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
Vito Scarola Maryland
Sankar Das Sarma Maryland
Eugene Demler Harvard
Thanks to: J. Schmiedmayer, M. Oberthaler, V. Vuletic,
M. Greiner, M. Oshikawa, Z. Hadzibabic

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. This talk:
Detection of many-body quantum phases by measuring correlation functions

4. Outline Measuring correlation functions in intereference experiments
1. Interference of independent condensates
2. Interference of interacting 1D systems
3. Interference of 2D systems
4. Full counting statistics of intereference experiments.
Connection to quantum impurity problem
Quantum noise interferometry in time of flight experiments
1. Detection of magnetically ordered Mott states in
optical lattices
2. Observation of fermion pairing

5. Measuring correlation functions
in intereference experiments

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

7. 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

8. Interference of one dimensional condensates
Experiments: Schmiedmayer et al., Nature Physics 1 (05) d
Amplitude of interference fringes, ,
contains information about phase fluctuations
within individual condensates x1 x2 y x

9. Interference amplitude and correlations
Polkovnikov, Altman, Demler, cond-mat/ L
For identical condensates
Instantaneous correlation function

10. Interference between Luttinger liquids
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

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

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

13. 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

14. Interference of two dimensional condensates
Interference of two dimensional condensates. Quasi long range order and the KT transition Ly Lx
Theory: Polkovnikov, Altman, Demler,
cond-mat/
Above KT transition
Below KT transition

15. Experiments with 2D Bose gas
Haddzibabic, Stock, Dalibard, et al. z
Time of
flight x
Typical interference patterns
low temperature
higher temperature

16. Experiments with 2D Bose gas
Haddzibabic, Stock, Dalibard, et al. 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

17. Experiments with 2D Bose gas
Haddzibabic, Stock, Dalibard, et al.
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 :

18. Experiments with 2D Bose gas
Haddzibabic, Stock, Dalibard, et al.
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

19. Experiments with 2D Bose gas
Experiments with 2D Bose gas. Proliferation of thermal vortices Haddzibabic, Stock, Dalibard, et al.
10%
20%
30%
central contrast
0.1
0.2
0.3
0.4
high T
low T
Fraction of images showing
at least one dislocation
Exponent a
0.5
0.1
0.2
0.3
0.4
central contrast
The onset of proliferation
coincides with a shifting to 0.5!
Z. Hadzibabic et al., in preparation

20. 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

21. Full counting statistics of interference between two interacting one
dimensional Bose liquids
Gritsev, Altman, Demler, Polkovnikov, cond-mat/

22. 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)

23. 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

24. 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

25. 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 a Schroedinger equation
Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999)
Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)
Spectral determinant

26. Evolution of the distribution function
Narrow distribution
for
Distribution width
approaches
Wide Poissonian
distribution for

27. 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

28. Quantum noise interferometry
in time of flight experiments

29. Atoms in an optical lattice. 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.

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

31. 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)

32. Hanburry-Brown-Twiss stellar interferometer

33. 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

34. 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)

35. Effect of parabolic potential on the second order coherence
Experiment: Spielman, Porto, et al.,
Theory: Scarola, Das Sarma, Demler, cond-mat/
Width of the correlation peak changes across the
Transition, reflecting the evolution of the Mott domains

36. Applications of quantum noise interferometry
Spin order in Mott states of atomic mixtures

37. Probing spin order of bosons
Correlation Function Measurements
Extra Bragg
peaks appear
in the second
order correlation
function in the
AF phase

38. Applications of quantum noise interferometry
Detection of fermion pairing

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

40. Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave pairing of fermions

41. Second order interference from a BCS superfluid
n(k)
n(r)
n(r’) k F k
BCS
BEC

42. Momentum correlations in paired fermions
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Greiner et al., PRL 94: (2005)

43. 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

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