1. 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

2. 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

3. Introduction Theory: 1D models –1D GPE: energetic/dynamical instability [Wu&Niu, Pethick et al.], Bogoliubov excitations, sound propagation [Kr ä mer 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

4. 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

5. 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

6. p=0: excitation spectrum, sound velocity 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 Radial breathing Axial phonons

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*n 1D <<1) and TF limits (a*n 1D >>1) GPE vs 1D effective model (s=0,5,10 from top to bottom)

8. P≠0: excitation spectrum, instabilities Real part of the excitation spectrum for p=0,0.25,0.5,0.55,0.75,1 (q B ) Phonon-antiphon resonance = a conjugate pair of complex frequencies appears -> resonance condition for two particles decaying into two different Bloch states E 1 (p+q) and E 1 (p-q) (non int. limit)

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 Excitation quasimomentum BEC quasimomentum stable energetic instab. en. + dyn. instab. Max growth rate

11. 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)

12. 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=0.2S=1.15

13. Beyond linear stability analysis: GPE dynamics Growth and (nonlinear) mixing of the dynamically unstable modes Density distribution after expansion: theory (top) vs experiment @LENS -> momentum peaks hidden in the background? Recently observed at MIT (G. Campbell et al.)

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 speckles ≈ 1D random potential intensity distribution ~ exp(-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: 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.

19. 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 (V 0 =2.5 —  z ): Sum rules vs GPE

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. -> 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

21. 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,   

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