10.6 Quantum Delta-Kicked Rotor
365
width n k = 2
KI
¯
h 2 T
. These estimates for the location and width of the primary
resonances will be useful in our subsequent discussion.
10.6.2 KAM-Like Behavior of the Quantum Delta-Kicked
Rotor
Geisel, Radons, and Rubner considered the quantum delta-kicked rotor (Geisel et al.
1986). for the parameter value K = 0.9716354, the value at which the last KAM tori
that block global diffusion of the angular momentum begin to break in the classical
system. At this value of K there are two remaining horizontal KAM tori in the
interval 0 < J < 2π (note that J = 2πp and θ = 2πx). There are also several
cantori that provide partial blockage at that parameter value.
They determined the extent to which these remnants of constants of motion in
the classical system also block flow of probability in Hilbert space. They take ¯
h =
0.01, T = 1, and I = 1. Since J = n ¯
h, there are 2π × 10 2 angular momentum
eigenstates in the interval 0 < J < 2π when ¯
h = 0.01. They start the system with
all probabilities concentrated on the state J 0 = 3.2 or n 0 = 320. They then compute
the asymptotic probability
P (J |J 0 ) = lim
n→∞
1
N
N −1
t=0
||n|ψ(t)|
2 .
(10.49)
Their results are shown in Fig. 10.11. The dot-dashed line indicates the last two
KAM tori in the interval 0 < J < 2π , and the dotted lines indicate cantori.
In Fig. 10.11a, the probability is plotted on a linear scale, while in Fig. 10.11b it
is plotted on a logarithmic scale. The KAM tori and cantori form barriers to the
probability. The probability appears to “tunnel” through them. Brown and Wyatt
(1986) have observed similar behavior in a driven oscillator model.
10.6.3 The Floquet Map
The delta-kicked rotor is particularly interesting to study quantum mechanically
because its dynamical evolution can be determined analytically in terms of a Floquet
matrix. The delta-kicks occur at times t = qT , where q is an integer. Between these
kicks, the system evolves as a free rotor. Let ψ(θ, 0 + ) denote the state of the system
at time t = 0 + (just after the kick at t = 0), and let
ψ(θ, 0
+ ) =
∞
n=−∞
ψ n (0
+ )e
inθ .
(10.50)
365
width n k = 2
KI
¯
h 2 T
. These estimates for the location and width of the primary
resonances will be useful in our subsequent discussion.
10.6.2 KAM-Like Behavior of the Quantum Delta-Kicked
Rotor
Geisel, Radons, and Rubner considered the quantum delta-kicked rotor (Geisel et al.
1986). for the parameter value K = 0.9716354, the value at which the last KAM tori
that block global diffusion of the angular momentum begin to break in the classical
system. At this value of K there are two remaining horizontal KAM tori in the
interval 0 < J < 2π (note that J = 2πp and θ = 2πx). There are also several
cantori that provide partial blockage at that parameter value.
They determined the extent to which these remnants of constants of motion in
the classical system also block flow of probability in Hilbert space. They take ¯
h =
0.01, T = 1, and I = 1. Since J = n ¯
h, there are 2π × 10 2 angular momentum
eigenstates in the interval 0 < J < 2π when ¯
h = 0.01. They start the system with
all probabilities concentrated on the state J 0 = 3.2 or n 0 = 320. They then compute
the asymptotic probability
P (J |J 0 ) = lim
n→∞
1
N
N −1
t=0
||n|ψ(t)|
2 .
(10.49)
Their results are shown in Fig. 10.11. The dot-dashed line indicates the last two
KAM tori in the interval 0 < J < 2π , and the dotted lines indicate cantori.
In Fig. 10.11a, the probability is plotted on a linear scale, while in Fig. 10.11b it
is plotted on a logarithmic scale. The KAM tori and cantori form barriers to the
probability. The probability appears to “tunnel” through them. Brown and Wyatt
(1986) have observed similar behavior in a driven oscillator model.
10.6.3 The Floquet Map
The delta-kicked rotor is particularly interesting to study quantum mechanically
because its dynamical evolution can be determined analytically in terms of a Floquet
matrix. The delta-kicks occur at times t = qT , where q is an integer. Between these
kicks, the system evolves as a free rotor. Let ψ(θ, 0 + ) denote the state of the system
at time t = 0 + (just after the kick at t = 0), and let
ψ(θ, 0
+ ) =
∞
n=−∞
ψ n (0
+ )e
inθ .
(10.50)