Pump-probe spectroscopy for Three-Level Atoms in Λ, Cascade, and V Configurations

This notebook has steady state solutions for two level atoms and the three types of three - level atoms - stationary atoms.

Two-leval atom absorption

**2- level equations**

eqns={0==-I x (2 r22-1)-(g-I d) r12,I x (2 r22-1)-(g+I d) r21==0,I x (r21-r12) -g2 r22==0}

Solve[eqns,{r12,r22,r21}]

r12[d_,g_,g2_,x_]:=-((d g2 x-I g g2 x)/((d^2) g2+(g^2) g2+4 g (x^2)))

Plot[Im[r12[d,2,1,.1]],{d,-5,5}]

3-Level Λ system - probe absorption

**3-level equations for lambda  system**

***gij stands for collision dephasing rates (Subscript[Γ, ij]) when i<j; g21 is partial decay rate from level 2 to level 1;  rij stands for density matrix element (Subscript[ρ, ij]) ***

tlelam={-I x r21+ I x r12 +g21 r22==0,I x r21-I x r12-I x1 r32+I x1 r23-g2 r22==0,-I x (r22-r11)+I x1 r13-(g2/2+g12-I d)r12==0,-I x1 (1-2 r22-r11)-I x r13-(g2/2+g23+I d1)r23==0,-I x r23+I x1 r12-(g13-I(d-d1)) r13==0,I x (r22-r11)-I x1 r31-(g2/2+g12+I d)r21==0,+I x1 (1-2r22-r11)+I x r31-(g2/2+g23-I d1)r32==0,+I x r32-I x1 r21-(g13+I(d-d1)) r31==0}

Solve[tlelam,{r32},{r23,r12,r22,r13,r11,r31,r21}]

Simplify[%]

r32lam[x_,x1_,d_,d1_,g2_,g21_,g12_,g23_,g13_]:=-(2 (x^2) x1 (2 (d1^3) (2 g12+g2) (g2-g21)-I (d1^2) (2 g12+g2) (g2-g21) (g2+2 g23)+(d^2) (2 g12+g2) (g2-g21) (2 d1-I (g2+2 g23))+2 d (-2 (d1^2) (2 g12+g2) (g2-g21)+I d1 (2 g12+g2) (g2-g21) (g2+2 g23)+(g2^2) (x^2)-g2 g21 (x^2)+2 g13 g21 (x1^2)+g2 g21 (x1^2)+2 g12 (g2 (x^2)-g21 (x^2)+g21 (x1^2)))+2 d1 ((g13^2) g2 (g2-g21)+2 g13 (g2-g21) (x1^2)+2 g12 ((g13^2) (g2-g21)-g2 (x^2)+g21 (x^2)-g21 (x1^2))-g2 (g2 (x^2)-g21 (x^2)+g21 (x1^2)))-I g13 (g2-g21) (g13 g2 (g2+2 g23)+2 g12 (g13 (g2+2 g23)+2 (x^2))+2 (2 g23 (x1^2)+g2 ((x^2)+(x1^2))))))/(4 (d1^4) (2 g12+g2) (g2-g21) (x^2)+4 (d^4) g21 (g2+2 g23) (x1^2)-8 (d^3) d1 g21 (g2+2 g23) (x1^2)+(g13 ((g2^2) (x^2)-g2 g21 (x^2)+2 g2 g23 (x^2)-2 g21 g23 (x^2)+2 g12 g21 (x1^2)+g2 g21 (x1^2)+12 (x^2) (x1^2))+2 ((x^2)+(x1^2)) (g2 (x^2)+g21 (-(x^2)+(x1^2)))) (g13 g2 (g2+2 g23)+2 g12 (g13 (g2+2 g23)+2 (x^2))+2 (2 g23 (x1^2)+g2 ((x^2)+(x1^2))))+(d^2) ((g2^4) (x^2)-(g2^3) g21 (x^2)+4 (g2^3) g23 (x^2)-4 (g2^2) g21 g23 (x^2)+4 (g2^2) (g23^2) (x^2)-4 g2 g21 (g23^2) (x^2)+4 (g13^2) g2 g21 (x1^2)+(g2^3) g21 (x1^2)+8 (g13^2) g21 g23 (x1^2)+2 (g2^2) g21 g23 (x1^2)+4 (g12^2) g21 (g2+2 g23) (x1^2)+8 (g2^2) (x^2) (x1^2)+8 g13 g21 (x^2) (x1^2)+16 g2 g23 (x^2) (x1^2)-8 g2 g21 (x1^4)-16 g21 g23 (x1^4)+4 (d1^2) ((g2^2) (x^2)+2 g12 (g2-g21) (x^2)+2 g21 g23 (x1^2)+g2 g21 (-(x^2)+(x1^2)))+2 g12 (g2+2 g23) ((g2^2) (x^2)-2 (x^2) (g21 g23-6 (x1^2))+g2 (-g21 (x^2)+2 g23 (x^2)+2 g21 (x1^2))))+(d1^2) (4 (g13^2) g2 (g2-g21) (x^2)+4 (g12^2) g21 (g2+2 g23) (x1^2)+8 g13 (g2-g21) (x^2) (x1^2)+g2 ((g2^3) (x^2)+4 (x^2) (-g21 (g23^2)+2 g21 (x^2)+6 g23 (x1^2))+(g2^2) (4 g23 (x^2)+g21 (-(x^2)+(x1^2)))+g2 (2 g21 g23 (-2 (x^2)+(x1^2))+4 (x^2) ((g23^2)-2 (x^2)+2 (x1^2))))+2 g12 ((g2^3) (x^2)+4 (g13^2) (g2-g21) (x^2)+(g2^2) (-g21 (x^2)+4 g23 (x^2)+2 g21 (x1^2))+4 (x^2) (-g21 (g23^2)+2 g21 (x^2)+6 g23 (x1^2))+g2 (4 g21 g23 (-(x^2)+(x1^2))+4 (x^2) ((g23^2)-2 (x^2)+2 (x1^2)))))-2 d d1 ((g2^4) (x^2)+4 (d1^2) (2 g12+g2) (g2-g21) (x^2)-(g2^3) g21 (x^2)+4 (g2^3) g23 (x^2)-4 (g2^2) g21 g23 (x^2)+4 (g2^2) (g23^2) (x^2)-4 g2 g21 (g23^2) (x^2)-4 (g2^2) (x^4)+4 g2 g21 (x^4)+(g2^3) g21 (x1^2)+2 (g2^2) g21 g23 (x1^2)+4 (g12^2) g21 (g2+2 g23) (x1^2)-4 g13 g2 (x^2) (x1^2)+8 (g2^2) (x^2) (x1^2)+20 g2 g23 (x^2) (x1^2)-4 g2 g21 (x1^4)-8 g21 g23 (x1^4)+2 g12 ((g2^3) (x^2)+(g2^2) (-g21 (x^2)+4 g23 (x^2)+2 g21 (x1^2))+4 (x^2) (g21 (-(g23^2)+(x^2))+6 g23 (x1^2))+g2 (4 g21 g23 (-(x^2)+(x1^2))+2 (x^2) (2 (g23^2)-2 (x^2)+5 (x1^2))))))

weak probe

Simplify[Series[r32lam[x,x1,d,d1,g2,g21,g12,g23,g13],{x1,0,1}]]

Simplify[Series[r32lam[x,x1,d,d1,g2,g21,0,0,0],{x1,0,1}]]

** *these are the equations for the weak probe limit ***

Absorption spectra with weak probe

Plot[Im[r32lam[1,.00001,0,d1,1,.5,0,0,0]],{d1,-3,3},PlotRange->All]

** *note absorption vanishes identically when δ'=δ, but not if collisions are present (below) ***

Plot[Im[r32lam[1,.00001,0,d1,1,.5,0,0,1]],{d1,-3,3},PlotRange->All]

In general for weak probe and non zero detuning and no collisions there are two resonances having different widths - these can be explained qualitatively using semiclassical dressed states, but not the exact zero when δ'=δ

Plot[Im[r32lam[0.6,.00001,3,d1,1,.5,0,0,0]],{d1,-4,7},PlotRange->All]

Plot[Im[r32lam[0.6,.00001,0,d1,1,.5,0,0,0]],{d1,-4,7},PlotRange->All]

Cascade system - probe absorption

***cascade sysytem****

tleclcasc={-I x r21+ I x r12 +g2 r22==0,I x r21-I x r12-I x1 r32+I x1 r23-(g2) r22+g3 (1-r22-r11)==0,-I x (r22-r11)+I x1 r13-(g2/2+g12-I d)r12==0,-I x1 (1-2 r22-r11)-I x r13-(g2/2+g3/2+g23-I d1)r23==0,-I x r23+I x1 r12-(g3/2+g13-I(d+d1)) r13==0,I x (r22-r11)-I x1 r31-(g2/2+g12+I d)r21==0,+I x1 (1-2r22-r11)+I x r31-(g2/2+g3/2+g23+I d1)r32==0,+I x r32-I x1 r21-(g3/2+g13+I(d+d1)) r31==0}

Solve[tleclcasc,{r23},{r21,r22,r13,r11,r31,r32,r12}]

Simplify[%]

r23[x_,x1_,d_,d1_,g2_,g3_,g12_,g23_,g13_]:=-(24 (2 d+2 d1+I (2 g13+g3)) (x^2) (x1^2) (-8 (d1^3) (2 g12+g2) g3+4 I (d1^2) (2 g12+g2) g3 (2 g23+g3)+8 I (d^2) g12 g3 (2 I d1+g2+2 g23+g3)+I g3 (2 g13+g2+g3) ((g2^2) g3+2 g2 g23 g3+g2 (g3^2)+2 g13 g2 (g2+2 g23+g3)+4 g2 (x^2)+2 g12 (g2 g3+2 g23 g3+(g3^2)+2 g13 (g2+2 g23+g3)+4 (x^2))+4 g2 (x1^2)+8 g23 (x1^2)+4 g3 (x1^2))-2 d1 (4 (g13^2) g2 g3+(g2^3) g3+2 (g2^2) g23 g3+2 (g2^2) (g3^2)+g2 (g3^3)-4 g2 g3 (x^2)+4 (g2^2) (x1^2)+4 g2 g3 (x1^2)+4 (g3^2) (x1^2)+2 g13 g3 ((g2^2)+2 g2 g3+4 (x1^2))+2 g12 (4 (g13^2) g3+(g2^2) g3+(g3^3)+2 g13 g3 (g2+2 g3)-4 g3 (x^2)+2 g2 (g23 g3+(g3^2)+2 (x1^2))))-2 d (4 (d1^2) (4 g12+g2) g3-4 I d1 g3 (g2 (-g13+g23)+g12 (g2+4 g23+2 g3))+g2 (2 g13+g2+g3) (g2 g3+2 g23 g3+(g3^2)+4 (x1^2))+2 g12 ((g2^2) g3-4 g3 (x^2)+g2 (2 g23 g3+(g3^2)+4 (x1^2))))))/(2 (I (2 d+2 d1+I (2 g13+g3)) (4 (d^2) g2+2 d g3 (2 d1-I (g2+2 g23+g3))+(2 g12+g2) (2 g12 g2+(g2^2)+2 I d1 g3+g2 g3+2 g23 g3+(g3^2)+12 (x^2)))-4 ((g2^2) (x^2)-4 I d1 g3 (x^2)-2 g2 g3 (x^2)-4 g23 g3 (x^2)-2 (g3^2) (x^2)+(g2^2) (x1^2)+2 I d1 g3 (x1^2)+g2 g3 (x1^2)+2 g23 g3 (x1^2)+(g3^2) (x1^2)+2 I d g2 ((x^2)+(x1^2))+2 g12 g2 ((x^2)+(x1^2)))) (2 I x1 (6 (-2 I d-2 I d1-2 g13+g3) (x^4)+((x^2)+(x1^2)) (-4 (d^2) g2+4 g12 g13 g2+2 g13 (g2^2)+2 g12 g2 g3+(g2^2) g3+12 g13 (x^2)+2 g3 (x^2)+2 I d1 (2 g12 g2+(g2^2)+6 (x^2))+d (-4 d1 g2+2 I (2 g12 g2+2 g13 g2+(g2^2)+g2 g3+6 (x^2)))+4 g2 (x1^2)))+(I (d+d1)-g13-(g3/2)) (-24 I x (-(1/4) (2 d+2 d1-2 I g13+I g3) (2 d1+I (g2+2 g23+g3)) x x1+x (x1^3))+(-2 I d1+g2+2 g23+g3) x1 (4 I (d^2) g2+2 d1 (2 g12 g2+(g2^2)+6 (x^2))+2 d (2 I d1 g2+2 g12 g2+2 g13 g2+(g2^2)+g2 g3+6 (x^2))-I ((g2^2) g3+2 g12 g2 (2 g13+g3)+2 g3 (x^2)+2 g13 ((g2^2)+6 (x^2))+4 g2 (x1^2)))))-x1 (16 (d^4) g2+32 (d^3) d1 g2+16 (g12^2) (g13^2) g2+16 g12 (g13^2) (g2^2)+4 (g13^2) (g2^3)+16 (g12^2) g13 g2 g3+16 g12 g13 (g2^2) g3+4 g13 (g2^3) g3+4 (g12^2) g2 (g3^2)+4 g12 (g2^2) (g3^2)+(g2^3) (g3^2)+96 g12 (g13^2) (x^2)+16 g12 g13 g2 (x^2)+48 (g13^2) g2 (x^2)+8 g13 (g2^2) (x^2)+80 g12 g13 g3 (x^2)+8 g12 g2 g3 (x^2)+40 g13 g2 g3 (x^2)+4 (g2^2) g3 (x^2)+16 g12 (g3^2) (x^2)+8 g2 (g3^2) (x^2)+8 I d1 (2 g12+g2) (g2+g3) (x^2)+32 g3 (x^4)+4 (d1^2) (4 (g12^2) g2+(g2^3)+12 g2 (x^2)+4 g12 ((g2^2)+6 (x^2)))+32 g12 g13 g2 (x1^2)+16 g13 (g2^2) (x1^2)+16 g12 g2 g3 (x1^2)+8 (g2^2) g3 (x1^2)+96 g13 (x^2) (x1^2)+16 g2 (x^2) (x1^2)+32 g3 (x^2) (x1^2)+16 g2 (x1^4)+4 (d^2) (4 (d1^2) g2+4 (g12^2) g2+4 (g13^2) g2+(g2^3)+4 g13 g2 g3+g2 (g3^2)+8 g2 (x^2)+4 g3 (x^2)+4 g12 ((g2^2)+6 (x^2))-8 g2 (x1^2))+8 d (I (g2+g3) (2 g12+2 g13+g2+g3) (x^2)+d1 (4 (g12^2) g2+(g2^3)+10 g2 (x^2)+2 g3 (x^2)+4 g12 ((g2^2)+6 (x^2))-4 g2 (x1^2)))) (16 (d1^3) g3-4 I (d^2) g2 (-2 I d1+g2+2 g23+g3)+4 I (d1^2) (2 g12 g2+(g2^2)+2 g3 (2 g13+g3))-2 d1 (2 g13 (g2^2)+(g2^3)+2 (g2^2) g23-8 g2 g23 g3-8 (g23^2) g3-4 g2 (g3^2)-8 g23 (g3^2)-2 (g3^3)+2 g12 g2 (2 g13+g2+2 (g23+g3))+8 g3 (x^2)-12 g2 (x1^2)-24 g23 (x1^2)-16 g3 (x1^2))+2 d (2 g13 (g2^2)-(g2^3)+4 g13 g2 g23-2 (g2^2) g23-4 (d1^2) (g2-2 g3)+4 I d1 g2 (g12-g13-g23-g3)+2 g13 g2 g3+2 (g2^2) g3+10 g2 g23 g3+8 (g23^2) g3+5 g2 (g3^2)+8 g23 (g3^2)+2 (g3^3)-2 g12 g2 (g2+2 g23+g3)+4 g2 (x^2)+16 g2 (x1^2)+24 g23 (x1^2)+12 g3 (x1^2))-I ((g2^3) g3+2 (g2^2) g23 g3-(g2^2) (g3^2)-8 g2 g23 (g3^2)-8 (g23^2) (g3^2)-4 g2 (g3^3)-8 g23 (g3^3)-2 (g3^4)+4 (g2^2) (x^2)-8 g2 g3 (x^2)-16 g23 g3 (x^2)-8 (g3^2) (x^2)+4 (g2^2) (x1^2)-8 g2 g3 (x1^2)-16 g23 g3 (x1^2)-8 (g3^2) (x1^2)+2 g12 g2 (g2 g3+2 g23 g3+(g3^2)+2 g13 (g2+2 g23+g3)+4 (x^2)+4 (x1^2))+2 g13 ((g2^3)+(g2^2) (2 g23-g3)-4 g2 (2 g23 g3+(g3^2)+3 (x1^2))-2 (4 (g23^2) g3+(g3^3)+6 g3 (x1^2)+4 g23 ((g3^2)+3 (x1^2)))))))

perturbation theory result for probe

Absorption spectra

Plot[Im[r23[.1,.1,4,d1,1,1,0,0,0]],{d1,-6,4}]

Autler Townes splitting

Plot[Im[r23[3,.1,0,d1,1,1,0,0,0]],{d1,-6,6}]

collision induced extra absorption

Plot[Im[r23[.1,.1,4,d1,1,1,.4,.4,.1]],{d1,-6,4}]

Some (unphysical) collision parameters lead to negative population

Plot[Im[r23[.01,.01,1.8,d1,1,.2,1000,.01,.01]],{d1,-5,3},AxesLabel->{"δ'","absorption, δ=1.8, cascade  unphysical collisions"}]

V system - probe absorption

***V system****

tlev={-I x r21+ I x r12 -g1 r11==0,I x r21-I x r12-I x1 r32+I x1 r23+(g1) r11+g3 (1-r22-r11)==0,-I x (r22-r11)+I x1 r13-(g1/2+g12+I d)r12==0,-I x1 (1-2 r22-r11)-I x r13-(g3/2+g23-I d1)r23==0,-I x r23+I x1 r12-(g3/2+g1/2+g13+I(d-d1)) r13==0,I x (r22-r11)-I x1 r31-(g1/2+g12-I d)r21==0,+I x1 (1-2r22-r11)+I x r31-(g3/2+g23+I d1)r32==0,+I x r32-I x1 r21-(g3/2+g1/2+g13-I(d-d1)) r31==0}

Solve[tlev,{r23},{r21,r22,r13,r11,r31,r32,r12}]

Simplify[%]

r23v[x_,x1_,d_,d1_,g1_,g3_,g12_,g23_,g13_]:=-(4 (-I d+I d1-(g1/2)-g13-(g3/2)) x1 (2 (2 I d g1-(g1^2)-2 g1 g12+g3 (2 I d1+2 g23+g3)) (16 (d^4) g1-32 (d^3) d1 g1+(g1^5)+4 (g1^4) g12+4 (g1^3) (g12^2)+4 (g1^4) g13+16 (g1^3) g12 g13+16 (g1^2) (g12^2) g13+4 (g1^3) (g13^2)+16 (g1^2) g12 (g13^2)+16 g1 (g12^2) (g13^2)+2 (g1^4) g3+8 (g1^3) g12 g3+8 (g1^2) (g12^2) g3+4 (g1^3) g13 g3+16 (g1^2) g12 g13 g3+16 g1 (g12^2) g13 g3+(g1^3) (g3^2)+4 (g1^2) g12 (g3^2)+4 g1 (g12^2) (g3^2)+4 (g1^3) (x^2)+8 (g1^2) g12 (x^2)+20 (g1^2) g13 (x^2)+40 g1 g12 g13 (x^2)+24 g1 (g13^2) (x^2)+48 g12 (g13^2) (x^2)-4 I d1 (g1+2 g12) (g1-g3) (x^2)+8 (g1^2) g3 (x^2)+16 g1 g12 g3 (x^2)+20 g1 g13 g3 (x^2)+40 g12 g13 g3 (x^2)+4 g1 (g3^2) (x^2)+8 g12 (g3^2) (x^2)+16 g3 (x^4)+4 (d1^2) ((g1^3)+4 (g1^2) g12+4 g1 (g12^2)+6 g1 (x^2)+12 g12 (x^2))+8 (g1^3) (x1^2)+16 (g1^2) g12 (x1^2)+16 (g1^2) g13 (x1^2)+32 g1 g12 g13 (x1^2)+8 (g1^2) g3 (x1^2)+16 g1 g12 g3 (x1^2)+16 g1 (x^2) (x1^2)+48 g13 (x^2) (x1^2)+16 g3 (x^2) (x1^2)+16 g1 (x1^4)+4 (d^2) (4 (d1^2) g1+2 (g1^3)+2 (g1^2) (2 g12+2 g13+g3)+2 (6 g12+g3) (x^2)+g1 (4 (g12^2)+4 (g13^2)+4 g13 g3+(g3^2)+8 (x^2)-8 (x1^2)))-4 d (-I (g1-g3) (2 g1+2 g12+2 g13+g3) (x^2)+2 d1 ((g1^3)+4 (g1^2) g12+(12 g12+g3) (x^2)+g1 (4 (g12^2)+7 (x^2)-4 (x1^2)))))+(-4 I (d^2) g1-2 d1 g1 (g1+2 g12)+2 d g1 (2 I d1+2 g1+2 g12+2 g13+g3)+I ((g1^3)+(g1^2) (2 g12+2 g13+g3)+4 g3 (x^2)+2 g1 (2 g12 g13+g12 g3+2 (x1^2)))) (16 (d^3) g1-4 (d^2) (2 d1 (2 g1+g3)+I (2 (g1^2)+2 g1 (2 g13+g3)-g3 (2 g23+g3)))+2 d (2 (g1^3)+8 (g1^2) g12+4 (d1^2) g3+4 I d1 (g12+g13-g23) g3+4 g12 g23 g3+4 g13 g23 g3+2 g12 (g3^2)+2 g13 (g3^2)+2 g23 (g3^2)+(g3^3)+24 g12 (x^2)+2 g1 (4 (g12^2)+2 I d1 g3+2 g23 g3+(g3^2)+8 (x^2)-4 (x1^2)))-I (2 (g1^4)+4 (d1^2) (g1+2 g12) g3+2 (g1^3) (4 g12+2 g13+g3)+(g1^2) (8 (g12^2)+2 g23 g3+(g3^2)+8 g12 (2 g13+g3)+8 (x^2)+8 (x1^2))-2 I d1 (2 (g1^3)+(g1^2) (8 g12-g3)-4 g12 g13 g3+4 g12 g23 g3+24 g12 (x^2)+8 g3 (x^2)+2 g1 (4 (g12^2)-g12 g3-g13 g3+g23 g3+6 (x^2))-4 g3 (x1^2))+2 (g12 (2 g13+g3) (2 g23 g3+(g3^2)+12 (x^2))-2 g3 (2 g23+g3) (2 (x^2)-(x1^2)))+g1 (8 (g12^2) (2 g13+g3)+(2 g13+g3) (2 g23 g3+(g3^2)+12 (x^2))+2 g12 (2 g23 g3+(g3^2)+8 ((x^2)+(x1^2))))))))/(((2 I d-2 I d1+g1+2 g13+g3) (8 (d^2) g1+2 I d g3 (2 I d1+2 g23+g3)+(g1+2 g12) (2 (g1^2)+4 g1 g12+2 I d1 g3+2 g23 g3+(g3^2)+12 (x^2)))+4 (2 I d g1 ((x^2)-2 (x1^2))-(g1^2) ((x^2)-2 (x1^2))-2 g1 g12 ((x^2)-2 (x1^2))-g3 (2 I d1+2 g23+g3) (2 (x^2)-(x1^2)))) (4 I (3 I (2 d-2 d1+I (g1+2 g13+3 g3)) (x^4)+((x^2)-2 (x1^2)) (-4 (d^2) g1+(g1^3)+2 (g1^2) g12+2 (g1^2) g13+4 g1 g12 g13+(g1^2) g3+2 g1 g12 g3+3 g1 (x^2)+6 g13 (x^2)+g3 (x^2)+2 I d1 ((g1^2)+2 g1 g12+3 (x^2))+d (4 d1 g1-2 I (2 (g1^2)+g1 (2 g12+2 g13+g3)+3 (x^2)))+4 g1 (x1^2)))+(2 d-2 d1-I (g1+2 g13+g3)) (-4 (d1^2) g1 (g1+2 g12)-2 (g1^3) g23-4 (g1^2) g12 g23-4 (g1^2) g13 g23-8 g1 g12 g13 g23-(g1^3) g3-2 (g1^2) g12 g3-2 (g1^2) g13 g3-4 g1 g12 g13 g3-2 (g1^2) g23 g3-4 g1 g12 g23 g3-(g1^2) (g3^2)-2 g1 g12 (g3^2)+4 (d^2) g1 (-2 I d1+2 g23+g3)+2 d g1 (2 I d1+2 g1+2 g12+2 g13+g3) (2 d1+I (2 g23+g3))+16 g23 g3 (x^2)+8 (g3^2) (x^2)-8 g1 g23 (x1^2)-4 g1 g3 (x1^2)+24 (x^2) (x1^2)+2 I d1 ((g1^3)+2 (g1^2) (g12+g13-g23)-8 g3 (x^2)+4 g1 (g12 g13-g12 g23+(x1^2)))))-2 (16 (d^4) g1-32 (d^3) d1 g1+(g1^5)+4 (g1^4) g12+4 (g1^3) (g12^2)+4 (g1^4) g13+16 (g1^3) g12 g13+16 (g1^2) (g12^2) g13+4 (g1^3) (g13^2)+16 (g1^2) g12 (g13^2)+16 g1 (g12^2) (g13^2)+2 (g1^4) g3+8 (g1^3) g12 g3+8 (g1^2) (g12^2) g3+4 (g1^3) g13 g3+16 (g1^2) g12 g13 g3+16 g1 (g12^2) g13 g3+(g1^3) (g3^2)+4 (g1^2) g12 (g3^2)+4 g1 (g12^2) (g3^2)+4 (g1^3) (x^2)+8 (g1^2) g12 (x^2)+20 (g1^2) g13 (x^2)+40 g1 g12 g13 (x^2)+24 g1 (g13^2) (x^2)+48 g12 (g13^2) (x^2)-4 I d1 (g1+2 g12) (g1-g3) (x^2)+8 (g1^2) g3 (x^2)+16 g1 g12 g3 (x^2)+20 g1 g13 g3 (x^2)+40 g12 g13 g3 (x^2)+4 g1 (g3^2) (x^2)+8 g12 (g3^2) (x^2)+16 g3 (x^4)+4 (d1^2) ((g1^3)+4 (g1^2) g12+4 g1 (g12^2)+6 g1 (x^2)+12 g12 (x^2))+8 (g1^3) (x1^2)+16 (g1^2) g12 (x1^2)+16 (g1^2) g13 (x1^2)+32 g1 g12 g13 (x1^2)+8 (g1^2) g3 (x1^2)+16 g1 g12 g3 (x1^2)+16 g1 (x^2) (x1^2)+48 g13 (x^2) (x1^2)+16 g3 (x^2) (x1^2)+16 g1 (x1^4)+4 (d^2) (4 (d1^2) g1+2 (g1^3)+2 (g1^2) (2 g12+2 g13+g3)+2 (6 g12+g3) (x^2)+g1 (4 (g12^2)+4 (g13^2)+4 g13 g3+(g3^2)+8 (x^2)-8 (x1^2)))-4 d (-I (g1-g3) (2 g1+2 g12+2 g13+g3) (x^2)+2 d1 ((g1^3)+4 (g1^2) g12+(12 g12+g3) (x^2)+g1 (4 (g12^2)+7 (x^2)-4 (x1^2))))) (16 (d1^3) g3-4 I (d1^2) ((g1^2)+2 g1 (g12-g3)-2 g3 (2 g13+g3))+4 (d^2) g1 (2 d1+I (2 g23+g3))-2 d (-4 I d1 g1 (g12-g13-g23-g3)+4 (d1^2) (g1+2 g3)-g1 (2 g23 g3+(g3^2)-2 g12 (2 g23+g3)+2 g13 (2 g23+g3)+4 (x^2)-8 (x1^2))+2 (4 (g23^2) g3+(g3^3)+6 g3 (x1^2)+4 g23 ((g3^2)+3 (x1^2))))+2 d1 ((g1^3)+4 g1 g12 (g13+g23+g3)+2 (g1^2) (g12+g13+g23+g3)+2 (4 (g23^2) g3+4 g23 ((g3^2)+3 (x1^2))+g3 ((g3^2)-4 (x^2)+8 (x1^2))))+I ((g1^3) (2 g23+g3)+(g1^2) (2 g23 g3+(g3^2)+2 g12 (2 g23+g3)+2 g13 (2 g23+g3)+4 (x^2)-8 (x1^2))+2 g1 (4 (g23^2) g3+(g3^3)+6 g3 (x1^2)+g12 (2 g23 g3+(g3^2)+2 g13 (2 g23+g3)+4 (x^2)-8 (x1^2))+4 g23 ((g3^2)+3 (x1^2)))+2 (2 g23+g3) (2 g13 (2 g23 g3+(g3^2)+6 (x1^2))+g3 (2 g23 g3+(g3^2)+4 ((x^2)+(x1^2)))))))

Weak probe field results

Simplify[Normal[Series[r23v[x,x1,d,d1,g1,g3,g12,g23,g13],{x1,0,1}]]]

Simplify[Normal[Series[%,{x,0,2}]]]

weak probe field results with no collisions

Simplify[r23v[0,x1,0,d1,g1,g3,0,0,0]]

Probe field absorption

Still probe absorption even without pump field

Plot[Im[r23v[0,.1,4,d1,1,1,0,0,0]],{d1,-3,10}]

modified slightly with pump field

Plot[Im[r23v[.2,.1,4,d1,1,1,0,0,0]],{d1,-3,10}]

Plot[Im[r23v[15,.1,60,d1,1,1,200,200,0]],{d1,40,80}]

difference between probe absorption with and without pump field in region of δ'=δ

Plot[Im[r23v[15,.1,60,d1,1,1,0,0,0]]-Im[r23v[0,.1,60,d1,1,1,0,0,0]],{d1,40,80}]

collisions change signal from absorption to gain ***

Plot[Im[r23v[15,.1,60,d1,1,1,200,200,0]]-Im[r23v[0,.1,60,d1,1,1,200,200,0]],{d1,40,80}]

Plot[Im[r23v[1,.1,60,d1,1,1,200,200,0]]-Im[r23v[0,.1,60,d1,1,1,200,200,0]],{d1,40,80}]

Autler-Townes like splitting

Plot[Im[r23v[5,.1,0,d1,1,1,0,0,0]]/.1,{d1,-60,60},PlotRange->All]