QUANTUM MANY‐BODY SYSTEMS OF ULTRACOLD ATOMSSYSTEMS OF ULTRACOLD ATOMSEugene Demler Harvard University

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

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

Supported by NSF, DARPA OLE, AFOSR MURI, ARO MURI

How cold are ultracold atoms?How cold are ultracold atoms?

keV MeV GeV TeVfeV peV µeV meV eVneV

pK nK µK mK K

He Nfirst BEC

roomtemperature

LHCcurrent experiments10-11 - 10-10 K

of alkali atoms

Bose-Einstein condensation of kl i t ti tweakly interacting atoms

Density 1013 cm-1

Typical distance between atoms 300 nmTypical scattering length 10 nm

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

yp g g

New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions

• Feshbach resonances

• Rotating systems

At i ti l l tti

• Low dimensional systems

• Atoms in optical lattices

• Systems with long range dipolar interactions• Systems with long range dipolar interactions

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

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

One dimensional systems

One dimensional systems in microtraps.Thywissen et al Eur J Phys D (99);

1D confinement in optical potentialWeiss et al., Science (05);Bloch et al Thywissen et al., Eur. J. Phys. D. (99);

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

Bloch et al., Esslinger et al.,

Strongly interactingStrongly interacting regime can be reached for low densities

Atoms in optical lattices

Th J k h t l PRL (1998)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);and many more …

Quantum simulations with ultracold atoms

Atoms in optical lattice

Antiferromagnetic and superconducting Tc

Atoms in optical lattice

Antiferromagnetism and pairing at nano Kelvin p g

of the order of 100 K temperatures

Same microscopic model

Strongly correated systemsStrongly correated systemsAtoms in optical latticesElectrons in Solids

Simple metalsPerturbation theory in Coulomb interaction applies. B d t t th d kBand structure methods work

Strongly Correlated Electron SystemsBand structure methods fail.

Novel phenomena in strongly correlated electron systems:Quantum magnetism, phase separation, unconventional superconductivity,Q g , p p , p y,high temperature superconductivity, fractionalization of electrons …

By studying strongly interacting systems of cold atoms we expect to get insights into the mysterious properties ofexpect to get insights into the mysterious properties of novel quantum materials: Quantum Simulators

BUTStrongly interacting systems of ultracold atoms :Strongly interacting systems of ultracold atoms :

are NOT direct analogues of condensed matter systemsThese are independent physical systems with their own “ liti ” h i l ti d th ti l h ll

Strongly correlated systems of ultracold atoms should

“personalities”, physical properties, and theoretical challenges

g y yalso be useful for applications in quantum information, high precision spectroscopy, metrology

First lecture: 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

L di i l d tLow dimensional condensates

Observing quasi-long range order in interference experimentsObserving quasi long range order in interference experimentsObservation of prethermolization

Second lecture: Ultracold fermions

Fermions in optical lattices. Fermi Hubbard model.C t t t f i tCurrent state of experiments

Lattice modulation experimentsLattice modulation experiments

Doublon lifetimes

Strongly interacting fermions in continuum. Stoner instability

Ultracold Bose atoms in optical lattices

Bose Hubbard model

Bose Hubbard model

UU

t

l f b hb lltunneling of atoms between neighboring wells

repulsion of atoms sitting in the same well

In the presence of confining potential we also need to include

Typically

Bose Hubbard model. Phase diagramU

n=3 Mott M.P.A. Fisher et al.,PRB (1989)2

1n

1

n=2

n=3

Superfluid

Mott

Mott

1

0

Mottn=1

Weak lattice Superfluid phase

Strong lattice Mott insulator phase

Bose Hubbard model

l kHamiltonian eigenstates are Fock statesSet .

U

0 1

U

Away from level crossings Mott states have a gap. Hence they should be stable to small tunneling. 

Bose Hubbard Model. Phase diagram

1n

U

n=3 Mott

2

1n=2 SuperfluidMott

0

Mottn=1

Particle‐hole

Mott insulator phase

Particle hole excitation 

Tips of the Mott lobesTips of the Mott lobes

z‐ number of nearest neighbors, n – filling factor

Gutzwiller variational wavefunction

Normalization

Kinetic energy

z – number of nearest neighbors

Interaction energy favors a fixed number of atoms per well.Kinetic energy favors a superposition of the number states.

Bose Hubbard Model. Phase diagram

U

n=3 Mott

21n

n=2

n=3

Superfluid

Mott

Mott

1

Mottn=1

n 2 Mott

0

Note that the Mott state only exists for integer filling factors.For                           even when       atoms are localized,

make a superfluid state. p

Nature 415:39 (2002)

Optical lattice and parabolic potentialParabolic potential acts as a “cut” through the phase diagram. Hence in a parabolic potential we find a “wedding cake” structureU potential we find a  wedding cake  structure.

21n

n=3 Mott

2

1

n=2 SuperfluidMott

0

Mottn=1

Jaksch et al.,Jaksch et al., PRL 81:3108 (1998)

Quantum gas microscopeQuantum gas microscopeBakr et al., Science 2010

y

y

density

x

Nature 2010

The Higgs (amplitude) mode in aThe Higgs (amplitude) mode in a trapped 2D superfluid on a lattice

Cold Atoms (Munich)Elementary Particles (CMS @ LHC)

Sherson et. al. Nature 2010

Theory: David Pekker, Eugene DemlerExperiments: Manuel Endres Takeshi Fukuhara Marc Cheneau Peter

Sherson et. al. Nature 2010

Experiments: Manuel Endres, Takeshi Fukuhara, Marc Cheneau, Peter Schauss, Christian Gross, Immanuel Bloch, Stefan Kuhr

Collective modes of strongly interactingsuperfluid bosons

Order parameter  Breaks U(1) symmetry

superfluid bosons

Figure from Bissbort et al.  (2010)

Phase (Goldstone) mode = gapless Bogoliubov mode

Gapped amplitude mode (Higgs mode)

Excitations of the Bose Hubbard modelExcitations of the Bose Hubbard model

U

22

1nn=3

Superfluid

Mott

2

1

n=2

p

Mott

Mott Superfluid

0

Mottn=1

Softening of the amplitude mode is the defining characteristicSoftening of the amplitude mode is the defining characteristicof the second order Quantum Phase Transition

Is there a Higgs mode in 2D ?Is there a Higgs mode in 2D ?neutron scattering

• Danger from scattering on phase modes

I 2D i f d di

Higgs

Higgs

• In 2D: infrared divergence 

• Different susceptibility has no divergencel tti d l ti

S. Sachdev, Phys. Rev. B 59, 14054 (1999) 

lattice modulation spectroscopy

W. Zwerger, Phys. Rev. Lett. 92, 027203 (2004) N. Lindner and A. Auerbach, Phys. Rev. B 81, 54512 (2010) Podolsky, Auerbach, Arovas, Phys. Rev. B 84, 174522 (2011)

Why it is difficult to observe the amplitude mode

Bissbort et al., PRL(2010) 

Stoferle et al., PRL(2004) 

Peak  at U dominates and does not change as the system goes through the SF/Mott transition

Exciting the amplitude mode

Absorbed energy

Exciting the amplitude modeManuel Endres, Immanuel Bloch and MPQ team

Mottn=1 Mottn=1 Mottn=1

Experiments: full spectrump pManuel Endres, Immanuel Bloch and MPQ team

Time dependent mean‐field: Gutzwiller

Si il t L d Lif hit ti i tiSimilar to Landau-Lifshitz equations in magnetism

Keep twoKeep twostates per siteonly

Threshold for absorption i t d llis captured very well

Plaquette Mean Field “B G ill ”“Better Gutzwiller”

• Variational wave functions better captures local physics– better describes interactions between quasi‐particles

• Equivalent to MFT on plaquettes

Time dependent cluster mean‐fieldh h ( )Lattice height 9.5 Er: (1x1 vs 2x2)

breathing mode

single amplitude mode excited multiple modes

excited?single amplitude mode excited

2x2 captures width of spectral feature

e c ted

breathing mode

mode excited

Comparison of experiments and Gutzwiller theories

Experiment 2x2 ClustersKey experimental facts:

• “gap” disappears at QCP• gap disappears at QCP• wide band• band spreads out deep in SF

Single site Gutzwiller Plaquette Gutzwiller

Captures gapDoes not capture width

Captures gapCaptures most of the width

Beyond Gutzwiller: Scaling at low frequenciessignature of Higgs/Goldstone mode couplingsignature of Higgs/Goldstone mode coupling

Higgs

2 Goldstonesw

External drive couples vacuum to HiggsHiggs can be excited only virtually

vacuum

Higgs can be excited only virtuallyHiggs decays into a pair of Goldstone modes with conservation of energy Matrix element w2/w=wDensity of states wyFermi’s golden rule: w2xw = w3

Open question: observing discreet modes

disappearing amplitude mode

B thi dBreathing mode

details at the QCP

spectrum remains gapped due to trap

Higgs Drum Modes1x1  calculation, 20 oscillationsEabs rescaled so peak heights coincide

Quantum magnetism with ultracoldQuantum magnetism with ultracoldatoms in optical lattices

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=1 Mott state with equal number of and atoms. We need to find spin arrangement in the ground state.

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

• FerromagneticA tif ti• Antiferromagnetic

Two component Bose Hubbard modelIn 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 into anisotropic Heisenberg model

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

HysteresisAltman et al., NJP (2003)

1st order

Two component Bose Hubbard model

+ infinitely large Uaa and Ubb

N f tNew feature:coexistence ofcheckerboard phasepand superfluidity

Exchange Interactions in SolidsExchange Interactions in Solidsantibonding

b dibonding

Kinetic energy dominates: antiferromagnetic state

Coulomb energy dominates: ferromagnetic stateCoulomb energy dominates: ferromagnetic state

Realization of spin liquid using cold atoms in an optical latticeusing cold atoms in an optical lattice

Theory: Duan, Demler, Lukin PRL (03)

Kitaev model Annals of Physics (2006)

H = - Jx ix j

x - Jy iy j

y - Jz iz j

z

y ( )

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

Superexchange interaction in experiments with double wellsp

Theory: A.M. Rey et al., PRL 2008Experiments: S. Trotzky et al., Science 2008

Observation of superexchange in a double well potentialTh A M R t l PRL 2008

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

J

Use magnetic field gradient to prepare a state g g p p

Observe oscillations between and states

Experiments:S. Trotzky et al.Science 2008

Preparation and detection of Mott statesof atoms in a double well potentialof atoms in a double well potential

Reversing the sign of exchange interaction

Comparison to the Hubbard model

Beyond the basic Hubbard model

Basic Hubbard model includes

y

only local interaction

Extended Hubbard modeltakes into account non-localinteractioninteraction

Beyond the basic Hubbard model

Probing low dimensional d t ith i t fcondensates with interference

experimentsp

Quasi long range orderQuasi long range order

Prethermalization

Interference of independent condensates

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

Theory: Javanainen, Yoo, PRL 76:161 (1996)Ci Z ll t l PRA 54 R3714 (1996)Cirac, Zoller, et al. PRA 54:R3714 (1996)Castin, Dalibard, PRA 55:4330 (1997)and many more

zExperiments with 2D Bose gas

Hadzibabic, Dalibard et al., Nature 2006

Time of

fli h

, ,

xflight

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

Interference of two independent condensates

rr’

1 r+dAssuming ballistic expansion

2d

Phase difference bet een clo ds 1 and 22 Phase difference between clouds 1 and 2is not well defined

Individual measurements show interference patternsIndividual measurements show interference patternsThey disappear after averaging over many shots

Interference of fluctuating condensatesPolkovnikov et al PNAS (2006); Gritsev et al Nature Physics (2006)

dAmplitude of interference fringes,

Polkovnikov et al., PNAS (2006); Gritsev et al., Nature Physics (2006)

x1For independent condensates Afr is finite frbut is random

x2

For identical condensates

Instantaneous correlation function

FDF of phase and contrast• Matter-wave interferometry

FDF of phase and contrast

phase, contrast

FDF of phase and contrast• Matter-wave interferometry

FDF of phase and contrast

phase, contrast

• Plot as circular statisticscontrast

phasephase

FDF of phase and contrast• Matter-wave interferometry: repeat

FDF of phase and contrast

many timesi>100phase, contrast

contrasti accumulate statistics

• Plot phase

Calculate average contrast

Fluctuations in 1d BECThermal fluctuations

Thermally energy of the superflow velocity

Quantum fluctuations

Interference between Luttinger liquids

Luttinger liquid at T=0

K Luttinger parameterK – Luttinger parameter

For non-interacting bosons and

For impenetrable bosons and

Finite temperature

Experiments: Hofferberth,Schumm SchmiedmayerSchumm, Schmiedmayer

Distribution function of fringe amplitudes for interference of fluctuating condensates

Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006

is a quantum operator. The measured value of

, , , , yImambekov, Gritsev, Demler, PRA (2007)

L

will fluctuate from shot to shot.

Higher moments reflect higher order correlation functions

We need the full distribution function ofWe need the full distribution function of

Distribution function of interference fringe contrastHofferberth et al Nature Physics 2009Hofferberth et al., Nature Physics 2009

Quantum fluctuations dominate: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

Comparison of theory and experiments: no free parametersComparison of theory and experiments: no free parametersHigher order correlation functions can be obtained

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

Distribution function of

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

2D quantum gravity,non-intersecting loopsnon intersecting loop model, growth of

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

Yang-Lee singularity

Fringe visibility and statistics of random surfaces

Distribution function of

Mapping between fringe pp g gvisibility and the problem of surface roughness for fluctuating random

)(h surfaces. Relation to 1/f Noise and Extreme Value Statistics

)(h

Roughness dh2

)(

Interference of two dimensional condensatesExperiments: Hadzibabic et al. Nature (2006)pe e ts ad bab c et a atu e ( 006)

Gati et al., PRL (2006)

Ly

LLxLx

Probe beam parallel to the plane of the condensates

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

Ly

Lx

Below BKT transitionAbove BKT transition

zExperiments with 2D Bose gas

Time of

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

xflight

low temperature higher temperature

Typical interference patterns

Experiments with 2D Bose gas

xfi t ti

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

z

Contrast afterintegration

0.4

integration

over x axis z

integration

i 0 2middle T

low T

integration

over x axisz

0.2

high Tg

over x axisz integration distance Dx

00 10 20 30

Dx(pixels)

Experiments with 2D Bose gasH d ib bi t l N t 441 1118 (2006)

fit by:

trast 0.4

l T

22

12 1~),0(1~

D

Ddxxg

DC

x

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

rate

d co

nt

0.2low T

middle T

Exponent

xx DD

integration distance D

Inte

gr

00 10 20 30

high T

0.5integration distance Dx

if g1(r) decays exponentially ith

0.4

0.3 high T low Twith :

if g1(r) decays algebraically or central contrast

0 0.1 0.2 0.3

g ow

if g1(r) decays algebraically or exponentially with a large :

central contrast

“Sudden” jump!?

Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al Nature (2006)thermal vortices Hadzibabic et al., Nature (2006)

i f i h i30% Fraction of images showing at least one dislocation

Exponent

0.520%

30%

0.4

0 3

10%hi h T low T

0 0.1 0.2 0.3

0.3

l

00 0.1 0.2 0.3 0.4

high T low T

central contrast

The onset of proliferation

central contrast

pcoincides with shifting to 0.5!

Quantum dynamics of splitQuantum dynamics of split one dimensional condensatesP th li tiPrethermalization

Theory: Takuya Kitagawa et al., PRL (2010)New J. Phys. (2011)

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

arXiv:1112 0013arXiv:1112.0013

Relaxation to equilibriumThermalization: an isolated interacting systems approaches thermalThermalization: an isolated interacting systems approaches thermal equilibrium at long times (typically at microscopic timescales). All memory about the initial conditions except energy is lost.

Bolzmann equation

U S h id t lU. Schneider et al., arXiv:1005.3545

Prethermalization

Heavy ions collisionsHeavy ions collisionsQCD

We observe irreversibility and approximate thermalization. At large time the system approaches stationary solution in the vicinity of, but not identical to thermal equilibrium The ensemble therefore retainsnot identical to, thermal equilibrium. The ensemble therefore retains some memory beyond the conserved total energy…This holds for interacting systems and in the large volume limit.

Prethermalization in ultracold atoms, theory: Eckstein et al. (2009); Moeckel et al. (2010), L. Mathey et al. (2010), R. Barnett  et al.(2010)

Measurements of dynamics of split condensate

Theoretical analysis of dephasingLuttinger liquid modelLuttinger liquid model

Luttinger liquid model of phase dynamics

Luttinger liquid model of phase dynamics

For each k-mode we have simple harmonic oscillators

Phase diffusion vs Contrast Decay

Segment size is smaller than the fluctuation lengthscale

y

Segment size is smaller than the fluctuation lengthscale

Segment si e is longer than the fl ct ation lengthscaleSegment size is longer than the fluctuation lengthscale

At long times the difference between the two regime occurs for

Length dependent phase dynamics

“Short segments” = phase diffusion

µm

10µm

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

µm

30

µm

20µ

m

61µ

m

41µ

“L t ” t t

110

µm

“Long segments” = contrast decay

Energy distributionAt t=0 system is in a squeezed state with large number fluctuationsAt t=0 system is in a squeezed state with large number fluctuations

Energy stored in each mode initially

Equipartition of energy For 2d also pointed out by Mathey, Polkovnikov in PRA (2010) o d a so po ted out by at ey, o o o ( 0 0)

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

Do we have thermal-like distributions at longer times

Prethermalization

Interference contrast is described by thermal distributions but at temperature much lowerdistributions but at temperature much lower than the initial temperature

Testing PrethermalizationTesting Prethermali ation

First lecture: 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

L di i l d tLow dimensional condensates

Observing quasi-long range order in interference experimentsObserving quasi long range order in interference experimentsObservation of prethermolization

Beyond Gutzwiller: Scaling at low frequencies

signature of Higgs/Goldstone mode coupling

Excite virtual Higgs excitationVirtual Higgs decays into a pair of Goldstone excitations Matrix element of Higgs to Goldstone coupling scales as w2

Phase space scales as 1/Phase space scales as 1/wFermi’s golden rule: (w2)2x(1/w) = w3