Thermometry and Cooling of a Bose-Einstein Condensate to 0.02 Times the Critical Temperature

Thermometry and cooling of a Bose-Einstein condensate to 0.02 times the critical temperature

1, 1 1, 1 1, 2 Ryan Olf, ∗ Fang Fang, G. Edward Marti, † Andrew MacRae, and Dan M. Stamper-Kurn 1Department of Physics, University of California, Berkeley, California 94720, USA 2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Dated: May 21, 2015)

Ultracold gases promise access to many-body termine indirectly the thermodynamic properties of the quantum phenomena at convenient length and strongly interacting system, the system is returned to time scales. However, it is unclear whether the the weakly interacting regime where relations between entropy of these gases is low enough to real- temperature, entropy, and other properties are known. ize many phenomena relevant to condensed mat- Therefore, methods to lower entropies and measure tem- ter physics, such as quantum magnetism. Here peratures of weakly interacting gases are important for we report reliable single-shot temperature mea- the study of both weakly and strongly interacting atomic- surements of a degenerate 87Rb gas by imag- gas systems. ing the momentum distribution of thermalized In this Letter, we report cooling a Bose gas to a few magnons, which are spin excitations of the atomic percent of the condensation temperature, Tc, correspond- 3 gas. We record average temperatures as low as ing to an entropy per particle of 10− kB (kB is the Boltz- 0.022(1)stat(2)sys times the Bose-Einstein conden- mann constant), nearly two orders of magnitude lower sation temperature, indicating an entropy per than the previous best in a dilute atomic gas [2, 11]. particle, S/N 0.001 kB at equilibrium, that is Surprisingly, we achieve this low entropy using a stan- well below the≈ critical entropy for antiferromag- dard technique: forced evaporation in an optical dipole netic ordering of a Bose-Hubbard system. The trap, which we find remains effective in a previously un- magnons themselves can reduce the temperature characterized regime. The lowest temperatures we re- of the system by absorbing energy during ther- port are achieved at very shallow final trap depths, as malization and by enhancing evaporative cooling, low as 20 nK, set by stabilizing the optical intensity with 2 allowing low-entropy gases to be produced within a long-term fractional reproducibility better than 10− . deep traps. In addition, we demonstrate and characterize a method Trapped quantum gases can be brought to impres- of cooling that lowers the entropy without changing the sively low temperatures. A single-component Bose gas trap depth, possibly allowing the low entropy regime to was cooled to around 500 pK by adiabatic expansion be reached or maintained in systems where the trap depth [1], and a two-component lattice-trapped Bose gas was is constrained. cooled to 350 pK using a spin demagnetization technique Both thermometry and cooling require a means of dis- [2]. However, the entropies achieved in these systems tinguishing thermal excitations. For example, forced are still much higher than would be required to observe evaporative cooling [12, 13] depends on the ability to se- many-body quantum effects such as magnetic ordering lectively expel high-energy excitations from the system. or d-wave superconductivity of atoms in optical lattices Similarly, thermometry of a degenerate quantum gas re- [3]. Estimated critical entropies required to achieve quan- quires one to identify the excitations that distinguish a tum magnetic ordering of atoms in optical lattices include zero- from a non-zero-temperature gas. Both these tasks S/N 0.3 kB and S/N 0.03 kB for bosons in state- become difficult when S/N, or, equivalently, the fraction ∼ ∼ dependent three- and two-dimensional lattices, respec- of thermal excitations Nth/N, is small [14]. Time-of- tively [4], and similar values for Néelordering of lattice- flight temperature measurements, in which the gas is re- trapped Fermi gases [5], with lower entropy needed for leased from the trap and allowed to expand before being less strongly interacting gases. imaged, have required Nth/N of at least several percent, In many experiments on strongly interacting atomic- limiting such thermometry of Bose gases to T 0.3 T ≥ c gas systems, the low-entropy regime is reached by first and of Fermi gases to T 0.05 TF , where Tc and TF are preparing a weakly interacting bulk Bose gas at the low- the Bose-Einstein condensation≥ and Fermi temperatures, arXiv:1505.06196v1 [cond-mat.quant-gas] 22 May 2015 est possible temperature, and then slowly transforming respectively. the system to become strongly interacting [6–10]. To dis- We extend thermometry to the deeply degenerate cern whether the transformation is adiabatic and to de- regime of a Bose gas by measuring the momentum distribution of a small number of spin excitations, similar to the co-trapped impurity thermometry commonly employed in Fermi gases [15–18]. Even in a highly degen- ∗ [email protected] † Present address: JILA, National Institute of Standards and erate Bose gas with a vanishingly small non-condensed Technology and University of Colorado, Boulder, Colorado fraction, the minority spin population can be made di- 80309-0440, USA; [email protected] lute enough to remain non-degenerate and thereby carry 2

a Create magnons age the momentum distribution of the magnons. Upon with RF extinguishing the trap light, the gas expands rapidly in Image with light the most tightly conﬁned (vertical) direction, quickly re- Trap depth Trap Time tevap thold ducing the outward pressure within the gas. We then Final trap depth/k (nK) use a sequence of microwave and optical pulses to drive B away atoms not in the m =0 Zeeman state. Finally, 590 ± 40 130 ± 20 35 ± 10 590 ± 40 130 ± 20 25 ± 10 | F i b c we transfer the remaining atoms to an atomic state ( F =2, m =1 ) suitable to two-dimensional magnetic fo- | F i 200 µm cusing [20] so that their transverse spatial distribution, which we image, closely reﬂects their initial transverse ve-

T=1.6 nK locity distribution. The majority gas is probed by imaging mF =0 atoms immediately after the application of an RF| pulse.i The advantage of using incoherent spin excitations to T=16.2 nK measure temperature is exhibited in Fig. 1, which com- pares the momentum distribution of the majority gas (b) to that of the magnons (c). At T . 0.3 Tc, the non- T=54.4 nK condensed fraction of the majority gas is obscured by the

Integrated column density column Integrated condensed fraction, the distribution of which is broad- -10 -5 0 5 10 -10 -5 0 5 10 Wavenumber (µm-1) ened owing to interactions and imaging resolution. In contrast, the temperature can be easily extracted from FIG. 1. Magnon thermometry. (a) Magnons are created the momentum distribution of the thermalized minority- and decohere rapidly in a non-degenerate spinor Bose gas at spin gas. Temperatures extracted from the magnons an intermediate trap depth. Forced evaporative cooling to a [Fig. 2(a)] agree with those from the majority-spin dis- final, variable, trap depth reduces the temperature and the tribution under conditions where both can be measured. majority gas undergoes Bose-Einstein condensation. Images At lower temperatures, the magnon thermometer allows and corresponding integrated line profiles of the momentum measurements in a hitherto unmeasurable regime. distribution of the majority (b) and magnon gas (c) are each Our measurements reveal that forced evaporative cool- shown at three different final trap depths. Line profiles are ing produces Bose gases with extremely low temper- shown offset for clarity. A condensate obscures the momen- ature and entropy. At the lowest trap-depth setting tum distribution of the non-condensed fraction of the majority gas, especially at low temperatures. In contrast, the magnons (dashed circles, Fig. 2), we measure an average gas tem- can have have little to no condensed fraction, allowing non- perature of T = 1.04(3)stat(7)sys nK, corresponding to condensed magnons to be identified and the temperature de- T/Tc = 0.022(1)(2) where Tc is calculated using the mea- termined. sured atom number, N = 8.1 105, and optical trap fre- × quencies. For this gas, kBT/µ 0.07, with µ the chemical potential, meaning that thermal≈ excitations in the a large entropy and energy per particle. Furthermore, the high density region of the condensed gas (in the major- minority spins that we use—magnons within a ferromag- ity spin state) are predominantly phonons. Measurement netic spinor Bose-Einstein condensate—support a higher procedures, error estimates, and calculations of Tc and µ Nth than the majority spins because of their free-particle are detailed in Methods. density of states [19], increasing the signal of the temper- We observe that evaporative cooling is highly efficient ature measurement. Performing spin-selective measure- with respect to particle loss in the regime µ/kB < T < Tc ments on the minority spins allows the gas temperature [Fig. 2(b)]. When T < µ/kB, the cooling efficiency re- to be readily determined. duces, consistent with the fact that the chemical poten- Our procedure to measure temperatures is illustrated tial accounts for a non-negligible amount of the energy in Fig. 1(a). Experiments begin with spin-polarized 87Rb carried away by each atom lost to evaporation. The rele- in the F =1, m = 1 state confined in an anisotropic vance of the chemical potential in evaporation is also indi- | F − i optical dipole trap and cooled to just above quantum cated in Fig. 2(a) by the observation that η U/(kBT ), degeneracy by forced evaporation to an intermediate trap the ratio of trap depth to thermal energy,≡ increases at depth. The spin quantization axis is defined by a 180 mG lower trap depths, and that the temperature at a fixed bias magnetic field. We tip the gas magnetization with a trap depth depends on the number of atoms. In an evap- brief radio-frequency (RF) pulse, coherently transferring oratively cooled gas, the temperature responds to the a small fraction (up to 15%) of the atoms primarily into effective trap depth, Ueff = U µ, the potential energy the F =1, m =0 state, creating magnons which rapidly depth minus the chemical potential,− rather than the trap | F i decohere and thermalize. The gas is then cooled further depth alone, and in the regime T < µ/kB, the difference by forced evaporation to a final trap depth where the U U = µ is significant. − eff temperature reaches a steady state. We calculate the Bogoliubov energy spectrum of the Next, we release the gas from the optical trap and im- confined quantum degenerate gas, including the effects 3

a 3 7 10 η b 10 1 14 20 30 45 0.8 2 10 T c 0.6 5 Number of impurities/10 c

0 1 2 T/T co µ/kB nstant 80 T 1 0.4 10 Temperature (nK) 70 µ/kBTc 0.2

Temperature (nK) Temperature 60 0 10 0 1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 10 10 10 10 Number of atoms/106 Trap depth/kB (nK)

FIG. 2. Thermometry results. Two runs (blue circles, green squares) differ in initial atom number. (a) T and (b) T/Tc are measured at various optical trap depths. Lower T and T/Tc are achieved in runs with larger initial atom number. In (a), thermometry using the majority spin population (light gray diamonds) agrees with thermalized magnon thermometry extrapolated to the zero-magnon values (circles and squares). Error bars show statistical uncertainty of extrapolated temperature (vertical) and systematic uncertainty in trap depth (horizontal). Thin diagonal gray lines show contours of η, the ratio of trapping potential depth to temperature. Thick gray lines show calculated Tc and µ/kB in (a) and µ/kB Tc, calculated using the frequencies of the kB 60 nK deep trap, in (b). (b) Efficiency of evaporation only (circles, squares), with error bars showing statistical × uncertainty of the extrapolated T/Tc, is compared to magnon-assisted evaporation (“x”). Solid lines connecting points are guides to the eye, with a steeper slope indicating more efficient evaporation. Dotted lines connect corresponding points at the same trap depth. Inset: the magnon-free temperature is determined by extrapolating single-shot temperatures (“x”) vs. total number of magnons imaged, here at trap depth of 800 nK, to the zero-magnon limit (circle). A similar extrapolation determines the value of T/Tc achieved by evaporation only.

of trap anharmonicity, and find its entropy at equilib- We observe that, for T & µ/kB, this magnon-assisted 3 rium (see Methods) to be S/N = 1 10− kB, the low- evaporation leads to T/Tc lower than that reached by est value ever reported for an atomic× gas. By com- single-component evaporation at the same final particle parison, the 500 pK Bose gas reported in Ref. [1] has number (“x” points in Fig. 2(b)). When T . µ/kB, we 3 S/N = 3.6 kB(T/Tc) 1.5 kB using relations for a non- do not find magnons to reduce the temperature, perhaps interacting gas at T/T∼ 0.75. Other reported values because of the small number of non-condensed magnons c ∼ include S/N 0.05 kB at the center of a resonantly in- and the decrease in evaporative cooling efficiency in gen- teracting Fermi∼ gas [11], S/N = 0.27 k [21] or 0.1 k [2] eral. Magnon-assisted evaporation is also evidenced by B ∼ B for bosons in a lattice, and T/Tc 0.15 in a double- the fact that the per-atom loss rate is higher for the well Bose-Einstein condensate [22].∼ Thermometry based magnon population than for the spin-majority popula- on imaging incoherent phonons indicated a temperature tion: For both populations, the atom loss reflects the around 7 times higher than reported here in a 87Rb con- selective evaporative loss of thermal excitations. For the densate of similar density [23]. spin-majority atoms, thermal excitations are just a small fraction of the total population, whereas they represent Having magnons present during evaporative cooling re- a much higher fraction of the magnons. duces the temperature of the trapped gas by increasing the efficacy of evaporative cooling. Forced evapo- Unbiased thermometry of the majority gas is per- rative cooling from a trap with an effective trap depth formed by extrapolating measurements with various numbers of magnons to the zero-magnon limit (Fig. 2(a) Ueff kBT has a cooling power proportional to the number of thermal excitations with excitation energies inset). In addition, by examining evaporative cooling in a state-dependent optical trap, we have confirmed that the above Ueff . In a weakly interacting single-component degenerate Bose gas, the number of thermal excitations is temperature indicated by the magnons varies with the determined by the temperature and is independent of the trap depth of the spin-majority gas, even for a fixed trap depth of atoms in the m =0 state. This observation total particle number, fixing the evaporative cooling rate. | F i By seeding the gas with additional spin excitations at gives further confidence that the magnon thermometer constant total particle number, the total number of ther- accurately records the temperature of the majority gas. mal excitations, and thus the evaporative cooling power, The same characteristics that make dilute magnons a increases. good thermometer—high relative energy and entropy per 4

a cycled cooling c 1 Make magnons Purge or image

Trap depth Trap 0.8 Time tevap thold b 120 Trap depth/k 0.6

: 1250 nK c B k T/µ 100 kBBT/µ 0 1 2 3 T/T c T onst T a 80 B nt T B 0.4 k 0 60 800 nK

600 nK −1.5 µ/kBTc 40 0.2 Temperature (nK) 200 nK −3

20 E per magnon/ ∆ ∆ E per magnon / k 0 0 0 2 4 6 8 10 0.5 1 1.5 2 2.5 6 Number of non−condensed magnons removed/105 Number of atoms/10

FIG. 3. Cycled decoherence cooling. (a) Magnons are created at the final trap depth and cool the gas as they thermalize. Thermalized magnons can be removed from the trap and the cooling process repeated before measuring the temperature by imaging the momentum distribution of the thermalized magnons. (b) Each non-condensed magnon removed from the trap takes energy from the gas. Decoherence cooling trajectories are plotted versus cumulative number of magnons removed. Closed circles show temperatures after a single cycle of decoherence, but with varying numbers of magnons. Some additional magnon- assisted evaporative cooling may also be present. Open triangles show repeated cycles of magnon creation, thermalization, and purge, with representative error bars on first and last triangles giving statistical error over several repetitions. Solid lines show zero-free-parameter theory predictions assuming each non-condensed magnon removes 3 kB T energy. Solid patches indicate the range of predictions included within the uncertainty in the cumulative number of magnons removed (we image 75% 10% ± of the magnons present). Inset: the amount of energy removed by each magnon, in units of kB T , is plotted versus the ratio of µ/kB T , where each point is based on an estimate of the slope over 4 consecutive triangle points. We observe that the net energy carried away by impurities vanishes when T . µ/kB . (c) Efficiency of evaporation only (black circles) is compared to cycled decoherence cooling (open triangles), with error bars representing statistical uncertainty. The solid line between circles is a guide to the eye. Dotted lines connect corresponding points at the same trap depth. Compared to evaporation, decoherence cooling can reach considerably lower T/Tc at the same trap depth in the regime T & µ/kB T and T/Tc . 0.9. particle—enable them to cool without lowering the trap tions. depth, beyond what can be achieved by magnon-assisted To demonstrate the capabilities of decoherence cooling, evaporation. Spin excitations created in a degenerate we introduce a variable number magnons at the final trap Bose gas decrease its temperature by a process known as depth and allow them to evolve toward thermal equilib- decoherence [24] or demagnetization [25, 26] cooling. Im- rium with the assistance of a weak magnetic field gradient mediately after the RF pulse is applied, the magnon pop- of 0.2–2 mG/cm (for more on magnon thermalization, ulation has the same energy and momentum distribution see Supplemental Discussion). Then, as illustrated in as the initially polarized degenerate Bose gas, with a large Fig. 3(a), we either measure the temperature or expel the condensed fraction, which carries no energy above the magnons using a spin-selective process, leaving a colder chemical potential, and a small normal fraction, which spin-polarized condensed gas. The decoherence cooling carries on the order of kBT of energy per particle. Upon process can then be repeated to further cool the sample, thermalization, the normal fraction within the magnon with the number of magnons, magnetic field gradient, gas increases, bounded from above by its critical num- and thermalization time optimized at each repetition. A ber for condensation, and the energy and entropy of the single pass of decoherence cooling, using pseudo-spins in magnons increases. The energy gained by the magnons a degenerate 87Rb gas, has previously been shown to is energy lost by the majority gas, as the entire process, slightly reduce the temperature, but not T/Tc, of the apart from the slight effects of magnetic field inhomo- resulting mixture [24]. geneities and dipolar interactions, occurs at constant en- Whereas the thermal energy k T achieved by forced ergy [27]. B evaporative cooling of a single-component gas is limited While the overall entropy rises during thermaliza- to around 1/η 1/10 of the trapping potential depth, the tion, the entropy per particle of the majority gas may temperatures reached∼ by cycled decoherence cooling are decrease. Considering a non-interacting harmonically far lower. For example, we produced a 20 nK gas within a trapped gas, we calculate that decoherence cooling would k 600 nK deep potential, corresponding to 1/η = 1/30, B × reduce T/Tc of the majority gas only when T/Tc < and reduced T/Tc substantially, from 0.58 to 0.37 [cyan 3 (3ζ(3))/(4ζ(4)) = 0.94, consistent with our observa- triangles, Fig. 3(b) and (c)]. On the other hand, de- p 5 coherence cooling is generally less efficient in terms of spin excitations involving a different hyperfine state can atom loss. An atom lost through evaporative cooling re- deposit energy into the condensate due to a mismatch of moves an energy ηk T , whereas a magnon brought to s-wave scattering lengths. ∼ B equilibrium and forcibly ejected only removes 3 kBT for In this work, we have demonstrated the use of spin ex- a non-interacting gas, consistent with our observations citations within a highly degenerate Bose gas to measure when kBT µ. At lower temperatures, we expect each and reduce the temperature. With our improved ther- 3 impurity to remove less energy, 1 to 1.5 kBT , as interac- mometry, we observe a record low entropy of 1 10− kB tions and quantization of the trap modes become more per particle for a spin-polarized Bose-Einstein× conden- important. Even so, cooling would be expected, in the sate, illustrating the power of standard cooling tech- ideal case, to become increasingly effective as the heat niques to reach the long-sought low entropy regime. capacity drops. In contrast, we find decoherence cool- Magnon-assisted evaporation and cycled decoherence ing to be ineffective at reducing both T and T/Tc in the cooling should work for other spin or pseudospin mixtures low temperature regime kBT . µ (Fig. 3(b, c), inset and of Bose gases so long as the gas thermalizes at constant red triangles), perhaps owing to energy deposited in the minority particle number, providing a flexible platform trapped gas during the magnon expulsion and by heating for creating or maintaining low entropies in deep traps. processes, which become more significant as thermalization times increase. In principle, the process of creating the spin mix- ACKNOWLEDGMENTS ture employed in decoherence cooling can heat the gas by changing its interaction energy. However, neglecting We thank Holger Kadau and Eric Copenhaver for as- small magnetic dipolar interactions, the magnon is a gap- sistance improving the experimental apparatus. We ac- less Goldstone excitation corresponding to the rotational knowledge the primary research support from NASA and symmetry of real spins [19]. Accordingly, the s-wave scat- the AFOSR through the MURI program, and also sec- tering lengths that characterize the interactions between ondary support for personnel through the NSF. G.E.M. two majority-spin atoms, a 1, 1, and between a majority acknowledges support from the Fannie and John Hertz − − and minority-spin atom, a 1,0, are identical. In contrast, Foundation. −

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[21] Bakr, W. S. et al. Orbital excitation blockade and algo- force on all Zeeman sublevels of F =1 87Rb . To ver- rithmic cooling in quantum gases. Nature 480, 500–503 ify that our temperatures vary with the trap depth of (2011). the majority gas, rather than that of the minority spins, [22] Gati, R. et al. A primary noise thermometer for ultracold we also performed experiments with circularly polarized Bose gases. New J. Phys. 8, 189–189 (2006). trap light, which applies a differential force to atoms in [23] Schley, R. et al. Planck Distribution of Phonons in a different Zeeman sublevels. Using this circularly polar- Bose-Einstein Condensate. Phys. Rev. Lett. 111, 055301 (2013). ized trap, we vary the trap depth of the majority gas [24] Lewandowski, H. J., McGuirk, J. M., Harber, D. M. & without affecting that of the mF =0 atoms by preparing the majority gas in either of| the mi =+1 or m = 1 Cornell, E. A. Decoherence-Driven Cooling of a Degener- | F i | F − i ate Spinor Bose Gas. Phys. Rev. Lett. 91, 240404 (2003). states. [25] Fattori, M. et al. Demagnetization cooling of a gas. Nat. The lifetime and heating rate of the atoms in the op- Phys. 2, 765–768 (2006). tical trap are consistent with spontaneous scattering of [26] Medley, P., Weld, D. M., Miyake, H., Pritchard, D. E. & the far-detuned trapping light. Ketterle, W. Spin Gradient Demagnetization Cooling of Ultracold Atoms. Phys. Rev. Lett. 106, 195301 (2011). [27] In Fermi gas systems, a similar process leads to heating as the energy and entropy per particle of the minority spins B. State purification and preparation is less than that of the majority gas at equilibrium. About one millisecond after extinguishing the optical trapping light, we initiate the imaging sequence by AUTHOR CONTRIBUTIONS purging atoms in the F =1, mF = 1 state—the majority component of our| gas in most− ofi our experiments— All authors provided experimental support and com- using a series of alternating microwave pulses (resonant mented on the manuscript. Experimental data were ac- with the F =1, mF = 1 F =2, mF =0 transition) | − i → | i quired by R.O. and F.F. and analyzed by R.O. G.E.M. and light pulses (resonant with the F =2 hyperfine man- conceived and performed preliminary experiments with ifold). cycled decoherence cooling. The manuscript was pre- The number and strength of the optical/microwave pared by R.O. and D.M.S-K. R.O. performed the calcu- pulses is optimized to reliably expel all of the lations of entropy per particle. D.M.S-K. supervised all F =1, m = 1 atoms with minimal impact on the num- | F − i work. ber of atoms in the F =1, mF =0 state. Regardless, the purge process removed| betweeni 10% and 40% of the F =1, m =0 atoms, depending on the number of | F i COMPETING FINANCIAL INTERESTS F =1, mF = 1 atoms being purged and other experimental| parameters,− i but it does not visibly impact the The authors declare no competing financial interests. momentum of the atoms that remain. We estimate the number of magnons from the images by accounting for this loss, allowing considerable uncertainty. METHODS Finally, we employ a calibrated microwave sweep to transfer atoms in the F =1, m =0 state to the mag- | F i A. Optical trapping netically trappable F =2, mF =1 state with greater than 95% efficiency. | i Experiments were performed on 87Rb gases in a single- beam optical dipole trap [M1]. Evaporative cooling was realized by gradually lowering the depth of the opti- C. Magnetic focusing cal trap. The optical trap was formed by light with a wavelength of 1064 nm, brought to a cylindrical fo- The magnetic focusing lens takes the form of a mag- cus at the location of the atoms. The optical trap was netic potential with negligible curvature along the verti- highly anisotropic, with trapping frequencies having a cal (imaging) axis and weak, harmonic curvature in the typical ratio 1:15:140, with the tightest confinement in image plane. The resulting magnetic trap causes atoms the vertical direction. At low optical powers, the optical with momenta ~p in the image plane to converge on the dipole trap is influenced by gravity so that the ratio of real-space point ~xp = ~p/mω after a time tfoc = 2π/4ω, trap frequencies differs somewhat. Trap frequencies were where m is the atomic mass and ω the trap frequency measured empirically at several optical powers, extrapo- for atoms in the F =2, mF =1 state. The vertical gra- lated between measurements, and confirmed by compar- dient is selected to| cancel thei effect of gravity, and the ing with a simple model of the trapping potential that trap frequency ω is chosen such that the thermal atomic accounts for the Gaussian focus and force of gravity. momentum-space cloud is resolvable by our imaging sys- The bulk of our experiments were performed with lin- tem at the lowest accessible temperatures while still al- early polarized trap light, which imparts equal dipole lowing us to image temperatures near Tc. 7

Uncertainty in the parameter ω is the largest source confined in-plane direction can be manifest. Also, the of systematic error in the temperatures reported in this tilt of the trap in the tight direction is more difficult work. Our measurements found ω = 2π 2.88 0.09 rad/s to calibrate, leading to a (fictitious) systematic upward × ± and the relation ~xp = ~p/mω was verified by comparing shift in the apparent Tt by the mechanism explained in the amplitude and phase of center-of-mass oscillations of Sec. C. Finally, at the lowest temperatures reported in our gas imaged both in situ and after application of the this work, the semiclassical condition kBT ~ω does magnetic focusing lens. not hold along the tight axis as it does along the weak Time-of-flight expansion of the gas in the unconfined axis (~ ωw, ωt /kB 0.05, 1 nK). Thus, in this work, { } ∼ { } vertical direction along with misalignment of the trap, we estimate temperatures by T = Tw alone. imaging, and focusing axes can lead to several types of All fits exclude regions of the column density that in- aberration. For example, any projection of the vertical clude condensed atoms, and each fit was performed many extent of the gas into the imaging plane will manifest times, varying both the size of the excluded central region as additional apparent momenta in the imaged column and the order j [0.5, 2] of the Bose function in order density. Such aberrations can lead to an overestimate of to look for systematic∈ shifts in the temperature. The the temperature of the gas and are most pertinent at low high-momentum tails of the momentum distribution np temperatures. Trap, imaging, and focusing axes are all are insensitive to the particular Bose function employed z z z aligned to gravity in our system, and can be verified by (gj(e ) e as e 0). We used only fits where the imaging highly degenerate gasses, which expand negligi- variation→ of the temperature→ with respect to the size of bly in the direction of weakest optical confinement. the exclusion region and the order of the Bose function was negligible compared to other sources of systematic error. D. The imaging system The uncertainty with which individual fits estimate the temperature is consistent with fundamental sources of After magnetic focusing, the atoms are imaged with a noise, photon and atom shot noise, and the short-time 40 to 100 µs pulse of light resonant with the F =2 F =3 (same day) shot-to-shot variation of estimated tempera- cycling transition. The magnification of our imaging→ sys- tures was consistent with the single-shot noise-limited er- tem was calibrated using an optical micrometer target ror estimated by the fitting routine. This justifies the use that was placed in a plane equivalent to the plane of the of an uncertainty-weighted average of many independent atoms, with respect to the imaging system. This cali- measurements to compute the precise low temperatures bration was repeated several times with fractional uncer- reported in this work. tainty of 1% and is included in the systematic error of our thermometer. The resolution of our imaging system, approximately F. Calculation of the entropy per particle 8 µm, corresponds to a temperature of about 0.2 nK. We do not correct extracted temperatures for finite imaging Calculations of the entropy per particle cannot rely resolution. on the local density approximation because the thermal phonon wavelength is far larger than the transverse extent of the condensed gas. We can, however, apply a E. Details of temperature fitting procedure one-dimensional local density approximation along the long axis of the condensate.

Atom column momentum densities np are fit to a Bose- To calculate the entropy, we numerically solve for the z enhanced momentum distribution np = b+Agj(e ), with Gross-Pitaevskii ground state Ψ and Bogoliubov spec- g the Bose function (polylogarithm) of order j and the trum i of small amplitude excitations in a weakly- j { } argument interacting 87Rb Bose gas with chemical potential µ in a trap with confinement in 2 dimensions, including the 2 2 z = α (pw pw0) /kBTw (pt pt0) /kBTt. effects of gravity and anharmonicity [M2], and no con- − − − − finement in the 3rd dimension. To do this calculation, For non-degenerate gasses, α = µ/kBT is a free parame- we discretize the 2-dimensional Gross-Pitaevskii equa- ter of the fit, along with the zero of momentum (pw0, pt0), tion and Bogoliubov equations on a Lagrange mesh based the background level b, peak level A, and temperatures on the Hermite polynomials [M3, M4]. The Bogoliubov 2 2 Tw and Tt along the weak and tight in-plane axes of the equations include the substitution µ µ ~ k /2m, optical trap, respectively. For degenerate gasses, α = 0. for relevant values of the wave vector →k, to− account for Generally, Tw = Tt within the error of the fit, however excitations in the unconfined longitudinal direction. We at low temperatures, several effects cause Tt be an un- verify that our calculation of the excitations i are ac- reliable estimate of the gas temperature. Although the curate by comparing to analytic results in regimes{ } where condensate expands primarily along the unfocused verti- they are available, for example in the harmonic non- cal direction, at the very low temperatures reported in interacting limit and in the Thomas-Fermi limit with this work, condensate expansion along the more tightly cylindrical symmetry. 8

The entropy and number of atoms per unit length can G. Calculating Tc and µ then be calculated using the equilibrium relations

The estimates of Tc and µ reported use standard for- mulas for Bose gases in the non-interacting and Thomas- S (i µ) /kBT (µ )/k T = − log 1 e − i B Fermi limits, respectively [M5]. The eﬀects of trap an- L e(i µ)/kB T 1 − − i − harmonicity and ﬁnite interactions on both µ and T were X − c N 1 considered and are on the order of the systematic uncer- = Nc + , L e(i µ)/kB T 1 tainty in our temperature measurements. i − X −

H. Theoretical limits for decoherence cooling with Nc = Ψ∗Ψ. By ﬁxing T and varying µ (Ψ and i are recalculated for each value of µ), we can consider{ the} entropy and number per unit length as functions of µ: Considering an ideal Bose gas in a harmonic trap with S N geometric-mean frequency ω, the following relations for L (µ) and L (µ). the saturated thermal atom number Nth, energy E, and We use the local density approximation and Thomas- heat capacity C, at the temperature T , apply: Fermi approximation along the long axis of our trap (with trap frequency ωz) to write the total entropy of the gas k T 3 as N = ζ(3) B (M1a) th ω ~ 4 (kBT ) 3ζ(4) ∞ S 1 2 2 E = 3ζ(4) = kBTNth (M1b) S = µmax ω z dz, ( ω)3 ζ(3) L − 2 z ~ Z−∞ dE ζ(4) C = = 12k N . (M1c) dT B ζ(3) th and the total number of atoms as Suppose we transfer dN > 0 atoms to the minority ∞ N 1 spin state (where dN is far below the critical number for N = µ ω2z2 dz, L max − 2 z magnon condensation), let them thermalize, and eject Z−∞ them. To lowest order, the majority gas loses energy dE = 3k T dN, and thus the temperature changes by − B where µmax is chosen to yield the experimentally mea- dT = dE/C = 3(kBT )/C dN. The non-condensed sured value of N. The entropy per particle of the gas is fraction changes as,− using Eqs. M1, then S/N. N 1 N At the lowest trap depths, the rate at which the d th = dN + th dN (M2a) N N th N magnons are lost by forced evaporation becomes com- parable to the rate at which non-condensed magnons are dN 3ζ(3) N = + th . (M2b) generated through thermalization. We speculate that the N −4ζ(4) N number of thermal excitations in the spin-majority gas under these conditions is similarly lower than at the ob- Thus, we must have Nth/N < (3ζ(3)) / (4ζ(4)) = 0.83 to served temperature in equilibrium. As such, our estimate reduce the non-condensed fraction through such cooling. 3 of S/N = 1 10− kB may be an underestimate of the This corresponds to, for the case under consideration, × 1/3 true steady-state entropy. T/Tc < (0.83) = 0.94.

[M1] Marti, G. E. et al. Coherent Magnon Optics in a Ferro- [M3] Baye, D. & Heenen, P.-H. Generalised meshes for quan- magnetic Spinor Bose-Einstein Condensate. Phys. Rev. tum mechanical problems. J. Phys. A. Math. Gen. 19, Lett. 113, 155302 (2014). 2041–2059 (1986). [M2] Neglecting gravity and approximating the trap as har- [M4] McPeake, D. Superﬂuidity, collective excitations and monic results in underestimating the entropy per parti- nonlinear dynamics of Bose-Einstein condensates. The- cle by a factor of 1.6. sis, Queen’s University, Belfast (2002). [M5] Pitaevskii, L. P. & Stringari, S. Bose-Einstein Conden- sation (Oxford University Press, 2003). Supplementary material: Thermometry and cooling of a Bose-Einstein condensate to 0.02 times the critical temperature

DISCUSSION found magnetic field inhomogeneity to facilitate decoherence and thermalization of spin populations, typically Magnon thermalization on timescales of several 10s–100s of milliseconds [S1–S3]. We have previously observed coherence between magnons We found that applying a magnetic field gradient was in different momentum states out to several hundred necessary to achieve thermalization between initially co- milliseconds [S4] with a negligible gradient. However, herent, quantum degenerate spin components, but can none of these experiments claim to have operated in the be chosen to be small enough to impart negligible energy. highly-degenerate phonon-dominated regime kBT < µ. A gradient of 0.2 mG/cm, typical in our experiments on With very small gradients, we find the zero-momentum decoherence cooling, corresponds to an energy of approx- magnons created in this work can remain coherent for many seconds. In strongly-interacting Fermi gases, de- imately h 10 Hz = kB 0.5 nK across the longest dimension of our× gas and was× sufficient to produce a saturated coherence of spin excitations has been well characterized and is limited by a fundamental lower-bound on the spin non-condensed fraction of the mF =0 state in 100–2000 ms, depending on the temperature| ofi the gas. Thermal- diffusivity, ~/m [S5, S6]. In this context, a gradient is ization of magnons created out of a degenerate gas in two essential to drive the diffusive spin transport. representative cases of kBT/µ < 1 and kBT/µ > 1 are When kBT < µ, the thermal excitations of a ho- shown in Fig. S1 (a) and (b). Fig. S1 (c) shows the ex- mogeneous Bose-Einstein condensate are phonon-like, tracted temperature of magnons created at a higher trap and thus thermalization of the free-particle-like magnons depth (in a non-degenerate gas) reaching a steady state with the majority gas requires collisions between quasi- following the final phase of evaporative cooling, for a rep- particles with very different dispersion. Such interactions resentative case kBT/µ < 1. Temperatures extracted are expected in the first order beyond the Bogoliubov from magnons created from a non-degenerate gas before mean-field theory (the Beliaev theory) [S7], though to completing evaporation and from magnons that are cre- our knowledge no one has produced an estimate or direct ated in the degenerate gas after evaporation agree with measurement of the thermal phonon-magnon cross sec- each other, when the same experimental parameters are tion. We speculate that thermalization of the magnons employed, and when extrapolated to the zero-magnon occurs primarily at the edges of the condensate where limit. low condensate density implies a better match between The physics behind the decoherence and thermaliza- majority spin and minority spin dispersion. In addition tion of spin excitations, and the variables that affect it, to facilitating decoherence, the applied gradient may aid remains a compelling direction of future research. Pre- in thermalization by driving the transport of low-energy vious experiments on degenerate F =1 87Rb gases have spin excitations to the edge of the condensate.

[S1] Chang, M.-S., Qin, Q., Zhang, W., You, L. & Chapman, magnetic Spinor Bose-Einstein Condensate. Phys. Rev. M. S. Coherent spinor dynamics in a spin-1 Bose con- Lett. 113, 155302 (2014). densate. Nat. Phys. 1, 111–116 (2005). [S5] Bardon, A. B. et al. Transverse demagnetization dynam- [S2] Lewandowski, H. J., McGuirk, J. M., Harber, D. M. & ics of a unitary Fermi gas. Science 344, 722–4 (2014). Cornell, E. A. Decoherence-Driven Cooling of a Degener- [S6] Koschorreck, M., Pertot, D., Vogt, E. & K¨ohl,M. Univer- arXiv:1505.06196v1 [cond-mat.quant-gas] 22 May 2015 ate Spinor Bose Gas. Phys. Rev. Lett. 91, 240404 (2003). sal spin dynamics in two-dimensional Fermi gases. Nat. [S3] Kronj¨ager, J. et al. Evolution of a spinor condensate: Phys. 9, 405–409 (2013). Coherent dynamics, dephasing, and revivals. Phys. Rev. [S7] Phuc, N. T., Kawaguchi, Y. & Ueda, M. Beliaev theory A 72, 063619 (2005). of spinor BoseEinstein condensates. Ann. Phys. (N. Y). [S4] Marti, G. E. et al. Coherent Magnon Optics in a Ferro- 328, 158–219 (2013).

[email protected] ∗ 80309-0440, USA; [email protected] † Present address: JILA, National Institute of Standards and Technology and University of Colorado, Boulder, Colorado 2

a b c 150 10 10

100 5 5 50 Temperature (nK) Temperature (nK) 0 0 0 250 1.5 15 3 200 3 1 10 150 100 0.5 5 magnons/10 magnons/10

50 Non−condensed Non−condensed 0 0 0 −2 −1 0 1 −2 −1 0 1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 Time after evaporation ramp (s) Thermalization time (s) Thermalization time (s)

FIG. S1. Magnon thermalization. Magnons created in a degenerate gas with (a) kB T 2µ and (b) kB T µ/3 thermalize at ∼ ∼ trap depths of approximately kB 1,200 and 35 nK, respectively, in the presence of a weak magnetic field gradient. Throughout, traces (colored points and connecting× lines) differ in initial number of magnons. In (a), the momentum distribution of the non- condensed magnons can be determined as soon as they are created. The indicated temperature reaches a steady state and the number of non-condensed magnons saturates after a few hundred milliseconds. The steady-state temperature depends on the number of magnons created owing to the effects of decoherence-driven cooling. In (b), temperatures and the number of non-condensed magnons cannot be determined until after nearly one second of thermalization, at which time the temperature has already reached steady state. (c) Magnons created before the final forced evaporation ramp are incoherent throughout the whole ramp and reach a steady-state temperature kB T µ/3 after two to three seconds at the final trap depth of approximately ∼ kB 25 nK. ×