Presentazione di PowerPoint

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

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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)

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Tuning the interactions with optical lattices
s-wave scattering length
bosonic field
tight-binding Ansatz Jaksch et al. PRL (1998)
For large enough barrier height
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)
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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-Thou
less 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
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Discrete dynamics
Gross-Pitaevskii approximation
tight-binding Ansatz for the Gross-Pitaevskii
equation with an optical lattice A.T. A.
Smerzi, PRL (1998)
When V0gtgtm
Bright localized solitons also with repulsion
(agt0) Repulsionnegative effective mass ?
effective attraction
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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. 2003 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
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Stabilization of solitons by an optical lattice
(I)
Recent proposals to engineer 3-body interactions
Paredes et al., PRA 2007 -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
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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
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• 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

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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
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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, arXiv0801.0009
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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
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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)
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3D lattice without disorder
Single particle energies
The relation between the number of particles and
the chemical potential is
For large filling
Watsons integrals
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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
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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 Greens functions for
  N ?0
• Computing the self-energy
• New chemical potential (effective t larger,
  larger density of states)

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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)
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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
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3D lattice with disorder Results for an
incommensurate potential
Two lattices
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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
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Thank you!
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Berezinskii-Kosterlitz-Thouless transition in a
2D lattice
thermally driven vortex proliferation
central peak of the momentum distribution Good
description at finite T by an XY model
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)
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Some details on the diagrammatic expansion (I)
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Some details on the diagrammatic expansion (II)
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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 (cgt0)
With
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2-Body Contact Interactions
N2 Lieb-Liniger model
it is integrable and the ground-state energy E
can be determined by Bethe ansatz
Mean-field works for
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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)