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

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

Overview1. Purpose of Study2. Introduction to BECs3. Mean-field model for coupled BEC4. Variational Approach and Reduction to System

of ODEs5. Introduction to Continuous Dynamical

Systems.6. Steady states of reduced model, and

Bifurcations of parameters.7. Further reduction of the system to a Newton

style Equation, its bifurcations, and comparison of dynamics.

8. Conclusion.

PurposeThe 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 BECDe 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 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 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)

Wave Function

External Potential Term

KineticEnergy Term

TimeDependence Term

Plank’s Constant

Coupling ConstantBetween Species 2 and 2

Atom’s Mass

Interaction Term: Like Species

AtomicDensity

Interaction Term:Unlike Species

Species #1

Species #2

Coupling ConstantBetween Species 2 and 1

Renormalized EquationsBy rescaling time, space, the wave function, and coupling constant, we can undimensionalize the system and get:

Variational 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 speciesL12 = Interaction Lagrangian E = Mechanical energy of system

Wave Packet Trial Function• 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• 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.

dt

dpp jj

Equations for the Parameters of the Trial Function

Analogy to Lorenz Equation 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.

0

y

v

y

v

x

u

ux

p

t

u 21

.

uy

p

t

v 21

uBz

p

t

w 21

Tct

p

t

Tc pp

2

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

Continuous Dynamical Systems The fixed points, p*, on continuous

dynamical system are obtained by:

0

,,,, 1

1 dt

ppdpppp njnj

The system can be linearized by

Jpp *ppk

jjk dp

pdJ

(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

Phase PortraitsIn a system of ODEs, we can plot the orbits on a phase portraitWe 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

Separated State Fixed Points:

0dt

dW

dt

dE

dt

dD

dt

dB

dt

dA dt

dC

Mixed State Fixed Points:

0ED tC

Trial Function Matches Full Solution

Mixed State

Separated State

Bifurcation of Position, B

D = supercritical pitchfork bifurcation pointsA = transcritical pitchfork bifurcation pointsB = saddle node bifurcation points

Bifurcation of Amplitude, A

Bifurcation of Width, W

Miscibility Criteria

Dynamics of Mixed State

PDE (solid line)ODE (dashed line)

Dynamics of Separated StatePDE (solid line)ODE (dashed line)

Reduced Dynamics 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)

dB

dU

dt

Bd eff2

2

.22

*25.02

*222 constgeABB WB

Three Orbits:(a)Potential Function(b)Phase Portrait(c)Position vs. Time

Amplitude and Period vs. Initial Velocity

Period vs. g

Future Work: Studying Other States

Conclusion 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.