# How do I add decoherence to an oscillating system

If a have an initial (two-qubit) system in the state

$$\rho_i= \begin{pmatrix} 0&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix}$$

and this state evolves according to the evolution, $$U=e^{-iHt}$$ with

$$H=A(\sigma_x\otimes\sigma_x+\sigma_y\otimes\sigma_y+\sigma_z\otimes\sigma_z)$$,

I can find the evolution of qubit $$1$$ after some time, $$t$$, by taking the partial trace, which, for some matrix

$$\rho_{AB}= \begin{pmatrix} a&b&c&d\\ e&f&g&h\\ i&j&k&l\\ m&n&o&p \end{pmatrix}$$

is given by

$$\rho_A= \begin{pmatrix} a+f&c+h\\ i+n&k+p \end{pmatrix}$$.

For $$\rho_i$$, plotting the first element of this gives the evolution of qubit $$1$$ (shown in the Figure).

I want to add a decoherence term to this first qubit only. I know the decay is exponential and happens on a $$\mu$$s timescale ($$T_1=1\mu$$s) and that the decay is such that the system approaches $$0.5$$ on the $$y$$ axis, however, I can't work out where to introduce this decay to give the result I am expecting. Any help would be appreciated!