Strongly correlated many-body systems: from electronic materials to ultracold atoms to photons • Introduction. Systems of ultracold atoms.

• Bogoliubov theory. Spinor condensates.

• Cold atoms in optical lattices. Band structure and semiclasical dynamics.

• Bose Hubbard model and its extensions • Bose mixtures in optical lattices Quantum magnetism of ultracold atoms.

Current experiments: observation of superexchange • Detection of many-body phases using noise correlations • Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions. Current experiments: Mott state • Experiments with low dimensional systems Interference experiments. Analysis of high order correlations • Non-equilibrium dynamics

Atoms in optical lattices. Bose Hubbard model

Bose Hubbard model

U t tunneling 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



U

2

n

n=3  1 Mott n=2 Mott 1 n=1 Mott 0 Weak lattice Bose Hubbard model. Phase diagram M.P.A. Fisher et al., PRB (1989) Superfluid Superfluid phase Strong lattice Mott insulator phase

Bose Hubbard model

Set .

Hamiltonian eigenstates are Fock states 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 

U

1 2 n=3

n

 1 n=2 Mott Mott n=1 Mott 0 Superfluid Tips of the Mott lobes Mott insulator phase Particle-hole excitation

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.

Gutzwiller variational wavefunction

Example: stability of the Mott state with

n

atoms per site Expand to order Take the middle of the Mott plateau Transition takes place when coefficient before becomes negative. For large n this corresponds to

Bose Hubbard Model. Phase diagram 

U

1 2 n=3

n

 1 Mott n=2 Mott n=1 Mott 0 Superfluid Note that the Mott state only exists for integer filling factors.

For even when atoms are localized, make a superfluid state.

Bose Hubbard model

Experiments with 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); many more …

Nature 415:39 (2002)

Shell structure in optical lattice

Optical lattice and parabolic potential 

U

1 2 n=3

n

 1 Mott n=2 Mott n=1 Mott 0 Parabolic potential acts as a “cut” through the phase diagram. Hence in a parabolic potential we find a “wedding cake” structure.

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

Shell structure in optical lattice

S. Foelling et al., PRL 97:060403 (2006) Observation of spatial distribution of lattice sites using spatially selective microwave transitions and spin changing collisions superfluid regime Mott regime n=1 n=2 Related work Campbell, Ketterle, et al.

Science 313:649 (2006)

arXive:0904.1532

Extended Hubbard model Charge Density Wave and Supersolid phases

Extended Hubbard Model

- on site repulsion nearest neighbor repulsion Checkerboard phase : Crystal phase of bosons.

Breaks translational symmetry

van Otterlo et al., PRB 52:16176 (1995) Variational approach Extension of the Gutzwiller wavefunction Supersolid – superfluid phase with broken translational symmetry

Quantum Monte-Carlo analysis

arXiv:0906.2009

Difficulty of identifying supersolid phases in systems with parabolic potential

Bose Hubbard model away from equilibrium.

Dynamical Instability of strongly interacting bosons in optical lattices

Moving condensate in an optical lattice. Dynamical instability Theory: Niu et al. PRA (01), Smerzi et al. PRL (02) Experiment: Fallani et al. PRL (04) v

Dynamical instability Classical limit of the Hubbard model. Discreet Gross-Pitaevskii equation Current carrying states Linear stability analysis: States with p> p /2 are unstable unstable unstable Amplification of density fluctuations r

Dynamical instability. Gutzwiller approximation Wavefunction Time evolution We look for stability against small fluctuations 0.5

Phase diagram. Integer filling Altman et al., PRL 95:20402 (2005) 0.4

0.3

d=1 d=2 0.2

0.1

stable 0.0

0.0

0.2

d=3 0.4

U/U c 0.6

unstable 0.8

1.0

Optical lattice and parabolic trap. Gutzwiller approximation 0.00

0.2

0.17

0.34

0.52

0.1

0.69

0.86

N=1.5

N=3 The first instability develops near the edges, where N=1 0.0

-0.1

-0.2

0 100 200 Time 300 U=0.01 t J=1/4 400 500 Gutzwiller ansatz simulations (2D)

PRL (2007)

Beyond semiclassical equations. Current decay by tunneling j j Current carrying states are metastable. They can decay by thermal or quantum tunneling j Thermal activation Quantum tunneling

Current decay by thermal phase slips Theory: Polkovnikov et al., PRA (2005) Experiments: De Marco et al., Nature (2008)

Current decay by quantum phase slips Theory: Polkovnikov et al., Phys. Rev. A (2005) Experiment: Ketterle et al., PRL (2007) Dramatic enhancement of quantum fluctuations in interacting 1d systems d=1 dynamical instability.

GP regime d=1 dynamical instability.

Strongly interacting regime

Engineering magnetic systems using cold atoms in an optical lattice

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

Two component Bose Hubbard model

Two component Bose Hubbard model In the regime of deep optical lattice we can treat tunneling as perturbation. We consider processes of the second order in

t

We can combine these processes into anisotropic Heisenberg model

Two component Bose Hubbard model

Quantum magnetism of bosons in optical lattices Duan et al., PRL (2003) • Ferromagnetic • Antiferromagnetic

Hysteresis Two component Bose mixture in optical lattice.

Mean field theory + Quantum fluctuations Altman et al., NJP (2003) 1 st order

Two component Bose Hubbard model + infinitely large

U aa

and

U bb

New feature: coexistence of checkerboard phase and superfluidity

Exchange Interactions in Solids

antibonding 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

= - J x S s i x s j x - J y S s i y s j y - J z S s i z s j z Questions: Detection of topological order Creation and manipulation of spin liquid states Detection of fractionalization, Abelian and non-Abelian anyons Melting spin liquids. Nature of the superfluid state

Superexchange interaction in experiments with double wells

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

Observation of superexchange in a double well potential Theory: 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 states of 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 only local interaction Extended Hubbard model takes into account non-local interaction

Beyond the basic Hubbard model

From two spins to a spin chain Spin oscillations ?

Data courtesy of S. Trotzky (group of I. Bloch)

1D: XXZ dynamics starting from the classical Neel state P. Barmettler et al, PRL 2009 ) Y (t=0 =

Ising-Order Quasi-LRO

Equilibrium phase diagram: D 1 • DMRG • Bethe ansatz • XZ model: exact solution Time,

Jt

XXZ dynamics starting from the classical Neel state D <1, XY easy plane anisotropy Oscillations of staggered moment, Exponential decay of envelope Except at solvable xx point where: D >1, Z axis anisotropy Exponential decay of staggered moment

Behavior of the relaxation time with anisotropy See also: Sengupta, Powell & Sachdev (2004) - Moment always decays to zero. Even for high easy axis anisotropy - Minimum of relaxation time at the QCP. Opposite of classical critical slowing.

Magnetism in optical lattices Higher spins and higher symmetries

F=1 spinor condensates

Spin symmetric interaction of F=1 atoms Ferromagnetic Interactions for Antiferromagnetic Interactions for

Antiferromagnetic spin F=1 atoms in optical lattices

Hubbard Hamiltonian Demler, Zhou, PRL (2003) Symmetry constraints Nematic Mott Insulator Spin Singlet Mott Insulator

Nematic insulating phase for N=1

Effective S=1 spin model Imambekov et al., PRA (2003) When the ground state is nematic in d=2,3.

One dimensional systems are dimerized: Rizzi et al., PRL (2005)

Nematic insulating phase for N=1. Two site problem

2 0 1 -2 1 4 Singlet state is favored when One can not have singlets on neighboring bonds.

Nematic state is a compromise. It corresponds to a superposition of and on each bond

SU(N) Magnetism with Ultracold Alkaline-Earth Atoms A. Gorshkov et al., Nature Physics (2010) Example: 87 Sr (I = 9/2) nuclear spin decoupled from electrons SU(N) spin models SU(N=2I+1) symmetry Example: Mott state with n A atoms in sublattice A and n B atoms in sublattice B Phase diagram for n A + n B = N There are also extensions to models with additional orbital degeneracy

Learning about order from noise

Quantum noise studies of ultracold atoms

Quantum noise

Classical measurement: collapse of the wavefunction into eigenstates of x Histogram of measurements of x

Probabilistic nature of quantum mechanics

Bohr-Einstein debate on spooky action at a distance Einstein-Podolsky-Rosen experiment Measuring spin of a particle in the left detector instantaneously determines its value in the right detector

Aspect’s experiments: tests of Bell’s inequalities + 1 q 1 S q 2 2 + S Correlation function Classical theories with hidden variable require Quantum mechanics predicts B=2.7 for the appropriate choice of q ‘s and the state Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982)

Hanburry-Brown-Twiss experiments

Classical theory of the second order coherence

Hanbury Brown and Twiss, Proc. Roy. Soc

. (

London

),

A

, 242, pp. 300-324 Measurements of the angular diameter of Sirius

Proc. Roy. Soc

. (

London

),

A

, 248, pp. 222-237

Quantum theory of HBT experiments

Glauber,

Quantum Optics and Electronics

(1965) For bosons For fermions HBT experiments with matter Experiments with neutrons Ianuzzi et al., Phys Rev Lett (2006) Experiments with electrons Kiesel et al., Nature (2002) Experiments with 4He, 3He Westbrook et al., Nature (2007) Experiments with ultracold atoms Bloch et al., Nature (2005,2006)

Shot noise in electron transport

Proposed by Schottky to measure the electron charge in 1918 e e Spectral density of the current noise Related to variance of transmitted charge When shot noise dominates over thermal noise Poisson process of independent transmission of electrons

Shot noise in electron transport

Current noise for tunneling across a Hall bar on the 1/3 plateau of FQE Etien et al. PRL 79:2526 (1997) see also Heiblum et al. Nature (1997)

Quantum noise analysis of time-of-flight experiments with atoms in optical lattices: Hanburry-Brown-Twiss experiments and beyond

Theory: Altman et al., PRA (2004) Experiment: Folling et al., Nature (2005); Spielman et al., PRL (2007); Tom et al. Nature (2006)

Time of flight experiments Cloud before expansion Cloud after expansion Quantum noise interferometry of atoms in an optical lattice Second order coherence

Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Experiment: Folling et al., Nature (2005)

Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Cloud before expansion Cloud after expansion Second order correlation function Here and are taken after the expansion time

t

.

Two signs correspond to bosons and fermions.

Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Relate operators after the expansion to operators before the expansion. For long expansion times use steepest descent method of integration TOF experiments map momentum distributions to real space images Second order real-space correlations after TOF expansion can be related to second order momentum correlations inside the trapped system

Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Example: Mott state of spinless bosons Only local correlations present in the Mott state

G

- reciprocal vectors of the optical lattice

Quantum noise in TOF experiments in optical lattices We get bunching when corresponds to one of the reciprocal vectors of the original lattice .

Boson bunching arises from the Bose enhancement factors. A single particle state with quasimomentum

q

is a supersposition of states with physical momentum

q+nG

. When we detect a boson at momentum

q

we increase the probability to find another boson at momentum

q+nG

.

Quantum noise in TOF experiments in optical lattices Another way of understanding noise correlations comes from considering interference of two independent condensates After free expansion When condensates 1 and 2 are not correlated We do not see interference in . Oscillations in the second order correlation function

Quantum noise in TOF experiments in optical lattices Mott state of spinless bosons

Interference of an array of independent condensates Hadzibabic et al., PRL 93:180403 (2004) 2.5

3 Smooth structure is a result of finite experimental resolution (filtering) 1.4

1.2

2 1 1.5

0.8

1 0.6

0.5

0.4

0 0.2

-0.5

0 -1 -1.5

0 200 400 600 800 1000 1200 -0.2

0 200 400 600 800 1000 1200

Second order correlations. Experimental issues.

Autocorrelation function Complications we need to consider: finite resolution of detectors - projection from 3D to 2D plane s – detector resolution - period of the optical lattice In Mainz experiments and The signal in is

Second order coherence in the insulating state of bosons.

Experiment: Folling et al., Nature (2005)

Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Example: Band insulating state of spinless fermions Only local correlations present in the band insulator state

Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Example: Band insulating state of spinless fermions We get fermionic antibunching . This can be understood as Pauli principle. A single particle state with quasimomentum

q

is a supersposition of states with physical momentum

q+nG

.

When we detect a fermion at momentum

q

we decrease the probability to find another fermion at momentum

q+nG

.

Second order coherence in the insulating state of fermions.

Experiment: Tom et al. Nature (2006)

Second order correlations as Hanburry-Brown-Twiss effect Bosons/Fermions

Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Bosons with spin. Antiferromagnetic order Second order correlation function New local correlations Additional contribution to second order correlation function - antiferromagnetic wavevector We expect to get new peaks in the correlation function when

Probing spin order in optical lattices Correlation function measurements after TOF expansion.

Extra Bragg peaks appear in the second order correlation function in the AF phase.

This reflects doubling of the unit cell by magnetic order.