Strongly correlated many-body systems: from electronic ...€¦ · • Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions. Current experiments:

• 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 latticesQuantum magnetism of ultracold atoms.

Current experiments: observation of superexchange

• Detection of many-body phases using noise correlations

• Fermions in optical latticesMagnetism 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

Strongly correlated many-body systems: from electronic

materials to ultracold atoms to photons

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 diagram

M.P.A. Fisher et al.,

PRB (1989)21+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

Particle-hole excitation

Mott insulator phase

21+n

Uµ

1

0

Mottn=1

n=2

n=3

Superfluid

Mott

Mott

Tips of the Mott lobes

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

Transition takes place when coefficient before becomes negative. For large n this corresponds to

Take the middle of the Mott plateau

Expand to order

Example: stability of the Mott state with n atoms per site

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.

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

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

Shell structure in optical latticeS. 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-Carloanalysis

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

v

Theory: Niu et al. PRA (01), Smerzi et al. PRL (02)Experiment: Fallani et al. PRL (04)

Linear stability analysis: States with p>p/2 are unstable

Classical limit of the Hubbard model. Discreet Gross-Pitaevskii equation

Current carrying states

r

Dynamical instability

Amplification ofdensity fluctuations

unstableunstable

Dynamical instability. Gutzwiller approximation

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

d=3

d=2

d=1

unstable

stable

U/Uc

p/π

Wavefunction

Time evolution

Phase diagram. Integer filling

We look for stability against small fluctuations

Altman et al., PRL 95:20402 (2005)

The first instability

develops near the edges,

where N=1

0 100 200 300 400 500

-0.2

-0.1

0.0

0.1

0.2

0.00 0.17 0.34 0.52 0.69 0.86

Ce

nte

r o

f M

ass M

om

en

tum

Time

N=1.5

N=3

U=0.01 t

J=1/4

Gutzwiller ansatz simulations (2D)

Optical lattice and parabolic trap.

Gutzwiller approximation

PRL (2007)

Beyond semiclassical equations. Current decay by tunneling

phase

jphase

jphase

j

Current carrying states are metastable. They can decay by thermal or quantum tunneling

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 enhancementof quantum fluctuations

in interacting 1d systems

d=1 dynamicalinstability.GP regime

d=1 dynamicalinstability.Strongly interactingregime

Engineering magnetic systems

using cold atoms in an optical lattice

tt

Two component Bose mixture in optical lattice

Two component Bose Hubbard model

Example: . Mandel et al., Nature (2003)

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.

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 into anisotropic Heisenberg model



Quantum magnetism of bosons in optical lattices

Duan et al., PRL (2003)

• Ferromagnetic

• Antiferromagnetic

Two component Bose mixture in optical lattice.

Mean field theory + Quantum fluctuations

2 ndorder line

Hysteresis

1st order

Altman et al., NJP (2003)


+ infinitely large Uaa and Ubb

New feature:coexistence of

checkerboard phase

and superfluidity

Exchange Interactions in Solids

antibonding

bonding

Kinetic energy dominates: antiferromagnetic state

Coulomb energy dominates: ferromagnetic state

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

Realization of spin liquid

using cold atoms in an optical latticeTheory: Duan, Demler, Lukin PRL (03)

H = - Jx Σ σix σj

x - Jy Σ σiy σj

y - Jz Σ σiz σj

z

Kitaev model Annals of Physics (2006)

Superexchange interaction

in experiments with double wells

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

J

J

Use magnetic field gradient to prepare a state

Observe oscillations between and states

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

Experiments:S. Trotzky et al.Science 2008

Reversing the sign of exchange interaction

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

Comparison to the Hubbard model

Basic Hubbard model includesonly local interaction

Extended Hubbard modeltakes into account non-localinteraction

Beyond the basic Hubbard model

Beyond the basic Hubbard model

From two spins to a spin chain

Spin oscillations ?

Data courtesy of Data courtesy of

S. S. TrotzkyTrotzky

(group of I. Bloch)(group of I. Bloch)

1D: XXZ dynamics starting from the classical Neel state

• DMRG• Bethe ansatz

• XZ model: exact solution

Time, Jt

∆Equilibrium phase diagram:

Ψ(t=0) =

Quasi-LRO

1

Ising-Order

P. Barmettler et al, PRL 2009

XXZ dynamics starting from the classical Neel state

∆<1, XY easy plane anisotropy

Oscillations of staggered moment, Exponential decay of envelope

∆>1, Z axis anisotropy

Exponential decay of staggered moment

Except at solvable xx point where:

Behavior of the relaxation time with anisotropy

- Moment always decays to zero. Even for high easy axis anisotropy

- Minimum of relaxation time at the QCP. Opposite of classical critical slowing.

See also: Sengupta,

Powell & Sachdev (2004)

Magnetism in optical lattices

Higher spins and higher symmetries

F=1 spinor condensates

Spin symmetric interaction of F=1 atoms

Antiferromagnetic Interactions for

Ferromagnetic Interactions for

Antiferromagnetic spin F=1 atoms in optical lattices

Hubbard Hamiltonian

Symmetry constraints

Demler, Zhou, PRL (2003)

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

12

0 -2 4

1

Singlet state is favored when

One can not have singlets on neighboring bonds.

Nematic state is a compromise. It correspondsto a superposition of and

on each bond

SU(N) Magnetism with Ultracold Alkaline-Earth Atoms

Example: 87Sr (I = 9/2)

nuclear spin decoupled from electrons SU(N=2I+1) symmetry

SU(N) spin models

A. Gorshkov et al., Nature Physics (2010)

Example: Mott state with nA atoms in sublattice A and nB atoms in sublattice B

Phase diagram for

nA + nB = N

There are also extensions to models with additional orbital degeneracy

Learning about order from noise

Quantum noise studies of ultracold atoms

Quantum noiseClassical 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

Measuring spin of a particle in the left detectorinstantaneously determines its value in the right detector

Einstein-Podolsky-Rosen experiment

Aspect’s experiments:tests of Bell’s inequalities

SCorrelation function

Classical theories with hidden variable require

Quantum mechanics predicts B=2.7 for the appropriate choice of θ‘s and the state

Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982)

+

-

+

-1 2θ1 θ2

S

Hanburry-Brown-Twiss experimentsClassical theory of the second order coherence

Measurements of the angular diameter of Sirius

Proc. Roy. Soc. (London), A, 248, pp. 222-237

Hanbury Brown and Twiss,

Proc. Roy. Soc. (London),

A, 242, pp. 300-324

Quantum theory of HBT experiments

For bosons

For fermions

Glauber,Quantum Optics and Electronics (1965)

HBT experiments with matter

Experiments with 4He, 3He

Westbrook et al., Nature (2007)

Experiments with neutrons

Ianuzzi et al., Phys Rev Lett (2006)

Experiments with electrons

Kiesel et al., Nature (2002)

Experiments with ultracold atoms

Bloch et al., Nature (2005,2006)

Shot noise in electron transport

e- e-

When shot noise dominates over thermal noise

Spectral density of the current noise

Proposed by Schottky to measure the electron charge in 1918

Related to variance of transmitted charge

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

Quantum noise interferometry of atoms in an optical lattice

Second order coherence

Cloud after expansion

Cloud before expansion

Experiment: Folling et al., Nature (2005)

Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices

Second order correlation function

Cloud after expansion

Cloud before expansion


Here and are taken after the expansion time t.Two signs correspond to bosons and fermions.

Relate operators after the expansion to operators before the expansion. For long expansion times use steepest descent

method of integration

Second order real-space correlations after TOF expansioncan be related to second order momentum correlations

inside the trapped system


TOF experiments map momentum distributions to real space images

Example: Mott state of spinless bosons

Only local correlations present in the Mott state

G - reciprocalvectors of the

optical lattice


Quantum noise in TOF experiments in optical lattices

We get bunching when corresponds to oneof 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 fromconsidering interference of two independent condensates

Oscillations in the second order correlation function

After free expansion

When condensates 1 and 2 are not correlated

We do not see interference in .

Quantum noise in TOF experiments in optical latticesMott state of spinless bosons

0 200 400 600 800 1000 1200-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Interference of an array of independent condensates

Hadzibabic et al., PRL 93:180403 (2004)

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

0 200 400 600 800 1000 1200-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Second order correlations. Experimental issues.

Complications we need to consider:- finite resolution of detectors

- projection from 3D to 2D plane

σ – detector resolution

Autocorrelation function

In Mainz experiments and

The signal in is

- period of the optical lattice

Second order coherence in the insulating state of bosons.

Experiment: Folling et al., Nature (2005)

Example: Band insulating state of spinless fermions


Only local correlations present in the band insulator state

Example: Band insulating state of spinless fermions


We get fermionic antibunching. This can be understoodas 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 theprobability 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


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 theunit cell by

magnetic order.

Documents

Strongly correlated many-body systems: from electronic ...€¦ · • Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions. Current experiments: