Eugene Demler Harvard University

Strongly correlated many-body systems: from electronic materials to ultracold atoms

to photons

“Conventional” solid state materials

Bloch theorem for non-interacting electrons in a periodic potential

B

VH

I

d

First semiconductor transistor

EF

Metals

EF

Insulators andSemiconductors

Consequences of the Bloch theorem

“Conventional” solid state materialsElectron-phonon and electron-electron interactions are irrelevant at low temperatures

kx

ky

kF

Landau Fermi liquid theory: when frequency and temperature are smaller than EF electron systems

are equivalent to systems of non-interacting fermions

Ag Ag

Ag

Strongly correlated electron systems

Quantum Hall systemskinetic energy suppressed by magnetic field

Heavy fermion materialsmany puzzling non-Fermi liquid properties

High temperature superconductorsUnusual “normal” state,Controversial mechanism of superconductivity,Several competing orders

UCu3.5Pd1.5CeCu2Si2

What is the connection between

strongly correlated electron systems

and

ultracold atoms?

Bose-Einstein condensation of weakly interacting atoms

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

Density 1013 cm-1

Typical distance between atoms 300 nmTypical scattering length 10 nm

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 resonance and fermionic condensates Greiner et al., Nature (2003); Ketterle et al., (2003)

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

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.,

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 …

Strongly correlated systemsAtoms in optical latticesElectrons in Solids

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

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 …

Strongly correlated systems of ultracold atoms shouldalso be useful for applications in quantum information, high precision spectroscopy, metrology

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

New Phenomena in quantum many-body systems of ultracold atoms

Long intrinsic time scales- Interaction energy and bandwidth ~ 1kHz- System parameters can be changed over this time scale

Decoupling from external environment- Long coherence times

Can achieve highly non equilibrium quantum many-body states

New detection methods

Interference, higher order correlations

Strongly correlated many-body systems of photons

Linear geometrical optics

Strong optical nonlinearities innanoscale surface plasmonsAkimov et al., Nature (2007)

Strongly interacting polaritons in coupled arrays of cavitiesM. Hartmann et al., Nature Physics (2006)

Crystallization (fermionization) of photonsin one dimensional optical waveguidesD. Chang et al., Nature Physics (2008)

Strongly correlated systems of photons

Outline of these lectures• Introduction. Systems of ultracold atoms.• Bogoliubov theory. Spinor condensates.• Cold atoms in optical lattices. • Bose Hubbard model and extensions• Bose mixtures in optical lattices Quantum magnetism of ultracold atoms. Current experiments: observation of superexchange• Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions. Current experiments: Mott state• Detection of many-body phases using noise correlations• Experiments with low dimensional systems Interference experiments. Analysis of high order correlations• Non-equilibrium dynamics

Emphasis of these lectures: • Detection of many-body phases• Dynamics

Ultracold atoms

Most common bosonic atoms: alkali 87Rb and 23Na

Most common fermionic atoms: alkali 40K and 6Li

Ultracold atoms

Other systems:

BEC of 133Cs (e.g. Grimm et al.)

BEC of 52Cr (Pfau et al.)

BEC of 84Sr (e.g. Grimm et al.), 87Sr and 88Sr (e.g. Ye et al.)

BEC of 168Yb, 170Yb, 172Yb, 174Yb, 176Yb

Quantum degenerate fermions 171Yb 173Yb (Takahashi et al.)

Single valence electron in the s-orbital and Nuclear spin

Magnetic properties of individual alkali atoms

Zero field splitting between and states

For 23Na AHFS = 1.8 GHz and for 87Rb AHFS = 6.8 GHz

Total angular momentum (hyperfine spin)

Hyperfine coupling mixes nuclear and electron spins

Magnetic properties of individual alkali atoms

Effect of magnetic field comes from electron spin

gs=2 and B=1.4 MHz/G

When fields are not too large one can use (assuming field along z)

The last term describes quadratic Zeeman effect q=h 390 Hz/G2

Magnetic trapping of alkali atoms

Magnetic trapping of neutral atoms is due to the Zeeman effect.The energy of an atomic state depends on the magnetic field.In an inhomogeneous field an atom experiences a spatially varying potential.

Example:

Potential:Magnetic trapping is limited by the requirement that the trapped atoms remain

in weak field seeking states. For 23Na and 87Rb there are three states

Optical trapping of alkali atoms

Based on AC Stark effect

- polarizability

Typically optical frequencies.

Potential:

Dipolar moment induced by the electric field

Far-off-resonant optical trap confines atoms regardlessof their hyperfine state

Bogoliubov theory of weakly interacting BEC. Collective modes

BEC of spinless bosons. Bogoliubov theory

We consider a uniform system first

For non-interacting atoms at T=0 all atoms are in k=0 state.Mean field equations

Minimizing with respect to N0 we find

- boson annihilation operator at momentum p,

- strength of contact s-wave interaction

- volume of the system

- chemical potential

BEC of spinless bosons. Bogoliubov theory

We now expand around the nointeracting solution for small U0 .

From the definition of Bose operators

When N0 >>1 we can treat b0 as a c-number and replace

by

- means that p,-p pairs should be counted only once

n0 = N0/V - density and = n0 U0

Mean-field Hamiltonian

Bogoliubov transformation

Bosonic commutation relations

are preserved when

Bogoliubov transformation

Mean-field Hamiltonian becomes

Bogoliubov transformation

Cancellation of non-diagonal terms requires

To satisfy take

Solution of these equations

Mean-field Hamiltonian

Bogoliubov modes

Dispersion of collective modes

Define healing length from

Long wavelength limit, , sound dispersion

Short wavelength limit, , free particles

Sound velocity

Probing the dispersion of BECby off-resonant light scattering

For details see cond-mat/0005001

Treat optical field as classical

Excitation rate out of the ground state |g>

Dynamical structure factor

For small q we find

PRL 83:2876 (1999)

ΩΩ

PRL 88:60402 (2002)

Rev. Mod. Phys. 77:187 (2005)

Gross-Pitaevskii equation

Gross-Pitaevskii equationHamiltonian of interacting bosons

Commutation relations

Equations of motion

This is operator equation. We can take classical limit by assuming that all atoms condense into the same state .The last equation becomes an equation on the wavefunction

Gross-Pitaevskii equation

Analysis of fluctuations on top of the mean-fieldGP equations leads to the Bogoliubov modes

Two-component mixtures

Two-component Bose mixture

Consider mean-field (all particles in the condensate)

Repulsive interactions. Miscible and immiscible regimes

System with afinite densityof both speciesis unstableto phase separation

What happens if we prepare a mixture of two condensates in the immiscible regime?

Two component GP equation

Assume equal densities

Analyze fluctuations

Bose mixture: dynamics of phase separation

Bose mixture: dynamics of phase separationDefine

density fluctuation

phase fluctuation

Equations of motion: charge conservation and Josephson relation

Equation on collective modes

Bose mixture: dynamics of phase separation

Here

When we get imaginary frequencies

Most unstable mode

Imaginary frequencies indicateexponential growth of fluctuations,i.e. instability. The most unstablemode sets the length for patternformation. Note that when . This is requiredby spin conservation.

PRL 82:2228 (1999)

Spinor condensates F=1

Spinor condensates. F=1

Three component order parameter: mF=-1,0,+1

Contact interaction depends on relative spin orientation

When g2>0 interaction is antiferromagnetic. Example 23Na

When g2<0 interaction is antiferromagnetic. Example 87Rb

F=1 spinor condensates. Hamiltonian

- spin operators for F=1

Total Fz is conserved so linear Zeeman term should (usually)

be understood as Lagrange multiplier that controls Fz.

Quadratic Zeeman effect causes the energy of mF=0

state to be lower than the energy of mF=-1,+1 states.

The antiferromagnetic interaction (g2>0) favors the nematic

(polar) state (mF=0 and its rotations). The ferromagnetic

interaction (g2<0) favors spin polarized state (mF=+1) and

its rotations).

Phase diagram of F=1 spinor condensates

g2>0

Antiferromagnetic

g2<0

Ferromagnetic

Shaded region: mixture of all three states. There is XY component of the spin.

Shaded region: mixture ofmF=-1,+1 states. This statelowers interaction energy.

Nature 396:345 (1998)

Using magnetic field gradient to explore the phase diagramof F=1 atoms with AF interactions23Na

Nature 443:312 (2006)

Magnetic dipolar interactionsin ultracold atoms

Magnetic dipolar interactions in spinor condensates

Comparison of contact and dipolar interactions.

Typical value a=100aB

For 87Rb =B and =0.007 For 52Cr =6B and =0.16

Bose condensationof 52Cr. T. Pfau et al. (2005)

Review: Menotti et al.,arXiv 0711.3422

Magnetic dipolar interactions in spinor condensates

Interaction of F=1 atoms

Ferromagnetic Interactions for 87Rb

Spin-depenent part of the interaction is small. Dipolar interaction may be important (D. Stamper-Kurn)

a2-a0= -1.07 aB

A. Widera, I. Bloch et al., New J. Phys. 8:152 (2006)

Spontaneously modulated textures in spinor condensates

Fourier spectrum of thefragmented condensate

Vengalattore et al.PRL (2008)

Energy scales: importance of Larmor precession

S-wave Scattering•Spin independent (g0n = kHz)•Spin dependent (gsn = 10 Hz)

Dipolar Interaction•Anisotropic (gdn=10 Hz)•Long-ranged

Magnetic Field•Larmor Precession (100 kHz)•Quadratic Zeeman (0-20 Hz)

BF

Reduced Dimensionality•Quasi-2D geometry spind

Stability of systems with static dipolar interactions

Ferromagnetic configuration is robust against small perturbations. Any rotation of the spins conflicts with the “head to tail” arrangement

Large fluctuation required to reach a lower energy configuration

XY components of the spins can lower the energy using modulation along z.

Z components of thespins can lower the energyusing modulation along x

X

Dipolar interaction averaged after precession

“Head to tail” order of the transverse spin components is violated by precession. Only need to check whether spins are parallel

Strong instabilities of systems with dipolar interactions after averaging over precession

X

Instabilities of F=1 Rb (ferromagnetic) condensate due to dipolar interactions

Theory: unstable modes in the regimecorresponding to Berkeley experiments.Cherng, Demler, PRL (2009)

Experiments.Vengalattore et al. PRL (2008)

Berkeley Experiments: checkerboard phase

Instabilities of magnetic dipolar interactions: general analysis

– angle between magneticfield and normal to the plane

dn – layer thickness

q measures the strengthof quadratic Zeeman effect

Instabilities of collective modes

Magnetoroton softening

Q measures the strengthof quadratic Zeeman effect

87Rb condensate: magnetic supersolid ?

Phase diagram of 4He

Possible supersolid phase in 4He

A.F. Andreev and I.M. Lifshits (1969):Melting of vacancies in a crystal due to strong quantum fluctuations.

Also G. Chester (1970); A.J. Leggett (1970)

T. Schneider and C.P. Enz (1971).Formation of the supersolid phase due tosoftening of roton excitations

Resonant period as a function of T

Interlayer coherence in bilayer quantum Hall systems at =1

Hartree-Fock predicts roton softeningand transition into a state with both interlayer coherence and stripe order. Transport experiments suggest first order transition into a compressible state.

L. Brey and H. Fertig (2000)Eisenstein, Boebinger et al. (1994)

Low energy effective model for ferromagnetic condensates

Mass superflow currentcaused by spin texture

Topological spin winding(Pontryagin index)

Magnetic dipolar interaction.Favors spin skyrmions

Requires net zero spin winding

Magnetic crystal phases in F=1 87Rbferromagnetic condensates

Experimental constraints:

1. Dihedral symmetry of the magnetic crystal2. Rectangular lattice3. Non-planar spin orientation

Magnetic crystal latticesoptimized within each class

Optimal spin configuration.

Cherng, Demler, unpublished

Spin correlation functionsfor spin componentsparallel and perpendicularto the magnetic field

Magnetic crystal phases in F=1 87Rbferromagnetic condensates

Ultracold atoms in optical lattices.Band structure. Semiclasical dynamics.

Optical lattice

The simplest possible periodic optical potential is formedby overlapping two counter-propagating beams. This resultsin a standing wave

Averaging over fast optical oscillations (AC Stark effect) gives

Combining three perpendicular sets of standing waves we get a simple cubic lattice

This potential allows separation of variables

Optical latticeFor each coordinate we have Matthieu equation

Eigenvalues and eigenfunctions are known

In the regime of deep lattice we get the tight-binding model

- bandgap

- recoil energy

Lowest band

Optical lattice

Effective Hamiltonian for non-interacting atoms in the lowest Bloch band

nearestneighbors

Band structure

Semiclassical dynamics in the lattice1. Band index is constant

2.

3.

Bloch oscillations

Consider a uniform and constant force

PRL 76:4508 (1996)

State dependent optical latticesHow to use selection rules for optical transitions to make different lattice potentials for different internal states.

Fine structure for 23Na and 87Rb

The right circularly polarized light couples to two

excited levels P1/2 and P3/2. AC Stark effects have opposite signs and cancel

each other for the appropriate frequency. At this frequency AC Stark effect for the state comes only from polarized light and gives the potential .

Analogously state will only be affected by , which gives the potential .

Decomposing hyperfine states we find

PRL 91:10407 (2003)

State dependent lattice