Semiclassical propagation of spin-coherent states

The semiclassical propagation of spin-coherent states is considered in complex phase space. For two time-independent systems, we find the appropriate classical trajectories and show that their combined contributions are able to describe quantum interference with great accuracy. Not only the modulus but also the phase of the quantum propagator, both dynamical and geometric terms combined, are accurately reproduced.

Figure 1

The normalized propagator as a function of scaled time $\tau =\hslash \nu T$ (in units of the classical period ${\tau }_c$) for $z_i=0.5$ and $j=50$. In (a) we see the separate contributions of six branches (each peak is due to a different branch). When they are added coherently we get (b), which is indistinguishable from the exact result, including the fast oscillations.

Figure 2

The different branches in the $\overline{z}\left(0\right)=x+iy$ plane contributing to the semiclassical return probability, for the same parameters as in the previous figure (quantities are dimensionless). These curves as parametrized by the time, and each one of them passes through the point (0.5, 0) at a different instant. Only the vicinity of this point is shown, for the sake of visibility.

Figure 3

The normalized propagator as a function of scaled time (in units of ${\tau }_r$) for $z_i=1$ and $j=10$. In (a) we see the separate contributions of 11 branches (all of them except one must be counted twice, see text). When they are added coherently we get the solid line in (b). The dashed line is the exact result.

Figure 4

The semiclassical (solid) and the exact (dashed) calculations of the real part of the normalized propagator, for $z_i=1$ and $j=10$. The imaginary part is also reproduced with great accuracy. Therefore the total phase, dynamical plus geometrical terms, can be recovered from the semiclassical expressions.

Figure 5

The normalized propagator as a function of scaled time (in arbitrary units) for $z_i=1+i$ and $j=10$ in the second example. In (a) we see the separate contributions of six branches. When they are added coherently we get the solid line in (b). The dashed line is the exact result.

Figure 6

The semiclassical (solid) and the exact (dashed) calculations of the real part of the normalized propagator, in the same situation as the previous figure. Once again the phase is accurately reproduced.