6 Coherent Control of Nonadiabatic Dynamics of Electron-Phonon …
127
between them turns to be weaker and the absorption of mode 1 photons is clearly
seen.
Comparing Figs. 6.2a with 6.3a, we found that electronic transition is suppressed
at t ∼ 2π and 4π also by the electron-phonon nonadiabaticity. We note that the
adiabatic PESs of the electron-phonon subsystem (PESSs) have an avoided crossing
at u = −ε/ν
for λ = 0. The photoexcited wavepackets bifurcate at the avoidedcrossing and thus the transition between |g and |e is more complicated for finite λ.
In this case, the interference between those processes affects the electronic transition,
and the resonance to the pump mode photons is blurred. Hence, the temporal change
of N (t) and n i (t) for t ∼ 10 is unclear as shown in Fig. 6.2a, b.
Since the wavepacket trajectory on the PESSs is an experimentally observable
quantity [20], we calculated the lattice displacement u(t) = =(t)| ˆ
u|(t) and focus
on the dynamics of the electron and phonons. Figure 6.4a–c show u(t) for the three
cases corresponding to Figs. 6.1, 6.2 and 6.3. As discussed previously, Fig. 6.4a, b
show that the semiclassical approximation is not valid for t > 2π for λ = 0. On the
contrary, the solid line and the dotted line in Fig. 6.4c are similar to each other, which
shows that the nonadiabaticity of the electron-phonon-photon dynamics is relevant
to the validity of the semiclassical approximation. As mentioned above, the (onedimensional) PESSs have an avoided-crossing at u = −ε/ν
, while the PESs for the
whole system has a CI. Since the semiclassical approximation takes into account only
the avoided-crossing, the wavepacket motion bifurcate in a different manner between
the quantum-mechanical calculation and the semiclassical calculation particularly in
the vicinity of the avoided-crossing or the CI, which results in the different dynamics
or trajectory shown in the figures. We stress that the quantum-mechanical nature
of the incident light plays an important role on the wavepacket motion, and that
detailed discussion will be possible by revealing the transient dynamics of coherent
phonons by ultrafast optical spectroscopy. To be more precise, the role of the CI on
the electronic transition should be revealed in order to determine the wavepackets
created by photons. In particular, as the irradiation of photons proceeds, deviation
from single-mode model becomes larger, the role of CI becomes more important.
As an example of coherent control by external light field, we show that the relative
phase between photons affects the created phonon states. Figure 6.5 shows the number
of phonons N n (t) = =(t)|a
† a|(t) for φ = −π, −π/2, 0, π/2 and π, where φ
denotes the phase difference between Stokes/anti-Stokes mode and the pump mode,
i.e., α 2 = |α 2 |e
iφ and α 3 = |α 3 |e
−iφ . Since the wavepacket trajectory varies with
φ, different phonon states are created in the material, which shows that the number
of phonons is able to be modified by φ. These results show that, when we consider
the control of phonon states by external light, not only the duration of the photon
pulses but also the relative phase between photon modes is important.
The quantum-mechanical nature of the electromagnetic field is reflected on the
entanglement between subsystems, i.e., photons, phonons, and electrons. Hence, we
calculated the bipartite entanglement entropy in which the whole system is divided
into (i) the electronic system and the phonon-photon system, and (ii) the electronphonon subsystem and the photons. Although it is not directly observable in experiments, we will have an important information on the quantum-mechanical nature
127
between them turns to be weaker and the absorption of mode 1 photons is clearly
seen.
Comparing Figs. 6.2a with 6.3a, we found that electronic transition is suppressed
at t ∼ 2π and 4π also by the electron-phonon nonadiabaticity. We note that the
adiabatic PESs of the electron-phonon subsystem (PESSs) have an avoided crossing
at u = −ε/ν
for λ = 0. The photoexcited wavepackets bifurcate at the avoidedcrossing and thus the transition between |g and |e is more complicated for finite λ.
In this case, the interference between those processes affects the electronic transition,
and the resonance to the pump mode photons is blurred. Hence, the temporal change
of N (t) and n i (t) for t ∼ 10 is unclear as shown in Fig. 6.2a, b.
Since the wavepacket trajectory on the PESSs is an experimentally observable
quantity [20], we calculated the lattice displacement u(t) = =(t)| ˆ
u|(t) and focus
on the dynamics of the electron and phonons. Figure 6.4a–c show u(t) for the three
cases corresponding to Figs. 6.1, 6.2 and 6.3. As discussed previously, Fig. 6.4a, b
show that the semiclassical approximation is not valid for t > 2π for λ = 0. On the
contrary, the solid line and the dotted line in Fig. 6.4c are similar to each other, which
shows that the nonadiabaticity of the electron-phonon-photon dynamics is relevant
to the validity of the semiclassical approximation. As mentioned above, the (onedimensional) PESSs have an avoided-crossing at u = −ε/ν
, while the PESs for the
whole system has a CI. Since the semiclassical approximation takes into account only
the avoided-crossing, the wavepacket motion bifurcate in a different manner between
the quantum-mechanical calculation and the semiclassical calculation particularly in
the vicinity of the avoided-crossing or the CI, which results in the different dynamics
or trajectory shown in the figures. We stress that the quantum-mechanical nature
of the incident light plays an important role on the wavepacket motion, and that
detailed discussion will be possible by revealing the transient dynamics of coherent
phonons by ultrafast optical spectroscopy. To be more precise, the role of the CI on
the electronic transition should be revealed in order to determine the wavepackets
created by photons. In particular, as the irradiation of photons proceeds, deviation
from single-mode model becomes larger, the role of CI becomes more important.
As an example of coherent control by external light field, we show that the relative
phase between photons affects the created phonon states. Figure 6.5 shows the number
of phonons N n (t) = =(t)|a
† a|(t) for φ = −π, −π/2, 0, π/2 and π, where φ
denotes the phase difference between Stokes/anti-Stokes mode and the pump mode,
i.e., α 2 = |α 2 |e
iφ and α 3 = |α 3 |e
−iφ . Since the wavepacket trajectory varies with
φ, different phonon states are created in the material, which shows that the number
of phonons is able to be modified by φ. These results show that, when we consider
the control of phonon states by external light, not only the duration of the photon
pulses but also the relative phase between photon modes is important.
The quantum-mechanical nature of the electromagnetic field is reflected on the
entanglement between subsystems, i.e., photons, phonons, and electrons. Hence, we
calculated the bipartite entanglement entropy in which the whole system is divided
into (i) the electronic system and the phonon-photon system, and (ii) the electronphonon subsystem and the photons. Although it is not directly observable in experiments, we will have an important information on the quantum-mechanical nature