262
8 Manifestations of Chaos in Quantum Scattering Processes
τ W S = −
i ¯
h
M
d
dE
ln(Det[ ¯
S]) = −
i ¯
h
M
d
dE
ln
Det[E ¯
1 N − ¯
H in − i ¯
w· ¯
w T ]
Det[E ¯
1 N − ¯
H in + i ¯
w· ¯
w T ]
.
(8.86)
The “effective” Hamiltonian, ¯
H eff = ¯
H in ±i ¯
w· ¯
w T , approximately determines the
location of poles of the S-matrix, as can be seen from Eq. (8.56). As described below
Eq. (8.56), the matrix ¯
w· ¯
w T has a slowly varying dependence on the energy E. If
we restrict ourselves to a small energy interval about some value of energy, E 0 , we
can set E = E 0 in the matrix, ¯
w· ¯
w T , so ¯
w o · ¯
w T
o = ( ¯
w· ¯
w T ) E=E o . We then obtain
the eigenvalues of ¯
H in ±i ¯
w o · ¯
w T
o in the neighborhood of energy E = E 0 . In this
approximation, the poles of the S-matrix are given by the eigenvalues, E n ±ii n , of
the “effective” Hamiltonian, ¯
H eff = ¯
H in ±i ¯
w o · ¯
w T
o , where E n is the real part of the
complex eigenenergy and n is the imaginary part. One must remember that this is
only true if evanescent modes can be neglected.
We can write the Wigner–Smith delay time in terms of the eigenvalues of ¯
H eff =
¯
H in ±i ¯
w o · ¯
w T
o . We then obtain
τ W S = −
i
M
d
dE
ln
N
n=1
E − E n − ii n
E − E n + ii n
=
2
M
N
n=1
n
(E − E n ) 2 + 2
n
.
(8.87)
Thus, as a function of real values of E, the Wigner–Smith delay time will have
Lorentzian-shaped peaks at values E = E n . The width of the nth peak is determined
by n , which is the distance of the nth pole of the S-matrix from the real energy axis.
In Example 8.4 below, we show the relation between Wigner–Smith delay times and
S-matrix poles for a simple example.
Example 8.4 (S-matrix Poles for a 1-d Scattering System)
Let us consider the 1-d scattering system described in Examples 8.1–8.3. In Example 8.3,
we showed a plot of the Wigner–Smith delay time as a function of energy. For the energy
interval considered, it had three Lorentzian-shaped peaks. It is possible to obtain values
of energy E, where Det[E ¯
1 N − ¯
H in ±i ¯
w· ¯
w T ] = 0. They occur at E 1 = 17.97 − i4.98,
E 2 = 44.34 − i17.34, and E 3 = 92.13 − i33.45 for a = 1 and V o = 10. These values are
in good agreement with the positions and widths of the Wigner–Smith delay time for this
example (Reichl and Akguc 2001). One reason for this good agreement is that systems with
one space dimension have only one propagating channel and no evanescent modes.
8.5 Scattering in the Ripple Waveguide
We will consider scattering of electrons in a two dimensional ripple waveguide
formed from a GaAs two-dimensional electron gas (2DEG). 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. We consider two different configurations. In the first
configuration, the left and right sides of the ripple billiard considered in Sect. 7.5.3
are opened and connected to infinitely long straight leads. Thus, in this first case,
electrons can enter from the left or right and scatter to the left or right. In the second
8 Manifestations of Chaos in Quantum Scattering Processes
τ W S = −
i ¯
h
M
d
dE
ln(Det[ ¯
S]) = −
i ¯
h
M
d
dE
ln
Det[E ¯
1 N − ¯
H in − i ¯
w· ¯
w T ]
Det[E ¯
1 N − ¯
H in + i ¯
w· ¯
w T ]
.
(8.86)
The “effective” Hamiltonian, ¯
H eff = ¯
H in ±i ¯
w· ¯
w T , approximately determines the
location of poles of the S-matrix, as can be seen from Eq. (8.56). As described below
Eq. (8.56), the matrix ¯
w· ¯
w T has a slowly varying dependence on the energy E. If
we restrict ourselves to a small energy interval about some value of energy, E 0 , we
can set E = E 0 in the matrix, ¯
w· ¯
w T , so ¯
w o · ¯
w T
o = ( ¯
w· ¯
w T ) E=E o . We then obtain
the eigenvalues of ¯
H in ±i ¯
w o · ¯
w T
o in the neighborhood of energy E = E 0 . In this
approximation, the poles of the S-matrix are given by the eigenvalues, E n ±ii n , of
the “effective” Hamiltonian, ¯
H eff = ¯
H in ±i ¯
w o · ¯
w T
o , where E n is the real part of the
complex eigenenergy and n is the imaginary part. One must remember that this is
only true if evanescent modes can be neglected.
We can write the Wigner–Smith delay time in terms of the eigenvalues of ¯
H eff =
¯
H in ±i ¯
w o · ¯
w T
o . We then obtain
τ W S = −
i
M
d
dE
ln
N
n=1
E − E n − ii n
E − E n + ii n
=
2
M
N
n=1
n
(E − E n ) 2 + 2
n
.
(8.87)
Thus, as a function of real values of E, the Wigner–Smith delay time will have
Lorentzian-shaped peaks at values E = E n . The width of the nth peak is determined
by n , which is the distance of the nth pole of the S-matrix from the real energy axis.
In Example 8.4 below, we show the relation between Wigner–Smith delay times and
S-matrix poles for a simple example.
Example 8.4 (S-matrix Poles for a 1-d Scattering System)
Let us consider the 1-d scattering system described in Examples 8.1–8.3. In Example 8.3,
we showed a plot of the Wigner–Smith delay time as a function of energy. For the energy
interval considered, it had three Lorentzian-shaped peaks. It is possible to obtain values
of energy E, where Det[E ¯
1 N − ¯
H in ±i ¯
w· ¯
w T ] = 0. They occur at E 1 = 17.97 − i4.98,
E 2 = 44.34 − i17.34, and E 3 = 92.13 − i33.45 for a = 1 and V o = 10. These values are
in good agreement with the positions and widths of the Wigner–Smith delay time for this
example (Reichl and Akguc 2001). One reason for this good agreement is that systems with
one space dimension have only one propagating channel and no evanescent modes.
8.5 Scattering in the Ripple Waveguide
We will consider scattering of electrons in a two dimensional ripple waveguide
formed from a GaAs two-dimensional electron gas (2DEG). 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. We consider two different configurations. In the first
configuration, the left and right sides of the ripple billiard considered in Sect. 7.5.3
are opened and connected to infinitely long straight leads. Thus, in this first case,
electrons can enter from the left or right and scatter to the left or right. In the second