Quantum oscillators

Consider an excited condensed-phase quantum oscillator Q, witli reduced mass p and nonnal coordinate q j. The batli exerts fluctuating forces on the oscillator. These fluctuating forces induce VER. The quantum mechanical Hamiltonian is [M, M]... 

Fig. 1. Total energy (in kj/mol) versus time (in fs) for different integrators for a collinear collision of a classical particle with a harmonic quantum oscillator (for details see [2]). Dashed line Nonsymplectic scheme. Dotted Symplectic integrator of first order. Solid PICKABACK (symplectic, second order).

The splitting of the quantum propagator negatively effects the efficiency of the scheme especially if m/M is small, i.e., if the quantum oscillation are much faster than the classical motion and the number n of substeps is becoming inefficiently large. 

These quantum effects, though they do not generally affect significantly the magnitude of the resistivity, introduce new features in the low temperature transport effects [8]. So, in addition to the semiclassical ideal and residual resistivities discussed above, we must take into account the contributions due to quantum localisation and interaction effects. These localisation effects were found to confirm the 2D character of conduction in MWCNT. In the same way, experiments performed at the mesoscopic scale revealed quantum oscillations of the electrical conductance as a function of magnetic field, the so-called universal conductance fluctuations (Sec. 5.2). 

T., Tsukerblat, B., Muller, A. and Barbara, B. (2008) Quantum oscillations in a molecular magnet. Nature,... 

Parameters Associated with the Two Elementary Quantum Oscillators Forming an Hydrogen Bond... 

The total Hamiltonian of Eq.(42) contains one oscillator, j=l, with energy E/fi = cos the subindex j is dropped in all equations. The behavior of the quantum oscillator is characterized by i) the natural frequency E/fi. = cos ii) the coupling strength /LE between the oscillator and the bath iii) the memory time xc= 1/y of the dissipation of oscillator energy by the heat bath and iv) the bath s temperatute T. The equation of motion is given by Eq.(51) without subindex j. 

Skrebkov, O. V. Diffusional description of vibrational realaxation in a binary mixture of diatomic molecules-quantum oscillators, Chem.Phys., 191(1995), 87-99... 

The Fermi surfaces of these salts have been studied by measuring the quantum oscillations [183] such as SdH (Shubnikov-de Haas) and dHvA and geometrical oscillations (AMRO, angle-dependent magnetoresistance oscillation) ([4], Appendix, pp 445 48). The Fermi surface of k-(ET)2Cu(NCS)2 (Fig. 14c) calculated based on the crystal structure is in good agreement with those observed data [225]. 

A one-level system e) that can exchange its population with the bath states [/) represents the case of autoionization or photoionization. However, the above Hamiltonian describes also a qubit, which can undergo transitions between the excited and ground states e) and g), respectively, due to its off-diagonal coupling to the bath. The bath may consist of quantum oscillators (modes) or two-level systems (spins) with different eigenfrequencies. Typical examples are spontaneous emission into photon or phonon continua. In the RWA, which is alleviated in Section 4.4, the present formalism applies to a relaxing qubit, under the substitutions... 

To avoid confusion, let us explain the difference between the probability Wlh determined by eqn. (19) and the probability Wfn nj), determined by eqn. (38). Wt is the probability of transition from the state with the complete set of quantum numbers for the nuclear motion (the index i characterizes this set). Wfn rif) is the probability of tunneling with the fixed quantum numbers Ti, and nt of one quantum oscillator while by all the other (classical) oscillators the averaging has been carried out. 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

Shubnikov - de Haas (SdH) and de Haas - van Alphen (dHvA) quantum oscillations were observed in the crystals studied at different magnetic field directions and temperatures. Fig. 6 (inset) shows an example of these SdH oscillations. It should be noted that no beating node occurs in these oscillations, suggesting again a strong 2D... 

The behaviour of the quantum oscillations in (BEDO-TTF)5[CsHg(SCN)4]2 seems to be in good agreement with the predictions of tight binding band structure calculations. The additional frequencies in the SdH oscillations spectrum are most probably caused by the quantum interference effect. Thus, we propose that Fig. 5 provides an adequate description of the Fermi surface of (BEDO-TTF)5[CsHg(SCN)4]2 and that the... 

Proust C., Audouard A., Brossard L., Pesotskii S.I., Lyubovskii R.B. and Lyubovskaya R.N. (2002) On competing types of quantum oscillations in the 2D organic conductor (ET)8[Hg4Cli2(C6H5Cl)2] Phys. 

Consider a single H-bond, for example, that is involved in a carboxylic acid-ether complex depicted in Fig. 1. It may be modelized by two quantum oscillators that are, respectively, the vx-h high frequency stretching mode and the low frequency mode corresponding to the H-bond bridge. 

Yu.A. Pashkin, T. Yamamoto, O. Ostafiev, Y. Nakamura, D.V. Averin, J.S. Tsai, Quantum oscillations in two coupled charge qubits, Nature 421 (2003) 823-826. 

Consider an excited condensed-phase quantum oscillator Q, with reduced mass /x and normal coordinate qQ. The bath exerts friction on the oscillator, which causes it to lose vibrational energy to its surroundings (Fig. 2). 

Magnetic quantum oscillations in bismuth were first observed in the field dependence of the electrical resistivity by Shubnikov and de Haas [246] shortly before the dHvA effect was discovered. Usually, however, the SdH effect is weak and hard to observe except in semimetals, like bismuth, and semiconductors. 

---