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