Universal Relations in an Ultracold Fermi Gas
Transcript of Universal Relations in an Ultracold Fermi Gas
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)