Matter-Wave Soliton Molecules

Transcription

1 Matter-Wave Soliton Molecules Usama Al Khawaja UAE University 6 Jan. 01 First International Winter School on Quantum Gases Algiers, January 1-31, 01

2 Outline Two solitons exact solution: new form Center-of-mass positions Phase difference Force of interaction Soliton molecule Scattering regimes: stability analysis Binding Mechanism Experimental realization Applications Conclusions 7 January 01

3 Soliton-Soliton force Bright solitons in a homogeneous medium e / cos( 1 ) 7 January 01 3

4 What is a soliton molecule? Realized experimentally with optical solitons: M. Stratmann et al. Phys. Rev. Lett 95, (005). 7 January 01 4

5 The Gross-Pitaevskii Equation (t) : arbitrary real function Integrability condition 0 - special cases : ( t) t c1 c cos( t) 7 January 01 5

6 Exact Two Solitons Solution Four parameters for each soliton: n ( x, t) : x x j j : cmj cmj ( t) : ( t) : normalization, phase, center-of-mass position, center-of-mass speed. j 1,, nonlinear shift [, ( t)], i 1,,3,4. i i n j 7 January 01 6

7 Exact Two Solitons Solution Main Features: solitons locations 7 January 01 7

8 Exact Two Solitons Solution Main Features: phases Note: phases are with respect to that of the first sech term 7 January 01 8

9 Solitons separation For resolvedsolitons: x d ( t) xcm( t) xcm 1( t) 1/ jj 7 January 01 9

10 Soliton-Soliton Force from the trapping potential phase contribution exponential decay Note : n s n 1 n 7 January 01 10

11 Soliton-Soliton Force Gordon s Formula - homogeneous background: - nearly equal solitons: (t) 0 n d 1 n n1 1 - small relative speed: v d v v1 1 nd ns g0 8 t 7 January 01 11

12 Soliton-Soliton Force Inhomogeneous background - Inhomogeneous background: - nearly equal solitons: n (t) 0 n d 1 n1 1 - small relative speed: v d v v1 1 d ns g0 n 8 t 7 January 01 1

13 Soliton Molecule for stationary solitons : v1 v 0 Eliminating t 7 January 01 13

14 Soliton Molecule equilibrium position: ( eq ) 0 for small oscillations : eq 7 January 01 14

15 Soliton Molecule spring constant: effective mass: coalescence condition : g0n s 1 7 January 01 15

16 Soliton Molecule g0n s.7 7 January 01 16

17 Soliton Molecule g0n s January 01 17

18 Soliton Molecule phase Arg[ ( x, t)] g0n s 1.5 g0n s. 7 7 January 01 18

19 Soliton Molecule Stability: Reflection from surfaces 7 January 01 19

20 Soliton Molecule Stability: Reflection from surfaces 7 January 01 0

21 Soliton Molecule Stability: Scattering by a potential barrier 7 January 01 1

22 Soliton Molecule Stability: Scattering by a potential barrier 7 January 01

23 Soliton Molecule Stability: potential barrier, changing velocity 7 January 01 3

24 Soliton Molecule Stability: Scattering by a potential well quantum reflection 7 January 01 4

25 Soliton Molecule Stability: Collision with other solitons 0 7 January 01 5

26 Soliton Molecule Stability: Collision with other solitons faster collision v=-0.04 slower collision v= January 01 6

27 Soliton Molecule Stability: Collision with other solitons 7 January 01 7

28 Binding Mechanism 7 January 01 8, ) cos( ) sin( tan ) cos( ) sin( tan ) (, ) cos( log 4 ) ( t t t t t n g t t d / 4 0,, 8 1,, 0 n n n n n n n g e g n d s d x s From the exact solution:

29 Binding Mechanism equations of motion 8 3 e 1 4 cos, 4 e 1 4 sin. 3 7 January 01 9

30 Binding Mechanism role of phase Solid curves: 0 = 4.0 Dashed curves: 0 = 4.5 n 1 = 0.1; n s = 1; g 0 = 1. 7 January 01 30

31 Binding Mechanism soliton-soliton interaction potential 7 January cos log 4, 4)cos / ( 0 0 ) ( 4 1 ) ( eq ss ss eq eq eq e e e e V d V Morse potential

32 Binding Mechanism soliton-soliton interaction potential Solid curve: numerical Dashed curve: analytical Parameters: n d = 0.; n s = 4.7; g 0 = 0.5; initial phase=0. 7 January 01 3

33 Binding Energy Binding energy= (energy of molecule)-(energy of infinitely separated solitons) Energy functional: E[ ] 1 x V ext ( x) 1 g 0 4 dx Binding energy: E b E[ molecule ] E[ single soliton] For the exact solution: E b 0 7 January 01 33

34 Stabilizing Soliton Molecules One method to stabilize the molecule is to oscillate the trapping potential strength: ( t) c c1 E b cos( ) 0 t E b [ ( t)] search for parameters of ( t) for which : E b 0. (under investigation) 7 January 01 34

35 Two-Dimensional Soliton Molecules Stability: energy scaling argument Let the typical size of the soliton be R. Normalizat ion: D 1 d r N ~ D / R Energy functional: Harmonic potential: E V ext 1 1 Vext ( r) g ( r) ~ r 7 January d D r

36 Two-Dimensional Soliton Molecules E 1 ~ R g 0 R 1 D R E3 D ~ 1 R R g 0 1 R 3 E D ~ 1 R R g 0 1 R 1 1 E1 D ~ R g 0 R R Not stable Metastable with trap Not stable need a trap Stable even without trap 7 January 01 36

37 Two-Dimensional Soliton Molecules force of interaction - at large separations D GPE: Asymptotic form of the wave function at large distance: D Interaction force: 3D Interaction force: 7 January 01 37

38 Two-Dimensional Soliton Molecules force of interaction - for all separations Locate two solitons off the trap center and record their separation. Two solitons separation Resultant force 7 January Interaction force

39 Two-Dimensional Soliton Molecules two solitons molecule in phase out of phase 7 January 01 39

40 Two-Dimensional Soliton Molecules seven solitons molecule in phase out of phase 7 January 01 40

41 Two-Dimensional Soliton Molecules soliton lattice in phase out of phase 7 January 01 41

42 Two-Dimensional Soliton Molecules soliton rings collision 7 January 01 4

43 Experimental proposal current experiment D. Dries, S. E. Pollack, J. M. Hitchcock, and R. G. Hulet, Phys. Rev. A 80, (010). 7 January 01 43

44 Experimental proposal our proposal 7 January 01 44

45 Experimental proposal 7 January 01 45

46 Experimental considerations 3D GPE: Scaled 1D GPE: Scaling: Length to Experimental values Time to 7 January 01 46

47 Simulation of the experiment 7 January 01 47

48 Effect of trap strength 7 January 01 48

49 Effect of switching time 7 January 01 49

50 Effect of initial speed 7 January 01 50

51 Gaussian barrier 7 January 01 51

52 Applications: Toda lattice of solitons Force on the jth soliton: Expanding up to second order in u j : To get the dispersion relation: The speed of phonons: 7 January 01 5

53 Applications: Soliton gas Equation of state: B = 7 January 01 53

54 Conclusions New form of the two solitons wavefunction allows for accurate analytic account of the force of interaction between two solitons for arbitrary solitons parameters Relative phase plays the role of the restoring force in the soliton molecule Soliton molecules are stable upon reflection from hard walls but less stable when reflected by softer ones The bond of the soliton molecule breaks upon collision with a slowly-incoming single soliton rather than by a faster one. Alternatively, a phase differnce of bewteen the incident soliton and the molecule leads to break up. 7 January 01 54

55 Conclusions Interaction potential between solitons is shown to be of molecular type and takes the form of a Morse potential. The equilibrium point depends on the initial separation and relative phase. Numerical simulations show that with a slight adjustment of the experiment of Dries et al., a soliton molecule can be realized experimentally. The new analytic form of solitons interaction potential opens the door for treating a host of intersting problems such as soliton gas or Toda lattice of solitons. 7 January 01 55