1. Static and dynamic probes of strongly interacting low-dimensional atomic systems. Anatoli Polkovnikov, Boston University Collaboration: Ehud Altman- The Weizmann Institute of Science Antonio Castro Neto-Boston University Eugene Demler - Harvard University Vladimir Gritsev - Harvard University Corinna Kollath-University of Geneva Ludwig Mattey-Harvard University

2. Why low dimensions? 1.Existence of many low-dimensional correlated phases: unconventional superconductivity, fractionalized and topological phases, QHE, Luttinger liquids, TG gas, etc. 2.Excellent laboratory for studying dynamics and thermalization in nonintegrable and integrable systems. 3.Realization of new accurate interference probes, which are not available in 3D systems.

3. This talk: 1.Interference between two 1D systems of interacting bosons: shot noiseshot noise noise due to phase fluctuationsnoise due to phase fluctuations full distribution functionfull distribution function 2.Quench experiments in 1D and 2D systems: excitations in coupled 1D condensatesexcitations in coupled 1D condensates dynamics after the quenchdynamics after the quench quenching coupled 2D systems: Kibble Zurek mechanism of topological defect formation.quenching coupled 2D systems: Kibble Zurek mechanism of topological defect formation.

4. Interference between two condensates. Interference between two condensates. d xTOF Free expansion: Andrews et. al. 1997

5. What do we observe? b) Uncorrelated, but well defined phases   int (x)  =0 Hanbury Brown-Twiss Effect xTOF c) Initial number state. Work with original bosonic fields: a)Correlated phases (  = 0) 

6. I. Casten and J. Dalibard (1997). Z. Hadzibabic et.al. (2004). Z. Hadzibabic et.al. (2004). The interference amplitude does not fluctuate at large N! Define an observable ( interference amplitude squared ): depends only on N

7. x z z1z1 z2z2 AQAQ Identical homogeneous condensates: Interference amplitude contains information about fluctuations within each condensate. Fluctuating Condensates.

8. Scaling with L: two limiting cases Ideal condensates: L x z Interference contrast does not depend on L. L x z Dephased condensates: Contrast scales as L -1/2.

9. Formal derivation: Ideal condensate: L Thermal gas: L

10. Intermediate case (quasi long-range order). z 1D condensates (Luttinger liquids): L Repulsive bosons with short range interactions: Finite temperature:

11. Angular Dependence. q is equivalent to the relative momentum of the two condensates (always present e.g. if there are dipolar oscillations). z x(z1)x(z1) x(z2)x(z2) (for the imaging beam orthogonal to the page, is the angle of the integration axis with respect to z.)

12. Higher Moments. is an observable quantum operator Identical condensates. Mean: Similarly higher moments Probe of the higher order correlation functions. Universal (size independent) distribution function: Shot noise contribution:  A 2n ~ L n(2-1/K) / L 1-1/K Shot noise is subdominant for K>1 at T=0.

13. Sketch of the derivation Action: With periodic boundary conditions we find: These integrals can be evaluated using Jack polynomials ( These integrals can be evaluated using Jack polynomials (Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995))

14. Two simple limits: Central limit theorem! Also at finite T. x z z1z1 z2z2 A Strongly interacting Tonks-Girardeau regime Weakly interacting BEC like regime.

15. Connection to the impurity in a Luttinger liquid problem. Boundary Sine-Gordon theory: Same integrals as in the expressions for (we rely on Euclidean invariance). P. Fendley, F. Lesage, H. Saleur (1995).

16. Experimental simulation of the quantum impurity problem 1.Do a series of experiments and determine the distribution function. T. Schumm, et. al., Nature Phys. 1, 57 (2005). Distribution of interference phases (and amplitudes) from two 1D condensates. 2.Evaluate the integral. 3.Read the result.

17. can be found using Bethe ansatz methods for half integer K. In principle we can find W: Difficulties: need to do analytic continuation. The problem becomes increasingly harder as K increases. Use a different approach based on spectral determinant: Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999); Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)

18. Evolution of the distribution function.

19. Universal Gumbel distribution at large K (  -1)/ 

20. Generalized extreme value distribution: Emergence of extreme value statistics on other instances: E. Bretin, Phys. Rev. Lett. 95, 170601 (2005) From independent random variables to correlated intervals Also 1/f noise Other examples of extreme value statistics.

21. Quench experiments in 1D and 2D systems: T. Schumm. et. al., Nature Physics 1, 57 - 62 (01 Oct 2005) Study dephasing as a function of time. What sort of information can we get?

22. Analyze dynamics of phase coherence: Idea: extract energies of excited states and thus go beyond static probes. Relevant model:

23. Excitations: solitons and breathers. Can create solitons only in pairs. Expect damped oscillations : solitons breathers (bound solitons) Can create isolated breathers. Expect undamped oscillations:

24. Analogy with a Josephson junction. EnEnEnEn  breathers soliton pairs (only with q  0)

25. Numerical simulations Hubbard model, 2 chains, 6 sites each b 02 b 24 b46b46b46b46 b04b04b04b04 b26b26b26b26 2s 01 2b 02 Fourier analysis of the oscillations is a way to perform spectroscopy.

26. Quench in 2D condensates Expect a very sharp change in T KT as a function of the layer separation. A simple entropic argument: r Energy ~ J  r 2  expect confinement KT argument J  =0: J  >0:

27. Sudden change in T/T KT can result in the Kibble- Zurek mechanism of the topological defect formation. Start at T>T KT  quench to T T KT  quench to T<T KT. If quench is fast  we expect that vortices do not thermalize. Have nonequilibrium vortex population. RG calculation for various values of vortex fugacity. Neglect dependence  c (T).

28. Conclusions. 1.Analysis of interference between independent condensates reveals a wealth of information about their internal structure. a)Scaling of interference amplitudes with L or  : correlation function exponents. b)Probability distribution of amplitudes: information about higher order correlation functions. c)Interference of two Luttinger liquids: partition function of 1D quantum impurity problem (also related to variety of other problems). 2.Quench experiments in 1D and 2D systems are a possible new way of performing spectroscopy in cold atom systems: a)Detecting solitons and breathers in 1D coupled condensates b)Kibble-Zurec mechanism in 2D condensates Such experiments are much simpler and more robust than e.g. parametric resonance. Such experiments are much simpler and more robust than e.g. parametric resonance.

29. Angular (momentum) Dependence. has a cusp singularity for K<1, relevant for fermions.