1. 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

2. Condensate diffraction from an optical grating
LENS, Florence

3. Energy and quasi-momentum are conserved

4. 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

5. 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

6. 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)

7. Array of Josephson junctions driven by a harmonic external field

8. 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)

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

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

11. 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 ?

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

14. The discrete nonlinear equation (DNL)

15. 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

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

17. 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

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

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

20. Bogoliubov spectrum

21. Sound

22. 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

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

24. 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:

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

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

27. 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

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

29. 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, (2002)

30. Newtonian Dynamics
Dynamical variational principle

31. Newtonian Dynamics
Group velocity
Effective force

33. 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

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

37. Quantum dynamics
Two-mode boson-Hubbard model

38. Number state representation

39. Coherent state representation

40. phase state representation

42. Two-modes base

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

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

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

50. Variational dynamics

55. Classical limit

56. Numerical solutions