1. Analysis of quantum noise
Quantum systems of ultracold atoms
New probes of many-body correlations
Analysis of quantum noise
Interference of fluctuating condensates and
correlation functions
From quantum noise to high
order correlation functions
Analysis of magnetization
fluctuations in lattice models
During the last few years theorists proposed many ideas for realizing interesting quantum many-body states
using ultracold atoms. However detection and characterization of such states remains an important
open problem. Today I will tell you some of the ideas that we have been working on during the last
few years. The people who did most of the work are Anatoli P and Ehud who have
been postdocs with us but have since moved to start their own groups and Vladimir Gritsev
who started as a postdoc last Fall. Special thanks goes to our experimental colleagues discussions with
whom started most of the projects that I will present today

2. Interference of two independent condensates
Andrews et al., Science 275:637 (1997)
Interference experiments are one of the trademarks of the field of ultracold atoms. The first set of experiments helped to understand
the question of how condensates that do not have a well defined relative phase give rise to an interference pattern.
Basically the argument
is that we can think of the system as if there was a certain phase difference between the condensates but we did not know what it was. So we can not predict the position of interference fringes even though we know they will be present. Such earlier experiments have been done with
large three dimensional condensates where phase fluctuations witin individual condensates could typically be neglected.

3. 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

4. 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

5. Interference between fluctuating condensates
high T
L [pixels]
0.4
0.2 10 20 30
middle T
low T
high T
BKT x z
Time of flight
low T
2d: BKT transition, Hadzibabic et al, 2006
1d: Luttinger liquid, Hofferberth et al., 2007

6. 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

7. 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

8. Studying coherent dynamics of strongly interacting systems
in interference experiments
So far we talked about applicaions of interference experiments to study correlation functions in equiulibrium.
A natural question to ask is whether they can be used to study time dependent phenomena?

9. Studying dynamics using interference experiments
Prepare a system by
splitting one condensate
Take to the regime of
zero tunneling
Measure time evolution
of fringe amplitudes

10. Dynamics of split condensates
Theory: Burkov et al., PRL 2007
Experiment: Hofferberth et al,. Nature 2007
Theoretical prediction

11. Probing spin systems using
distribution function of magnetization

12. Probing spin systems using distribution function of magnetization
Cherng, Demler, New J. Phys. 9:7 (2007)
Magnetization in a finite system
Average magnetization
Higher moments of contain information about higher order
correlation functions

13. Distribution Functions
x-Ferromagnet
polarized or g 1 ?
So now I can fill in the missing distribution functions which turn out to be gaussian which we rescale to collapse distribution functions for different block sizes. When physical domain walls are bound there is symmetry breaking described by the presence of two peaks in the Mx distribution. At the same time, dual domain walls proliferate and the Mz distribution has a single peak. On the other hand, when physical domain walls proliferate there is no symmetry breaking and the Mx distribution only has a single peak centered at zero. At the same time, dual domain walls are bound which leads to different distributions for even and odd values of the magnetization along z leading to a distinctive splitting. Away from the critical point, this splitting tends to a constant, zero for small g and finite for large g. At the critical point, the splitting actually decays with a power-law in the number of sites. ? ?

14. Using noise to detect spin liquids
Spin liquids have no broken symmetries
No sharp Bragg peaks
Algebraic spin liquids have long range
spin correlations A
No static magnetization
Noise in magnetization exceeds shot noise

15. Work in progress on theoretical milestones for MURI
Investigate coherent quantum dynamics of strongly
interacting spin systems and superfluids
Investigate experimental requirements for reaching and
detecting the antiferromagnetic phase
(Long lived doublon states)
Investigate theoretically new approaches to realizing and
detecting of spin liquid states and anyon statistics
(Measuring spin loop operators using coupling to cavity)

16. MURI quantum simulation project
Phase I
Validation and Verification
Simulate solvable Hamiltonians. Compare with calculations.
Examples: low dimensional systems, precision study of Mott insulator phases, superexchange interactions, fermionic superfluidity in optical lattice.
New tools for detection and caracterization of strongly correlated states
Optical addressability  Quantum gas microscope
Critical velocity in moving superfluids
Bragg spectroscopy in optical lattices
Quantum noise analysis
Phase II
Combine all tools and methods developed during phase I to tackle goals of this MURI project:
Quantum magnetism
Fermionic superfluidity in systems with repulsive interactions
Ultimate goal: use the results of quantum analogue simulations to
identify new solid state systems with favorable properties

17. Quantum Simulations of Condensed Matter Systems using Ultracold Atomic Gases
FY07 MURI Topic #18
Markus Greiner (principal investigator), Eugene Demler, John Doyle, Luming Duan, Mark Kasevich, Wolfgang Ketterle, Mikhail Lukin, Subir Sachdev, Martin Zwierlein, Joseph Thywissen,Immanuel Bloch, Peter Zoller
Collaborating Universities: Harvard, MIT, Stanford, Michigan, Toronto, University of Mainz, Germany, University of Innsbruck, Austria