1. ultracold atomic gases
Optical lattices for
ultracold atomic gases
Andrea Trombettoni
(SISSA, Trieste)
with:
G. Mussardo, E. Fersino (SISSA)
L. Dell’Anna, S. Fantoni (SISSA), P. Sodano (Perugia)
Firenze, GGI, 10 September 2008

2. Why using optical lattices?
Outlook
Why using optical lattices?
Effective tuning of the interactions
Experimental realization of interacting/integrable lattice Hamiltonians
Discrete dynamics
Stabilization of solutions unstable in the continuous limit
Ultracold bosons on a lattice with disorder: the shift of the critical temperature

3. Ultracold bosons in an optical lattice
a 3D lattice
It is possible to control:
- barrier height
interaction term
the shape of the network
the dimensionality (1D, 2D, …)
the tunneling among planes or among tubes
(in order to have a layered structure) …

4. Tuning the interactions with
optical lattices
s-wave scattering length
bosonic field
For large enough barrier height
tight-binding Ansatz
[Jaksch et al. PRL (1998)]
Bose-Hubbard
Hamiltonian
increasing the scattering length or
increasing the barrier height
 the ratio U/t increases
Ultracold fermions in an optical lattice  (Fermi-)Hubbard Hamiltonian
[Hofstetter et al., PRL (2002) – Chin et al., Nature (2006)]

5. Phase transitions in bosonic arrays
Increasing V, one passes from a
superfluid to a Mott insulator
[Greiner et al., Nature (2001)]
Similar phase transitions studied in superconducting arrays
[see Fazio and van der Zant, Phys. Rep. 2001]:
At finite temperature, Berezinskii-Kosterlitz-Thouless transitions in a 2D bosonic lattice is quantitatively well described by the XY model [A.T. et al., New J. Phys (2005)] and it has been recently observed in Schweikhard et al., PRL (2007) – see also Hadzibabibc et al., Nature (2006) continuous 2D Bose gas

6. Discrete dynamics Gross-Pitaevskii approximation:
tight-binding Ansatz for the Gross-Pitaevskii equation with an optical lattice [A.T. & A. Smerzi, PRL (1998)]
When V0>>m:
Bright localized solitons also
with repulsion (a>0):
Repulsion+negative effective
mass  effective “attraction”

7. Discreteness vs. Nonlinearity
Josephson oscillations in a bosonic array [Cataliotti et al., Science 2001]
New mechanisms for breakdown of superfluidity [Cataliotti et al., New J. Phys – Fallani et al., PRL 2004]
New Josephson regimes: self-trapping [Anker et al., PRL 2005]
Anderson localization vs. nonlinearity [previous talk by G. Modugno]
Stabilization of solitons by an optical lattice
LENS, Florence

8. Stabilization of solitons by an optical lattice (I)
Recent proposals to engineer 3-body interactions
[Paredes et al., PRA Buchler et al., Nature Pysics 2007]
In 1D with attractive 3-body contact interactions:
no Bethe solution is available – in mean-field [Fersino et al., PRA 2008]:
in order to have a finite energy per particle

9. Stabilization of solitons by an optical lattice (II)
Problem: a small (residual) 2-body interaction make unstable
such soliton solutions
Adding an optical lattice :
Soliton solutions stable for
for small q

10. Why using optical lattices?
Outlook
Why using optical lattices?
Effective tuning of the interactions
Experimental realization of interacting/integrable lattice Hamiltonians
Study of discrete dynamics: negative mass, solitons, dynamical instabilities
Stabilization of solutions unstable in the continuous limit
Ultracold bosons on a lattice with disorder: the shift of the critical temperature

11. Bosons on a lattice with disorder: shift of the critical temperature
total number of particles
filling
number of sites
random variables: produced by a speckle or
by an incommensurate bichromatic lattice
From the replicated action 
disorder is similar to an attractive interaction

12. Shift of the critical temperature
in a continuous Bose gas due to the repulsion
For an ideal Bose gas, the Bose-Einstein critical temperature is
What happens if a repulsive interaction is present?
The critical temperature increases
for a small (repulsive) interaction…
…and finally decreases
[see Blaizot, arXiv: ]

13. Long-range limit (I) Without random-bond disorder
The matrix to diagonalize is
where
The relation between the number of particles and the chemical potential is
The critical temperature is then

14. Long-range limit (II) With random-bond disorder
Using results from the theory of random matrices
[in agreement with the results for the spherical spin glass
by Kosterlitz, Thouless, and Jones, PRL (1976)]

15. 3D lattice without disorder
Single particle energies:
The relation between the number of particles and the chemical potential is
For large filling
Watson’s integrals

16. Connection with the spherical model
The ideal Bose gas is in the same universality class of the spherical model [Gunton-Buckingham, PRL (1968)]
For large filling, the critical temperature coincides with the critical temperature of the spherical model
with the (generalized) constraint

17. 3D lattice with disorder
3D lattice, with random-bond and on-site disorder:
Introducing N replicas of the system and computing the effective replicated action
Disorder (both on links and on-sites) is equivalent to an effective
attraction among replicas
Diagram expansion for the Green’s functions for N 0
Computing the self-energy
New chemical potential (effective t larger, larger density of states)

18. 3D lattice with disorder: Results for random-bond disorder
For large filling
results for the continuous (i.e., no optical lattice)
Bose gas [Vinokur & Lopatin, PRL (2002)]

19. 3D lattice with disorder: Results for on-site disorder
When a random on-site disorder (average zero, variance v02) is present
When both random-bond and random on-site disorder are present

20. 3D lattice with disorder: Results for an incommensurate potential
Two lattices:

21. A (very) qualitative explanation
Continuous Bose gas:
Repulsion  critical temp. Tc
increases
Disorder  “attraction” 
Tc decreases
Lattice Bose gas: Disorder  “attraction”
Small filling  continuous limit  Tc decreases
Large filling  all the band is occupied  effective “repulsion” 
Tc increases

22. Thank you!

23. Berezinskii-Kosterlitz-Thouless transition in a 2D lattice
central peak of the momentum distribution:
Good description at finite T by an XY model
thermally driven vortex proliferation
[Schweikhard et al., PRL (2007)]
In the continuous 2D Bose gas BKT transition
observed in the Dalibard group in Paris,
see Hadzibabibc et al., Nature (2006)
[A. Trombettoni, A. Smerzi and P. Sodano,
New J. Phys. (2005)]

24. Some details on the diagrammatic expansion (I)

25. diagrammatic expansion (II)
Some details on the
diagrammatic expansion (II)

26. N-Body Attractive Contact Interactions
We consider an effective attractive 3-body contact interaction and, more generally,
an N-body contact interaction:
contact interaction
N-body
attractive (c>0)
With

27. 2-Body Contact Interactions
N=2 Lieb-Liniger model
it is integrable and the ground-state energy E
can be determined by Bethe ansatz:
Mean-field works for
[3]:
is the ground-state of the nonlinear Schrodinger equation
in order to have a finite
energy per particle
with energy
[3] F. Calogero and A. Degasperis, Phys. Rev. A 11, 265 (1975)