Measuring correlation functions in interacting systems of cold atoms

1. Measuring correlation functions in interacting systems of cold atoms Anatoli PolkovnikovHarvard/Boston University Ehud AltmanHarvard/Weizmann Vladimir GritsevHarvard Mikhail LukinHarvard Eugene Demler Harvard Thanks to: M. Greiner , Z. Hadzibabic, M. Oberthaler, J. Schmiedmayer, V. Vuletic

2. Correlation functions in condensed matter physics Most experiments in condensed matter physics measure correlation functions Example: neutron scattering measures spin and density correlation functions Neutron diffraction patterns for MnO Shull et al., Phys. Rev. 83:333 (1951)

3. Outline Lecture I: Measuring correlation functions inintereferenceexperiments Lecture II: Quantum noiseinterferometryin time of flight experiments Emphasis of these lectures: detection and characterization of many-body quantum states

4. Lecture I Measuring correlation functions inintereferenceexperiments 1. Interference of independent condensates 2. Interference of interacting 1Dsystems 3. Interference of 2D systems 4. Full distribution function of the fringe amplitudes in intereference experiments. 5. Studying coherent dynamics of strongly interacting systems in interference experiments

5. Lecture II Quantum noiseinterferometry in time of flight experiments 1. Detection of spin order in Mott states ofatomic mixtures 2. Detection of fermion pairing

6. Measuring correlation functions in intereference experiments Analysis of high order correlation functions in low dimensional systems Polkovnikov, Altman, Demler,PNAS (2006)

7. Interference of two independent condensates Andrews et al., Science 275:637 (1997)

8. Interference of two independent condensates r’ r 1 r+d d 2 Clouds 1 and 2 do not have a well defined phase difference. However each individual measurement shows an interference pattern

9. x y Interference of one dimensional condensates Experiments: Schmiedmayer et al., Nature Physics (2005) d Amplitude of interference fringes, , contains information about phase fluctuations within individual condensates x1 x2

10. Interference amplitude and correlations L For identical condensates Instantaneous correlation function

11. Interacting bosons in 1d at T=0 Low energy excitations and long distance correlation functions can be described by the Luttinger Hamiltonian. K – Luttinger parameter Connection to original bosonic particles Small K corresponds to strong quantum fluctuations

12. For non-interacting bosons For impenetrable bosons Luttinger liquids in 1d Correlation function decays rapidly for small K. This decay comes from strong quantum fluctuations

13. L For non-interacting bosons and For impenetrable bosons and Analysis of can be used for thermometry Interference between 1d interacting bosons Luttinger liquid at T=0 K – Luttinger parameter Luttinger liquid at finite temperature

14. For large imaging angle, , Rotated probe beam experiment Luttinger parameter K may be extracted from the angular dependence of q

15. Interference between two-dimensional BECs at finite temperature. Kosteritz-Thouless transition

16. Ly Lx Lx Interference of two dimensional condensates Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/0506559 Gati, Oberthaler, et al., cond-mat/0601392 Probe beam parallel to the plane of the condensates

17. Ly Lx Below KT transition Above KT transition Interference of two dimensional condensates.Quasi long range order and the KT transition Above Kosterlitz-Thouless transition: Vortices proliferate. Short range order Below Kosterlitz-Thouless transition: Vortices confined. Quasi long range order

18. Experiments with 2D Bose gas low temperature higher temperature Hadzibabic et al., Nature (2006) Time of flight z x Typical interference patterns

19. Experiments with 2D Bose gas Hadzibabic et al., Nature (2006) Contrast after integration integration over x axis z 0.4 low T z middle T 0.2 integration over x axis high T z 0 0 Dx 10 20 30 integration distance Dx (pixels) x integration over x axis z

20. Experiments with 2D Bose gas Hadzibabic et al., Nature (2006) 0.4 low T 0.2 Exponent a middle T 0 0 10 20 30 high T if g1(r) decays exponentially with : high T low T 0.5 0.4 if g1(r) decays algebraically or exponentially with a large : central contrast 0.3 “Sudden” jump!? 0 0.1 0.2 0.3 fit by: Integrated contrast integration distance Dx

21. Experiments with 2D Bose gas Hadzibabic et al., Nature (2006) c.f. Bishop and Reppy 0.5 1.0 0.4 T (K) 0 1.1 1.0 1.2 0.3 0 0.1 0.2 0.3 Exponent a high T low T central contrast Ultracold atoms experiments: jump in the correlation function. KT theory predicts a=1/4 just below the transition He experiments: universal jump in the superfluid density

22. 30% Fraction of images showing at least one dislocation 20% 10% low T high T 0 0 0.1 0.2 0.3 0.4 central contrast Experiments with 2D Bose gas. Proliferation of thermal vortices Haddzibabic et al., Nature (2006)

23. Rapidly rotating two dimensional condensates Time of flight experiments with rotating condensates correspond to density measurements Interference experiments measure single particle correlation functions in the rotating frame

24. Interference between two interacting one dimensional Bose liquids Full distribution function of the amplitude of interference fringes Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475

25. L Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995) Higher moments of interference amplitude is a quantum operator. The measured value of will fluctuate from shot to shot. Can we predict the distribution function of ? Higher moments Changing to periodic boundary conditions (long condensates)

26. Impurity in a Luttinger liquid Expansion of the partition function in powers of g Partition function of the impurity contains correlation functions taken at the same point and at different times. Moments of interference experiments come from correlations functions taken at the same time but in different points. Euclidean invariance ensures that the two are the same

27. Distribution function can be reconstructed from using completeness relations for the Bessel functions Relation between quantum impurity problemand interference of fluctuating condensates Normalized amplitude of interference fringes Distribution function of fringe amplitudes Relation to the impurity partition function

28. is related to the single particle Schroedinger equation Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999) Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001) Spectral determinant can be obtained from the Bethe ansatz following Zamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95) Making analytic continuation is possible but cumbersome Bethe ansatz solution for a quantum impurity Interference amplitude and spectral determinant

30. Evolution of the distribution function Narrow distribution for . Approaches Gumble distribution. Width Wide Poissonian distribution for

31. correspond to vacuum eigenvalues of Q operators of CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999 2D quantum gravity, non-intersecting loops on 2D lattice Yang-Lee singularity From interference amplitudes to conformal field theories When K>1, is related to Q operators of CFT with c<0. This includes 2D quantum gravity, non-intersecting loop model on 2D lattice, growth of random fractal stochastic interface, high energy limit of multicolor QCD, …

32. Studying coherent dynamics of strongly interacting systems in interference experiments

33. J Coupled 1d systems Motivated by experiments of Schmiedmayer et al. Interactions lead to phase fluctuations within individual condensates Tunneling favors aligning of the two phases Interference experiments measure only the relative phase

34. J Coupled 1d systems Conjugate variables Relative phase Particle number imbalance Small K corresponds to strong quantum flcutuations

35. Quantum Sine-Gordon model Hamiltonian Imaginary time action Quantum Sine-Gordon model is exactly integrable Excitations of the quantum Sine-Gordon model soliton antisoliton breather

36. J Coherent dynamics of quantum Sine-Gordon modelMotivated by experiments of Schmiedmayer et al. Prepare a system at t=0 Take to the regime of finite tunneling and let evolve for some time Measure amplitude of interference pattern

37. Coherent dynamics of quantum Sine-Gordon model Amplitude of interference fringes time Oscillations or decay?

38. From integrability to coherent dynamics At t=0 we have a state with for all This state can be written as a “squeezed” state Matrix can be constructed using connection to boundary SG model Calabrese, Cardy (2006); Ghoshal, Zamolodchikov (1994) Time evolution can be easily written Interference amplitude can be calculated using form factor approach Smirnov (1992), Lukyanov (1997)

39. J Coherent dynamics of quantum Sine-Gordon model Prepare a system at t=0 Take to the regime of finite tunneling and let evolve for some time Measure amplitude of interference pattern

40. Coherent dynamics of quantum Sine-Gordon model Amplitude of interference fringes time Amplitude of interference fringes shows oscillations at frequencies that correspond to energies of breater

41. Conclusions for part I Interference of fluctuating condensates can be used to probe correlation functions in one and two dimensional systems. Interference experiments can also be used to study coherent dynamics of interacting systems

42. Lecture II Measuring correlation functions in interacting systems of cold atoms Quantum noiseinterferometry in time of flight experiments 1. Time of flight experiments. Second order coherence in Mott states of spinless bosons 2. Detection of spin order in Mott states ofatomic mixtures 3. Detection of fermion pairing Emphasis of these lectures: detection and characterization of many-body quantum states

43. Bose-Einstein condensation Cornell et al., Science 269, 198 (1995) Ultralow density condensed matter system Interactions are weak and can be described theoretically from first principles

44. Mott insulator Superfluid t/U Superfluid to Insulator transition Greiner et al., Nature 415:39 (2002)

45. Time of flight experiments Quantum noise interferometry of atoms in an optical lattice Second order coherence

46. Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment Theory: Altman et al., PRA 70:13603 (2004) Experiment: Folling et al., Nature 434:481 (2005)

47. Hanburry-Brown-Twiss stellarinterferometer

48. Hanburry-Brown-Twissinterferometer

49. Bosons at quasimomentum expand as plane waves with wavevectors Second order coherence in the insulating state of bosons First order coherence: Oscillations in density disappear after summing over Second order coherence: Correlation function acquires oscillations at reciprocal lattice vectors

50. Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment Theory: Altman et al., PRA 70:13603 (2004) Experiment: Folling et al., Nature 434:481 (2005)