1. Learning about order from noise Quantum noise studies of ultracold atoms
Eugene Demler Harvard University
Collaborators:
Ehud Altman, Robert Cherng, Adilet Imambekov,
Vladimir Gritsev, Mikhail Lukin, Anatoli Polkovnikov,
Ana Maria Rey
Funded by NSF, Harvard-MIT CUA, AFOSR, DARPA, MURI

2. Outline Introduction. Historical review
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices:
HBT experiments and beyond
Quantum noise in interference experiments
with independent condensates
Observation of superexchange interactions
in optical lattices

3. Quantum noise Classical measurement:
collapse of the wavefunction into eigenstates of x
Quantum noise – the process of making a measurement introduces a noise.
Individual measurements do not give us the average value of x. Quantum noise
contains information about the quantum mechanical wavefunction
Histogram of measurements of x

4. Probabilistic nature of quantum mechanics
Bohr-Einstein debate on spooky action at a distance
Einstein-Podolsky-Rosen experiment
Measuring spin of a particle in the left detector
instantaneously determines its value in the right detector

5. Aspect’s experiments: tests of Bell’s inequalities+ - 1 2 q1 q2 S S
Correlation function
Classical theories with hidden variable require
Cascade two photon transition in Ca-40. J=0->J=1->J=0
s- source of two polarized photons
1,2 – polarization analyzers set at angles theta 1,2
+/- horizontal/vertical channels
Quantum mechanics predicts B=2.7 for the appropriate choice of q‘s and the state
Experimentally measured value B= Phys. Rev. Let. 49:92 (1982)

6. Hanburry-Brown-Twiss experiments
Classical theory of the second order coherence
Hanbury Brown and Twiss,
Proc. Roy. Soc. (London),
A, 242, pp
Measurements of the angular diameter of Sirius
Proc. Roy. Soc. (London), A, 248, pp

7. Quantum theory of HBT experiments
Glauber,
Quantum Optics and Electronics (1965)
HBT experiments with matter
Experiments with neutrons
Ianuzzi et al., Phys Rev Lett (2006)
For bosons
Experiments with electrons
Kiesel et al., Nature (2002)
Experiments with 4He, 3He
Westbrook et al., Nature (2007)
For fermions
Experiments with ultracold atoms
Bloch et al., Nature (2005,2006)

8. Shot noise in electron transport
Proposed by Schottky to measure the electron charge in 1918 e-
Spectral density of the current noise
Related to variance of transmitted charge
When shot noise dominates over thermal noise
Poisson process of independent transmission of electrons

9. Shot noise in electron transport
Current noise for tunneling across a Hall bar on the 1/3 plateau of FQE
Etien et al. PRL 79:2526 (1997)
see also Heiblum et al. Nature (1997)
points with error bars – experiment
open circles include a correction for finite tunneling

10. Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices:
Hanburry-Brown-Twiss experiments
and beyond
Theory: Altman, Demler, Lukin, PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005);
Spielman et al., PRL 98:80404 (2007);
Tom et al. Nature 444:733 (2006)

11. Atoms in optical lattices
Theory: Jaksch et al. PRL (1998)
Experiment: Kasevich et al., Science (2001);
Greiner et al., Nature (2001);
Phillips et al., J. Physics B (2002)
Esslinger et al., PRL (2004);
Ketterle et al., PRL (2006)

12. Bose Hubbard model U t tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well

13. Bose Hubbard model Superfluid phase Mott insulator phase
M.P.A. Fisher et al.,
PRB40:546 (1989)
N=3
Mott 4
Superfluid
N=2
Mott 2
N=1
Mott
Superfluid phase
Weak interactions
Mott insulator phase
Strong interactions

14. Superfluid to insulator transition in an optical lattice
M. Greiner et al., Nature 415 (2002)
t/U
Superfluid
Mott insulator

15. Why study ultracold atoms in optical lattices

16. Fermionic atoms in optical latticesU
Experiments with fermions in optical lattice, Kohl et al., PRL 2005

17. Same microscopic model
Antiferromagnetic and superconducting Tc
of the order of 100 K
Atoms in optical lattice
Antiferromagnetism and
pairing at sub-micro Kelvin
temperatures
Same microscopic model

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

19. Same microscopic model
Atoms in optical lattice
Same microscopic model
Quantum simulations of strongly correlated
electron systems using ultracold atoms
Detection?

20. Quantum noise analysis as a probe of many-body states of ultracold atoms

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

22. Second order coherence in the insulating state of bosons
Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment
Experiment: Folling et al., Nature 434:481 (2005)

23. Hanburry-Brown-Twiss stellar interferometer

24. 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

25. Second order coherence in the insulating state of bosons
Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment
Experiment: Folling et al., Nature 434:481 (2005)

26. Second order coherence in the insulating state of fermions
Second order coherence in the insulating state of fermions. Hanburry-Brown-Twiss experiment
Experiment: Tom et al. Nature 444:733 (2006)

27. How to detect antiferromagnetism

28. Probing spin order in optical lattices
Correlation Function Measurements
Extra Bragg
peaks appear
in the second
order correlation
function in the
AF phase

29. How to detect fermion pairing
Quantum noise analysis of TOF images
is more than HBT interference

30. Second order interference from the BCS superfluid
Theory: Altman et al., PRA 70:13603 (2004)
n(k)
n(r)
n(r’) k F k
BCS
BEC

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

32. 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

33. Superexchange interaction in experiments with double wells
Refs:
Theory: A.M. Rey et al., Phys. Rev. Lett. 99: (2007)
Experiment: S. Trotzky et al., Science 319:295 (2008)

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

35. Quantum magnetism of bosons in optical lattices
Duan, Demler, Lukin, PRL 91:94514 (2003)
Ferromagnetic
Antiferromagnetic

36. Observation of superexchange in a double well potential
Theory: A.M. Rey et al., arXiv: J J
Use magnetic field gradient to prepare a state
Observe oscillations between and states

37. Preparation and detection of Mott states
of atoms in a double well potential

38. Comparison to the Hubbard model
Experiments: I. Bloch et al.

39. Beyond the basic Hubbard model
Basic Hubbard model includes
only local interaction
Extended Hubbard model
takes into account non-local
interaction

40. Beyond the basic Hubbard model

41. Interference experiments with cold atoms

42. 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

43. Nature 4877:255 (1963)

44. Experiments with 2D Bose gasz
Time of
flight
Experiments with 2D Bose gas
Hadzibabic, Dalibard et al., Nature 441:1118 (2006)
Experiments with 1D Bose gas S. Hofferberth et al. arXiv

45. 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

46. Interference of fluctuating condensates
Polkovnikov, Altman, Demler, PNAS 103:6125(2006) d
Amplitude of interference fringes, x1
For independent condensates Afr is finite
but Df is random x2
However recently experiments have been done on low dimensional systems where we know that fluctuations may not be neglected.
For example here I consider interference between one dimensional condensates. This geometry follows experiments
done with elongated condensates in microtraps in Jerg Schmidmayer’s group.
If we look at the interference pattern coming from point x1, we get a perfect contrast. When we look at another point x2, we also
find a perfect contrast. However two interference patterns will not be in phase with each other when condensates have fluctuations.
So when we image by sending a laser beam along the axis we will find a reduced contrast. The lesson that we learn is that
the reduction of the contrast contains information about phase fluctuations. Let us see how we can make this statement more
Rigorous.
For identical
condensates
Instantaneous correlation function

47. Fluctuations in 1d BEC Thermal fluctuations Quantum fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations

48. Interference between Luttinger liquids
Luttinger liquid at T=0
K – Luttinger parameter
For non-interacting bosons and
For impenetrable bosons and
Finite
temperature
Experiments: Hofferberth,
Schumm, Schmiedmayer

49. Distribution function of fringe amplitudes for interference of fluctuating condensates
Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006
Imambekov, Gritsev, Demler, cond-mat/ L
is a quantum operator. The measured value of
will fluctuate from shot to shot.
Higher moments reflect higher order correlation functions
We need the full distribution function of

50. Distribution function of interference fringe contrast
Experiments: Hofferberth et al., arXiv
Theory: Imambekov et al. , cond-mat/
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
Comparison of theory and experiments: no free parameters
Higher order correlation functions can be obtained

51. Interference of two dimensional condensates
Experiments: Hadzibabic et al. Nature (2006)
Gati et al., PRL (2006) Lx Ly Lx
Probe beam parallel to the plane of the condensates

52. Interference of two dimensional condensates
Interference of two dimensional condensates. Quasi long range order and the KT transition Ly Lx
Above KT transition
Below KT transition

53. Experiments with 2D Bose gas
Hadzibabic, Dalibard et al., Nature 441:1118 (2006) z
Time of
flight x
Typical interference patterns
low temperature
higher temperature

54. Experiments with 2D Bose gas
Hadzibabic et al., Nature 441:1118 (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

55. Experiments with 2D Bose gas
Hadzibabic et al., Nature 441:1118 (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 :

56. Experiments with 2D Bose gas
Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature 441:1118 (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
The onset of proliferation
coincides with a shifting to 0.5!

57. Summary Experiments with ultracold atoms provide a new
perspective on the physics of strongly correlated
many-body systems. Quantum noise is a powerful
tool for analyzing many body states of ultracold atoms
Thanks to: