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Bose-Einstein condensatesin optical lattices and speckle potentials
Michele ModugnoLens & Dipartimento di Matematica Applicata, FlorenceCNR-INFM BEC Center, Trento
BEC Meeting, 2-3 May 2006
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).
Part I: Effect of the transverse confinement on the dynamics of BECs in 1D optical lattices
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.]
• 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)
• Experiment: Burger et al. [PRL 86,4447 (2001)]:– breakdown of superfluidity under dipolar oscillations interpreted as Landau (energetic) instability
Energetic (Landau) vs dynamical instability
Negative eigenvalues of M(p) -> (Landau) instability (takes place in the presence of dissipation, not accounted by GPE)
Stationary solution + fluctuations: Time dependent fluctuations:
Linearized GPE -> Bogoliubov equations:
Imaginary eigenvalues -> modes that grow exponentially with time
A cylindrical condensate in a 1D lattice
-> Bloch description in terms of periodic functions
Bogliubov equations -> excitation spectrum
3D Gross-Pitaevskii eq.
harmonic confinement + lattice
p=0: excitation spectrum, sound velocity
Excitation spectrum (s=5): the lowesttwo Bloch bands, 20 radial branches
Bogoliubov sound velocity of the lowestphononic branch vs the analytic predictionc=(m*)-1/2
Radial breathing
Axial phonons
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Velocity of sound from a 1D effective model
•Factorization ansatz: -> two effective 1D GP eqs:
axial -> m*, g*
radial -> µ(n)
g*
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 spectrumfor 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 statesE1(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
Excitation quasimomentum
BEC quasimomentum
stable
energetic instab.
en. + dyn. instab.
Max growth rate
Revisiting the Burger et al. experiment
-> Quantitative analisys of the unstable regimes
+ 3D dynamical simulations (GPE)
-> Breakdown of superfluidity (in the experiment) driven by dynamical instability
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
Center-of-mass velocity vs BEC quasimomentum. Dashed line: experimental critical velocity
Center-of-mass velocity vs time.Density distribution as in experiments(in 1D the disruption is more dramatic)
BECs in a moving lattice
The (theoretical) growth rates show a peculiar behavior as a function of the band index and lattice heigth
By adiabatically raising a moving lattice -> project the BEC on a selected Bloch state-> explore dynamically unstable states not accessibile by dipole motion
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
Growth and (nonlinear) mixing of thedynamically unstable modes
Density distribution after expansion:theory (top) vs experiment @LENS-> momentum peaks hidden in thebackground?
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 instability3D 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:
Dipole and quadrupole frequency shifts for 100 different realizations of the speckle potential
-> uncorrelated shifts
random vs periodic: correlated shifts (top), but uncorrelated frequencies (bottom) that depend on the position of the condensate in the potential.
GPE dynamics
Small amplitudes: coherent undamped oscillations. Large amplitudes: the motion is damped and a breakdown of superfluidity occur.
Dipole oscillations in the speckle potential (V0=2.5 —z):
Sum rules vs GPE
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.
-> localization as a classical effect due to the actual shape of the potential
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
Quantum behavior of a single defect
(a)-(b): potential well, (c)-(d): barrier (a)-(c) =0.2, (b)-(d) =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.
Single defect ~ -> analytic solution (Landau&Lifschitz)
Incident wavepacket of momentum k: quantum behaviour signalled by 2|0.5-T(k,
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