Optical lattices for ultracold atomic gases Andrea Trombettoni

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

Outlook Why using optical lattices? ü ü 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

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

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

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 2 D 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 2 D Bose gas

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

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 • Anderson localization vs. nonlinearity [Anker et al. , PRL 2005] [previous talk by G. Modugno] • Stabilization of solitons by an optical lattice LENS, Florence

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 1 D 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

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 small q

Outlook Why using optical lattices? ü ü 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

Bosons on a lattice with disorder: shift of the critical temperature filling total number of particles 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

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, ar. Xiv: 0801. 0009]

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

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

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

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

3 D lattice with disorder 3 D 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)

3 D 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)]

3 D lattice with disorder: Results for on-site disorder When a random on-site disorder (average zero, variance v 02) is present When both random-bond and random on-site disorder are present

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

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

Thank you!

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

Some details on the diagrammatic expansion (I)

Some details on the diagrammatic expansion (II)

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

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