probing fluctuating orders luttinger liquids to psuedogap states ashvin vishwanath uc berkeley with:...
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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
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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.
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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)
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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
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Feshbach Resonance: Two Atom Problem
JILA Expts on K40
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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).
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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.
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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
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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]
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• 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
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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
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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
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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)
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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
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/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.
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• 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
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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
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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)
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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
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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
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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
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Luttinger Parameters from Noise Correlations
• Correlations of Atom Shot Noise: ')',( qkqk FFnnqqG
-kF kF
X X
-kF kF
X X
q-q
q
q’CDW
SC
Calculate using Bosonization:
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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
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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
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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