Abstract
Neutral fermions present new opportunities for testing models of manybody quantum systems, realizing precision atom interferometry, producing ultracold molecules, and investigating fundamental forces. However, since they were first observed^{1}, quantum degenerate Fermi gases (DFGs) have continued to be challenging to produce, and have been realized in only a handful of laboratories^{2,3,4,5,6,7,8,9,10}. In this letter, we report the production of a DFG using a simple apparatus based on a microfabricated magnetic trap. Similar approaches applied to Bose–Einstein condensation of ^{87}Rb (refs 11,12) have accelerated evaporative cooling and eliminated the need for multiple vacuum chambers. We demonstrate sympathetic cooling for the first time in a microtrap, and cool ^{40}K to Fermi degeneracy in just six seconds—faster than has been possible in conventional magnetic traps. To understand our sympathetic cooling trajectory, we measure the temperature dependence of the ^{40} K–^{87}Rb crosssection and observe its Ramsauer–Townsend reduction.
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Main
Microfabricating the electromagnets used to trap ultracold atoms leads to a series of experimental benefits. Decreasing the radius R of a surfacemounted 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, collisioninsensitive clocks, and precision interferometry^{14,15}. 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 magnetooptical traps (MOTs).
In our system^{16}, the entire experimental cycle takes place in a single vapour cell (Fig. 1a). Counterpropagating laser beams collect, cool, and trap 2×10^{7} ^{40}K and 10^{9} ^{87}Rb 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 ^{40}K radial (longitudinal) oscillation frequency is (ω_{l}/2π=46.2±0.7 Hz). The corresponding ^{87}Rb trap frequencies are a factor of smaller, where m_{Rb} and m_{K} are the atomic masses of ^{87}Rb and ^{40}K, respectively.
After loading, the 1.1mKdeep chip trap holds approximately 2×10^{5} ^{40}K and 2×10^{7} ^{87}Rb doubly spinpolarized atoms, at a temperature ∼300 μK. Lower temperatures are achieved by forced evaporative cooling of ^{87}Rb. A transverse magnetic field oscillating at radiofrequency (RF) ν_{RF} (typically swept from 30 to 3.61 MHz) selectively removes the highest energy ^{87}Rb atoms by driving spinflip transitions to untrapped states. The ^{40}K atoms, with smaller Zeeman splittings, are not ejected but are sympathetically cooled^{2,17,18} by thermalizing with the ^{87}Rb reservoir by means of elastic ^{40}K–^{87}Rb collisions^{6,8,10,19}.
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 ^{40}K and ^{87}Rb 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 nearunity 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 nondegenerate limit. Owing to the tight confinement of the μEM trap, cooling increases the ^{40}K fugacity by 10^{12} 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 ^{40}K is a rare isotope, and is therefore more difficult to collect from vapour than ^{87}Rb. To our knowledge, this is the first observation of sympathetic cooling, of Fermi degeneracy, and of dual degeneracy in a μEM trap.
Below T≈1 μK, we observe two independent signatures of Fermi degeneracy. First, we compare the r.m.s. cloud size of ^{40}K and ^{87}Rb (or its noncondensed 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, gaussianestimated) ^{40}K temperature approaches a finite value, whereas the ^{87}Rb 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 statistics^{2}: 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, E_{F}≈k_{B}×1.1 μK. We plot data with thermal and Bosecondensed ^{87}Rb separately, to show that the densitydependent attractive interaction between ^{40}K and ^{87}Rb 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 χ^{2} three times lower than the gaussian fit. After all of the ^{87}Rb atoms have been evaporated, we use Fermi–Dirac fits to measure temperature, and find k_{B}T/E_{F} as low as 0.09±0.05 with as many as 4×10^{4} ^{40}K atoms.
We empirically optimize the sympathetic cooling trajectory, and find that RF sweep times faster than 6 s are not successful, whereas ^{87}Rb alone can be cooled to degeneracy in 2 s. This indicates that ^{40}K and ^{87}Rb rethermalize more slowly than ^{87}Rb with itself. Measuring the temperature ratio during sympathetic cooling (Fig. 4a) reveals that ^{40}K lags behind ^{87}Rb at high temperatures, despite the fact that our optimal frequency ramp starts slowly (when the atoms are hottest), and accelerates at lower temperatures.
In the lowtemperature limit, we do not expect the crossspecies thermalization to lag the ^{87}Rb–^{87}Rb thermalization, as the ^{40}K–^{87}Rb crosssection σ_{KRb}=1,480±70 nm^{2} (ref. 20) exceeds the ^{87}Rb–^{87}Rb crosssection, σ_{RbRb}=689.6±0.3 nm^{2} (ref. 21). However, several conflicting values for σ_{KRb} have recently been presented^{10,19,20,22,23}.
We investigate σ_{KRb} further by measuring the crossspecies thermalization rate^{24} at several temperatures. Starting from equilibrium, we abruptly cool ^{87}Rb by reducing ν_{RF}, wait for a variable hold time to allow crossthermalization, and then measure the ^{40}K 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 crosssection 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 deltafunction contact potential. Figure 4b shows that the swave scattering crosssection of this ‘naive’ model (further described in the Methods section) would predict a higher σ_{KRb} than σ_{RbRb} throughout the cooling cycle, in stark contrast to our measurements. Better agreement is given by an effectiverange model^{26}, which includes a reduction in scattering phase (and thus crosssection) below the naive expectation. Our highest temperature data point lies below the effectiverange prediction, however a moresophisticated 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 ^{40}K–^{87}Rb crosssection is reduced well below the ^{87}Rb–^{87}Rb crosssection 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, σ_{KRb} exceeds σ_{RbRb}.
We attribute the observed reduction in scattering crosssection to the onset of the Ramsauer–Townsend effect, in which the swave scattering phase and crosssection approach zero for a particular value of relative energies between particles^{27}. At higher temperatures, the scattering crosssection 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 pwave threshold of 110 μK. Despite the hightemperature reduction in crosssection, ^{40}K and ^{87}Rb remain relatively good sympathetic cooling partners. For instance, recent measurements of ^{87}Rb–^{6}Li sympathetic cooling^{9} suggest a zerotemperature crosssection approximately 100 times smaller than σ_{KRb}, that is, a maximum crosssection 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 alloptical trapping and cooling, which has been used with ^{6}Li 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 ^{40}K 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 fermions^{29,30}. The high μEM trap frequencies boost the ratio of Fermi energy E_{F} to the recoil energy ħ^{2}k^{2}/2m_{K} 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 405nm light desorbs ^{87}Rb and ^{40}K atoms from the Pyrex vacuum cell walls, boosting the MOT atom number 100fold compared with loading from the background vapour. Potassium alone is first loaded into the MOT for 25 s, after which ^{87}Rb is loaded for an additional 3–5 s, while maintaining the ^{40}K population. Both MOTs operate with a detuning of −26 MHz, until the last 10 ms, when ^{40}K is compressed with a −5 MHz detuning. After MOT loading, 3 ms of optical molasses cooling is applied to the ^{87}Rb atoms, and the ^{40}K atoms are optically pumped into the F=9/2,m_{F}=9/2〉 hyperfine ground state.
Microelectromagnet trap
7μmthick gold wires are patterned lithographically and electroplated on a silicon substrate. Two defects are present near the centre of the principal Zwire, 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 Uwire 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 σ_{λ} is the resonant absorption crosssection. T/T_{F} can be extracted directly from the fugacity using . Nondegenerate clouds are fitted to a gaussian distribution Aexp[−ϱ^{2}/2r^{2}], 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 20nK 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 onedimensional radial data set is binned into 2pixel bins, and fitted as described.
Scattering theories
The ‘naive’ interaction model discussed in the text gives σ_{KRb}=4πa^{2}/(1+a^{2}k^{2}), where a is the swave 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 nextorder correction in the swave scattering amplitude f(k)=−[1/a+i k+k^{2}r_{e}/2+⋯]^{−1} requires an effective range, which we calculate using ref. 26 to be r_{e}=20.2±0.3 nm, for a_{KRb}=−10.8±0.3 nm (ref. 20).
Analysis of thermalization data
When the ^{87}Rb atom number N_{Rb} is much larger than the ^{40}K atom number, the relaxation of the ^{40}K temperature T to T_{Rb} is described by u φ φ̇=−u τ^{−1}(1+m_{Rb}u/(m_{Rb}+m_{K}))^{1/2}(1+u/2)^{−(3/2)}, where u≡(T/T_{Rb})−1, and thermalization time τ given by
in which trap frequencies are for ^{87}Rb (ref. 25). Fitting for τ allows us to extract σ_{KRb}. Note that all thermalization data is taken with N_{K} below 4% of N_{Rb}.
The data in Fig. 4b is analysed assuming a temperatureindependent crosssection within the range of initial to final temperature. To check this assumption, we reanalyse the data using a selfconsistent method that assumes an effectiverange temperature dependence, and find a small upward shift of the bestfit crosssection values. Using this shift as an estimate of the methodologydependent systematic error, we fit our four lowest temperature measurements with the effectiverange model, and find a_{KRb}=−9.9±1.4±2.2 nm, in agreement with ref. 20. The second uncertainty reported is systematic, and also includes uncertainty in the ^{87}Rb number calibration.
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Acknowledgements
We would like to thank D. Jin, J. Dalibard, J. Bohm, and D. GueryOdelin for helpful conversations about scattering theory, and A. Simoni for sending us unpublished ^{40}K–^{87}Rb crosssection calculations. We also thank N. Bigelow, A. Aspect, T. Schumm, and H. Moritz for stimulating conversations, P. Bouyer and R. Nyman for providing a tapered amplifier used in this work, and J. Estève for fabricating the chip used in this work. This work is supported by the NSERC, CFI, OIT, PRO, CRC, and Research Corporation. S.A., L.J.L. and D.M. acknowledge support from NSERC. M.H.T.E. acknowledges support from OGS.
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Aubin, S., Myrskog, S., Extavour, M. et al. Rapid sympathetic cooling to Fermi degeneracy on a chip. Nature Phys 2, 384–387 (2006). https://doi.org/10.1038/nphys309
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DOI: https://doi.org/10.1038/nphys309
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