Clock Shifts

Clock Shifts

Erich Mueller

Cornell University

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

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?



QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

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


are needed to see this picture.QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.

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

Documents

Clock Shifts