1. Theory of RF spectroscopy in Strongly Interacting Fermi Gases
Erich Mueller --Cornell University
Sourish Basu
Stefan Baur
Theja De Silva (Binghampton)
Dan Goldbaum
Kaden Hazzard
Thanks: Chin, Fumarola, Ho, Ketterle, Levin, Shin, Stoof, Strinati, Torma, Leggett, Zwerger

2. Outline Summary Spectroscopy Pseudogap Final State interactions
Trap inhomogeneities

3. Take Home Message I
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:

4. Take-Home Message - II
RF/Microwave spectroscopy 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
Mueller, ArXiv:
Ketterle Group: Science 316, (2007)
“Pairing without Superfluidity: The Ground State of an Imbalanced Fermi Mixture”
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:

5. Take-Home Message - IIIN S
Forbidden
Polarization
Pseudogap feature moves up in energy as polarization increases

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

7. 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]

8. 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: ]

9. 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)

10. Fermi gases: What to explore?
What is interesting/new about unitary Fermi gases?
A: Pseudogap
[Levin, Randeria, Sa de Melo, Stoof…]
Gap: Superfluid No low energy fermionic excitations
Atoms bound in condensed pairs
Colloquial pic: energy cost of breaking pairs gives gap
More precise: quantum interference of particle and hole states
Pseudogap: Normal Few low energy fermionic excitations
Atoms bound in non-condensed pairs:
Gap “blurred out” by incoherently adding contributions from pairs with different momenta

11. What is pseudogap? (in BCS-BEC crossover literature)
BCS spectral density (what is energy of excitations with momentum k?)
Idea: two ways to add particle
Simply insert particle

12. What is pseudogap? (in BCS-BEC crossover literature)
BCS spectral density (what is energy of excitations with momentum k?)
Idea: two ways to add particle
Simply insert particle
Insert pair, and a hole
(adding pair leaves state unchanged -- Condensate)

13. What is pseudogap? (in BCS-BEC crossover literature)
BCS spectral density (what is energy of excitations with momentum k?)
Idea: two ways to add particle
Simply insert particle
Insert pair, and a hole
States hybridize

14. What is pseudogap? (in BCS-BEC crossover literature)
BCS spectral density (what is energy of excitations with momentum k?)
Idea: two ways to add particle
Simply insert particle
Insert pair, and a hole
States hybridize
Add “coherence factors”

15. What is pseudogap? Superfluid (in BCS-BEC crossover literature)
Pseudogap spectral density (what is energy of excitations with momentum k?)
Normal
Insert particle

16. What is pseudogap? Superfluid (in BCS-BEC crossover literature)
Pseudogap spectral density (what is energy of excitations with momentum k?)
Normal
Simply insert particle
Insert hole and pair
Many ways to do this

17. What is pseudogap? Superfluid (in BCS-BEC crossover literature)
Pseudogap spectral density (what is energy of excitations with momentum k?)
Normal
Simply insert particle
Insert hole and pair
Many ways to do this
Hybridize + +
+ …

18. What is pseudogap? Superfluid (in BCS-BEC crossover literature)
Pseudogap spectral density (what is energy of excitations with momentum k?)
Normal
Structures persist at weaker coupling: (less broadening)

19. How does pseudogap evolve with polarization?
Fumarola and Mueller, arXiv: T N S
Forbidden
Sharp fermions at Fermi energy
Pseudogap smearing pushed to finite energy
Pairs exist as excitations
Polarization

20. How does pseudogap evolve with polarization?
Fumarola and Mueller, arXiv: T N S
Forbidden
Sharp fermions at Fermi energy
Pseudogap smearing pushed to finite energy
Pairs exist as excitations
Polarization

21. RF Spectroscopy Simple limit: 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
Should reveal pseudogap!!

22. Problem: Strong Final State Interactions
Innsbruck expt grp +NIST theory grp, PRL 94, (2005)
Strongly interacting superfluid
BCS-BEC crossover

23. 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]

24. Result: initial state at unitarity
Bound-Free
Bound-Bound
Experiment
Many-body

25. “Phase Diagram”

26. Typical spectra
Missing in this calculation: finite lifetime of final states
Experiment: significant broadening
Ketterle group: Phys. Rev. Lett. 99, (2007)

27. What does trap averaging do?
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)

28. Back-of envelope argument
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

29. 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

30. 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)

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