Strongly Interacting Atoms in Optical Lattices

Javier von Stecher

JILA and Department of Physics , University of Colorado

Support

INT 2011“Fermions from Cold Atoms to Neutron Stars:…

arXiv:1102.4593to appear in PRL

In collaboration withVictor Gurarie, Leo Radzihovsky,Ana Maria Rey

Strongly interacting Fermions:…Benchmarking the Many-Body Problem.”

BCS-BEC crossover

a0<0a0>0 a0=±

Degenerate Fermi Gas

(BCS)

Molecular BEC

Strongly interacting Fermions + Lattice:…Understanding the Many-Body Problem?”

?More challenging:

-Band structure, nontrivial dispersion relations, …

-Single particle?, two-particle physics??

Not unique:

- different lattice structure and strengths.

Interaction EnergyHopping Energy

J

U

i+1

i

Fermi-Hubbard modelMinimal model of interacting fermions in the tight-binding regime

Fermi-Hubbard modelSchematic phase diagram for the Fermi Hubbard model

Esslinger, Annual Rev. of Cond. Mat. 2010

• half-filling • simple cubic lattice •3D

Experiments:R. Jordens et al., Nature (2008)U. Schneider et al., Science (2008).

Open questions:- d-wave superfluid phase?- Itinerant ferromagnetism?

Many-Body Hamiltonian (bosons):

Hamiltonian parameters:

Beyond the single band Hubbard Model

Extension of the Fermi Hubbard Model:

Zhai and Ho, PRL (2007)Iskin and Sa´ de Melo, PRL(2007)Moon, Nikolic, and Sachdev, PRL (2007)…

Very complicated…But, what is the new physics?

T. Muller,…, I. Bloch PRL 2007

Populating Higher bands:

Scattering in Mixed Dimensions with Ultracold Gases G. Lamporesi et al. PRL (2010)JILA KRb Experiment

New Physics: Orbital physicsExperiments:

Ramanpulse

Long lifetimes ~100 ms (10-100 J)

G. Wirth, M. Olschlager, Hemmerich

“Orbital superfluidity”:

New Physics: Resonance PhysicsExperiments:

T. Stöferle , …,T. Esslinger PRL 2005

Tuning interactions in lattices:

Molecules of Fermionic Atoms in an Optical Lattice

tune interaction

Two-body spectrum in a single site: Theory and Experiment

Lattice induced resonances

Resonance

Tight Binding + Short range interactions:

• Strong onsite interactions.

Good understanding of the onsite few-body physics.

• weak nonlocal coupling

New degree of freedom: internal and orbital structure of atoms and molecules

Separation of energy scales

Independent control of onsite and nonlocal interactions

Feshbach resonance in free spaceFeshbach resonance in free space

Two-body level: weakly bound molecules

Many-body level: BCS-BEC crossover…

1D Feschbach resonance

Interaction λ

Ener

gy

0bound state

scatteringcontinuum

Lattice induced resonances

Feshbach resonance + LatticeFeshbach resonance + Lattice

What is the many-body behavior?

Interaction λ

Ener

gy

1D Feschbach resonance

bands

P.O. Fedichev, M. J. Bijlsma, and P. Zoller PRL 2004G. Orso et al, PRL 2005X. Cui, Y. Wang, & F. Zhou, PRL 2010H. P. Buchler, PRL 2010N. Nygaard, R. Piil, and K. Molmer PRA 2008…

L. M. Duan PRL 2005, EPL 2008Dickerscheid , …, Stoof PRA, PRL 2005K. R. A. Hazzard & E. J. Mueller PRA(R) 2010…

Two-body physics: Many-body physics (tight –binding):

What is the two-body behavior? Resonances

Lattice induced resonances

Our strategy

• Start with the simplest case– Two particles in 1D + lattice.

• Benchmark the problem:– Exact two-particle solution

• Gain qualitative understanding– Effective Hamiltonian description

Two-body calculations are valid for two-component Fermi systems and bosonic systems .Below, we use notation assuming bosonic statistics.

Two 1D particles in a lattice

y

xz

+ a weak lattice in the z-direction+ a weak lattice in the z-direction

Hamiltonian:Hamiltonian:

1D interaction:

Confinement induced resonance

One Dimension:

Vx=Vy=200-500 Er, Vz=4-20 Er

Two 1D particles in a lattice

Hamiltonian:Hamiltonian:

1D interaction:

Confinement induced resonance

One Dimension:

1D dimers with 40K

H. Moritz, …,T. Esslinger PRL 2005

Bound States in 1D:Form at any weak attraction.

Non interacting lattice spectrumEnergy

k

+

+

k=0

Single particleSingle particle Two particlesTwo particles

Tight-binding limit:Tight-binding limit:

k1=K/2+k, k2=K/2-k

Energy

K=(k1+k2)

K=0

(1,0)

(0,0)0

1

2

Non interacting lattice spectrum

V0=4 Er

K a/(2 π)

(0,0)

(0,1)

(0,2)

(1,1)

V0=20 Er

K a/(2 π)

(0,0)

(0,1)

(0,2)

(1,1)

Two-body scattering continuum bands

Two particles in a lattice, single band Hubbard model

Tight-binding approximation

Nature 2006 Grimm, Daley, Zoller…

J

U

i+1i

U>0, repulsive bound pairs U<0, attractive bound pairs

Calculations in a finite lattice with periodic boundary conditions

Exact two-body solution

Plane wave expansion:Plane wave expansion:

Single particle basis functions:

Two particles:

Very large basis set to reach convergence ~ 104-105

Bloch Theorem:Bloch Theorem:

Two-atom spectrum

Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er

(0,0)

(0,1)

(0,0)

Two-atom spectrum

Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er

(0,0)

(0,1)

(0,0)

Two-atom spectrum

Tight-binding regime

Two-atom spectrum

Tight-binding regime

Avoided crossing between a molecular band and the two-atom continuum

Interaction

Ener

gy

dimer

continuum

K=0

Interaction

Ener

gy

K=π/a

First excited dimer crossing

How can we understand this qualitatively change in the atom-dimer coupling?

Interaction En

ergy

Interaction

Ener

gy

K=0

K=π/a

Second excited dimer crossing

Two-atom spectrum

Tight-binding regime

Effective HamiltonianEnergy

K

ΔE

Atoms and dimers are in the tight-binding regime.

They are hard core particles (both atoms and dimers).

Leading terms in the interaction are produced by hopping of one particle.

L. M. Duan PRL 2005, EPL 2008

wa,i(r) Wm,i(R,r)

Effective HamiltonianEnergy

K

ΔE

Ja, Jd, gex, g and εd are input parameters

d†

a†

ggex

JaJd

Parity effects The atomic and dimer wannier functions are symmetric or antisymmetric with respect to the center of the site.

Parity effects on the atom-dimer interaction:

S coupling

g-1 g+1

g+1= g-1

Parity effects The atomic and dimer wannier functions are symmetric or antisymmetric with respect to the center of the site.

Parity effects on the atom-dimer interaction:

AS coupling

g-1 g+1

g+1= -g-1

Parity effects

Atom-dimer interaction in quasimomentum space:

K = center of mass quasi momentum

Prefer to couple at :

K=π/a (max K)

K=0 (min K)

Energy

k

atoms

Energy

k

molecules

Energy

k

molecules

Comparison model and exact solution

(1,0) molecule: 1st excited (2,0) molecule: 2nd excited

21 sites and V0=20Er

Molecules above and below!Molecules above and below!

Dimer Wannier Function

Effective Hamiltonian matrix elements:

Jd, g and εd fitting parameters to match spectrum?

i+1i

gex

How to calculate gex?is a three-body term

Neglected terms:

wa,i(r)

Wannier function for dimers:

di†

ai†

Wm,i(R,r)

Prescription to calculate all eff. Ham. Matrix elements

Dimer Wannier Function

(0,1) dimer Wannier Function

0

Energy

K

bound state

bare dimer

Extraction of the bare dimer:

Extraction of Jd, g and εd : excellent agreement with the fitting values. (g1.7 J for (0,1) dimer)

Effective Hamiltonian parameters

•Construct dimer Wannier function•Extract eff. Hamiltonian parameters

Single band Hubbard model:

… and symmetric couplingEnhanced assisted tunneling!

P=pd+p1+p2

Atoms in different bands or species:

More dimensions:

extra degeneracies…more than one dimer

Parity effects

Positive parity

Neg

ative

pa

rity

+ +

+

_

Rectangular latticeRectangular lattice

gb

ga

Experimental observation:

Initialize system in dimer state.

Change interactions with time.

Measure molecule fraction as a function of quasimomentum.

Ramp Experiment:

Time En

ergy

dimer state

Scattering continuum

dimer fraction

Observe quasimomentum dependence of atom-dimer coupling

N. Nygaard, R. Piil, and K. Molmer PRA 2008

Also K-dependent quantum beats…

Dimer fraction (Landau-Zener):

Summary Lattice induced resonances (Lattice + Resonance + Orbital Physics)can be used to tuned lattice systems in new regimes.

The orbital structure of atoms and dimer plays a crucial role in the qualitative behavior of the atom-dimer coupling.

The momentum dependence of the molecule fraction after a magnetic ramp provides an experimental signature of the lattice induced resonances.

Outlook:Outlook: What is the many-body physics of the effective Hamiltonian?