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Bose-Einstein condensates in optical lattices and speckle potentials

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.

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Bose-Einstein condensates in optical lattices and speckle potentials

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  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. 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.] Introduction • 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 -> 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

  5. 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

  6. 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

  7. 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)

  8. P≠0: excitation spectrum, instabilities 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) Real part of the excitation spectrum for p=0,0.25,0.5,0.55,0.75,1 (qB)

  9. 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

  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 (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 + 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 -> Breakdown of superfluidity (in the experiment) driven by dynamical instability

  12. 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 S=0.2 S=1.15 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 -> the most unstable mode imprints the dynamics well beyond the linear regime

  13. Beyond linear stability analysis: GPE dynamics Density distribution after expansion:theory (top) vs experiment @LENS -> 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 @ LENS • Attractive condensates: dynamically unstable at p=0, can be stabilized for p>0? • 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 -> 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 < correlation length (10 µm) -> speckles ≈ 1D random potential intensity distribution ~ exp(-I/<I>) A typical BEC ground state in the harmonic+speckle potential

  18. Dipole and quadrupole modes Sum rules approach, the speckles potential as a small perturbation: -> uncorrelated shifts Dipole and quadrupolefrequency shifts for 100 different realizations of the speckle potential randomvs 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 (V0=2.5—wz): Sum rulesvsGPE 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 -> localization as a classical effect due to the actual shape of the potential

  21. 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.

  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) ->reduce the correlation length in order to observe Anderson-like localization effects -> two-colored (quasi)random lattices

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