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arXiv:1606.07021

Taming the divergent terms that occur during adiabatic switching in perturbation theory

Dan Solomon

Published Jun 21, 2016• 0 citations
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Abstract

A potential problem with adiabatic switching in perturbation theory is that divergent terms appear in the series solution. An example of this was presented by C. Brouder et al [4] for a simple 2 state system where the evolution of system in the presence of a time dependent perturbation was considered. One of their results is that the evolution operator has no well-defined limit for adiabatic switching. We will rework this problem to show that for adiabatic switching the evolved states are well-defined with any divergences being absorbed in a time independent phase factor which can be removed. These results will then be applied to the more general problem of a system with an arbitrary number of states. It will be shown that for this case, also, the potentially divergent terms all appear in a time-independent phase factor.

DOI
10.48550/arxiv.1606.07021

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A potential problem with adiabatic switching in perturbation theory is that divergent terms appear in the series solution. An example of this was presented by C. Brouder et al [4] for a simple 2 state system where the evolution of system in the presence of a time dependent perturbation was considered. One of their results is that the evolution operator has no well-defined limit for adiabatic switching. We will rework this problem to show that for adiabatic switching the evolved states are well-defined with any divergences being absorbed in a time independent phase factor which can be removed. These results will then be applied to the more general problem of a system with an arbitrary number of states. It will be shown that for this case, also, the potentially divergent terms all appear in a time-independent phase factor.
A nondegenerate perturbation theory is studied using the adiabatic-theorem formalism. A new factorization of the adiabatic time-evolution operator analogous to that obtained by Morita in his derivation of the linked-cluster theorem is given by carrying out a series of intermediate representations. This leads to a new expansion for the perturbed eigenvectors \ensuremath{\Vert}\ensuremath{\alpha}〉 and energies ${E}_{\ensuremath{\alpha}}$ which are analytically expressed with respect to two matrix series scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n). The nth term of each series (scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n)) includes perturbational contributions of power p, p\ensuremath{\ge}n. An explicit recurrence, scrV(n){\mathrm{scrV}}^{(n)}scrV(n)=V(${\mathrm{scrV}}^{n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$), E(n){E}^{(n)}E(n)=E(${\mathrm{scrV}}^{(n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$) is derived and allows us to calculate these matrices to an arbitrary iteration order (n). This is a central feature of our approach since the eigenvectors and eigenvalues can be generated, to this arbitrary order, by a simple numerical program.
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.
Abstract Within the framework of the method of variation of constants, time-dependent perturbation theory is presented in a form that naturally eliminates the undesirable ‘secular ’ terms for a time-independent perturbation, permits an order-by-order calculation of the exact wavefunction for a ‘resonant’ harmonic perturbation and offers a simple route to establish the adiabatic hypothesis in quantum mechanics. The formulation rests on the introduction of a flexibility in the choice of the phase factors associated with the varying amplitudes, together with development of a perturbation procedure that closely resembles the Brillouin-Wigner formalism in the static case, and which exploits the flexibility of the phase factors at each order of the theory.
We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field.
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.