Complex saddle trajectories for multidimensional quantum wave packet and coherent state propagation: Application to a many-body system

A practical search technique for finding the complex saddle points used in wave packet and coherent state propagation is developed which works for a large class of Hamiltonian dynamical systems with many degrees of freedom. The method can be applied to problems in atomic, molecular, and optical physics and other domains. A Bose-Hubbard model is used to illustrate the application to a many-body system where discrete symmetries play an important and fascinating role. For multidimensional wave packet propagation, locating the necessary saddles involves the seemingly insurmountable difficulty of solving a boundary value problem in a high-dimensional complex space, followed by determining whether each particular saddle found actually contributes. In principle, this must be done for each propagation time considered. The method derived here identifies a real search space of minimal dimension, which leads to a complete set of contributing saddles up to intermediate times much longer than the Ehrenfest timescale for the system. The analysis also gives a powerful tool for rapidly identifying the various dynamical regimes of the system.

Figure 1

Typical saddle family characteristics. The oscillating curve in the upper panel is the real part of $\mathcal{A}\left(t\right)$ for one particular saddle family, and 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 with time. Each saddle family member has an energy and total particle number ($n_T$) surface to which it belongs. At short times, the real parts of the energy and particle number of the saddle trajectory are greater than the energy and particle number expectation values of the wave packet, and at longer times they are less than the expectation values. The saddle family's peak contribution occurs near where the real parts of the energy and $n_T$ equals the energy and total particle number (here $\langle n_T\rangle =40$) expectation values of the wave packet. This saddle family is taken from an example of the Bose-Hubbard model defined ahead in Sec. 3.

Figure 2

View of seed trajectory locations in time and initial conditions. For 4000 initial conditions, the distance function is calculated as a function of time. The initial conditions are chosen uniformly along the eigenvector mentioned in the text across the interval $[-4\sigma ,4\sigma ]$ corresponding to the Wigner transform of the coherent state density wave. The points are blackened where $D_{\gamma }\le 20$. The initial conditions are labeled by an index on the $y$ axis. The full discrete symmetry is encapsulated by a reflection symmetry with respect to the $x$ axis. The multiplicity 1 saddles are found using the $y=0$ line seed trajectories, and the rest have multiplicity 2.

Figure 3

Total number of saddles as a function of time. The saddles counted are those found with seed trajectories with a $D_{\gamma }\le 20$. A dashed quadratic curve is shown as a guide.

Figure 4

Equivalent of Fig. 2 for the eigenvector perpendicular to the energy surface. This eigenvector is associated with the second largest eigenvalue. For 1000 initial conditions, the distance function is calculated as a function of time. The initial conditions are chosen uniformly along the eigenvector across the interval $[-4\sigma ,4\sigma ]$ corresponding to the Wigner transform of the coherent state density wave. The points are blackened where $D_{\gamma }\le 20$. The initial conditions are labeled by an index on the $y$ axis.

Figure 5

$D_{\gamma }$ near the revival time. Each black spot gives rise to a single seed trajectory, and hence there is a one-to-one correspondence between spots and saddles. A $300\times 300$ grid of initial conditions were used to calculate $D_{\gamma }\le 20$ at ${\tau }_2$. The six dotted lines correspond to the six symmetry-related vectors of $(\delta n,-\delta n,0)$. The three long-dashed and three medium-dashed lines correspond to cyclic permuations of $(\delta n,\delta n,-2\delta n)$ and its negative. They separate the plane into twelve domains. The domains $I^{-}$ and $I^{+}$ are mirror images of each other about the dashed line between them, similarly for $I{I}^{-}$ and $I{I}^{+}$. The three copies of each domain, $I^{\pm }$ and $I{I}^{\pm }$, are related by ${120}^{\circ }$ rotation. One choice for a fundamental domain would be the sum of the regions $I^{-}$ and $I{I}^{+}$ adjacent on the right side of the figure.

Figure 6

$\mathcal{M}$ as a function of time. The effective multiplicity of the saddles begins at one and increases toward the maximum possible of six in this case as time increases. At long times, it saturates at six and remains there as these saddles come from a higher dimensional space of initial conditions than the others.

Figure 7

Spectrum, $\left\{{log}_{10}\left({\lambda }_{j,+}\right)\right\}$, of ${\mathbf{M}}_{\tau }·{\mathbf{A}}_{\alpha }^{-1}·{\mathbf{M}}_{\tau }^T$ as a function of $\theta$. The initial state is a coherent state density wave for a ring with eight sites whose parameters are given in the main text. All $8{\lambda }_{j,+}$ are shown, but degeneracies make it appear as though fewer are plotted. There is a strong realignment of the associated eigenvectors, $\left\{{\left(\genfrac{}{}{0pt}{}{\delta {\vec{p}}_0}{\delta {\vec{q}}_0}\right)}_j\right\}$, at the transition point in the spectrum. To the right of the transition, the eigenvectors essentially do not involve the initially unoccupied sites, whereas to the left, all the sites are involved and there is a double repetition around the ring in the structure of the eigenstates.

Figure 8

Largest eigenvalue $\left\{{log}_{10}\left({\lambda }_{1,+}\right)\right\}$ of ${\mathbf{M}}_{\tau }·{\mathbf{A}}_{\alpha }^{-1}·{\mathbf{M}}_{\tau }^T$ as a function of $\theta$. The initial state is a coherent state density wave for rings with 4,6,8,10,12,14,16,18 sites with populated sites of mean number $n=5$. All eight cases of ${\lambda }_{1,+}$ are shown, but portions of the curves with fewer sites are copied in rings with greater numbers of sites. For example, the 12-site ring follows partly the 4-site ring and partly the 6-site ring results. That makes it appear as though fewer examples are plotted. The upper panel is a magnification of the transition region, which is magnified again in the lower panel near the sharp peak.

Figure 9

Spectrum, $\left\{{log}_{10}\left({\lambda }_{j,+}\right)\right\}$, of ${\mathbf{M}}_{\tau }·{\mathbf{A}}_{\alpha }^{-1}·{\mathbf{M}}_{\tau }^T$ as a function of $\theta$. The upper panel shows the $\left\{{log}_{10}\left({\lambda }_{j,+}\right)\right\}$ of a four-site ring with equal populations and phase relations; see text for details. The lower panel shows the $\left\{{log}_{10}\left({\lambda }_{j,+}\right)\right\}$ for an eight-site ring with an alternating phase relationship from site to site; see text for details. There is no abrupt dynamical transition for these examples as there is for coherent state density waves. There is a transition to chaotic dynamics in the lower panel and the greatest degree of instability seen for any trajectories.