for out-of phase electronic coherence ða À b 1 Þ, ða À b 2 Þ and ðb 1 À b 2 Þ each current
can be expressed by that corresponding in-phase coherent currents with π-phase shift.
3.5 Time Dependent Angular Momentum for Three Types
of Electron Coherence
Figure 4 shows the results of a numerical calculation of time-dependent angular
momentum for three types of electronic coherence with in-phase, ða þ b 1 Þ, ða þ b 2 Þ
and ðb 1 þ b 2 Þ. The same laser pulses in Fig. 3 are adopted for Fig. 4. For ða þ b 1 Þ
and ða þ b 2 Þ electronic coherences, the generated angular momenta are along
X-axis with π-phase shift, while for ðb 1 þ b 2 Þ electronic coherence the angular
momentum is generated along Z-axis. Directions of total angular momentum can be
understood from Fig. 2b.
3.6 Design of Ultrafast Multi-dimensional Quantum
Switching
As a more developed application of our method to a design of ultrafast multi-dimensional quantum switching, we proposed a quantum sequential switching of angular
momentum along X and Z-axis [20]. Figure 5 (which is cited from Fig. 3 of Ref. [20])
shows a three-dimensional plot of the resultant angular momentum switching based on
the sequential four-step scheme. It can be seen from Fig. 5a that the angular momenta
were successfully controlled by the pulses depicted in Fig. 5b, i.e., both the rotational
axis (parallel to the Z or X axis) and the rotational direction around the axis (clockwise
or anticlockwise) were satisfactorily controlled by the sequential four-step process. In
Fig. 5b, the quantum control at each switching step was carried out by using overlapped
pump and dump pulses with specific polarization directions. A pulsed laser with
amplitude of F = 1.2 GV/m was used in the second and fourth steps, while F = 0.3 GV/
m was used in the first and third steps. The pulses shown in Fig. 5b have two features:
first, the pump (dump) pulse for each step has~ e
ðþÞ
ab ~ e
ðÀÞ
ab
or~ e
ðÀÞ
ab ~ e
ðþÞ
ab
polarization.
Second, the pump and dump pulses partially overlap. For the first step (i.e., creation of
CC rotation), for example, the electric field of the pump pulse was
~ E
ðþÞ
b1b2 ðtÞ ¼ ~ e
ðþÞ
b1b2 E
0
b1b2 sin
2
ðpt=T b1b2 Þ sinðx c;b1b2 tÞ, while that of the dump pulse was
~ E
ðÀÞ
b1b2 ðtÞ ¼ ~ e
ðÀÞ
b1b2 E
0
b1b2 sin
2
ðpðt À t
pd
b1b2 Þ=T b1b2 Þ sinðx c;b1b2 t þ p=2Þ, where E
0
b1b2 is the
amplitude of the pulse, T b1b2 (here equal to 60.9 fs) is the oscillation period between the
two excited states b 1 and b 2 , x c;b1b2 is the central frequency between the two excited
states, and t
pd
b1b2 is the time interval between the pump and dump pulses, which was set
to T b1b2 =2.
170
H. Mineo and Y. Fujimura
can be expressed by that corresponding in-phase coherent currents with π-phase shift.
3.5 Time Dependent Angular Momentum for Three Types
of Electron Coherence
Figure 4 shows the results of a numerical calculation of time-dependent angular
momentum for three types of electronic coherence with in-phase, ða þ b 1 Þ, ða þ b 2 Þ
and ðb 1 þ b 2 Þ. The same laser pulses in Fig. 3 are adopted for Fig. 4. For ða þ b 1 Þ
and ða þ b 2 Þ electronic coherences, the generated angular momenta are along
X-axis with π-phase shift, while for ðb 1 þ b 2 Þ electronic coherence the angular
momentum is generated along Z-axis. Directions of total angular momentum can be
understood from Fig. 2b.
3.6 Design of Ultrafast Multi-dimensional Quantum
Switching
As a more developed application of our method to a design of ultrafast multi-dimensional quantum switching, we proposed a quantum sequential switching of angular
momentum along X and Z-axis [20]. Figure 5 (which is cited from Fig. 3 of Ref. [20])
shows a three-dimensional plot of the resultant angular momentum switching based on
the sequential four-step scheme. It can be seen from Fig. 5a that the angular momenta
were successfully controlled by the pulses depicted in Fig. 5b, i.e., both the rotational
axis (parallel to the Z or X axis) and the rotational direction around the axis (clockwise
or anticlockwise) were satisfactorily controlled by the sequential four-step process. In
Fig. 5b, the quantum control at each switching step was carried out by using overlapped
pump and dump pulses with specific polarization directions. A pulsed laser with
amplitude of F = 1.2 GV/m was used in the second and fourth steps, while F = 0.3 GV/
m was used in the first and third steps. The pulses shown in Fig. 5b have two features:
first, the pump (dump) pulse for each step has~ e
ðþÞ
ab ~ e
ðÀÞ
ab
or~ e
ðÀÞ
ab ~ e
ðþÞ
ab
polarization.
Second, the pump and dump pulses partially overlap. For the first step (i.e., creation of
CC rotation), for example, the electric field of the pump pulse was
~ E
ðþÞ
b1b2 ðtÞ ¼ ~ e
ðþÞ
b1b2 E
0
b1b2 sin
2
ðpt=T b1b2 Þ sinðx c;b1b2 tÞ, while that of the dump pulse was
~ E
ðÀÞ
b1b2 ðtÞ ¼ ~ e
ðÀÞ
b1b2 E
0
b1b2 sin
2
ðpðt À t
pd
b1b2 Þ=T b1b2 Þ sinðx c;b1b2 t þ p=2Þ, where E
0
b1b2 is the
amplitude of the pulse, T b1b2 (here equal to 60.9 fs) is the oscillation period between the
two excited states b 1 and b 2 , x c;b1b2 is the central frequency between the two excited
states, and t
pd
b1b2 is the time interval between the pump and dump pulses, which was set
to T b1b2 =2.
170
H. Mineo and Y. Fujimura