oscillating spinor solitons
DESCRIPTION
Oscillating Spinor Solitons. 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). - PowerPoint PPT PresentationTRANSCRIPT
Quantum Technologies Conference, Toruń1 2010-09-02
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
Quantum Technologies Conference, Toruń 2
OSCILLATING SPINOR SOLITONSPiotr SzańkowskiMarek TrippenbachEryk Infeld
2010-09-02
Quantum Technologies Conference, Toruń
3
Outline
2010-09-02
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
2010-09-02Quantum Technologies Conference, Toruń
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.
We will consider atoms with hyperfine spin F=1 (e.g. 87Rb).
The Hamiltonian of the system:
the field operator of each spin componenttrapping potentialspin 1 operatorswhere af is s-wave scattering length of total spin f channel
Quantum Technologies Conference, Toruń
5
Spinor Gross-Pitaevskii equation
2010-09-02
Mean-field approximation and variational principleleads to the spinor Gross-Pitaevskii equation
the order parameter (spinor wave function)
where
and
Quantum Technologies Conference, Toruń
6
Solitons
2010-09-02
Generally, spinor GP equation,
supports two kinds of soliton solutions:
Polar soliton
Ferromagneticsoliton
rotation operator
The spin of polar and ferromagnetic solitons
Quantum Technologies Conference, Toruń
7
Complete Integrability of the spinor GP equation
2010-09-02
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) )
What happens when γ is not equal to 1?
Quantum Technologies Conference, Toruń
8
Soliton collision: creation of the Oscillatons
2010-09-02
Target:Polar Soliton
Bullet:Ferromagnetic Soliton
Quantum Technologies Conference, Toruń
9
Spin transfer
2010-09-02
In:PolarSoliton
In: FerromagneticSoliton
Out:Post - polarOscillaton
Out: Post - ferromagneticOscillaton
What is the spin after the collision of two polar solitons?
In:PolarSoliton
In: PolarSoliton
Out:Post - polarOscillaton
Out: Post - polarOscillaton
Quantum Technologies Conference, Toruń
10
Mathematical model ofthe Oscillatons
2010-09-02
Oscillations of the components:
The frequency of the oscillations:
Spin:
Density:The Oscillaton equations:
Quantum Technologies Conference, Toruń
11
Approximate solutions:Post-polar Oscillatons
2010-09-02
Quantum Technologies Conference, Toruń
12
Approximate solutions:Post-ferromagnetic Oscillatons
2010-09-02
Quantum Technologies Conference, Toruń
13
Solitons as the special case of Oscillatons
2010-09-02
Polar soliton:
Ferromagnetic soliton:
14
Oscillaton collisions
2010-09-02Quantum Technologies Conference, Toruń
IN:Two Oscillatons created in different soliton collisions
Oscillaton Alice Oscillaton Bob
OUT:Two, altered Oscillatons!
Alice (lesser spin)Bob (greater spin)
15
Summary
2010-09-02Quantum Technologies Conference, Toruń
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 andretain their character after colliding with each other
Solitons are special cases of the Oscillatons