# Distribution of density operators under Stochastic Master Equation

Stochastic master equations (SME) are used in studies of open quantum systems. The general form of an SME is: \begin{align} \tag{1} d\tilde{\sigma}(t) = - i [H, \tilde{\sigma}(t) ]dt + \frac{1}{2}\sum_{j=1}^d \left([L_j \tilde{\sigma}(t), L_j^*] + [L_j, \tilde{\sigma}(t) L_j^*] \right) dt \\ +\sum_{j=1}^d \left( L_j \tilde{\sigma}(t) + \tilde{\sigma}(t) L_j^* \right)dW_j, \end{align} where $$\tilde{\sigma}(t)$$ could be viewed as non-normalized density operator, $$L_j$$ is a Lindblad operator and $$dW_j$$ is a stochastic differential.

If we drop the stochastic term in Eq (1) then we get the usual master equation in the Lindblad form.

It is possible to obtain the density operator $$\tilde{\rho}(t)$$ from $$\tilde{\sigma}(t)$$ as follows: $$\tilde{\rho}(t) = \tilde{\sigma}(t)/Tr(\tilde{\sigma}(t)).$$

Since Eq (1) is a stochastic differential equation, $$\tilde{\rho}(t)$$ is a solution which is also random.

Could anyone tell me what's the distribution of $$\tilde{\rho}(t)$$ is?
Any beginner-level relevant resources are also very much appreciated!

In short, I would like to know the following: $$\tilde{\rho}(t) \sim \ ?$$

What I have figured out so far: If we take the expectation of Eq (1) we get the Lindblad master equation, and its solution $$\rho(t)$$ is the mean of $$\tilde{\rho}(t)$$, i.e. $$\mathbb{E}\left( \tilde{\rho}(t) \right) = \rho(t).$$

• For a pedagogical, beginner-level, derivation of the Lindblad master equation without the stochastic term see this. I never worked with stochastic master equations, but Pierre Rouchon is a name which I associate them to. I hope this helps. Jan 4, 2023 at 10:54
• @GFrazao thanks for the links, although the first link doesn't work. I've seen the paper by Pierre Rouchon and unfortunately, it doesn't have the info I need. Jan 4, 2023 at 22:33
• It seems that there are algorithms solving it numerically(I think searching stochastic master equation will find the ref easily? Not sure if this link helps), but I'm not sure if there is a method to solve this analytically for the general case. Jan 5, 2023 at 0:53
• this looks very interesting thanks @narip Jan 5, 2023 at 1:55
• MonteNero: the first link missed a ":" after "https". @narip: cool ref, it will help me too. Jan 5, 2023 at 15:06