Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

140 C.W.J. Beenakker
A stroboscopic model of an Andreev billiard was constructed by Jacquod
et al. [27], building on an existing model for open normal billiards called the
open kicked rotator [28]. The Andreev kicked rotator possesses the same phenomenology
as the Andreev billiard, but is much more tractable numerically. 3
In this subsection we discuss how it is formulated. Some results obtained by
this numerical method will be compared in subsequent sections with results
obtained by analytical means.
A compact unitary map is represented in quantum mechanics by the Floquet
operator F , which gives the stroboscopic time evolution u(pτ0) =F pu(0) of an initial wave function u(0). (We set the stroboscopic period τ0 =1in
most equations.) The unitary M × M matrix F has eigenvalues exp(−iεm),
with the quasi-energies εm ∈ (−π, π) (measured in units of �/τ0). This describes
the electron excitations above the Fermi level. Hole excitations below
the Fermi level have Floquet operator F ∗ and wave function v(p) =(F∗ ) pv(0). The mean level spacing of electrons and holes separately is δ =2π/M.
An electron is converted into a hole by Andreev reflection at the NS interface,
with phase shift −i for ε ≪ τ0∆/� [cf. (20)]. In the stroboscopic
description one assumes that Andreev reflection occurs only at times which
are multiples of τ0. TheN × M matrix P projects onto the NS interface. Its
elements are Pnm =1ifm = n ∈{n1,n2,...nN } and Pnm = 0 otherwise. The
dwell time of a quasiparticle excitation in the normal metal is τdwell = M/N,
equal to the mean time between Andreev reflections.
Putting all this together one constructs the quantum Andreev map from
the matrix product
�
F 0
F = P
0 F ∗
� �
T T 1 − P P −iP P
, P =
−iP TP 1 − P T �
. (23)
P
(The superscript “T” indicates the transpose of a matrix.) The particle-hole
wave function Ψ =(u, v) evolves in time as Ψ(p) =F pΨ(0). The Floquet
operator can be symmetrized (without changing its eigenvalues) by the unitary
transformation F→P−1/2FP1/2 , with
� √ �
1 T 1
1 − (1 − 2)P P −i
P 1/2 =
−i 1
2
√
T
2
2 2P P
√ √
T 1 T 2P P 1 − (1 − 2 2)P P
. (24)
The quantization condition det(F −e−iε ) = 0 can be written equivalently
as [27]
Det [1 + S(ε)S(−ε) ∗ ]=0, (25)
in terms of the N × N scattering matrix [28, 29]
S(ε) =P [e −iε − F (1 − P T P )] −1 FP T . (26)
3 The largest simulation to date of a two-dimensional Andreev billiard has N =30,
while for the Andreev kicked rotator N =10 5 is within reach, cf. Fig. 24.