second term in Eq. (21a) representing the magnitude and direction of the angular
momentum created on each ring is symmetric with respect to a rotation of 180°
around the Z-axis, whereas the first term in Eq. (21a) gives asymmetric contribution
of angular momentum by a rotation of 180° around the Z-axis.
By combining with the laser optimal control theory [24–26] we can selectively
localize the angular momentum or ring current on a particular aromatic ring in a
covalently linked molecular system, i.e., ~ l m 6 ¼ ~ 0; ~ l 1 ¼ Á Á Á ¼ ~ l mÀ1 ¼ ~ l mþ1 ¼ Á Á Á
¼ ~ l n ¼ 0.
Finally we remark that Eq. (21a) can also be applied to a ring current by
replacing l
m
ab to an expectation value of ring current J
m
ab .
4 Summary and Conclusion
In conclusion we have developed a density matrix theory for a coherent π electron
angular momentum and current induced by the ultrafast linearly polarized UV
pulses, and applied the theory to (P)-2,2′-biphenol. The expressions of coherent
ring current and angular momentum are derived analytically within LCAO MO
approximation. We showed time-dependent coherent ring currents and angular
momentum for three types of in-phase electronic coherent excited states ða þ b 1 Þ,
ða þ b 2 Þ with B symmetry and ðb 1 þ b 2 Þ with A symmetry in the point group C 2 .
As a characteristic feature for electronic coherent states ða þ b 1 Þ and ða þ b 2 Þ, a
non-zero bridge bond current appears and oscillates between two phenol rings,
while for electronic coherence ðb 1 þ b 2 Þ the bridge bond current disappears permanently. The direction of resultant angular momentum for electronic coherences
ða þ b 1 Þ and ða þ b 2 Þ is parallel to Z-axis, while for electronic coherence ðb 1 þ b 2 Þ
is parallel to X-axis. The initial directions of ring currents and angular momenta are
determined from the initial phases of electronic coherences.
As an application of our study we demonstrated an ultrafast two-dimensional
switching of π-electron angular momentum in (P)-2,2′-biphenol using a sequence of
overlapped pump-dump pulses with different phase and polarization direction. The
key to multi-dimensional switching is to coherently excite π-electronic states of a
nonplanar aromatic ring molecules with covalently bonded aromatic rings. Finally it
would be fascinating to explore a method for directly measuring the coherent
electronic dynamics of a nonplanar chiral aromatic molecule.
Acknowledgments This work was supported by the National Science Council of Taiwan (Grant
No. 102-2112-M-001-003-MY3). H.M. would like to thank Professor J.-L. Kuo for his critical
comments and support.
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H. Mineo and Y. Fujimura
momentum created on each ring is symmetric with respect to a rotation of 180°
around the Z-axis, whereas the first term in Eq. (21a) gives asymmetric contribution
of angular momentum by a rotation of 180° around the Z-axis.
By combining with the laser optimal control theory [24–26] we can selectively
localize the angular momentum or ring current on a particular aromatic ring in a
covalently linked molecular system, i.e., ~ l m 6 ¼ ~ 0; ~ l 1 ¼ Á Á Á ¼ ~ l mÀ1 ¼ ~ l mþ1 ¼ Á Á Á
¼ ~ l n ¼ 0.
Finally we remark that Eq. (21a) can also be applied to a ring current by
replacing l
m
ab to an expectation value of ring current J
m
ab .
4 Summary and Conclusion
In conclusion we have developed a density matrix theory for a coherent π electron
angular momentum and current induced by the ultrafast linearly polarized UV
pulses, and applied the theory to (P)-2,2′-biphenol. The expressions of coherent
ring current and angular momentum are derived analytically within LCAO MO
approximation. We showed time-dependent coherent ring currents and angular
momentum for three types of in-phase electronic coherent excited states ða þ b 1 Þ,
ða þ b 2 Þ with B symmetry and ðb 1 þ b 2 Þ with A symmetry in the point group C 2 .
As a characteristic feature for electronic coherent states ða þ b 1 Þ and ða þ b 2 Þ, a
non-zero bridge bond current appears and oscillates between two phenol rings,
while for electronic coherence ðb 1 þ b 2 Þ the bridge bond current disappears permanently. The direction of resultant angular momentum for electronic coherences
ða þ b 1 Þ and ða þ b 2 Þ is parallel to Z-axis, while for electronic coherence ðb 1 þ b 2 Þ
is parallel to X-axis. The initial directions of ring currents and angular momenta are
determined from the initial phases of electronic coherences.
As an application of our study we demonstrated an ultrafast two-dimensional
switching of π-electron angular momentum in (P)-2,2′-biphenol using a sequence of
overlapped pump-dump pulses with different phase and polarization direction. The
key to multi-dimensional switching is to coherently excite π-electronic states of a
nonplanar aromatic ring molecules with covalently bonded aromatic rings. Finally it
would be fascinating to explore a method for directly measuring the coherent
electronic dynamics of a nonplanar chiral aromatic molecule.
Acknowledgments This work was supported by the National Science Council of Taiwan (Grant
No. 102-2112-M-001-003-MY3). H.M. would like to thank Professor J.-L. Kuo for his critical
comments and support.
174
H. Mineo and Y. Fujimura