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Universal Thermodynamics of a Unitary Fermi gas

Universal Thermodynamics of a Unitary Fermi gas. Takashi Mukaiyama. University of Electro-Communications. outline. Introduction. Thermodynamics of a unitary Fermi gas - from global thermodynamics quantities to local ones - . Toward further understanding of a unitary gas

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Universal Thermodynamics of a Unitary Fermi gas

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  1. Universal Thermodynamics of a Unitary Fermi gas Takashi Mukaiyama University of Electro-Communications

  2. outline Introduction Thermodynamics of a unitary Fermi gas - from globalthermodynamics quantities to local ones- Toward further understanding of a unitary gas (Fermi liquid or pseudogap?) Tan relation new, universal description of thermodynamics for interacting fermions

  3. BEC in a cold atom system Cold atoms are • very dilute (1011~1014 cm-3), • with no impurities, no defects. Amenable to simple theoretical description J. R. Ensher, et al., Phys. Rev. Lett. 77, 4984 (1996). Condensate fraction 5% deviation of critical temperature from theoretical predictions ・3% shift due to finite size correction ・2% shift due to interaction

  4. Inter-atomic interaction is tunable !! Feshbachresonance There are two channels corresponding to different spin states. Resonance occurs when open and closed channel are energetically degenerate. E Closed (bound) channel bound state R Open (scattering) channel S. Inouye, et al., Nature 392, 151 (1998).

  5. Loss near a Feshbach resonance Number of atoms scattering length loss due to vibrational quenching E R Vibrational quenching S. Inouye et al., Nature, 392, 151 (1998).

  6. JILA ultracold fermionic atoms 1999Fermi degenerate gas Collision channel Identical bosons : l=0(s-wave), l=2 (d-wave), … Identical fermions: l=1 (p-wave),l=3 (f-wave), … ultracold : s-wave is the dominant collision channel. Identical fermions do not collide. Think about two-component fermions At the Feshbach resonance for and , no loss occurs due to Pauli exclusion principle. Therefore two-component fermions are stable even at a Feshbach resonance.

  7. We are able to prepare an interacting (reasonably stable) two-component Fermi gas of atoms with an arbitrary interaction strength !!

  8. Ultracold, dilute, interacting Fermi gases as T n-1/3 R ・ ultracold : s-wave is the dominant channel. collide only with ・ dilute : details of the potential is much smaller than n-1/3 The collision process can be described by a single parameter, so-called scattering length as.

  9. Ultracold dilute Fermi gas as T n-1/3 Remember the fact that as is tunable!! Then, what happens when… |as|∞ This situation is called unitarity limit.

  10. Unitarity limit and Universality as n-1/3 T T n-1/3 as drops out of the description of the thermodynamics. Thermodynamics depends only on the density nand temperature T. … like a non-interacting case.

  11. Universal thermodynamics According to universal hypothesis, all thermodynamics should obey the universal functions: Internal energy : Helmholtz free energy : Dimensionless universal functions, (shape of the function is different from those for an ideal gas) Chemical potential : Entropy : System looks like a non-interacting Fermi gas. (no other dimensional parameters involved in the problem)

  12. Universal thermodynamics Bertsch’s Many-Body X challenge, Seattle, 1999 What are the ground state properties of the many-body system composed of spin ½ fermions interacting via a zero-range, infinite scattering-length contact interaction. pure number

  13. b value At T=0 Fermi gas profile in a 3D harmonic trap at ideal gas unitary gas L. Luo et al. (2009) -0.61(2) T. Bourdel et al. (2004) -0.64(15) C. A. Regal et al. (2005) -0.62(7) +0.13 M. Bartenstein et al. (2004) -0.68 -0.10 (from kinetic energy) J. Kinast et al. (2005) -0.49(4) J. Joseph et al. (2007) -0.565(15) +0.05 (from sound velocity) J. Stewart et al. (2006) -0.54 -0.12 Y.-I Shin et al. (2007) -0.50(7)

  14. Conventional thermometry Temperature is determined by fitting the profile. This scheme is good only when interaction energy << kinetic energy.

  15. virial theorem at unitarity take delivative L. Luo and J. E. Thomas, J Low Temp. Phys. 154, 1 (2009). universal thermodynamic function (in a harmonic trap!!) entropy : can be measured after adiabatic magnetic field sweep to weakly-interacting regime J. E. Thomas et al. PRL 95, 120402 (2005)

  16. Universal thermodynamics H. Hu, P. D. Drummond & X.-J. Liu, Nature Physics 3, 469 - 472 (2007)

  17. trap inhomogeneity T is constant over the cloud (thermal equilibrium). EF depends on the position (local density). is position-dependent. Global measurement only gives the integration of all the different phases.

  18. Goal of this experiment Measurement of local thermodynamic quantities and the determination of the universal thermodynamic function.

  19. Deceleration and trapping Li oven T~700K MOT（magneto-optical trap） T = 200 mK N > 108

  20. Evaporative cooling in an optical dipole trap Optical dipole trap Number Number Number Energy Energy Energy

  21. molecular BEC B = 650 - 800 G 6Li原子のs波散乱長(a0) degenerate Fermi gas 磁場 [G] B = 0-450 G 6Li in quantum degenerate regime I B I

  22. Determination of local energye(r) density profile Useful equations : ・ Equation of state of unitary gas : ・ mechanical equilibrium (eq. of force balance) :

  23. Determination of temperature T and Adiabatic B-field sweep to turn off the interaction entropy Le Luo and J.E. Thomas, J Low Temp Phys 154, 1 (2009).

  24. Our scheme

  25. Experimental determination of fE[T/TF] Ideal Unitary M. Horikoshi, S. Nakajima, M. Ueda and T. Mukaiyama, Science, 327, 442 (2010). About 800 images are analyzed.

  26. Verification of the determined fE[T/TF] • Energy comparison • Potential energy par particle : Comparison • Internal energy par particle : Epot = Eint Nice consistency !!

  27. Verification of the determined fE[T/TF] • 2. Effective speed of the first sound 6Li Light pulse to make density perturbation

  28. Verification of the determined fE[T/TF] • 2. Effective speed of the first sound Propagation time 0.1ms 1.1ms 2.1ms 3.1ms 4.1ms 5.1ms 6.1ms 7.1ms

  29. Verification of the determined fE[T/TF] • 2. Effective speed of the first sound Unitary gas shows hydrodynamic behavior due to the large collision rate Effective speed of the first sound : [ P. Capuzzi, PRA 73, 021603(R) (2006) ] Comparison Experiment

  30. Verification of the determined fE[T/TF] • 2. Effective speed of the first sound Experimental values vs. calculated values from fE[q] u1,Meas. = u1,Calc Nice consistency !!

  31. JILA absorption image after expansion signature of BEC = bimodal distribution BEC bimodal thermal • strong interaction • pairing by many-body physics

  32. “Projection” projection… convert correlated pair of atoms into tightly-bound molecules by sweeping the magnetic field toward BEC side of the resonance W. Ketterleet al.,arXiv:0801.2500 Magnetic field sweep has to be … - slow enough to satisfy the adiabatic condition of two-body binding process - fast enough so that one can neglect collisions

  33. Bimodal distribution of a fermion pair condensate Bimodal distribution Unitarity limit BEC side BCS side Preformed pair Bound molecule 650 700 750 800 834 900 Magnetic field [Gauss]

  34. Condensate fraction vs Temperature

  35. Universal thermodynamic functions Internal energy Helmholtz free energy Chemical potential Entropy

  36. A. Bulgac et al. PRL, 96, 090404 (2006). A. Bulgac et al. PRA, 78, 023625(2008).

  37. Different approach to obtain local thermodynamic quantities T.-L. Ho and Q. Zhou, Nature Physics 6, 131 (2010). Ho’s scheme to obtain the local pressure 1D profile 2D profile absorption imaging atom cloud Gibbs-Duhem equation :

  38. Different approach to obtain local thermodynamic quantities S. Nascimbène et al. Nature 463, 1057 (2010). S. Nascimbène et al. New Journal of Physics 12, 103026 (2010). 7Li (bosonic isotope) thermometer Assume second-order virial expansion is correct at high T region.

  39. S. Nascimbène et al. Nature 463, 1057 (2010). • First experimental determination of virial-3 and virial-4 coefficient. contribution of 3-body and 4-body physics to the unitary gas thermodynamics • Observation of Fermi liquid behavior above superfluid Tc.

  40. Observation of Fermi liquid behavior S. Nascimbène et al. Nature 463, 1057 (2010). Fermi liquid theory : Monte-Calro simulation C. Lobo et al, Phys. Rev. Lett. 97, 200403 (2006) by our result from projection

  41. momentum resolved spectroscopy RF J. Stewart et al. Nature 454, 744 (2008). Make a transition from to with an RF pulse. Turn off the trap potential immediately after applying RF pulse. • RF pulse shorter than the trap period • no interaction effect (simple dispersion in the excited state) • no collision during expansion

  42. Observation of pseudogap behavior J. Stewart et al. Nature 454, 744 (2008). ideal gas BEC side (atoms + dimers) on resonance (unitary gas)

  43. Observation of pseudogap behavior J. P. Gaebler et al. Nature Physics 6, 569 (2010). back bending universal behavior of fermions Fermi liquid (no finite-energy excitation above Tc) or Pseudogap (preformed pair formation above Tc)

  44. Momentum distribution and “Contact” experimental verification 1 momentum distribution [kF] k J. Stewart et al. PRL 104, 235301 (2010) L.Viveritet. al,PRA69, 013607 (2004) C = “Contact”

  45. Tan relations by Shina Tan 2005 - Energy relation - Virial theorem in a harmonic trap - Inelastic two-body loss - Adiabatic relation - density-density correlation - Pressure relation universal relations : relations which hold any system consisting of two-component fermions with a large scattering length S. Tan, Ann. Phys. 323, 2952 2008). S. Tan, Ann. Phys. 323, 2971 (2008). S. Tan, Ann. Phys. 323, 2987 (2008). E. Braaten arXiv:1008.2922

  46. Experimental verification of Tan relations J. Stewart et al. PRL 104, 235301 (2010) Testing adiabatic sweep relation Testing virial theorem determined directly determined from C measured separately

  47. Summary • Thermodynamicfunctions of a unitary Fermi gas • was determined at the unitarity limit. M. Horikoshi, S. Nakajima, M. Ueda and T. Mukaiyama, Science, 327, 442 (2010). Ideal • Microscopic mechanism of superfluidity in a unitary Fermi gas is now under intense research. Unitary • Tan relation - new universal thermodynamic formalism -

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