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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).

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Oscillating Spinor Solitons

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Oscillating spinor solitons

Oscillating Spinor Solitons

The project „Photonic implementations of quantum-enhanced technologies” isrealizedwithin the TEAM programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund (ProgrammeInnovativeEconomy 2007-2013)

Quantum Technologies Conference, Toruń


Oscillating spinor solitons

Quantum Technologies Conference, Toruń

Oscillating Spinor Solitons

Piotr SzańkowskiMarek TrippenbachEryk Infeld


Outline

Outline

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

Quantum Technologies Conference, Toruń


The system

The system

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:

We will consider atoms with hyperfine spin F=1 (e.g. 87Rb).

the field operator of each spin component

where af is s-wave scattering length of total spin f channel

trapping potential

spin 1 operators

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Spinor gross pitaevskii equation

Spinor Gross-Pitaevskii equation

Mean-field approximation and variational principleleads to the spinor Gross-Pitaevskii equation

where

the order parameter (spinor wave function)

and

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Solitons

Solitons

Generally, spinor GP equation,

Changethesign of c0withFeshbachresonance

Lift trap adiabatically and obtainthesoliton

Theground state for c0 > 0

supports two kinds of soliton solutions:

Polar soliton

Ferromagneticsoliton

The spin of polar and ferromagnetic solitons

rotation operator

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Complete integrability of the spinor gp equation

Complete Integrability of the spinor GP equation

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.

What happens when γ is not equal to 1?

( T. Tsuchida and M. Wadati, J. Phys. Soc. Jpn. 67, 1175 (1998) )

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Soliton collision creation of the oscillatons

Soliton collision: creation of the Oscillatons

Bullet:Ferromagnetic Soliton

Target:Polar Soliton

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Spin transfer

Spin transfer

Out:Post - polarOscillaton

Out:Post - polarOscillaton

Out: Post - ferromagneticOscillaton

Out: Post - polarOscillaton

Before the collision:

Whatisthe spin afterthecollisionof two polar solitons?

Whatisthe spin afterthecollision?

In:PolarSoliton

In:PolarSoliton

In: FerromagneticSoliton

In: PolarSoliton

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Mathematical model of the oscillatons

Mathematical model ofthe Oscillatons

Spin:

Oscillations of the components:

Density:

The Oscillaton equations:

Amplitude of theoscillationsdepends on α!

The frequency of the oscillations:

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Approximate solutions post polar oscillatons

Approximate solutions:Post-polar Oscillatons

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Approximate solutions post ferromagnetic oscillatons

Approximate solutions:Post-ferromagnetic Oscillatons

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Solitons as the special case of oscillatons

Solitons as the special case of Oscillatons

Polar soliton:

Ferromagnetic soliton:

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Oscillaton collisions

Oscillatoncollisions

Alice (lesser spin)

Bob (greater spin)

IN:Two Oscillatons created in different soliton collisions

Oscillaton Alice

Oscillaton Bob

OUT:Two, altered Oscillatons!

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Summary

Summary

Polar and ferromagnetic solitons in spinor Bose-Einstein condensate

Inelastic collisions of solitons leads to creation of new kind of solutions – the Oscillatons

Solitons are special cases of 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

Quantum Technologies Conference, Toruń


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