Probing Fluctuating OrdersLuttinger Liquids to Psuedogap States

Ashvin Vishwanath

UC Berkeley

With: Ehud Altman (Weizmann)and Ludwig Mathey (Harvard)

E. Altman and A.V. PRL 2005 and L. Mathey, E. Altman and A. V. cond-mat/0507108

Order Vs. Disorder• Phases distinguished by order

parameters – spontaneous symmetry breaking.

• Order can be destroyed via:– Thermal Fluctuations– Quantum Fluctuations

• D=1 systems, Mermin Wagner theorem• Phases with Topological Order (eg. Fractional

Quantum Hall States)• Mott Insulators, charged superconductors,

integer quantum Hall systems.

Measuring Order• Supefluid Order:

– nboson(k=0)

• Fermion Pair Superfluid– nfermion(k) similar to Fermi Gas at

finite T.– BUT, signature in Noise

correlations

1 trap

Anderson etal, Science (95)

(k,-k) pairs

n(r)

n(-r)

Probing Fluctuating Order• Part 1.

– Thermally Fluctuating paired superconductor, near resonance. Probed via dynamics.

0V0

pairingcV-

T

Tc

Eb

Phase fluctuation induced psuedogap

conventional sc

• Part 2.– 1 D quantum systems probed via

Noise correlations

Feshbach Resonance: Two Atom Problem

JILA Expts on K40

Feshbach resonance: Many body problem

• For gs/ Ef >>1, ‘wide resonance’. Can integrate out the molecules to get theory of just atoms [c].

• Effective interaction leads to a scattering length a.

)..cc(bgbbBm4

kcc

2m

kH

q-kqkq

kkB

2

k-k

2

ch

Now N atoms; density n

Fermi Energy Ef (~10kHz=100nK for K40 expts)

Coupling gs=g√n

Ratio gs/ Ef = 8 (K40)

= 200 (Li6).

Ramping Through a Freshbach Resonance

Timescales:

1. Adiabatic

2. Non-Equilibrium growth (Anderson; Barankov,

Levitov,Spivak, Altshuler)

3. Fast (considered here)

2Min2

A

1

2A

2s

1g

2sg

a

• In conventional superconductors, typical gap ~ 1Kelvin => Time scale 1010Hz.

• Here, gap ~ 100nK => Time scale in kHz. + Long relaxation times– highly non-equilibrium quantum

many body states.

Adiabatic ramp through resonance

• Slow sweep across resonance. Rate ≈1msec/Gauss

• No start position dependence.

M. Greiner, C. Regal, D. Jin Nature 426, 537 (2003)

N0 Molecular

condensate

t=0

Measurement: Probe Molecules

Fast ramp through the resonance

C. Regal, M. Greiner, D. Jin PRL (2004)

B

aMeasure

Molecules Atoms

Start position dependence on final state molecular condensate

Is this a faithful reflection of initial eqlbrm properties?

Ramp rate =50μsec/Gauss‘BCS’‘BEC’

Also Zwierlein et al. PRL (2004). [6Li]

• Sudden approx: final state=initial state.Evaluate molecule population nm(q)

Sudden Approximation to Ramping

Diener and Ho, cond-mat/0404517 :

0|)ccv(u|k-kkk

i

• Assume variational initial state (fix N, a in initial state)

with:

Molecular wavefn. in final state

i|and

Sudden Approximation

• Naïve Expectation:

– Final molecule size :

– Cooper pair size:

– Therefore expect:

– BUT

kkk-kc vucc)k(

Cooper pair wavefn:

N0 =condensed mol.

→Cooper pair/Mol. overlap

2

c3

0 )k()k(dkN

0a

430

3

00 10)a(

aN

fk

Cooper pair size

0a

Sudden Approximation

• While normal molecule number Nn:

• Condensed molecules (from the integral):

• Reason: – short distance singularity of Cooper pair wavefn.

30

n )a(N

Nfk

)a(EN

N0

2

f

0fk

kkkc E2

vu)k(

2c k

m)k(

hence

r

1

r

1)0r(c

Cooper pairs can be efficiently converted to molecules

Nn

N0

1. No Dynamics – no dependence on ramp rate

Effective dynamics for fast sweeps

Include fluctuations with RPA (Not all Cooper Pairs are condensed)

Limitations of the Sudden Approximation

Altman and A.V., PRL (2005)

Effective dynamics for fast sweeps

• For finite sweep rates, if molecule binding energy is large, ramping not sudden. Changes character when:

*)(*)( 2 aEadt

dEb

b

• Approximate subsequent evolution as adiabatic.

(eg. Kibble-Zurek, defects generated in a quench)

• Project onto Molecules of size– Correct parametric dependencies– Checked against exact numerics in Dicke model– Assumes – Dynamics (2 body). Initial state (many body)

)(a

a

~Sudden~Adiabatic

a*a0

2s

2b g*)(E a (fast)

3/1

3/2g*

s

f ak

/1)a(N 3m

fk

Conversion efficiency vs ramp rate I

• Projection effectively onto molecules of size

• Cooper pair conversion efficiency– Slow dependence on ramp rate

• Incoherent conversion (non-Cooper pair)

– Strong dependence on ramp rate

3

1

/1a

3

1

0 /1aN fk

verified by: Barankov and Levitov, Pazy et al.

• Only q=0 molecules – no phase fluctuations.• Similar to BCS pairing Hamiltonian. • Anderson spin representation – classical spin dynamics• Ramp in time T:

– Solve evolution numerically and count molecules at the end

Numerical check: dynamics of Dicke model

Paired

(far from resonance)

Scaling consistent with 2 stage dynamics!

fi

Unpaired

Conversion Efficiency vs. Ramp Rate II

• Preliminary check against

experimental data: – fast sweep molecule number vs.

cubic root of inverse ramp speed.

– Most data not in fast sweep regime (eg. 50μsec/Gauss)

Data: JILA exp 40K. M. Greiner (private comm)

Cooper Pairs (?)

0 1 2 3 40

2

4

6

8

10x 104

8 37 s/G

311

x104

Regime of Validity in K40 JILA expts.

• Requires

[Inv. Ramp Speed] < 60μsec/Gauss

2sg

Nm

ol (

104 )

JILA expt. 40K:

Nm

Effect of Fluctuations

• Take fluctuations into account using RPA (Engelbrecht, Randeria, de Melo)

Phase fluctuations (finite q Cooper pairs) in ground state.V(x)

x

`BCS’

RPA

RPABCS

1. condensed cooper pairs

2. uncorrelated pairs AND

3. uncondensed cooper pairs(phase fluctuations)

Summary of Part 1

• Fast ramping across resonance - sensitive probe of pairing.– Identify by ramp rate dependence.

– Sensitive to pairs both condensed and not.

• Study pairing in the psuedogap state?

• Momentum dependence of pairs? [npair(k)]

• Useful to study polarized Fermi systems? – finite center of mass pair fluctuations.

Nm

1x

3/1~ x

x~

paired

unpaired

B

T

psuedogap

SF FL

Nature (March 2005)

PRL (2005)

Noise correlations in Mott insulator of Bosons:

n(k)Foelling et. al. (Mainz) G(k-k’)

Recent Shot Noise Experiments

Luttinger Parameters from Noise Correlations

• Simplest Case – spinless fermions on a line.– Direct realization: single species of fermions,

interactions via Bose mixture/p-wave Feschbach res.– Single phase: Luttinger liquid. – Asymptotics: power law correlations characterized by

(vF,K), with K<1 (repulsive).

-kF kF

xkiLR

FexxO 2)()( CDW

)()( xxOLR SC

K

xki

x

eOxO

F

2

2

~)0()( CDWCDW

Kx

OxO 2

1~)0()(

SCSC

Fluctuating Orders

Typically -> can measure CDW power law from scattering. Noise measurement sensitive to both CDW/SC. fk21

Luttinger Parameters from Noise Correlations

• Correlations of Atom Shot Noise: ')',( qkqk FFnnqqG

-kF kF

X X

-kF kF

X X

qq

q-q

q

q’CDW

SC

Calculate using Bosonization:

Luttinger Parameters From Noise Correlations

K=0.4

K=2.5

q

CDW

SC

q’

K<1/2

K>2

For ½<K<2

)'

1,)1(~)',( 2

|q||q|

1Min()q'Sgn(q)Sgn( KqqG

K=0.8

K=1.25

Noise Correlations – Fermions with spin in D=1

• Fermions on a line – two phases, Luttinger Liquid and spin-gapped Luther-Emery liquid (depending on the sign of backscattering g)

SDW / CDW T-SC /SSC

CDW S-SCS-SC/CDW (cusp)

K21/2

CDW/S-SC (cusp)

q’ q

g

Spin-gap

Molecular condensate fraction

• (condensed)/(total molecules) independent of ramp speed for fast ramps. (both arise from Cooper pairs).

-1 1

JILA Expts

Expect non monotonic condensate fraction at very fast sweeps

Probe of uncondensed pairs