Corrected Maslov index for complex saddle trajectories

Saddle-point approximations, extremely important in a wide variety of physical contexts, require the analytical continuation of canonically conjugate quantities to complex variables in quantum mechanics. An important component of this approximation's implementation is arriving at the phase correction attributable to caustics, which involves determinantal prefactors. The common prescription of using the inverse of half a certain determinant's total accumulated phase sometimes leads to sign errors. The root of this problem is traced to the zeros of the determinants at complex times crossing the real time axis. Deformed complex time contours around the zeros can repair the sign errors that sometimes occur, but a much more practical way is given that links saddles back to associated real trajectories and avoids the necessity of locating the complex time zeros of the determinants.

Figure 1

The real part of $\mathcal{A}\left(t\right)$ for one particular saddle family; the envelope is the absolute value. There is a faster phase oscillation at short times decreasing as time increases corresponding to changes in the complex saddle trajectory. At short times, the real parts of the saddle trajectory's energy are greater than the wave packet's energy expectation value, and at longer times they are less. The saddle family's peak contribution occurs near where the two are equal.

Figure 2

Contribution to the evolution of a wave packet from a single continuous family of saddles as a function of position; the saddles are linked to a classical foliation of real initial conditions labeled by $⑦$ in Fig. 5 ahead. The initial wave packet, Eq. (2), is centered at $(q_{\alpha },p_{\alpha })=(0,20)$ of width parameter $b_{\alpha }=32$ and propagated with Eq. (12). The propagation time is equal to $3\tau$ where $\tau$ is the period of motion for the centroid initial condition $(q_{\alpha },p_{\alpha })$.

Figure 3

The evolution of the total accumulated phase for a time $t=3\tau$. In the left panel, the real and imaginary parts of ${\mathcal{D}}_1$ are plotted for all times $0\le t\le 3\tau$. Real trajectory initial conditions are used. At $t=0$, the ${\mathcal{D}}_1$ curve starts at the point (1,0) and proceeds always in a counterclockwise direction as time increases up to the final time $3\tau$. In the right panel, the phase for the critical saddle begins similarly, but the complexification of the stability matrix elements generates a point at which ${\mathcal{D}}_1=0$.

Figure 4

The location in time of zeros for three saddles from foliation $⑦$. The middle zero at each of the arrows' origination point is for $x=0.0$. Following the up arrows are the zeros for $x=1.9008$, and likewise, following the down arrows $x=-1.9008$. For this latter point, the zero a bit before $t=2\tau$ has just crossed the real axis, the point $x=-1.9008$ is just to the left of a sign flip. The next zero is very close to crossing, but has not yet. A small further decrease in the $x$ value leads to a second sign flip. There is one zero not pictured for $0\le t\le \tau$, but its imaginary value is quite large and positive too high to be seen on the figure's scale.

Figure 5

Illustration of distinct classical transport pathways. The vertically oriented ellipses are contours of the Wigner transform of the initial wave packet defined using a single degree of freedom version of Eq. (2) and the parameters given in the Fig. 2 caption. They are propagated classically for three periods of motion of the centroid, and they generate the distorted spirals. The $5\sigma$ contour can be divided into nine foliations, a trajectory member of which can be used to generate initial conditions for a Newton-Raphson scheme to locate unique saddles. The blackened dots show the locations of the real parts of the nine associated saddles' complex final phase-space points contributing to $\varphi (x=0,t)$ (the initial conditions follow by back propagating a time $3\tau$).

Figure 6

The real parts of $M_{22}$ and $M_{21}$ for the saddle of foliation $⑧$ at $x=0$ and its associated real trajectory. If the stability matrix elements were not so similar, the Newton-Raphson search for the saddle could not converge since they are essential to the method.

Figure 7

The imaginary parts of $M_{22}$ for the saddles of foliations $⑥$ and $⑧$ at $x=0$. For the associated real trajectories, the imaginary parts vanish for all times. The stability matrix elements' imaginary parts using the saddles are orders of magnitude smaller than the real parts of Fig. 6. As noted the saddle of foliation $⑦$ is equivalent to its associated real orbit, and its imaginary parts have vanished for this case. Foliations $⑥$ and $⑧$ are the closest neighboring foliations on opposite sides. Their stability matrix elements' imaginary parts are reflected, which is a trend that continues. All the lower energy foliations possess an $\text{Im}\left(M_{11}\right)$ that goes negative first, which is the opposite of the higher-energy foliations.

Figure 8

Rotation sense of the total accumulated phase of the foliation $⑥$ ($x=0$) determinant ${\mathcal{D}}_1$. For roughly half a cycle of the motion it rotates counterclockwise, and then reverses itself to a clockwise rotation sense for the remainder of its propagation. Nevertheless, its final point is very close to the final point of its associated real trajectory, which rotated exclusively counterclockwise.