ddτˆρ(τ)=−i(ω)[ˆa†ˆa,ˆρS(τ)]+Γ0(N(ω)+1)(ˆaˆρS(τ)ˆa†−12ˆa†ˆaˆρS(τ)−12ˆρS(τ)ˆa†ˆa)++Γ0N(ω)(ˆa†ˆρS(τ)ˆa−12ˆaˆa†ˆρS(τ)−12ˆρS(τ)ˆaˆa†)
From master equation we find the probabilities
P(n,τ)=<n|ρ|n> for the oscillator to be in n-th energy eigenstate, namely the Pauli master equation
˙P(n,τ)=Γ0(N(ω)+1)((n+1)P(n+1,τ)−nP(n,τ))++Γ0N(ω)(nP(n−1,τ)−(n+1)P(n,τ))
according to Petruccione the stationary solution is
Ps(n)=1N(ω)+1(N(ω)N(ω)+1)n
I am trying to show the above result as well as to find the
ρ(τ)but i find different solution. The method that i use is discrete equations second order. First I solve
0=Γ0(N(ω)+1)((n+1)P(n+1,τ)−nP(n,τ))++Γ0N(ω)(nP(n−1,τ)−(n+1)P(n,τ))
and I find
Pt(n)=c1(N(ω)N(ω)+1)n+c2(n+1n+2)n
then
˙P(n,τ)=Pt(n)
but nothing