Solution of the PDE I: zero temperature $T=0\leadsto n_B=0$

In this case, we only have first order derivatives. There is a (more or less) complete theory of first order PDEs: they are solved by the method of characteristics (cf. Courant/Hilbert).

We write the PDE as

$$\left\{ \frac{\partial}{\partial t} - i\left[ \bar{\Omega} - i\ka... ...a\right]z^*\frac{\partial}{\partial z^*} \right\} P(z,z^*t) = 2\kappa P(z,z^*t)$$ (80)

and consider the function $P(z,z^*,t)$ on trajectories $z=z(t)$ and $z^*=z^*(t)$ where $P(z,z^*,t)=P(z(t),z^*(t),t)$. We regard the l.h.s. of Eq.(7.81) as a total differential. Along the trajectories, the temporal change of $P$ is

$$\frac{d}{dt} P(z(t),z^*(t),t) = \left\{ \dot{z}(t) \partial_z +\dot{z}^*(t) \partial_{z^*} +\partial_t \right\} P(z(t),z^*(t),t)$$

$$= 2\kappa P(z(t),z^*(t),t)$$ (81)

Comparison yields

$$\dot{z}(t) = - i\left[ \bar{\Omega} - i\kappa\right]z(t)\leadsto z(t)=z_0e^{- i\left[ \bar{\Omega} - i\kappa\right]t}$$

$$\dot{z}^*(t) = i\left[ \bar{\Omega} + i\kappa\right]z^*(t)\leadsto z^*(t)=z_0^*e^{ i\left[ \bar{\Omega} + i\kappa\right]t}.$$ (82)

On the other hand, $\frac{d}{dt} P = 2\kappa P$ yields

$$P(z(t),z^*(t),t) = e^{2\kappa t}P_0(z_0,z_0^*).$$ (83)

Here, $P_0$ is the initial condition for $P$, with $z_0=z(t=0)$ and $z_0^*=z^*(t=0)$. This looks very innocent but has a deep physical (and geometrical) meaning: we can trace back our trajectories $z(t)$,$z^*(t)$ to their origin $z_0$, $z_0^*$, writing

$$z_0 = z(t)e^{+i\left[ \bar{\Omega} - i\kappa\right]t},\quad z_0^*= z^*(t)e^{-i\left[ \bar{\Omega} + i\kappa\right]t}.$$ (84)

We thus have expressed the inital values $z_0$, $z_0^*$ in terms of the `final' values $z(t)$,$z^*(t)$. Insertion into Eq.(7.84) yields

$$P(z(t),z^*(t),t) = e^{2\kappa t}P_0\left( z(t)e^{+i\left[ \bar{\Omega} - i\kappa\right]t}, z^*(t)e^{-i\left[ \bar{\Omega} + i\kappa\right]t}\right).$$ (85)

We now write again $z$ and $z^*$ instead of $z(t)$, $z^*(t)$, and therefore have

$$P(z,z^*,t) = e^{2\kappa t}P_0\left( ze^{+i\left[ \bar{\Omega} - i\kappa\right]t}, z^*e^{-i\left[ \bar{\Omega} + i\kappa\right]t}\right).$$ (86)