Bose-Einstein condensates in optical lattices and speckle potentials

1
Bose-Einstein condensatesin optical lattices
and speckle potentials

• Michele Modugno
• Lens Dipartimento di Matematica Applicata,
  Florence
• CNR-INFM BEC Center, Trento

BEC Meeting, 2-3 May 2006
2
Part I Effect of the transverse confinement on
the dynamics of BECs in 1D optical lattices

• A) Energetic/dynamical instability
• M. Modugno, C. Tozzo, and F. Dalfovo, Phys. Rev.
  A 70, 043625 (2004)
• Phys. Rev. A 71, 019904(E) (2005).
• L. Fallani, L. De Sarlo, J. E. Lye, M. Modugno,
  R. Saers, C. Fort, and M. Inguscio,
• Phys. Rev. Lett. 93, 140406 (2004).
• L. De Sarlo, L. Fallani, C. Fort, J. E. Lye, M.
  Modugno, R. Saers, and M. Inguscio,
• Phys. Rev. A 72, 013603 (2005).

B) Sound propagation M. Kraemer, C. Menotti, and
M. Modugno, J. Low Temp. Phys 138, 729 (2005).
3
Introduction

• Theory 1D models
• 1D GPE energetic/dynamical instability WuNiu,
  Pethick et al., Bogoliubov
  excitations, sound propagation Krämer et al.
• DNLSE (tight binding) modulational (dynamical)
  instability Smerzi et al.

• Experiment Burger et al. PRL 86,4447 (2001)
• breakdown of superfluidity under dipolar
  oscillations interpreted as Landau (energetic)
  instability

• Effect of the transverse confinement ?
• Need for a framework for quantitative comparison
  with experiments both in weak anf tight binding
  regimes
• Clear indentification of dynamical vs energetic
  instabilities
• Role of dimensionality on the dynamics (3D vs 1D)

4
Energetic (Landau) vs dynamical instability

• Stationary solution fluctuations

• Time dependent fluctuations

• Linearized GPE -gt Bogoliubov equations

• Negative eigenvalues of M(p) -gt (Landau)
  instability (takes place in the presence of
  dissipation, not accounted by GPE)

• Imaginary eigenvalues -gt modes that grow
  exponentially with time

5
A cylindrical condensate in a 1D lattice
3D Gross-Pitaevskii eq.
harmonic confinement lattice
-gt Bloch description in terms of periodic
functions
Bogliubov equations -gt excitation spectrum
6
p0 excitation spectrum, sound velocity
Radial breathing Axial phonons
Excitation spectrum (s5) the lowest two Bloch
bands, 20 radial branches
Bogoliubov sound velocity of the lowest phononic
branch vs the analytic prediction c(?m)-1/2
7
Velocity of sound from a 1D effective model

• Factorization ansatz

-gt two effective 1D GP eqs
axial -gt m, g
radial -gt µ(n)
g
Exact in the 1D meanfield (an1D ltlt1) and TF
limits (an1D gtgt1)
GPE vs 1D effective model (s0,5,10 from top to
bottom)
8
P?0 excitation spectrum, instabilities
Phonon-antiphon resonance a conjugate pair of
complex frequencies appears -gt resonance
condition for two particles decaying into two
different Bloch states E1(pq) and E1(p-q) (non
int. limit)
Real part of the excitation spectrum for
p0,0.25,0.5,0.55,0.75,1 (qB)
9
NPSE a 1D effective model
3D-gt1D factorization z-dependent Gaussian
ansatz for the radial component
-gt change in the functional form of
nonlinearity (works better that a simple
renormalization of g) Effect of the transverse
trapping through a residual axial-to-radial
coupling
Same features of the ?0 branch of GPE
10
Stability diagrams
stable
energetic instab.
Excitation quasimomentum
en. dyn. instab.
Max growth rate
BEC quasimomentum
11
Revisiting the Burger et al. experiment

• Dipole oscillations of an elongated BEC in
  magnetic trap optical lattice (s1.6)
• lattice spacing ltlt axial size of the condensate
  infinite cylinder
• small amplitude oscillations well-defined
  quasimomentum states

• -gt Quantitative analisys of the unstable regimes

3D dynamical simulations (GPE)
Center-of-mass velocity vs time. Density
distribution as in experiments (in 1D the
disruption is more dramatic)
Center-of-mass velocity vs BEC quasimomentum.
Dashed line experimental critical velocity
-gt Breakdown of superfluidity (in the experiment)
driven by dynamical instability
12
BECs in a moving lattice
By adiabatically raising a moving lattice -gt
project the BEC on a selected Bloch state -gt
explore dynamically unstable states not
accessibile by dipole motion
S0.2
S1.15
The (theoretical) growth rates show a peculiar
behavior as a function of the band index and
lattice heigth
Similar shapes are found in the loss rates
measured in the experiment
-gt the most unstable mode imprints the dynamics
well beyond the linear regime
13
Beyond linear stability analysis GPE dynamics
Density distribution after expansiontheory
(top) vs experiment _at_LENS -gt momentum peaks
hidden in the background?
Recently observed at MIT (G. Campbell et al.)
Growth and (nonlinear) mixing of thedynamically
unstable modes
14
Conclusions perspectives

• Effects of radial confinement on the dynamics of
  BECs
• Proved the validity of a 1D approch for sound
  velocity
• Dynamical vs Energetic instability
• 3D GPE linear stability analysis framework for
  quantitave comparison with experiments
• Description of past and recent experiments _at_ LENS
• Attractive condensates dynamically unstable at
  p0, can be stabilized for pgt0?
• Periodic vs random lattices

15
Part II BECs in random (speckle) potentials

• M. Modugno, Phys. Rev. A 73 013606 (2006).
• J. E. Lye, L. Fallani, M. Modugno, D. Wiersma, C.
  Fort, and M. Inguscio, Phys. Rev. Lett. 95,
  070401 (2005).
• C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M.
  Modugno, D. S. Wiersma, and M. Inguscio, Phys.
  Rev. Lett. 95, 170410 (2005).

16
Introduction

• Disordered systems rich and interesting
  phenomenology
• Anderson localization (by interference)
• Bose glass phase (from the interplay of
  interactions and disorder)
• BECs as versatile tools to revisit condensed
  matter physics -gt promising tools to engineer
  disordered quantum systems
• Recent experiments with BECs speckles
• Effects on quadrupole and dipole modes
• localization phenomena during the expansion in a
  1D waveguide
• Effects of disorder for BECs in microtraps

17
A BEC in the speckle potential
BEC radial size lt correlation length (10 µm) -gt
speckles 1D random potential
intensity distribution exp(-I/ltIgt)
A typical BEC ground state in the
harmonicspeckle potential
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Dipole and quadrupole modes
Sum rules approach, the speckles potential as a
small perturbation
-gt uncorrelated shifts
Dipole and quadrupole frequency shifts for 100
different realizations of the speckle potential
random vs periodic correlated shifts (top), but
uncorrelated frequencies (bottom) that depend on
the position of the condensate in the potential.
19
GPE dynamics
Dipole oscillations in the speckle potential
(V02.5 wz)
Sum rules vs GPE
Small amplitudes coherent undamped oscillations.
Large amplitudes the motion is damped and a
breakdown of superfluidity occur.
20
Expansion in a 1D waveguide

• red-detuned speckles vs periodic
• almost free expansion of the wings (the most
  energetic atoms pass over the defects)
• the central part (atoms with nearly vanishing
  velocity) is localized in the initially occupied
  wells
• intermediate region acceleration across the
  potential wells during the expansion
• The same picture holds even in case of a single
  well.

• blue-detuned speckles (Aspect experiments)
• reflection from the highest barriers that
  eventually stop the expansion
• the central part gets localized, being trapped
  between high barriers

-gt localization as a classical effect due to the
actual shape of the potential
21
Quantum behavior of a single defect
Single defect
-gt analytic solution (LandauLifschitz)
Incident wavepacket of momentum k quantum
behaviour signalled by 20.5-T(k, a?b??
(a)-(b) potential well, (c)-(d) barrier
(a)-(c) a0.2, (b)-(d) a1. Dark regions
indicate complete reflection or transmission,
yellow corresponds to a 50 transparency.
Current experiments (ß1) quantum effects only
in a very narrow range close to the top of the
barrier or at the well border. By reducing the
length scale of the disorder by an order of
magnitude (ß0.1) quantum effects may eventually
become predominant.
22
Conclusions perspectives

• BECs in a shallow speckle potentials
• Uncorrelated shifts of dipole and quadrupole
  frequencies
• Classical localization effects in 1D expansion
• (no quantum reflection)
• -gtreduce the correlation length in order to
  observe
• Anderson-like localization effects
• -gt two-colored (quasi)random lattices