ORIGINAL ARTICLE: Gerhard C. Hegerfeldt, Time-Optimal Transport of a Harmonic Oscillator: Analytic Solution, Phys. Scr. 98 (2023) 095101, DOI: https://doi.org/10.1088/1402-4896/ace999

Introduction

Motivated by the experimental transport of a trap with a quantum mechanical system modeled as a harmonic oscillator (h.o.) the corresponding classical problem is investigated. We consider a classical one-dimensional harmonic oscillator (h.o.) without friction in the center of a long wagon, where a small mass
m
is attached to a spring on the wagon. When the wagon is accelerated the h.o. will start to perform oscillations. In this case the frequency Ω of the h.o. depends on the spring constant and on
m
.

The problem:

1
.
Initially the wagon is at rest and the h.o. is in its equilibrium position, also at rest.
2
.
Then the wagon undergoes an acceleration
a(t)
, where
a(t)
can vary between
±
a
max
, until it has traveled a prescribed distance
d
.
3
.
Upon arrival at the end point the wagon should be at rest, and the h.o. should again be in its equilibrium position and at rest.
Is this achievable? And if so what is the shortest time possible? Can this time be further lowered by allowing the h.o. frequency Ω to be time dependent, i.e.
Ω(t)
? These questions are answered in the affirmative in [1].

Fixed oscillator frequency

x
h
= position of mass point relative to the wagon center
​
x
w
= position of wagon center
̈
x
h
-
2
Ω
x
h
-a ​​
̈
x
w
a .
Pontryagin Maximum Principle (PMP) [2] yields:
a(t)
is piecewise constant,
a(t)
=
±
a
max
. Then the solution
x
h
(t)
is obtained from
x
h
(t)+i
-1
Ω

x
h
(t)exp[-iΩ(t-
t
0
)]·
x
h
(
t
0
)+i
-1
Ω

x
h
(
t
0
)±
a
max

2
Ω
∓
a
max

2
Ω
Optimal procedure (“protocol”) for constant Ω [1] :
1
.
For given
d
determine the unique optimal time
t
f
by the equation
d
1
4
a
max
2
t
f
1-
8
2
(Ω
t
f
)
2
(arccos(
2
cos
(Ω
t
f
/4)))

.
2
.
With wagon and oscillator at rest at
t=0
, accelerate with
a
max
until time
1
2
t
f
-
t
1
where
t
1
,
0≤Ω
t
1
≤π/2
, is given by
t
1

1
Ω
arccos(
2
cos
(Ω
t
f
/4))
.
3
.
Decelerate with
-
a
max
until time
1
2
t
f
.
4
.
Accelerate with
a
max
until time
1
2
t
f
+
t
1
.
5
.
Finally decelerate with
-
a
max
until time
t
f
.
t
f
can not be shorter than
T
abs
2
d/
a
max
, when no h.o. is present.
In[]:=
<<MaTeX`
In[]:=
g=InverseFunction
1
4
2
#
-8
2
ArcCos
2
Cos
#
4


&;
In[]:=
Plot[{2*Sqrt[4Pi^2*s],g[4Pi^2*s]},{s,0,4},PlotLegendsPlaced[{MaTeX["T_{\\rm abs}"],MaTeX["t_{\\rm f}(d)"]},Scaled[{0.8,0.5}]],Axes->True,PlotStyle->{Dashed,Solid},AxesLabel{MaTeX["d/d_\\Omega",Magnification1.1],MaTeX[" {\\rm time}/\\Omega^{-1}",Magnification1.1]},AspectRatio1]​​
With
Ω
res
(d)4π/
T
abs
one obtains
t
f
as a function of Ω:
In[]:=
Plot[1/4/Pi/s*g[4Pi^2*s^2],{s,.05,2.5},PlotLegendsPlaced[{MaTeX["t_{\\rm f}(\\Omega)"]},Scaled[{0.8,0.5}]],AxesLabel{MaTeX["\\Omega /\\Omega_{\\rm res}(d) ",Magnification1.1],MaTeX["t_{\\rm f}/T_{\\rm abs}(d)",Magnification1.1]},PlotRange{{.4,2.5},{1,1.2}},AspectRatio1]​​
Out[]=

Time-dependent oscillator frequency

0≤
Ω
-
≤Ω(t)≤
Ω
+
<∞
,
​
|a(t)|≤
a
max
.
Scaled variables:
d
0
a fixed length unit, ​
2
Ω
0

a
max

d
0
,           ω(t)Ω(t)
Ω
0
​​
τ
Ω
0
t,                       u(τ)a(t)/
a
max
​​
ξ
1
(τ)
x
h
(t)
d
0
,         
ξ
2
(τ)
d
dτ
ξ
1
(τ)
​​
ξ
3
(τ)
x
w
(t)
d
0
,        
ξ
4
(τ)
d
dτ
ξ
3
(τ)
​so that
u(τ)
can vary between -1 and 1. Then​
̈
ξ
1
≡
2
d
d
2
τ
ξ
1
-
2
ω
(τ)
ξ
1
-u(τ)
​​
̈
ξ
3
u(τ) 
.We choose
d
0
=d
. Then
τ
abs
=2
and
ω
res
=2π
.
If
n
Ω
res
(d)
lies in the interval
[
Ω
-
,
Ω
+
]
for some
n
then choosing
n
Ω
res
(d)
will give the minimal time
T
abs
(d)
. Otherwise, PMP yields (with possible exceptions):
u(t)
is piecewise constant,
u(t)±1
,
ω(t)
is piecewise constant,
ω(t)
ω
±
.
There are switches in
u
at
τ
f
/2
and
τ
f
/2±
τ
1
where both
τ
f
and
τ
1
are yet to be determined. The distance traveled in time
τ
f
is
1
4
2
τ
f
-2
2
τ
1
and if
τ
f
is to be optimal it must satisfy
d
d
0
1
1
4
2
τ
f
-2
2
τ
1
(1) where
τ
f
=
τ
f
(
ω
-
,
ω
+
)
.

Case
0<
ω
-
<
ω
+
<
ω
res
2π

(i) If
ω
+
≤

ω
+
π
2
only
ω
+
enters and there is no switch in
ω
. This is the case of constant
ω
from above and the corresponding
t
f
(
Ω
+
,
Ω
-
)
is depicted in the next plot.
In[]:=
Pab=ContourPlot3D[{(2*tauf)^2/4-2*1/(2*Pi*ompl)^2*ArcCos[Cos[2*Pi*ompl*2*tauf/4]^2]^2==1},{ompl,0,1},{ommi,0,1},{tauf,1,1.52},RegionFunctionFunction[{ompl,ommi,tauf},0<ommi<ompl<1&&ompl<=Sqrt[2]/4],RegionBoundaryStyleNone,ContourStyle{Directive[Green,Opacity[0.3]]},AxesLabel{MaTeX["\\Omega_+/\\Omega_{\\rm res}(d)",Magnification.9],MaTeX["\\Omega_-/\\Omega_{\\rm res}(d)",Magnification.9],MaTeX["t_{\\rm f}/T_{\\rm abs}(d)",Magnification.9]}]​​​​
Out[]=
(ii) If
ω
+
>π
2
one starts with
ω
-
, continues with two complete
ω
+
periods, and finally switches to
ω
-
.
τ
f
/2
satisfies
π/
ω
+
≤
τ
f
/2≤π/
ω
+
+π/
ω
-
. Then
τ
1
has to satisfy ​
cos[
ω
+
τ
1
]
2
ω
+
2
2
ω
-
1+cos
ω
-
τ
f
2+
ω
+
-
ω
-
ω
+
π
.One now solves this equation together with Eq. (1) for
τ
f
under the condition that
τ
f
/2
lies in the last
ω
-
interval.
These plots can be combined together with the upper, lower and diagonal boundary:
These plots can be combined together with the lower and diagonal boundary:

References

[1] G.C. Hegerfeldt, Phys. Scr. 98 (2023) 095101
[2] L.M. Hocking, Optimal Control: An Introduction to the Theory With Applications (Oxford: Clarendon Press 1991)

CITE THIS NOTEBOOK

Optimal Transport of a Classical Harmonic Oscillator​
by Gerhard C. Hegerfeldt​
Wolfram Community, STAFF PICKS, October 13, 2023
​https://community.wolfram.com/groups/-/m/t/3035725