1. Interference experiments with ultracold atoms Collaborators: Ehud Altman, Anton Burkov, Robert Cherng, Adilet Imambekov, Serena Fagnocchi, Vladimir Gritsev, Mikhail Lukin, David Pekker, Anatoli Polkovnikov Funded by NSF, Harvard-MIT CUA, AFOSR, DARPA, MURI Eugene Demler Harvard University

2. Outline Introduction. Interference of fluctuating low dimensional condensates. Systems of mixed dimensionality Interference of fermions: probing paired states Detection of s-wave pairing Detection of FFLO Detection of d-wave pairing Interference experiments and non-equilibrium dynamics Decoherence of uniformly split condensates Ramsey interference of one dimensional systems Splitting condensates on Y-junctions

3. Interference of independent condensates Experiments: Andrews et al., Science 275:637 (1997) Theory: Javanainen, Yoo, PRL 76:161 (1996) Cirac, Zoller, et al. PRA 54:R3714 (1996) Castin, Dalibard, PRA 55:4330 (1997) and many more

4. x z Time of flight Experiments with 2D Bose gas Hadzibabic, Dalibard et al., Nature 441:1118 (2006) Experiments with 1D Bose gas Hofferberth et al. arXiv0710.1575

5. x1x1 Amplitude of interference fringes, Interference of fluctuating condensates For identical condensates Instantaneous correlation function For independent condensates A fr is finite but Df is random x2x2 Polkovnikov, Altman, Demler, PNAS 103:6125(2006) L

6. Interference between fluctuating condensates 1d: Luttinger liquid, Hofferberth et al., 2007 x z L [pixels] 0.4 0.2 0 0 1020 30 middle T low T high T 2d: BKT transition, Hadzibabic et al, 2006 Time of flight low T high T BKT

7. Distribution function of interference fringe contrast Experiments: Hofferberth et al., arXiv0710.1575 Theory: Imambekov et al., cond-mat/0612011 Comparison of theory and experiments: no free parameters Higher order correlation functions can be obtained Quantum fluctuations dominate : asymetric Gumbel distribution (low temp. T or short length L) Thermal fluctuations dominate: broad Poissonian distribution (high temp. T or long length L) Intermediate regime : double peak structure

8. Systems of mixed dimensionality Weakly coupled 2D condensates Experiments: M. Kasevich et al., Interplay of two dimensional physics of the BKT transition and coupling along the 3 rd direction Connection to quasi-2D and 1D condensed matter systems: quantum magnets, organic superconductors, high Tc cuprates, and many more

9. Berezinskii – Kosterlitz – Thouless transition T=T BKT temperature T=0 Fisher & Hohenberg, PRB 37, 4936 (1988)

10. Temperature scales for weakly coupled pancakes 3D XY 3D Phonons temperature T=0 T=T BKT T=T C T=2t

11. Interference of a stack of coupled pancakes Pekker, Gritsev, Demler

12. Interference experiments with fermions: probing paired states

13. Interference of fermionic systems Time of flight X Y Z 1 2 A pair of independent fermionic systems

14. Interference as a probe of fermionic pairing 1 2 W W Pairing correlations Time of flight Expectation value vanishes for independent systems due to random relative phase between D 1 and D 2

15. Interference as a probe of pairing 1 2 WIWI W II WIWI Experimental procedure Interfere two independent systems Measure and in the same shot Extract and from Fourier transforms in the z-direction Calculate and Find from averaging over many shots Polkovnikov, Gritsev, Demler

16. FFLO phase Pairing at finite center of mass momentum Theory: Fulde, Ferrell (1964); Larkin, Ovchinnikov (1965); Bowers, Rajagopal (2002); Liu, Wilczek (2003); Sheehy, Radzihovsky (2006); Combescot (2006); Yang, Sachdev (2006); Pieri, Srinati (2006); Parish et al., (2007); and many others Experiments: Zwierlein, Ketterle et al., (2006) Hulet et al., (2006)

17. Interference as a probe of FFLO phase 1 2 WIWI W II WIWI when q matches one of the wavevectors of D (r) of FFLO phase Integration by a laser beam Manual integration x y

18. d-wave pairing Fermionic Hubbard model Possible phase diagram of the Hubbard model D.J.Scalapino Phys. Rep. 250:329 (1995)

19. Phase sensitive probe of d-wave pairing in high Tc superconductors Superconducting quantum interference device (SQUID) F Van Harlingen, Leggett et al, PRL 71:2134 (93)

20. Other signatures of d-wave pairing: dispersion of quasiparticles ++ - - Quasiparticle energies Superconducting gap Normal state dispersion of quasiparticles Low energy quasiparticles correspond to four Dirac nodes Observed in: Photoemission Raman spectroscopy T-dependence of thermodynamic and transport properties, c V, k, l L STM and many other probes

21. Phase sensitive probe of d-wave pairing in high Tc superconductors Superconducting quantum interference device (SQUID) F Van Harlingen, Leggett et al, PRL 71:2134 (93)

22. Interference as a probe of d-wave pairing 2 W II 1 WIWI W III WIWI W II W III System 1 is an s-wave superfluid System 2 is a d-wave superfluid Regions II and III differ only by 90 o rotation Phase sensitive probe of d-wave pairing d-wave superfluid s-wave superfluid

23. Interference experiments and non-equilibrium dynamics

24. Uniform splitting of the condensates Prepare a system by splitting one condensate Take to the regime of zero tunneling Measure time evolution of fringe amplitudes

25. Long phase coherence implies squeezing factor of 10. Squeezing due to finite time of splitting. Leggett, Sols, PRL (1998) Burkov et al., PRL (2007) Squeezing factor

26. 1d BEC: Decay of coherence Experiments: Hofferberth, Schumm, Schmiedmayer, Nature (2007) double logarithmic plot of the coherence factor slopes: 0.64 ± 0.08 0.67 ± 0.1 0.64 ± 0.06 get t 0 from fit with fixed slope 2/3 and calculate T from T 5 = 110 ± 21 nK T 10 = 130 ± 25 nK T 15 = 170 ± 22 nK

27. Relative phase dynamics beyond single mode approximation Conjugate variables Hamiltonian can be diagonalized in momentum space A collection of harmonic oscillators with Need to solve dynamics of harmonic oscillators at finite T Coherence Initial state f q = 0

28. Quantum regime 1D systems 2D systems Classical regime 1D systems 2D systems Relative phase dynamics beyond single mode approximation Bistritzer, Altman, PNAS (2007) Burkov, Lukin, Demler, PRL (2007)

29. Dynamics of condensate splitting and Ramsey interference

30. Working with N atoms improves the precision of frequency spectroscopy by. Ramsey interference t 0 1

31. Interaction induced collapse of Ramsey fringes Experiments in 1d tubes: A. Widera, et al. arXiv 0709:2094

32. Spin echo. Time reversal experiments Single mode approximation predicts full revival Experiments in 1d tubes : A.Widera, et al. arXiv 0709:2094. Need to analyze multi-mode model in 1d Only q=0 mode shows complete spin echo Finite q modes continue decay The net visibility is a result of competition between q=0 and other modes

33. Splitting condensates on Y-junctions: quantum zipper problem

34. Splitting condensates on Y-junctions Partial splitting stage: new physics Full splitting stage: same as before Earlier work: Non-interacting atoms: Scully and Dowling, PRA (1993) Analysis of transverse modes: Jaasekelatnen and Stenholm, PRA (2003) Tonks-Girardeau regime: Girardeau et al. PRA (2002)

35. Splitting condensates on Y-junctions: beyond mean-field x Wave equation in both arms of the interferometer. c is the speed of sound Relative phase Time dependent boundary conditions in the frame of the condensate. v is the condensate velocity 1 st arm 2 nd arm

36. Moving mirror problem in optics Moore, J. Math. Phys. 9:2679 (1970) Exciting photons in a cavity with a moving mirror Experimentally always in the adiabatic regime c is the speed of light

37. Splitting condensate on Y-junction: beyond mean-field K – Luttinger parameter v is the condensate velocity d This is similar to the usual 1d condensates Extra suppression due to finite velocity of splitting

38. Splitting with acceleration: Unruh like effect Fagnocchi, Altman, Demler Splitting condensates with relativistic acceleration gives rise to thermal correlations. This is analogous to the Unruh effect in field theory and quantum gravity Unruh (1974), Fulling and Davies (1976)

39. Summary Interference experiments with ultracold atoms provide a powerful tool for analyzing equilibrium properties and dynamics of many-body systems. Analysis beyond mean-field and single mode approximation is needed important Thanks to: