Clock Shifts

Clock Shifts

Erich Mueller

Cornell University

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Sourish BasuStefan Baur

Theja De Silva (Binghampton)Dan GoldbaumKaden Hazzard

Outline

• What we want to measure• A tool: Doppler free

spectroscopy– Capabilities

– Challenges

• Probing fermionic superfluidity near Feshbach resonance





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

Ex: modeling condensed matter systems

Profound increase in complexity



How to probe?

Big Question:

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?



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Atomic Spectroscopy

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

Possibly very powerful

E

I() [transfer rate]

0

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

Interaction energy in Fermi gas experiments: 100 kHz

Sharp Spectral linesHyperfine 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

Solid: condensedOpen: non-condensed

Spectrum gives histogram of density



BEC Density bump

Exp: (Kleppner group) PRL 81, 3811 (1998)Theory: Killian, PRA 61, 033611 (2000)

[OSU connection -- Oktel]

Center of cloud

Why is density histogram useful?Optical absorption: column density

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

Bose-Mott physicsOptical lattice:

Weak interactions: atoms delocalize -- superfluid-- Poisson number distribution

Strong interactions: suppress hopping -- insulator

Kinetic energy from hopping dominates

Energy cost of creating particle-hole pair exceeds hopping

Phase Diagram

Incommensurate:

(lines of fixed density)

“extra” particles delocalize

Wedding CakesTrap: spatially dependent

Hard to see terraces in column densities

r

1

2

3

4

5

n

Discontinuities Cusps

r

nc

RF Spectroscopy



12345

Exp: Ketterle group [Science, 313, 649 (2006)] Thy: Hazzard and Mueller [arXiv:0708.3657]

Discrete bumps: density plateaus

Spectral shift proportional to density

Sensitivity

Significant peaks, even in superfluid

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

Spatially resolved





Column densities



So simple?Spectrum knows about more than density!

Jin group [Nature 424, 47 (2003)]Ex: RF dissociation - Potassium Molecules

Initially weakly bound pairs in

(and free atoms in these states)

-9

2

-72

-5

2

-3

2

-1

2

12

32

52

7

2

9

2

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

Free atoms

pairs

(Thermal, non-superfluid fermionic gas)


are needed to see this picture.Transfer

[kHz]

Ex: RF dissociation - Lithium Molecules


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B [Gauss]

All 1+2 atoms in molecular bound state

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

Related work

(note reversal of sign of shift)

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

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 ksingle free b-atom

Example: BCS state -- darker = larger spectral density

k

k k



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 RuleMehmet O. Oktel, Thomas C. Killian, Daniel Kleppner, L. S. Levitov,Phys. Rev. A 65, 033617 (2002)

Mean clock shift

Ex: Born approximation point interaction

Problem

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

Tails dominate sum rule

Pethick and Stoof, PRA 64, 013618 (2001)

w

I

(unmeasurable)

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



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

Strongly interacting superfluid

BCS-BEC crossover -- Randeria

Outline

• Homogeneous lineshapes within BCS model of superfluid

• Crude model for trapped gas– Highly polarized limit (normal state)– Demonstrates universality of line shape

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

Variational ModelIdea: 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]

Result

Bound-Free

Bound-Bound

Many-body

Typical spectra

1-2 paired drive 2-3 1-2 paired drive 1-3

(most spectral weight is in delta function)

Experiment

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

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

Perali, Pieri, StrinatiarXiv:0709.0817

Summary: Homogeneous Lineshape

• Final state interactions crucial:– Is there a bound state?– Distorted spectrum if resonance in continuum– Sets scale

Next: trap

Inhomogeneous line shapesMost experiments show trap averaged lineshape

Grimm group, Science 305, 1128 (2004)

Bimodality:due to trap

Where spectral weight comes fromMassignan, Bruun, and Stoof, ArXiv:0709.3158

Edge of cloud

Calculation in normal state: Ndown<NupMore particles at center

Generic propertiesHighly 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

Bimodalitynup

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 singularity

Quantitative

Nozieres and Schmidt-Rink

(no adjustable params)

Calculating Free Energy(Only if asked)

If ndown is small, is only function of up and x=-up-dn.

Arctan vanishes for negative x [so is large]

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

Dec

reas

ing

T/T

F

Decreasing a/

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)

Fin

Clock Shifts

Documents

density plateausspectral

density histogram useful

density clock shiftapplication

weak interactions

particlehole pair

ground state

body statepossibly

elementary particles