I am trying to solve the optical block equations for three level system which are time dependent, which i generated using the "Atomic Density Matrix" package.
When i use the command to solve this 9 linear coupled differential equations with some initial conditions it is showing me no enough memory is available.
So what should be the system requirements to solve them Analytically ?? i have to increase the RAM or hard disk memory or anything else ?
Similarly i want to work out for five level system where there will be 25 such equations.
here are the equations with initial conditions.
variable = {Subscript[ρ, 1, 1][t], Subscript[ρ, 1, 2][t],Subscript[ρ, 1, 3][t], Subscript[ρ, 2, 1][t], Subscript[ρ, 2, 2][t], Subscript[ρ, 2, 3][t],Subscript[ρ, 3, 1][t], Subscript[ρ, 3, 2][t], Subscript[ρ, 3, 3][t]};
init = {Subscript[ρ, 1, 1][0] == 1, Subscript[ρ, 1, 2][0] == 0, Subscript[ρ, 1, 3][0] == 0, Subscript[ρ, 2, 1][0] == 0, Subscript[ρ, 2, 2][0] == 0, Subscript[ρ, 2, 3][0] == 0, Subscript[ρ, 3, 1][0] == 0, Subscript[ρ, 3, 2][0] == 0, Subscript[ρ, 3, 3][0] == 0};
equations = {Derivative[1][Subscript[ρ, 1, 1]][t] == γ/
2 - γ Subscript[ρ, 1, 1][t] -
I (1/2 ΩRa Subscript[ρ, 1, 3][t] -
1/2 ΩRa Subscript[ρ, 3, 1][t]) +
1/2 Γ Subscript[ρ, 3, 3][t],
Derivative[1][Subscript[ρ, 1, 2]][
t] == -γ Subscript[ρ, 1, 2][t] -
I (Δa Subscript[ρ, 1, 2][
t] - Δb Subscript[ρ, 1, 2][t] +
1/2 ΩRb Subscript[ρ, 1, 3][t] -
1/2 ΩRa Subscript[ρ, 3, 2][t]),
Derivative[1][Subscript[ρ, 1, 3]][t] ==
1/2 (-γ Subscript[ρ, 1, 3][
t] - (γ + Γ) Subscript[ρ, 1, 3][
t]) - I (1/2 ΩRa Subscript[ρ, 1, 1][t] +
1/2 ΩRb Subscript[ρ, 1, 2][
t] + Δa Subscript[ρ, 1, 3][t] -
1/2 ΩRa Subscript[ρ, 3, 3][t]),
Derivative[1][Subscript[ρ, 2, 1]][
t] == -γ Subscript[ρ, 2, 1][t] -
I (-Δa Subscript[ρ, 2, 1][
t] + Δb Subscript[ρ, 2, 1][t] +
1/2 ΩRa Subscript[ρ, 2, 3][t] -
1/2 ΩRb Subscript[ρ, 3, 1][t]),
Derivative[1][Subscript[ρ, 2, 2]][t] == γ/
2 - γ Subscript[ρ, 2, 2][t] -
I (1/2 ΩRb Subscript[ρ, 2, 3][t] -
1/2 ΩRb Subscript[ρ, 3, 2][t]) +
1/2 Γ Subscript[ρ, 3, 3][t],
Derivative[1][Subscript[ρ, 2, 3]][t] ==
1/2 (-γ Subscript[ρ, 2, 3][
t] - (γ + Γ) Subscript[ρ, 2, 3][
t]) - I (1/2 ΩRa Subscript[ρ, 2, 1][t] +
1/2 ΩRb Subscript[ρ, 2, 2][
t] + Δb Subscript[ρ, 2, 3][t] -
1/2 ΩRb Subscript[ρ, 3, 3][t]),
Derivative[1][Subscript[ρ, 3, 1]][t] ==
1/2 (-γ Subscript[ρ, 3, 1][
t] - (γ + Γ) Subscript[ρ, 3, 1][
t]) - I (-(1/2) ΩRa Subscript[ρ, 1, 1][
t] - 1/2 ΩRb Subscript[ρ, 2, 1][
t] - Δa Subscript[ρ, 3, 1][t] +
1/2 ΩRa Subscript[ρ, 3, 3][t]),
Derivative[1][Subscript[ρ, 3, 2]][t] ==
1/2 (-γ Subscript[ρ, 3, 2][
t] - (γ + Γ) Subscript[ρ, 3, 2][
t]) - I (-(1/2) ΩRa Subscript[ρ, 1, 2][
t] - 1/2 ΩRb Subscript[ρ, 2, 2][
t] - Δb Subscript[ρ, 3, 2][t] +
1/2 ΩRb Subscript[ρ, 3, 3][t]),
Derivative[1][Subscript[ρ, 3, 3]][
t] == -I (-(1/2) ΩRa Subscript[ρ, 1, 3][t] -
1/2 ΩRb Subscript[ρ, 2, 3][t] +
1/2 ΩRa Subscript[ρ, 3, 1][t] +
1/2 ΩRb Subscript[ρ, 3, 2][
t]) - (γ + Γ) Subscript[ρ, 3, 3][
t]};
TableForm[solution = DSolve[Join[equations, init], variable, t]]
If it has to be solved numerically then how to do that ?
can any one help me out with this ?