Phase Separation and Dynamics of a Two Component Bose-Einstein Condensate
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Phase Separation and Dynamics of a Two Component Bose-Einstein Condensate. December 8 th , 2008 Rafael Navarro and Ron Caplan. Overview. Purpose of Study Introduction to BECs Mean-field model for coupled BEC Variational Approach and Reduction to System of ODEs

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Phase Separation and Dynamics of a Two Component Bose-Einstein Condensate

December 8th, 2008

Rafael Navarro and Ron Caplan


Overview Bose-Einstein Condensate

  • Purpose of Study

  • Introduction to BECs

  • Mean-field model for coupled BEC

  • Variational Approach and Reduction to System of ODEs

  • Introduction to Continuous Dynamical Systems.

  • Steady states of reduced model, and Bifurcations of parameters.

  • Further reduction of the system to a Newton style Equation, its bifurcations, and comparison of dynamics.

  • Conclusion.


Purpose Bose-Einstein Condensate

  • The goal of this work is to attain a global description of how two condensates different spin interact with each other.

  • We wish to give a criteria for miscibility (mixing) and frequency of oscillation between the two condensates.

  • Also, the stability of the system is studied.


Introduction to BEC Bose-Einstein Condensate

De Broglie: All particles are wave-like, with wavelength depended on momentum.

Heisenberg: Uncertainty in momentum and position related: x p = h

In a gas, there is an average distance between atoms called the scattering length, d.

x p= h

xp= h

x p = h


Properties of a bec
Properties of a BEC Bose-Einstein Condensate

  • In BECs, bosons occupy the same quantum mechanical ground state.

  • All atoms in the BEC act as one and move in unison.

  • The condensate displays wave properties which can be modeled using the Nonlinear Schrödinger Equation (NLS).

  • Two component BECs are when two sets of atoms, each in a different spin state are formed into a BEC together. They interact with each other, and typically repel each other.

  • The dynamics of a two-component BEC can be modeled using two coupled NLS equations.


BEC in a Quasi 1-D trap Bose-Einstein Condensate

  • The external is generated by a magnetic trap and is very narrow in the transverse direction x< y=z.

  • The system is quasi 1-dimensional and two degrees of freedom can be integrated out of the NLS.


Coupled Nonlinear Schrodinger Equations (NLS) Bose-Einstein Condensate

Time

Dependence

Term

Kinetic

Energy

Term

External

Potential

Term

Interaction Term: Like Species

Interaction Term:

Unlike Species

Species

#1

Species

#2

Wave Function

Atom’s

Mass

Coupling

Constant

Between

Species

2 and 2

Coupling

Constant

Between

Species

2 and 1

Plank’s

Constant

Atomic

Density


Renormalized equations
Renormalized Equations Bose-Einstein Condensate

By rescaling time, space,

the wave function, and

coupling constant,

we can undimensionalize

the system and get:


Variational model
Variational Bose-Einstein Condensate Model

  • The Lagrangian is a functional that represents the energy of the system.

  • When the variation of the Lagrangian is minimized, the optimum solution is obtained.

L1 = Lagrangian of first species L2 = Lagrangian second species

L12 = Interaction LagrangianE = Mechanical energy of system


Wave packet trial function
Wave Packet Trial Function Bose-Einstein Condensate

  • We propose a wave-packet trial function composed of a carrier wave packet.

  • The wave packet has amplitude A, position B, width W, phase C, frequency D, and frequency modulation E.


Euler lagrange equations
Euler-Lagrange Equations Bose-Einstein Condensate

  • The Lagrangian is evaluated for the trial function yielding:

  • Equations of motion (ODEs) can be obtained for each parameter of the trial function through the Euler Lagrange equations

  • p1= A, p2= B, p3= C, p4= D, p5= E, p6= W.



Analogy to lorenz equation
Analogy to Lorenz Equation Bose-Einstein Condensate

.

  • Lorentz’s system of three coupled nonlinear ordinary differential equations were obtained by approximating the Navier-Stokes equations, a set of five coupled partial differential equations.

  • In our case, we used the VA to obtainODEs describing the motion of our solution form the system of PDEs


Continuous dynamical systems
Continuous Dynamical Systems Bose-Einstein Condensate

  • The system can be linearized by

(J is called the Jacobian of the system.)

  • A fixed point is stable when real(i)<0, for all i=1,2,…,n and unstable otherwise. Here, i represents the eigenvalues of J.

  • Note the difference here compared to discrete dynamics - that our critical eigenvalue is 0 rather than 1

  • The fixed points, p*, on continuous dynamical system are obtained by:


Phase Portraits Bose-Einstein Condensate

In a system of ODEs, we can plot the orbits on a phase portrait

We plot a arrow-field which shows the direction that an orbit will take.

As an example, we look at different possibilities in a 2-dimentional system (J2x2):


Fixed points of our system
Fixed Points of our System Bose-Einstein Condensate

  • Mixed State Fixed Points:

  • Separated State Fixed Points:


Trial function matches full solution
Trial Function Matches Full Solution Bose-Einstein Condensate

Separated State

Mixed State


Bifurcation of position b
Bifurcation of Position, B Bose-Einstein Condensate

D = supercritical pitchfork bifurcation points

A = transcritical pitchfork bifurcation points

B = saddle node bifurcation points


Bifurcation of amplitude a
Bifurcation of Amplitude, A Bose-Einstein Condensate


Bifurcation of width w
Bifurcation of Width, W Bose-Einstein Condensate


Miscibility criteria
Miscibility Criteria Bose-Einstein Condensate


Dynamics of mixed state
Dynamics of Mixed State Bose-Einstein Condensate

PDE (solid line)ODE (dashed line)


Dynamics of separated state
Dynamics of Separated State Bose-Einstein Condensate

PDE (solid line)ODE (dashed line)


Reduced dynamics
Reduced Dynamics Bose-Einstein Condensate

  • It is possible to obtain a simple equation for the dynamics of the position of the condensate.

  • If we assume that the variations in the amplitude and width are small, the time dependent amplitude and width can be replaced by the steady state amplitude and width: A(t)A* and B(t) B*.

  • The motion can be described in terms of a potential, Ueff.

  • Ueff = (atom-atom interaction) + (atom-potential interaction)


  • Three Orbits: Bose-Einstein Condensate

  • Potential Function

  • Phase Portrait

  • Position vs. Time



Period vs g
Period vs. g Bose-Einstein Condensate


Future work studying other states
Future Work: Studying Other States Bose-Einstein Condensate


Conclusion
Conclusion Bose-Einstein Condensate

  • We have shown the reduction of a coupled system of PDEs into a system of ODEs using the variational approach with a trial function.

  • We performed stability analysis on the steady state solutions to the system of ODEs.

  • We found bifurcations and stability of different parameters, and found miscibility conditions.

  • The point of phase transition is predicted well by the reduced model, but the equilibrium value of the parameters near the phase transition point deviate from the PDE’s solution.

  • We then reduced the model to a Newton-style equation and numerically compared the reduced model, the system of ODEs and the PDEs and showed that generally, the ODEs and Newton equations are very good at describing the dynamics of the system for fully separated states and fully mixed states.


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