1. Probing Fluctuating Orders Luttinger 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

2. 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.

3. Measuring Order Supefluid Order: –n boson (k=0) Fermion Pair Superfluid –n fermion (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)

4. Probing Fluctuating Order Part 1. –Thermally Fluctuating paired superconductor, near resonance. Probed via dynamics. pairing TcTc EbEb Phase fluctuation induced psuedogap conventional sc Part 2. –1 D quantum systems probed via Noise correlations

5. Feshbach Resonance: Two Atom Problem JILA Expts on K 40

6. Feshbach resonance: Many body problem For g s / E f >>1, ‘wide resonance’. Can integrate out the molecules to get theory of just atoms [c]. Effective interaction leads to a scattering length a. Now N atoms; density n Fermi Energy E f (~10kHz=100nK for K 40 expts) Coupling g s =g√n Ratiog s / E f = 8 (K 40 ) = 200 (Li 6 ).

7. Ramping Through a Freshbach Resonance Timescales: 1.Adiabatic 2.Non-Equilibrium growth (Anderson; Barankov, Levitov,Spivak, Altshuler) 3.Fast (considered here) In conventional superconductors, typical gap ~ 1Kelvin => Time scale 10 10 Hz. Here, gap ~ 100nK => Time scale in kHz. + Long relaxation times –highly non-equilibrium quantum many body states.

8. 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) N 0 Molecular condensate t=0 Measurement: Probe Molecules

9. Fast ramp through the resonance C. Regal, M. Greiner, D. Jin PRL (2004) B a MeasureMeasure 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). [ 6 Li]

10. Sudden approx: final state=initial state. Evaluate molecule population n m (q) Sudden Approximation to Ramping Diener and Ho, cond-mat/0404517 : Assume variational initial state (fix N, a in initial state) with: Molecular wavefn. in final state and

11. Sudden Approximation Naïve Expectation: –Final molecule size : –Cooper pair size: –Therefore expect: –BUT Cooper pair wavefn : N 0 =condensed mol. →Cooper pair/Mol. overlap Cooper pair size

12. Sudden Approximation While normal molecule number N n : Condensed molecules (from the integral): Reason: –short distance singularity of Cooper pair wavefn. hence Cooper pairs can be efficiently converted to molecules NnNn N0N0

13. 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)

14. Effective dynamics for fast sweeps For finite sweep rates, if molecule binding energy is large, ramping not sudden. Changes character when: 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 ~Sudden ~ Adiabatic a* a0a0 (fast)

15. 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 verified by: Barankov and Levitov, Pazy et al.

16. 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! Unpaired

17. 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 40 K. M. Greiner (private comm) Cooper Pairs (?) x10 4 Regime of Validity in K 40 JILA expts. Requires [Inv. Ramp Speed] < 60μsec/Gauss N mol (10 4 ) JILA expt. 40 K: NmNm

18. 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 1.condensed cooper pairs 2. uncorrelated pairs AND 3.uncondensed cooper pairs (phase fluctuations)

19. 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? [n pair (k)] Useful to study polarized Fermi systems? – finite center of mass pair fluctuations. NmNm paired unpaired psuedogap

20. 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

21. 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 (v F,K), with K<1 (repulsive). -k F kFkF Fluctuating Orders Typically -> can measure CDW power law from scattering. Noise measurement sensitive to both CDW/SC.

22. Luttinger Parameters from Noise Correlations Correlations of Atom Shot Noise: -k F kFkF X X kFkF X X qq q -q q q’ CDW SC Calculate using Bosonization:

23. Luttinger Parameters From Noise Correlations K=0.4 K=2.5 q CDW SC q’ K<1/2 K>2 For ½<K<2 K=0.8 K=1.25

24. 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-SC S-SC/ CDW (cusp) 21/2 CDW/ S-SC (cusp) q’ q g Spin -gap

26. Molecular condensate fraction (condensed)/(total molecules) independent of ramp speed for fast ramps. (both arise from Cooper pairs). 1 JILA Expts Expect non monotonic condensate fraction at very fast sweeps Probe of uncondensed pairs