8.5 Scattering in the Ripple Waveguide
267
family of resonances associated with scattering states sitting on the chaotic layer
are very broad and can all be seen in Fig. 8.8. These broad resonances also appear
periodically in energy.
An interesting phenomenon occurs in multi-ripple waveguides. The multi-ripple
waveguide behaves like a sequence of quantum dots, and quasibound states in the
waveguide correspond to the excited states of an atomic system. As was shown in
(Lee and Reichl 2008), the poles of the ripple-waveguide scattering matrix show a
kind of Dicke effect (Dicke 1953). One of the S-matrix poles withdraws further from
the real axis as each quantum dot is added, indicating that the pole is behaving like
that of a super-radiant quasibound state. The lifetime of the super-radiant state, for
N quantum dots, decreases as
1
N . This behavior of the lifetime of the super-radiant
state is a signature of the Dicke effect, and has been observed in arrays of identical
independent excited atoms in a common radiation field.
8.5.2 Wigner–Smith Delay Times for a Chaotic Scattering
System
We next consider the scattering of an electron in the waveguide shown in Fig. 8.10.
The electron enters from the left with energy E along an infinitely long straight lead
that has infinitely hard walls. The electron wave is reflected back to the left by an
infinitely hard wall located at x = L. The scattering is strongly affected by the
region 0 < x < L (the reaction region) in which the upper wall is rippled, and the
dynamics inside the cavity can be chaotic.
The Schrodinger equation that describes propagation of the particle wave,
(x, y, t), for all times t is given Eq. (8.88). The potential V(x,y) has the following
properties: V (x, y) = ∞ for (L≤x≤∞); V (x, 0) = ∞ for (−∞≤x≤L); V (x, y =
g(x)) = ∞ for (0 < x < L); and V (x, y = d) = ∞ for (−∞ < x < 0); where
g(x) = d + asin(5πx/L) gives the contour of the ripple, d is the average width of
the cavity, L is the length, and a is the ripple amplitude. We take the electron mass
to be the effective mass of an electron in GaAs, m = 50.067m e , where m e is the
free electron mass.
Fig. 8.10 Two dimensional
rippled electron wave guide is
the region defined with solid
lines. The dotted dashed line
shows the entrance between
leads and scattering region.
Here a is the width of the
ripple, d is the width of the
leads. The scattering region
extents from x = 0 to x = L
267
family of resonances associated with scattering states sitting on the chaotic layer
are very broad and can all be seen in Fig. 8.8. These broad resonances also appear
periodically in energy.
An interesting phenomenon occurs in multi-ripple waveguides. The multi-ripple
waveguide behaves like a sequence of quantum dots, and quasibound states in the
waveguide correspond to the excited states of an atomic system. As was shown in
(Lee and Reichl 2008), the poles of the ripple-waveguide scattering matrix show a
kind of Dicke effect (Dicke 1953). One of the S-matrix poles withdraws further from
the real axis as each quantum dot is added, indicating that the pole is behaving like
that of a super-radiant quasibound state. The lifetime of the super-radiant state, for
N quantum dots, decreases as
1
N . This behavior of the lifetime of the super-radiant
state is a signature of the Dicke effect, and has been observed in arrays of identical
independent excited atoms in a common radiation field.
8.5.2 Wigner–Smith Delay Times for a Chaotic Scattering
System
We next consider the scattering of an electron in the waveguide shown in Fig. 8.10.
The electron enters from the left with energy E along an infinitely long straight lead
that has infinitely hard walls. The electron wave is reflected back to the left by an
infinitely hard wall located at x = L. The scattering is strongly affected by the
region 0 < x < L (the reaction region) in which the upper wall is rippled, and the
dynamics inside the cavity can be chaotic.
The Schrodinger equation that describes propagation of the particle wave,
(x, y, t), for all times t is given Eq. (8.88). The potential V(x,y) has the following
properties: V (x, y) = ∞ for (L≤x≤∞); V (x, 0) = ∞ for (−∞≤x≤L); V (x, y =
g(x)) = ∞ for (0 < x < L); and V (x, y = d) = ∞ for (−∞ < x < 0); where
g(x) = d + asin(5πx/L) gives the contour of the ripple, d is the average width of
the cavity, L is the length, and a is the ripple amplitude. We take the electron mass
to be the effective mass of an electron in GaAs, m = 50.067m e , where m e is the
free electron mass.
Fig. 8.10 Two dimensional
rippled electron wave guide is
the region defined with solid
lines. The dotted dashed line
shows the entrance between
leads and scattering region.
Here a is the width of the
ripple, d is the width of the
leads. The scattering region
extents from x = 0 to x = L