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

Outline Summary Spectroscopy Pseudogap Final State interactions Trap inhomogeneities

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:0712.1007

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:0711.0182 Ketterle Group: Science 316, 867-870 (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, 012713; Ohashi and Griffin, PRA 72, 013601 (2005); He, Chen, and Levin, PRA 72, 011602 (2005); Yu and Baym, PRA 73, 063601 (2006); He, Chien, Chen, Levin, arXiv:0707.2625; Baym, Pethick, Yu, and Zwierlein, arXiv:0707.0859; Punk and Zwerger, arXiv:0707.0792; Perali, Pieri, Strinati, arXiv:0709.0817; Massignan, Bruun, and Stoof, arXiv:0709.3158

Take-Home Message - III N S Forbidden Polarization Pseudogap feature moves up in energy as polarization increases

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?

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]

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:0708.3657]

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)

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

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

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)

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

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”

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

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

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 + + + …

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)

How does pseudogap evolve with polarization? Fumarola and Mueller, arXiv:0706.1205 T N S Forbidden Sharp fermions at Fermi energy Pseudogap smearing pushed to finite energy Pairs exist as excitations Polarization

How does pseudogap evolve with polarization? Fumarola and Mueller, arXiv:0706.1205 T N S Forbidden Sharp fermions at Fermi energy Pseudogap smearing pushed to finite energy Pairs exist as excitations Polarization

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!!

Problem: Strong Final State Interactions Innsbruck expt grp +NIST theory grp, PRL 94, 103201 (2005) Strongly interacting superfluid BCS-BEC crossover

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:0709.0817 + [Exact if (final int)=(initial int) or if (final int)=0]

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

“Phase Diagram”

Typical spectra Missing in this calculation: finite lifetime of final states Experiment: significant broadening Ketterle group: Phys. Rev. Lett. 99, 090403 (2007)

What does trap averaging do? Massignan, Bruun, and Stoof, ArXiv:0709.3158 (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, 011602 (2005)

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, 160404 (2004)] High density: Different a

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

Quantitative Nozieres and Schmidt-Rink (no adjustable params) Ketterle Group: Science 316, 867-870 (2007) Nozieres and Schmidt-Rink Mueller, ArXiv:0711.0182 Agreement: fortuitous, but gives scaling with parameters (no adjustable params)

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