Experimental determination of Universal Thermodynamic Functions for a

Experimental determination of Universal Thermodynamic Functions for a Unitary Fermi Gas Takashi Mukaiyama University of Electro-Communications Japan Science Technology Agency, ERATO M. Horikoshi, S. Nakajima, M. Ueda and T. Mukaiyama, Science, 327, 442 (2010).

ultracold dilute Fermi gas 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.

ultracold dilute Fermi gas as T n-1/3 R one interesting feature of cold atoms: as is tunable!!

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

ultracold dilute Fermi gas as T n-1/3 What happens when… |as| ∞ This situation is called unitarity limit.

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. Neutron star Besides pure theoretical curiosity, this problem is relevant to neutron stars! Wikipedia

Unitarity limit and Universality as T n-1/3 as drops out of thermodynamic description. Thermodynamics depends only by the density n and temperature T. Universal hypothesis (UH)

Universal thermodynamics According to the UH, all thermodynamics should obey the universal functions: Internal energy : Helmholtz free energy : Chemical potential : Entropy : Fermi energy : reduced temperature : Dimensionless universal functions

inhomogeneous density profile of a trapped gas T is constant over the cloud (thermal equilibrium). TF depends on the density. is position-dependent. Global measurement only gives you the integration of all the different phases.

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

Experimental scheme 834 Gauss （ Resonance magnetic field of Feshbach resonance ） Optical dipole trap (1064 nm) 6 Li Equal mixture of N ~ 106 : |F=1/2, m. F=+1/2> : |F=1/2, m. F=-1/2>

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

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

Our scheme This scheme works not only for a unitary Fermi gas but also an ideal Fermi gas!!

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

Verification of the determined f. E [T/TF ] 1. Energy comparison ü Potential energy particle : Comparison ü Internal energy particle : Epot = Eint ! ! cy n e is st on c ce i N

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

Verification of the determined f. E [T/TF ] 2. Effective speed of the first sound Propagation time 0. 1 ms 1. 1 ms 2. 1 ms 3. 1 ms 4. 1 ms 5. 1 ms 6. 1 ms 7. 1 ms

Verification of the determined f. E [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

Verification of the determined f. E [T/TF ] 2. Effective speed of the first sound Experimental values vs. calculated values from f. E [q ] u 1, Meas. = u 1, Calc ! ! cy n e is st on c ce i N

The universal function of the internal energy f. E [T/TF ] Universal hypothesis : Ideal Equation of state : Unitary Mechanical equilibrium : Energy comparison Epot = Eint ce Ni !! y nc e t sis n co Speed of the first sound u 1, Meas. = u 1, Calc cy n e ist s on c ce Ni !!

momentum distribution measurement absorption imaging Bosonic case thermal bimodal lens pure BEC CCD Fermion pair condensate BCS limit BEC limit spatially correlated pair momentum correlated pair

“projection” magnetic field sweep If we sweep the magnetic field BCS - slow enough to convert atom pairs into molecules - fast enough such that the momentum distribution of the projected molecules reflects that of pairs prior to the sweep B BEC C. A. Regal et al. Phys. Rev. Lett. , 92, 040403 (2004) We can convert correlated pairs into tightly-bound molecules. MIT JILA C. A. Regal et al. , PRL 92, 040403 (2004) M. W. Zwierlein et al. , PRL 92, 120403 (2004)

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

Condensate fraction vs Temperature

Universal thermodynamic functions Internal energy Helmholtz free energy Chemical potential Entropy

Summary • The universal function of the internal energy was determined at the unitarity limit Ideal Unitary • The othermodynamic functions were derived from thermodynamic relationship • The critical parameters were determined at the superfluid transition temperature M. Horikoshi, S. Nakajima, M. Ueda and T. Mukaiyama, Science, 327, 442 (2010).

The team (ERATO project) Masahito Ueda (project leader) T. Mukaiyama (Group leader ) M. Horikoshi (Postdoc) S. Nakajima (Ph. D student)