1. Probing phases and phase transitions in cold atoms using interference experiments. Anatoli Polkovnikov, Boston University Collaboration: Ehud Altman- The Weizmann Institute of Science Eugene Demler - Harvard University Vladimir Gritsev - Harvard University AFOSR APS March Meeting. New Orleans, Mar. 2008

2. Interference between independent sources. (Hanbury-Brown Twiss effect)

3. Interference between independent sources. (Hanbury-Brown Twiss effect) x1x1x1x1 x2x2x2x2 Origin of interference – superposition principle. Interference term drops out for uncorrelated sources Intensity-intensity correlator survives! HBT effect is the classical wave phenomenon!

4. From classical waves to quantum particles x1x1x1x1 x2x2x2x2 Origin of interference – superposition principle. Interference term drops out for uncorrelated sources Density-Density correlator survives! Note: at this level it is not important whether sources have random phases or have fixed number of particles. (Castin Dalibard 1997)

5. Interference between two condensates. Interference between two condensates. d xTOF Free expansion:

6. 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) 

8. Define an observable ( interference amplitude squared ): depends only on N Shot (counting) noise distinguishes quantum particles from classical waves (only affects fluctuations of the interference amplitude): The interference amplitude does not fluctuate at large N! Castin, Dalibard 1997. A.P. 2006 Imambekov, Gritsev, Demler, 2007

9. Interference from multiple sources d x Having many particles in each condensate does not reduce noise in the interference amplitude. Redefine the observable Utilize higher momentum harmonics. Noiseless HBT signal for large number of sources, true both for bosons and for fermions. We can have classical Grassman waves for fermions!

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

11. 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.

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

13. Observing the Kosterlitz-Thouless transition Above KT transition LyLy LxLx Below KT transition L x  L y Universal jump of  at T KT Experiment: Hadzibabic et. al. (2006): J14.00004 11:15 AM–2:15 PM, Tuesday, March 11, 2008

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

15. 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. ~

16. Connection to the impurity in a Luttinger liquid problem. Experimental simulation of the quantum impurity problem 1.Do a series of experiments and determine the distribution function of the interference amplitude. 2.Evaluate the integral. 3.Read the result. One-dimensional electron gas + impurity.

17. Evolution of the distribution function. Universal Gumbel distribution at large K: distribution at large K:

18. 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.

19. S. Hofferberth, I. Lesanovsky, T. Schumm, A. Imambekov, V. Gritsev, E. Demler, J. Schmiedmayer, arXiv:0710.1575, to appear in Nature Physics. Small T High T

20. Detecting fermionic superfluidity (D-wave…) d TOF Before Good for shot noise but not for superfluidity. Now Problem with gauge invariance (undefined relative phase)

21. Can go to 4 th order correlation functions. Shot noise will kill us. Two ways around Introduce weak tunneling coupling such that macroscopic phases are locked but correlation functions are not yet affected.2 1. 2. Guage invariance is restored because of long range coherence. Still measure local gap!

22. Fermions on a lattice. Different orientations of imaging beams give strong angular dependence of the signal in D-wave case. Can detect even the sign of the pairing gap using two different imaging beams for B L and B R and having one D-condensate and one S-condensate.

23. Conclusions. Interference is a powerful tool for detecting properties of correlated cold atom systems. Two sources of noise in the interference: a) thermal or quantum fluctuations b) shot noise Mean amplitude of interference contains information on two-particle correlation functions. Higher moments contain additional information.