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, , …