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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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