# Single-Qubit Density Matrix Evolution

In this paper general form of reduced single-qubit density matrix evolution interacting with bosonic reservoir can be cast in the following form:

$$\rho^s(t)=\begin{pmatrix} \rho^s_{11}(t) & \rho^s_{10}(t)\\ \rho^s_{01}(t) & \rho^s_{11}(t) \end{pmatrix}$$

Where

$$\rho^s_{11}(t)=u^s_t\rho^s_{11}(0)+v^s_t\rho^s_{00}(0)$$

$$\rho^s_{00}(t)=(1-u^s_t)\rho^s_{11}(0)+(1-v^s_t)\rho^s_{00}(0)$$

$$\rho^s_{10}(t)=\rho^{s*}_{10}(t)=z^s_t\rho^s_{10}(0)$$

With $$u^s_t$$, $$v^s_t$$,$$z^s_t$$ a function of time

Is there an intuitive reason why they use the following form ? Is it general enough ? I can see a little resemblance of evolution of state by amplitude damping (population and coherence term is separated) but the paper state want to study more general term of qubit interaction with the environment (Markovian and Non-Markovian). Why diagonal element of the density matrix is not a function of off diagonal matrix ?