Optical pumping and optical lattice potentials

This program lets you obtain the optical pumping equations and the optical lattice potentials.

Input the subroutines replaceing the locations by the location where you store the subroutines

Program

<<c:\pberman\mathematica\5.2\clebsh.m

<<c:\pberman\mathematica\5.2\threej.m

<<c:\pberman\mathematica\5.2\ninej.m

<<c:\pberman\mathematica\5.2\sixj.m

<<c:\pberman\mathematica\5.2\prate.m

***value of  (γ^((K)))(H;G)  -  not needed in this program but would be needed for Eqs.(17.70)***

Optical Pumping

gam[G_,H_,K_]:=(2 H+1) (-1)^(G+H+1+K)sixj[{H,G},{H,G},{K,1}](Subscript[γ, HG])

gam[1,2,2]

***Kronecker delta function***

kron[G_,G1_]:=If[G==G1,1,0]

*** The A function given in Eq. (17.28)******

A[K_,Kp_,Kb_,Q_,Qp_,G_,H_]:=(-1)^Qp χ^2Sum[ϵ[Kb,Qb]  T3[{Kp,-Qp},{K,Q},{Kb,Qb}] (((Subscript[γ, HG])-I δ)^(-1)+(-1)^(K+Kb+Kp)((Subscript[γ, HG])+I δ)^(-1))  ,{Qb,-Kb,Kb}]

ϵ (K, Q) for z polarized light

ϵ[K_,Q_]:=-kron[K,0] kron[Q,0]/(Sqrt[3])+kron[K,2] (kron[Q,0](Sqrt[2/3]))

Clear[G]

ρd is derivative of ρ given in Eq. (17.27)- (Subscript[dρ, Q]^K)(G)/dt in terms of (Subscript[ρ, Q]^K)(G)  (it depends implicitly on the value of H) - you must define the  ϵ(K,Q) for your specific polarizations

ρd[G_,H_,K_,Q_]:=Simplify[Sum[A[K,Kp,Kb,Q,Qp,G,H]p[G,H,K,Kp,Kb]ρ[G,Kp,Qp],{Kp,0,2 G},{Qp,-2 G,2 G},{Kb,0,2 }]]

output for z - polarized light with G = 1 and H = 2

ρd[1,2,0,0]

ρd[1,2,2,0]

**In other words, this output is (Subscript[dρ, 0]^2)(1)/dt in terms of (Subscript[ρ, 0]^0)(1) and (Subscript[ρ, 0]^2)(1)***

ρd[1,2,1,0]

** these are only equations you need if you start from an unpolarized state***

ρd[1,2,1,1]

ρd[1,2,2,1]

ρd[1,2,2,2]

x - polarized fields

ϵ[K_,Q_]:=-kron[K,0] kron[Q,0]/(Sqrt[3])+kron[K,2] (-kron[Q,0]/(Sqrt[6])+(kron[Q,2]+kron[Q,-2])/2)

ρd[1,2,0,0]

ρd[1,2,1,0]

ρd[1,2,2,0]

ρd[1,2,2,2]

ρd[1,2,2,-2]

***Solve in steady state Γ=2(Subscript[γ, HG])****

Solve[{6 I (Sqrt[2]) δ ρ[1,1,0]+Γ(5 -2 (Sqrt[6]) ρ[1,2,0]+30 ρ[1,2,2])==0,5 (Sqrt[2]) /Sqrt[3]+2 (Sqrt[6]) ρ[1,2,-2]-22 ρ[1,2,0]+2 (Sqrt[6]) ρ[1,2,2]==0,-6 I (Sqrt[2]) δ ρ[1,1,0]+Γ(5 -2 (Sqrt[6]) ρ[1,2,0]+30 ρ[1,2,2])==0,Γ ρ[1,1,0]+I (Sqrt[2]) δ (ρ[1,2,-2]-ρ[1,2,2])==0},{ρ[1,2,2],ρ[1,2,-2],ρ[1,2,0],ρ[1,1,0]}]

A for two fields given in Eq. (17.39)

A[K_,Kp_,Kb_,Q_,Qp_,G_,H_,j_,jp_]:=(-1)^Qp χ[j] Conjugate[χ[jp]]Sum[ϵ[Kb,Qb,j,jp]  T3[{Kp,-Qp},{K,Q},{Kb,Qb}] Exp[I( k[j]-k[jp]) z](((Subscript[γ, HG])-I δ)^(-1)+(-1)^(K+Kb+Kp)((Subscript[γ, HG])+I δ)^(-1))  ,{Qb,-Kb,Kb}]

**density matrix equation (17.37)***

ρd2[G_,H_,K_,Q_]:=Simplify[Sum[A[K,Kp,Kb,Q,Qp,G,H,j,jp]p[G,H,K,Kp,Kb]ρ[G,Kp,Qp],{Kp,0,2 G},{Qp,-2 G,2 G},{Kb,0,2 },{j,1,2},{jp,1,2}]]

cross polarized, oppositely propagating fields

ϵ[K_,Q_,1,1]:=-kron[K,0] kron[Q,0]/(Sqrt[3])+kron[K,2] (-kron[Q,0]/(Sqrt[6])+(kron[Q,2]+kron[Q,-2])/2)

ϵ[K_,Q_,2,2]:=-kron[K,0] kron[Q,0]/(Sqrt[3])+kron[K,2] (-kron[Q,0]/(Sqrt[6])-(kron[Q,2]+kron[Q,-2])/2)

ϵ[K_,Q_,1,2]:=I kron[K,1] kron[Q,0]/(Sqrt[2])+I kron[K,2] (kron[Q,2]-kron[Q,-2])/2

ϵ[K_,Q_,2,1]:=-I kron[K,1] kron[Q,0]/(Sqrt[2])-I kron[K,2] (kron[Q,2]-kron[Q,-2])/2

χ[1]=χ

χ[2]=χ

k[1]=k

k[2]=-k

ρd2[1/2,1/2,1,0]

***which is Eq. (17.51)***

ρd2[1/2,3/2,1,0]

***which is Eq. (17.52)***

optical lattice potentials Eq. (17.61)

h[m_,mp_,G_,H_]:=-3 hbar /(δ-I (Subscript[γ, H])/2) Sum[χ[j]Conjugate[χ[jp]]Exp[I(k[j]-k[jp])z](-1)^(K+H+G+Q+G-mp)ϵ[K,Q,j,jp]CG[{G,m},{G,-mp},{K,Q}]sixj[{G,1},{G,1},{K,H}],{j,1,2},{jp,1,2},{K,0,2 G},{Q,-2 G, 2 G}]

cross polarized fields

ϵ[K_,Q_,1,1]:=-kron[K,0] kron[Q,0]/(Sqrt[3])+kron[K,2] (-kron[Q,0]/(Sqrt[6])+(kron[Q,2]+kron[Q,-2])/2)

ϵ[K_,Q_,2,2]:=-kron[K,0] kron[Q,0]/(Sqrt[3])+kron[K,2] (-kron[Q,0]/(Sqrt[6])-(kron[Q,2]+kron[Q,-2])/2)

ϵ[K_,Q_,1,2]:=I kron[K,1] kron[Q,0]/(Sqrt[2])+I kron[K,2] (kron[Q,2]-kron[Q,-2])/2

ϵ[K_,Q_,2,1]:=-I kron[K,1] kron[Q,0]/(Sqrt[2])-I kron[K,2] (kron[Q,2]-kron[Q,-2])/2

χ[1]=χ

χ[2]=χ

k[1]=k

k[2]=-k

h[1/2,-1/2,1/2,3/2]

h[-1/2,-1/2,1/2,3/2]

h[1/2,1/2,1/2,3/2]

**which is Eq. (17.63)**

h[-1/2,-1/2,1/2,1/2]

h[1/2,1/2,1/2,1/2]

**which is Eq. (17.64)**