Outline of these lectures

1
Outline of these lectures

• Introduction. Systems of ultracold atoms.
• Cold atoms in optical lattices.
• Bose Hubbard model. Equilibrium and dynamics
• Bose mixtures in optical lattices.
• Quantum magnetism of ultracold atoms.
• Detection of many-body phases using noise
  correlations
• Experiments with low dimensional systems
• Interference experiments. Analysis of high
  order correlations
• Fermions in optical lattices
• Magnetism and pairing in systems with
  repulsive interactions. Current experiments
  paramgnetic Mott state, nonequilibrium dynamics.
• Dynamics near Fesbach resonance. Competition of
  Stoner instability and pairing

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Learning about order from noiseQuantum noise
studies of ultracold atoms
3
Quantum noise
Classical measurement collapse of
the wavefunction into eigenstates of x
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
Aspects experimentstests of Bells inequalities
S
Correlation function
Classical theories with hidden variable require
Quantum mechanics predicts B2.7 for the
appropriate choice of qs and the state
Experimentally measured value B2.697. Phys. Rev.
Let. 4992 (1982)
6
Hanburry-Brown-Twiss experiments
Classical theory of the second order coherence
Hanbury Brown and Twiss, Proc. Roy. Soc.
(London), A, 242, pp. 300-324
Measurements of the angular diameter of
Sirius Proc. Roy. Soc. (London), A, 248, pp.
222-237
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
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
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Shot noise in electron transport
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)
10
Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices Hanburry-Brown-
Twiss experiments and beyond
Theory Altman et al., PRA (2004)
Experiment Folling et al., Nature (2005)
Spielman et al., PRL (2007)
Tom et al. Nature (2006)
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Time of flight experiments
Quantum noise interferometry of atoms in an
optical lattice
Second order coherence
12
Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Experiment Folling et al., Nature (2005)
13
Hanburry-Brown-Twiss stellar interferometer
14
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
15
Second order correlations asHanburry-Brown-Twiss
effect
Bosons/Fermions
16
Second order coherence in the insulating state of
fermions.
Experiment Tom et al. Nature (2006)
17
Second order correlations asHanburry-Brown-Twiss
effect
Bosons/Fermions
Tom et al. Nature (2006)
Folling et al., Nature (2005)
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Probing spin order in optical lattices
Correlation function measurements after TOF
expansion.
Extra Bragg peaks appear in the second order
correlation function in the AF phase.
This reflects
doubling of
the
unit cell by
magnetic order.
19
Interference experimentswith cold atoms
Probing fluctuations in 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 1D Bose gas Hofferberth et al.
Nat. Physics 2008
22
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 et al., PNAS (2006) Gritsev et al.,
Nature Physics (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
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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, PRA
(2007)
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 2009
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
28
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
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Interference of two dimensional condensates
Experiments Hadzibabic et al. Nature (2006)
Gati et al., PRL (2006)
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
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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
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Experiments with 2D Bose gas. Proliferation of
thermal vortices Hadzibabic et al.,
Nature (2006)
Fraction of images showing at least one
dislocation
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Spin dynamics in 1d systemsRamsey
interference experiments
A. Widera, V. Gritsev et al, PRL 2008, Theory
Expt
T. Kitagawa et al., PRL 2010, Theory
J. Schmiedmayer et al., unpublished expts
37
Ramsey interference
Atomic clocks and Ramsey interference
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Ramsey Interference with BEC
Single mode approximation
Interactions should lead to collapse and revival
of Ramsey fringes
Amplitude of Ramsey fringes
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Ramsey Interference with 1d BEC
1d systems in microchips
1d systems in optical lattices
Two component BEC in microchip

• Ramsey interference in 1d tubes
• Widera et al.,
• PRL 100140401 (2008)

Treutlein et.al, PRL 2004, also Schmiedmayer,
Van Druten
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Ramsey interference in 1d condensates
A. Widera, et al, PRL 2008
Collapse but no revivals
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Spin echo. Time reversal experiments
Single mode approximation
The Hamiltonian can be reversed by changing a12
Predicts perfect spin echo
42
Spin echo. Time reversal experiments
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?
43
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
44
Interaction induced collapse of Ramsey
fringes.Multimode analysis
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
Luttinger liquid provides good agreement with
experiments.
Technical noise could also lead to the absence
of echo
Need smoking gun signatures of many-body
decoherece
45
Probing spin dynamics using distribution
functions
Distribution contains information about higher
order correlation functions
For longer segments shot noise is not
important. Joint distribution functions for
different spin components can also be obtained!
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Distribution function of fringe contrastas a
probe of many-body dynamics
Short segments
Radius Amplitude
Angle Phase
Long segments
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Distribution function of fringe contrastas a
probe of many-body dynamics
Splitting one condensate into two.
Preliminary results by J. Schmiedmayers group
48
Long segments
Short segments
l 110 mm
l 20 mm
Expt
Theory
Data Schmiedmayer et al., unpublished
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Summary of lecture 2

• Detection of many-body phases using noise
  correlations
• AF/CDW phases in optical lattices, paired
  states
• Experiments with low dimensional systems
• Interference experiments as a probe of BKT
  transition in 2D,
• Luttinger liquid in 1d. Analysis of high
  order correlations

Quantum noise is a powerful tool for analyzing
many body states of ultracold atoms
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Lecture 3

• Introduction. Systems of ultracold atoms.
• Cold atoms in optical lattices.
• Bose Hubbard model
• Bose mixtures in optical lattices
• Detection of many-body phases using noise
  correlations
• Experiments with low dimensional systems
• Fermions in optical lattices
• Magnetism
• Pairing in systems with repulsive
  interactions
• Current experiments Paramagnetic Mott state
• Experiments on nonequilibrium fermion dynamics
• Lattice modulation experiments
• Doublon decay
• Stoner instability

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Second order correlations. Experimental issues.
Autocorrelation function

• Complications we need to consider
• finite resolution of detectors
• projection from 3D to 2D plane

s detector resolution
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Second order coherence in the insulating state of
bosons.
Experiment Folling et al., Nature (2005)
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Second order correlation function
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Relate operators after the expansion to operators
before the expansion. For long expansion times
use steepest descent method of integration
TOF experiments map momentum distributions to
real space images
Second order real-space correlations after TOF
expansion can be related to second order momentum
correlations inside the trapped system
56
Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Example Mott state of spinless bosons
Only local correlations present in the Mott state
G - reciprocal vectors of the optical lattice
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Quantum noise in TOF experiments in optical
lattices
Boson bunching arises from the Bose enhancement
factors. A single particle state with
quasimomentum q is a supersposition of states
with physical momentum qnG. When we detect a
boson at momentum q we increase the probability
to find another boson at momentum qnG.
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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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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Example Band insulating state of spinless
fermions
Only local correlations present in the band
insulator state
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Quantum noise analysis of time-of-flight experimen
ts with atoms in optical lattices
Example Band insulating state of spinless
fermions
We get fermionic antibunching. This can be
understood as Pauli principle. A single particle
state with quasimomentum q is a supersposition of
states with physical momentum qnG. When we
detect a fermion at momentum q we decrease
the probability to find another fermion at
momentum qnG.