Is the dynamics of open quantum systems always linear?

We study the influence of the preparation of an open quantum system on its reduced time evolution. In contrast to the frequently considered case of an initial preparation where the total density matrix factorizes into a product of a system density matrix and a bath density matrix the time evolution generally is no longer governed by a linear map nor is this map affine. Put differently, the evolution is truly nonlinear and cannot be cast into the form of a linear map plus a term that is independent of the initial density matrix of the open quantum system. As a consequence, the inhomogeneity that emerges in formally exact generalized master equations is in fact a nonlinear term that vanishes for a factorizing initial state. The general results are elucidated with the example of two interacting spins prepared at thermal equilibrium with one spin subjected to an external field. The second spin represents the environment. The field allows the preparation of mixed density matrices of the first spin that can be represented as a convex combination of two limiting pure states, i.e., the preparable reduced density matrices make up a convex set. Moreover, the map from these reduced density matrices onto the corresponding density matrices of the total system is affine only for vanishing coupling between the spins. In general, the set of the accessible total density matrices is nonconvex.

Figure 1

The Bloch vector component $S_{z1}$ as a function of the field $\beta F_z$ resulting from Eq. (28) for $\beta e=1$. With increasing coupling parameter $\beta g=0.5,1,1.5$ the slope at $F_z=0$ decreases. Apparently, the functions $S_{z1}\left(F\right)$ are monotonic and hence possess a unique inverse.

Figure 2

The Bloch vector component $S_{z2}$ as a function of $S_{z1}$ resulting from Eq. (28) for $\beta e=1$ and different values $\beta g=0,0.5,1,1.5$. Note that a strictly linear dependence results only for the case of vanishing coupling $g=0$. An approximate linear regime exists in a neighborhood of $S_{1z}=0$.

Figure 3

The nonvanishing correlation functions $C_{xx},C_{yy},C_{zz}$ as a function of the Bloch vector component $S_z$ for $\beta e=1$ and $\beta g=1.5$. The correlation functions $C_{xx}$ and $C_{yy}$ clearly deviate from straight lines following from a convex blow-up map, see Eq. (31). For the correlation functions also an approximately linear regime exists in a neighborhood of $S_{1z}=0$.