LETTERS PUBLISHED ONLINE: 20 JULY 2015 | DOI: 10.1038/NPHYS3408 Thermometry and cooling of a Bose gas to 0.02 times the condensation temperature Ryan Olf1*, Fang Fang1, G. Edward Marti1†, Andrew MacRae1 and Dan M. Stamper-Kurn1,2 Trapped quantum gases can be cooled to impressively low depends on the ability to selectively expel high-energy excitations temperatures1,2, but it is unclear whether their entropy is from the system. Similarly, thermometry of a degenerate quantum low enough to realize phenomena such as d-wave supercon- gas requires one to identify the excitations that distinguish a ductivity and magnetic ordering3. Estimated critical entropies zero-temperature from a non-zero-temperature gas. Both these = per particle for quantum magnetic ordering are 0.3kB and tasks become difficult when S N, or, equivalently, the fraction of ∼ = 14 0.03kB for bosons in three- and two-dimensional lattices, thermal excitations Nth N, is small . Time-of-flight temperature respectively∼ 4, with similar values for Néel ordering of lattice- measurements, in which the gas is released from the trap and 5 = trapped Fermi gases . Here we report reliable single-shot allowed to expand before being imaged, have required Nth N of temperature measurements of a degenerate Rb gas by imaging at least several percent, limiting such thermometry of Bose gases the momentum distribution of thermalized magnons, which are to T ≥ 0.3 Tc and of Fermi gases to T ≥ 0.05 TF , where TF is the spin excitations of the atomic gas. We record average temper- Fermi temperature. atures fifty times lower than the Bose–Einstein condensation We extend thermometry to the deeply degenerate regime of temperature, indicating an entropy per particle of 0.001kB a Bose gas by measuring the momentum distribution of a small at equilibrium, nearly two orders of magnitude lower∼ than the number of spin excitations, similar to the co-trapped impurity previous best in a dilute atomic gas2,6 and well below the critical thermometry commonly employed in Fermi gases15–18. Even in entropy for antiferromagnetic ordering of a Bose–Hubbard a highly degenerate Bose gas with a vanishingly small non- system. The magnons can reduce the temperature of the condensed fraction, the minority spin population can be made system by absorbing energy during thermalization and by dilute enough to remain non-degenerate and thereby carry a large enhancing evaporative cooling, allowing the production of entropy and energy per particle. Furthermore, the minority spins low-entropy gases in deep traps. that we use—magnons within a ferromagnetic spinor Bose–Einstein In many experiments on strongly interacting atomic-gas systems, condensate—support a higher number of thermal excitations than the low-entropy regime is reached by first preparing a weakly the majority spins because of their free-particle density of states19, interacting bulk Bose gas at the lowest possible temperature, increasing the signal of the temperature measurement. Performing and then slowly transforming the system to become strongly spin-selective measurements on the minority spins allows the gas interacting7–11. To discern whether the transformation is adiabatic temperature to be readily determined. and to determine indirectly the thermodynamic properties of the Our procedure to measure temperatures is illustrated in Fig.1 a. 87 strongly interacting system, the system is returned to the weakly Experiments begin with spin-polarized Rb in the jF D1,mF D−1i interacting regime where relations between temperature, entropy state confined in an anisotropic optical dipole trap and cooled and other properties are known. Therefore, methods to lower to just above quantum degeneracy by forced evaporation to an entropies and measure temperatures of weakly interacting gases are intermediate trap depth. The spin quantization axis is defined by important for the study of both weakly and strongly interacting a 180 mG magnetic field. We tip the gas magnetization with a brief atomic-gas systems. radiofrequency (RF) pulse, coherently transferring a small fraction In this Letter, we report cooling a Bose gas to a few per cent (up to 15%) of the atoms primarily into the jF D 1, mF D 0i state, of the condensation temperature, Tc, corresponding to an entropy creating magnons that rapidly decohere and thermalize. The gas is = −3 per particle S N ≈1×10 kB, where kB is the Boltzmann constant. then cooled further by forced evaporation to a final trap depth where Surprisingly, we achieve this low entropy using a standard technique: the temperature reaches a steady state. forced evaporation in an optical dipole trap, which we find remains Next, we release the gas from the optical trap and image the effective in a previously uncharacterized regime. The lowest tem- momentum distribution of the magnons. On extinguishing the trap peratures we report are achieved at very shallow final trap depths, light, the gas expands rapidly in the most tightly confined (vertical) as low as 20 nK, set by stabilizing the optical intensity with a long- direction, quickly reducing the outward pressure within the gas. We term fractional reproducibility better than 10−2. In addition, we then use a sequence of microwave and optical pulses to drive away demonstrate and characterize a method of cooling that lowers the atoms not in the jmF D 0i Zeeman state. Finally, we transfer the entropy without changing the trap depth, possibly allowing the low- remaining atoms to an atomic state (jF D 2, mF D1i) suitable for entropy regime to be reached or maintained in systems where the two-dimensional magnetic focusing20 so that their transverse spatial trap depth is constrained. distribution, which we image, closely reflects their initial transverse Both thermometry and cooling require a means of distinguishing velocity distribution. The majority gas is probed by imaging jmF D0i thermal excitations. For example, forced evaporative cooling12,13 atoms immediately after the application of an RF pulse. 1Department of Physics, University of California, Berkeley, California 94720, USA. 2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA. †Present address: JILA, National Institute of Standards and Technology and University of Colorado, Boulder, Colorado 80309-0440, USA. *e-mail: [email protected] 720 NATURE PHYSICS | VOL 11 | SEPTEMBER 2015 | www.nature.com/naturephysics © 2015 Macmillan Publishers Limited. All rights reserved NATURE PHYSICS DOI: 10.1038/NPHYS3408 LETTERS a Create magnons depth to thermal energy, increases at lower trap depths, and that with RF the temperature at a fixed trap depth depends on the number of Image with light atoms. In an evaporatively cooled gas, the temperature responds to µ Trap depth the effective trap depth, Ueff D U − , the potential energy depth Time tevap thold minus the chemical potential, rather than the trap depth alone, and Final trap depth/k (nK) <µ/ µ B in the regime T kB, the difference U −Ueff D is significant. b 590 ± 40 130 ± 20 25 ± 10 c 590 ± 40 130 ± 20 25 ± 10 We calculate the Bogoliubov energy spectrum of the confined quantum degenerate gas, including the effects of trap anharmonicity, and find its entropy per particle at equilibrium µ = −3 200 m (see Methods) to be S N D 1 × 10 kB, the lowest value ever reported for an atomic gas. By comparison, the 500 pK Bose gas T = 1.6 nK T = 1.6 nK = D . = /3 ∼ Integrated column density reported in ref.1 has S N 3.6 kB T Tc 1.5 kB using relations = for a non-interacting gas at T Tc ∼ 0.75. Other reported values = include S N ∼0.05kB at the centre of a resonantly interacting Fermi 6 = gas , S N D0.27kB (ref. 21) or ∼0.1kB (ref.2 ) for bosons in a lattice, T = 16.2 nK T = 16.2 nK = 22 and T Tc ∼ 0.15 in a double-well Bose–Einstein condensate . Thermometry based on imaging incoherent phonons indicated a temperature around seven times higher than reported here in a 87 23 T = 54.4 nK T = 54.4 nK Rb condensate of similar density . Having magnons present during evaporative cooling reduces the −10 −5 0510 −10 −5 0510 temperature of the trapped gas by increasing the efficacy of evapora- Wavenumber (µm−1) tive cooling. Forced evaporative cooling from a trap with an effective trap depth Ueff kBT has a cooling power proportional to the Figure 1 | Magnon thermometry. a, Magnons are created and decohere number of thermal excitations with excitation energies above Ueff. rapidly in a non-degenerate spinor Bose gas at an intermediate trap depth. In a weakly interacting single-component degenerate Bose gas, the Forced evaporative cooling to a final, variable, trap depth reduces the number of thermal excitations is determined by the temperature and temperature and the majority gas undergoes Bose–Einstein condensation. is independent of the total particle number, fixing the evaporative b,c, Images and corresponding integrated line profiles of the momentum cooling rate. By seeding the gas with additional spin excitations at distribution of the majority (b) and magnon gas (c) are each shown at three constant total particle number, the total number of thermal excita- dierent final trap depths. Line profiles are shown oset for clarity. tions, and thus the evaporative cooling power, increases. & µ/ A condensate obscures the momentum distribution of the non-condensed We observe that, for T kB, this magnon-assisted evaporation = fraction of the majority gas, especially at low temperatures. In contrast, the leads to T Tc lower than that reached by single-component magnons can have little to no condensed fraction, allowing non-condensed evaporation at the same final particle number (`×' points in Fig.2 b). µ/ magnons to be identified and the temperature determined. When T kB, we find magnons not to reduce the temperature, perhaps because of the small number of non-condensed magnons The advantage of using incoherent spin excitations to measure and the decrease in evaporative cooling efficiency in general.