Interacting Bosons and Interacting Bosons and Fermions in 3D Optical Fermions in 3D Optical Lattice PotentialsLattice PotentialsSebastian Will, Thorsten Best, Simon Braun, Ulrich Schneider, KC Fong, Lucia Hackermüller,

Stefan Trotzky, Yuao Chen,Ute Schnorrberger,

Stefan Kuhr, Jacob Sherson, Christof Weitenberg, Manuel Endres,

Theory: Belén Paredes, Mariona Moreno

Immanuel Bloch

Johannes Gutenberg-Universität, Mainz

funding by€ DFG, European Union,$ AFOSR, DARPA (OLE) www.quantum.physik.uni-

mainz.de

Fermions in a 3D Lattice

with repulsive interactions

Quantum Phase Diffusionand Bose-Fermi Mixtures

Our starting point: Ultracold Quantum Gases

Bose-Einstein condensate

e.g. 87Rb atoms

ground states at T=0Parameters: Densities: 1015 cm-3

Temperatures: nanoKelvinNumber of Atoms: about 106

Degenerate Fermi gas

e.g. 40K atoms

Optical Lattice in 3D

3D lattice: array of quantum dots

Hubbard Hamiltonian

Restriction to single (lowest) band and expansion in localized wannier functions yields:

Tunneling matrix element: Onsite interaction matrix element:

Bose-

Hubbard

Hamiltonian

Fermionswith repulsive interactions

U. Schneider, L. Hackermüller, S. Will, Th. Best, I.Bloch &A. Rosch, Th. Costi, D. Rasch, R. Helmes (Science, 322, 1520 (2008))

Strongly Interacting Fermions in Optical Lattices

• Phases predicted at half filling for strong interactions U/12J > 1:

Related experimental work at ETHZ (T. Esslinger)

e.g. M. Köhl et al., PRL 94, 080403 (2005), R. Jördens et al. Nature 455, 204 (2008)

maximal entropy:

S/N = kB 2 ln(2)

Hubbard Model and High-Tc

Can we help identifying the phase diagram of the Hubbard model?

W. Hofstetter, J.I. Cirac, P. Zoller, E. Demler, M.D. Lukin, PRL 89, 220407 (2002),

P. A. Lee, N. Nagosa, X. G. Wen, Rev. Mod. Phys. 78 , 17 (2008)

Experimental Setup: Fermions in the Optical Lattice

• Crossed Dipole Trap

1030nm (elliptical beams)

Spin mixture of K atoms in F=9/2, mF=-9/2 and F=9/2, mF=-7/2:

• Blue Detuned Lattice Beams

738nm (160 µm waist)

T=0.06 to 0.13 TF with about 3 x 105 atoms!

Compression of the Quantum Gas

Total Potential for Atoms: Optical Lattice combined with Dipole Trap!

Independent control of

Lattice Depth

and

Dipole Trap Depth

Compression Range:

+

Hubbard Hamiltonian: All Parameters Tunable!

40K Feshbach resonance:

+

(JILA parametrization)

Experimental Observables:

• Global Observable: Compressibility

• Local Observable:

For example: in-situ cloud size

with phase-constrast imaging

For example: pair fraction with Feshbach ramp

or central occupation

(see L. De Leo et al., 2008, alternative method: see

Zürich experiment R. Jördens et al., Nature 455, 204 (2008))

U. Schneider, L. Hackermüller, S. Will, Th. Best, I.Bloch &

A. Rosch, Th. Costi, D. Rasch, R. Helmes (Science, 322, 1520 (2008))

2R

Quantum Phases of Repulsive Fermions in Trap

compressible!

incompressible!

incompressible!

Comparison with Theory (I)

Dynamical Mean Field Theory (DMFT)Metzner, Vollhardt, Georges, Kotliar

e.g. A. Georges et al. Rev. Mod. Phys. 68, 13 (1996)

Real Space Adaptation (Inhomogeneous Systems)

Achim Rosch, Theo Costi (here LDA + DMFT)

see also: L. De Leo et al. PRL, 101, 210403 (2008)

and work by W. Hofstetter

Calculations at Forschungszentrum Jülich:

JUGENE, IBM Blue-Gene Supercomputer

# 1 in Germany

# 6 on TOP 500 list worldwide

First test bed for DMFT in 3D!

Measuring the Cloud Size…

Measuring the Compressibility (I)

U. Schneider, et al. (Science, 322, 1520 (2008))

Theory: R. W. Helmes et al. (PRL 100, 056403(2008))

Measuring the Compressibility (II)

Pair Fraction versus Compression

Entropy Distibution in the Trap

Entropy of non-interacting

gas in harmonic trap

T/TF = 0.15

S/N > kB 2 ln(2)

While entropy of MI is only

S/N = kB ln(2) !

U/12J = 1.5

Summary: Pair Fraction & Compression Measurements

In-situ cloud size / Compression Measurements:

Pair fraction measurements:

• Very good quantitative agreement with ab-initio DMFT

for weak and strong compressions!

• Direct measurement of the (in-)compressibility of the

many-body system.

• Deviations beyond U/6J = 4 in low compression regime

• Good agreement with ab-initio DMFT theory (T approx. 0.15 TF)

• But note: Melted MI and strongly interacting metallic phases

can also show suppression of pairs!

Multi-Orbital Quantum Phase Diffusion

Sebastian Will, Thorsten Best, Simon Braun, Ulrich Schneider, KC Fong, Lucia Hackermüller, Dirk-Sören Lühmann, Immanuel Bloch

From BEC to a Superfluid in an Optical Lattice…

BEC in a harmonic trap…

Onsite picture:

Coherent State

Poisson distribution

Non-interacting, homogeneous case:

…plus a weak lattice

Dynamics of a coherent state:

In the limit of zero tunneling (J = 0)

evolution is determined by:

The matterwave field on a lattice site… experimentally observable as

time-evolution

of coherent state

Visibility

Phase Diffusion Dynamics: Collapse and Revival

1. Matterwave field collapses and revives after multiple times of h/U

2. Collapse time depends on the variance of the atom number distribution

Theory: Yurke & Stoler, 1986, F. Sols 1994; Wright et al. 1997; Imamoglu, Lewenstein & You et al. 1997,

Castin & Dalibard 1997, E. Altman & A. Auerbach 2002, Exp: M. Greiner et al. 2002, G.-B. Jo et al. 2006, J. Sebby-Strabley et al. 2007, A. Widera et al., 2007,

M. Oberthaler et al. 2008

Dynamical Evolution of the Interference Pattern

t=50µs t=150µs t=200µs t=300µs

t=400µs t=450µs t=600µs

Dynamics after potential jump from 8Erec to 22Erec!

Collapse & Revival under Optimal Harmonic Confinement

• Up to 70 revivals can be detected!

• And: Multiple frequency components!

Why Multiple Frequencies?

Here U is assumed to be

constant, independent of filling…

Breakdown of the single-band approximation!

Admixture of higher-band orbitals!

for a differential measurement, see also: G. Campbell et. al., Science (2006)

n = 2

U(2)

n = 3

U(3)

n = 4

U(4)

Fourier Spectrum

Strong signal of

small contributions due to

heterodyning effect!

E(2) + E(4) – 2E(3)

2 E(2) - E(3)

E(2)

c2 · c32 · c4

c1 · c22 · c3

c0 · c12 · c2

Comparison with Exact Diagonalization

Theory: D.-S. Lühmann, Hamburg University

of order 2∙U

of order U

Atom distribution along the SF to MI transition:

BOSE-FERMI Mixtures in the Optical Lattice

Lattice site

with 1 FermionEffective Onsite Potential for Boson at Ubf < 0:

nboson = 1 nboson = 2

Theory: D.-S. Lühmann et al., PRL 2008R. Lutchyn, S. Tewari, S. Das Sarma, arxiv:0812.0815v2

Self-Trapping of 87Rb due to 40K

Increasing Boson Repulsion due

to Self-Trapping of Fermions!

Increasing Boson Filling due

to Bose-Fermi Attraction!

Shift of SF-MI Transition in Bose-Fermi Mixtures

Experiment: Th. Best, S. Will et al., arXiv:0807.4504 (in press)S. Ospelkaus et al., PRL 2006, K. Günter et al. PRL 2006, J. Catani et al. PRA 2008Theory: D.-S. Lühmann et al., PRL 2008

Conclusion

Global Compressibility Measurements on Repulsively Interacting Fermi Gases in a 3D Optical Lattice

• Evidence for Incompressible Mott Core

• Good Agreement with ab-initio DMFT calculations

Quantum Phase Diffusion as a Probe in Strongly Interacting Quantum Gas Mixtures

• Quantum Phase Diffusion with Fock State Resolution

• Renormalized Hubbard Parameters

• Self-Trapping in Bose-Fermi Mixtures (Multi-Band Physics)

THANK YOU!

Useful Variables:

Interactions versus Kinetic Energy

Confinement versus Kinetic Energy, where

Initial Temperature (Entropy)

characteristic trap energy

= Fermi energy at T=0, J=0

and no interaction

trap aspect ratio:

Doubly occupied sites, compressibility, , …