Parametric cooling of a degenerate Fermi gas in an optical trap
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec Parametric cooling of a degenerate Fermi gas in an optical trap
Jiaming Li, Ji Liu, Wen Xu, Leonardo de Melo, and Le Luo ∗ Department of Physics, Indiana University Purdue University Indianapolis, Indianapolis, IN 46202 (Dated: December 7, 2015)We demonstrate a novel technique for cooling a degenerate Fermi gas in a crossed-beam opticaldipole trap, where high-energy atoms can be selectively removed from the trap by modulatingthe stiffness of the trapping potential with anharmonic trapping frequencies. We measure thedependence of the cooling effect on the frequency and amplitude of the parametric modulations.It is found that the large anharmonicity along the axial trapping potential allows to generate adegenerate Fermi gas with anisotropic energy distribution, in which the cloud energy in the axialdirection can be reduced to the ground state value.
Evaporative cooling in an optical dipole trap (ODT)has remained a key technique for producing Bose-Einstein condensates and degenerate Fermi gases formore than a decade [1–3]. The most common approachfor evaporation is to reduce the optical trapping potentialcontinuously by decreasing the intensity of the trappingbeams, so called the “weakening” scheme. The weaken-ing scheme results in a reduction of trapping frequenciesinevitably, which not only decreases the collision rate butalso limits the maximum phase space density availablein an optical trap. To overcome this drawback, severalauxiliary techniques have been implemented to maintaintrapping frequencies during evaporation, including a dim-ple trap [4], moving traps [5], time-delay traps [6], and amagnetic field tilting trap [7]. These techniques increasethe evaporation speed and the final phase space densitysubstantially, but require a more experimental setting.Alternatively, it is desirable to develop an “expelling”scheme for an ODT, an analogy of the radio-frequencyknife for a magnetic trap [8], where high-energy atomscan be selectively removed from optical traps while keep-ing the trapping potential intact. Since both the colli-sion rate and the phase space density scale with the cubeof the average trapping frequency [9], such an expellingscheme has the potential to improve evaporative coolingin optical traps significantly, which will be essential forexperiments with ultracold polar molecules. In those ex-periments, the coldest sample is close to the Fermi tem-perature T F in an ODT, but cooling into deep quantumdegeneracy has yet to be realized [10]. Developing anexpelling scheme may pave the way for the final stagecooling in the degenerate regime.In this letter, we report an “expelling” scheme tocool a degenerate Fermi gas by parametric excitation ofhigh-energy atoms out of an optical trap. Our schemeemploys the intrinsic anharmonicity of a crossed-beamODT, where high-energy atoms experience smaller trap-ping frequencies than low-energy atoms. The spatial dif-ferential trapping frequencies turn parametric excitationof atomic motion from a well-established laser-inducedheating and loss source [12, 13] into a robust coolingmechanism, in which high-energy atoms can be selec-tively removed from the trap when the modulation fre- FIG. 1: The local radial trap frequency of a crossed-beamoptical trap. The x-axis trap frequency ω x ( x, , z ) is plottedin the x-z plane in term of the harmonic frequency ω x =760 Hz (the calculated value from the trapping potential).The radial and axial atom densities n ( x ) and n ( z ) are plottedin the left and bottom frames for a Fermi gas of 1 . × atoms per spin state at T /T F = 0 . σ x ) and axial ( σ z ) directions,where the local trapping frequency drops to ω x ( σ x , ,
0) =0 . ω x and ω x (0 , , σ z ) = 0 . ω x . quency is tuned to resonance with the trapping frequen-cies of high-energy atoms. Parametric modulation in-duced cooling has previously been observed for bosonicatoms either in a magnetic trap [14] or in a standing wavelattice [15]. However, in both cases, the bosonic atomswere in the thermal states with phase space densities of10 − ∼ − . When approaching the quantum regimewith a phase space density close to one, bosonic atomstend to occupy the lowest vibrational states, resulting ina negligible differential trapping frequency between high-energy and low-energy atoms. It becomes very difficult toparametrically cool a Bose gas at very low temperatures,which has not yet been reported, to the best of our knowl-edge. In contrast, fermionic atoms, indebted to the Pauliexclusion principle, occupy a significant fraction of thevibrational states even at the degenerate temperature,making parametric cooling much more feasible. Here weuse a noninteracting degenerate Fermi gas for a proof ofprinciple study, in which other cooling mechanisms areminimized and the excited high-energy atoms will leavethe trap quickly without colliding with the low-energyatoms, manifesting the parametric cooling effect.Anharmonicity of the trapping potential plays a cen-tral role in parametric cooling. In our experiment, weload Li atoms into a crossed-beam ODT with a cross-ing angle 2 θ = 12 ◦ in the x-z plane, which is de-scribed by U ( x, y, z ) = − U [(1 + Z − /z ) − Exp[ − y + X ) /w ] + (1 + Z /z ) − Exp[ − y + X − ) /w ]], with X ± = x cos θ ± z sin θ and Z ± = z cos θ ± x sin θ . Thesingle-beam trap depth U is 2.8 µ K which is formed bya 100 mW Gaussian beam at 1.06 µ m wavelength witha focused beam waist w = 37 µ m. The local radial trapfrequency along the x-axis ω x ( x, y, z ) can be calculatedfrom anharmonic radial motions of the atoms [15], givenby ω x ( x, y, z ) = π p /m Z x − x [ U ( x, y, z ) − U ( e x, y, z )] − / d e x . (1)From Eq.1, the harmonic frequency ω x ,y ,z is readilyobtained by approximating the center of the trap with aharmonic potential of m ( ω x x + ω y y + ω z z ) /
2, where m is the mass of Li atom. We plot the dispersion of thelocal frequency ω x ( x, , z ) in the y = 0 plane in Fig. 1,showing that the local frequency decreases significantlyfrom the center to the edge of the atom cloud even at T /T F = 0 .
6. The large frequency drops along the z-axis allows applying parametric excitation to selectivelyremove high-energy atoms in a degenerate Fermi gas.In the noninteracting regime, parametric modulationsusually induce temperature (energy) anisotropy since theatomic motions along the different axes of the cloudsare uncoupled. Instead of
T /T F , we use E/E F , theratio between the energy per particle and the Fermienergy, as an effective thermometry, which provides aconvenient way to characterize temperature (energy)anisotropy due to the parametric modulation. The to-tal energy per particle is given by E = E x + E y + E z based on uncoupled atomic motions in different direc-tions. For a noninteracting gas, the viral theorem gives E x,y,z = 2 U x,y,z by using an harmonic approximationfor the trapping potential. U x is the potential energy perparticle along the x-axis, which can be determined by U x = N / mω x h x i /
2. The number-independent meansquare size (NIMS) h x i = R x n ( x ) dx/N / can be ob-tained directly from the 1D density profile of the atomcloud [16]. Finally the energy in the x-direction is givenby E x /E F = mω x h x i / (6 / ¯ hω ), where ω = ( ω x ω y ω z ) / is the average trap frequency. It is noted that we ignorethe anharmonic correction of the potential energy since itis small for a degenerate Fermi gas at low temperatures.We prepare a gas of Li atoms in the two lowest hy-perfine states of F = 1 / , m F = ± / | i and | i states) in a magneto-optical trap. The precooled atoms w/w x0 x z N E x / E F E z / E F (a)(b)(c)(d)(e) FIG. 2: The dependence of parametric excitation on the mod-ulation frequency. The radial and axial NIMS, the normalizedatom number, and the radial and axial energies are shownfrom the top to the bottom. All the data have a modulationtime t m = 500 ms and a modulation amplitude δ = 0 . ω x = 740 Hz is the measured value.The dashed lines indicate the average value without paramet-ric modulation ( δ =0). The solid lines show the simulationresults. are then transferred into a crossed-beam ODT made by a100 W IPG fiber laser. The bias magnetic field is quicklyswept to 330 G to implement evaporative cooling. Thetrap potential is lowered to 0.1% of the full trap depth in2.6 s, giving a final trap depth of 5 . µ K for the crossed-beam trap. A noisy radio-frequency pulse is then appliedto prepare a 50:50 spin mixture. To prepare a noninter-acting Fermi gas, the magnetic field is swept to 527.3 G,where the s-wave scattering length of | i and | i states iszero [17]. Typically we have a noninteracting Fermi gasof N = 1 . × atoms per spin state at T /T F ≈ . T F ≈ . µ K to start parametric modulation. The tem-perature of a noninteracting Fermi gas is measured byfitting the 1D density profile with the finite temperatureThomas-Fermi distribution [11]. In the noninteractingregime, the temperature from the density profile is alsoconfirmed by fitting the time-of-flight cloud sizes withballistic expansion dynamics [18].The parametric excitation is applied to the atomclouds by modulating the optical intensity of the trap-ping beams with an acousto-optic modulator. The trapdepth is modulated as U ( t ) = U [1 + δ cos( ω m t + θ )],where U is the trap depth without modulation, δ is themodulation depth, ω m is the modulation frequency, and θ is the modulation phase. The modulation is turned onfor a time t m . After that, the atoms are allowed to stay inthe trap for 100 ms to reach a steady state before the trapis turned off. Subsequently the atom cloud ballisticallyexpands for 800 µ s before a 10 µ s resonant optical pulse isapplied for absorption imaging. We first extract both theradial and axial NIMSs of the time-of-flight clouds fromthe absorption images, and then determine the NIMSsand energies of the in-situ clouds from ballistic expan-sion. We avoid taking in-situ images to eliminate thesystematic error due to the high column density.We first use small δ = 0 .
05 to measure the harmonictrap frequencies, where modulations of twice the har-monic trap frequency induces parametric heating [12].We measure ω x = 2 π × (740 ±
10) Hz, ω y = 2 π × (750 ±
10) Hz, ω z = 2 π × (75 ±
5) Hz, which agreevery well with the theoretical calculation based on theparameters of the trapping potential. We next examinethe dependence of the parametric excitation effect on themodulation frequency using large modulation amplitude δ = 0 .
15. The radial (x-axis) and axial (z-axis) NIMSsand energies of the clouds are shown in Fig. 2, where ω m varies from a near zero value to about 2 . ω x . We findthat the NIMSs of the axial and radial directions showquite different frequency dependence, resulting in the en-ergy anisotropy after modulation. The radial NIMSsincrease significantly around 2 ω x , indicating the usualparametric heating effect [12, 19]. In contrast, the axialNIMSs barely change at 2 ω x , showing the modulation at2 ω x frequency mainly excites atomic motion along theradial direction and does not couple to the motion alongthe axial direction. The most striking feature evidentin our measurements is that the axial NIMSs decreasesignificantly in a wide range between 0 . ω x to 1 . ω x ,indicating a reduction of the axial cloud energy E z . Thisaxial parametric cooling effect can be explained by thefact that the atoms along the z-axis experience differentlocal radial frequency ω x ( z ), such that the high-energyatoms at the edge of the trap are selectively excited out ofthe trap. As shown in Fig. 1, anharmonicity plays a muchmore significant role in the axial direction than that inthe radial one, therefore parametric cooling mainly takesplace in the axial direction.We develop a simple model of a group of anharmonicoscillators distributed in a 2D plane to simulate the para-metric modulation process. We assume the initial columndensity is described by a 2D Thomas-Fermi n ( x, z ) distri-bution [11], and the local densities n ( x, z ) are associatedto different anharmonic oscillators. During the paramet-ric modulation, the atoms associated to a specific n ( x, z )oscillate along the radial (x-axis) direction only, whoseequation of motion is given by d xdt + 1 + δ cos( ω m t + θ ) m dU ( x, z ) dx = 0 . (2)Here U ( x, z ) = U ( x, , z ) by approximating the col- δ (a) xz d =0 0.08 0.15 0.23 0.30 0.38 E x / E F (b) E z / E F (c) δ FIG. 3: The dependence of the radial and axial energieson the modulation amplitudes. (a) The absorption imagesof the atoms clouds show a dramatic decrease of the axialcloud sizes with an increase of modulation amplitudes, where ω m = 1 . ω x . (b) The dependence of the radial energies onthe modulation amplitude. (c) The dependence of the radialenergies on the modulation amplitude. In both (b) and (c),blue triangles are at the modulation frequency 1 . ω x , and redsquares are at 2 . ω x . The solid lines represent the simulationof the anharmonic oscillator model using the same simulationparameters for the frequency dependence. umn density in the 2D plane of y = 0. The initialatom position and velocity are given by x (0) = x and( dx/dt ) t =0 = (2 U ( x , , z ) − U (0 , , z )) / m ) / . Wesolve Eq. 2 numerically for t ∈ { , t m } to obtain the posi-tion and kinetic energy of atoms after parametric excita-tion. In this model, atoms associated with a local density n ( x, z ) can be selectively excited when the parametricoscillation frequency ω m is tuned to resonance with thelocal frequency ω x ( x, z ). When ω m ≈ ω x , the mod-ulation will excite the atoms in the center of the trap,inducing a loss of the low-energy atoms and an increaseof the energy per particle. In comparison, when the ω m is tuned to 1 . ω x close to the local trapping frequencyat the edge of the trap, the parametric process will ex-cite the high-energy atoms out of the trap and result ina cooling effect. The simulation results of the atom num-ber, E x , and E z are shown by the solid lines in Fig. 2,where we only adjust the harmonic frequency ω x to 825Hz in our simulation for the best fitting of the experimen-tal data, while keeping all other simulation parameters asthe experimental values. This model manifests the anal-ogy between parametric excitation in an optical trap andrf-knife in a magnetic trap, both of which depend on thespatial variation of frequency (either trapping frequencyor rf transition frequency) between high-energy and low-energy atoms.We also study the dependence of the parametric excita-tion on the modulation amplitude with the results shownin Fig. 3. For the modulation ω m = 2 ω x = 2 π × E x /E F increases dramatically when the modulation am-plitude increases, which is consistent with the parametricheating effect along the radial direction. For the modu-lation ω m = 1 . ω x , E z /E F decreases significantly withan increase of the modulation amplitude, showing thata stronger cooling effect takes place when larger mod-ulation expels more high-energy atoms out of the trap.The cooling effect saturates when δ increases to 0.25 dueto the fact that the modulation becomes so strong thatmost atoms in the anharmonic region have already beenexpelled from the trap. We simulate the dependence onthe modulation amplitude shown by the solid lines inFig.3 (b) and (c). The simulations exhibit both heatingand cooling features, which agree with the experimentalresults reasonably well.After parametric modulation, the anisotropic energydistributions between the axial and radial directions areobserved. We find that these anisotropic energy distri-butions are quasi-steady around 528 G, showing a lackof cross-dimensional thermalization [20–22] in the nonin-teracting regime. For a noninteracting Fermi gas, s-waveelastic collisions are absent, and the cross-dimensionalterm of the trapping potential is negligible for a degen-erate Fermi gas at very low temperatures. To obtainisotropic temperature, the magnetic field is adiabaticallyswept to 330 G, where a finite scattering length of -280 a [23] assists a fast cross-dimensional thermalization.After that, the atom clouds are swept back to 527.3 G,and the energy distributions are measured by taking ab-sorption images. We verify that the equipartition energydistribution is retrieved within a 100 ms holding time at330 G. By subtracting the finite heating rate during thesweeping and holding time [24], we find that the energyper particle E of a noninteracting Fermi gas is decreasedfrom 1 . E F to 1 . E F by parametric cooling. It is worthnoting that this parametric cooling effect is almost op-timal since the cloud energy decreases only in the axialdirection. For a noninteracting Fermi gas in a harmonictrap, the ground state energy in each direction is 0 . E F .Starting with the initial cloud energy E = 1 . E F , thecoldest sample that can be obtained by axial parametriccooling is limited by (2 / . E F + 0 . E F = 1 . E F .In conclusion, we report parametric cooling of a nonin-teracting degenerate Fermi gas. The parametric coolingmethod provides a selective way to remove high-energyatoms from optical traps in the noninteracting regime. Italso provides a convenient method to generate temper-ature anisotropy for studying nonequilibrium thermody-namics in quantum gases. This method can be imple-mented in many existing optical traps, such as a power-law trap [25] and a box trap [26], in which large anhar-monicity exists in all three directions. With speciallydesigned anharmonic traps, parametric cooling has po-tential to be implemented in a large temperature scale,where a Fermi gas may be directly cooled from the ther-mal state to the degenerate regime by parametric modu- lations.Le Luo is a member of the Indiana University Centerfor Spacetime Symmetries (IUCSS). Le Luo thanks sup-ports from Indiana University RSFG and Purdue Uni-versity PRF. ∗ [email protected][1] M. D. Barrett, J. A. Sauer, and M. S. Chapman, Phys.Rev. Lett. , 010404 (2001).[2] S. R. Granade, M. E. Gehm, K. M. O’Hara, and J. E.Thomas, Phys. Rev. Lett. , 120405 (2002).[3] R. Grimm., M. Weidem¨uller, and Y. B. Ovchinnikov,Adv. At. Mol. Opt. Phys. , 95 (2000).[4] J.-F. Cl´ement, J.-P. Brantut, M. Robert-de Saint-Vincent, R. A. Nyman, A. Aspect, T. Bourdel, andP. Bouyer, Phys. Rev. A , 061406 (2009).[5] T. Kinoshita, T. Wenger, and D. S. Weiss, Phys. Rev. A , 011602 (2005).[6] K. Arnold and M. Barrett, Optics Communications ,3288 (2011).[7] C.-L. Hung, X. Zhang, N. Gemelke, and C. Chin, Phys.Rev. A , 011604 (2008).[8] W. Ketterle and N. J. V. Druten, Adv. At. Mol. Opt.Phys. , 181 (1996).[9] L. Luo, B. Clancy, J. Joseph, J. Kinast, A.Turlapov, andJ. E. Thomas, New Journal of Physics , 213 (2006).[10] S. A. Moses, J. P. Covey, M. T. Miecnikowski, B. Yan,B. Gadway, J. Ye, and D. S. Jin, Science , 659 (2015).[11] L. Luo, Ph.D. thesis, Duke University (2008).[12] T. A. Savard, K. M. O’Hara, and J. E. Thomas, Phys.Rev. A , R1095 (1997).[13] M. E. Gehm, K. M. O’Hara, T. A. Savard, and J. E.Thomas, Phys. Rev. A , 3914 (1998).[14] M. Kumakura, Y. Shirahata, Y. Takasu, Y. Takahashi,and T. Yabuzaki, Phys. Rev. A , 021401(R) (2003).[15] N. Poli, R. J. Brecha, G. Roati, and G. Modugno, Phys.Rev. A , 021401 (2002).[16] L. Luo and J. E. Thomas, Journal of Low TemperaturePhysics , 1 (2008).[17] G. Z¨urn, T. Lompe, A. N. Wenz, S. Jochim, P. S. Juli-enne, and J. M. Hutson, Phys. Rev. Lett. , 135301(2013).[18] The ballistic expansion of a noninteracting Fermi gas pro-vides h x ( t ) i = h x i [1+2 k B T t / ( m h x i )], where t is time-of flight and h x ( t ) i is the mean square cloud size. Thetemperature T can be determined by the time depen-dence of the mean-square size.[19] S. Friebel, C. DAndrea, J. Walz, M. Weitz, and T. W.H¨ansch, Phys. Rev. A , R20 (1998).[20] C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt,and C. E. Wieman, Phys. Rev. Lett. , 414 (1993).[21] K. Aikawa, A. Frisch, M. Mark, S. Baier, R. Grimm,J. Bohn, D. Jin, G. Bruun, and F. Ferlaino, Phys. Rev.Lett. , 263201 (2014).[22] C. Hahn, Master’s thesis, Ludwig Maximilians UniversityMunich (2009).[23] P. M. Duarte, R. A. Hart, J. M. Hitchcock, T. A. Corcov-ilos, T.-L. Yang, A. Reed, and R. G. Hulet, Phys. Rev.Lett. , 061406(R) (2011).[24] The trap lifetime is around 20 seconds, where the heating and loss is due to the background collisions and/or trap-ping potential noise. To manifest the rethermalization,we subtract this heating effect due to the finite magneticfield sweeping and holding time.[25] G. D. Bruce, S. L. Bromley, G. Smirne, L. Torralbo-Campo, and D. Cassettari, Phys. Rev. A , 053410 (2011).[26] A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P.Smith, and Z. Hadzibabic, Phys. Rev. Lett.110