2012 Kourovka Lecture 1

7/29/2019 2012 Kourovka Lecture 1

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QUANTUMMANY

BODY

Eugene Demler Harvard University

Grad students: A. Imambekov (->Rice),Takuya KitagawaPostdocs: E. Altman ->Weizmann A. Polkovnikov ->U. BostonA.M. Rey (->U. Colorado), V. Gritsev (-> U. Fribourg),D. Pekker (-> Caltech), R. Sensarma (-> J QI Maryland)

Collaborations with experimental groups ofI. Bloch (MPQ),T. Esslinger (ETH), J .Schmiedmayer (Vienna)

Supportedby

NSF,

DARPA

OLE,

AFOSR

MURI,

ARO

MURI


2/89

keV MeV GeV TeVfeV peV eV meV eVneV

pK nK K mK K

He Nfirst BEC

roomtemperature

LHCcurrentexperiments

10-11 - 10-10 K

of alkali atoms


3/89

Bose-Einstein condensation ofwea y n erac ng a oms

Density 1013 cm-1

Typical distance between atoms 300 nmT ical scatterin len th 10 nm

Scattering length is much smaller than characteristic interparticle distances.Interactions are weak


4/89

New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions

Feshbach resonances

Rotating systems

Low dimensional systems

oms n op ca a ces


5/89

Feshbach resonanceGreiner et al., Nature 2003 ; Ketterle et al., 2003

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


6/89

One dimensional systems

One dimensional systems in microtraps.

1D confinement in optical potential

Weiss et al., Science (05);., . . . .

Hansel et al., Nature (01);Folman et al., Adv. At. Mol. Opt. Phys. (02)

.,Esslinger et al.,

regime can be reached

for low densities


7/89

Atoms in optical lattices

eory: a sc e a .

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

Esslinger et al., PRL (2004);and many more


8/89

Quantum simulations with ultracold atoms

Antiferromagnetic andsu erconductin Tc

Antiferromagnetismandpairing at nano Kelvin

of the order of 100 K temperatures

Same microscopic model


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Atoms in optical latticesElectrons in Solids

Simple metals

Perturbation theory in Coulomb interaction applies.an s ruc ure me o s wor

Strongly Correlated Electron SystemsBand structure methods fail.

Novel phenomena in strongly correlated electron systems:

uantum ma netism hase se aration unconventional su erconductivithigh temperature superconductivity, fractionalization of electrons


10/89

By studying strongly interacting systems of cold atoms we

novel quantum materials: Quantum Simulators

BUT

are NOT direct analogues of condensed matter systemsThese are independent physical systems with their own

Stron l correlated s stems of ultracold atoms should

persona es , p ys ca proper es, an eore ca c a enges

also be useful for applications in quantum information,

high precision spectroscopy, metrology


11/89

First lecture:

experiments with ultracold bosons

Cold atoms in optical lattices

Bose Hubbard model. Superfluid to Mott transition

Looking for Higgs particle in the Bose Hubbard modelQuantum magnetism with ultracold atoms in optical lattices

ow mens ona con ensa es

-Observation of prethermolization


12/89

Second lecture:Ultracold fermions

Fermions in optical lattices. Fermi Hubbard model.urren s a e o expermen s

Doublon lifetimes

Strongly interacting fermions in continuum.Stoner instability


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Ultracold Bose atoms in optical lattices

Bose Hubbard model


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Bose Hubbard model

t

tunnelingofatomsbetweenneighboringwells

repulsionofatomssittinginthesamewell

Inthe

presence

of

confining

potential

we

also

need

to

include

Typically


15/89

Bose Hubbard model. Phase diagramU

M.P.A.Fisheretal.,

PRB(1989)21n

n=2 SuperfluidMott

0

Mottn=1

Weaklattice Superfluidphase

Stronglattice Mottinsulatorphase


16/89

Bose Hubbard model

Hami tonianeigenstatesareFoc statesSet .

0 1

AwayfromlevelcrossingsMottstateshave

agap.

Hence

they

should

be

stable

to

small

tunneling.


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Bose Hubbard Model. Phase diagram

1n

U

n=3 Mott

2

1

n=2 SuperfluidMott

0

Mottn=1

Particlehole

Mottinsulatorphase

excitation

z

numberof

nearest

neighbors,

n fillingfactor


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Gutzwiller variational wavefunction

Normalization

Kineticenergy

z numberofnearestneighbors

Interactionenergyfavorsafixednumberofatomsperwell.

Kineticenergyfavorsasuperpositionofthenumberstates.


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Bose Hubbard Model. Phase diagram

U

21n

= Superfluid

1

Mottn=1

0

Notethat

the

Mott

state

only

exists

for

integer

filling

factors.

For evenwhen atomsarelocalized,

makeasu erfluidstate.


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Nature415:39(2002)


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Optical lattice and parabolic potential

Parabolicpotentialactsasacutthrough

thephasediagram.Henceinaparabolic

U

.

1nn=3 Mott

1

n=2 SuperfluidMott

0

Mottn=1

.,

PRL81:3108(1998)


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Bakr et al., Science 2010

ydensi

t

x


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Nature2010


24/89

trapped 2D superfluid on a lattice

ColdAtoms(Munich)ElementaryParticles(CMS@LHC)

Sherson et. al. Nature 2010

Theory: David Pekker, Eugene Demler

, , ,Schauss, Christian Gross, Immanuel Bloch, Stefan Kuhr


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Collective modes of strongly interacting

Orderparameter BreaksU(1)symmetry

FigurefromBissbort etal. (2010)

Phase(Goldstone)mode=gaplessBogoliubov mode

Gappedamplitudemode(Higgsmode)


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U

21nn=3

Su erfluid

Mott

1

n=2 Mott

Mott Superfluid

0

Mottn=1

ofthesecondorderQuantumPhaseTransition


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neutronscattering

Dangerfromscatteringonphasemodes

Higgs

Higgs

n : n rare vergence

Differentsusceptibilityhasnodivergence

S.Sachdev,Phys.Rev.B59,14054(1999)

a cemo u a on

spectroscopy

W.Zwerger,Phys.Rev.Lett.92,027203(2004)

N.LindnerandA.Auerbach,Phys.Rev.B81,54512(2010)

Podolsky,

Auerbach,

Arovas, Phys.

Rev.

B

84,

174522

(2011)


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Whyitisdifficulttoobservetheamplitudemode

Bissbort etal.,PRL(2010)

Stoferle etal.,PRL(2004)

Peak atUdominatesanddoesnot

changeas

the

system

goes

through

theSF/Motttransition


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Excitingtheamplitudemode

Absorbedenergy


30/89

ExcitingtheamplitudemodeManuel

Endres,

Immanuel

Bloch

and

MPQ

team

Mottn=1 Mottn=1 Mottn=1


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Experiments:fullspectrumManuel

Endres,

Immanuel

Bloch

and

MPQ

team


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Timedependentmeanfield:Gutzwiller

m ar o an au- s z equa ons n magne sm

statespersite

only

Thresholdforabsorption

scap ure verywe


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Plaquette MeanField

Better

utzw er

Variational wavefunctionsbettercaptureslocalphysics betterdescribesinteractionsbetweenquasiparticles

EquivalenttoMFTonplaquettes


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Timedependentclustermeanfield

Latticeheight

9.5

Er:

1x1

vs 2x2

breathingmode

singleamplitude

modeexcited multiplemodes

excited?singleamplitude

2x2captureswidthofspectralfeature

breathingmode

C i f i t


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Comparison of experimentsand Gutzwille theories

Experiment 2x2ClustersKey experimental facts:

wide bandband spreads out deep in SF

Single site Gutzwiller Plaquette Gutzwiller

Captures gap

Does not capture width

Captures gap

Captures most of the width


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Beyond Gutzwiller: Scaling at low frequencies

Higgs

2Goldstonesw

External drive couples vacuum to Higgs

vacuum

Higgs decays into a pair of Goldstone modes with conservation of energyMatrix element w2/w=wDensit of states wFermis golden rule: w2xw = w3


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Openquestion:observingdiscreetmodes

disappearingamplitudemode

reat ngmo e

detailsattheQCP

spectrumremains

gappeddue

to

trap


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HiggsDrumModes

1x1 calculation,20oscillations

Eabs rescaledsopeakheights

coincide


39/89

atoms in optical lattices

T t B i t i ti l l tti


40/89

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

tt

Two component Bose Hubbard model

We consider two component Bose mixture in the n=1Mott state with equal number of and atoms.

We need to find spin arrangement in the ground state.

Q i f b i i l l i


41/89

Quantum magnetism of bosons in optical latticesDuan et al., PRL (2003)

Ferromagnetic

n erromagne c


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In the regime of deep optical lattice we can treat tunnelingas perturbation. We consider processes of the second order in t

We can combine these processes intoanisotropic Heisenberg model

Two component Bose mixture in optical lattice


43/89

Two component Bose mixture in optical lattice.Mean field theory + Quantum fluctuations

Hysteresis

Altman et al., NJ P (2003)

1st order

bb d d l


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+ infinitely large Uaa and Ubb

ew ea ure:coexistence ofcheckerboard phase

and superfluidity


45/89

antibonding

on ng

Kinetic energy dominates: antiferromagnetic state

Realization of spin liquid


46/89

Realization of spin liquid

Theory: Duan, Demler, Lukin PRL (03)

Kitaev model Annals of Ph sics (2006)

H =- J x ixjx - J y iyjy - J z iz jz

Questions:Detection of topological orderCreation and manipulation of spin liquid statesDetection of fractionalization, Abelian and non-Abelian anyonsMelting spin liquids. Nature of the superfluid state


47/89

Superexchange interactionin ex eriments with double wells

Theory: A.M. Rey et al., PRL 2008

Experiments: S. Trotzky et al., Science 2008

Ob ti f h i d bl ll t ti l


48/89

Observation of superexchange in a double well potential

Jeory: . . ey e a .,

J

Use ma netic field radient to re are a state

Observe oscillations between and states

Experiments:S. Trotzky et al.Science 2008

Preparation and detection of Mott states


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Preparation and detection of Mott states

Reversing the sign of exchange interaction

C i t th H bb d d l


50/89

Comparison to the Hubbard model


51/89

Beyond the basic Hubbard model

Basic Hubbard model includesonly local interaction

Extended Hubbard modeltakes into account non-local



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53/89

Probing low dimensionalcon ensa es w n er erence

ex eriments

Prethermalization

f f i d d d


54/89

Interference of independent condensates

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

Theory: J avanainen, Yoo, PRL 76:161 (1996)rac, o er, e a . :

Castin, Dalibard, PRA 55:4330 (1997)

and many more

zExperiments with 2D Bose gas


55/89

zHadzibabic, Dalibard et al., Nature 2006

Time of

xg t

Experiments with 1D Bose gas Hofferberth et al. Nat. Physics 2008

Interference of two independent condensates


56/89

Interference of two independent condensates

r

r

1 r+dAssuming ballistic expansion

2

d

ase erence e ween c ou s anis not well defined

They disappear after averaging over many shots

Interference of fluctuating condensates


57/89

Interference of fluctuating condensates

dAmplitude of interference fringes,

., .,

x1For independent condensates Afris finitebut is random

x2

For identicalcondensates

Instantaneous correlation function


58/89

Matter-wave interferometry

phase,contrast


59/89

Matter-wave interferometry

phase,contrast

Plot

as

circular

statisticscontrast


60/89

Matter-wave interferometry: repeatmany times

i>100phase,contrast

contrasti accumulatestatistics

Plot phase

Calculate average contrast

Fluctuations in 1d BEC


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Fluctuations in 1d BEC

Thermal fluctuations

Thermally energy of the superflow velocity

Quantum fluctuations

Interference between Luttinger liquids


62/89

g q

Luttinger liquid at T=0

For non-interacting bosons and

For impenetrable bosons and

Finitetemperature

Experiments: Hofferberth,,

Distribution function of fringe amplitudes


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Distribution function of fringe amplitudes

for interference of fluctuating condensatesGritsev, Altman, Demle , Polkovnikov, Nature Ph sics 2006

is a quantum operator. The measured value of

Imambekov, Gritsev, Demler, PRA (2007)

L

will fluctuate from shot to shot.

Higher moments reflect higher order correlation functions

Distribution function of interference fringe contrast


64/89

g.,

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

Higher order correlation functions can be obtained

Interference between interacting 1d Bose liquids.Distribution function of the interference am litude


65/89

Distribution function of the interference am litude

Distribution function of

Quantum impurity problem: interacting one dimensional

electrons scattered on an impurity

Conformal field theories with negativecentral charges: 2D quantum gravity,

-

2D quantum gravity,non-intersecting loops

,random fractal stochastic interface,high energy limit of multicolor QCD,

Yang-Lee singularity

Fringe visibility and statistics of random surfaces


66/89

Distribution function of

Mapping between fringevisibility and the problemof surface roughness forfluctuating randomsurfaces.

Relation to 1/f Noise andExtreme Value Statistics

Roughness dh2

)(

Interference of two dimensional condensatesE i t H d ib bi t l N t 2006


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Ex eriments: Hadzibabic et al. Nature 2006

Gati et al., PRL (2006)

Ly

x

Lx

Probe beam parallel to the plane of the condensates

Interference of two dimensional condensates.


68/89

Ly

Lx

Below BKT transitionAbove BKT transition

Experiments with 2D Bose gas


69/89

Time of

Hadzibabic, Dalibard et al., Nature 441:1118 (2006)

xflight

low temperature higher temperature

Typical interference patterns



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x

Hadzibabic et al., Nature 441:1118 (2006)

z

Contrast a ter

integration0.4

negra on

overxaxis z

integrationmiddleT

lowT

inte ration

overxax sz

.

highT

overxaxisz integration distanceDx

00 10 20 30

Dx(pixels)



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fit by:rast 0.4

2

2

12 1~),0(

1~

D

dxxgCx

a z a c e a ., a ure :

rated

con

0.2

owmiddleT

Exponent

Integ

00 10 20 30

high

0.5

ifg1(r) decays exponentially

0.4

0.3hi h low

w :

if r deca sal ebraicall or central contras

0 0.1 0.2 0.3

exponentially with a large :

Sudden jump!?

Experiments with 2D Bose gas. Proliferation of


72/89

.,

Fract on o mages s ow ng

at least one dislocation

Exponent

0.520%

0.410%

0 0.1 0.2 0.3

.

00 0.1 0.2 0.3 0.4

g

centra contrast

The onset of roliferationcentral contrast

coincides withshifting to 0.5!


73/89

one dimensional condensatesret erma zaton

Theory: Takuya Kitagawa et al., PRL (2010)

New J . Phys. (2011)

Experiments: D. Smith, J . Schmiedmayer, et al.

.

Relaxation to equilibrium


74/89

equilibriumatlongtimes(typicallyatmicroscopictimescales).All

memoryabouttheinitialconditionsexceptenergyislost.

Bolzmann equation

.

c ne ere

a .,

arXiv:1005.3545

Prethermalization


75/89

QCD

Weobserveirreversibilityandapproximatethermalization.Atlarge

timethesystemapproachesstationarysolutioninthevicinityof,but

, .

somememory

beyond

the

conserved

total

energyThis

holds

for

interactingsystemsandinthelargevolumelimit.

Prethermalization inultracold atoms,theory:Ecksteinetal.(2009);

Moeckel etal.(2010),L.Mathey etal.(2010),R.Barnett etal.(2010)

Measurements of dynamics of split condensate


76/89


77/89

Theoretical analysis of dephasing

Luttinger liquid model of phase dynamics


78/89

Luttinger liquid model of phase dynamics


79/89

For each k-mode we have simple harmonic oscillators

Phase diffusion vs Contrast Deca


80/89

egmen s ze s onger an e uc ua on eng sca e

At long times the difference between

the two regime occurs for

Length dependent phase dynamics


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Short segments = phase diffusion

m

10

m

15 ms 15.5 16 16.5 17 19 21 24 27 32 37 47 62 77 107 137 167 197

m

30m

20

m

61m

41

110

ong segmen s = con rasdecay

Energy distribution


82/89

Energy stored in each mode initially

Equipartition of energyFor 2d also ointed out b Mathe Polkovnikov in PRA 2010

The system should look thermal like after different modes dephase.

Effective temperature is not related to the physical temperature

Comparison of experiments and LL analysis


83/89

Do we have thermal-like distributions at longer time

Prethermalization


84/89

Interference contrast is described by thermali tri ti n t t t m r t r m h l w r

than the initial temperature

Testin Prethermalization


85/89

First lecture:


86/89

experiments with ultracold bosons

Cold atoms in optical lattices

Bose Hubbard model. Superfluid to Mott transitionLooking for Higgs particle in the Bose Hubbard modelQuantum magnetism with ultracold atoms in optical lattices

ow mens ona con ensa es

-Observation of prethermolization


87/89

Beyond Gutzwiller: Scaling at low frequencies


88/89

signature of Higgs/Goldstone mode coupling

Excite virtual Higgs excitation

Virtual Higgs decays into a pair of Goldstone excitationsMatrix element of Higgs to Goldstone coupling scales as w2

ase space sca es as wFermis golden rule: (w2)2x(1/w) = w3


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2012 Kourovka Lecture 1