D. Jin

JILA, NIST and the University of Colorado

$ NSF, NIST

Universal Relations in an Ultracold Fermi Gas

Investigate many-body quantum physics with a model system

•  low density, low temperature •  unique expt tools for probing, manipulating •  well understood microscopics •  controllable interactions

Why study atomic gases?

Fermi superfluid

Interactions in an ultracold atom gas

Ultracold atoms interact via a short-range, or contact, interaction.

40K spin ↑

spin ↓

T = 50 nK, n = 1013 cm-3, T/TF = 0.1

Ener

gy

distance between the atoms

deBroglie wavelength (≈d, spacing between particles)

Interactions in an ultracold atom gas

r0

Interactions characterized by the s-wave scattering length

Interactions can be controlled using a Feshbach resonance.

Interactions in an ultracold atom gas

The contact and universal relations

S. Tan, Annals of Physics 323, 2952 (2008); Ibid., p. 2971; Ibid., p. 2987

a breakthrough in our understanding of interacting quantum gases (with short-range interactions)

In 2005, two papers by Shina Tan appeared on the arXiv.org preprint server.

Energetics of a strongly correlated Fermi gas

Large momentum part of a strongly correlated Fermi gas

These papers introduced “the contact”.

What is “the contact”?

•  Units: length-1 •  Extensive property of the sy stem •  Central to exact universal relations, which are

Theory papers: Tan, Leggett, Braaten, Combescot, Baym, Blume, Werner, Castin, Randeria, Strinati,…

few-body or many-body T=0 or finite T homogeneous or trapped gas superfluid or normal strong or weak interactions 50/50 or imbalanced spin mixture

independent of details of the interaction applicable to Fermi or Bose gases independent of the state of the system:

for

1.Momentumdistribu0on

2.Energy

3.Localpairdensity

4.Adiaba0csweeptheorem

5.Pressure

6.Virialtheorem

7.RFlineshape

Tan’s universal relations for

for

What is “the contact”?

C and 1/a are conjugate thermodynamic variables (like P and V, or µ and N)

1/a is the “generalized force” C is the “generalized displacement”

similar to enthalpy, H=U-PV

Consider the adiabatic relation

Measurements of an interacting Fermi gas

Other work: Photoassociation (Rice),

Bragg spectroscopy (Swinburne)

H. Hu et al., arXiv 1001.3200 (2009)

These experiments extract C and compare to theory, but do not directly test any of the universal relations.

Partridge et al., Phys. Rev. Lett. 95, 020404 (2005) F. Werner, L. Tarruell, and Y. Castin, Euro. Phys. J. B 68, 401 (2009)

With 40K atoms, we can measure momentum distribution of atoms by expanding at a=0.

a=0

40K

Measurements of an interacting Fermi gas

Momentum distribution

(in units of kF)

(nor

mal

ized

)

T/TF = 0.11

Momentum distribution

C/k4

k4 n

(k)

k

Momentum distribution

the Contact

(in units of kF)

(kFa)-1 ≈ 0 T/TF = 0.11

Rf spectroscopy

(in units of EF/ħ)

(nor

mal

ized

)

Rf spectroscopy

Γ(ω

)

ω

1/ω3/2

23/2

π2

ω3/

2 Γ

(ω)

ω

the Contact

Rf spectroscopy

(kFa)-1 ≈ 0 T/TF = 0.11

(in units of kF)

The contact from n(k) and Γ(ω)

weak coupling

strong coupling

T/TF = 0.11

1/(kFa)

the

cont

act,

C

Comparison with theory

weak coupling

strong coupling

T/TF = 0.11

T=0 theory line from F. Werner, L. Tarruell, & Y. Castin, Euro. Phys. J. B 68, 401 (2009)

1/(kFa)

the

cont

act,

C

Energy measurements

kinetic energy

E = T + I + V

cloud size in trap cloud size after expansion (release energy)

interaction energy

potential energy

ENS, Innsbruck, Duke, JILA

Energy Measurements E = T+I+V

T/TF=0.11

Adiabatic Sweep Theorem

2π d

E/d

(1/k

Fa)

1/(kFa)

Testing the virial relation T+

I-V

(kFa)-1

(in u

nits

of E

F)

Conclusion •  The “contact” is an important new development

in the understanding of interacting gases.

•  We’ve measured the contact and directly verified 1.  the high-k tail of the momentum distribution 2.  the high-ω tail of rf spectroscopy 3.  the adiabatic relation 4.  the virial relation

•  Future work: use this to probe the physics of strongly interacting gases

J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, PRL104, 235301 (2010)

Thanks.

Con

tact

(kFa)-1

Comparing C with theory

T=0 theory line from F. Werner, L. Tarruell, & Y. Castin, Euro. Phys. J. B 68, 401 (2009)

Con

tact

(kFa)-1

Comparing C with theory

MomentumDistribu0on

Momentum-resolved RF Spectroscopy

With 40K atoms, we can measure momentum distribution of atoms by expanding at a=0.

a=0

40K

Momentum Distribution

1.  Suddenly turn off the trap. 2.  Suddenly turn off interactions.

(fast B ramp to a=0) 3.  Let the cloud expand for 6 ms. 4.  Take an absorption image: OD(x,y) 5.  Take the azimuthal average. 6.  Take inverse Abel transform to get n(k).

Energy measurements

1.  Suddenly turn of the trap. 2.  Allow the cloud to expand for 16 ms. 3.  Take an absorption image: OD(x,z) 4.  Find the release energy from the mean squared cloud widths in

x and z.

1.  Suddenly turn of the trap. 2.  Allow the cloud to expand for 1.6 ms. 3.  Take an absorption image: OD(x, z) 4.  Find the potential energy from the mean squared cloud width in z.

Release energy

Potential energy

RF spectroscopy

1.  Apply a pulse of rf. 2.  Take spin-selective absorption image (at high B). 3.  Count how many atoms appear in the new spin state. 4.  Vary the rf frequency to obtain an rf spectrum.

Turning off the interactions C

onta

ct

ramp rate (G/µs)