We suggest a model of q-deformed rotating oscillator which possesses the symmetry of quantum group and point out that this model can be used to describe the vibrational and rotational structures of many microscopic systems. We construct the representation of q-oscilla or algebra and give energy values of q-oscillaor in this representation. It is shown that the q-oscillator spectra can be used to describe infrared spectra, Raman spectra and vibrational structure of electronic transitions of diatomic molecules.
We develop a model with
Hq(4) and
SUq(2) quantum group symmetries to describe the vibrational and rotational spectra of diatomic molecules. The representations of
Hq(4) and
SUq(2) and solutions to the quantum systems of q oscillators and q rotators are provided. We emphasize that the energy spectra of the q oscillator and q rotator exhibit remarkably the properties of the infrared, vibrational, and rotational Raman spectra as well as vibrational and rotational structures of electronic transitions of the diatomic molecule. When the parameters are properly selected, the spectra of this model coincide with the observed spectra of diatomic molecules. It is suggested that
Hq(4) \ensuremath{\bigotimes}
SUq(2) is an essential symmetry of vibrations and rotations in diatomic molecules.
We have numerically examined optimization of the experimentally achievable shapes of mid-infrared femtosecond laser pulses to construct elementary quantum gates using molecular rovibrational states of a single molecule. Instead of optimizing the electric field by optimal control theory, the method generally used in theoretical studies of coherent control, the transmittance and the phase shift of conventional pulse shapers were optimized in the frequency domain by a genetic algorithm. The target molecular system we examined was the rotation-vibration states of CO in the $X({^{1}\ensuremath{\Sigma}}^{+})$ ground electronic state. Although the existence of the rotational degrees of freedom makes the quantum gate operation complicated, high fidelity of over 0.95 was achieved for one-rovibrational-qubit systems, which indicates that a molecular quantum computer may be feasible for at least one-qubit calculations on a diatomic molecule. On the other hand, two-qubit calculations are more difficult to achieve in a single CO molecule using rotational and vibrational degrees of freedom as different qubits. The importance of the control of the rotational in addition to the vibrational wave packet in order to realize a molecular quantum computer is discussed.