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Crossover from a molecular Bose-Einstein condensate to a degenerate Fermi gas

We demonstrate a reversible conversion of a Li2 molecular Bose-Einstein condensate to a degenerate Fermi gas of atoms by adiabatically crossing a Feshbach resonance. By optical in situ imaging, we observe a smooth change of the cloud size in the crossover regime. On the Feshbach resonance, the ensemble is strongly interacting and the measured cloud size is 75(7)% of the one of a non-interacting zero-temperature Fermi gas. The high condensate fraction of more than 90% and the adiabatic crossover suggest our Fermi gas to be cold enough to form a superfluid.

We demonstrate a reversible conversion of a 6 Li2 molecular Bose-Einstein condensate to a degenerate Fermi gas of atoms by adiabatically crossing a Feshbach resonance. By optical in situ imaging, we observe a smooth change of the cloud size in the crossover regime. On the Feshbach resonance, the ensemble is strongly interacting and the measured cloud size is 75(7)% of the one of a non-interacting zero-temperature Fermi gas. The high condensate fraction of more than 90% and the adiabatic crossover suggest our Fermi gas to be cold enough to form a superfluid. Bose-Einstein condensation (BEC) of molecules formed by fermionic atoms was recently demonstrated [1,2,3,4]. The tunability of interactions in such systems provides a unique possibility to explore the Bose-Einstein condensate to Bardeen-Cooper-Schrieffer (BEC-BCS) crossover [5], an intriguing interplay between the superfluidity of bosons and Cooper pairing of fermions. While the BEC and BCS limits are both well understood, the crossover takes place in a strongly interacting regime, which represents a challenge for many-body theory.
Feshbach resonances [6] play a central role to control two-body interaction and have been used for conversion between fermionic atoms and bosonic molecules [7,8,9,10]. They are also the experimental key to investigate phenomena related to the BEC-BCS crossover. For example, it has been predicted in Ref. [11] that a pure molecular BEC can be converted into a superfluid Fermi gas by an adiabatic passage over the Feshbach resonance. Moreover, in the crossover regime where the interactions are unitarity limited, a universal behavior is expected [12,13]. Ultracold gases in that regime may provide new insights into other strongly-interacting systems such as high-T c superconductors, 3 He superfluids, and neutron stars.
A spin-mixture of 6 Li atoms in the lowest two hyperfine sub-levels is an excellent system to investigate the crossover [14,15] based on a broad Feshbach resonance at a magnetic field of B = 850G [16,17,18]. An efficient formation of ultracold molecules has been realized by three-body recombination [10,19], or by sweeping the magnetic field across the resonance [8]. The long lifetime of the molecules permits efficient evaporation [1,8,10] and facilitates slow, adiabatic changes of the system.
In this work, we explore the regime where the BEC-BCS crossover is expected by analyzing the density profiles of the trapped cloud at different magnetic fields. Our experimental setup is described in Ref. [1]. We load 2 × 10 6 precooled 6 Li atoms into a single focusedbeam dipole trap, which is generated by a 10W Yb:YAG laser operating at a wavelength of 1030 nm. We evaporatively cool the cloud by exponentially lowering the trap depth with a time constant of 460ms. The radial and axial trap frequencies are ω r /2π = 110Hz(P/mW) 1/2 and ω z /2π = (600B/kG + 0.94P/mW) 1/2 Hz, respectively, where P is the laser power. The curvature of the magnetic field that we use for Feshbach tuning results in a magnetic contribution to the axial trapping. In the low power range where the molecular BEC is formed (P < 50 mW), the axial confinement is predominantly magnetic. During the whole evaporation process the magnetic field is kept at B = 764G. At this field the molecular binding energy is ∼ k B × 2µK, where k B is Boltzmann's constant. For the scattering length of elastic molecule-molecule collisions we expect a mol = 2200 a 0 , based on the predicted relation of a mol = 0.6a [20] and an atomic scattering length of a = 3500 a 0 [17]. Here a 0 is Bohr's radius. Using radio-frequency spectroscopy which allows us to distinguish signals from atoms and molecules [7], we observe a complete atom to molecule conversion when the thermal energy of the particles is reduced to values well below the molecular binding energy.
For detection we apply in situ absorption imaging to record spatial density profiles of the trapped ensemble. To image at high magnetic fields, we illuminate the cloud for 20µs with a probe beam (intensity 0.5mW/cm 2 ) tuned to the atomic |2S 1/2 , m J = −1/2, m I = 0 → |2P 3/2 , m ′ J = −3/2, m ′ I = 0 transition. The probe beam dissociates the molecules and is used to image the resulting atom cloud [3]. Compared to the absorption imaging of unbound atoms, we found that the detection efficiency of the molecules approaches 100% at fields higher than 750G and ∼ 50% at 650G. The difference is due to the Frank-Condon wavefunction overlap, which favors fields closer to the resonance where the interatomic separation in the molecular state is larger. In our cigar-shaped trap, the radial cloud size is on the order of our imaging resolution of 10µm, while the axial cloud size of typically ∼ 100µm can be accurately measured. We therefore obtain axial density distributions from images integrated radially.
To measure the condensate fraction, we adiabatically reduce the magnetic field from 764 G to 676 G in a 200ms linear ramp after completion of the evaporation ramp. This reduces the scattering length a mol and thus in- creases the visibility of the characteristic bimodal distribution. Fig. 1(a) shows a bimodal profile observed in this way with N mol = N/2 = 4×10 5 molecules remaining at a final evaporation ramp power of 28 mW. A Gaussian fit to the thermal wings (dashed line) yields a temperature of T=430 nK, which is a factor of 7.5 below the calculated trap depth of 3.2 µK. The observed condensate fraction of ∼ 20% is consistent r ω z ) 1/3 is the mean vibration frequency, and the factor of 0.8 takes into account the ∼20% down-shift in T c due to interactions [21].
We obtain pure molecular condensates when we continue the evaporation process down to final power levels of a few mW. Fig. 1(b) shows an essentially pure condensate of N mol = 2.0 × 10 5 molecules obtained at a final power of 3.8 mW, where the trap depth is 450nK. The density profile is well fit by a Thomas-Fermi density distribution ∝ (1 − z 2 /z 2 TF ) 2 with a radius z TF = 105 µm. The corresponding peak molecular density is 1.2 × 10 13 cm −3 . In the image a thermal component is not discernable. A careful analysis of the profile provides us with a lower bound of 90% for the condensate fraction. For the chemical potential of the BEC we obtain µ = 1 2 m mol ω 2 z z 2 TF = k B × 130 nK. Here m mol = 2m is the mass of the 6 Li dimer. Based on the prediction a mol = 0.6a = 650a 0 , the calculated chemical potential of 1 2 (15 2 N molω 3 a mol √ m mol ) 2/5 = k B ×155 nK is consistent with the observed value of k B × 130 nK considering the experimental uncertainty. In particular, the particle number is calibrated to within a factor of 1.5 through fluorescence imaging [10].
The pure molecular BEC at 764 G serves as our starting point for exploring the crossover to the degenerate Fermi gas. Before we change the magnetic field, we first adiabatically increase the trap power from 3.8 mW to 35 mW in a 200-ms exponential ramp. The higher power provides a trap depth of ∼k B × 2 µK for the atoms, which is roughly a factor of two above the Fermi energy, and avoids spilling of the Fermi gas produced at magnetic fields above the resonance [1]. The compression increases the peak density of the condensate by a factor of 2.5. All further experiments reported here are performed in the recompressed trap with ω r /2π = 640 Hz and ω z /2π = (600B/kG + 32) 1/2 Hz.
We measure the lifetime of the BEC in the compressed trap at 764G to be 40s. The peak molecular density is estimated to be n mol = (15/8π)(ω r /ω z ) 2 N mol /z 3 T F = 1.0(5) × 10 13 cm −3 . This provides an upper bound for the binary loss coefficient of 1 × 10 −14 cm 3 /s, and is consistent with previous measurements in thermal molecular gases [8,10] together with the predicted scattering length scaling [20] and the factor-of-two suppression of binary collision loss in a condensate.
For exploring the crossover to a Fermi gas we apply slow magnetic field ramps. To ensure their adiabaticity we performed several test experiments. In one series of measurements we ramped up the field from 764G to 882G and back to 764G with variable ramp speed. This converts the molecular BEC into a strongly interacting Fermi gas and vice versa. Therefore substantial changes are expected in the cloud size. After the up-and-down ramp we observe an axial oscillation of the ensemble at the quadrupolar excitation frequency [1,22]. This collective oscillation is the lowest excitation mode of the system and is thus sensitive to non-adiabaticity effects. We observe axial oscillations with relative amplitudes of > 5% for ramp speeds above 1.2G/ms. For ramp speeds of 0.6G/ms and lower, the axial oscillation was no longer visible.
We also checked the reversibility of the crossover process by linearly ramping up the magnetic field from 764 G to 1176 G and down again to 764 G within 2 s (ramp speed of ±0.41 G/ms). In Fig. 2, we compare the axial profile taken after this ramp (•) with the corresponding profile obtained after 2 s at fixed magnetic field (•). The comparison does not show any significant deviation. This highlights that the conversion into a Fermi gas and its back-conversion into a molecular BEC are lossless and proceed without noticeable increase of the entropy.
To investigate the spatial profile of the trapped gas in different regimes we start with the molecular BEC at 764 G and change the magnetic field in 1-s linear ramps to final values between 740 G and 1440 G. Images are then taken at the final ramp field. To characterize the size of the trapped gas, we determine the root-mean-squared axial size z rms . This rms-size is related to the axial radius z TF by z rms = z TF / √ 7 in the case of a pure BEC in the Thomas-Fermi limit and by z rms = z TF / √ 8 in the cases of zero-temperature non-interacting or strongly interact- ing Fermi gases [23]. Fig. 3(b) shows how the measured axial size z rms changes with the magnetic field. For comparison, Fig.  3(a) displays the magnetic-field dependence of the atomic scattering length a. Up to 950 G an increase in z rms is due to the crossover from the molecular BEC to the degenerate Fermi gas. For higher magnetic fields the axial cloud size of the Fermi gas shrinks with B as the axial magnetic confinement increases (ω z ∝ √ B). For the following discussions, we normalize the observed size to the one expected for a non-interacting Fermi gas. In particular, this removes the explicit trap dependence. In Fig. 3(c) we show the normalized axial size ζ = z rms /z 0 , where z 0 = (E F /4mω 2 z ) 1/2 is the rms axial size of a non-interacting zero-temperature Fermi gas with N = 4 × 10 5 atoms. The Fermi energy E F = 2 k 2 F /2m = ω(3N ) 1/3 amounts to k B × 1.1 µK at 850 G, and the Fermi wave number k F corresponds to a length scale of k −1 F = 3600 a 0 . Below the Feshbach resonance, the observed dependence of the cloud size agrees well with the meanfield behavior of a BEC in the Thomas-Fermi limit. In this regime, the normalized size is given by ζ = 0.688(a mol /a) 1/5 (E F /E b ) 1/10 , where E b = 2 /ma 2 is the molecular binding energy. Fig. 3(c) shows the corresponding curve (solid line) calculated with a mol /a = 0.6 [20]. This BEC limit provides a reasonable approximation up to ∼800 G; here the molecular gas interaction parameter is n mol a 3 mol ≈ 0.08. Alternatively, the interaction strength can be expressed as k F a ≈ 1.9.
The crossover to the Fermi gas is observed in the vicinity of the Feshbach resonance between 800 G and 950 G; here ζ smoothly increases with the magnetic field until it levels off at 950 G, where the interaction strength is characterized by k F a ≈ −1.9. Our results suggest In (a) the atomic scattering length a is shown according to [17]; the resonance at 850 G is marked by the vertical dashed line. The data in (b) display the measured rms cloud sizes. In (c), the same data are plotted after normalization to a non-interacting Fermi gas. The solid line shows the expectation from BEC mean-field theory with a mol = 0.6 a. In (b) and (c), the error bars show the statistical error of the size measurements from typically five individual images.
that the crossover occurs within the range of −0.5 (k F a) −1 0.5, which corresponds to the stronglyinteracting regime. The smoothness of the crossover is further illustrated in Fig. 4. Here the spatial profiles near the resonance show the gradually increasing cloud size without any noticeable new features.
On resonance a universal regime is realized [12,13,14], where scattering is fully governed by unitarity and the scattering length drops out of the description. Here the normalized cloud size can be written as ζ = (1 + β) 1/4 , where β parameterizes the mean-field contribution to the chemical potential in terms of the local Fermi energy [14]. At 850 G our measured value of ζ = 0.75 ± 0.07 provides β = −0.68 +0. 13 −0.10 . Here the total error range includes all statistic and systematic uncertainties with the particle number giving the dominant contribution. Note that the uncertainty in the Feshbach resonance position is not included in the errors [18]. Our experimental results reveal a stronger interaction effect than previous measurements that yielded β = −0.26(7) at T = 0.15T F [14] and β ≈ −0.3 at T = 0.6T F [15]. Our value of β lies within the range of the theoretical predictions for a zero temperature Fermi gas: −0.67 [12,24], −0.43 [24] and, in particular, −0.56(1) from a recent quantum Monte Carlo calculation [25].
Beyond the Feshbach resonance, in the Fermi gas regime above 950 G we observe an essentially constant normalized cloud size of ζ = 0.83 ± 0.07. In this regime the interaction parameter k F a is calculated to vary be-tween −2 (at 950 G) and −0.8 (at 1440 G), which allows us to estimate ζ to vary between 0.90 and 0.95 based on the interaction energy calculations in Ref. [12]. Our observed values are somewhat below this expectation, which requires further investigation.
In summary, we have demonstrated the smooth crossover from a molecular condensate of 6 Li dimers to an atomic Fermi gas. Since the conversion is adiabatic and reversible, the temperature of the Fermi gas can be estimated from the conservation of entropy [11]. Our high condensate fraction of > 90% suggests a very small entropy which in the Fermi gas limit corresponds to an extremely low temperature of k B T < 0.04E F . In this scenario, superfluidity can be expected to extend from the molecular BEC regime into the strongly interacting Fermi gas regime above the Feshbach resonance where k F a −1. Our experiment thus opens up intriguing possibilities to study atomic Cooper pairing and superfluidity in resonant quantum gases.