Parametric Oscillators: Two Case Studies
Thiago Barros − tmachado@sfu.ca
Pouya Bastani − pbastani@cs.sfu.ca
*+,-./.+0123456718941:,+;.63<71=>3,+.9?
%
Introduction
"
Equation (9), combined with (7) and (8), leads to the following homogeneous system of equations:



2
2
1 0 S2 − C
−2C1C2
A1
 0 1 2kC2S2 k(S 2 − C 2)   B1 

=0
2
2 
(10)
 C1 S1 −C2



A2
−S2
B2
−S1 C1 kS2
−kC2
#
where k = ω2/ω1, Si = sin(ωiT /2) and Ci = cos(ωiT /2) for i = 1, 2. This system has a non-trivial
(non-zero) solution only if its determinant is zero:
!"
2k + S1S2(k 2 + 1) − 2C1C2 = 0
$
(1)
An important difference between parametric excitation and forced oscillations is related to the dependence of the growth of energy on the energy already stored in the system. While for parametric
excitation the increment in energy is proportional to the energy stored in the system, in forced oscillation the increment is proportional to the amplitude of oscillations, i.e., to the square root of the
energy.
B6+,
ẍ + G(t)x = 0, G(t + τ ) = G(t)
!
This gives an expression in terms of and T . To find the transition curves, we used the Matlab
fsolve procedure. The result is shown on Figure 2.
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In this poster, we present two particular instances of Hill’s equation that are known as the Mathieu
and Meissner equations. We’ll also describe an application of the Mathieu equation known as Ion
trapping.
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Application: Ion Trapping
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#
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!
The Mathieu Equation
(2)
Although no analytical solutions of this equation exists in terms of standard functions, the periodicity of G(t) allows the use Floquet Thoery, the theory of linear differential equations with periodic
coefficients. The theory predicts that for certain values of α and β there exist solutions to (2) that
are unstable, i.e. grow exponentially with time.
The curves in the (α, β)-space which separate the stable solutions from the unstable ones are called
transition curves. It can be shown that these curves represent 2π and 4π periodic solutions. For the
former case, we can represent the solutions by the complex Fourier series:
x(t) =
∞
X
cneint
n=−∞
which can only be satisfied if all the coefficients are zero. This gives us an infinite set of homogenous equations for {cn}, for which a non-zero solution exists only if the determinant formed by the
coefficients is zero. When α 6= n2, this becomes
. . . . . . .
. γ1 1 γ1 0 0 . . 0 γ0 1 γ0 0 . = 0
(5)
. 0 0 γ1 1 γ1 . . . . . . . .
where γn = 21 β/(α − n2). Following a similar procedure, we can obtain a determinant equation in
α and β for the case of 4π-periodic solutions. The curves in the (α, β)-space can be obtained by
specifying α (or β) in the determinants and solving for the corresponding β (or α). For the purposes
of this work, we implemented a routine in Matlab using the Gauss-Newton method programmed in
the fsolve procedure.
The stability diagram for Mathieu Equation is shown in Figure 1.
As can be seen, the transition curves corresponding to 2π-periodic solutions pass through the points
β = 0, α = n2, while the curves of 4π-periodic solutions pass through β = 0, α = 41 (2n − 1)2.
'
(
x1(t) = A1 cos(ω1t) + B1 sin(ω1t), 0 < t < T /2
where r and z are the polar coordinates of the ion and d0 is a characteristic internal dimension of
the trap:
q if the ring electrode has minimum radius r0 and the endcaps have smallest separations 2z0,
d0 =
r02 + 2z02.
z
x2(t) = A2 cos(ω2t) + B2 sin(ω2t), T /2 < t < T
(8)
√
√
where ω1 = 1 + and ω2 = 1 − . We need the conditions on α and T that would result in
periodic solutions. At each instant of abrupt change, we must make a transition from one of these
equations to the other. To do so, we specify boundary conditions that must hold true between the
transitions. This can be provided by requiring the continuity of x(t) and ẋ(t) at such instances:
x1(0) = x2(T )
ẋ1(0) = ẋ2(T )
x1(T /2) = x2(T /2) ẋ(T /2) = ẋ2(T /2)
(9)
1
0.9
"m icro m o tio n "
1 .0
0 .5
cos2p
VV0ocos
!fTt t
0 .0
- 0.5
"s ec u lar" m o tio n
- 1.0
0
(a)
2
4
tim e
6
8
10
(b)
Figure 3: Quadruple Paul Ion Trap
Figure 2.1: (a) Electrode structure for an rf (Paul) ion trap. The electrodes are hyper
bolic
and,
when
rf potential
is applied
to!the electrodes
as indicated
The equation of motion for an
ionin
ofcross-section
charge Q and
mass
m an
in the
Paul Trap
can be written
as
"
2
2
2
it gives rise to a!potential of the form V =!V0 cos(ΩT t) x +yd2−2z . (b) The ion’
0
motion
considered
to be made up of two parts. The secular mo
4QV0 in a Paul trap may be 2QV
0
z̈ = tion represents
cos (Ωt)the z,
r̈movement
=
cos
(Ωt) r
(13)of frequencie
ion’s
in
a
three-dimensional
harmonic
well
2
2
√
md0
md
ωx = ωy = ωz /2 = 2QV0 /(md20 Ω0T ). Here Q is the charge on the ion, m is its mass
d0 is a characteristic
the(2).
trap. The micromotion, which occur
both of which are parametricand
oscillator
equations thatinternal
can be dimension
convertedofinto
at the drive frequency ΩT , is of small amplitude, vanishing at the null point of the r
field (in the center of the trap) .
References
0.8
[1] John Bechhoefer and Brad Johnson. A simple model for Faraday waves. American Journal of
trapped in a three-dimensional simple harmonic oscillator potential with “secular fre
Physics, 64:1482–1487, 1996.
0.7
√ oscillator2 at square-wave modulation.1
[2] Eugene I Butikov. Parametric
resonance
in
a
linear
quencies” ωx = ωy = ωz /2 ≈ 2QV0 /(md0 ΩT ), where m is the ion’s mass .
European Journal of Physics, 26:157–174, 2005.
0.6
At this point, the following mechanical analogy may help [66]. Consider a marbl
[3] D.W Jordan and P Smith. Nonlinear
Ordinary Differential Equations - An Introduction to Dynamical Systems. Oxfordresting
University
third edition,
1999.
on a Press,
saddle-shaped
surface
(under the effect of gravity). The saddle is a stabl
0.5
0.4
[4] Hans Christian Karlsen. A Study of Analytical and Numerical Methods Regarding Parametric
potential in one direction, but is unstable in the other. Thus, under the influence o
Resonance. PhD thesis, Norwegian University of Science and Technology, December 1999.
0.3
gravity,
the marbleand
tends
to roll down
the unstable
of Ions.
the saddle.
[5] Brian E King. Quantum State
Engineering
Information
Processing
with“sides”
Trapped
PhD However, i
thesis, University of Colorado, 1999.
we rotate the saddle about the vertical at the proper frequency, we may create a stabl
0.1
0
x
y
(7)
0.2
From the plot it can be seen that there is a small region of stability for negative values of α. One
physical interpretation of this is that the motion of an inverted pendulum whose pivot is driven in the
vertical direction with frequency f and amplitude A, can become stable for certain values of these
parameters.
According to Earshaw’s Theorem, it is impossible to confine an isolated charge in free space using
only static electric fields. One technique used for Ion Trapping employs the Quadruple Paul Ion
Trap, which produces a potential between the electrodes of the form
!
r2 − 2z 2
V = V0 cos(Ωt)
(12)
2
d0
)
where T is the period of the square-wave. Unlike the Mathieu Equation, it is possible in this case to
obtain simple analytical solultions. During the time interval (0, T /2) and (T /2, T ), the value of f (t)
is constant and so during each half-period x(t) is some harmonic oscillation:
Amplitude of modulation (epsilon)
(4)
&
When the sinusoidal parametric modulation in the Mathieu Equation is replaced by a square-wave
function, the Meissner Equation is obtained (after some change of variables):
1, if 0 < t < T /2
ẍ + (1 + f (t))x = 0,
f (t) =
(6)
-1, if T /2 < t < T
n=−∞
∞ X
1
1
2
βcn+1 + (α − n )cn + βcn−1 eint = 0
2
2
%
The Meissner Equation
(3)
Substituting (3) into (2) leads to
$
@/A;,
Figure 1: Stability Diagram for the Mathieu Equation
When the periodic coefficient in (1) is a simple sinusoidal function, one obtains the Mathieu Equation:
ẍ + (α + β cos t)x = 0
(11)
p o sitio n
Parametric Oscillators have frequencies depending periodically on time and are commonly modeled
by Hill’s Equation:
trap for the marble: as the marble begins to roll downhill in some direction, the saddl
0
2
4
6
Period of modulation (T)
8
10
12
rotates so that what was previously downhill now becomes uphill. With the prope
rotation frequency (which depends upon the marble’s mass and the curvature of th
Figure 2: Transition Curves for the Meissner Equation
1
This result only holds in the so-called “pseudopotential approximation,” which I will discuss below