Learning about order from noise Quantum noise studies of ultracold atoms

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Learning about order from noiseQuantum noise
studies of ultracold atoms
Eugene Demler Harvard
University
Collaborators Takuya Kitagawa, Susanne Pielawa,
Adilet Imambekov, Ehud Altman, Vladimir
Gritsev, Anatoli Polkovnikov, Mikhail Lukin
Experiments Bloch et al., Dalibard et al.,
Schmiedmayer et al.
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Quantum noise
Classical measurement collapse of
the wavefunction into eigenstates of x
Histogram of measurements of x
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Aspects experiments with correlated photon
pairs tests of Bells inequalities (1982)
Probabilistic nature of quantum mechanics
Bohr-Einstein debate EPR thought experiment
(1935)
Spooky action at a distance
Analysis of correlation functions can be used to
rule out hidden variables theories
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Second order coherence HBT experiments
Classical theory Hanburry Brown and Twiss
(1954)
Quantum theory Glauber (1963)
For bosons
For fermions
Used to measure the angular diameter of Sirius
HBT experiments with matter
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Shot noise in electron transport
Shot noise Schottky (1918)
Variance of transmitted charge
Measurements of fractional charge
Current noise for tunneling across a Hall bar on
the 1/3 plateau of FQE
Etien et al. PRL 792526 (1997) see also Heiblum
et al. Nature (1997)
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Analysis of quantum noisepowerful experimental
toolCan we use it for cold atoms?
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Outline
Quantum noise in interference experiments with
independent condensates
Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices HBT
experiments and beyond
Quantum noise in Ramsey interference experiments
Goal new methods of detection of
quantum many-body phases of ultracold atoms
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Interference experimentswith cold atomsAnalysis
of thermal and quantum noisein low dimensional
systems
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Interference of independent condensates
Experiments Andrews et al., Science 275637
(1997)
Theory Javanainen, Yoo, PRL 76161
(1996) Cirac, Zoller, et al. PRA 54R3714
(1996) Castin, Dalibard, PRA 554330 (1997) and
many more
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Experiments with 2D Bose gas
z
Hadzibabic, Kruger, Dalibard et al., Nature
4411118 (2006)
x
Experiments with 1D Bose gas Hofferberth et
al., Nature Physics 4489 (2008)
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Interference of two independent condensates
r
r
Assuming ballistic expansion
1
rd
d
2
Phase difference between clouds 1 and 2 is not
well defined
Individual measurements show interference
patterns They disappear after averaging over many
shots
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Interference of fluctuating condensates
Polkovnikov, Altman, Demler, PNAS 1036125(2006)
d
x1
For independent condensates Afr is finite but Df
is random
x2
Instantaneous correlation function
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Fluctuations in 1d BEC
Thermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
Weakly interacting atoms
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Interference between Luttinger liquids
Luttinger liquid at T0
K Luttinger parameter
Finite temperature
Experiments Hofferberth, Schumm, Schmiedmayer
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Distribution function of fringe amplitudes for
interference of fluctuating condensates
Gritsev, Altman, Demler, Polkovnikov, Nature
Physics 2006 Imambekov, Gritsev, Demler, Varenna
lecture notes, c-m/0703766
Higher moments reflect higher order correlation
functions
We need the full distribution function of

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Distribution function of interference fringe
contrast
Hofferberth et al., Nature Physics 4489 (2008)
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
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Interference between interacting 1d Bose
liquids. Distribution function of the
interference amplitude
Quantum impurity problem interacting one
dimensional electrons scattered on an impurity
Conformal field theories with negative central
charges 2D quantum gravity, non-intersecting
loop model, growth of random fractal stochastic
interface, high energy limit of multicolor QCD,

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Fringe visibility and statistics of random
surfaces
Mapping between fringe visibility and the
problem of surface roughness for fluctuating
random surfaces. Relation to 1/f Noise and
Extreme Value Statistics
Analysis of sine-Gordon models of the type
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Time-of-flight experiments with atoms in optical
lattices
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Atoms in optical lattices
Theory Jaksch et al. PRL (1998)
Experiment Greiner et al., Nature (2001) and
many more
Motivation quantum simulations of strongly
correlated electron systems including quantum
magnets and unconventional superconductors
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Fermionic Hubbard model
From high temperature superconductors to
ultracold atoms
Atoms in optical lattice
Antiferromagnetism and pairing at sub-micro
Kelvin temperatures
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Superfluid to insulator transition in an optical
lattice
M. Greiner et al., Nature 415 (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
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
Experiment Folling et al., Nature 434481 (2005)
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Second order coherence in the insulating state of
fermions.Hanburry-Brown-Twiss experiment
Experiment Tom et al. Nature 444733 (2006)
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Probing spin order in optical lattices
Correlation Function Measurements
Extra Bragg peaks appear in the second order
correlation function in the AF phase
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Detection of fermion pairing
Quantum noise analysis of TOF images is more
than HBT interference
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Second order interference from the BCS superfluid
Theory Altman et al., PRA 7013603 (2004)
n(k)
k
BCS
BEC
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Momentum correlations in paired fermions
Experiments Greiner et al., PRL 94110401 (2005)
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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
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Quantum noise as a probe of non-equilibrium
dynamics Ramsey interferometry and many-body
decoherence
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Ramsey interference
Atomic clocks and Ramsey interference
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Interaction induced collapse of Ramsey fringes
Two component BEC. Single mode approximation
Ramsey fringe visibility
time
Experiments in 1d tubes A. Widera et al. PRL
100140401 (2008)
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Spin echo. Time reversal experiments
Single mode approximation
The Hamiltonian can be reversed by changing a12
Predicts perfect spin echo
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Spin echo. Time reversal experiments
Expts A. Widera et al., PRL (2008)
Experiments done in array of tubes. Strong
fluctuations in 1d systems. Single mode
approximation does not apply. Need to analyze the
full model
No revival?
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Interaction induced collapse of Ramsey
fringes.Multimode analysis
Low energy effective theory Luttinger liquid
approach
Luttinger model
Changing the sign of the interaction reverses the
interaction part of the Hamiltonian but not the
kinetic energy
Time dependent harmonic oscillators can be
analyzed exactly
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Time-dependent harmonic oscillator
See e.g. Lewis, Riesengeld (1969)
Malkin, Manko (1970)
Explicit quantum mechanical wavefunction can be
found
From the solution of classical problem
We solve this problem for each momentum component
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Interaction induced collapse of Ramsey fringesin
one dimensional systems
Only q0 mode shows complete spin echo Finite q
modes continue decay The net visibility is a
result of competition between q0 and other modes
Decoherence due to many-body dynamics of low
dimensional systems
Fundamental limit on Ramsey interferometry
How to distinquish decoherence due to many-body
dynamics?
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Interaction induced collapse of Ramsey fringes
Single mode analysis Kitagawa, Ueda, PRA 475138
(1993)
Multimode analysis evolution of spin distribution
functions
T. Kitagawa, S. Pielawa, A. Imambekov, et al.
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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
Harvard-MIT
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Interference experimentswith cold atomsAnalysis
of thermal and quantum noisein low dimensional
systems
Theory For review see Imambekov et al., Varenna
lecture notes, c-m/0612011
Experiment 2D Hadzibabic, Kruger, Dalibard,
Nature 4411118 (2006) 1D Hofferberth et al.,
Nature Physics 4489 (2008)
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Time-of-flight experiments with atoms in optical
lattices
Theory Altman, Demler, Lukin, PRA 7013603
(2004)
Experiment Folling et al., Nature 434481
(2005) Spielman et al., PRL
9880404 (2007) Tom et al.
Nature 444733 (2006)
Guarrera et al., PRL 100250403 (2008)
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z
x
Typical interference patterns
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Experiments with 2D Bose gas
Hadzibabic et al., Nature 4411118 (2006)
x
integration over x axis
z
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Experiments with 2D Bose gas
Hadzibabic et al., Nature 4411118 (2006)
fit by
Integrated contrast
integration distance Dx
Sudden jump BKT transition
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Experiments with 2D Bose gas. Proliferation of
thermal vortices Hadzibabic et al., Nature
4411118 (2006)
The onset of proliferation coincides with a
shifting to 0.5!
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Fringe contrast in two dimensions. Evolution of
distribution function
Experiments Kruger, Hadzibabic, Dalibard