Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

3.2 Introduction to quantum dots and various derivations 67
quantum system is generally non-stationary when the system is initialised in a
non-eigenstate, the experimental observation of such oscillations highlights our
ability to manipulate nature on a microscopic level.
Whenever we instantaneously change the parameters δ, λ, ω of the system we
may expect CO because, after the change, the system is generally not in a stationary
state. In the following, we omit the ‘−’ index for the frequency in Eqs. (3.27)
and (3.28) and include the explicit √ n + 1 dependence by setting ˜λ := λ √ n + 1.
To prepare the system in a non-eigenstate we propose a non-adiabatic change of
λ, simply by altering the strength of the SO coupling by means of an additional
gate voltage pulse. In terms of the JCM such a pulse leads to oscillations in the
{|n,↑〉,|n + 1,↓〉} subspace of the Hilbert space for fixed ω. In contrast, every
change in ω would destroy the strict decomposition of the Hilbert space, leading
to a mixing of different ω − modes. A change of magnetic field would lead to a
simultaneous modification of all parameters ω, δ, λ.
This relation between the parameters due to an external magnetic field is a
significant difference to the JCM in quantum optics where all parameters can be
tuned individually.
In the following, we solve the time-dependent Schrödinger equation for the
experimental scheme which has been proposed in chapter 3.1. To prepare a nonstationary
state we perform the following: (i) Assume the system is in the eigenstate
of H JC (α 1 ), |ψ − α 1
〉, for times t < 0. (ii) At t = 0 we instantaneously switch
from α 1 → α 2 while keeping all other parameters fixed. Thus, in the basis of
eigenstates of H JC (α 2 ) we have
where
|ψ − α 1
〉 = cosθ − α 1
|n,↑〉 + sinθ − α 1
|n + 1,↓〉 = c + |ψ + α 2
〉 + c − |ψ − α 2
〉 (3.30)
c ± = 〈ψ ± α 2
|ψ − α 1
〉 = cos ( θ ± α 2
− θ − α 1
)
=: cos∆θ± . (3.31)
The probability to find the system in state |ψ + α 1
〉 after pulse time t p > 0 is
P(t p ) = |〈ψ + α 1
|Ψ(t p )〉| 2 = |〈ψ + α 1
|e −iH(α 2)t p / |ψ − α 1
〉| 2 (3.32)
= 1 − |cos 2 ∆θ + e −2iΩt p
+ cos 2 ∆θ − | 2 , (3.33)
where Ω := (E + α 2
− E − α 2
)/2 is the frequency of CO (in units of ω 0 ).
From Eq. (3.28) follows
Ω = 1 2 ˜λ 2
√ ( δ
˜λ 2
)
+ 4. (3.34)