1. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 20081 2D S o l it o n s in Dipolar BECs 1 I. Tikhonenkov, 2 B. Malomed, and 1 A. Vardi 1 Department of Chemistry, Ben-Gurion University 2 Department of Physical Electronics, School of Electrical Engineering, Tel-Aviv University

2. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 20082 Dilute Bose gas at low T Contact pseudopotential

3. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 20083 Gross-Pitaevskii description Lowest order mean-field theory: Condensate order-parameter Gross-Pitaevskii energy functional: minimize E GP under the constraint: Gross-Pitaevskii (nonlinear Schr ö dinger) equation:

4. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 20084 Variational Calculation Evaluation of the E GP in an harmonic trap, using a gaussian solution with varying width b. Kinetic energy per-particle varies as 1/b 2 - dispersion. Nonlinear interaction per-particle varies as gn - g/b 3 in 3D, g/b in 1D. In 1D with g<0, kinetic dispersion can balance attraction and arrest collapse.

5. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 20085 Solitons Localized solutions of nonlinear differential equations. Result in from the interplay of dispersive terms and nonlinear terms. Propagate long distances without dispersion. Collide without radiating. Not affected by their excitations.

6. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 20086 Zero-temperature BEC solitons NLSE in 1D with attractive interactions (g<0), no confinement Posesses self-localized sech soliton solutions: Bright soliton: Healing length at x=0Chemical potential of a bright soliton

7. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 20087 Zero-temperature BEC solitons Attractive interactions, (self-focusing nonlinearity) x time No interactions, matter wave dispersion x time

8. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 20088 (1)Prepare BEC (static) in the trap (2) Turn off the trap and let evolve (3) Turn off both the trap and interactions (Feshbach mechanism) L. Khaykovich et al. Science 296, 1290 (2002). Observation of BEC bright solitons

9. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 20089 Observation of BEC solitons Dark solitons by phase imprinting: J. Denschlag et al., Science 287, 5450 (2000). Bright solitons L. Khaykovich et al. Science 296, 1290 (2002). Bright soliton train: K. E. Strecker et al., Nature 417, 150 (2002).

10. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200810 Instability of 2D solitons without dipolar-interaction - characteristic width of a 2D BEC wavefunction expansioncollapse is monotonic in

11. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200811 Dipole-dipole interaction   vacuum permittivity d - magnetic/electric dipole moment

12. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200812 Units

13. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200813 2D Bright solitons in dipolar BECs P. Pedri and L. Santos, PRL 95, 200404 (2005)

14. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200814 Manipulation of dipole-dipole interaction In order to stabilize 2D solitary waves in the PS configuration, it is necessary to reverse dipole- dipole behavior, so that side-by-side dipoles attract each other and head-to-tail dipoles repell one another. The total dipolar interaction is attractive at L  L z. There is a maximum in E(L  , hence no soliton.

15. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200815 The magnetic dipole interaction can be tuned, using rotating fields from + V d at , to - V d /2 at  The maximum becomes a minimum and 2D bright SWs can be found, provided that the dipole term is sufficiently strong to overcome the kinetic+contact terms, i.e. Or, for Manipulation of dipole-dipole interaction S. Giovanazzi, A. Goerlitz, and T. Pfau, PRL 89, 130401 (2002)

16. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200816 E  for confinement along the dipolar axis z, gaussian ansatz, g=500

17. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200817 Dipolar axis in the 2D plane I. Tikhonenkov, B. A. Malomed, and AV, PRL 100, 090406 (2008)

18. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200818 Dipolar axis in the 2D plane For g d > 0 stable self trapping along the dipolar axis z: y z x

19. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200819 For g d > 0, what happens along x ? Self trapping along x is enabled by the interplay of 1/L x 2 kinetic dispersion and -1/L x dipolar attraction y z x y z x

20. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200820 E  for confinement perpendicular to the dipolar axis

21. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200821 3D Propagation and stability

22. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200822 Driven Rotation Deviation from  /2 rotated soliton at t=  /2 

23. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200823 Experimental realization 52 Cr (magnetic dipole moment d=6  B ) Dipolar molecules (electric dipole of ~0.1-1D) For g,g d > 0 :

24. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200824 Conclusions 2D bright solitons exist for dipolar alignment in the free-motion plane. For this configuration, no special tayloring of dipole-dipole interactions is called for. The resulting solitary waves are unisotropic in the 2D plane, hence interesting soliton collision dynamics.

25. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200825 I n c o h e r e n t matter-wave S o l it o n s 1,2 H. Buljan, 1 M. Segev, and 3 A. Vardi 1 Department of Physics, The Technion 2 Department of Physics, Zagreb Univesity 3 Department of Chemistry, Ben-Gurion University

26. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200826 Prepared (static) BEC partially condensed Condensed particles Thermal cloud Trap OFF → nonequilibrium dynamics ? (1) Thermal cloud (and vice versa) (2) Condensate depletion during dynamics BEC-soliton dynamics affected by What about quantum/thermal fluctuations ?

27. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200827 T=0 - Bogoliubov theory (ask Nir) Want to calculate zero temperature fluctuations. Separate: condensatefluctuations retain quadratic fluctuation terms and add N 0 constraint:

28. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200828 T=0 - Bogoliubov theory Bogoliubov transformation: v(x)

29. 29 Bogoliubov spectrum of a bright soliton linearize about a bright soliton solution:

30. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200830 Transmittance: Bogoliubov spectrum of a bright soliton Scattering without reflection Bogoliubov quasiparticles scatter without reflection on the soliton (B. Eiermann et al., PRL 92, 230401 (2004), S. Sinha et al., PRL 96, 030406 (2006)).

31. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200831 Limitations on Bogoliubov theory The condensate number is fixed - no backreaction The GP energy is treated separately from the fluctuations Due to exchange energy in collisions between condensate particles and excitations, it may be possible to gain energy By exciting pairs of particles from the condensate ! direct + exchange no exchange ! pair production

32. 32 TDHFB approximation  Heisenberg eq. of motion for the Bose field operator  separate, like before  retain quadratic terms in the fluctuations, to obtain coupled equations for: Fluctuations Condensate order-parameter Pair correlation functions - single particle normal and anomalous densities

33. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200833 TDHFB approximation Condensate density Normal noncondensate density Anomalous noncondensate density (e.g., Proukakis, Burnett, J. Res. NIST 1996, Holland et al., PRL 86 (2001))

34. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200834 Initial Conditions - static HFB solution in a trap Bose distribution Fluctuations do not vanish even at T=0, quantum fluct.

35. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200835 Dynamics - TDHFB equations Initial conditions:

36. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200836  Quasi 1D geometry x   N = 2.2 10 4 7 Li atoms  ω  = 4907 Hz ; a  = 1.3 μm  ω x = 439 Hz ; a x = 4.5 μm  Na 3D = -0.68 μm  Parameters close to experiment:  TDHFB can be used only for limited time-scales:  T evolution ω  << T collisional ω  ~ 10 4 System Parameters

37. 37 GPE evolution, mechanical stability Without interactions matter wave dispersion TDHFB vs. GP Dynamical condensate depletion PRL 80, 180401 (2005) TDHFB: pairing

38. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200838 Correlations Mixture of condensed and noncondensed atoms Re μ(x 1,x 2,t) Im μ(x 1,x 2,t) Re μ(x 1,x 2,t=0) Incoherent matter-wave solitons

39. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200839 Number conservation Energy conservation condensate fraction thermal population condensate kinetic energy thermal cloud kinetic energy total interaction energy Number and energy conservation

40. Ultrafast-UltracoldEin Gedi, Feb. 24-29, 200840 Conclusions Dynamics of a partially condensed Bose gas calculated via a nonlinear TDHFB model Noncondensed particles (thermal/quantum) affect the dynamics of BEC solitons Pairing instability - dynamical depletion of a BEC with attractive interactions Incoherent matter-wave solitons constituting both condensed and noncondensed particles Analogy with optics: Coherent light in Kerr media Ξ zero-temperature BEC Partially (in)coherent light in Kerr media Ξ partially condensed BEC