Download Learning about order from noise Quantum noise studies of

Outline of these lectures
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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
Learning about order from noise
Quantum noise studies of ultracold atoms
Quantum noise
Classical measurement:
collapse of the wavefunction into eigenstates of x
Histogram of measurements of x
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
Aspect’s experiments:
tests of Bell’s inequalities
+
+
1
-
q1
S
q2
2
-
S
Correlation function
Classical theories with hidden variable require
Quantum mechanics predicts B=2.7 for the appropriate choice of q‘s and the state
Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982)
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
Quantum theory of HBT experiments
Glauber,
Quantum Optics and
Electronics (1965)
HBT experiments with matter
For bosons
Experiments with neutrons
Ianuzzi et al., Phys Rev Lett (2006)
Experiments with electrons
Kiesel et al., Nature (2002)
For fermions
Experiments with 4He, 3He
Westbrook et al., Nature (2007)
Experiments with ultracold atoms
Bloch et al., Nature (2005,2006)
Shot noise in electron transport
Proposed by Schottky to measure the electron charge in 1918
e-
e-
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
Shot noise in electron transport
Current noise for tunneling
across a Hall bar on the 1/3
plateau of FQE
Etien et al. PRL 79:2526 (1997)
see also Heiblum et al. Nature (1997)
Quantum noise analysis of time-of-flight
experiments 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)
Time of flight experiments
Cloud before expansion
Cloud after expansion
Quantum noise interferometry of atoms in an optical lattice
Second order coherence
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Experiment: Folling et al., Nature (2005)
Hanburry-Brown-Twiss stellar interferometer
Second order coherence in the insulating state of bosons
Bosons at quasimomentum
expand as plane waves
with wavevectors
First order coherence:
Oscillations in density disappear after summing over
Second order coherence:
Correlation function acquires oscillations at reciprocal lattice vectors
Second order correlations as
Hanburry-Brown-Twiss effect
Bosons/Fermions
Second order coherence in the insulating state of fermions.
Experiment: Tom et al. Nature (2006)
Second order correlations as
Hanburry-Brown-Twiss effect
Bosons/Fermions
Folling et al., Nature (2005)
Tom et al. Nature (2006)
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.
Interference experiments
with cold atoms
Probing fluctuations in low dimensional systems
Interference of independent condensates
Experiments: Andrews et al., Science 275:637 (1997)
Theory: Javanainen, Yoo, PRL 76:161 (1996)
Cirac, Zoller, et al. PRA 54:R3714 (1996)
Castin, Dalibard, PRA 55:4330 (1997)
and many more
Experiments with 2D Bose gas
Hadzibabic, Dalibard et al., Nature 2006
z
Time of
flight
x
Experiments with 1D Bose gas Hofferberth et al. Nat. Physics 2008
Interference of two independent condensates
r’
r
Assuming ballistic expansion
1
r+d
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
Interference of fluctuating condensates
d
Polkovnikov et al., PNAS (2006); Gritsev et al., Nature Physics (2006)
Amplitude of interference fringes,
x1
x2
For independent condensates Afr is finite
but Df is random
For identical
condensates
Instantaneous correlation function
Fluctuations in 1d BEC
Thermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
Interference between Luttinger liquids
Luttinger liquid at T=0
K – Luttinger parameter
For non-interacting bosons
For impenetrable bosons
Finite
temperature
Experiments: Hofferberth,
Schumm, Schmiedmayer
and
and
Distribution function of fringe amplitudes
for interference of fluctuating condensates
Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006
Imambekov, Gritsev, Demler, PRA (2007)
is a quantum operator. The measured value of
will fluctuate from shot to shot.
L
Higher moments reflect higher order correlation functions
We need the full distribution function of
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
Interference between interacting 1d Bose liquids.
Distribution function of the interference amplitude
Distribution function of
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, …
2D quantum gravity,
non-intersecting loops
Yang-Lee singularity
Fringe visibility and statistics of random surfaces
Distribution function of
Mapping between fringe
visibility and the problem
of surface roughness for
fluctuating random
surfaces.
Relation to 1/f Noise and
Extreme Value Statistics
h ( )
2
Roughness
  h( ) d
Interference of two dimensional condensates
Experiments: Hadzibabic et al. Nature (2006)
Gati et al., PRL (2006)
Ly
Lx
Lx
Probe beam parallel to the plane of the condensates
Interference of two dimensional condensates.
Quasi long range order and the KT transition
Ly
Lx
Above KT transition
Below KT transition
Experiments with 2D Bose gas
z
Hadzibabic, Dalibard et al., Nature 441:1118 (2006)
Time of
flight
x
Typical interference patterns
low temperature
higher temperature
Experiments with 2D Bose gas
Hadzibabic et al., Nature 441:1118 (2006)
x
integration
over x axis z
z
Contrast after
integration
0.4
low T
integration
middle T
0.2
over x axis
z
high T
integration
over x axis
Dx
0
z
0
10
20
30
integration distance Dx
(pixels)
Experiments with 2D Bose gas
Integrated contrast
Hadzibabic et al., Nature 441:1118 (2006)
0.4
fit by:
C2 ~
low T
1
Dx
 1 

 Dx 
Dx
2


g
(
0
,
x
)
dx ~ 
 1
middle T
0.2
Exponent a
high T
0
0
10
20
30
integration distance Dx
if g1(r) decays exponentially
with
:
0.5
0.4
0.3
high T
0
if g1(r) decays algebraically or
exponentially with a large
:
0.1
low T
0.2
0.3
central contrast
“Sudden” jump!?
2a
Experiments with 2D Bose gas. Proliferation of
thermal vortices
Hadzibabic et al., Nature (2006)
30%
Fraction of images showing
at least one dislocation
Exponent a
20%
0.5
10%
0.4
low T
high T
0
0
0.1
0.2
0.3
central contrast
The onset of proliferation
coincides with a shifting to 0.5!
0.4
0.3
0
0.1
0.2
central contrast
0.3
Spin dynamics in 1d systems:
Ramsey 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
Ramsey interference
1
0
Atomic clocks and Ramsey interference:
Working with N atoms improves
the precision by
.
t
Ramsey Interference with BEC
Single mode
approximation
Amplitude of
Ramsey fringes
Interactions should
lead to collapse and
revival of Ramsey fringes
time
Ramsey Interference with 1d BEC
1d systems in optical
lattices
Ramsey interference in 1d tubes:
A.Widera et al.,
B. PRL 100:140401 (2008)
1d systems in microchips
Two component BEC
in microchip
Treutlein et.al, PRL 2004,
also Schmiedmayer, Van Druten
Ramsey interference in 1d condensates
A. Widera, et al, PRL 2008
Collapse but no revivals
Spin echo. Time reversal experiments
Single mode approximation
The Hamiltonian can be reversed by changing a12
Predicts perfect spin echo
Spin echo. Time reversal experiments
A. Widera et al., PRL 2008
No revival?
Experiments done in array of tubes.
Strong fluctuations in 1d systems.
Single mode approximation does not apply.
Need to analyze the full model
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
Interaction induced collapse of Ramsey fringes.
Multimode analysis
Only q=0 mode shows complete spin echo
Finite q modes continue decay
The net visibility is a result of competition
between q=0 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
Distribution
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!
Distribution function of fringe contrast
as a probe of many-body dynamics
Short segments
Radius =
Amplitude
Angle =
Phase
Long segments
Distribution function of fringe contrast
as a probe of many-body dynamics
Splitting one
condensate
into two.
Preliminary results
by J. Schmiedmayer’s group
Short segments
Long segments
l =20 mm
l =110 mm
Expt
Theory
Data: Schmiedmayer et al.,
unpublished
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
Lecture 3
• 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
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
- period of the optical lattice
In Mainz experiments
The signal in
and
is
Second order coherence in the insulating state of bosons.
Experiment: Folling et al., Nature (2005)
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Cloud before expansion
Cloud after expansion
Second order correlation function
Here
and
are taken after the expansion time t.
Two signs correspond to bosons and fermions.
Quantum noise analysis of time-of-flight
experiments 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
Quantum noise analysis of time-of-flight
experiments 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
Quantum noise in TOF experiments in optical lattices
We get bunching when
corresponds to one
of the reciprocal vectors of the original lattice.
Boson bunching arises from the Bose enhancement factors.
A single particle state with quasimomentum q is a
supersposition of states with physical momentum q+nG.
When we detect a boson at momentum q we increase
the probability to find another boson at momentum q+nG.
Interference of an array of independent condensates
Hadzibabic et al., PRL 93:180403 (2004)
Smooth structure is a result of finite experimental resolution (filtering)
3
1.4
2.5
1.2
2
1
1.5
0.8
1
0.6
0.5
0.4
0
-0.5
0.2
-1
0
-1.5
0
200
400
600
800
1000
1200
-0.2
0
200
400
600
800
1000
1200
Quantum noise analysis of time-of-flight
experiments with atoms in optical lattices
Example: Band insulating state of spinless fermions
Only local correlations present in the band insulator state
Quantum noise analysis of time-of-flight
experiments 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 q+nG.
When we detect a fermion at momentum q we decrease the
probability to find another fermion at momentum q+nG.