Approximate solutions to Mathieu's equation

doi:10.1016/j.physe.2018.02.019

Physica E: Low-dimensional Systems and Nanostructures

Volume 100, June 2018, Pages 24-30

https://doi.org/10.1016/j.physe.2018.02.019 Get rights and content

Highlights

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A step-by-step guide to the various analytical approximations that can be used when dealing with Mathieu's equation.
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Particular attention is paid to quantities relevant to the physics of Josephson junctions.
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Analytic approximations are compared with numerical solutions.
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Second-order corrections to some common approximations can be trivially implemented and greatly improve accuracy.

Abstract

Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrödinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various approximations which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.

Introduction

Mathieu's equation, $\frac{d^{2} ψ}{d z^{2}} + (a - 2 η \cos (2 z)) ψ = 0 .$ has appeared in theoretical physics in many different contexts. Mathieu originally formulated the equation to describe the vibration modes of an elliptical membrane [1], but the equation has since been applied to the theory of quadrupole ion traps [[2], [3], [4]], ultracold atoms [5] and quantum rotor models [6,7]. This equation has also found attention as a simplified model of a particle moving in a periodic potential [8].

Although Mathieu's equation is easy to solve numerically, and although exact results are achievable in certain limits, a general analytic solution of Mathieu's equation has not yet been achieved. Instead, there exists throughout the literature, both on physics and mathematics, a myriad of approximations and numerical methods which may be used to extract quantities of interest. It is the goal of this paper to collect these approximations together in one place for easy reference, to review them explicitly and explore their regimes of validity. The focus is to illustrate and compare the results found in the vast body of literature on this topic.

This manuscript will focus primarily on applications of Mathieu's equation to the physics of Josephson junctions [[9], [10], [11], [12]], however we will keep the notation general as the results presented herein may be of use across diverse fields. Josephson junctions are elements in superconducting circuits, which are of great interest due to potential applications in quantum technology [9,13,14].

A single Josephson junction is governed by the Hamiltonian $H = - 4 E_{C} \frac{\partial^{2}}{\partial ϕ^{2}} - E_{J} \cos (ϕ)$ where $E_{C} = e^{2} / 2 C$ is the charging energy, $C$ is the junction capacitance, $E_{J}$ the Josephson energy and $ϕ$ is the phase difference of the superconducting condensate across the junction. With this Hamiltonian, the time-independent Schödinger equation becomes $[- 4 E_{C} \frac{\partial^{2}}{\partial ϕ^{2}} - E_{J} \cos (ϕ)] ψ = E ψ .$ This reduces to Mathieu's equation upon making the substitutions $ϕ / 2 \to z$ , $E / E_{C} \to a$ , $E_{J} / 2 E_{C} \to η$ . To maintain generality, we will retain the notation of Mathieu's equations, but we will bear these substitutions in mind and make frequent reference to results obtained in the theory of Josephson junctions.

The focus will be on quantities corresponding to physical observables in Josephson junctions. We will therefore not be concerned with the details of the Mathieu functions themselves (physically, the wavefunctions of the Josephson junction array), but primarily on the characteristic value $a$ , the floquet exponent $ν$ , and related quantities depicted in Fig. 1.

Each of these quantities will be discussed in detail below, but each can be understood loosely as follows: $t = b_{1} - a_{0}$ is the difference between the lowest characteristic value of an odd-parity Mathieu function and the lowest characteristic value of an even-parity Mathieu function. Physically it corresponds to the bandwidth of the lowest energy band of a Josephson junction.

For characteristic values between $a_{1}$ and $b_{1}$ , stable Mathieu functions do not exist. $δ = a_{1} - b_{1}$ represents a gap in characteristic values of stable Mathieu functions. Physically, $δ$ corresponds to the band gap in the energy spectrum of the Josephson junction.

$V (ν) = d a / d ν$ is a quantity little discussed in the mathematics literature, but in the physics of Josephson junctions it is known as the effective voltage [9].

In experiments on Josephson junctions the quantity $η$ is often a controlled parameter. In fact, if one adopts a SQUID geometry, $E_{J}$ , and by extension $η$ , can be tuned in real time by adjusting the applied magnetic flux [15]. We are therefore primarily interested with how these various parameters vary with $η$ . In Fig. 1 we have ploted $a (ν)$ and $V (ν)$ for $η = 0$ and $η = 0.2$ .

The limits of both strong coupling ( $η ≫ 1$ ) and weak coupling ( $η ≪ 1$ ) are relatively straightforward. In both cases the characteristic values can be expressed as asymptotic expansions in powers of $η$ or $1 / η$ respectively. Below we will explore both of these extreme limits of the model, and investigate the region $η \sim 1$ where the approximations are expected to break down. We will also examine properties of Mathieu's equation which may be deduced from periodicity arguments, as these are expected to be valid for any value of $η$ .

Section snippets

Small $η$

In the limit that $η \to 0$ , Mathieu's equation becomes $\frac{d^{2} ψ}{d ϕ^{2}} + a ψ = 0 .$ This differs from Schrödinger's equation for a particle moving in free space only in that the co-ordinate $ϕ$ has the topology of a circle. In this limit, the eigenvalues are continuous and do not form separate energy bands or levels. The Mathieu functions themselves are simply $\pm \cos (\sqrt{a_{n}} z)$ , $\pm \sin (\sqrt{b_{n + 1}} z)$ (as can be trivially verified). By convention we take the sign to be positive. The characteristic value of the sine solution is denoted $b_{n +}$

Large $η$

When $η ≫ 1$ , $z$ remains close to the minima of $\cos 2 z$ , so that when expanded as a Taylor series only the second order term is relevant. This reduces the Mathieu equation to the form of Schrödinger's equation for a harmonic oscillator, so that the Mathieu functions may be approximated by the wavefunctions of a harmonic oscillator $ψ_{n}^{H O} (z) = c_{n} H_{n} ({(2 η)}^{1 / 4} z) e^{- \frac{1}{2} \sqrt{2 η} z^{2}}$ with energy levels $a_{n} = 4 \sqrt{η} (n + \frac{1}{2}) - 2 η$ where $H_{n} (x)$ are Hermite polynomials familiar from the theory of the quantum harmonic oscillator, $c_{n}$ is a

Floquet theory and the characteristic exponent

In physical applications, often the characteristic value is a desired output of the theory (for example in the physics of Josephson junctions it corresponds to an energy eigenvalue). We have seen that at a given value of $η$ there exist continuous bands of characteristic values which give stable solutions to Mathieu's equation. Therefore, an addition parameter is required to uniquely determine the characteristic value for a particular $η$ . To this end we turn to Floquet theory, where we will see

Comparison to numerical solutions

Despite the lack of exact analytic results, numerically solving Mathieu's equation is quite straight-forward. In the application of Mathieu's equation to Josephson junction arrays we make use of the fact that phase and charge are canonical conjugate variables, and re-write Schrödinger's (Mathieu's) equation (Eq. (23)) in the charge basis: $\begin{align} \sum_{n} & [- 4 E_{C} {(\hat{n} - q)}^{2} - \frac{E_{J}}{2} (| n + 1 〉〈 n | + | n - 1 〉〈 n |)] | ψ_{m} 〉 \\ = E_{m} | ψ_{m} 〉, \end{align}$ where $\hat{n}$ is the Cooper pair number operator, $m$ labels the different energy levels and $q$ is the quasicharge

Effective voltage

When $E_{C} \sim E_{J}$ , it is convenient to describe the Josephson junction not in terms of discrete charges $n$ or in terms of the Josephson phase $ϕ$ , but rather in terms of the quasicharge $q$ [9] (equivalent to the characteristic exponent $ν$ of Mathieu's equation). In this case, the effective voltage across a junction is $d E_{0} / d q$ , or, in the Mathieu equation notation used above, $d a / d ν$ . Our asymptotic formulae above approximate the bands of $a$ as infinitely thin in $ν$ , and therefore do not include explicit $ν$

Conclusion

Mathieu's equation appears in many problems within theoretical physics. In most situations, it is convenient to simply solve the equation numerically. In some cases, however, an analytic approximation may be desired. We have gathered here several analytic approximations for various quantities relating to Mathieu's equation and compared them to numerical results (which may, in principle, be evaluated to arbitrary accuracy).

One results of particular interest is that of the gap between stable

Acknowledgements

This work was supported in part by the Australian Research Council under the Discovery and Centre of Excellence funding schemes (project numbers DP140100375 and CE170100039). Computational resources were provided by the NCI National Facility systems at the Australian National University through the National Computational Merit Allocation Scheme supported by the Australian Government.

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Approximate solutions to Mathieu's equation

Highlights

Abstract

Introduction

Section snippets

Small η

Large η

Floquet theory and the characteristic exponent

Comparison to numerical solutions

Effective voltage

Conclusion

Acknowledgements

J. Am. Soc. Mass Spectrom.

J. Am. Soc. Mass Spectrom.

Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique

J. Math. Pure Appl.

An introduction to quadrupole ion trap mass spectrometry

J. Mass Spectrom.

Ultracold atoms confined in an optical lattice plus parabolic potential: a closed-form approach

Phys. Rev.

Atom optics quantum pendulum

J. Russ. Laser Res.

The physical pendulum in quantum mechanics

A soluble problem in energy bands

Phys. Rev.

Theory of the Bloch-wave oscillations in small Josephson junctions

J. Low Temp. Phys.

Charge-insensitive qubit design derived from the Cooper pair box

Phys. Rev.

Model for a Josephson junction array coupled to a resonant cavity

Phys. Rev. B

Small $η$

Large $η$