Generalized method of Feynman-Pines diagram technique in the theory of energy spectrum of two-level quasiparticle renormalized due to multi-phonon processes at cryogenic temperature

Abstract

Theory of the spectrum of localized two-level quasi-particle renormalized due to interaction with polarization phonons at cryogenic temperature is developed using the generalized method of Feynman-Pines diagram technique. Using the procedure of partial summing of infinite ranges of the main diagrams, mass operator is obtained as a compact branched chain fraction, which effectively takes into account multi-phonon processes. It is shown that multi-phonon processes and interlevel interaction of quasiparticle and phonons cardinally change the renormalized spectrum of the system depending on the difference of energies of two states, which either resonates with phonon energy or does not. The spectrum of non-resonant system contains renormalized energies of the main states and two similar infinite series of groups of phonon satellite levels. The spectrum of a resonant system contains a renormalized ground state and infinite series of satellite groups.

Method of successive separation and summing of multiplicative diagrams of mass operator for the multi-level quasiparticle interacting with polarization phonons

“…To obtain a renormalized spectrum of the system consisting of a multi-level localized quasiparticle interacting with polarization phonons, we use the Fröhlich Hamiltonian, like in [39,41,42]…”

“…The energy spectrum of the system renormalized due to the interaction at cryogenic temperature (formally T = 0 K) is obtained within the method of Feynman-Pines diagram technique [42][43][44] for the Fourier image of quasiparticle casual Green's function G µµ (ω). Moreover, we use the approach proposed in papers [41,42], modified for the case of a quasiparticle [with an arbitrary number (τ) of levels] interacting with phonons. Taking into account the Hamiltonian (1), the Green's functions G µµ (ω) satisfy the system of τ 2 Dyson equations (2) where M µµ 1 is the complete matrix MO and E µ=2,...,τ = E 1 + ∆E µ=2,...,τ .…”

“…In our previous paper [41], we studied only the first (m 1,m µµ ) approximation for a complete MO and revealed the main property of a renormalized spectrum of two-level localized quasiparticle interacting with phonons, in a wide range of energies. In this paper, using the obtained MO in third approximation…”

“…In our paper [41], we studied the renormalized spectrum of a two-level quasiparticle interacting with dispersionless phonons within zero (m 0 ) and first (m I ) approximation for the MO. In this paper, using the analytic results presented in the previous section, we shall take into account the terms of MO in the second (m II ) and the third (m III ) approximations, which essentially affect some properties of the energy spectrum.…”

“…For the simplest model of an extractor with the "torn phonon ladder", created by two energy levels of a quasiparticle and by their phonon satellites, in the papers [41,42] there was presented a theory of an energy spectrum of two-level localized quasiparticle interacting with polarization phonons at T = 0 K. Using the modified method of Feynman-Pines diagram technique, it was shown that in the first approximation of renormalized mass operator (MO) (where all multiplicative diagrams are taken into account completely), the satellite mini-bands are really observed in the spectrum.…”