1. Measuring correlation functions in interacting systems of cold atoms
Anatoli Polkovnikov Harvard/Boston University
Ehud Altman Harvard/Weizmann
Vladimir Gritsev Harvard
Mikhail Lukin Harvard
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 in intereference experiments
Lecture II:
Quantum noise interferometry in time of flight experiments
Emphasis of these lectures:
detection and characterization of many-body quantum states

4. Lecture I Measuring correlation functions in intereference experiments
1. Interference of independent condensates
2. Interference of interacting 1D systems
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 noise interferometry in time of flight experiments
1. Detection of spin order in Mott states of atomic 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 condensates1
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. 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 y x

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

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

14. Rotated probe beam experiment
For large imaging angle, ,
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. Interference of two dimensional condensates
Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/
Gati, Oberthaler, et al., cond-mat/ Lx Ly Lx
Probe beam parallel to the plane of the condensates

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

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

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

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

21. Experiments with 2D Bose gas
Hadzibabic et al., Nature (2006)
Exponent a
T (K)
1.0
1.1
1.2
c.f. Bishop and Reppy
0.5
0.1
0.2
0.3
0.4
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. Experiments with 2D Bose gas
Experiments with 2D Bose gas. Proliferation of thermal vortices Haddzibabic et al., Nature (2006)
Fraction of images showing
at least one dislocation
10%
20%
30%
central contrast
0.1
0.2
0.3
0.4
high T
low T

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

25. 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)
Explicit expressions for are available but cumbersome
Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)

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

28. Bethe ansatz solution for a quantum impurity
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
Interference amplitude and spectral determinant
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

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

31. From interference amplitudes to conformal field theories
correspond to vacuum eigenvalues of Q operators of CFT
Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999
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, …
Yang-Lee singularity
2D quantum gravity,
non-intersecting loops on 2D lattice

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

33. Coupled 1d systems J 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. Coupled 1d systems J 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. Coherent dynamics of quantum Sine-Gordon model Motivated by experiments of Schmiedmayer et al.J
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
time
Amplitude of
interference fringes
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. Coherent dynamics of quantum Sine-Gordon modelJ
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
Lecture II
Quantum noise interferometry
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 of atomic 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. Superfluid to Insulator transition
Greiner et al., Nature 415:39 (2002)
t/U
Superfluid
Mott insulator
What if the ground state cannot be described by single particle matter waves? Current experiments are reaching such regimes. A nice example is the experiment by Markus and collaborators in Munich where cold bosons on an optical lattice were tuned across the SF-Mott transition.
Explain experiment.
SF well described by a wavefunction in which the zero momentum state is macroscopically occupied.
Mott state is rather well described by definite RS occupations.

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
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 stellar interferometer

48. Hanburry-Brown-Twiss interferometer

49. Second order coherence in the insulating state of bosons
Bosons at quasimomentum expand as plane waves
with wavevectors
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
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)

51. Effect of parabolic potential on the second order coherence
Experiment: Spielman, Porto, et al.,
Theory: Scarola, Das Sarma, Demler, PRA (2006)
Width of the correlation peak changes across the
transition, reflecting the evolution of Mott domains

52. Width of the noise peaks

53. Interference of an array of independent condensates
Hadzibabic et al., PRL 93: (2004)
Smooth structure is a result of finite experimental resolution (filtering)

54. Applications of quantum noise interferometry
in time of flight experiments
Detection of spin order in Mott states
of boson boson mixtures

55. Engineering magnetic systems using cold atoms in an optical lattice
See also lectures by A. Georges and I. Cirac in this school

56. Spin interactions using controlled collisions
Experiment: Mandel et al., Nature 425:937(2003)
Theory: Jaksch et al., PRL 82:1975 (1999)

57. Two component Bose mixture in optical lattice
Example: Mandel et al., Nature 425:937 (2003) t t
Two component Bose Hubbard model

58. Quantum magnetism of bosons in optical lattices
Kuklov and Svistunov, PRL (2003)
Duan et al., PRL (2003)
Ferromagnetic
Antiferromagnetic

59. Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state

60. Two component Bose mixture in optical lattice
Two component Bose mixture in optical lattice. Mean field theory + Quantum fluctuations
Altman et al., NJP 5:113 (2003)
Hysteresis
1st order
2nd order line

61. Probing spin order of bosons
Correlation Function Measurements

62. Engineering exotic phases
Optical lattice in 2 or 3 dimensions: polarizations & frequencies
of standing waves can be different for different directions YY ZZ
Example: exactly solvable model
Kitaev (2002), honeycomb lattice with
Can be created with 3 sets of
standing wave light beams !
Non-trivial topological order, “spin liquid” + non-abelian anyons
…those has not been seen in controlled experiments

63. Applications of quantum noise interferometry
in time of flight experiments
Detection of fermion pairing

64. Fermionic atoms in optical lattices
Pairing in systems with repulsive interactions.
Unconventional pairing. High Tc mechanism

65. Fermionic atoms in a three dimensional optical lattice
Kohl et al., PRL 94:80403 (2005)
See also lectures of T. Esslinger and W. Ketterle in this school

66. Fermions with repulsive interactions
Possible d-wave pairing of fermions

67. High temperature superconductors
Picture courtesy of UBC
Superconductivity group
High temperature superconductors
Superconducting Tc 93 K
Hubbard model – minimal model for cuprate superconductors
P.W. Anderson, cond-mat/
After many years of work we still do not understand
the fermionic Hubbard model

68. Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor

69. Second order correlations in the BCS superfluid
n(k)
n(r)
n(r’) k F k
BCS
BEC
Expansion of atoms in TOF maps k into r

70. Momentum correlations in paired fermions
Greiner et al., PRL 94: (2005)

71. Fermion pairing in an optical lattice
Second Order Interference
In the TOF images
Normal State
Superfluid State
measures the Cooper pair wavefunction
One can identify unconventional pairing

72. Simulation of condensed matter systems: Hubbard Model and high Tc superconductivity
Personal opinion:
The fermionic Hubbard model contains 90% of the physics of
cuprates. The remaining 10% may be crucial for getting high Tc
superconductivity. Understanding Hubbard model means finding
what these missing 10% are. Electron-phonon interaction?
Mesoscopic structures (stripes)?
Using cold atoms to go beyond “plain vanilla” Hubbard model
a) Boson-Fermion mixtures: Hubbard model + phonons
b) Inhomogeneous systems, role of disorder

73. Boson Fermion mixtures
Fermions interacting with phonons

74. Boson Fermion mixtures
See lectures by T. Esslinger and G. Modugno in this school
BEC
Bosons provide cooling for fermions
and mediate interactions. They create
non-local attraction between fermions
Charge Density Wave Phase
Periodic arrangement of atoms
Non-local Fermion Pairing
P-wave, D-wave, …

75. Boson Fermion mixtures
“Phonons” :
Bogoliubov (phase) mode
Effective fermion-”phonon” interaction
Fermion-”phonon” vertex
Similar to electron-phonon systems

76. Boson Fermion mixtures in 1d optical lattices
Cazalila et al., PRL (2003); Mathey et al., PRL (2004)
Spin ½ fermions
Spinless fermions

77. Boson Fermion mixtures in 2d optical lattices
Wang et al., PRA (2005)
40K -- 87Rb
40K -- 23Na
(a)
=1060nm
=1060 nm
(b)
=765.5nm

78. Conclusions Interference of extended condensates is a powerful
tool for analyzing correlation functions in one and
two dimensional systems
Noise interferometry can be used to probe
quantum many-body states in optical lattices