Main

Microfabricating the electromagnets used to trap ultra-cold atoms leads to a series of experimental benefits. Decreasing the radius R of a surface-mounted wire increases the maximum magnetic field gradient as R−1/2 (ref. 13). As the oscillation frequency ω of the trapped atoms increases linearly with transverse field gradient, decreasing R from centimetres to micrometres can increase the confinement frequency by orders of magnitude. In addition, one can imagine a ‘lab on a chip’, in which multiple devices are integrated on a single device, expediting applications for complex manipulation of fermionic atoms for simulations of strongly correlated systems, quantum transport experiments, collision-insensitive clocks, and precision interferometry14,15. The strong confinement provided by a microfabricated electromagnet (μEM) trap also has a practical advantage: it facilitates faster cooling, which relaxes constraints on vacuum quality and leads to a tremendous simplification over traditional DFG experiments that require multiple ovens, Zeeman slowers, or two magneto-optical traps (MOTs).

In our system16, the entire experimental cycle takes place in a single vapour cell (Fig. 1a). Counter-propagating laser beams collect, cool, and trap 2×107 40K and 109 87Rb atoms in a MOT. Atoms are transferred to a purely magnetic trap formed by external quadrupole coils and transported to the chip 5 cm away. Figure 1b shows several microscopic gold wires supported by the substrate. In the presence of uniform magnetic fields, current flowing through the central ‘Z’-shaped wire creates a magnetic field minimum above the chip. At the centre of this trap, the 40K radial (longitudinal) oscillation frequency is (ωl/2π=46.2±0.7 Hz). The corresponding 87Rb trap frequencies are a factor of smaller, where mRb and mK are the atomic masses of 87Rb and 40K, respectively.

Figure 1: A simple apparatus for Fermi degeneracy.
figure 1

a, The dual-species MOT (red sphere) is formed at the intersection of six laser beams. The cloud is then magnetically trapped using external quadrupole coils (blue), transported 5 cm vertically using an offset coil (purple), and compressed in the μEM trap. b, Schematic diagram of the central region of the μEM chip. A magnetic trap is formed 180 μm above the surface at the location marked with a white ‘X’ by applying IZ=2.0 A, IU=30 mA, BBIAS=21.4 G, and BIOFFE=5.2 G. Wire widths from left to right are 20, 60, and 420 μm.

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After loading, the 1.1-mK-deep chip trap holds approximately 2×105 40K and 2×107 87Rb doubly spin-polarized atoms, at a temperature ∼300 μK. Lower temperatures are achieved by forced evaporative cooling of 87Rb. A transverse magnetic field oscillating at radiofrequency (RF) νRF (typically swept from 30 to 3.61 MHz) selectively removes the highest energy 87Rb atoms by driving spin-flip transitions to untrapped states. The 40K atoms, with smaller Zeeman splittings, are not ejected but are sympathetically cooled2,17,18 by thermalizing with the 87Rb reservoir by means of elastic 40K–87Rb collisions6,8,10,19.

The evolution of temperature T and atom number N during sympathetic cooling is measured by releasing atoms from the trap and observing their expansion with absorptive imaging. Figure 2 shows the cooling of 40K and 87Rb to quantum degeneracy. In the degenerate regime, bosons accumulate in the ground state (forming a Bose–Einstein condensate), whereas fermions fill the lowest energy levels of the trap with near-unity occupation. Fermi degeneracy can be quantified with the fugacity : the ground state has occupation , which approaches 1 in the high- degenerate limit and in the non-degenerate limit. Owing to the tight confinement of the μEM trap, cooling increases the 40K fugacity by 1012 in only 6 s. The steep ascent of fermion fugacity in Fig. 2 also demonstrates the efficiency of sympathetic cooling. The inherent efficiency of sympathetic cooling is significant, as 40K is a rare isotope, and is therefore more difficult to collect from vapour than 87Rb. To our knowledge, this is the first observation of sympathetic cooling, of Fermi degeneracy, and of dual degeneracy in a μEM trap.

Figure 2: Sympathetic cooling in a chip trap.
figure 2

Spin-polarized fermions without a bosonic bath cannot be successfully evaporatively cooled (blue diamond). However, if bosonic 87Rb (red squares) is evaporatively cooled, then fermionic 40K is sympathetically cooled (blue dots) to quantum degeneracy (grey area). For bosonic 87Rb, the vertical axis is the occupation of the ground state; for fermionic 40K, the vertical axis is the fugacity, as discussed in the text. These two quantities are equivalent in the non-degenerate limit. A typical run-to-run spread in atom number is shown on the right-most point; all vertical error bars are smaller than the marker size.

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Below T≈1 μK, we observe two independent signatures of Fermi degeneracy. First, we compare the r.m.s. cloud size of 40K and 87Rb (or its non-condensed fraction) by fitting the density profiles to a gaussian profile. As described in the Methods section, this is an appropriate method for finding the temperature of a classical Boltzmann gas. Figure 3 shows that the apparent (that is, gaussian-estimated) 40K temperature approaches a finite value, whereas the 87Rb temperature approaches zero, even though the two gases are in good thermal contact. In fact, this deviation is evidence of the ‘Pauli pressure’ expected of a gas obeying Fermi statistics2: at zero temperature, fermions fill all available states up to the Fermi energy , where N is the number of fermions, and ħ is the reduced Planck’s constant. For our typical parameters, EF≈kB×1.1 μK. We plot data with thermal and Bose-condensed 87Rb separately, to show that the density-dependent attractive interaction between 40K and 87Rb does not significantly affect the release energy. A second signature of Fermi statistics is evident in the shape of the cloud. Figure 3, bottom inset, compares the residuals of a gaussian fit (which assumes Boltzmann statistics) with the residuals of a fit which assumes Fermi–Dirac statistics. The Fermi distribution describes the data well, with a χ2 three times lower than the gaussian fit. After all of the 87Rb atoms have been evaporated, we use Fermi–Dirac fits to measure temperature, and find kBT/EF as low as 0.09±0.05 with as many as 4×104 40K atoms.

Figure 3: Observation of Fermi statistics.
figure 3

Owing to Pauli pressure, Fermi degenerate 40K clouds seem to stop getting colder, even when the reservoir temperature approaches zero. The apparent temperature of the fermions, as measured by gaussian fits to images of 40K clouds, is plotted versus temperature of both thermal (diamonds) and Bose-condensed (circles) 87Rb. Data is compared with its classical expectation (dashed line) and with a gaussian fit of theoretically generated ideal Fermi distribution (solid line). Both temperatures are scaled by the Fermi energy EF of each 40K cloud. Error bars are statistical (one standard deviation), with uncertainty smaller than the sizes of symbols for lower temperature data. Top insets: Absorption images for kBT/EF=0.35 (left) and 0.95 (right), including a black circle indicating the Fermi energy EF. Bottom inset: At kBT/EF=0.13−0.07+0.04 with the 87Rb atoms completely evaporated, a closer look at the fermion cloud shape reveals that it does not follow a Boltzmann distribution. The fit residuals of a radially averaged cloud profile show a strong systematic deviation when assuming Boltzmann (blue circles) instead of Fermi (red diamonds) statistics. A degenerate Fermi cloud is flatter at its centre than a Boltzmann distribution, and falls more sharply to zero near its edge.

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We empirically optimize the sympathetic cooling trajectory, and find that RF sweep times faster than 6 s are not successful, whereas 87Rb alone can be cooled to degeneracy in 2 s. This indicates that 40K and 87Rb rethermalize more slowly than 87Rb with itself. Measuring the temperature ratio during sympathetic cooling (Fig. 4a) reveals that 40K lags behind 87Rb at high temperatures, despite the fact that our optimal frequency ramp starts slowly (when the atoms are hottest), and accelerates at lower temperatures.

Figure 4: Cross-species thermalization.
figure 4

a, The ratio of the temperature of 40K to the temperature of 87Rb approaches unity as the 40K temperature is lowered during sympathetic cooling. b, Measurements of σKRb (diamonds) are compared with the ‘naive’ model (dashed) and an effective-range model (solid), both described in the text. Inset: We measure cross-thermalization by abruptly reducing the temperature of 87Rb and watching the temperature of 40K relax versus time. The data shown has an asymptotic 40K temperature of 27 μK. The highest temperature point in b (open diamond) did not completely thermalize, and is discussed further in the text. For reference, the s-wave σRbRb is also shown (dotted). The error bars shown in all parts of the figure are statistical (one standard deviation), with the exception of the horizontal temperature error bars in b, which show the spread between initial and final 40K temperature.

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In the low-temperature limit, we do not expect the cross-species thermalization to lag the 87Rb–87Rb thermalization, as the 40K–87Rb cross-section σKRb=1,480±70 nm2 (ref. 20) exceeds the 87Rb–87Rb cross-section, σRbRb=689.6±0.3 nm2 (ref. 21). However, several conflicting values for σKRb have recently been presented10,19,20,22,23.

We investigate σKRb further by measuring the cross-species thermalization rate24 at several temperatures. Starting from equilibrium, we abruptly cool 87Rb by reducing νRF, wait for a variable hold time to allow cross-thermalization, and then measure the 40K temperature, as shown in the inset of Fig. 4b. We repeat this measurement at several temperatures, and fit each to the model of ref. 25. We find that the cross-section has a dramatic dependence on temperature (see Fig. 4b), decreasing over an order of magnitude between 10 and 200 μK.

The simplest model for atom–atom scattering uses a delta-function contact potential. Figure 4b shows that the s-wave scattering cross-section of this ‘naive’ model (further described in the Methods section) would predict a higher σKRb than σRbRb throughout the cooling cycle, in stark contrast to our measurements. Better agreement is given by an effective-range model26, which includes a reduction in scattering phase (and thus cross-section) below the naive expectation. Our highest temperature data point lies below the effective-range prediction, however a more-sophisticated analysis may be required to extract a quantitative measurement for this point, due to severe trap anharmonicity at high temperature. Overall, both data and theory show that the 40K–87Rb cross-section is reduced well below the 87Rb–87Rb cross-section for a large range of temperatures, explaining the requirement for a slow initial RF frequency sweep for sympathetic cooling. Below 20 μK, where no temperature lag is observed, σKRb exceeds σRbRb.

We attribute the observed reduction in scattering cross-section to the onset of the Ramsauer–Townsend effect, in which the s-wave scattering phase and cross-section approach zero for a particular value of relative energies between particles27. At higher temperatures, the scattering cross-section should increase again, however free evaporation from our trap limits our measurements to below 300 μK. Additional partial waves may also affect scattering above the p-wave threshold of 110 μK. Despite the high-temperature reduction in cross-section, 40K and 87Rb remain relatively good sympathetic cooling partners. For instance, recent measurements of 87Rb–6Li sympathetic cooling9 suggest a zero-temperature cross-section approximately 100 times smaller than σKRb, that is, a maximum cross-section roughly equal to the lowest value we measure here.

The high collision rates in mixtures trapped with a μEM allow us to cool fermions sympathetically to quantum degeneracy in 6 s, faster than previously possible. Our method is an alternative to all-optical trapping and cooling, which has been used with 6Li to achieve Fermi degeneracy in 3.5 s (ref. 28). However, magnetic traps allow cooling of fermions without direct evaporative loss, which is critical in the case of 40K because of its low isotopic abundance. In conclusion, we have achieved simultaneous quantum degeneracy of bosonic and fermionic atoms in a μEM trap and demonstrated an approach that can simplify future research with cold fermions. One prospect is the observation of Pauli blocking in light scattering off degenerate fermions29,30. The high μEM trap frequencies boost the ratio of Fermi energy EF to the recoil energy ħ2k2/2mK to ∼2.5, within the range necessary to explore such quantum optical effects.

Methods

Loading

Our experimental cycle is similar to that described in ref. 16, with several key modifications emphasized here and in the main text. Approximately 600 mW of incoherent 405-nm light desorbs 87Rb and 40K atoms from the Pyrex vacuum cell walls, boosting the MOT atom number 100-fold compared with loading from the background vapour. Potassium alone is first loaded into the MOT for 25 s, after which 87Rb is loaded for an additional 3–5 s, while maintaining the 40K population. Both MOTs operate with a detuning of −26 MHz, until the last 10 ms, when 40K is compressed with a −5 MHz detuning. After MOT loading, 3 ms of optical molasses cooling is applied to the 87Rb atoms, and the 40K atoms are optically pumped into the |F=9/2,mF=9/2〉 hyperfine ground state.

Microelectromagnet trap

7-μm-thick gold wires are patterned lithographically and electroplated on a silicon substrate. Two defects are present near the centre of the principal Z-wire, which result in the formation of three ‘dimples’ in the trapping potential. We use the magnetic gradient generated by 30 mA of current through the U-wire to centre the magnetic trap on one of these dimples.

Fitting absorption image data

ϱ Degenerate Fermi clouds are fitted using a semiclassical expression for the optical density: , where ϱ is the radial coordinate, is the peak optical density, the fugacity, and is the Fermi–Dirac function. The temperature is given by , where r is the fit width and t is the time of flight. The atom number is extracted using , where σλ is the resonant absorption cross-section. T/TF can be extracted directly from the fugacity using . Non-degenerate clouds are fitted to a gaussian distribution Aexp[−ϱ2/2r2], with the same interpretation of r. Probes along both and (see Fig. 1) were used for imaging. Comparison of temperature measurements along axes of expansion suggest a 20-nK kick (possibly magnetic) is given to clouds along , and that other temperatures agree systematically at the 5% level. Data for residuals shown in Fig. 3, bottom inset, are radially averaged about an ellipse defined by the two trap frequencies of the image plane. This one-dimensional radial data set is binned into 2-pixel bins, and fitted as described.

Scattering theories

The ‘naive’ interaction model discussed in the text gives σKRb=4πa2/(1+a2k2), where a is the s-wave scattering length and k is the relative wave vector in the centre of mass frame. Figure 4b shows the thermally averaged theory curves. Including the next-order correction in the s-wave scattering amplitude f(k)=−[1/a+i k+k2re/2+⋯]−1 requires an effective range, which we calculate using ref. 26 to be re=20.2±0.3 nm, for aKRb=−10.8±0.3 nm (ref. 20).

Analysis of thermalization data

When the 87Rb atom number NRb is much larger than the 40K atom number, the relaxation of the 40K temperature T to TRb is described by u φ φ̇=−u τ−1(1+mRbu/(mRb+mK))1/2(1+u/2)−(3/2), where u≡(T/TRb)−1, and thermalization time τ given by

in which trap frequencies are for 87Rb (ref. 25). Fitting for τ allows us to extract σKRb. Note that all thermalization data is taken with NK below 4% of NRb.

The data in Fig. 4b is analysed assuming a temperature-independent cross-section within the range of initial to final temperature. To check this assumption, we re-analyse the data using a self-consistent method that assumes an effective-range temperature dependence, and find a small upward shift of the best-fit cross-section values. Using this shift as an estimate of the methodology-dependent systematic error, we fit our four lowest temperature measurements with the effective-range model, and find aKRb=−9.9±1.4±2.2 nm, in agreement with ref. 20. The second uncertainty reported is systematic, and also includes uncertainty in the 87Rb number calibration.