Phonons creation operator

Here ak a ) is the annihilation (creation) operator of an exciton with the momentum k and energy Ek, operator an(a ) annihilates (creates) an exciton at the n-th site, 6,(6lt,) is the annihilation (creation) operator of a phonon with the momentum q and energy u) q), x q) is the exciton-phonon coupling function, N is the total number of crystal molecules. The exciton energy is Ek = fo + tfcj where eo is the change of the energy of a crystal molecule with excitation, and tk is the Fourier transform of the energy transfer matrix elements. 

Or is the frequency of the harmonic oscilator and b) are boson (phonon) creation (annihilation) operators. In order to use the perturbation theory we have to split the Hamiltonian (16) onto the unperturbed part Hq and the perturbation H ... 

Here d ,dl and airaj are annihilation and creation operators for the QD electrons and phonons, respectively. As in case (1), Mq is a semiconductor electron-phonon constant and a>fD is a phonon frequency. A-D is the energy of noninteracting electrons and 3 is a Coulomb integral. 

Here Efn(0) refers to the /th excited state of a free molecule in the crystal a n + (aQ is the Bose operator of creation (annihilation) of an intramolecular vibrational excitation in the nth molecule M2(k) refers to the energy of an optical phonon with the wave vector k connected with proton oscillations in the O H O bridge (bk) is the Bose operator of phonon creation (annihilation) and is the coupling energy between the molecular excitation and phonons. 

The interaction in non-metals (e.g. ionic crystals or covalent semiconductors) will be now expressed in terms of phonon creation ay and destruction a j operators related to the pnonon amplitudes in (2.2) by... 

The vibrational factors f0, gi and g j of (4.5) can be expressed in terms of creation operators at, of SSANMV operating on the ground state function of noninteracting phonons namely... 

Considering Equation 6.38 again, we need to transform the Hamiltonian expression. Thus, if cos(k) and ss(k) are the frequency and the polarization vector for the classic modes with polarization s and wave vector k, respectively, we can define the phonon creation (aks+) and annihilation ( /,s ) operators as... 

Expanding the quantity q in (3.90) with respect to deviations from equilibrium up to quadratic terms and introducing normal coordinates the Hamiltonian Hl can be written as a sum of Hamiltonians which correspond to harmonic oscillators in their normal coordinates. Then we use the phonon creation and annihilation operators, i.e. the operators 6 r and 5qr (q is the phonon wavevector and r indicates the corresponding frequency branch) and obtain the Hamiltonian Hl in the form... 

The first two terms are diagonalized by introducing the phonon creation and annihilation operators (see (Cohen-Tannoudji et al. 1977)) ... 

A real nanometric material is composed not only of electrons but also of a crystal lattice. In this case, after a dressed photon is generated on an illuminated nanometric particle, its energy can be exchanged with the crystal lattice, as shown by the Feynman diagram of Fig. 1.3a. By this exchange, the crystal lattice can excite the vibration mode coherently, creating a coherent phonon state. As a result, the dressed photon and the coherent phonon can form a coupled state, as is schematically explained by Fig. 1.3b. The creation operator a] of this novel form of elementary excitation is expressed as... 

Introducing the electron and phonon annihilation and creation operators, Oa, al bj, b], we can rewrite the Hamiltonian in the notation of second quantization after a canonical transformation... 

We shall always be interested in longitudinal vibrations of quasi-one-dimensional systems, so the vector notation can mostly be omitted in subsequent discussions if one refers to the component of a vector parallel to the chain axis. The phonon amplitudes m, can be expressed in terms of phonon creation and annihilation operators as ... 

In the phonon picture, the relations (2.122,123) and (2.120) are interpreted in the following way the creation operator creates an additional... 

We now quantize the vibrational modes r and write the normal mode coordinates in terms of the phonon creation/annihilation operators... 

In terms of the creation-annihilation electron and phonon operators the Hamiltonian can be cast as follows ... 

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