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 (Fermi liquid or pseudogap?) Tan relation new, universal description of thermodynamics for interacting fermions

Cold atoms are very dilute (10 11 ~10 14 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 BEC in a cold atom system

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

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

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

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

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

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

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

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

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. Universal thermodynamics pure number

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

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

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

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

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

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

MOT （ magneto-optical trap ） T = 200  K N > 10 8 Deceleration and trapping Li oven T~700K

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

6 Li 原子の s 波散乱長 ( a 0 ) 磁場 [G] B = G molecular BEC B = G degenerate Fermi gas 6 Li in quantum degenerate regime B I I

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

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

Our scheme

Experimental determination of f E [  F ] M. Horikoshi, S. Nakajima, M. Ueda and T. Mukaiyama, Science, 327, 442 (2010). Ideal Unitary About 800 images are analyzed.

Verification of the determined f E [  F ] 1.Energy comparison Potential energy par particle : Internal energy par particle : Comparison E pot = E int

Verification of the determined f E [  F ] 2. Effective speed of the first sound 6 Li Light pulse to make density perturbation

Verification of the determined f E [  F ] 0.1ms 1.1ms 2.1ms 3.1ms 4.1ms 5.1ms 6.1ms 7.1ms Propagation time 2. Effective speed of the first sound

Verification of the determined f E [  F ] Unitary gas shows hydrodynamic behavior due to the large collision rate Effective speed of the first sound : Comparison Experiment [ P. Capuzzi, PRA 73, (R) (2006) ] 2. Effective speed of the first sound

Verification of the determined f E [  F ] Experimental values vs. calculated values from f E [  ] 2. Effective speed of the first sound

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

“Projection” projection … convert correlated pair of atoms into tightly-bound molecules by sweeping the magnetic field toward BEC side of the resonance W. Ketterle et al.,arXiv: 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

Bimodal distribution of a fermion pair condensate

Condensate fraction vs Temperature

Internal energy Universal thermodynamic functions Helmholtz free energy Chemical potential Entropy

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

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

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

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 T c.

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

momentum resolved spectroscopy RF - RF pulse shorter than the trap period - no interaction effect (simple dispersion in the excited state) - no collision during expansion 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.

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

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 T c ) or Pseudogap (preformed pair formation above T c )

momentum distribution 1 k L.Viverit et. al,PRA 69, (2004) J. Stewart et al. PRL 104, (2010) [kF][kF] experimental verification C = “Contact” Momentum distribution and “Contact”

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

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

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