measuring correlation functions in interacting systems of cold...

Measuring correlation functions in

interacting systems of cold atoms

Thanks to: M. Greiner , Z. Hadzibabic, M. Oberthaler, J. Schmiedmayer, V. Vuletic

Anatoli Polkovnikov Harvard/Boston UniversityEhud Altman Harvard/Weizmann

Vladimir Gritsev Harvard

Mikhail Lukin HarvardEugene Demler Harvard

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. 83:333 (1951)

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

Lecture I

Measuring correlation functions in intereference experiments

1. Interference of independent condensates2. Interference of interacting 1D systems

3. Interference of 2D systems4. Full distribution function of the fringe amplitudes

in intereference experiments. 5. Studying coherent dynamics of strongly interacting

systems in interference experiments

Lecture II

Quantum noise interferometryin time of flight experiments

1. Detection of spin order in Mott states of atomic mixtures2. Detection of fermion pairing

Analysis of high order correlation

functions in low dimensional systems

Polkovnikov, Altman, Demler, PNAS (2006)

Measuring correlation functions in intereference experiments

Interference of two independent condensates

Andrews et al., Science 275:637 (1997)

Interference of two independent condensates

1

2

r

r+d

d

r’

Clouds 1 and 2 do not have a well defined phase difference.However each individual measurement shows an interference pattern

Amplitude of interference fringes, , contains information about phase fluctuations within individual condensates

y

x

Interference of one dimensional condensates

x1

d Experiments: Schmiedmayer et al., Nature Physics (2005)

x2

Interference amplitude and correlations

For identical condensates

Instantaneous correlation function

L

Interacting bosons in 1d at T=0

K – Luttinger parameter

Low energy excitations and long distance correlation functions can be described

by the Luttinger Hamiltonian.

Small K corresponds to strong quantum fluctuations

Connection to original bosonic particles

Luttinger liquids in 1d

For non-interacting bosons

For impenetrable bosons

Correlation function decays rapidly for small K. This decay comes from strong quantum fluctuations

For impenetrable bosons and

Interference between 1d interacting bosons

Luttinger liquid at T=0

K – Luttinger parameter

Luttinger liquid at finite temperature

For non-interacting bosons and

Analysis of can be used for thermometry

L

Luttinger parameter K may beextracted from the angular

dependence of

Rotated probe beam experiment

θ

For large imaging angle, ,

Interference between two-dimensional

BECs at finite temperature.

Kosteritz-Thouless transition

Interference of two dimensional condensates

Ly

Lx

Lx

Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/0506559

Probe beam parallel to the plane of the condensates

Gati, Oberthaler, et al., cond-mat/0601392

Interference of two dimensional condensates.Quasi long range order and the KT transition

Ly

Lx

Below KT transitionAbove KT transition

Below Kosterlitz-Thouless transition:Vortices confined. Quasi long range order

Above Kosterlitz-Thouless transition:Vortices proliferate. Short range order

x

z

Time of

flight

low temperature higher temperature

Typical interference patterns

Experiments with 2D Bose gasHadzibabic et al., Nature (2006)

integration

over x axis

Dx

z

z

integration

over x axisz

x

integration distance Dx

(pixels)

Contrast after

integration

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 (2006)

fit by:

integration distance Dx

Inte

gra

ted

con

tras

t 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 α

central contrast

0.5

0 0.1 0.2 0.3

0.4

0.3high T low T

[ ]α2

2

1

2 1~),0(

1~

∫

x

D

x Ddxxg

DC

x

“Sudden” jump!?

Experiments with 2D Bose gasHadzibabic et al., Nature (2006)

Exponent α

central contrast

0.5

0 0.1 0.2 0.3

0.4

0.3 high T low T

He experiments:

universal jump in

the superfluid density

T (K)

1.0 1.1 1.2

1.0

0

c.f. Bishop and Reppy

Experiments with 2D Bose gasHadzibabic et al., Nature (2006)

Ultracold atoms experiments:jump in the correlation function.

KT theory predicts α=1/4

just below the transition

Experiments with 2D Bose gas. Proliferation of

thermal vortices Haddzibabic et al., Nature (2006)

Fraction of images showing

at least one dislocation

0

10%

20%

30%

central contrast

0 0.1 0.2 0.3 0.4

high T low T

Rapidly rotating two dimensional condensates

Time of flight experiments with rotating condensates correspond to density measurements

Interference experiments measure singleparticle correlation functions in the rotatingframe

Interference between two interacting

one dimensional Bose liquids

Full distribution function

of the amplitude of

interference fringes

Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475

Higher moments of interference amplitude

L Higher moments

Changing to periodic boundary conditions (long condensates)

Explicit expressions for are available but cumbersomeFendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)

is a quantum operator. The measured value of will fluctuate from shot to shot.

Can we predict the distribution function of ?

Impurity in a Luttinger liquid

Expansion of the partition function in powers of g

Partition function of the impurity contains correlation functionstaken at the same point and at different times. Momentsof interference experiments come from correlations functionstaken at the same time but in different points. Euclidean invarianceensures that the two are the same

Relation between quantum impurity problem

and interference of fluctuating condensates

Distribution function

of fringe amplitudes

Distribution function can be reconstructed fromusing completeness relations for the Bessel functions

Normalized amplitude

of interference fringes

Relation to the impurity partition function

is related to the single particle Schroedinger equationDorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999)

Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)

Spectral determinant

Bethe ansatz solution for a quantum impurity

can be obtained from the Bethe ansatz followingZamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95)

Making analytic continuation is possible but cumbersome

Interference amplitude and spectral determinant

0 1 2 3 4

P

rob

ab

ility

P(x

)

x

K=1

K=1.5

K=3

K=5

Evolution of the distribution function

Narrow distributionfor .Approaches Gumbledistribution. Width

Wide Poissoniandistribution for

When K>1, is related to Q operators of CFT with c<0. This includes 2D quantum gravity, non-intersecting loop model on 2D lattice, growth of randomfractal stochastic interface, high energy limit of multicolor QCD, …

Yang-Lee singularity

2D quantum gravity,non-intersecting loops on 2D lattice

correspond to vacuum eigenvalues of Q operators of CFTBazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999

From interference amplitudes to conformal field theories

Studying coherent dynamics

of strongly interacting systems

in interference experiments

Coupled 1d systems

J

Interactions lead to phase fluctuations within individual condensates

Tunneling favors aligning of the two phases

Interference experiments measure only the relative phase

Motivated by experiments of Schmiedmayer et al.

Coupled 1d systems

J

Relative phase Particle number imbalance

Conjugate variables

Small K corresponds to strong quantum flcutuations

Quantum Sine-Gordon model

Quantum Sine-Gordon model is exactly integrable

Excitations of the quantum Sine-Gordon model

Hamiltonian

Imaginary time action

soliton antisoliton breather

Coherent dynamics of quantum Sine-Gordon model

Motivated by experiments of Schmiedmayer et al.

J

Prepare a system at t=0 Take to the regime of finite

tunneling and let evolve

for some time

Measure amplitude

of interference pattern

time

Amplitude of

interference fringes

Oscillations or decay?

Coherent dynamics of quantum Sine-Gordon model

From integrability to coherent dynamics

At t=0 we have a state with for all

This state can be written as a “squeezed” state

Time evolution can be easily written

Matrix can be constructed using connection to boundary SG model

Calabrese, Cardy (2006); Ghoshal, Zamolodchikov (1994)

Interference amplitude can be calculated using form factor approach

Smirnov (1992), Lukyanov (1997)

Coherent dynamics of quantum Sine-Gordon model

J

Prepare a system at t=0 Take to the regime of finite

tunneling and let evolve

for some time

Measure amplitude

of interference pattern

time

Amplitude of

interference fringes

Amplitude of interference fringes shows

oscillations at frequencies that correspond

to energies of breater

Coherent dynamics of quantum Sine-Gordon model

Conclusions for part I

Interference of fluctuating condensates can be usedto probe correlation functions in one and two

dimensional systems. Interference experiments can

also be used to study coherent dynamics of interactingsystems

Measuring correlation functions in interacting systems of cold atoms

Lecture II

Quantum noise interferometryin time of flight experiments

1. Time of flight experiments.

Second order coherence in Mott states of spinless bosons2. 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

Bose-Einstein condensation

Cornell et al., Science 269, 198 (1995)

Ultralow density condensed matter systemInteractions are weak and can be described theoretically from first principles

Superfluid to Insulator transitionGreiner et al., Nature 415:39 (2002)

U

µ

1−n

t/U

SuperfluidMott insulator

Time of flight experiments

Quantum noise interferometry of atoms in an optical lattice

Second order coherence

Second order coherence in the insulating state of bosons.

Hanburry-Brown-Twiss experiment

Theory: Altman et al., PRA 70:13603 (2004)

Experiment: Folling et al., Nature 434:481 (2005)

Hanburry-Brown-Twiss stellar interferometer

Hanburry-Brown-Twiss 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 coherence in the insulating state of bosons.

Hanburry-Brown-Twiss experiment

Theory: Altman et al., PRA 70:13603 (2004)

Experiment: Folling et al., Nature 434:481 (2005)

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

Width of the noise peaks

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

Applications of quantum noise interferometryin time of flight experiments

Detection of spin order in Mott states of boson boson mixtures

Engineering magnetic systems

using cold atoms in an optical lattice

See also lectures by A. Georges and I. Cirac in this school

Spin interactions using controlled collisions

Experiment: Mandel et al., Nature 425:937(2003)

Theory: Jaksch et al., PRL 82:1975 (1999)

t

t

Two component Bose mixture in optical latticeExample: . Mandel et al., Nature 425:937 (2003)

Two component Bose Hubbard model

Quantum magnetism of bosons in optical lattices

Duan et al., PRL (2003)

• Ferromagnetic

• Antiferromagnetic

Kuklov and Svistunov, PRL (2003)

Exchange Interactions in Solids

antibonding

bonding

Kinetic energy dominates: antiferromagnetic state

Coulomb energy dominates: ferromagnetic state

Two component Bose mixture in optical lattice.

Mean field theory + Quantum fluctuations

2 ndorder line

Hysteresis

1st order

Altman et al., NJP 5:113 (2003)

Probing spin order of bosons

Correlation Function Measurements

Engineering exotic phases

• Optical lattice in 2 or 3 dimensions: polarizations & frequencies

of standing waves can be different for different directions

ZZ

YY

• Example: exactly solvable model

Kitaev (2002), honeycomb lattice with

H = Jx

< i, j>∈x

∑ σ i

xσ j

x + Jy

< i, j>∈y

∑ σ i

yσ j

y + Jz

< i, j>∈z

∑ σ i

zσ j

z

• 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

Applications of quantum noise interferometryin time of flight experiments

Detection of fermion pairing

Fermionic atoms in optical lattices

Pairing in systems with repulsive interactions. Unconventional pairing. High Tc mechanism

Fermionic atoms in a three dimensional optical lattice

Kohl et al., PRL 94:80403 (2005)

See also lectures of T. Esslinger and W. Ketterle in this school

Fermions with repulsive interactions

t

U

t

Possible d-wave pairing of fermions

Picture courtesy of UBC

Superconductivity group

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

Positive U Hubbard model

Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)

Antiferromagnetic insulator

D-wave superconductor

Second order correlations in the BCS superfluid

)'()()',( rrrr nnn −≡∆

n(r)

n(r’)

n(k)

k

0),( =Ψ−∆ BCSn rr

BCS

BEC

kF

Expansion of atoms in TOF maps k into r

Momentum correlations in paired fermionsGreiner et al., PRL 94:110401 (2005)

Fermion pairing in an optical lattice

Second Order InterferenceIn the TOF images

Normal State

Superfluid State

measures the Cooper pair wavefunction

One can identify unconventional pairing

Simulation of condensed matter systems:Hubbard Model and high Tc superconductivity

t

U

t

Using cold atoms to go beyond “plain vanilla” Hubbard modela) Boson-Fermion mixtures: Hubbard model + phononsb) Inhomogeneous systems, role of disorder

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)?

Boson Fermion mixtures

Fermions interacting with phonons

Boson Fermion mixtures

BEC

See lectures by T. Esslinger and G. Modugno in this school

Bosons provide cooling for fermionsand mediate interactions. They createnon-local attraction between fermions

Charge Density Wave Phase

Periodic arrangement of atoms

Non-local Fermion Pairing

P-wave, D-wave, …

Boson Fermion mixtures

q/kF

/EF

ω

4

3

2

1

0 2 1 0

“Phonons” :Bogoliubov (phase) mode

Effective fermion-”phonon” interaction

Fermion-”phonon” vertex Similar to electron-phonon systems

Boson Fermion mixtures in 1d optical lattices

Cazalila et al., PRL (2003); Mathey et al., PRL (2004)

Spinless fermions Spin ½ fermions

Boson Fermion mixtures in 2d optical lattices

40K -- 87Rb 40K -- 23Na

=1060 nm

(a) =1060nm

(b) =765.5nm

Wang et al., PRA (2005)

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 probequantum many-body states in optical lattices