Equation of State of Strongly Interacting Fermi Gas

Transcription

1 he 19 th Particle and Nuclei International Conference (PANIC11) Equation of State of Strongly Interacting ermi Gas Mark Ku, Ariel Sommer, Lawrence Cheuk, Andre Schirotzek, Martin Zwierlein heory collaborators (Ghent, UMass Amherst, EH): Kris Van Houcke, elix Werner, Evgeny Kozik, Nikolay Prokofev, Boris Svistunov Massachusetts Institute of echnology Center for Ultracold Atoms at MI and Harvard

2 Unitary ermi Gases Dilute, strongly-interacting, 2-components ermi gas: Unitary regime Scattering length >> interparticle spacing>> interaction range All properties depends only on density & temperature Low viscosity, high c, high v c Realized in: quark gluon plasma, neutron star, cold ermi gases. M.W. Zwierlein et al, Nature 45, (2005)

3 Ultracold atomic ermi Gases Ideal test-bed for Many-Body physics Highly controllable unable, resonant interactions Realize idealized models of many-body physics Benchmarking the many-body problem Need high precision to discriminate between theories Precision measurement of thermodynamics across the superfluid phase transition in strongly interacting ermi gases

4 hermodynamics of the Unitary ermi Gas Normal state: Is it a ermi liquid? Are there preformed pairs (pseudogap regime)? High- Classical gas ermi Liquid Superfluid properties: Preformed pairs? ransition temperature Critical Entropy Energy of the superfluid, /E = at =0 (Bertsch Many-Body X challenge, Seattle, 1999) c Low- Superfluid

5 Equation of State of a ermi gas at Unitarity or a balanced ermi gas at unitarity (a ) the only energy scales are the ermi energy E =k B and the temperature E universal function of Or equivalently: n f( ) Density equation of state Need n as function of and βμ

6 Energy [E Energy Equation of State of a ermi gas at Unitarity In a trap, provided the local density approximation holds: V() r 0 V () r Position Position [R 0 n( r ) f ( ( r )) ( r ) V ( r ) Local chemical potential Experiments measure column density: n (, ) (,, ) 2 D x z dy n x y z 0 Imaging light Atom cloud CCD

7 Obtaining the Density Equation of State n [10 11 /cm ] 1. Absorption image of a trapped ermi gas Column density n2 D ( x, y, z) 2. Inverse Abel ransform (requires cylindrical symmetry) D density n(, z) n( V(, z)). Equipotential averaging (requires accurate knowledge of V) n(v) V [ K]

8 Obtaining the Density Equation of State n [10 11 /cm ] 1. Absorption image of a trapped ermi gas Column density n2 D ( x, y, z) 2. Inverse Abel ransform (requires cylindrical symmetry) D density n(, z) n( V(, z)). Equipotential averaging (requires accurate knowledge of V) n(v) 4. emperature and chemical potential from fit to known EOS n f ( ) V [ K] 1 n ( z) e 2b e b e V 2 2 V V D 2 Initially: Virial Expansion Virial Expansion b2=/2 5/2, b= Ho and Mueller 2004; Liu, Hu, and Drummond 2009

9 Obtaining the Density Equation of State n [10 11 /cm ] 1. Absorption image of a trapped ermi gas Column density n2 D ( x, y, z) 2. Inverse Abel ransform (requires cylindrical symmetry) D density n(, z) n( V(, z)). Equipotential averaging (requires accurate knowledge of V) n(v) 4. emperature and chemical potential from fit to known EOS n f ( ) V [ K] 1 n ( z) e 2b e b e V 2 2 V V D 2 Initially: Virial Expansion Virial Expansion b2=/2 5/2, b= Ho and Mueller 2004; Liu, Hu, and Drummond 2009

10 Constructing the Density Equation of State Virial Expansion

11 Normalized Density Equation of State Normalized by EOS of non-interacting ermi gas How can we rule out systematics? Study a gas that we know! he non-interacting ermi gas! Virial Expansion

12 Normalized Density Equation of State Normalized by EOS of non-interacting ermi gas Measurement of the non-interacting ermi gas EOS using only its Virial expansion as input

13 Compressibility Equation of State n(, ) So far: requires from fitting 2 parameter fit We know - V hus d - dv Can measure 1 n Compressibility 2 Local compressibility dn 1 dn d 2 n dv Equation of State ( n, ) Requires only 1- parameter fit to get

14 Compressibility Equation of State

15 Compressibility Equation of State Sudden rise and fall

16 Compressibility Equation of State Divergence of Compressibility

17 Compressibility Equation of State Divergence of Compressibility rounded off by finite resolution (res. m, typical cloud size 60 m interparticle spacing ~0.5 m) Direct observation of the superfluid transition at C / = 0.17(1)

18 Compressibility Equation of State Going back to Density Equation of State i / i / d( / ) ~ with i and i known at high temperatures

19 Normalized Density Equation of State

20 Uncertainty in the Resonance Position he dominant error in f is the uncertainty in the position of the eshbach resonance G +/- 1.5 G Bartenstein et al., PRL % -5% Correction is known from the emperature-dep. Contact:

21 Normalized Equation of State At low : EOS must attain limiting value: n (4 ) /2 /(6 2 /2 ) New value for 0.80(15)

22 Normalized Equation of State At low : EOS must attain limiting value: n (4 ) /2 /(6 2 /2 ) New value for 0.80(15) Recent upper bound (after experiment!) orbes, Gandolfi, Gezerlis arxiv:1011:2197 (2011) 0.8(1)

23 Normalized Equation of State Agreement with Diagrammatic Monte-Carlo K. V. Houcke,. Werner, E. Kozik, B. Svistunov, N. Prokof ev

24 Normalized Equation of State irst order eynman diagrams Haussmann, Rantner, Cerrito, Zwerger, PRA 75, (2007)

25 Normalized Equation of State Deviation from 8 auxiliary field QMC (on a lattice, non-zero range) (Bulgac, Drut, Magierski, 2006)

26 Normalized Equation of State Critical Point from Determinant Diagrammatic Monte-Carlo Burovski et al., 2006

27 Normalized Equation of State Critical Point from Determinant Diagrammatic Monte-Carlo Goulko, Wingate, 2010 Burovski et al., 2006

28 Normalized Pressure Gibbs-Duhem relation:

29 Influence of Resonance Position +/- %

30 Normalized Pressure Pressure at low temperatures =0 limit: P /k B (4 ) 5/2 /(60 /2 )

31 Normalized Pressure Very good agreement with Diagrammatic Monte-Carlo K. V. Houcke,. Werner, E. Kozik, B. Svistunov, N. Prokof ev

32 Normalized Pressure Deviates from auxiliary field QMC and Determinant Diagrammatic Monte-Carlo irst order eynman diagrams Haussmann, Rantner, Cerrito, Zwerger, PRA 75, (2007) Burovski et al., 2006 Goulko, Wingate, 2010 Bulgac, Drut, Magierski

33 Normalized Pressure Deviates at low temperatures from ENS Experimental Data (which does not determine the density)

34 Normalized Pressure Deviates at low temperatures from ENS Experimental Data (which does not determine the density) Reason: Pressure is calibrated using an independently measured = 0.415(10) 0.415

35 Directly follows from EOS: Chemical Potential 4 1 (6 ) ( n ) 2 2/ 2/ Experimental data

36 Directly follows from EOS: Chemical Potential 4 1 (6 ) ( n ) 2 2/ 2/ n 0.46(1) Experimental data New value for s 0.80(15) E C Superfluid Normal Maximum occurs at / =0.175 Goulko, Wingate, 2010: C / =0.17(6)

37 On resonance: Entropy vs emperature 2.5 E 1 S E PV N S PV So: 2 Nk B 5 2 P nk B 2.0 S/Nk B MI Data Non-Interacting ermi Gas /

38 Specific Heat C V /N At unitarity: C N Heat capacity V 2 P P Noninteracting ermi Gas /

39 Specific Heat C V /N At unitarity: C N Heat capacity V 2 P P Unitary ermi Gas /

40 At unitarity: C N Heat capacity V 2 P P 0 0 A lambda-like feature in the specific heat

41 At unitarity: C N Heat capacity V 2 P P 0 0 BCS-solution + Phonons C / = 0.17(1)

42 At unitarity: C N Heat capacity V 2 P P 0 0 m Resolution C / = 0.17(1)

43 Conclusion Precision measurement of thermodynamics across the superfluid transition Obtain thermodynamic quantities across c Chemical potential, Entropy, Energy etc. all as a function of the bulk / Measure the superfluid energy =0.80(15) Observe transition at c =0.17(1)

44

45 d / d d d ~ 2 2 Compressibility Equation of State Going back to Density Equation of State ~ 1 ) / ( / / 2 2 i i d with i i and known at high temperatures

46 Do we have a ermi Liquid? Specific heat is NO linear above C, not even for a normal ermi gas

47 Chemical Potential Ideal ermi gas E Monotonically decreasing 2

48 Chemical Potential ermi Liquid Interaction shifts 2 m* E n 12 m effective mass comes in Monotonically decreasing 2...

49 Directly follows from EOS: 1 Chemical Potential 4 1 (6 ) ( n ) 2 2/ 2/ 0 Experimental data /E /

50 Bounds and convergence on Energy: E( N,, V ) E( N,0, V ) 5 NE ree Energy: ( N,, V ) ( N,0, V ) 5 NE Upper Bound: Lower Bound: Upper Lower ( / / ) 5 5 E NE NE P ne E 5 P ne Also: P( )>P( 0) see Castin, Werner, 2011

51 Bounds and convergence on 5 E NE E 5 NE

52 Do we have a ermi Liquid? Normalized Pressure

53 n(,)/n 0 (,0) Do we have a ermi Liquid? 5.5 Normalized Density MI Linear fit 1/ (/ )

54 Entropy vs emperature Haussmann, Rantner, Cerrito, Zwerger, PRA 75, (2007)

55 Superfluid transition C =0.165(10) S C =0.65(15) Nk B

56 Other orms of the Equation of State Each only requires 1- param. fits So far: requires from fitting 2 parameter fit We know - V hus d - dv Can measure Local compressibility 1 n 2 n(, ) dn d Compressibility 1 n 2 dn dv P Equations of State Local pressure d n V dv Pressure ' n( V ') ( n, ) P( n, )

57 Experiments on hermodynamics of the Bulk Measurements of Energy / Pressure: Integration over cloud profile okyo: Horikoshi et al., Science 2010 Assumptions: Harmonic trapping Expansion is hydrodynamic Profiles fit by the shape of a non-interacting ermi gas Resulting difficulties: Disagreement with high-temperature Virial expansion Required a calibrated thermometer (E vs from Duke) ENS: Nascimbène et al., Nature 2010, Science 2010 Assumptions: Harmonic trapping Resulting difficulties: Independent thermometer ( 7 Li) condenses at low temperatures Calibration relied on independently measured value of = 0.415(10)

58 omography of a ermi gas Challenge: How to go from n2 D( x, z ) to f ( )? omography via Inverse Abel transform Only assumes cylindrical symmetry z z x n2d(x,y) obtained from CCD Reconstructed D density n(,z) ρ

59 Energy [E Energy Equation of State of a ermi gas at Unitarity Density [10 11 /cm ] In a trap, provided the local density approximation holds: V() r 0 V () r Position Position [R Low noise thanks to equipotential averaging Works for ANY (reasonable) potential! 0 n( r ) f ( ( r )) ( r ) V ( r ) Local chemical potential Experimental n vs V from single profile 0.8 V [ K] 1.2

60 Density [10 11 /cm ] hermometry of a strongly interacting ermi gas Good news: Known high-temp. virial expansion 2 f ( ) e 2b2e be... 2 b b Ho, Mueller, PRL 92, (2004); Liu, Hu, Drummond, PRL 102, (2009) Blume et al. (2010?) Solves the thermometry problem for unitary ermi gases n ( z) e 2b e b e V 2 2 V V D V [ K] =198 nk βμ=0.06

61 C V (/ )/C V0 (/ ) Normalized heat capacity Crossing at / = 0.19(1) However: hat s not c ( -Anomaly?) /

62 Pioneering Experiments on hermodynamics in a trap Measurements of Energy: Integration of r 2 over cloud profile Duke group, O Hara et al., Science 2002 ENS, Bourdel et al., PRL 200 Innsbruck, Bartenstein et al., PRL 2004 Energy vs emperature: JILA, Stewart et al., PRL 2006 Kinast et al., Science 2005 Entropy vs Energy: Luo et al., PRL 2007 See also Luo & homas, JLP 154, 1 (2009)

63 S / V n Back to the harmonic trap n( r ) f ( m r ) m 2 r 2 /2k B m 2 r 2 /2k B.0

64 Back to the harmonic trap n( r ) f ( m r 2 2 ) Let s introduce Non-interacting case: 1 1 f2( ) dx f ( x) x 2 f( ) dx x f ( x) n. i. f 2 ) ( e ( 2 n. i. f ) ( e ( ) ) 4 / 2 f4( ) dx x f ( x) n. i. f 4 ) ( e ( 4 )

65 Back to the harmonic trap ) 2 1 ( 1 ) ( 2 2 r m f r n ) ( ) ( f k r n r d N B trap E, 1/, ) ( f trap ) ( ) ( ) ( ) ( f f k r V r n r d N N E B ) ( ) ( f f k Nk E Nk S B B B ) ( ) ( 9 ) ( ) ( f f f f Nk C B V

66 Bulk vs rap hermometry C,trap 0.227(5) /,rap C, trap, bulk 0.165(10) /,bulk

67 2.5 Energy vs emperature E NE C, trap, trap 0.694(10) 2.0 E/NE /,trap

68 5 Entropy vs emperature S C, trap Nk B 1.67(5) S/Nk B 4 S Nk n. i. B 2, trap 2 1 S Nk 0.77 B, trap /

69 Energy vs Entropy 2.0 MI Duke E/NE,trap Luo & homas, JLP S/Nk B

70 Specific Heat in the rap C V /Nk B /,trap

71 C V (/,trap )/C V0 (/,trap ) Normalized Specific Heat Crossing at C, trap, trap C,trap 0.227(5) /,trap

72 Obtaining the Potential from Cloud Profiles or every single cloud, we have n(,z) = n(v(,z)) We know the potential along z with extremely high accuracy: V ( 0, z) m z (2) 2 rom n(0,z) get V(n) and thus V(,z) Hz

73 Obtaining the Potential from Cloud Profiles or every single cloud, we have n(,z) = n(v(,z)) Stack up many images

74 Hints of superfluidity in the Cloud Profiles Integration over cloud profiles reduces sensitivity Superfluid transition is not easily observed However, we know that density profiles can reveal the Superfluid transition: Around c ar below c Ketterle, Zwierlein, Varenna Lectures (2008), e-print: arxiv:

75 Hints of superfluidity in the Cloud Profiles Profiles: Hardly any signature Rapid Ramp: Condensates Signature in the curvature! and fit-residuals Very faint signature in the cloud profiles Avoid integration Obtain the density equation of state Ketterle, Zwierlein, Varenna Lectures (2008), e-print: arxiv:

76 S V Normalized entropy density vs / Shows distinct peak Single-fermion excitaitons Phonon S 1 / 2 excitations / 2 / 2 V Peak at = 0.175(10)

77 Ultracold atomic ermi Gases Ideal test-bed for Many-Body physics I. Realize idealized models of many-body physics Benchmarking the many-body problem Need high precision to discriminate between theories Precision measurement of thermodynamics across the superfluid phase transition II. Beyond equilibrium physics Explore many-body dynamics in real time Universal Spin ransport in a resonant ermi gas alk by Ariel Sommer on hursday

78 f( ) Constructing the Equation of state

79

80 Atoms as probe for the trapping potential Our trap: Axially magnetic curvature parabolic radially a laser beam gaussian power, waist, frequency prone to systematics Instead: In each experimental run, the atoms experience the same trapping potential Simply add up many density profiles (may be at varying temperature and total atom number) low noise LDA: Equidensity lines are Equipotential lines We know the potential along the axial direction perfectly We know it everywhere! z

81 Outlook hermodynamics across the superfluid transition inite-temperature Equation of State + across the BEC-BCS crossover + as a function of spin imbalance + as a function of the dimensionality