System

Assume a system with Hilbert space ${\cal H}=C^2$ with basis vectors $(1,0)^{\dagger}$ and $(0,1)^{\dagger}$. In general, a `System'-Hamiltonian will have the form of the Hamiltonian of a Pseudo Spin $\frac{1}{2}$ in a (time-dependent) classical pseudo magnetic field ${\bf B}(t)$ (c-number),

$$H_S(t)\equiv {\bf B}(t){\vec{\sigma}},\quad \vec{\sigma}=\lef... ...array}[h]{c} \sigma_x\\ \sigma_y\\ \sigma_z \end{array}\right).$$ (109)

• Note that for a time-dependent ${\bf B}(t)$, the free Schrödinger equation with $H_S(t)$ only is in general not analytically solvable. For an isolated two-level system ($H_S(t)$ only), this is not a problem because one can easily solve a two-by two differential equation on a computer. However, problems start when it comes to system-bath Hamiltonians. Many of the `simpler' system bath theories implicitely assume that the time-evolution under $H_s$ is trivial (which it is for constant ${\bf B}(t)={\bf B}$).
• Some special cases are analytically solvable: Landau-Zener-Rosen tunneling (Landau 1932).
• For a periodic time-dependence of ${\bf B}(t)$: Floquet theory (Shirley 1965).
• The wave function can aquire a geometrical phase (Berry phase, Berry 1984).