Driven Quantum Systems - Institut für Physik

282 Driven Quantum Systems
This control by a periodic continuous-wave driving can be generalized by recourse to
more complex perturbations. The goal by which a pre-assigned task for the output of
a quantum dynamics is imposed from the outside by applying a sequence of properly
designed (in phase and/or shape) pulse perturbations is known as quantum control
[8]. For example, a primary goal in chemical physics is to produce desired product
yields or to manipulate the atomic and molecular properties of matter [8, 42, 43]. As
an archetype situation, we present the control of the quantum dynamics of two coupled
electronic surfaces
⎛
i¯h ∂ Ψ ⎞ ⎡
⎤ ⎛
g
H g −µ(R)E(t)
⎝ ⎠ =
Ψ ⎞
g
⎣
⎦ ⎝ ⎠ , (5.167)
∂t Ψ e −µ(R)E ∗ (t) H e Ψ e
where R denotes the nuclear coordinates and H g,e are the Born-Oppenheimer Hamiltonians
for the ground- (g) and excited- (e) field free surfaces, respectively. The surfaces
are coupled within the dipole approximation by the transition dipole operator µ(R)and
the generally complex-valued radiation field E(t). Notice that the structure in (5.167)
is identical to that obtained in the driven TLS. Following Kosloff, Hammerich, and
Tannor [43], the rate of change to the ground-state population n g (t) =〈Ψ g (t)|Ψ g (t)〉
is readily evaluated to read
dn g
dt =2Re〈Ψ g|˙Ψ g 〉
If we set with C(t) a real-valued function
=− 2¯h Im ( 〈Ψ g (t)|µ(R)|Ψ e (t)〉E(t) ) . (5.168)
E(t) → E 0 (t) =〈Ψ e (t)|µ(R)|Ψ g (t)〉C(t), (5.169)
we can freeze the population transfer (null-population transfer), with (5.169),
dn g
=0, (5.170)
dt
for all times t. In other words, the population in the ground electronic surface, and
necessarily also the population of the excited surface, remains fixed. If we were to
chose E(t) → iE 0 (t), it would cause population to be transferred to the upper state,
while E(t) →−iE 0 (t) would dump population down to the groundstate. Hence, by
controlling the phase of a laser, we can control the population transfer at will. This
phenomenon applies equally well to driven tunneling in a TLS. What can we achieve
if we manipulate the amplitude C(t)? The change in the energy of the ground state
surface, which can be varied by exciting specific ground-state vibrational modes, is
obtained as
dE g
dt
= d dt
〈Ψ g (t)|H g |Ψ g (t)〉
. (5.171)
〈Ψ g (t)|Ψ g (t)〉