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Mode transformation for a Schrödinger type equation: avoided and unavoidable level crossings. (English) Zbl 1461.81031

In quantum mechanics, the dynamics of a system is described by the Schrödinger equation \[ -i \hbar \partial_t \Psi(t) = \mathcal{H}(t) \Psi(t), \] with \(\mathcal{H}(t)\) being the self-adjoint Hamiltonian operator with instantaneous eigenvalues \(\varepsilon_n(t)\) and corresponding eigenvectors \(\varphi_n(t)\). For a time-independent Hamiltonian, if the system starts in an eigenstate \(\varphi_n\) at \(t=0\), then its time-evolution is simply \(\Psi_n(t) = e^{i\varepsilon_n t/\hbar} \varphi_n\), i.e., a system initialized in an eigenstate stays in the same state. The adiabatic theorem tells us that a similar statement holds for time-dependent Hamiltonians under certain conditions: if the system is initialized in an instantaneous eigenstate \(\varphi_n(t)\) at \(t=0\), then its time-evolution is given by \(\Psi_n(t) = \text{exp}\left\{ \frac{i}{\hbar} \int_0^t \varepsilon_n(t') dt'\right\} \varphi_n(t)\), provided that \(\varepsilon_n(t)\) stays away from all other eigenvalues. If this condition is violated at some time \(t = t_0\), i.e., if \(\left\lvert \varepsilon_n(t_0) - \varepsilon_m(t_0) \right\rvert\) is “small” for some \(\varepsilon_m(t)\), then the system can transition to \(\varphi_m(t)\) with a finite probability. One is then interested in the transition matrix \(\mathcal{T}\) that connects the adiabatic modes at different sides of \(t=t_0\), which is given by the Landau-Zener formula.
In the present article, the authors extend the Landau-Zener formula to a more general class of systems, whose dynamics is given by \[ -i \hbar \, \Gamma \partial_t \Psi(t) = \widehat{\mathcal{K}}(t) \Psi(t), \] with self-adjoint \(\widehat{\mathcal{K}}(t)\) and \(\Gamma\). This can also be thought of as the Schrödinger equation with Hamiltonian \(\mathcal{H}(t) = \Gamma^{-1} \widehat{\mathcal{K}}(t)\), which, in general, is not self-adjoint. In quantum mechanics, such “Hamiltonians” are used to describe open systems. They are also useful in other problems of physical interest, where \(t\) is a spatial coordinate. Examples of such systems include the scattering problem for a Dirac electron (wherein the Hamiltonian has first order spatial derivatives) and wave propagation through a slowly irregular waveguide.
For the computation of the transition matrices, the authors consider systems where \(\widehat{\mathcal{K}}(t) = \mathcal{K}(t) + \sqrt\hbar \mathcal{B}(t)\), such that the generalized eigenvalues satisfying \(\mathcal{K}(t) \varphi_n(t) = \varepsilon_n(t) \Gamma \varphi_n(t)\) exhibit a twofold degeneracy at \(t=0\) such that \(\varepsilon_2(t) - \varepsilon_1(t) \sim 2Qt\) for some \(Q>0\) as \(t\to0\). The small perturbation \(\sqrt\hbar \mathcal{B}(t)\) then lifts this degeneracy, yielding the Landau-Zener setup. The transition matrix for such systems – with certain constraints on the analyticity and boundedness of various operators – is computed explicitly. This involves construction of separate asymptotic solutions inside and outside neighborhood of \(t=0\), whose matching then yields the transition matrix.
Away from \(t = 0\), the adiabatic theorem holds and an asymptotic solution was constructed by substituting the ansatz \(\Psi(t) = \Phi(t) \exp\left\{ \frac{i}\hbar \int^t \vartheta(t') dt' \right\}\), expanding both \(\Phi(t)\) and \(\vartheta(t)\) in powers of \(\sqrt{\hbar}\) and solving the Schrödinger equation order by order. On the other hand, in a small neighborhood of \(t=0\), an asymptotic solution was constructed by first expanding the operators in a Taylor series in \(t\) and switching to the slow variable \(\tau \equiv t/\sqrt\hbar\). This renders the Schrödinger equation in a form where an asymptotic solution \(\Psi(t)\) can again be obtained by expanding its magnitude and phase separately as a power series in \(\sqrt\hbar\). The transition matrix was then obtained by matching the asymptotics of the two solutions obtained above, the former in the limit of \(t\to0\) and the latter in the limit of \(\tau\to\infty\).

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
81Q37 Quantum dots, waveguides, ratchets, etc.
35C20 Asymptotic expansions of solutions to PDEs
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
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