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Introduction of the F=1 spinor BEC

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Introduction of the F=1 spinor BEC

郭西川

國立彰化師範大學物理系

Spinor BEC

hyperfine spin

electron spin

nuclear spin

magnetic trapping (one-component, scalar)

BEC

optical trapping (multi-component, vector)

F=1 spinor BEC :23Na, 87Rb,...

Grand canonical energy-functional for the spinor BEC (Ho, Ohmi)

N is a fixed number

(gs<0: ferromagnetic; gs>0: antiferromagnetic)

order parameter:

global phase

rotational symmetry in spin space

spin operators:

Ground state structure of spinor BEC

Define the normalized spinor by

Such that all spinors are degenerate with the transformation

where

Euler angles

Define

then the free energy can be expressed as

The free energy of K is minimized by:

Coupled GPE for spinor BEC

Time-dependent coupled GPE

Time-independent coupled GPE

Modification of GPE – conservation of magnetization

The spin-exchange interaction also preserves the magnetization M, so we have two constraints for the GPE

The conservation of particle number and magnetization is equivalent to introduce the Lagrange multipliers and B in the free energy

N and M areboth fixed numbers

The corresponding GPE’s are modified to

Some interesting results

Ferromagnetic:

3 identical decoupled equations

anti-ferromagnetic:

2 coupled-mode equations

Field-induced phase segregation of the condensate configuration

the spin configuration of

Using the inequality

the ground state can be constructed by minimizing the spin-dependent part of the Hamiltonian

Case 1:

Choose

However, since

(ferromagnetic state)

Let

(Larmor frequency)

the Hamiltonian is invariant under the gauge transformation

Case 2:

so that we have

Choose

The minimum is achieved if

We may assume that

0 and is real but since

which is in contraction with the condition

and we must conclude that

For

(polar state )

where

the condition

Note that when

cannot be fulfilled and we must chooseFsuch that

is as close to

as possible. This implies that

and thus the ground state is described by

(ferromagnetic state )

For

the two different configurations coexist:

(polar region)

(ferro region)

The phase boundary rb is determined by

The free energy is given by

In the Thomas-Fermi limit, the minimization of the free energy,

leads to

Example:

phase boundary

radius of atomic cloud

The total particle number and the chemical potential is related by

Derivation of hydrodynamic equations

collective excitations

polar region

hydrodynamic-like mode

ferro region

fluctuation of number density

Furthermore, we let

fluctuation of spin density

Substituting

into the time-dependent GP equation

Upon linearization we obtain the hydrodynamic equations in different regions:

ferro region

(Stringari)

polar region

Example:

where

Now let

To solve the coupled equations, let

and we obtain the recursion relation

Denote u as the eigenvectors of A with eigenvalues

and let

Boundary condition requires that the series must terminate at some interger k=2n

and the solution is

The dispersion relations do not depend on the magnetic field!

L-independence of solutions of polynomial-type

Consider a general quadratic potential

where the 33 matrix (ij) is positively definite.

Consider a polynomial solution

where Pk(r) and Qk(r) are homogeneous polynomials of degree k.

Note that if Pk(r) is a polynomial of x,y,z with degree k

is a polynomial of degree k-2

is a polynomial of degree k

Clearly, terms of even degree are decoupled from terms of odd degree. So we may assume

Collecting terms of degree n on both sides

The obtained frequency does not depend on L

In memory of Prof. W.-J. Huang (黃文瑞)

—A friend and a tutor

Gross-Pitaeviski Hamlitonian for the spinor BEC(Ho, Ohmi)

(gs<0: ferromagnetic; gs>0: antiferromagnetic)