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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
Learning about order from noise
Quantum noise studies of ultracold atoms
Quantum noiseClassical 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
Measuring spin of a particle in the left detectorinstantaneously determines its value in the right detector
Einstein-Podolsky-Rosen experiment
Aspect’s experiments:tests of Bell’s inequalities
SCorrelation 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)
+
-
+
-1 2q1 q2
S
Hanburry-Brown-Twiss experimentsClassical theory of the second order coherence
Measurements of the angular diameter of SiriusProc. Roy. Soc. (London), A, 248, pp. 222-237
Hanbury Brown and Twiss, Proc. Roy. Soc. (London), A, 242, pp. 300-324
Quantum theory of HBT experiments
For bosons
For fermions
Glauber,Quantum Optics and Electronics (1965)
HBT experiments with matter
Experiments with 4He, 3HeWestbrook et al., Nature (2007)
Experiments with neutronsIanuzzi et al., Phys Rev Lett (2006)
Experiments with electronsKiesel et al., Nature (2002)
Experiments with ultracold atomsBloch et al., Nature (2005,2006)
Shot noise in electron transport
e- e-
When shot noise dominates over thermal noise
Spectral density of the current noise
Proposed by Schottky to measure the electron charge in 1918
Related to variance of transmitted charge
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-flightexperiments 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
Quantum noise interferometry of atoms in an optical lattice
Second order coherence
Cloud after expansion
Cloud before expansion
Experiment: Folling et al., Nature (2005)
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
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 asHanburry-Brown-Twiss effect
Bosons/Fermions
Second order coherence in the insulating state of fermions.
Experiment: Tom et al. Nature (2006)
Second order correlations asHanburry-Brown-Twiss effect
Bosons/Fermions
Tom et al. Nature (2006)Folling et al., Nature (2005)
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 experimentswith 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
x
z
Time of
flight
Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 2006
Experiments with 1D Bose gas Hofferberth et al. Nat. Physics 2008
Interference of two independent condensates
1
2
r
r+d
d
r’
Phase difference between clouds 1 and 2is not well defined
Assuming ballistic expansion
Individual measurements show interference patternsThey disappear after averaging over many shots
x1
d
Amplitude of interference fringes,
Interference of fluctuating condensates
For identical condensates
Instantaneous correlation function
For independent condensates Afr is finite but Df is random
x2
Polkovnikov et al., PNAS (2006); Gritsev et al., Nature Physics (2006)
Fluctuations in 1d BEC
Thermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
For impenetrable bosons and
Interference between Luttinger liquids
Luttinger liquid at T=0
K – Luttinger parameter
Finite temperature
Experiments: Hofferberth,Schumm, Schmiedmayer
For non-interacting bosons and
Distribution function of fringe amplitudes for interference of fluctuating condensates
L
is a quantum operator. The measured value of will fluctuate from shot to shot.
Higher moments reflect higher order correlation functions
Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006Imambekov, Gritsev, Demler, PRA (2007)
We need the full distribution function of
Distribution function of interference fringe contrastHofferberth et al., Nature Physics 2009
Comparison of theory and experiments: no free parametersHigher order correlation functions can be obtained
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
Quantum impurity problem: interacting one dimensionalelectrons 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, …
Interference between interacting 1d Bose liquids.Distribution function of the interference amplitude
Distribution function of
Yang-Lee singularity
2D quantum gravity,non-intersecting loops
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
)(h
Roughness dh2
)(
Distribution function of
Interference of two dimensional condensates
Ly
Lx
Lx
Experiments: Hadzibabic et al. Nature (2006)
Probe beam parallel to the plane of the condensates
Gati et al., PRL (2006)
Interference of two dimensional condensates.Quasi long range order and the KT transition
Ly
Lx
Below KT transitionAbove KT transition
x
z
Time of
flight
low temperature higher temperature
Typical interference patterns
Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (2006)
integration
over x axis
Dx
z
z
integration
over x axisz
x
integration distance Dx
(pixels)
Contrast afterintegration
0.4
0.2
00 10 20 30
middle Tlow T
high T
integration
over x axis z
Experiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)
fit by:
integration distance Dx
Inte
grat
ed c
ontr
ast 0.4
0.2
00 10 20 30
low Tmiddle T
high T
if g1(r) decays exponentially with :
if g1(r) decays algebraically or exponentially with a large :
Exponent a
central contrast
0.5
0 0.1 0.2 0.3
0.4
0.3high T low T
2
21
2 1~),0(
1~
x
D
x Ddxxg
DC
x
“Sudden” jump!?
Experiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)
Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature (2006)
Fraction of images showing at least one dislocation
Exponent a
0.5
0 0.1 0.2 0.3
0.4
0.3
central contrast
The onset of proliferation coincides with a shifting to 0.5!
0
10%
20%
30%
central contrast
0 0.1 0.2 0.3 0.4
high T low T
Spin dynamics in 1d systems:Ramsey interference experiments
T. Kitagawa et al., PRL 2010, Theory
A. Widera, V. Gritsev et al, PRL 2008, Theory + Expt
J. Schmiedmayer et al., unpublished expts
Working with N atoms improves the precision by .
Ramsey interference
t0
1
Atomic clocks and Ramsey interference:
Ramsey Interference with BEC
Single modeapproximation
time
Am
plit
ude
of
Ra
mse
y fr
ing
es
Interactions shouldlead to collapse andrevival of Ramsey fringes
1d systems in microchips
Treutlein et.al, PRL 2004, also Schmiedmayer, Van Druten
Two component BEC in microchip
Ramsey Interference with 1d BEC
1d systems in opticallattices
Ramsey interference in 1d tubes: A.Widera et al.,B. PRL 100:140401 (2008)
Ramsey interference in 1d condensates
Collapse but no revivals
A. Widera, et al, PRL 2008
Spin echo. Time reversal experiments
Single mode approximation
Predicts perfect spin echo
The Hamiltonian can be reversed by changing a12
Spin echo. Time reversal experiments
No revival?
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
Interaction induced collapse of Ramsey fringes.Multimode analysis
Luttinger model
Changing the sign of the interaction reverses the interaction part of the Hamiltonian but not the kinetic energy
Time dependent harmonic oscillatorscan be analyzed exactly
Low energy effective theory: Luttinger liquid approach
Technical noise could also lead to the absence of echo
Need “smoking gun” signaturesof many-body decoherece
Luttinger liquid provides good agreement with experiments.
Only q=0 mode shows complete spin echoFinite q modes continue decay
The net visibility is a result of competition between q=0 and other modes
Interaction induced collapse of Ramsey fringes.Multimode analysis
Probing spin dynamics using distribution functions
Distribution contains informationabout higher order correlation functions
For longer segments shot noise is not important.Joint distribution functions for different spin components
can also be obtained!
Dis
trib
uti
on
Distribution function of fringe contrastas a probe of many-body dynamics
Short segments
Long segments
Radius =Amplitude
Angle =Phase
Distribution function of fringe contrastas a probe of many-body dynamics
Preliminary results by J. Schmiedmayer’s group
Splitting one condensate into two.
Short segments Long segments
l =20 mm l =110 mm
Expt Theory Data: Schmiedmayer et al., unpublished
Summary of lecture 2
Quantum noise is a powerful tool for analyzing many body states of ultracold atoms
• 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
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
Second order correlations. Experimental issues.
Complications we need to consider:- finite resolution of detectors- projection from 3D to 2D plane
s – detector resolution
Autocorrelation function
In Mainz experiments and
The signal in is
- period of the optical lattice
Second order coherence in the insulating state of bosons.
Experiment: Folling et al., Nature (2005)
Second order correlation function
Cloud after expansion
Cloud before expansion
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
Here and are taken after the expansion time t.Two signs correspond to bosons and fermions.
Relate operators after the expansion to operators before the expansion. For long expansion times use steepest descent method of integration
Second order real-space correlations after TOF expansioncan be related to second order momentum correlationsinside the trapped system
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
TOF experiments map momentum distributions to real space images
Example: Mott state of spinless bosons
Only local correlations present in the Mott state
G - reciprocalvectors of theoptical lattice
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
Quantum noise in TOF experiments in optical lattices
We get bunching when corresponds to oneof 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.
0 200 400 600 800 1000 1200-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Interference of an array of independent condensates
Hadzibabic et al., PRL 93:180403 (2004)
Smooth structure is a result of finite experimental resolution (filtering)
0 200 400 600 800 1000 1200-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Example: Band insulating state of spinless fermions
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
Only local correlations present in the band insulator state
Example: Band insulating state of spinless fermions
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices
We get fermionic antibunching. This can be understoodas Pauli principle. A single particle state with quasimomentumq is a supersposition of states with physical momentum q+nG.When we detect a fermion at momentum q we decrease theprobability to find another fermion at momentum q+nG.
