Measuring correlation functions in interacting systems of cold atoms

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Measuring correlation functions in interacting
systems of cold atoms
Anatoli Polkovnikov Harvard/Boston
University Ehud Altman
Harvard/Weizmann Vladimir Gritsev
Harvard Mikhail Lukin
Harvard Eugene Demler
Harvard
Thanks to M. Greiner , Z. Hadzibabic, M.
Oberthaler, J. Schmiedmayer,
V. Vuletic
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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. 83333 (1951)
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Outline
Lecture I Measuring correlation functions in
intereference experiments
Lecture II Quantum noise interferometry in
time of flight experiments
Emphasis of these lectures detection and
characterization of many-body quantum states
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Lecture I
Measuring correlation functions
in intereference experiments 1.
Interference of independent condensates 2.
Interference of interacting 1D systems 3.
Interference of 2D systems 4. Full
distribution function of the fringe amplitudes
in intereference experiments. 5.
Studying coherent dynamics of strongly
interacting systems in interference
experiments
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Lecture II
Quantum noise interferometry
in time of flight experiments 1.
Detection of spin order in Mott states of atomic
mixtures 2. Detection of fermion pairing
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Measuring correlation functions in intereference
experiments
Analysis of high order correlation functions in
low dimensional systems
Polkovnikov, Altman, Demler, PNAS (2006)
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Interference of two independent condensates
Andrews et al., Science 275637 (1997)
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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
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Interference of one dimensional condensates
Experiments Schmiedmayer et al., Nature Physics
(2005)
d
Amplitude of interference fringes, ,
contains information about phase
fluctuations within individual condensates
x1
x2
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Interference amplitude and correlations
L
For identical condensates
Instantaneous correlation function
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Interacting bosons in 1d at T0
Low energy excitations and long distance
correlation functions can be described by the
Luttinger Hamiltonian.
K Luttinger parameter
Connection to original bosonic particles
Small K corresponds to strong quantum fluctuations
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Luttinger liquids in 1d
Correlation function decays rapidly for small K.
This decay comes from strong quantum fluctuations
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Interference between 1d interacting bosons
Luttinger liquid at T0
K Luttinger parameter
Luttinger liquid at finite temperature
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Rotated probe beam experiment
Luttinger parameter K may be extracted from the
angular dependence of
q
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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
Above Kosterlitz-Thouless transition Vortices
proliferate. Short range order
Below Kosterlitz-Thouless transition Vortices
confined. Quasi long range order
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z
x
Typical interference patterns
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x
integration over x axis
z
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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 Haddzibabic et al.,
Nature (2006)
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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
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Interference between two interacting one
dimensional Bose liquids Full distribution
function of the amplitude of interference
fringes
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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(No Transcript)
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Evolution of the distribution function
Narrow distribution for
. Approaches Gumble distribution. Width
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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Studying coherent dynamics of strongly
interacting systems in interference experiments
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Coupled 1d systems
Motivated by experiments of Schmiedmayer et al.
Interactions lead to phase fluctuations within
individual condensates
Tunneling favors aligning of the two phases
Interference experiments measure only the
relative phase
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Coupled 1d systems
Conjugate variables
Relative phase
Particle number imbalance
Small K corresponds to strong quantum
flcutuations
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Quantum Sine-Gordon model
Hamiltonian
Imaginary time action
Quantum Sine-Gordon model is exactly integrable
Excitations of the quantum Sine-Gordon model
soliton
antisoliton
breather
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Coherent dynamics of quantum Sine-Gordon model
Motivated by experiments of Schmiedmayer et al.
Prepare a system at t0
Take to the regime of finite tunneling and let
evolve for some time
Measure amplitude of interference pattern
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Coherent dynamics of quantum Sine-Gordon model
Oscillations or decay?
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From integrability to coherent dynamics
At t0 we have a state with
for all
This state can be written as a squeezed state
Matrix can be constructed using
connection to boundary SG model Calabrese, Cardy
(2006) Ghoshal, Zamolodchikov (1994)
Time evolution can be easily written
Interference amplitude can be calculated using
form factor approach Smirnov (1992), Lukyanov
(1997)
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Coherent dynamics of quantum Sine-Gordon model
Prepare a system at t0
Take to the regime of finite tunneling and let
evolve for some time
Measure amplitude of interference pattern
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Coherent dynamics of quantum Sine-Gordon model
Amplitude of interference fringes
time
Amplitude of interference fringes
shows oscillations at frequencies that correspond
to energies of breater
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Conclusions for part I
Interference of fluctuating condensates can be
used to probe correlation functions in one and
two dimensional systems. Interference
experiments can also be used to study coherent
dynamics of interacting systems
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Lecture II
Measuring correlation functions in interacting
systems of cold atoms

Quantum noise interferometry
in time of flight experiments 1. Time
of flight experiments. Second order
coherence in Mott states of spinless bosons 2.
Detection of spin order in Mott states of atomic
mixtures 3. Detection of fermion pairing
Emphasis of these lectures detection and
characterization of many-body quantum states
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Bose-Einstein condensation
Cornell et al., Science 269,
198 (1995)
Ultralow density condensed matter system
Interactions are weak and can be described
theoretically from first principles
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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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Hanburry-Brown-Twiss 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, PRA (2006)
Width of the correlation peak changes across the
transition, reflecting the evolution of Mott
domains
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Width of the noise peaks
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Interference of an array of independent
condensates
Hadzibabic et al., PRL 93180403 (2004)
Smooth structure is a result of finite
experimental resolution (filtering)
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Applications of quantum noise interferometry
in time of flight experiments
Detection of spin order in Mott states of
boson boson mixtures
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Engineering magnetic systems using cold atoms in
an optical lattice
See also lectures by A. Georges and I. Cirac in
this school
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Spin interactions using controlled collisions
Experiment Mandel et al., Nature 425937(2003)
Theory Jaksch et al., PRL 821975 (1999)
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Two component Bose mixture in optical lattice
Example . Mandel et al., Nature
425937 (2003)
Two component Bose Hubbard model
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Quantum magnetism of bosons in optical lattices
Kuklov and Svistunov, PRL (2003)
Duan et al., PRL (2003)

• Ferromagnetic
• Antiferromagnetic

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Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates antiferromagnetic state
Coulomb energy dominates ferromagnetic state
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Two component Bose mixture in optical
lattice.Mean field theory Quantum fluctuations
Altman et al., NJP 5113 (2003)
Hysteresis
1st order
2nd order line
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Probing spin order of bosons
Correlation Function Measurements
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Engineering exotic phases

• Optical lattice in 2 or 3 dimensions
  polarizations frequencies
• of standing waves can be different for different
  directions

YY
ZZ

• Can be created with 3 sets of
• standing wave light beams !

• Non-trivial topological order, spin liquid
  non-abelian anyons
• those has not been seen in
  controlled experiments
• 
  

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Applications of quantum noise interferometry
in time of flight experiments
Detection of fermion pairing
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Fermionic atoms in optical lattices
Pairing in systems with repulsive interactions.
Unconventional pairing. High Tc mechanism
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Fermionic atoms in a three dimensional optical
lattice
Kohl et al., PRL 9480403 (2005)
See also lectures of T. Esslinger and W. Ketterle
in this school
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Fermions with repulsive interactions
U
t
t
Possible d-wave pairing of fermions
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High temperature superconductors
Superconducting Tc 93 K
Hubbard model minimal model for cuprate
superconductors
P.W. Anderson, cond-mat/0201429
After many years of work we still do not
understand the fermionic Hubbard model
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Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep.
250329 (1995)
Antiferromagnetic insulator
D-wave superconductor
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Second order correlations in the BCS superfluid
n(k)
k
BCS
BEC
Expansion of atoms in TOF maps k into r
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Momentum correlations in paired fermions
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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Simulation of condensed matter systems Hubbard
Model and high Tc superconductivity
Personal opinion The fermionic Hubbard model
contains 90 of the physics of cuprates. The
remaining 10 may be crucial for getting high
Tc superconductivity. Understanding Hubbard model
means finding what these missing 10 are.
Electron-phonon interaction? Mesoscopic
structures (stripes)?
Using cold atoms to go beyond plain vanilla
Hubbard model a) Boson-Fermion mixtures Hubbard
model phonons b) Inhomogeneous systems, role
of disorder
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Boson Fermion mixtures
Fermions interacting with phonons
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Boson Fermion mixtures
See lectures by T. Esslinger and G. Modugno in
this school
Bosons provide cooling for fermions and mediate
interactions. They create non-local attraction
between fermions
Charge Density Wave Phase
Periodic arrangement of atoms
Non-local Fermion Pairing
P-wave, D-wave,
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Boson Fermion mixtures
Phonons Bogoliubov (phase) mode
Effective fermion-phonon interaction
Fermion-phonon vertex
Similar to electron-phonon systems
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Boson Fermion mixtures in 1d optical lattices
Cazalila et al., PRL (2003) Mathey et al., PRL
(2004)
Spin ½ fermions
Spinless fermions
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Boson Fermion mixtures in 2d optical lattices
Wang et al., PRA (2005)
40K -- 87Rb
40K -- 23Na
(b)
765.5nm
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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