a = 0 Density profile Relative phase Momentum distribution We would like to explore the transport properties of a BEC in a periodic potential. Therefore we better start From the simplest experiment we can think about: a condensate expanding in the lattice (NO Harmonic Field) This is a GPE numerical simulation of a condensate wave-packet, here is the density …. Notice that here we consider non-interacting atoms Momentum distribution

Condensate diffraction from an optical grating LENS, Florence

Energy and quasi-momentum are conserved

a> 0 Let’s try to understand what is happening to the expanding condensate. This is a numerical simulation with the inter-atomic interaction turned on. Notice that in the momentum space there are three peaks, that are what it would be seen if we switch off the trap, and observe the expansion of the condensate

Array of weakly coupled BEC Let’s came back to the GPE simulation. This is a zoom over few wells. There are a couple of important comments that are in order here: The density really seems to oscillate randomly, BUT, you see that the shape of the condensate trapped in each well does not change. In other words, there is here an indication that there are not internal excitations: each condensate in each well can be described by a Gaussian envelope, and the important dynamical variables are N and \phi. If this is true, as it happens to be, than we should introduce a further parameter in the problem, controlling the transfer of atoms among different well, as a function of the phase differences, the Tunneling rate…. Before to study this problem, let’s consider a different experiment

BEC expanding in a 1D optical lattice width of the wave-packet versus time .... we have here a paradox A. Trombettoni and A. Smerzi, PRL 86, 2353 (2000)

Array of Josephson junctions driven by a harmonic external field

The array is governed by a pendulum equation Oscillations of the three peaks of the interferogram. Blue circles: no periodic potential The array is governed by a pendulum equation F.S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Science 293, 843 (2001)

Small amplitude pendulum oscillations Triangles: GPE; stars: variational calculation of K Circles: experimental results Relation between the oscillation frequency and the tunneling rate

Breakdown of Josephson oscillations The interwell phase coherence breaks down for a large initial displacement of the BEC center of mass

Questions: 1) Why the interaction can break the inter-well phase coherence of a condensate at rest confined in a periodic potential ? 2) Why a large velocity of the BEC center of mass can break the inter-well phase coherence of a condensate confined in a periodic potential and driven by a harmonic field ? ...of course transport phenomena are complicate stuff to study, as Gary Larson has nicely stated Which are the transport properties of BEC in periodic potentials ?

Nonlinear tight-binding approximation Replace in GPE and integrate over spatial degrees of freedom Dynamical equations for atom numbers and phases in each well

The discrete nonlinear equation (DNL)

Array of weakly coupled pancakes, cigars, ellipsoidal condensates The shape and effective dimensionality of each condensate depends on a balance between the local interaction chemical potential and trapping frequencies Array of weakly coupled pancakes, cigars, ellipsoidal condensates

Bloch energies & effective masses Effective masses depend on the height of the inter-well barriers and on the density

Bloch energy and chemical potential Bloch states: Bloch energy and chemical potential Masses Effective masses depend on the height of the inter-well barriers and on the density

Bloch energies, effective masses & velocities How/which mass and velocity enter in the dynamics ?

Bogoliubov spectrum Replace in the DNL After linearization, retrieve the dispersion relation

Bogoliubov spectrum

Sound

Dynamical instability The amplitude of the perturbation modes grows exponentially fast, dissipating the energy of the large amplitude wave-packet No dynamical instabilities New mechanism for the breakdown of superfluidity of a BEC in a periodic potential

Energetic instability cfr. with the free (V=0) limit: Landau criterion for breakdown of superfluidity

Landau criterion for Superfluidity Gross-Pitaevskii equation with a defect: Vdef =V0 θ(t)δ(x) Expansion of the wave-function in terms of the quasi-particle basis:

Orthogonality and Symmetry conditions with up and vp satisfying the Bogoliubov-De Gennes equations Bogoliubov frequency

Quasi-particle amplitude For small defects the quasi-particles occupations are small compared to the condensate mode: Landau critical velocity vc=c

Landau criterion for Superfluidity Normal fluid: The presence of the defect causes dissipation and quasi-particles creation: growing of the thermal fraction. Laser beam BEC Critical velocity Superfluid the defect does not affect the motion of the condensate which moves without dissipations. C.Raman et al., Phys. Rev. Lett., Vol. 83, No. 13

Energetic vs. dynamical instability EI always sets in before the DI EI DI & EI stable DI & EI stable EI

Breakdown of superfluidity for a BEC driven by a harmonic field Quasi-momentum vs. time for three different initial displacements: 40, 80, 90 sites Density at t=0,20,40 ms as a function of the Position. Initial displacements: 50, 120 sites A. Smerzi, A. Trombettoni, P.G. Kevrekidis, A.R. Bishop, PRL 89, 170402 (2002)

Newtonian Dynamics Dynamical variational principle

Newtonian Dynamics Group velocity Effective force

Bloch oscillations Green line: a=0 Blue line: a>0 Atoms are condensed in the optical and magnetic fields. The harmonic confinement is instantaneously removed along the x direction. 3. A linear potential is superimposed to the system Green line: a=0 Blue line: a>0

Damping of Bloch Oscillations Solid line: Analytical Dashed line: Numerical A. Trombettoni and A. Smerzi, PRL 86, 2353 (2000)

Quantum dynamics Two-mode boson-Hubbard model

Number state representation

Coherent state representation

phase state representation

Two-modes base

Fock states can be seen as a superposition of phase states with random phase

The lowest N+1 eigenenergies are exactly the N+1 eigenenergies of the two-mode boson-Hubbard Hamiltonian

Quantum phase model The QPM describes the Fock regime and part of the Josephson regime

Variational dynamics

Classical limit

Numerical solutions