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Physical Review A

Eigenvalue problem of the square of the pulse area for two-level systems

Edward J. Robinson

Published Apr 1, 1984• 20 citations
Publisher

Abstract

This paper analyzes the dynamics of two-level systems from a new perspective. The notion of eigenvalues (EV) of the pulse area squared, which was introduced in a previous paper, is developed further. Integral expressions for the eigenvalues in terms of eigenfunctions are exhibited, and it is pointed out that these formulas also enable one to estimate EV variationally. Examples are given of how the method may be used to predict qualitative features of two-level spectra without solving equations of motion. Finally, an expression for the transition amplitude in terms of an eigenfunction expansion is derived. It is suggested that this expansion may be useful for both numerical determinations of transition probabilities and for analytic, variational calculations of these parameters. As an application, an expression for the transition probability at small detunings for a rather general class of coupling functions is derived, and its predictions compared to exact numerical calculations in a specific case.

DOI
10.1103/physreva.29.1665

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Physical Review A•Jan 1, 1998•28 citations
This paper analyzes the dynamics of two-level systems from a new perspective. The notion of eigenvalues (EV) of the pulse area squared, which was introduced in a previous paper, is developed further. Integral expressions for the eigenvalues in terms of eigenfunctions are exhibited, and it is pointed out that these formulas also enable one to estimate EV variationally. Examples are given of how the method may be used to predict qualitative features of two-level spectra without solving equations of motion. Finally, an expression for the transition amplitude in terms of an eigenfunction expansion is derived. It is suggested that this expansion may be useful for both numerical determinations of transition probabilities and for analytic, variational calculations of these parameters. As an application, an expression for the transition probability at small detunings for a rather general class of coupling functions is derived, and its predictions compared to exact numerical calculations in a specific case.
Previous results obtained for transition probabilities in two-level systems are simplified and generalised. It was shown in an earlier paper by Robinson (1984) that one could express the transition amplitude as the product of its first-order approximant and a factor which is determined by the eigenvalues and eigenfunctions of a certain nonHermitian operator. In the present work, it is demonstrated that explicit knowledge of the eigenfunctions is not needed-it is sufficient to know the ensemble of eigenvalues. As an illustration, the transition probability induces by a Lorentzian pulse slightly detuned from resonance is calculated. Unlike most pulse shapes, which exhibit power broadening, it is found that the Lorentzian is neutral to first order in the detuning, and exhibits power narrowing when quadratic terms in the detuning are included.
The problem of calculating transition probabilities in two-level systems is studied in the limit where the detuning is large compared to the inverse duration of the interaction. Coupling potentials whose Fourier transforms $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(\ensuremath{\omega})$ are of the form $f(\ensuremath{\omega}){e}^{\ensuremath{-}(|b\ensuremath{\omega}|)}$ for large frequencies give rise to solutions which may be classified into families according to the form of $f(\ensuremath{\omega})$. Within each family transition probabilities may be calculated from formulas that differ only in the numerical value of a scaling parameter. In cases where the coupling function has a pole in the complex time plane, the families are identified with the order of this singularity. In particular, for poles of first order, a connection with the Rosen-Zener solution can be made. The analysis is performed via high-order perturbation expansions which are shown to always converge for two-level systems driven by coupling potentials of finite pulse area.
Under a time-dependent perturbation, it is common to calculate the probability of a transition from one eigenstate to another eigenstate of a quantum system. Here we study the transition from a \textit{linear superposition of eigenstates} to an eigenstate under a delta-function pulse. We consider a two-level system with energy levels E1E_1E1​ and E2E_2E2​ and obtain exact analytical expressions for the coefficients c1c_1c1​ and c2c_2c2​ of the final state. The expressions are general since the coefficients α1α_1α1​ and α2α_2α2​ of the initial superposition state are free parameters constrained only by ∣α1∣2+∣α2∣2=1|α_1|^2+ |α_2|^2=1∣α1​∣2+∣α2​∣2=1. This opens up new possibilities and in particular allows for an abrupt transition to a definite eigenstate with unit probability. We obtain a general analytical expression for the probability Pα1,α2→2P_{α_1,α_2 \to 2}Pα1​,α2​→2​ of an initial superposition state to transition to the second eigenstate. Armed with this general expression we study some interesting special cases. With a delta-function pulse, the transitions are abrupt/instantaneous and we show that they do not depend on the energy gap E2−E1E_2-E_1E2​−E1​ and hence on the relative phase between the two eigenstates. For specific values of the interaction strength βββ, the initial superposition transitions abruptly to a definite eigenstate with unit probability. We discuss the similarities and differences such a transition has with the collapse of the wavefunction familiar in the context of a measurement.
The problem of the evolution of a two-level quantum system in a radiation field of Gaussian shape with detuning is addressed by seeking an approximate expression for the transformation function governing the interaction. The transformation operator requires the evaluation of two independent matrix elements: the former is obtained by the Rosen-Zener conjecture; the latter is based on an expansion using the main parameters involved in the first one. The expressions obtained, which are supposed to hold in the case of small detunings, have been tested by comparison with a numerical integration. Then the solutions are modified to take into account decay phenomena to other levels. As a final example, the expressions dealing with the transition probability in the case of two separated oscillatory fields are given.
The problem of a two-level atom interacting with a radiation pulse is studied in the limit that the atom-field detuning times the pulse duration is much greater than unity. Owing to the large atom-field detuning, transitions result from nonadiabatic coupling of the states by the field. The transition probability for the atom to be excited following the pulse is studied as a function of field strength for five different pulse shapes: hyperbolic secant, Lorentzian, hyperbolic secant squared, Lorentzian squared, and Gaussian. It is shown that the behavior of the transition probability differs qualitatively for these pulses. An explanation of this qualitative difference is given in terms of the Massey parameter. Numerical solutions are compared with asymptotic solutions and several anomalies are noted. In the limit of large field strength, a universal expression for the transition probability is found. An interesting feature of the solutions is that, in the limit of very large field strengths, the transition probability for a Gaussian pulse can approach unity despite the fact that the pulse has an exponentially small Fourier amplitude at the atom-field detuning. This apparent violation of the energy-time uncertainty principle is explained in terms of the nonlinear atom-field interactions.