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arXiv:quant-ph/9507007

The Leading Order of the Theory of Strong Perturbations in Quantum Mechanics

Marco Frasca

Published Jul 12, 1995• 0 citations
arXivHTMLPDF

Abstract

We prove that, for a quantum system that undergoes a strong perturbation, the solution of the leading order equation of the strong field approximation (M.Frasca, Phys. Rev. A, {\bf 45}, 43 (1992)) can be derived by the adiabatic approximation. In fact, it is shown that greatest is the perturbation and more similar the quantum system is to an adiabatic one, the solution being written as a superposition of eigenstates of the time-dependent perturbation.A direct consequence of this result is that the solution of the Schrödinger equation in the interaction picture, in the same approximation for the perturbation, coincides with the one of the leading order of the strong field approximation. The limitation due to the requirement that the perturbation has to commute at different times is so overcome. Computational difficulties could arise to go to higher orders. Beside, the method is not useful for perturbations that are constant in time. In such a case a small time series is obtained, indicating that this approximation is just an application to quantum mechanics of the Kirkwood-Wigner expansion of statistical mechanics. The theory obtained in this way is applied to a time-dependent two-level spin model, already considered for the study of the Berry's phase, showing that a geometrical phase could arise if a part of the hamiltonian is considered as a strong perturbation. No adiabatic approximation is taken on the parameters of the hamiltonian, while their cyclicity is retained.

DOI
10.48550/arxiv.quant-ph/9507007

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We prove that, for a quantum system that undergoes a strong perturbation, the solution of the leading order equation of the strong field approximation (M.Frasca, Phys. Rev. A, {\bf 45}, 43 (1992)) can be derived by the adiabatic approximation. In fact, it is shown that greatest is the perturbation and more similar the quantum system is to an adiabatic one, the solution being written as a superposition of eigenstates of the time-dependent perturbation.A direct consequence of this result is that the solution of the Schrödinger equation in the interaction picture, in the same approximation for the perturbation, coincides with the one of the leading order of the strong field approximation. The limitation due to the requirement that the perturbation has to commute at different times is so overcome. Computational difficulties could arise to go to higher orders. Beside, the method is not useful for perturbations that are constant in time. In such a case a small time series is obtained, indicating that this approximation is just an application to quantum mechanics of the Kirkwood-Wigner expansion of statistical mechanics. The theory obtained in this way is applied to a time-dependent two-level spin model, already considered for the study of the Berry's phase, showing that a geometrical phase could arise if a part of the hamiltonian is considered as a strong perturbation. No adiabatic approximation is taken on the parameters of the hamiltonian, while their cyclicity is retained.
A perturbation series is derived for the Schr\"odinger equation where the perturbation is permitted to go to \ensuremath{\infty}. Such an asymptotic series can be obtained in a representation where the time evolution of the states is due to the unperturbed part of the Hamiltonian of the problem and the time evolution of the observables is determined by the perturbation, contrary to the interaction picture. The equivalence between the series given by such a picture, and an asymptotic perturbation scheme is given. The method is applied to a spin-1/2 particle in a two-component magnetic field, one component of which, considered as a perturbation, can be time varying. Choosing a constant perturbation we show the equivalence between the exact solution and the approximate one when a component of the magnetic field is much larger than the other. In this case the limit t\ensuremath{\rightarrow}\ensuremath{\infty} cannot be taken, as mixed-secular terms appear in the asymptotic series. Taking a linear time-varying dependence for a component of the magnetic field, we get a nonanalytic asymptotic series.
We introduce a perturbative approach to solving the time-dependent Schr\"odinger equation, named adiabatic perturbation theory (APT), whose zeroth-order term is the quantum adiabatic approximation. The small parameter in the power series expansion of the time-dependent wave function is the inverse of the time it takes to drive the system's Hamiltonian from the initial to its final form. We review other standard perturbative and nonperturbative ways of going beyond the adiabatic approximation, extending and finding exact relations among them, and also compare the efficiency of those methods against the APT. Most importantly, we determine APT corrections to the Berry phase by use of the Aharonov-Anandan geometric phase. We then solve several time-dependent problems, allowing us to illustrate that the APT is the only perturbative method that gives the right corrections to the adiabatic approximation. Finally, we propose an experiment to measure the APT corrections to the Berry phase and show, for a particular spin-1∕21∕21∕2 problem, that to first order in APT the geometric phase should be two and a half times the (adiabatic) Berry phase.
A precise definition of an adiabaticity parameter \ensuremath{\nu} of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(\ensuremath{\tau})=${\ensuremath{\sum}}_{\mathrm{\ensuremath{\ell}}}$${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) with ${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) being at least of the order ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\ell}}}$. In particular, U(0){\mathrm{U}}^{(0)}U(0)(\ensuremath{\tau}) corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the 1/\ensuremath{\tau} expansion of U(\ensuremath{\tau}). It is also shown that the nonadiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. This suggests the introduction of an adiabatic product expansion for U(\ensuremath{\tau}) which turns out to yield exact expressions for U(\ensuremath{\tau}) for a large number of quantum systems. In particular, a simple application of the adiabatic product expansion is used to show that for the Hamiltonian describing the dynamics of a magnetic dipole in an arbitrarily changing magnetic field, there exists another Hamiltonian with the same eigenvectors for which the Schr\"odinger equation is exactly solvable. Some related issues concerning geometric phases and their physical significance are also discussed.
We present details and expand on the framework leading to the recently introduced degenerate adiabatic perturbation theory [Phys. Rev. Lett. 104, 170406 (2010)], and on the formulation of the degenerate adiabatic theorem, along with its necessary and sufficient conditions [given in Phys. Rev. A 85, 062111 (2012)]. We start with the adiabatic approximation for degenerate Hamiltonians that paves the way to a clear and rigorous statement of the associated degenerate adiabatic theorem, where the non-Abelian geometric phase (Wilczek-Zee phase) plays a central role to its quantitative formulation. We then describe the degenerate adiabatic perturbation theory, whose zeroth-order term is the degenerate adiabatic approximation, in its full generality. The parameter in the perturbative power-series expansion of the time-dependent wave function is directly associated to the inverse of the time it takes to drive the system from its initial to its final state. With the aid of the degenerate adiabatic perturbation theory we obtain rigorous necessary and sufficient conditions for the validity of the adiabatic theorem of quantum mechanics. Finally, to illustrate the power and wide scope of the methodology, we apply the framework to a degenerate Hamiltonian, whose closed-form time-dependent wave function is derived exactly, and also to other nonexactly solvable Hamiltonians whose solutions are numerically computed.
A new and intuitive perturbative approach to time-dependent quantum mechanics problems is presented, which is useful in situations where the evolution of the Hamiltonian is slow. The state of a system which starts in an instantaneous eigenstate of the initial Hamiltonian is written as a power series which has a straightforward diagrammatic representation. Each term of the series corresponds to a sequence of ``adiabatic'' evolutions, during which the system remains in an instantaneous eigenstate of the Hamiltonian, punctuated by transitions from one state to another. The first term of this series is the standard adiabatic evolution, the next is the well known first correction to it, and subsequent terms can be written down essentially by inspection. Although the final result is perhaps not terribly surprising, it seems to not be widely known, and the interpretation is new, as far as we know. Application of the method to the adiabatic approximation is given, and some discussion of the validity of this approximation is presented.
We present a full introduction to the recent devised perturbation theory for strong coupling in quantum mechanics. In order to put the theory in a proper historical perspective, the approach devised in quantum field theory is rapidly presented, showing how it implies a kind of duality in perturbation theory, from the start. The approach of renormalization group in perturbation theory is then presented. This method permits to resum secularities in perturbation theory and makes fully algorithmical the resummation, transforming the perturbation calculations in a step by step computational procedure. The general theorem on which is founded a proper application of the strong coupling expansion, based on a result in the quantum adiabatic theory, is then exposed. This theorem gives the leading order of a strong coupling expansion. Then, after the introduction of the principle of duality in perturbation theory that puts in a proper context the quantum field theory method, the resulting theory of the strong coupling expansion and the free picture are presented. An algorithm for the computation of the perturbation series is finally given. This approach has a lot of applications in fields as quantum optics, condensed matter and so on, extending the original expectations of the quantum field theory method. So, we give some examples of application for a class of two-level systems that, in recent years, proved to be extremely important. One of the most interesting concepts that can be obtained in this way is that of a Quantum Amplifier (QAMP) that permits to obtain an amplification to the classical level of the quantum fluctuations of the ground state of a single radiation mode.