Kink escape from a potential well created by an external

perturbation

LENCOS, July , 14-17 2009

Monica A. Garcia Ñustes

This talk is on based on a joint work with J. A. González, A. Sánchez and P. V. E. McClintock.

New Journal of Physics, 10 113015 (2008)

Kink escape from a potential

Outline

Outline

Introduction Stability conditionso Effective Potentialo Internal Modes

Tunneling Kink-Antikink pair creation All the ingredients together: Escape from a potential well Some experiments Conclusions

Kink escape from a potential Introduction

Introduction

Classically, escape over a finite barrier can occur trough the action of external perturbations: Noise assisted barrier crossing.

In quantum physics, a particle can escape from a potential well with sub-barrier energies by the mechanism of tunneling.

Escape from a metastable state of a dynamical system plays a important role in many classes of physical phenomena as

Stochastic ResonanceDirect diffusion in ratchets

Tunneling of fluxons in condensed matter

Kink escape from a potential Introduction

The model equation is:

If F(x)=0 we have the well known kink and antikink solutions.

Klein-Gordon modelParticle physics

Domain walls in ferromagnetsDislocations in crystals

Fluxons in Josephson Junctions

Kink escape from a potential

Effective PotentialIt is known that the external force creates an effective potential for the kink soliton. In fact, the zeros of F(x) are equilibrium positions for the kink.

Stability Conditions

Stability conditions

J. A. González and J. A. Holyst, Phys. Rev. B45 10338 (1992)A. Sanchez and A. R. Bishop, SIAM Rev. 40 579 (1998)

Kink escape from a potential

Following this idea, we can consider that the zero of F(x) represents a stable position for the kink if

For the antikink we have the opposite, the zero of F(x) a stable position if

Stability conditions

Intrinsically, this analysis describes the kink as a point particle. The external force create an effective potential of type V(xCM) where xCM is the coordinate of the kink center of mass.

Kink escape from a potential Stability conditions

Internal modesLet us consider perturbations over a static kink solution placed at an equilibrium position,

This analysis leads to a spectral problem of the form:

The eigenvalues Г corresponding to the soliton internal modes : Г0 represents the translational modeГi represents internal shape modesand a continuous spectrum that represents phonon modes.

J. A. González and J. A. Holyst, Phys. Rev. B45 10338 (1992)

Kink escape from a potential Stability conditions

In general, the stability condition for kink internal modes is given by

Now, let us compare both considerations about stability conditions.

By contrast, this analysis considerer the kink soliton as an extended object with an complicated internal behavior.

Kink escape from a potential Stability conditions

By example, if F(x) is given by,

The model equation has the following static solution,

Stability ConditionsPoint Particle Extended Object

4B2>1(1) 4B2>1

(2) Λ(Λ+1)<3/2B2

(3) Λ>1/2B2

When condition (2) is not fulfill the force has two additional zeros and when Λ<1/2B2 the translational mode Г0 is unstable. Condition (1) is not sufficient.

Kink escape from a potential Stability conditions

A physical meaning of previous results is the following: if the additional zeros of the force are closer to the kink center and interactions of kink wings with these zeros are sufficiently strong to make the whole kink move.

J. A. González A. Bellorín and L. E. Guerrero, Phys. Rev. E60 R37 (1999)O. M. Braun and Y. S. Kivshar, The Frenkel-Kontorova Model,

Kink escape from a potential Tunneling

TunnelingNow, we consider a force that creates an effective potential with two equilibrium points: an unstable position at x=0, and a stable one at x=-d (this force can be obtained in terms of FAB(x)) .

If the translational mode is unstable, the soliton will move to the right, crossing the barrier even if its center of mass is placed in the minimum of the potential and its initial velocity is zero.

Kink escape from a potential Tunneling

This phenomenon is only possible if the distance between the minimum and the maximum (where d is given by the expression below) of the potential well is less than the kink’s width.

Kink escape from a potential Kink-Antikink pair creation

Kink-Antikink pair creation

If this conditions is fulfill, the first internal shape mode is unstable

Kink escape from a potential Kink-Antikink pair creation

The development of the instability of the first internal shape mode (Г1) of the soliton leads to the break up of the kink and to a creation of a kink- antikink pair.

Kink escape from a potential Escape from a potential well

All the ingredients together:

Escape from a potential well

Kink escape from a potential Escape from a potential well

Let us put together all the ingredients (all the previous analytical results) and would have a qualitative scenario of the dynamics under the effect of the force.

Due the properties of the force, the stability problem can be reduced to three simpler problems that similar to those already discuss.

Therefore, in the neighborhood of an equilibrium position, the stability problem can be solved exactly.

Kink escape from a potential Escape from a potential well

d> kink’s width

Tunneling No tunneling

d< kink’s width

d >2 d < 2 and 1/10<B2<1/4

Kink escape from a potential Escape from a potential well

If 4B2<1 the position x=0 is unstable but if 10B2>1, the kink can move away without large deformations. But if 10B2<1, the first internal shape mode is unstable leading to decay of the kink into an antikink and two kinks. Let us consider the situation where d>2, so tunneling is impossible.

Kink escape from a potential Escape from a potential well

There is not escape by tunneling mechanism or by noise-assisted emission. The kink escapes by kink-antikink pair creation!!!

Kink escape from a potential Some experiments

Josephson Junctions are good physical objects for the observation of soliton dynamics.

There has been constructed devices in which details of the dynamics of individual fluxons could be observed.

In order to produce the effective potential, a Josephson junction can possess inhomogeneities (microshort) that act as a potential well where the fluxon is trapped.

Some experimental setups using Josephson junctions create a double-well potential. The height of the barrier and deepness of wells are controlled by the experimentalist.

Some experiments

H. Akoh, S. Sakai, A. Yagi and H. Hayakawa, IEEE Trans. Magn., 21 737 (1985)

Kink escape from a potential Some experiments

We expect that kink escape by kink-antikink pair creation could be observed in a similar experimental setup.

P.D. Shaju et al., Phys. Lett. A332 326 (2004)A. N. Price et al. , preprint 0807-0488v1

Kink escape from a potential

Kink escape from a potential Conclusions

We have shown a new mechanism of escape of particles from a potential via an antikink-kink pair creation. Our theory of the process is dynamical and we can follow in detail what happens in simulations.

We point to an experiment with Josephson Junction where we believe that the phenomenon can be observed.

Conclusions

Thank you