Download Chapter 4 The Classical Delta

Chapter 4
The Classical Delta-Kicked Harmonic
Oscillator
4.1
Introduction
The test system that we propose in order to experimentally investigate quantum chaos is the
the delta-kicked harmonic oscillator. In this chapter we go into considerable detail to explain
the properties of the classical delta kicked harmonic oscillator, in order to provide context,
and to give us something to compare the expected quantum dynamics to. The delta-kicked
harmonic oscillator has has been extensively investigated in a series of papers by Zaslavsky
and co-workers [1, 2, 3], and is also dealt with in depth in his book [4]. What follows is mostly
an amalgam of these various publications; certain aspects have been elaborated upon, and
others deemed less immediately relevant have been omitted.
The most extensively studied test system of all is of course the delta-kicked rotor, covered
extensively in [5]; there have even been experiments on an equivalent system in the quantum
regime, carried out in atom-optical systems by the group of Raizen [6]. The delta-kicked
harmonic oscillator offers the advantage that when dealing with cold atoms or ions, where
experiments on quantum chaotic systems could be (and have been [6]) carried out, there
is essentially always some kind of trapping potential, which is harmonic to some degree of
accuracy. As this potential is present in any case, it is convenient to simply take it into
account, and the delta-kicked harmonic oscillator has some interesting properties on its own.
4.2
Derivation of a Kick to Kick Mapping
We consider a point particle held in a harmonic potential, periodically kicked by a cosine
potential. The classical Hamiltonian function of this delta-kicked harmonic oscillator is given
by
∞
X
p2
mω 2 x2
H=
+
+ K cos(kx)
δ(t − nτ ),
(4.1)
2m
2
n=−∞
41
42
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
where x, p is the momentum, m is the mass of the particle, ω is the harmonic frequency,
k = 2π/λ is the wavenumber of the cosine potential (where λ is the wavelength). The most
important parameter is the kick strength K, which has the dimension of action (kgm2 s−1 );
the delta function has the dimension of a frequency, where τ is the time between kicks.
Hamilton’s equations of motion for this system are:
dx
p
= ,
dt
m
∞
X
dp
= −mω 2 x + kK sin(kx)
δ(t − nτ ).
dt
n=−∞
(4.2)
(4.3)
Clearly, between kicks, the time evolution is simply that of a conventional harmonic oscillator, given by
p(0)
sin(ωt),
mω
p(t) = p(0) cos(ωt) − mωx(0) sin(ωt).
(4.4)
x(t) = x(0) cos(ωt) +
(4.5)
We now consider Eq. (4.3), the equation of motion for the momentum, integrating over the
neighbourhood of the time when some arbitrary kick occurs n0 τ , to get
0
0
p(n τ + ) − p(n τ − ) = −mω
2
= −mω 2
Z
n0 τ +
Z
n0 τ +
dtx(t) + kK
n0 τ −
Z
n0 τ +
dt sin[kx(t)]
n0 τ −
∞
X
n=−∞
dtx(t) + kK sin[kx(n0 τ )].
δ(t − nτ )
(4.6)
n0 τ −
As we let → 0, the remaining integral shrinks to nothing, leaving us with
p(n0 τ + ) = p(n0 τ − ) + kK sin[kx(n0 τ )].
(4.7)
We can now combine this with the equations of motion for the harmonic oscillator to derive
a mapping from just before one kick to just before the next. We introduce the convenient
shorthand xn , pn to stand for x(nτ − ), p(nτ − ).
1
[pn + kK sin(kxn )] sin(ωτ ),
mω
= [pn + kK sin(kxn )] cos(ωτ ) − mωxn sin(ωτ ).
xn+1 = xn cos(ωτ ) +
(4.8)
pn+1
(4.9)
4.3. RESCALING TO DIMENSIONLESS VARIABLES
4.3
4.3.1
43
Rescaling to Dimensionless Variables
Dimensionless x and p.
We can rescale the variables x and p to dimensionless quantities in order to reduce the
number of free parameters:
kx
x0 = √ ,
2
kp
.
p0 = √
2mω
(4.10)
(4.11)
We also let t0 = ωt, τ 0 = ωτ , so that we have a dimensionless time. This means that the
kick to kick mapping can be rewritten to give
√
xn+1 = xn cos(τ ) + [pn + κ sin( 2xn )] sin(τ ),
(4.12)
√
pn+1 = [pn + κ sin( 2xn )] cos(τ ) − xn sin(τ ),
(4.13)
√
where κ = k 2 K/ 2mω, and we have dropped the primes for convenience. It is now apparent
that we have exactly two free parameters: the dimensionless kicking strength κ, and the
dimensionless kick to kick period τ . We can also write a dimensionless rescaled Hamiltonian
function in terms of the rescaled dimensionless x and p: H 0 = Hk 2 /2mω 2 ,
∞
X
√
κ
p2 x2
√
+
+
cos( 2x)
δ(t − nτ ),
H =
2
2
2
n=−∞
0
(4.14)
where we will drop the prime in future.
The dynamics produced by the mapping described in Eqs. (4.12,4.13) can be grouped
into two broad categories: one where τ /2π is a rational number, which is to say that the
particle is kicked a rational number of times per harmonic oscillator period; and where τ /2π
is an irrational number. We will concentrate on the case where τ /2π is rational, which
has some interesting particular properties. Sample stroboscopic Poincaré sections of the
dynamics described by the mapping of Eqs. (4.12,4.13) for τ /2π = 1/4, 1/5, 1/6, and 1/7
are displayed in Figs. (4.1,4.2,4.3,4.4).
The initial condition in each of the cases displayed in Figs. (4.1,4.2,4.3,4.4) are all unstable, iterated over 40000 kicks. There is obviously a high degree of rotational symmetry
in the trajectories taken, which may seem incompatible with the idea of chaotic dynamics.
The symmetry is essentially rotational in phase space however, and the unpredictable part
of the point particle’s p
dynamics is more in how it moves “in and out” through phase space,
i.e. what the value of x2 + p2 is. The symmetric structure brought out by these unstable
dynamics is called a stochastic web, as there is an interconnected web of channels of unstable
dynamics spread through all of phase space.
Note also that in the cases of τ /2π = 1/4 and 1/6 displayed in Figs. (4.1,4.3) there
appears to be a translational symmetry in the phase space structure described by the plotted
trajectory. The significance of this will be dealt with in what follows, in particular in
Section 4.7.
44
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
60
40
20
p
0
−20
−40
−60
−80
−60
−40
−20
0
x
20
40
60
80
Figure 4.1: τ /2π = 1/4, κ = −0.8
60
40
20
p
0
−20
−40
−60
−100
−50
0
x
50
Figure 4.2: τ /2π = 1/5. κ = −0.8
100
45
4.3. RESCALING TO DIMENSIONLESS VARIABLES
60
40
20
p
0
−20
−40
−60
−80
−60
−40
−20
0
x
20
40
60
80
Figure 4.3: τ /2π = 1/6. κ = −0.8
80
60
40
20
p
0
−20
−40
−60
−80
−100
−50
0
x
50
Figure 4.4: τ /2π = 1/7. κ = −0.7
100
46
4.3.2
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
Phase Variable α
The mapping described in Eqs. (4.12,4.13) can be rewritten as a single equation
√ in terms of
a single complex variable α and its complex conjugate, where α = (x + ip)/ 2:
κ
∗
αn+1 = αn + i √ sin(αn + αn ) e−iτ .
(4.15)
2
This more compact description is often advantageous when investigating symmetry properties of this system.
4.4
Perturbation Expansion
From now on we consider only τ = 2πr/q, where r/q is an irreducible rational. Bearing this
in mind, from the mapping described by Eq. (4.15) we can determine an expression for the
value of α at a time q kicks (= r oscillation periods) later:
q−1
αn+q
κ X
∗
= αn + i √
sin(αn+j + αn+j
)ei2πjr/q .
2 j=0
(4.16)
From this one can directly determine expressions to describe the position and momentum at
this time:
xn+q = xn − κ
pn+q = pn + κ
q−1
X
√
sin( 2xn+j ) sin(2πjr/q),
(4.17)
√
sin( 2xn+j ) cos(2πjr/q).
(4.18)
j=0
q−1
X
j=0
As can be seen, this means that for κ = 0, the position and momentum have cycled back to
their original values, which they continue to do, every q kick periods.
We now consider the case of small perturbations to the harmonic oscillator dynamics, i.e.
small κ. Taking Eq. (4.16) and keeping terms up to first order in κ only, we express αn+q
purely in terms of αn , as
q−1
αn+q
κ X
= αn + i √
sin(αn e−i2πjr/q + αn∗ ei2πjr/q )ei2πjr/q ,
2 j=0
(4.19)
which can be written in terms of x and p as
xn+q = xn − κ
pn+q = pn + κ
q−1
X
√
sin{ 2[xn cos(2πjr/q) + pn sin(2πjr/q)]} sin(2πjr/q),
(4.20)
√
sin{ 2[xn cos(2πjr/q) + pn sin(2πjr/q)]} cos(2πjr/q).
(4.21)
j=0
q−1
X
j=0
4.5. CANONICAL TRANSFORMATION
47
This mapping possesses a q-fold rotational symmetry in phase space. This can be see as
follows. We substitute αn with βn = αn ei2πl/q [this is equivalent to replacing xn with yn =
xn cos(2π/q) − pn sin(2π/q) and pn with sn = xn sin(2π/q) + pn cos(2π/q)]. Thus, we have
#
"
q−1
κ X
(4.22)
sin(αn e−i2π(jr−l)/q + αn∗ ei2π(jr−l)/q )ei2π(jr−l)/q ei2πl/q .
βn+q = αn + i √
2 j=0
We have as the imaginary argument to all the exponentials inside the square brackets 2π(jr−
l)/q. Now, e−i2πjr/q has exactly q possible values, all of which are represented in the sum
in Eq. (4.16). If we add in an l/q to the argument, this clearly still holds for the sum in
Eq. (4.22). The point of this is that, because of the cyclic nature of the variable terms being
summed over, both sums contain the same terms, and we can replace one sum with another.
Thus
#
"
q−1
κ X
(4.23)
βn+q = αn + i √
sin(αn e−i2πjr/q + αn∗ ei2πjr/q )ei2πjr/q ei2πl/q ,
2 j=0
from which we can immediately see that if βn = αn ei2πl/q , then βn+q = αn+q ei2πl/q , which
describes the q-fold rotational symmetry in phase space. As this is a result coming from
a perturbation expansion around κ, this is only an approximate symmetry, most valid for
small κ. This symmetry can be expressed in terms of x and p as
yn = xn cos(2πl/q) − pn sin(2πl/q) ⇒ yn+q = xn+q cos(2πl/q) − pn+q sin(2πl/q), (4.24)
sn = pn cos(2πl/q) + xn sin(2πl/q) ⇒ sn+q = pn+q cos(2πl/q) + xn+q sin(2πl/q), (4.25)
√
where (y + is)/ 2 = β. This explains the observed rotational symmetry of the phase space
structure shown by the Poincaré sections of Figs. (4.1,4.2,4.3,4.4).
4.5
Canonical Transformation
In the previous section we saw that there is a rotational q symmetry that makes itself
apparent every q kicks. This motivates us to examine the dynamics of rotating variables.
Thus, we now wish to make a canonical transformation into a rotating reference frame [7],
using the new variables X and P , defined by
X = x cos(t) − p sin(t),
P = p cos(t) + x sin(t).
(4.26)
(4.27)
Note that the new variables X and P coincide with x and p whenever t = 2πn, where n ∈ ,
and that q kick periods corresponds to t = 2πr. Thus, if one observes the variables just
before every kick, the two pairs of variables coincide every q kicks, and only every q kicks.
In the absence of kicks, these transformations have the effect of fixing any initial condition
in X, P phase space, for all time. Similarly, when kicks are periodically added, there will be
no movement through X, P phase space between kicks; movement will only occur when a
48
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
kick is applied. The hope is that examining the transformed Hamiltonian will bring out in
it some underlying structure, fixed in X, P space, with the rotational symmetry observed in
the perturbation expansion. As the evolution of x and p coincides with that for X and P
every q kicks, this should tell us something about the averaged dynamics of x and p over a
time period of q kicks.
The transformed Hamiltonian H̃ is given by [7]
H̃ = H +
∂F
,
∂t
(4.28)
where
∂
F (x, X) = p,
∂x
∂
F (x, X) = −P.
∂X
(4.29)
(4.30)
We thus need to express p and P exclusively in terms of x and X. With a little algebra, it
is easy to see that
x cos(t) − X
,
sin(t)
x − X cos(t)
.
P =
sin(t)
p=
(4.31)
(4.32)
Substituting Eqs. (4.31) into Eqs. (4.29) and integrating, we see that
x2 cos(t)/2 − xX
+ f (X).
F (x, X) =
sin(t)
(4.33)
Substituting this expression for F (x, X) with Eq. (4.32) back into Eq. (4.30), we arrive at
−
∂
x − X cos(t)
x
+
f (X) = −
sin(t) ∂X
sin(t)
(4.34)
which implies directly that
∂
f (X) = X cot(t).
(4.35)
∂X
We thus see immediately that f (X) = X 2 cot(t)/2 [plus an arbitrary constant, which can be
seen to be irrelevant on examining Eq. (4.28)], and that therefore
F (x, X) =
x2 + X 2
cot(t) − xXcosec(t).
2
(4.36)
We now need to take the partial derivative with respect to time of F (x, X):
x2 + X 2
∂
F (x, X) = −
cosec2 (t) + xXcosec(t) cot(t).
∂t
2
(4.37)
49
4.6. AVERAGED HAMILTONIAN
If we substitute in x = X cos(t) + P sin(t), we can write this expression terms of X and P
only. After some algebra, we arrive at
X2 P 2
∂
F (x, X) = −
−
.
∂t
2
2
As x2 + p2 = X 2 + P 2 , we can now rewrite Eq. (4.28) in terms of X and P as
∞
X
√
κ
√
cos{ 2[X cos(t) + P sin(t)]}
H̃ =
δ(t − nτ ),
2
n=−∞
(4.38)
(4.39)
and we have the transformed Hamiltonian, written in terms of the canonically conjugate
rotating variables X and P .
4.6
Averaged Hamiltonian
4.6.1
Derivation
We substitute the following identity for the train of delta kicks
∞
X
n=−∞
into Eq. (4.39), and get
δ(t − nτ ) =
q−1
∞
X
X
j=0 m=−∞
δ[t − (mq + j)τ ]
(4.40)
q−1
∞
√
κ X X
H̃ = √
δ[t − (mq + j)τ ] cos( 2{X cos[(mq + j)τ ] + P sin[(mq + j)τ ]}). (4.41)
2 j=0 m=−∞
Considering now explicitly the case where τ = 2πr/q, this simplifies down to
q−1
∞
X
√
κ X
cos{ 2[X cos(2πjr/q) + P sin(2πjr/q)]}
δ[t − (mq + j)2πr/q]. (4.42)
H̃ = √
2 j=0
m=−∞
The infinite train of delta kicks is periodic in time, and can thus be expanded as a Fourier
series,
∞
∞
X
1 X −in2πj/q int/r
e
e
.
(4.43)
δ[t − (mq + j)2πr/q] =
2πr
n=−∞
m=−∞
We substitute this back into Eq. (4.42), and after rearranging somewhat, the Hamiltonian
can be expressed as
H̃ = Hr/q + Vr/q ,
(4.44)
q−1
Hr/q =
√
κ X
√
cos{ 2[X cos(2πjr/q) + P sin(2πjr/q)]},
2πr 2 j=0
q−1
Vr/q
∞
(4.45)
X
√
κ X
= √
cos{ 2[X cos(2πjr/q) + P sin(2πjr/q)]}
cos[n(t/r − 2πj/q)]. (4.46)
πr 2 j=0
n=1
50
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
All of the time dependence in H̃ is in Vr/q , and we note that over a time of q kick periods
(i.e. t = 2πr), the average value of Vr/q is zero. We thus call Hr/q the averaged Hamiltonian,
as it is all that is left over after this averaging procedure.
4.6.2
Interpretation
The averaged Hamiltonian Hr/q clearly has a q-fold rotational symmetry through phase
space, in the same sense as that described in Section 4.4, i.e. if we replace X with Y =
X cos(2πl/q) − P sin(2πl/q) and P with S = P cos(2πl/q) + X sin(2πl/q), Hr/q remains
unchanged. This rotational symmetry can be seen graphically in Figs. (4.5,4.6,4.7,4.8,4.9),
where we have displayed contour plots of rHr/q for q = 1 − 7. Note that except for a factor
of two, q = 1 or 2 gives the same result, as does q = 3 or 6, and so we do not plot H r/q
specifically for q = 2 or 6.
A number of things become clear upon examining the plots for Hr/q . Firstly, certain of the
Hr/q display a translational, or crystal, symmetry, corresponding to that seen for the Poincaré
sections where q = 4 and 6 in Figs. (4.1,4.3). This takes place when q ∈ qc = {1, 2, 3, 4, 6}.
Given that it is known that it is only possible to combine a translational with a rotational
symmetry when q ∈ qc [8], equivalent to tiling the plane without gaps with regular q-gons,
it is inevitable that this be the case. When q ∈
/ qc , rHr/q displays a quasicrystal symmetry
[2, 3]. Secondly, in the cases q = 3, 4, 6 there is a network of interconnected separatrices
covering all of phase space. The contour plots of Hr/q show an energy surface; thus this
separatrix net describes an interconnected region of constant energy extended throughout
phase space.
One would therefore expect point particles whose dynamics are governed by Hr/q starting
on this separatrix net, to travel freely along this net through a wide area of phase space. In
the case of q ∈
/ qc , shown in Figs. (4.8,4.9) for the examples q = 5 and 7, there is no such
an interconnected network, but one can see that there are many closed loops which lie very
close together, and that it would take only a small disturbance to connect them, and we
would again have a structure spread throughout phase space.
Such a disturbance is provided for by Vr/q for small κ. Remember that Hr/q is the
averaged Hamiltonian, and that perturbations due to Vr/q average out to zero over any given
cycle of q kicks (a q cycle). At any one time during a q cycle there are of course non-zero
contributions to H̃ from Vr/q , although if κ is small, one expects these contributions to be
slight. We can thus say that the stochastic webs observed in Figs. (4.1,4.2,4.3,4.4) are based
around the separatrix nets or web skeletons defined by Hr/q . We emphasize that this is valid
for small κ only; if this is the case then the dynamics of a point particle are mostly accounted
for by Hr/q , but small perturbations due to Vr/q link nearly adjacent loops in phase space
together in the case that q ∈
/ qc , meaning that a point particle can diffuse out through phase
space along the stochastic web, and in general these perturbations cause the stochastic web
to have a finite thickness.
Those a little more expert may notice this derivation appears to contradict the KAM [9]
theorem, which essentially states that if an integrable system is perturbed so as to become
non-integrable, then the KAM tori are mostly expected to be similar, apart from some
51
4.6. AVERAGED HAMILTONIAN
10
8
6
4
2
p
0
−2
−4
−6
−8
−10
−15
−10
−5
0
x
5
10
15
Figure 4.5: Contour plot of Hr/q when q = 1 or 2.
10
8
6
4
2
p
0
−2
−4
−6
−8
−10
−15
−10
−5
0
x
5
10
Figure 4.6: Contour plot of Hr/q when q = 3 or 6.
15
52
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
10
8
6
4
2
p
0
−2
−4
−6
−8
−10
−15
−10
−5
0
x
5
10
15
Figure 4.7: Contour plot of Hr/q when q = 4.
30
20
10
p
0
−10
−20
−30
−50
0
x
Figure 4.8: Contour plot of Hr/q when q = 5.
50
53
4.6. AVERAGED HAMILTONIAN
60
40
20
p
0
−20
−40
−60
−80
−60
−40
−20
0
x
20
40
60
80
Figure 4.9: Contour plot of Hr/q when q = 7.
distortion, to the invariant tori of the integrable system. If the plots shown in Figs. 4.6–4.9
are supposed to approximate the phase space structure for the limit of small κ, this is clearly
not true. Even in this limit the harmonic oscillator invariant tori (exact circles in this case)
are all crossed over with the separatrix net. The KAM theorem has some rather hedging
conditions however. In particular there should be no degeneracy of the frequencies of the
orbits of the unperturbed system. The phase space orbits of the harmonic oscillator clearly
all have exactly the same frequency, an extreme case of degeneracy which means that the
KAM theorem is no longer valid. In the case of rational kicks, i.e. τ = 2πr/q there is a
resonance between the kicking frequency and the frequency of every single periodic orbit in
phase space. Consequently, nearly every invariant torus is destroyed, in the cases of q = 3,
4 or 6 with infinitesimal κ. Arnol’d diffusion [10] takes place along the separatrix net which
would be impossible for a one dimensional system subject to the KAM theorem.
4.6.3
Equations of Motion
We now show explicitly that the dynamics produced by Hr/q coincide exactly with the
perturbation expansion to first order in κ described by Eqs. (4.20,4.21). To do this we take
Hamilton’s equations of motion for Hr/q considered in isolation. As Hr/q is time independent,
the equations of motion for X and P must be integrable, and we can in fact find an exact
54
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
solution.
q−1
√
κ X
dX
=−
sin{ 2[X cos(2πjr/q) + P sin(2πjr/q)]} sin(2πjr/q),
dt
2πr j=0
(4.47)
q−1
√
dP
κ X
sin{ 2[X cos(2πjr/q) + P sin(2πjr/q)]} cos(2πjr/q).
=
dt
2πr j=0
(4.48)
From this we can immediately see that
dX
dP
= − sin(2πjr/q) .
dt
dt
We now take the second time derivative of X,
cos(2πjr/q)
(4.49)
q−1
√
d2 X
κ X
√
=
−
sin{
2[X cos(2πjr/q) + P sin(2πjr/q)]} sin(2πjr/q)
dt2
2πr j=0
dP
dX
cos(2πjr/q) +
sin(2πjr/q) (4.50)
×
dt
dt
which, because of Eq. (4.49), is equal to zero. If we differentiate Eq. (4.49) with respect to
time, we immediately see that this in turn also immediately implies that d2 P/dt2 = 0. The
derivatives dX/dt and dP/dt must then both be constant, and we conveniently choose to
evaluate these constants at the initial time t = 0. The variables X and P thus evolve like
X(t) = X(0) + dX/dt|t=0 t and P (t) = P (0) + dP/dt|t=0 t, giving
q−1
X(t) = X(0) −
√
κ X
sin{ 2[X(0) cos(2πjr/q) + P (0) sin(2πjr/q)]} sin(2πjr/q)t, (4.51)
2πr j=0
q−1
√
κ X
P (t) = P (0) +
sin{ 2[X(0) cos(2πjr/q) + P (0) sin(2πjr/q)]} cos(2πjr/q)t. (4.52)
2πr j=0
When considering t = 2πr, i.e. q kick periods, this reduces to
X(2πr) = X(0) − κ
P (2πr) = P (0) + κ
q−1
X
√
sin{ 2[X(0) cos(2πjr/q) + P (0) sin(2πjr/q)]} sin(2πjr/q), (4.53)
j=0
q−1
X
√
sin{ 2[X(0) cos(2πjr/q) + P (0) sin(2πjr/q)]} cos(2πjr/q). (4.54)
j=0
As we are considering t = 2πr, this means that X and P coincide with x and p. Making this
simple substitution, along with x(0), p(0) = xn , pn and x(2πr), p(2πr) = xn+q , pn+q , we find
that we have reproduced Eqs. (4.20,4.21). Thus the averaged Hamiltonian Hr/q produces the
dynamics of the delta kicked harmonic oscillator, when the time evolution is taken to only
first order in the kicking parameter κ, which is to say that Hr/q produces the dynamics of
the harmonic oscillator undergoing small perturbations due to a rational fraction of cosine
kicks every oscillator period.
55
4.7. TRANSLATIONAL SYMMETRY
4.7
Translational Symmetry
We have already observed that in Hr/q , which gives the dynamics up to first order in κ, there
is a translational or crystal symmetry for a certain subset of cases where τ = 2πr/q, namely
where q ∈ qc = {1, 2, 3, 4, 6}. In other words, there is a translational or symmetry in the
phase space, expressed over a time period of q kicks. We will now see that this translational
symmetry is in fact exact, for any value of κ.
To do this we return to the phase variable notation introduced in Section 4.3.2. Again using Eq. (4.16), we will determine the conditions for such a translational symmetry, expressed
as
βn = αn + γ ⇒ βn+q = αn+q + γ, γ ∈ .
(4.55)
Substituting this into Eq. (4.16) results in
q−1
κ X
∗
)ei2πjr/q ,
αn+q + γ = αn + γ + i √
sin(βn+j + βn+j
2 j=0
(4.56)
from which it can immediately be inferred that
q−1
X
sin(αn+j +
∗
)ei2πjr/q
αn+j
=
q−1
X
∗
)ei2πjr/q .
sin(βn+j + βn+j
(4.57)
j=0
j=0
This in turn provides the necessary condition
∗
∗
+ 2πlj ∀ j, lj ∈ .
= αn+j + αn+j
βn+j + βn+j
Taking this condition, Eq. (4.15) for βn gives
κ
∗
βn+1 = αn + γ + i √ sin(αn + αn ) e−i2πr/q ,
2
(4.58)
(4.59)
which states that if βn = αn + γ, then βn+1 = αn+1 + γe−i2πr/q . This can be carried out
iteratively, so that defining γj = βn+j − αn+j , we can state that γj = γe−i2πjr/q . Using this,
Eq. (4.16) for βn+q can be simplified to
q−1
βn+q
κ X
∗
+ γj + γj∗ )ei2πjr/q .
sin(αn+j + αn+j
= αn + γ + i √
2 j=0
(4.60)
Our condition for translational symmetry is thus reduced to γj + γj∗ = 2πlj , which can be
re-expressed as
γ + γ∗
γ − γ∗
πlj =
cos(2πjr/q) − i
sin(2πjr/q).
(4.61)
2
2
It is immediately apparent that l0 = (γ + γ ∗ )/2π, and so this can be simplified to
lj = l0 cos(2πjr/q) − i
γ − γ∗
sin(2πjr/q).
2π
(4.62)
56
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
If we now let j± = q/2 ± m or (q ± m)/2, depending on whether q is even or odd, we can
add or subtract lj+ and lj− . Bearing in mind that cos θ is symmetric and sin θ antisymmetric
around θ = nπ, we get
lj + l j −
(4.63)
cos(2πj+ r/q) = +
∈ ,
2l0
This implies that cos(2π/q) ∈ , and it is known that this can only be true if q ∈ qc =
{1, 2, 3, 4, 6} [11]. It is then easy to see that cos(2πjr/q) ∈ ∀ k, r ∈ , the only possible
values being 0, ±1/2, ±1. Thus there is is indeed a translational symmetry in phase space,
for q ∈ qc only, as expected. It should be emphasized that this is an exact symmetry, for any
value of κ, unlike the rotational q symmetry. There are an infinite number of values of γ for
which this translational symmetry applies, the values of which can be determined from the
following formulae derived in the same way as Eq. (4.63):
γ + γ∗
cos(2πjr/q) ∈
π
γ − γ∗
sin(2πjr/q) ∈
i
π
,
(4.64)
.
(4.65)
From this, one can easily determine the allowed symmetry
in po√ preserving displacements √
sition and momentum
= 2πn;
√ ∆x and ∆p,
√ where (∆x + i∆p)/ 2 = γ: for q√= 1 or 2, ∆xp
for q = 4, ∆x = 2πn, ∆p = 2πm; and for q = 3 or 6, ∆x = 2πn, ∆p = 3/2πm,
where in each case n, m ∈ . One can also think of these equations as determining the size
of the basic cell of the crystal structure, where the dynamics in equivalent locations in any
two cells are also equivalent, when viewed over a time period of q kicks. A point particle
may of course move from one cell to another, in which case the dynamics of other equivalent
point particles would dictate that they move in lockstep into corresponding cells.
There is thus an exact crystal symmetry in the mapping of Eq. (4.16) for q ∈ qc . A
direct implication of this is that we can confine ourselves to studying the dynamics of point
particles initially in just one cell. There are also important implications in the case of the
quantum delta-kicked harmonic oscillator.
4.8
Poincaré Sections
Our test example from now on will be the case where r/q = 1/6. There is thus one remaining
free parameter, the dimensionless kick strength κ. We investigate the effect of varying this
by studying Poincaré sections for κ from -0.2 to -4, displayed in Figs. (4.10–4.18).
The
sign of κ does make a difference to the overall phase space dynamics, it so happens that in
this case for negative κ the transition to chaos is more rapid.
In each case the same set of initial conditions was chosen, and each initial condition was
subjected to 10000 kicks. We show a region of phase space based around the central island
stable for small κ, with parts of the next circle of stable islands also depicted.
As expected, the most exact rotational symmetry is visible in Fig. (4.10), where κ =
−0.2 is the smallest studied. The stochastic web appears very thin, and while present, the
57
4.8. POINCARÉ SECTIONS
3
2
1
p
0
−1
−2
−3
−5
0
x
5
Figure 4.10: Poincaré section of the delta-kicked harmonic oscillator when κ = −0.2.
3
2
1
p
0
−1
−2
−3
−5
0
x
5
Figure 4.11: Poincaré section of the delta-kicked harmonic oscillator when κ = −0.8.
58
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
3
2
1
p
0
−1
−2
−3
−5
0
x
5
Figure 4.12: Poincaré section of the delta-kicked harmonic oscillator when κ = −1.
3
2
1
p
0
−1
−2
−3
−5
0
x
5
Figure 4.13: Poincaré section of the delta-kicked harmonic oscillator when κ = −1.2.
59
4.8. POINCARÉ SECTIONS
3
2
1
p
0
−1
−2
−3
−5
0
x
5
Figure 4.14: Poincaré section of the delta-kicked harmonic oscillator when κ = −1.6.
3
2
1
p
0
−1
−2
−3
−5
0
x
5
Figure 4.15: Poincaré section of the delta-kicked harmonic oscillator when κ = −2.2.
60
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
3
2
1
p
0
−1
−2
−3
−5
0
x
5
Figure 4.16: Poincaré section of the delta-kicked harmonic oscillator when κ = −2.8.
3
2
1
p
0
−1
−2
−3
−5
0
x
5
Figure 4.17: Poincaré section of the delta-kicked harmonic oscillator when κ = −3.4.
61
4.8. POINCARÉ SECTIONS
3
2
1
p
0
−1
−2
−3
−5
0
x
5
Figure 4.18: Poincaré section of the delta-kicked harmonic oscillator when κ = −4; no
obvious structure remains.
phase space appears to be dominated by stable dynamics, as signified by the many closed
curves, bound inside cells which are either hexagonal or triangular. One can clearly see the
correspondence of the contour plot of H1/q shown in Fig. (4.6) to Fig. (4.10). In Fig. (4.11) we
see that the stochastic web has substantially thickened, and the stable islands within the web
are now significantly distorted. In Fig. (4.12) we see bifurcations of the stable islands, and in
Fig. (4.13) we see how the boundaries between the newly formed “sub-islands” become filled
with chaotic dynamics. In Fig. (4.16) the central stable cell has bifurcated at the origin (and
correspondingly so in the next outer ring of cells). In Fig. (4.17) these cells have drastically
shrunk, and in Fig. (4.18) global chaos reigns.
There is an important difference between the central stable island and the surrounding
“rings” of islands. The closed curves observed inside the central cell describe stable dynamics
which really are confined within that cell. In the case of the next ring of corresponding stable
cells, the closed curves describe stable dynamics which are confined within that ring of cells;
point particles hop anticlockwise from one cell in that ring to the next. For this reason we
say that the stable orbits in the central cell are based around a stable fixed point of order
one, whereas the orbits of the next outer ring are based around a stable fixed point of order
six. These fixed points make up a periodic orbit.
62
CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR
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