1. Quantum simulation of low-dimensional systems using interference experiments. Anatoli Polkovnikov, Boston University Collaboration: Ehud Altman- The Weizmann Institute of Science Eugene Demler - Harvard University Vladimir Gritsev - Harvard University

2. Quantum Simulations 1.Universal: can simulate any unitary evolution 2.Simulating specific (interacting) Hamiltonians a)direct: simulate the system we realize b)Indirect: simulate one system realizing another one. This talk: simulating a quantum impurity model in a 1D interacting Fermi gas using interference between homogeneous 1D bosons.

3. This talk: Interference between two systems of interacting bosons: measurements and interferencemeasurements and interference shot noiseshot noise noise due to phase fluctuationsnoise due to phase fluctuations full distribution function and quantum simulationsfull distribution function and quantum simulations outlook.outlook.

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)  Y. Castin and J. Dalibard, 1997

7. P. Anderson, 1984, Do two superfluids which have never seen one another possess a definitive phase? Y. Castin and J. Dalibard, 1997 for a specific gedanken experiment: yes but this phase is spontaneously generated by measurement!!! A.P., E. Altman and E. Demler, 2005: yes for ToF experiments.

8. How do we analyze the image? This analysis does not take into account quantum effects, i.e. Need to correct for this.

9. Define an observable ( interference amplitude squared ): The interference amplitude does not fluctuate at large N! depends only on N

10. Decrease the effect of shot noise. Extended condensates. Reduce the interference contrast. This talk: how to analyze this reduction of the contrast.

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

12. 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.

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

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

15. 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.)

16. Two-dimensional condensates at finite temperature CCD camera x z Time of flight z x y imaging laser (picture by Z. Hadzibabic) Elongated condensates: L x >>L y.

17. Observing the Kosterlitz-Thouless transition Above KT transition LyLy LxLx Below KT transition Universal jump of  at T KT Always algebraic scaling, easy to detect.

18. Zoran Hadzibabic, Peter Kruger, Marc Cheneau, Baptiste Battelier, Sabine Stock, and Jean Dalibard (2006). integration over x axis X z z integration over x axis z x integration distance X (pixels) Contrast after integration 0.4 0.2 0 0 102030 middle T low T high T Interference contrast:

19. Exponent  central contrast 0.5 00.10.20.3 0.4 0.3 high Tlow T T (K) 1.0 1.1 1.2 1.0 0 “universal jump in the superfluid density” c.f. Bishop and Reppy Z. Hadzibabic et. al. Vortex proliferation Fraction of images showing at least one dislocation: 0 10% 20% 30% central contrast 0 0.1 0.20.30.4 high T low T

20. 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 ~ A 2n / L 1-1/K Shot noise is subdominant for K>1 at T=0.

21. 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))

22. 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.

23. 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).

24. 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.

25. Relevance of the boundary SG model to other problems. An isolated impurity in a 1D Fermi gas. An isolated impurity in a 1D Fermi gas. K<1, attractive interactions – impurity is relevant interacting electron gas scattering on impurity K>1, repulsive interactions – impurity is irrelevant Kane and Fisher, 1992 We can directly simulate the partition function for this problem in interference experiments.

26. h  plays the role of the relative momentum q I. Affleck, W. Hofstetter, D. R. Nelson, U. Schollwock, J.Stat.Mech. 0410 P003 (2004) Interacting flux lines in 2D superconductors.

27. 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)

28. Evolution of the distribution function.

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

30. 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.

31. Extension: direct probing of fermionic superfluidity BCS model

32. Conclusions. Analysis of interference between independent condensates reveals a wealth of information about their internal structure. a)Shot noise and phase fluctuations are responsible for decrease of the interference contrast. Shot noise is subdominant in large systems with (quasi) long range order. b)Scaling of interference amplitudes with L or  reveals correlation function exponents. c)Probability distribution of amplitudes gives the information about higher order correlation functions. d)Interference of two Luttinger liquids allows one to obtain partition function of a 1D quantum impurity problem (also related to variety of other problems) and thus to simulate it. Extensions to other cases: fermions, out of equilibrium systems, spin systems, etc.

33. 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?

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

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

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

37. 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.

38. 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:

39. 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).

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