Transcript VonStecher
Strongly Interacting Atoms in Optical Lattices
Javier von Stecher
JILA and Department of Physics , University of Colorado
In collaboration with Victor Gurarie, Leo Radzihovsky, Ana Maria Rey INT 2011
“Fermions from Cold Atoms to Neutron Stars:…
arXiv:1102.4593
to appear in PRL Support
Strongly interacting Fermions:
…Benchmarking the Many-Body Problem.”
BCS-BEC crossover
a 0 >0
Molecular BEC
a 0 = ±
a 0 <0
Degenerate Fermi Gas (BCS)
Strongly interacting Fermions + Lattice:
…Understanding the Many-Body Problem?”
Not unique: - different lattice structure and strengths.
More challenging: -Band structure, nontrivial dispersion relations, … -Single particle?, two particle physics??
Fermi-Hubbard model
Minimal model of interacting fermions in the tight-binding regime
Hopping Energy Interaction Energy
J
i i+1
U
Fermi-Hubbard model
Schematic phase diagram for the Fermi Hubbard model
• half-filling • simple cubic lattice • 3D
Esslinger, Annual Rev. of Cond. Mat. 2010
Experiments: R. Jordens et al., Nature (2008) U. Schneider et al., Science (2008).
Open questions: - d-wave superfluid phase?
- Itinerant ferromagnetism?
Beyond the single band Hubbard Model
Many-Body Hamiltonian (bosons):
Hamiltonian parameters:
Very complicated… But, what is the new physics?
Extension of the Fermi Hubbard Model:
Zhai and Ho, PRL (2007) Iskin and Sa … ´ de Melo, PRL(2007) Moon, Nikolic, and Sachdev, PRL (2007)
New Physics: Orbital physics
Experiments:
“Orbital superfluidity”:
Populating Higher bands:
Raman pulse
Long lifetimes ~100 ms (10-100 J)
T. Muller,…, I. Bloch PRL 2007 Scattering in Mixed Dimensions with Ultracold Gases G. Lamporesi et al. PRL (2010) JILA KRb Experiment G. Wirth, M. Olschlager, Hemmerich
New Physics: Resonance Physics
Experiments:
Two-body spectrum in a single site: Theory and Experiment Tuning interactions in lattices:
tune interaction Molecules of Fermionic Atoms in an Optical Lattice T. Stöferle , …,T. Esslinger PRL 2005
Lattice induced resonances
Tight Binding + Short range interactions: Good understanding of the onsite few body physics.
Resonance
New degree of freedom: internal and orbital structure of atoms and molecules
•
weak nonlocal coupling
•
Strong onsite interactions.
Separation of energy scales Independent control of onsite and nonlocal interactions
Lattice induced resonances
Feshbach resonance in free space
1D Feschbach resonance scattering continuum Two-body level: weakly bound molecules Many-body level: BCS-BEC crossover…
Interaction λ
bound state
0
Lattice induced resonances
1D Feschbach resonance
Feshbach resonance + Lattice
bands What is the many-body behavior?
Resonances
What is the two-body behavior?
Interaction λ Two-body physics:
P.O. Fedichev, M. J. Bijlsma, and P. Zoller PRL 2004 G. Orso et al, PRL 2005 X. Cui, Y. Wang, & F. Zhou, PRL 2010 H. P. Buchler, PRL 2010 N. Nygaard, R. Piil, and K. Molmer PRA 2008 …
Many-body physics (tight –binding):
L. M. Duan PRL 2005, EPL 2008 Dickerscheid , …, Stoof PRA, PRL 2005 K. R. A. Hazzard & E. J. Mueller PRA(R) 2010 …
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
One Dimension:
y z x
Hamiltonian: + a weak lattice in the z-direction
V x =V y =200-500 E r , V z =4-20 E r
1D interaction:
Confinement induced resonance
Two 1D particles in a lattice
One Dimension:
Bound States in 1D: Form at any weak attraction.
1D dimers with 40 K
Hamiltonian:
1D interaction:
Confinement induced resonance
H. Moritz, …,T. Esslinger PRL 2005
Non interacting lattice spectrum
Single particle
k=0
Two particles
K=0 2 1 0 (1,0) + (0,0) + K=(k 1 +k 2 )
Tight-binding limit:
k
k 1 =K/2+k, k 2 =K/2-k
Non interacting lattice spectrum
Two-body scattering continuum bands
V 0 =4 E r V 0 =20 E r (1,1) (0,2) (0,2) (1,1) (0,1) (0,1) (0,0) (0,0)
K a/(2 π) K a/(2 π)
Two particles in a lattice, single band Hubbard model Nature 2006 Grimm, Daley, Zoller… Tight-binding approximation
J
U
i i+1
U<0, attractive bound pairs U>0, repulsive bound pairs
Exact two-body solution
Calculations in a finite lattice with periodic boundary conditions
Bloch Theorem: Plane wave expansion:
Single particle basis functions: Two particles:
Very large basis set to reach convergence ~ 10 4 -10 5
Two-atom spectrum
(0,1) (0,0) (0,0)
Two body spectrum as a function of the interaction strength for a lattice with V 0 =4 E r
Two-atom spectrum
(0,1) (0,0) (0,0)
Two body spectrum as a function of the interaction strength for a lattice with V 0 =4 E r
Two-atom spectrum
Tight-binding regime
Two-atom spectrum
Tight-binding regime
First excited dimer crossing
Avoided crossing between a molecular band and the two-atom continuum
K=0
dimer continuum
K=π/a Interaction Interaction
Second excited dimer crossing
K=0 Interaction K=π/a Interaction How can we understand this qualitatively change in the atom-dimer coupling?
Two-atom spectrum
Tight-binding regime
Effective Hamiltonian
K ΔE
w
a,i (r)
W
m,i (R,r)
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
Effective Hamiltonian
J d
K d † a † ΔE
g ex g J a
J a , J d , g ex , g and ε d are input parameters
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:
Prefer to couple at : K=π/a (max K) K=0 (min K) K = center of mass quasi momentum atoms molecules molecules k k k
Comparison model and exact solution (1,0) molecule: 1 st excited (2,0) molecule: 2 nd excited
21 sites and V 0 =20E r
Molecules above and below!
Dimer Wannier Function
J d , g and ε d fitting parameters to match spectrum?
g ex
How to calculate g ex ?
is a three-body term
i i+1 Wannier function for dimers: d i † a i †
W w
m,i a,i (r) (R,r)
Effective Hamiltonian matrix elements: Neglected terms:
Prescription to calculate all eff. Ham. Matrix elements
Dimer Wannier Function
Extraction of the bare dimer: bound state bare dimer
(0,1) dimer Wannier Function
0 K
(g Extraction of J d , g and ε d
1.7 J for (0,1) dimer) : excellent agreement with the fitting values.
Effective Hamiltonian parameters
•
Construct dimer Wannier function
•
Extract eff. Hamiltonian parameters
Single band Hubbard model: Enhanced assisted tunneling!
… and symmetric coupling
Parity effects
Atoms in different bands or species:
P=p d +p 1 +p 2 g a g b
Rectangular lattice
More dimensions:
extra degeneracies… more than one dimer
+ _ +
Positive parity
+
Experimental observation:
Observe quasimomentum dependence of atom-dimer coupling Initialize system in dimer state.
Change interactions with time.
Measure molecule fraction as a function of quasimomentum.
Dimer fraction (Landau-Zener):
Ramp Experiment:
dimer state Scattering continuum dimer fraction
Time Also K-dependent quantum beats…
N. Nygaard, R. Piil, and K. Molmer PRA 2008
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:
What is the many-body physics of the effective Hamiltonian?