Beyond the quantum adiabatic approximation: Adiabatic perturbation theory

Gustavo Rigolin, Gerardo Ortiz, and Victor Hugo Ponce

Phys. Rev. A 78, 052508 – Published 18 November 2008

Abstract

We introduce a perturbative approach to solving the time-dependent Schrödinger 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- $1 ∕ 2$ problem, that to first order in APT the geometric phase should be two and a half times the (adiabatic) Berry phase.

Received 8 July 2008

DOI:https://doi.org/10.1103/PhysRevA.78.052508

Authors & Affiliations

Gustavo Rigolin^1,*, Gerardo Ortiz^1,†, and Victor Hugo Ponce²

¹Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
²Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energía Atómica and Universidad Nacional de Cuyo, 8400 Bariloche, Argentina

^*rigoling@indiana.edu
^†ortizg@indiana.edu

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Issue

Vol. 78, Iss. 5 — November 2008

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Images

Figure 1
Different approximation methods to solving the time-dependent Schrödinger equation. APT: adiabatic perturbation theory (Garrison, Ponce, this paper). IRBM: iterative rotating-basis method (Kato, Garrido, Nenciu, Berry). TDPT: time-dependent perturbation theory (Dirac). SA: sudden approximation (Messiah). AA: adiabatic approximation (Born and Fock).Reuse & Permissions
Figure 2
(Color online) A beam of particles prepared in the g.s. is split into two equal parts. One (upper beam) goes through a region of constant magnetic field whose strength $B_{2}$ is such that at the end it acquires the dynamical phase $α^{(1)} (τ_{s})$ of the lower beam. The latter beam goes through a region where the magnetic field $B_{1}$ rotates around the $z$ axes until it returns to itself. (In the original proposal [4], the field strengths are the same, $B_{1} = B_{2}$ .) Finally, the beams are recombined and the intensity measured, allowing us to determine the geometric phase $β^{(1)} (τ_{s})$ . See text for more details.Reuse & Permissions
Figure 3
(Color online) Here $θ_{j}^{0} = 1$ , $E = 2$ , and $v = w_{j} = 0.5$ , which gives $ϵ^{- 1} = 0.25$ $(ℏ = 1)$ . At the top we have $θ_{j} (s)$ , $j = 1, 2$ , and at the bottom $j = 3$ . The black dotted curves represent the infidelity between the zeroth-order correction, Eq. (196), and the exact solution, Eq. (191), as a function of the rescaled time $s$ . Both quantities are adimensional. The blue dashed curves are the infidelity when we go up to first order [Eq. (197)] and the red solid ones when we include the second-order term [Eq. (199)]. For $j = 1$ , the first- and second-order curves are indistinguishable and the solid and dotted curves go as high as 0.004.Reuse & Permissions
Figure 4
(Color online) The same parameters of Fig. 3, but with different gaps. Top: $ϵ^{- 1} = 0.125$ and 0.25. Bottom: 0.5 and 5. All curves represent the infidelity between the exact solution and the adiabatic approximation corrected up to second order [Eq. (199)]. The solid curve represents $θ_{3} (s)$ , the dashed one $θ_{2} (s)$ , and the dotted one $θ_{1} (s)$ . In the first panel all curves coincide, while at the next two the dashed and dotted curves are indistinguishable. Note the difference of scale at the bottom panels. For the first three, the APT works beautifully and the results are better the lower $ϵ^{- 1}$ . At the last panel we see the three curves and the break down of the APT since $ϵ^{- 1} > 1$ .Reuse & Permissions
Figure 5
(Color online) The same parameters and notation of Fig. 3 for the case $θ_{2} (s)$ . At the top panels we used, for expressions coming from the APT and throughout the whole range of $s$ , $\ddot{θ} (0) = 0$ for the first panel and $\ddot{θ} (0) = 2 w_{2}$ for the second one. At the bottom panel we used $\ddot{θ} (0) = 0$ for $s < 0$ and $\ddot{θ} (0) = 2 w_{2}$ for $s ⩾ 0$ plus the continuity of the wave function at $s = 0$ .Reuse & Permissions

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