Introduction of the F1 spinor BEC

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Introduction of the F1 spinor BEC
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Spinor BEC
hyperfine spin
electron spin
nuclear spin
magnetic trapping (one-component, scalar)
BEC
optical trapping (multi-component, vector)
F1 spinor BEC 23Na, 87Rb,...
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Grand canonical energy-functional for the spinor
BEC (Ho, Ohmi)
N is a fixed number
(gslt0 ferromagnetic gsgt0 antiferromagnetic)
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order parameter
global phase
rotational symmetry in spin space
spin operators
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Ground state structure of spinor BEC
Define the normalized spinor by
Such that all spinors are degenerate with the
transformation
where
Euler angles
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Define
then the free energy can be expressed as
The free energy of K is minimized by
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Coupled GPE for spinor BEC
Time-dependent coupled GPE
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Time-independent coupled GPE
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Modification of GPE conservation of
magnetization
The spin-exchange interaction also preserves the
magnetization M, so we have two constraints for
the GPE
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The conservation of particle number and
magnetization is equivalent to introduce the
Lagrange multipliers ? and B in the free energy
N and M are both fixed numbers
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The corresponding GPEs are modified to
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Some interesting results
Ferromagnetic
3 identical decoupled equations
anti-ferromagnetic
2 coupled-mode equations
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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)
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Let
(Larmor frequency)
the Hamiltonian is invariant under the gauge
transformation
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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
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For
(polar state )
where
the condition
Note that when
cannot be fulfilled and we must choose F such
that
is as close to
as possible. This implies that
and thus the ground state is described by
(ferromagnetic state )
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For
the two different configurations coexist
(polar region)
(ferro region)
The phase boundary rb is determined by
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The free energy is given by
In the Thomas-Fermi limit, the minimization of
the free energy,
leads to
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Example
phase boundary
radius of atomic cloud
The total particle number and the chemical
potential is related by
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Derivation of hydrodynamic equations
collective excitations
polar region
hydrodynamic-like mode
ferro region
fluctuation of number density
Furthermore, we let
fluctuation of spin density
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Substituting
into the time-dependent GP equation
Upon linearization we obtain the hydrodynamic
equations in different regions
ferro region
(Stringari)
polar region
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Example
where
Now let
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To solve the coupled equations, let
and we obtain the recursion relation
Denote u? as the eigenvectors of A with
eigenvalues
and let
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Boundary condition requires that the series must
terminate at some interger k2n
and the solution is
The dispersion relations do not depend on the
magnetic field!
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?L-independence of solutions of polynomial-type
Consider a general quadratic potential
where the 3?3 matrix (?ij) is positively
definite.
Consider a polynomial solution
where Pk(r) and Qk(r) are homogeneous polynomials
of degree k.
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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
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In memory of Prof. W.-J. Huang (???) A friend
and a tutor
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Gross-Pitaeviski Hamlitonian for the spinor BEC
(Ho, Ohmi)
(gslt0 ferromagnetic gsgt0 antiferromagnetic)