QUANTUM MANY‐BODY SYSTEMS OF ULTRACOLD ATOMS

Eugene 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

keV MeV GeV TeVfeV peV µeV meV eV

pK nK µK mK K

neV

roomtemperature

LHC

He N

current experiments10-11 - 10-10 K

How cold are ultracold atoms?

first BECof alkali atoms

Bose-Einstein condensation of weakly 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

New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions

• Atoms in optical lattices

• Feshbach resonances

• Low dimensional systems

• Systems with long range dipolar interactions

• Rotating systems

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

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

One dimensional systems

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

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)

Strongly interacting regime can be reached for low densities

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);and many more …

Quantum simulations with ultracold atoms

Antiferromagnetic and superconducting Tc of the order of 100 K

Atoms in optical lattice

Antiferromagnetism and pairing at nano Kelvin temperatures

Same microscopic model

Strongly correated systemsAtoms in optical latticesElectrons in Solids

Simple metalsPerturbation theory in Coulomb interaction applies. Band structure methods work

Strongly Correlated Electron SystemsBand structure methods fail.

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

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

BUTStrongly interacting systems of ultracold atoms :

are NOT direct analogues of condensed matter systemsThese are independent physical systems with their own “personalities”, physical properties, and theoretical challenges

Strongly correlated systems of ultracold atoms shouldalso 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

Low dimensional condensates

Observing quasi-long range order in interference experimentsObservation of prethermolization

Second lecture: Ultracold fermions

Fermions in optical lattices. Fermi Hubbard model.Current state of experiments

Lattice modulation experiments

Doublon lifetimes

Strongly interacting fermions in continuum. Stoner instability

Ultracold Bose atoms in optical lattices

Bose Hubbard model

Bose Hubbard model

tunneling of atoms between neighboring wells

repulsion of atoms sitting in the same well

U

t

In the presence of confining potential we also need to include

Typically

Bose Hubbard model. Phase diagramM.P.A. Fisher et al.,PRB (1989)2

1+n

Uµ

1

0

Mottn=1

n=2

n=3

Superfluid

Mott

Mott

Weak lattice Superfluid phase

Strong lattice Mott insulator phase

Bose Hubbard model

Hamiltonian eigenstates are Fock states

Uµ

0 1

Set .

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

Bose Hubbard Model. Phase diagram

21+n

Uµ

1

0

Mottn=1

n=2

n=3

Superfluid

Mott

Mott

Mott insulator phase

Particle‐hole excitation 

Tips 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

21+n

Uµ

1

0

Mottn=1

n=2

n=3

Superfluid

Mott

Mott

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

make a superfluid state. 

Nature 415:39 (2002)

Optical lattice and parabolic potential

Jaksch et al., PRL 81:3108 (1998)

Parabolic potential acts as a “cut” through the phase diagram. Hence in a parabolic potential we find a “wedding cake” structure.

21+n

Uµ

1

0

Mottn=1

n=2

n=3

Superfluid

Mott

Mott

Quantum gas microscopeBakr et al., Science 2010

xy

density

Nature 2010

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

Sherson et. al. Nature 2010

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

Theory: David Pekker, Eugene DemlerExperiments: 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

Figure from Bissbort et al.  (2010)

Phase (Goldstone) mode = gapless Bogoliubov mode

Gapped amplitude mode (Higgs mode)

Excitations of the Bose Hubbard model

2

Mott Superfluid

21+n

Uµ

1

0

Mottn=1

n=2

n=3

Superfluid

Mott

Mott

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

Is there a Higgs mode in 2D ?neutron scattering

lattice modulation spectroscopy

• Danger from scattering on phase modes

• In 2D: infrared divergence 

• Different susceptibility has no divergence

φHiggs

φHiggs

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

Stoferle et al., PRL(2004) 

Bissbort et al., PRL(2010) 

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

Exciting the amplitude mode

Absorbed energy

Mottn=1 Mottn=1 Mottn=1

Exciting the amplitude modeManuel Endres, Immanuel Bloch and MPQ team

Experiments: full spectrumManuel Endres, Immanuel Bloch and MPQ team

Time dependent mean‐field: Gutzwiller

Similar to Landau-Lifshitz equations in magnetism

Keep twostates per siteonly

Threshold for absorption is captured very well

Plaquette Mean Field “Better Gutzwiller”

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

• Equivalent to MFT on plaquettes

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

2x2 captures width of spectral feature

breathing mode

single amplitude mode excited multiple modes

excited?single amplitude mode excited

breathing mode

Comparison of experiments and Gutzwiller theories

Experiment 2x2 ClustersKey experimental facts:

• “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 coupling

2 Goldstones

Higgs

wvacuum

External drive couples vacuum to HiggsHiggs can be excited only virtuallyHiggs decays into a pair of Goldstone modes with conservation of energy Matrix element w2/w=wDensity of states wFermi’s golden rule: w2xw = w3

Open question: observing discreet modes

disappearing amplitude mode

Breathing 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 ultracold atoms in optical lattices

tt

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

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)

• Ferromagnetic• 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

2 ndorder line

Hysteresis

1st order

Altman et al., NJP (2003)

Two component Bose Hubbard model

+ infinitely large Uaa and Ubb

New feature:coexistence ofcheckerboard phaseand superfluidity

Exchange Interactions in Solidsantibonding

bonding

Kinetic energy dominates: antiferromagnetic state

Coulomb energy dominates: ferromagnetic state

Realization of spin liquid using cold atoms in an optical lattice

Theory: Duan, Demler, Lukin PRL (03)

Kitaev model Annals of Physics (2006)

H = - Jx Σ σix σj

x - Jy Σ σiy σj

y - Jz Σ σiz σj

z

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 wells

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

Observation of superexchange in a double well potentialTheory: A.M. Rey et al., PRL 2008

J

J

Use magnetic field gradient to prepare a state

Observe oscillations between and states

Experiments:S. Trotzky et al.Science 2008

Preparation and detection of Mott statesof 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 includesonly local interaction

Extended Hubbard modeltakes into account non-localinteraction

Beyond the basic Hubbard model

Probing low dimensional condensates with interference

experiments

Quasi long range order

Prethermalization

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

r’r

Assuming ballistic expansion1 r+d

d

2 Phase difference between clouds 1 and 2is not well defined

Individual measurements show interference patternsThey disappear after averaging over many shots

Interference of fluctuating condensates

x1

dAmplitude of interference fringes,

For independent condensates Afr is finite but ∆φ is random

x2

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

For identical condensates

Instantaneous correlation function

FDF of phase and contrast• Matter-wave interferometry

phase, contrast

FDF of phase and contrast• Matter-wave interferometry

contrast

phase, contrast

phase

• Plot as circular statistics

• Matter-wave interferometry: repeat many times

• Plot

i>100

contrasti

phase

phase, contrast

FDF of phase and contrast

accumulate statistics

Calculate average contrast

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

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 parametersHigher order correlation functions can be obtained

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

Distribution function of

Quantum impurity problem: interacting one dimensionalelectrons scattered on an impurity

Yang-Lee singularity

2D quantum gravity,non-intersecting loops

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, …

Fringe visibility and statistics of random surfaces

Distribution function of

)(ϕh

Roughness ϕϕ dh2

)(∫=

Mapping between fringe visibility and the problem of surface roughness for fluctuating random surfaces. Relation to 1/f Noise and Extreme Value Statistics

Interference of two dimensional condensatesExperiments: Hadzibabic et al. Nature (2006)

Gati et al., PRL (2006)

Ly

LxLx

Probe beam parallel to the plane of the condensates

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

Ly

Lx

Above BKT transition Below BKT transition

x

z

Time of

flight

Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (2006)

Typical interference patterns

low temperature higher temperature

Experiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)

integration

over x axis

Dx

z

z

integration

over x axisz

xintegration

over x axis z

integration distance Dx (pixels)

Contrast afterintegration

0.4

0.2

00 10 20 30

middle Tlow T

high T

Experiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)

[ ]α2

21

2 1~),0(1~ ⎟⎟⎠

⎞⎜⎜⎝

⎛∫

x

D

x Ddxxg

DC

x0.4

0.2

00 10 20 30

low Tmiddle T

high T

fit by:

Inte

grat

ed c

ontra

st

Exponent α

central contrast

0.5

0 0.1 0.2 0.3

0.4

0.3 high T low T

“Sudden” jump!?

integration distance Dx

if g1(r) decays exponentially with :

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

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

Fraction of images showing at least one dislocation

Exponent α

0.5

0 0.1 0.2 0.3

0.4

0.3

central contrast

The onset of proliferation coincides with α shifting to 0.5!

0

10%

20%

30%

central contrast0 0.1 0.2 0.3 0.4

high T low T

Quantum dynamics of split one dimensional condensatesPrethermalization

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

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

arXiv:1112.0013

Relaxation to equilibriumThermalization: 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. Schneider et al., arXiv:1005.3545

Prethermalization

Heavy 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 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 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

Segment 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

110

µm

61µ

m

41µ

m

30

µm

20µm

10

µm

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

“Long segments” = contrast decay

Energy distributionAt 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)

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 lower than the initial temperature

Testing Prethermalization

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

Low dimensional condensates

Observing 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/wFermi’s golden rule: (w2)2x(1/w) = w3