1. 2010-09-02 Quantum Technologies Conference, Toruń 1 The project „Photonic implementations of quantum-enhanced technologies” is realized within the TEAM programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund (Programme Innovative Economy 2007-2013) Oscillating Spinor Solitons

2. OSCILLATING SPINOR SOLITONS Piotr Szańkowski Marek Trippenbach Eryk Infeld 2010-09-02 Quantum Technologies Conference, Toruń 2

3. Outline 2010-09-02 Quantum Technologies Conference, Toruń 3 Bose-Einstein Condensate in the optical trap Solitons in the spinor BEC Soliton collisions – creation of the Oscillatons Mathematical description of the Oscillatons Collisions of the Oscillatons

4. The system 2010-09-02 Quantum Technologies Conference, Toruń 4 Bose-Einstein Condensate at T=0 confined in the optical trap. Spins of trapped atoms are not frozen like in the magnetic trap! The spinor nature of the condensate can be manifested. The Hamiltonian of the system: the field operator of each spin component trapping potential spin 1 operators where a f is s-wave scattering length of total spin f channel

5. Spinor Gross-Pitaevskii equation 2010-09-02 Quantum Technologies Conference, Toruń 5 Mean-field approximation and variational principle leads to the spinor Gross-Pitaevskii equation the order parameter (spinor wave function) where and

6. Solitons 2010-09-02 Quantum Technologies Conference, Toruń 6 Generally, spinor GP equation, supports two kinds of soliton solutions: Polar soliton Ferromagnetic soliton rotation operator The spin of polar and ferromagnetic solitons

7. Complete Integrability of the spinor GP equation 2010-09-02 Quantum Technologies Conference, Toruń 7 For γ = 1 spinor GP equation is Completely Integrable. Solitons collide with each other elastically: There is no transfer of spin or momentum. Solitons retain their character after the collision. ( T. Tsuchida and M. Wadati, J. Phys. Soc. Jpn. 67, 1175 (1998) )

8. Soliton collision: creation of the Oscillatons 2010-09-02 Quantum Technologies Conference, Toruń 8 Target: Polar Soliton Bullet: Ferromagnetic Soliton

9. Spin transfer 2010-09-02 Quantum Technologies Conference, Toruń 9 In: Polar Soliton In: Ferromagnetic Soliton Out: Post - polar Oscillaton Out: Post - ferromagnetic Oscillaton In: Polar Soliton Out: Post - polar Oscillaton

10. Mathematical model of the Oscillatons 2010-09-02 Quantum Technologies Conference, Toruń 10 Oscillations of the components: Spin: Density: The Oscillaton equations:

11. Approximate solutions: Post-polar Oscillatons 2010-09-02 Quantum Technologies Conference, Toruń 11

12. Approximate solutions: Post-ferromagnetic Oscillatons 2010-09-02 Quantum Technologies Conference, Toruń 12

13. Solitons as the special case of Oscillatons 2010-09-02 Quantum Technologies Conference, Toruń 13 Polar soliton: Ferromagnetic soliton:

14. Oscillaton collisions 2010-09-02 Quantum Technologies Conference, Toruń 14 IN: Two Oscillatons created in different soliton collisions Oscillaton AliceOscillaton Bob Alice (lesser spin)Bob (greater spin)

15. Summary 2010-09-02 Quantum Technologies Conference, Toruń 15 Polar and ferromagnetic solitons in spinor Bose-Einstein condensate Inelastic collisions of solitons leads to creation of new kind of solutions – the Oscillatons The Oscillatons are similiar to solitons in Completely Integrable systems: they propagate in time without dispersion and retain their character after colliding with each other Solitons are special cases of the Oscillatons

16. Oscillations of the wave function components 2010-09-02 Quantum Technologies Conference, Toruń 16

17. Radiation 2010-09-02 Quantum Technologies Conference, Toruń 17

18. How to describe emerging Oscillatons with the model? 2010-09-02 Quantum Technologies Conference, Toruń 18 Step I: Find the angles Step II: Rotate the reference frame Step III: Find chemical potentials Step IV: Solve the Oscillaton equations

19. Oscillaton collisions 2010-09-02 Quantum Technologies Conference, Toruń 19 Case I: post-polar Oscillaton vs. post-polar Oscillaton Oscillaton (1):Oscillaton (2): Case II: post-polar Oscillaton vs. Post-ferro Oscillaton Post-polar Oscillaton (1):Post-ferro Oscillaton (2):