1. Probes of Pairing in Strongly Interacting Fermions
Erich Mueller --Cornell University
Sourish Basu
Stefan Baur
Theja De Silva (Binghampton)
Dan Goldbaum
Kaden Hazzard

2. Outline A tool: Doppler free spectroscopy
Capabilities
Challenges
Probing fermionic superfluidity near Feshbach resonance

3. Take-Home Message - I
RF/Microwave spectroscopy does tell you details of the many-body state
Weak coupling -- density
Strong coupling -- complicated by final-state effects
Bimodal RF spectra in trapped Fermi gases not directly connected to pairing (trap effect)
Chin et al. Science 20, 1128 (2004)
Abs
dn [kHz]
Decreasing T
Ketterle Group: Science 316, (2007)
“Pairing without Superfluidity: The Ground State of an Imbalanced Fermi Mixture”

4. Take Home Message II
Homogeneous RF spectrum: two components -- bound-bound and bound-free
Final state effects are crucial -- qualitative role
Bound-Free
Bound-Bound
Basu and Mueller, arXiv:

5. Context: Upcoming Cold Atom Physics
Profound increase in complexity
Ex: modeling condensed matter systems
Big Question:
How to probe?

6. Atomic Spectroscopy I(w) [transfer rate] E w w w0
Narrow hyperfine spectral line in vacuum (Hz): in principle sensitive to details of many-body state (Eint~100kHz)
(weak coupling)
(weak coupling)
Line shift proportional to density [Clock Shift]

7. Applications Spectrum gives histogram of density BEC: Density bump
Solid: condensed Open: non-condensed
Exp: (Kleppner group) PRL (1998)
Theory: Killian, PRA (2000)
Mott Shells:
Exp: Ketterle group [Science, 313, 649 (2006)]
Density Plateaus
Thy: Hazzard and Mueller [arXiv: ]

8. Pairs Spectrum knows about more than density!
Jin group [Nature 424, 47 (2003)]
Ex: RF dissociation - Potassium Molecules
(Thermal, non-superfluid fermionic gas)
Free atoms
Initially weakly bound pairs in
(and free atoms in these states)
pairs
Drive mf=-5/2 to mf=-7/2
Also see: Grimm group [Science 305, 1128 (2004)] Ketterle group [Science 300, 1723 (2003)

9. Many-Body Simple limit I: Final state does not interact (V(ab)=0)
analogous to momentum resolved tunneling (or in some limits photoemission)
probe all single particle excitations
Initial: ground state
Final: single a-quasihole of momentum k single free b-atom
Simple Limits II: Final state interacts same as initial (V(ab)=V(bb)),
Ladder operator
General Case: System specific

10. Lithium near Feshbach resonance
Innsbruck expt grp +NIST theory grp, PRL 94, (2005)
Strongly interacting superfluid
Strong final state interactions!!!
BCS-BEC crossover

11. Basu and Mueller, arXiv:0712.1007
Variational Model
Idea: include all excitations consisting of single quasiparticles quasiholes
“coherent contribution” -- should capture low energy structure
a-b pairs -- excite from b to c
Neglects multi-quasiparticle intermediate states
(equivalent to BCS-RPA A. Perali, P. Pieri, G.C. Strinati, arXiv: )
[Exact if (final int)=(initial int) or if (final int)=0]

12. Result
Bound-Free
Bound-Bound
Experiment
Many-body

13. Typical spectra

14. What about the trap? Most experiments show trap averaged lineshape
Grimm group, Science 305, 1128 (2004)
Bimodality: due to trap
Thy: Kinnunen, Rodriguez, and Torma, Science 305, 1131 (2004); Heiselberg NJP 6, 137 (2004); Chin and Julienne, PRA 71, ; Ohashi and Griffin, PRA 72, (2005); He, Chen, and Levin, PRA 72, (2005); Yu and Baym, PRA 73, (2006); He, Chien, Chen, Levin, arXiv: ; Baym, Pethick, Yu, and Zwierlein, arXiv: ; Punk and Zwerger, arXiv: ; Perali, Pieri, Strinati, arXiv: ; Massignan, Bruun, and Stoof, arXiv:

15. Where spectral weight comes from
Massignan, Bruun, and Stoof, ArXiv: (Neglects Final state interaction)
Edge of cloud
Calculation in normal state: Ndown<Nup
More particles at center
No superfluidity!! Pairing?
Also see He, Chen and Levin, PRA 72, (2005)

16. Generic properties Highly polarized limit: only one down-spin particle
Assumption: local clock shift =
(homogeneous spectrum peaks there)
High temp:
[Virial expansion: Ho and Mueller, PRL 92, (2004)]
High density:
Different a

17. Bimodality
nup
ndn r
Center of trap: highest down-spin density -- gives broad peak
Edge of trap: low density, but a lot of volume
-- All contribute at same detuning
-- Gives power law-log singularity

18. Quantitative Nozieres and Schmidt-Rink (no adjustable params)
Ketterle Group: Science 316, (2007)
Nozieres and Schmidt-Rink
Mueller, ArXiv:
Agreement: fortuitous, but
gives scaling with parameters
(no adjustable params)

19. Summary
Trap leads to bimodal spectrum (model independent) regardless of pairing
Homogeneous spectrum can reveal pairing (bound-bound transition) but final state interactions are crucial