Exact results for models of multichannel quantum nonadiabatic transitions

We consider nonadiabatic transitions in explicitly time-dependent systems with Hamiltonians of the form $\widehat{H}\left(t\right)=\widehat{A}+\widehat{B}t+\widehat{C}/t$, where $t$ is time and $\widehat{A},\widehat{B},\widehat{C}$ are Hermitian $N\times N$ matrices. We show that in any model of this type, scattering matrix elements satisfy nontrivial exact constraints that follow from the absence of the Stokes phenomenon for solutions with specific conditions at $t\to -\infty$. This allows one to continue such solutions analytically to $t\to +\infty$, and connect their asymptotic behavior at $t\to -\infty$ and $t\to +\infty$. This property becomes particularly useful when a model shows additional discrete symmetries. In particular, we derive a number of simple exact constraints and explicit expressions for scattering probabilities in such systems.

Figure 1

Contours of complex time for evolution with Eq. (1) connecting real points at $t\to \pm \infty$. (a) Arbitrary contour ${\mathit{C}}^{\prime }$ laying in the upper half-plane can be deformed without encountering singularities into the contour $\mathit{C}$ shown in (b) that has the shape of a semicircle with radius $R$. This contour crosses two rays, $t=R{e}^{i\varphi }$, at $\varphi =3\pi /4$ and $\varphi =\pi /4$. Along those rays, the highest slope amplitude has the highest rate of, respectively, decay and growth with increasing $R$. (c) Without changing asymptotic values of the solution at real $t\to \pm \infty$, the contour $\mathit{C}$ can be deformed to the contour that is placed almost everywhere on the real time axis, except an infinitely small semicircle that avoids the singularity of the Hamiltonian (3) at $t=0$. (d) Applying the same arguments to the extremal amplitude with the lowest slope of diabatic energy, we arrive at a contour $\mathit{C}$ at the lower complex half-plane that can be deformed to a contour along the real axis except an infinitely small semicircle around $t=0$ in the lower half-plane.

Figure 2

Plot of diagonal elements of the matrix $\widehat{B}t+\widehat{A}$ in the diabatic basis can be used to identify extremal states and counterintuitive transitions. For the shown system of six states, levels 1 and 2 have the highest slope, levels 5 and 6 have the lowest slope, and transitions from level 6 to level 5 and from level 1 to level 2 are counterintuitive.

Figure 3

(a) Numerical test of Eq. (38). Solid curves are theoretical predictions and discrete points are numerical results. Parameters: $\beta =2,k=0.5$. Solution of Eq. (1) was obtained for the time interval $t\in (0.001,1000)$ with initial condition $|\psi \rangle =|+\rangle$. Details of the numerical program are discussed in the supplementary file for Ref. [22]. (b) Plot of eigenvalues of the Hamiltonian for the model (32) as functions of time at $g=0.4$. At $t\to 0$, eigenstates coincide with states $|\pm \rangle$, while at $t\to +\infty$, eigenstates approach the diabatic states $|1\rangle$ and $|2\rangle$.

Figure 4

(a) Plot of eigenvalues of the Hamiltonian (73). (b) Numerical test of Eq. (72). Parameters for numerical simulations: $g_1=0.5,g_2=0.7,\epsilon =0.5$. The evolution time interval is $t\in (0.001,1000)$. Solid curves are theoretical predictions and discrete points are numerical results.

Figure 5

(a) Plot of eigenvalues of the Hamiltonian (82) as functions of time at $g=0.3,\epsilon =0.5$. (b)–(d) Numerical test of Eqs. (91) and (96) at $\epsilon =0.5$ for different couplings $g$. Evolution time interval is $t\in (0.001,1000)$. Solid curves are theoretical predictions and discrete points are results of the numerical solution of the evolution equation with the Hamiltonian (82).

Figure 6

(a) Plot of eigenvalues of the Hamiltonian (97) as functions of time at $g=\epsilon =0.5,k=2$. (b)–(d) Numerical test of Eqs. (104) at $\epsilon =0.5,k=0.3$ for different couplings $g$. Evolution time interval is $t\in (0.001,1000)$. Solid curves are theoretical predictions and discrete points are results of the numerical solution of the evolution equation with the Hamiltonian (97).

Figure 7

(a) The integration contour $\mathit{A}$ enclosing all branch cuts (dashed lines) from a large distance. (b) Each integral over ${\mathit{\gamma }}_n$ can be transformed into the integral over the contour $\mathit{C}$ by a change of variables.