Measuring correlation functions in interacting systems of cold atoms

1
Measuring correlation functions in interacting
systems of cold atoms
Detection and characterization of many-body
quantum phases

• Anatoli Polkovnikov Harvard/Boston
  University
• Ehud Altman
  Harvard/Weizmann
• Vladimir Gritsev Harvard
• Mikhail Lukin Harvard
• Vito Scarola
  University of Maryland
• Sankar Das Sarma University of
  Maryland
• Eugene Demler Harvard

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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

• References
• 1. Polkovnikov, Altman, Demler,
  cond-mat/0511675
• Gritsev, Altman, Demler, Polkovnikov,
  cond-mat/0602475
• Altman, Demler, Lukin, PRA 7013603 (2004)
• Scarola, Das Sarma, Demler, cond-mat/0602319

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Measuring correlation functions in intereference
experiments
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Interference of two independent condensates
Andrews et al., Science 275637 (1997)
5
Interference of two independent condensates
r
r
1
rd
d
2
Clouds 1 and 2 do not have a well defined phase
difference. However each individual measurement
shows an interference pattern
6
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
7
Interference amplitude and correlations
Polkovnikov, Altman, Demler, cond-mat/0511675
For identical condensates
Instantaneous correlation function
8
Interference between Luttinger liquids
Luttinger liquid at T0
K Luttinger parameter
Luttinger liquid at finite temperature
9
Interference between two-dimensional BECs at
finite temperature. Kosteritz-Thouless transition
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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
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Interference of two dimensional
condensates.Quasi long range order and the KT
transition
Theory Polkovnikov, Altman, Demler,
cond-mat/0511675
12
z
x
Typical interference patterns
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x
integration over x axis
z
14
fit by
Integrated contrast
integration distance Dx
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Exponent a
high T
low T
central contrast
Ultracold atoms experiments jump in the
correlation function. KT theory predicts a1/4
just below the transition
He experiments universal jump in the superfluid
density
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Experiments with 2D Bose gas. Proliferation of
thermal vortices Hadzibabic, Stock,
Dalibard, et al.
Fraction of images showing at least one
dislocation
Z. Hadzibabic et al., in preparation
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Full counting statistics of interference between
two interacting one dimensional Bose liquids
Gritsev, Altman, Demler, Polkovnikov,
cond-mat/0602475
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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)
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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
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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
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Bethe ansatz solution for a quantum impurity
Interference amplitude and spectral determinant
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Evolution of the distribution function
Narrow distribution for
. Distribution width approaches
Wide Poissonian distribution for
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From interference amplitudes to conformal
field theories
When Kgt1, is related
to Q operators of CFT with clt0. This includes 2D
quantum gravity, non-intersecting loop model on
2D lattice, growth of random fractal stochastic
interface, high energy limit of multicolor QCD,
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Outlook
Full counting statistics of interference between
fluctuating 2D condensates
One and two dimensional systems with tunneling.
Competition of single particle tunneling,
quantum, and thermal fluctuations. Full counting
statistics of the phase and the amplitude
Expts Shmiedmayer et al. (1d),
Oberthaler et al. (2d)
Time dependent evolution of distribution functions
Extensions, e.g. spin counting in Mott states of
multicomponent systems
Ultimate goal Creation, characterization, and
manipulation of quantum
many-body states of atoms
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Quantum noise interferometry in time of flight
experiments
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Atoms in an optical lattice.Superfluid to
Insulator transition
Greiner et al., Nature 41539 (2002)
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Time of flight experiments
Quantum noise interferometry of atoms in an
optical lattice
Second order coherence
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Second order coherence in the insulating state of
bosons.Hanburry-Brown-Twiss experiment
Theory Altman et al., PRA 7013603 (2004)
Experiment Folling et al., Nature 434481 (2005)
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Hanburry-Brown-Twiss stellar interferometer
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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
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Second order coherence in the insulating state of
bosons.Hanburry-Brown-Twiss experiment
Theory Altman et al., PRA 7013603 (2004)
Experiment Folling et al., Nature 434481 (2005)
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Effect of parabolic potential on the second order
coherence
Experiment Spielman, Porto, et al., Theory
Scarola, Das Sarma, Demler, cond-mat/0602319
Width of the correlation peak changes across the
transition, reflecting the evolution of Mott
domains
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Potential applications of quantum noise
intereferometry
Altman et al., PRA 7013603 (2004)
Detection of magnetically ordered Mott states
Detection of paired states of fermions
Experiment Greiner et al. PRL
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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