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Clock Shifts. Sourish Basu Stefan Baur Theja De Silva (Binghampton) Dan Goldbaum Kaden Hazzard. Erich Mueller Cornell University. Outline. What we want to measure A tool: Doppler free spectroscopy Capabilities Challenges Probing fermionic superfluidity near Feshbach resonance.

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clock shifts

Clock Shifts

Sourish Basu

Stefan Baur

Theja De Silva (Binghampton)

Dan Goldbaum

Kaden Hazzard

Erich Mueller

Cornell University

  • What we want to measure
  • A tool: Doppler free spectroscopy
    • Capabilities
    • Challenges
  • Probing fermionic superfluidity near Feshbach resonance
take home message
Take-Home Message
  • 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)

Ketterle Group: Science 316, 867-870 (2007)

“Pairing without Superfluidity: The Ground State of an Imbalanced Fermi Mixture”

context upcoming cold atom physics
Context: Upcoming Cold Atom Physics

Profound increase in complexity

Ex: modeling condensed matter systems

Big Question:

How to probe?

what we want to know
What we want to know
  • Is the system ordered? (crystaline, magnetic, superconducting, topological order)
  • What are the elementary excitations?
  • How are they related to the elementary particles?
atomic spectroscopy
Atomic Spectroscopy

I(w) [transfer rate]





Narrow spectral line in vacuum: in principle sensitive to details of many-body state

Measured hyperfine linewidth ~ 2 Hz [PRL 63, 612, 1989]

Interaction energy in Fermi gas experiments: 100 kHz

Possibly very powerful

sharp spectral lines
Sharp Spectral lines

Hyperfine spectrum: nuclear spin flips (cf. NMR)

“Forbidden” optical transitions: Hydrogen 1S-2S

Couple weakly to environment: influenced by interactions? Does internal structure of atom depend on many-body state?

(weak coupling)

(weak coupling)

Line shift proportional to density [Clock Shift]

application detecting bec

Center of cloud

Application -- Detecting BEC


Density bump

Spectrum gives histogram of density

Solid: condensedOpen: non-condensed

Exp: (Kleppner group) PRL 81, 3811 (1998)

Theory: Killian, PRA 61, 033611 (2000)

[OSU connection -- Oktel]

why is density histogram useful
Why is density histogram useful?

Optical absorption: column density

obscures interesting features -- ex. Mott Plateaus -- digression

bose mott physics
Bose-Mott physics

Optical lattice:

Kinetic energy from hopping dominates

Weak interactions: atoms delocalize -- superfluid

-- Poisson number distribution

Energy cost of creating particle-hole pair exceeds hopping

Strong interactions: suppress hopping -- insulator

phase diagram
Phase Diagram

(lines of fixed density)


“extra” particles delocalize

wedding cakes

Discontinuities Cusps

Wedding Cakes

Trap: spatially dependent m

Hard to see terraces in column densities

rf spectroscopy
RF Spectroscopy

Exp: Ketterle group [Science, 313, 649 (2006)]

Thy: Hazzard and Mueller [arXiv:0708.3657]






Spectral shift proportional to density

Discrete bumps: density plateaus


Significant peaks, even in superfluid

Q: could this be used to detect other corrugations? FFLO? CDW?

spatially resolved
Spatially resolved

Column densities

so simple
So simple?

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)


Drive mf=-5/2 to mf=-7/2

related work


n [kHz]

Related work

Ex: RF dissociation - Lithium Molecules

B [Gauss]

All 1+2 atoms in molecular bound state

(note reversal of sign of shift)

Grimm group [Science 305, 1128 (2004)]Background: Ketterle group [Science 300, 1723 (2003)

what is probed by rf spectroscopy
What is probed by RF spectroscopy?

Single Component Bose system:

Excite with perturbation

Final state has Hamiltonian

Fermi’s Golden Rule

(pseudospin susceptibility)

simple limits i
Simple Limits 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

Example: BCS state -- darker = larger spectral density





simple limits ii
Simple Limits II

Final state interacts same as initial (V(ab)=V(bb)), and dispersion is same

  • Coherent spin rotation

Formally can see from X acts as ladder operator

general case sum rule
General Case -- Sum Rule

Mehmet O. Oktel, Thomas C. Killian, Daniel Kleppner, L. S. Levitov,

Phys. Rev. A 65, 033617 (2002)

Mean clock shift

Ex: Born approximation point interaction


Not a low energy observable!!!!!! -- dif potentials = dif results

Tails dominate sum rule


Pethick and Stoof, PRA 64, 013618 (2001)

summary of spectroscopy
Summary of spectroscopy
  • Weak coupling
    • peak mostly shifted (proportional to density)
    • long tails (probably unobservable)
  • final interaction = initial
    • Peak sharp and unshifted
  • General
    • No simple universal picture
      • sum rules are ambiguous
    • Important for experiments on strongly interacting fermionic Lithium atoms
lithium near feshbach resonance
Lithium near Feshbach resonance

Innsbruck expt grp +NIST theory grp, PRL 94, 103201 (2005)

Strongly interacting superfluid

BCS-BEC crossover -- Randeria


(what is RF lineshape -- and what does it tell us)

  • Homogeneous lineshapes within BCS model of superfluid
  • Crude model for trapped gas
    • Highly polarized limit (normal state)
    • Demonstrates universality of line shape
variational model
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

[Exact if (final int)=(initial int) or if (final it)=0]





typical spectra
Typical spectra

1-2 paired drive 2-3

1-2 paired drive 1-3

(most spectral weight is in delta function)


Perali, Pieri, StrinatiarXiv:0709.0817


Ketterle group: Phys. Rev. Lett. 99, 090403 (2007)

Sant-Feliu update: has seen “bound-bound”

summary homogeneous lineshape
Summary: Homogeneous Lineshape
  • Final state interactions crucial:
    • Is there a bound state?
    • Distorted spectrum if resonance in continuum
    • Sets scale

Next: trap

inhomogeneous line shapes
Inhomogeneous line shapes

Most experiments show trap averaged lineshape

Grimm group, Science 305, 1128 (2004)

Bimodality:due to trap

where spectral weight comes from
Where spectral weight comes from

Massignan, Bruun, and Stoof, ArXiv:0709.3158

Edge of cloud

Calculation in normal state: Ndown<Nup

More particles at center

generic properties

Highly polarized limit: only one down-spin particle

Generic properties

Assumption: local clock shift =

(homogeneous spectrum peaks there)

High temp:

[Virial expansion: Ho and Mueller, PRL 92, 160404 (2004)]

High density:

Different a





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 singularity


Nozieres and Schmidt-Rink

(no adjustable params)

calculating free energy
Calculating Free Energy

(Only if asked)

If ndown is small, q is only function of mup and x=w-mup-mdn.

Arctan vanishes for negative x [so w is large]

summary trap
Summary: Trap
  • Trap leads to bimodal spectrum (model independent)
  • Simple model using NSR energy: energy scales work, temp scales seem a bit off
  • Final state interactions: mostly scale spectrum

Decreasing T/TF

Decreasing a/l

summary spectroscopy
Summary -- Spectroscopy
  • Powerful probe of local properties
    • Density: SF-Mott
  • Simple when interactions are weak
  • Open Q’s when interactions are strong
  • Bimodal RF spectra are not directly related to pairing (implicit in works of Torma and Levin)