Transcript Bose-Einstein condensates in optical lattices and speckle

Bose-Einstein condensates
in optical lattices
and speckle potentials
Michele Modugno
Lens & Dipartimento di Matematica Applicata, Florence
CNR-INFM BEC Center, Trento
BEC Meeting, 2-3 May 2006
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).
Introduction
• Theory: 1D models
– 1D GPE: energetic/dynamical instability [Wu&Niu, 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)
Energetic (Landau) vs dynamical instability
 Stationary solution + fluctuations:
 Time dependent fluctuations:
 Linearized GPE -> Bogoliubov equations:
 Negative eigenvalues of M(p) -> (Landau)
instability (takes place in the presence of
dissipation, not accounted by GPE)
 Imaginary eigenvalues -> modes that
grow exponentially with time
A cylindrical condensate in a 1D lattice
3D Gross-Pitaevskii eq.
harmonic confinement + lattice
-> Bloch description in terms of periodic functions
Bogliubov equations -> excitation spectrum
p=0: excitation spectrum, sound velocity
Radial breathing
Axial phonons
Excitation spectrum (s=5): the lowest
two Bloch bands, 20 radial branches
Bogoliubov sound velocity of the lowest
phononic branch vs the analytic prediction
c=(m*)-1/2
Velocity of sound from a 1D effective model
•Factorization ansatz:
-> two effective 1D GP eqs:
axial -> m*, g*
radial -> µ(n)
g*
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Exact in the 1D meanfield (a*n1D <<1)
and TF limits (a*n1D >>1)
GPE vs 1D effective model
(s=0,5,10 from top to bottom)
P≠0: excitation spectrum, instabilities
Real part of the excitation spectrum
for p=0,0.25,0.5,0.55,0.75,1 (qB)
Phonon-antiphon resonance = a conjugate
pair of complex frequencies appears
-> resonance condition for two particles
decaying into two different Bloch states
E1(p+q) and E1(p-q) (non int. limit)
NPSE: a 1D effective model
3D->1D: factorization + z-dependent Gaussian ansatz for the radial component
-> 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
Stability diagrams
stable
Excitation quasimomentum
energetic instab.
en. + dyn. instab.
Max growth rate
BEC quasimomentum
Revisiting the Burger et al. experiment
 Dipole oscillations of an elongated BEC in magnetic trap + optical lattice (s=1.6)
– lattice spacing << axial size of the condensate ~ infinite cylinder
– small amplitude oscillations: well-defined quasimomentum states
-> Quantitative analisys of the unstable regimes
Center-of-mass velocity vs BEC quasimomentum.
Dashed line: experimental critical velocity
-> Breakdown of superfluidity (in the experiment)
driven by dynamical instability
+ 3D dynamical simulations (GPE)
Center-of-mass velocity vs time.
Density distribution as in experiments
(in 1D the disruption is more dramatic)
BECs in a moving lattice
By adiabatically raising a moving lattice -> project the BEC on a selected Bloch state
-> explore dynamically unstable states not accessibile by dipole motion
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
-> the most unstable mode imprints
the dynamics well beyond the linear regime
S=0.2
S=1.15
Beyond linear stability analysis: GPE dynamics
Density distribution after expansion:
theory (top) vs experiment @LENS
-> momentum peaks hidden in the
background?
Growth and (nonlinear) mixing of the
dynamically unstable modes
Recently observed at MIT
(G. Campbell et al.)
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 @ LENS
• Attractive condensates: dynamically unstable at p=0, can be
stabilized for p>0?
• Periodic vs random lattices……
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).
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
-> 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
A BEC in the speckle potential
BEC radial size < correlation length (10 µm) -> speckles ≈ 1D random potential
intensity distribution
~ exp(-I/<I>)
A typical BEC ground state in the harmonic+speckle potential
Dipole and quadrupole modes
Sum rules approach, the speckles potential as a small perturbation:
-> 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.
GPE dynamics
Dipole oscillations in the speckle potential (V0=2.5 —wz):
Sum rules vs GPE
Small amplitudes: coherent undamped oscillations.
Large amplitudes: the motion is damped and a breakdown of superfluidity occur.
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
-> localization as a classical effect due to the actual shape of the
potential
Quantum behavior of a single defect
Single defect ~
-> analytic solution (Landau&Lifschitz)
Incident wavepacket of momentum k:
quantum behaviour signalled by 2|0.5-T(k, a b
(a)-(b): potential well, (c)-(d): barrier
(a)-(c) a=0.2, (b)-(d) a=1.
Dark regions indicate complete reflection or
transmission, yellow corresponds to a 50%
transparency.
Current experiments (ß~1) : 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 (ß~0.1) quantum effects may eventually become predominant.
Conclusions & perspectives
• BECs in a shallow speckle potentials:
– Uncorrelated shifts of dipole and quadrupole frequencies
– Classical localization effects in 1D expansion
(no quantum reflection)
->reduce the correlation length in order to observe
Anderson-like localization effects
-> two-colored (quasi)random lattices