Measuring correlation functions in interacting systems of cold atoms

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Measuring correlation functions in interacting systems of cold atoms Anatoli Polkovnikov Harvard/Boston University Ehud Altman Harvard/Weizmann Vladimir Gritsev Harvard Mikhail Lukin Harvard Luming Duan Michigan Vito Scarola Maryland Sankar Das Sarma Maryland Eugene Demler Harvard to: J. Schmiedmayer, M. Oberthaler, V. Vuletic, M. Greiner, M. Oshikawa, Z. Hadzibabic

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Measuring correlation functions in interacting systems of cold atoms. Anatoli Polkovnikov Harvard/Boston University Ehud Altman Harvard/Weizmann Vladimir Gritsev Harvard Mikhail Lukin Harvard Luming Duan Michigan Vito Scarola Maryland - PowerPoint PPT Presentation

Transcript of Measuring correlation functions in interacting systems of cold atoms

Page 1: Measuring correlation functions in interacting systems of cold atoms

Measuring correlation functions in interacting systems of cold atoms

Anatoli Polkovnikov Harvard/Boston UniversityEhud Altman Harvard/WeizmannVladimir Gritsev HarvardMikhail Lukin HarvardLuming Duan MichiganVito Scarola MarylandSankar Das Sarma MarylandEugene Demler Harvard

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

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Bose-Einstein condensation

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

Ultralow density condensed matter system

Interactions are weak and can be described theoretically from first principles

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New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions

• Optical lattices

• Feshbach resonances

• Low dimensional systems

• Systems with long range dipolar interactions (magnetic dipolar interactions for atoms, electric dipolar interactions for molecules)

• Rotating systems

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Atoms in optical lattices

Theory: Jaksch et al. PRL (1998)

Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004);

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Feshbach resonance and fermionic condensates Greiner et al., Nature 426:537 (2003); Ketterle et al., PRL 91:250401 (2003)

Ketterle et al.,Nature 435, 1047-1051 (2005)

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One dimensional systems

Strongly interacting regime can be reached for low densities

One dimensional systems in microtraps.Thywissen et al., Eur. J. Phys. D. (99);Hansel et al., Nature (01);Folman et al., Adv. At. Mol. Opt. Phys. (02)

1D confinement in optical potentialWeiss et al., Science (05);Bloch et al., Esslinger et al.,

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New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions

Goals

• Resolve long standing questions in condensed matter physics (e.g. origin of high temperature superconductivity)

• Resolve matter of principle questions (e.g. existence of spin liquids in two and three dimensions)

• Study new phenomena in strongly correlated systems (e.g. coherent far from equilibrium dynamics)

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This talk:

Detection of many-body quantum phases by measuring correlation functions

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Outline

Measuring correlation functions in intereference experiments 1. Interference of independent condensates 2. Interference of interacting 1D systems 3. Interference of 2D systems 4. Full counting statistics of intereference experiments. Connection to quantum impurity problem Quantum noise interferometry in time of flight experiments

1. Detection of magnetically ordered Mott states in optical lattices 2. Observation of fermion pairing

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Measuring correlation functions in intereference experiments

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Interference of two independent condensates

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

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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

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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 1 (05)

x2

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Interference amplitude and correlations

For identical condensates

Instantaneous correlation function

L

Polkovnikov, Altman, Demler, cond-mat/0511675

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For impenetrable bosons and

Interference between Luttinger liquidsLuttinger 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

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Luttinger parameter K may beextracted from the angular dependence of

Rotated probe beam experiment

For large imaging angle, ,

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Interference between two-dimensional BECs at finite temperature.Kosteritz-Thouless transition

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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

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

Ly

Lx

Below KT transitionAbove KT transition

Theory: Polkovnikov, Altman, Demler, cond-mat/0511675

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x

z

Time of

flight

low temperature higher temperature

Typical interference patterns

Experiments with 2D Bose gasHaddzibabic, Stock, Dalibard, et al.

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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 gasHaddzibabic, Stock, Dalibard, et al.

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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

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 gasHaddzibabic, Stock, Dalibard, et al.

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Exponent

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 gasHaddzibabic, Stock, Dalibard, et al.

Ultracold atoms experiments:jump in the correlation function.

KT theory predicts =1/4 just below the transition

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Experiments with 2D Bose gas. Proliferation of thermal vortices Haddzibabic, Stock, Dalibard, et al.

Fraction of images showing at least one dislocation

Exponent

0.5

0 0.1 0.2 0.3

0.4

0.3

central contrast

The onset of proliferation coincides with shifting to 0.5!

0

10%

20%

30%

central contrast

0 0.1 0.2 0.3 0.4

high T low T

Z. Hadzibabic et al., in preparation

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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

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Full counting statistics of interference between two interacting one dimensional Bose liquids

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

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Higher moments of interference amplitude

L Higher moments

Changing to periodic boundary conditions (long condensates)

Explicit expressions for are available but cumbersome Fendley, 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 ?

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Impurity in a Luttinger liquid

Expansion of the partition function in powers of g

Partition function of the impurity contains correlation functions taken 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

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Relation between quantum impurity problemand 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

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is related to a Schroedinger equation Dorey, 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

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0 1 2 3 4

P

roba

bilit

y P

(x)

x

K=1 K=1.5 K=3 K=5

Evolution of the distribution function

Narrow distributionfor .Distribution widthapproaches

Wide Poissoniandistribution for

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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 CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999

From interference amplitudes to conformal field theories

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Quantum noise interferometry in time of flight experiments

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Atoms in an optical lattice.Superfluid to Insulator transition

Greiner et al., Nature 415:39 (2002)

U

1n

t/U

SuperfluidMott insulator

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Time of flight experiments

Quantum noise interferometry of atoms in an optical lattice

Second order coherence

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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)

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Hanburry-Brown-Twiss stellar interferometer

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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

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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)

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Effect of parabolic potential on the second order coherence

Experiment: Spielman, Porto, et al.,Theory: Scarola, Das Sarma, Demler, cond-mat/0602319

Width of the correlation peak changes across the Transition, reflecting the evolution of the Mott domains

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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

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Applications of quantum noise interferometry

Spin order in Mott states of atomic mixtures

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t

t

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

Two component Bose Hubbard model

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Two component Bose mixture in optical lattice.Magnetic order in an insulating phase

Insulating phases with N=1 atom per site. Average densities

Easy plane ferromagnet

Easy axis antiferromagnet

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Quantum magnetism of bosons in optical lattices

Duan, Lukin, Demler, PRL (2003)

• Ferromagnetic• Antiferromagnetic

Kuklov and Svistunov, PRL (2003)

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Probing spin order of bosons

Correlation Function Measurements

Extra Braggpeaks appearin the secondorder correlationfunction in theAF phase

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Applications of quantum noise interferometry

Detection of fermion pairing

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Fermions with repulsive interactions

t

U

tPossible d-wave pairing of fermions

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Positive U Hubbard model

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

Antiferromagnetic insulator

D-wave pairing of fermions

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Second order interference from a BCS superfluid

)'()()',( rrrr nnn

n(r)

n(r’)

n(k)

k

0),( BCSn rr

BCS

BEC

kF

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Momentum correlations in paired fermionsTheory: Altman et al., PRA 70:13603 (2004)Experiment: Greiner et al., PRL 94:110401 (2005)

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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

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Conclusions

Interference of extended condensates can be usedto probe correlation functions in one and two dimensional systems

Noise interferometry is a powerful tool for analyzingquantum many-body states in optical lattices

Spectral determinants provide a new approach to noise calculations in systems of strongly interacting particles. Extension to full counting statistics of current in electron tunneling should be possible.