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Spinor BECs are BECs with spin internal degrees of freedom.As a result of which the ground state is a vector. For Spin-1 BEC = _
1 0 1
The state that a spinor-BEC takes upon being cooled below the critcal temperature depends upon the kind of interaction between the spins and external parameters such as the magnetic _eld that a_ect the order parameter
Arthur, Doe About Beamer September 2, 2017 2 / 28
Spin-1 BEC
The ground states possible for a spin-1 BEC are: Nematic state (NS) : ~u =
0 u
0
0
_
Arthur, Doe About Beamer September 2, 2017 3 / 28
Spin-1 BEC
The ground states possible for a spin-1 BEC are: Nematic state (NS) : ~u =
0 u
0
0
_ Magnetic state (MS) : ~u =
u
1
0 0
_ ; ~u =
0 0 u
1
_
Arthur, Doe About Beamer September 2, 2017 3 / 28
Spin-1 BEC
The ground states possible for a spin-1 BEC are: Nematic state (NS) : ~u =
0 u
0
0
_ Magnetic state (MS) : ~u =
u
1
0 0
_ ; ~u =
0 0 u
1
_ Mixed state (2C) : ~u =
u
1
0 u
1
_
Arthur, Doe About Beamer September 2, 2017 3 / 28
Spin-1 BEC
The ground states possible for a spin-1 BEC are:
Nematic state (NS) : ~u =
0 u
0
0
_ Magnetic state (MS) : ~u =
u
1
0 0
_ ; ~u =
0 0 u
1
_ Mixed state (2C) : ~u =
u
1
0 u
1
_ Three component state (3C) : ~u =
u
1
u
0
u
1
_
Arthur, Doe About Beamer September 2, 2017 3 / 28
Aim
Determine the ground state patterns and their phase transitions in the parameter space (p,q) for di_erent kinds of interaction between the spins and magnetic _eld. Here, 'p' and 'q' are linear and quadratic Zeeman energy in the presence of a uniform magnetic _eld B^z
Arthur, Doe About Beamer September 2, 2017 4 / 28
Zeemann Energy
The interaction of atoms with the applied magnetic _eld, B^z,introduces an additional energy, called the Zeeman energy: HZ
ee
= q(j
1
j
2
+j
1
j
2
)+p(j
1
j
2
j
1
j
2
)+E
0
(j
1
j
2
+j
0
j
2
+j
1
j
2
) where; p = 1=2(E
1
E
1
) q = 1=2(E
1
+ E
1
2E
0
)
Arthur, Doe About Beamer September 2, 2017 5 / 28
Hamiltonian of the system
H( ) = H
kinetic
( ) + H
pot
( ) + H
n
( ) + H
s
( ) where, H
n
is the spin independent interaction H
s
is the spin dependent interaction H( ) =
h
2
2m
(5 )
2
+ V(x) j j
2
+
c
n
2
j j
4
+
c
s
2
__
y
F __
2
where, V(x) is the trapping potential F is the spin operator
Arthur, Doe About Beamer September 2, 2017 6 / 28
Spin Operator for spin-1 systems
~F = (F
x
; F
y
; F
z
) F
x
=
p1 2
2 4 0 1 0 1 0 1 0 1 0 3 5F
y
=
p
i 2
2 4 0
1 0 1 0
1 0 1 0 3 5F
z
= 2 4 1 1 0 0 0 0 0 0
1 3 5
Arthur, Doe About Beamer September 2, 2017 7 / 28
Complete Hamiltonian
The complete Hamiltonian of a spinor BEC in the presence of a magnetic _eld: H( ) = R
h
2
2m
(5 )
2
+ V(x) j j
2
+
c
n
2
j j
4
+
c
s
2
__
y
F __
2
+ q(j
1
j
2
+ j
1
j
2
) + pM + E
0
N M = R (j
1
j
2
j
1
j
2
) N = R (j
1
j
2
+ j
0
j
2
+ j
1
j
2
)
Arthur, Doe About Beamer September 2, 2017 8 / 28
Invariants in the system
The number of particles in the system remain constant.i.e, N = R (j
1
j
2
+ j
0
j
2
+ j
1
j
2
) Magnetization of the system also remains unchanged: M = R
(j
1
j
2
j
1
j
2
)
Arthur, Doe About Beamer September 2, 2017 9 / 28
Finding the ground states
H ( ) = ( ) Finding the ground state implies we have to minimize the Hamiltonian constrained to the invariant quantities min f [ ] : N[ ];M[ ]g H
TF
= @H where H( ) = R
h
2
2m
(5 )
2
+V(x) j j
2
+
c
n
2
j j
4
+
c
s
2
__
y
F __
2
+q(j
1
j
2
+j
1
j
2
) In the low energy limit the potential and kinetic energy terms can be ignored and H
TF
=
1 2
(ju
1
j
2
+ ju
0
j
2
+ ju
1
j
2
) +
_2
_ 2u
2 0
(u
1
sgn(_)u
1
)
2
+ (u
2 1
u
2
1
) _ + q(u
2 1
+ u
2
1
)
Arthur, Doe About Beamer September 2, 2017 10 / 28
Finding the ground state
H
TF
=
12
(ju
1
j
2
+ ju
0
j
2
+ ju
1
j
2
) +
_2
_ 2u
2 0
(u
1
sgn(_)u
1
)
2
+ (u
2 1
u
2
1
) _ + q(u
2 1
+ u
2
1
) where, _ =
c
s
c
n
q =
q
0
c
n
m
= u
m
e
i_
N = R u
2 1
+ u
2

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