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 mechanicsBohr-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

dAmplitude 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 BECThermal 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

ontra

st 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

0.3 high T low T

a2

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 contrast0 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

plitu

de o

fR

amse

y fri

nges

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!

Dist

ribut

ion

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.