finally, theory predicts saturation to Kp= 2.0 × 10−8cm3-s−1, which agrees very well with
the measured value.
The theory of Refs. [8] and [9] show that Kpgoes to a constant value, independent of
T, in the low-T and low-I limits. However, the free-bound matrix element depends on T in
this theory. We find it convenient to define the free-bound matrix element as T-independent
from the start, and use this to obtain a completely T-independent analytic approximation
for Kp. Therefore, we scale the two-atom unbound wave function φk(r) such that outside
of the interior of the ground-state molecular potential φk(r) =
sin(kr+δ)
kr
, where δ is the
s-wave scattering phase shift. The overlap between the unity-normalized molecular wave
function ψv(r) and φk(r) is Mkv = 4π?∞
of (length)3/2and reaches a constant value as T → 0. In our experiment, for example, at
0r2φk(r)ψ∗
v(r)dr. This matrix element has units
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T ≃ 600 nK and for v′= 83, the wave number k ≃ 2 × 10−4a−1
point Rc= 103 aoand the s-wave scattering length a = −27.6 ao[29], where aois the Bohr
radius. The value of |Mkv|2for these conditions differs from the T → 0 value by only 0.02%,
showing that the system is well within the quantum threshold regime. At low intensities
o, the classical outer turning
and temperatures, we find Kp= γ
I
Isat|Mkv|2, where Isatand γ are the molecular saturation
intensity and spontaneous decay rate, respectively. For this transition Isat = 6Ia, where
Ia= 5.1 mW/cm2is the atomic saturation intensity.
Saturation comes about in the theory of Ref. [11] because of the quantum mechanical
unitarity limit on the rate of two-body collisions. Accordingly, the upper bound for the rate
constant is K(u)
p
= σ?v?, where σ =
2
πλ2is the maximum cross section between identical,
non-condensed bosons, ?v? is the thermally averaged velocity, and λ is the thermal de Broglie
wavelength. For T = 600 nK, K(u)
p
= 2.5 × 10−8cm3s−1, in good agreement with the full
theoretical calculation and experiment. The saturation observed in previous experiments
[22, 23] is consistent with the unitarity limit, although at rates many orders of magnitude
lower than observed here. It is interesting to speculate on how our results relate to the
other proposed saturation mechanisms. The reverse process of dissociation can result in
the formation of pairs of “hot” atoms that are not returned to the original translational
state of the cold atomic gas [12, 14, 15, 16]. The energy width of the dissociated pairs is
proportional to the PA rate, and results in a rate limit of ∼¯ hn2/3/m [16]. Interestingly, at
Tc, where λ ∝ n−1/3, the unitarity limit gives the same maximum rate nK(u)
Since association must involve atom pairs in close proximity, depletion of the pair correlation
p
≃ ¯ hn2/3/m.
function at short range may also limit the rate. From a classical, particle-like perspective, the
maximum association rate is reached when atoms cannot move quickly enough to replenish
the necessary close-range pairs, leading to a maximum rate of ∼v/d, where d ≃ n−1/3is
the mean separation between atoms and v is their mean velocity. But since v ≃ ¯ h/mλ and
n ≃ N0/λ3, the resulting limit is ∼N−1/3
limit when T ≃ Tc. The limit imposed by this classical argument, however, has already
been violated in the BEC experiment reported in Ref. [21], in which a PA rate greatly
0
¯ hn2/3/m, which is again the same as the unitarity
in excess of v/d was observed. Although no saturation was observed in that experiment,
the rate was well below the unitarity limit, and ¯ hn2/3/m. Our experiment cannot clearly
distinguish between the mechanisms because their predicted limits are all of the same order
at Tc. However, the overall good quantitative agreement over a large range of intensity with
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a theory whose rate is limited by unitarity in the two-body scattering amplitude, strongly
supports this explanation for our data. Finally, we point out that mechanisms limiting the
PA rate will also limit the rate of molecule formation by magnetic Feshbach resonances.
In summary, we report the results of precise measurements of the rate and spectral shift
of photoassociation resonances of a quantum degenerate gas. The overall agreement between
theory, with no free parameters, and our measurement is good. The agreement attests to
the general validity of the theory and also points to quantum mechanical unitarity as the
limitation of the rate for a gas at Tc. The distinction between the unitarity-limited model
and the hot dissociation model could nevertheless be unambiguously tested with a lithium
gas cooled significantly below Tc, an experiment we hope to perform in the future.
We thank Robin Cˆ ot´ e, Juha Javanainen and Paul Julienne for helpful discussions. This
work was partially funded by grants from the National Science Foundation, the Office of
Naval Research, the NASA, and the Welch Foundation.
[1] J. Weiner, V. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod. Phys. 71, 1 (1999).
[2] W. C. Stwalley and H. Wang, J. Mol. Spectrosc. 195, 194 (1999).
[3] P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, Phys. Rev A 58, R797 (1998).
[4] D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, Phys. Rev. Lett. 84,
5029 (2000).
[5] M. Mackie, R. Kowalski, and J. Javanainen, Phys. Rev. Lett. 84, 3803 (2000).
[6] J. Javanainen and M. Mackie, Phys. Rev. A 59, R3186 (1999).
[7] F. H. Mies, E. Tiesinga, and P. S. Julienne, Phys. Rev. A 61, 022721 (2000).
[8] R. Napolitano, J. Weiner, C. J. Williams, and P. S. Julienne, Phys. Rev. Lett. 73, 1352 (1994).
[9] P. Pillet, et al., J. Phys. B 30, 2801 (1997).
[10] R. Cˆ ot´ e and A. Dalgarno, Phys. Rev. A 58, 498 (1998).
[11] J. L. Bohn and P. S. Julienne, Phys. Rev. A 60, 414 (1999).
[12] M. Koˇ strun, M. Mackie, R. Cˆ ot´ e, and J. Javanainen, Phys. Rev. A 62, 063616 (2000).
[13] E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, Phys. Rep. 315, 199 (1999).
[14] K. Goral, M. Gajda, and K. Rzazewski, Phys. Rev. Lett. 86, 1397 (2001).
[15] M. Holland, J. Park, and R. Walser, Phys. Rev. Lett. 86, 1915 (2001).
7

Page 8

[16] J. Javanainen and M. Mackie, Phys. Rev. Lett. 88, 090403 (2002).
[17] P. O. Fedichev, Y. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 77,
2913 (1996).
[18] J. L. Bohn and P. S. Julienne, Phys. Rev. A 56, 1486 (1997).
[19] J. Javanainen and M. Mackie, Phys. Rev. A 58, R789 (1998).
[20] J. M. Gerton, B. J. Frew, and R. G. Hulet, Phys. Rev. A 64, 053410 (2001).
[21] C. McKenzie, et al., Phys. Rev. Lett. 88, 120403 (2002).
[22] C. Drag, et al., IEEE J. Quantum Electron. 36, 1378 (2000).
[23] U. Schl¨ oder, C. Silber, T. Deuschle, and C. Zimmermann, Phys. Rev. Lett 66, 061403 (2002).
[24] C. A. Sackett, C. C. Bradley, M. Welling, and R. G. Hulet, Appl. Phys. B 65, 433 (1997).
[25] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997).
[26] C. A. Sackett, J. M. Gerton, M. Welling, and R. G. Hulet, Phys. Rev. Lett. 82, 876 (1999).
[27] E. R. I. Abraham, N. W. M. Ritchie, W. I. McAlexander, and R. G. Hulet, J. Chem. Phys.
103, 7773 (1995).
[28] In Ref. [20], we had erroneously assumed that dm was quantized along the magnetic field
direction, and considering laser polarization, overestimated dmby a factor of√2.
[29] E. R. I. Abraham, W. I. McAlexander, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 74,
1315 (1995).
[30] W. I. McAlexander, E. R. I. Abraham, N. W. M. Ritchie, C. J. Williams, H. T. C. Stoof, and
R. G. Hulet, Phys. Rev. A 51, R871 (1995).
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FIG. 1: Photoassociation resonance curve for v′= 83. The solid circles are experimental data
points and the solid line is a Lorentzian fit to them. The signal is the normalized number of atoms
remaining in the trap after the photoassociation pulse of intensity 0.185 W/cm2and duration 2.0
ms. The initial number of atoms is 5 × 105at a temperature of ∼520 nK. The uncertainty in
relative frequency is ∼2 MHz, caused by drift of the PA and reference laser frequencies. The data
fit to a rate constant with a Lorentzian width of 14 MHz, in good agreement with the expected
natural width of 11.7 MHz, which is twice the atomic natural width [10]. The thermal contribution
to the lineshape is negligible at this temperature.
FIG. 2: Light-induced frequency shift of the photoassociation resonance. The solid circles are data.
The error bars are uncertainty estimates for the identification of the resonance peak positions. The
dashed line is a fit to a straight line, where each data point is weighted by the inverse of the error
bar length.
FIG. 3: On-resonance photoassociation rate coefficient Kpas a function of laser intensity. The open
circles are experimental data, obtained from trap loss analysis. The solid line is the theoretical
prediction, with no adjustable parameters. The dashed line is an extrapolation of the theoretical
result in the low-intensity limit.
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-80-60 -40
Relative?Detuning?(MHz)
-200 2040 60 80
0.75
0.80
0.85
0.90
0.95
1.00
Figure?1?of??Prodan?et?al.
?
Relative?Trap?Loss
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0 10 20304050
-100
-80
-60
-40
-20
0
Figure?2?of?Prodan?et?al.
?
Frequency?Shift?(MHz)
Laser?Intensity?(W/cm
2)
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0 10 20 3040 50 607080
0
1
2
3
4
5
Figure?3?of?Prodan?et?al.
?
?
KP?(10
-8?cm
3?s
-1)
Laser?Intensity?(W/cm
2)
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