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Physics of Particles and Nuclei

Kinetic equations in polaron theory for the case of spatial inhomogeneity

Nikolai Nikolaevich Bogoliubov, A. A. Kasarian

Published Nov 25, 2010• 0 citations
Publisher

Abstract

Construction of a kinetic equation for a dynamical system interacting with a boson field in the case of spatial inhomogeneity is based on a method expounded in [1–3]. In the present article, it is shown that approaches considered in [1, 2] can be generalized for the case of spatial inhomogeneity. An arbitrary operator function that depends on the momentum and spatial variable is applied for derivation of a kinetic equation. We consider a method of studying an electron-phonon system by means of exclusion of boson operators from corresponding operator equations. In particular, interaction of an electron with a boson field is described by a kinetic equation for a polaron in the case of spatial inhomogeneity. In the relevant limit, a Boltzmann equation for the polaron is obtained.

DOI
10.1134/s1063779610070051

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Construction of a kinetic equation for a dynamical system interacting with a boson field in the case of spatial inhomogeneity is based on a method expounded in [1–3]. In the present article, it is shown that approaches considered in [1, 2] can be generalized for the case of spatial inhomogeneity. An arbitrary operator function that depends on the momentum and spatial variable is applied for derivation of a kinetic equation. We consider a method of studying an electron-phonon system by means of exclusion of boson operators from corresponding operator equations. In particular, interaction of an electron with a boson field is described by a kinetic equation for a polaron in the case of spatial inhomogeneity. In the relevant limit, a Boltzmann equation for the polaron is obtained.
This chapter generalizes works by N.N. Bogolubov and N.N. Bogolubov, Jr. [1,6], and \nis written with the aim of formulating methods of investigation of the electron-phonon \nsystem and the elimination of phonon operators from the corresponding kinetic equations. In particular, a kinetic equation is derived for the interaction of an electron \nwith a phonon field. Moreover, under the proper approximation, the exact Boltzmann equation for the polaron system follows from this kinetic equation. Methods of \ncalculation of the response functions (impedance and admittance), based on the "approximating" Hamiltonian with linear interaction, are also proposed. The equilibrium \ndensity probability function of the particle is also calculated.
Kinetics of a quantum system immersed in an equilibrium medium and completely described by its statistical operator is investigated. The case of a separable interaction is considered in details. A concept of the effective Hamilton operator of the system is discussed. The kinetic equation for a point particle in a spatially uniform external field placed in a medium of point particles is obtained in terms of the Wigner distribution function of a particle. The kinetic equation is applicable for strong spatially inhomogeneous (localized) states of the particle and strong external field. The elaborated theory is used also for a system of two-level emitters and photons interacting with an equilibrium phonon medium. The kinetic equation for the statistical operator of the system of the emitters and photons that describes the effect of emitter motion on the super radiance phenomenon is obtained.
The inversion method to derive the quantum kinetic equation is extended to the case of an inhomogeneous system. Within the closed-time-path formalism, the source is introduced to probe the (Fourier transform of the) Wigner distribution function and the equation of motion for the Wigner function is derived perturbatively by the inversion formulae. The equation of motion has the form of the generalized Boltzmann equation, which is non-local in space and time. For the obtained kinetic equation, the local conservation laws of number, energy and momentum are confirmed.
The polaron model is a basic model of quantum field theory describing a single particle interacting with a bosonic field. It arises in many physical contexts. We are mostly concerned with models applicable in the context of an impurity atom in a Bose-Einstein condensate as well as the problem of electrons moving in polar crystals. The model has a simple structure in which the interaction of the particle with the field is given by a term linear in the field’s creation and annihilation operators. In this work, we investigate the properties of this model by providing rigorous estimates on various energies relevant to the problem. The estimates are obtained, for the most part, by suitable operator techniques which constitute the principal mathematical substance of the thesis. The first application of these techniques is to derive the polaron model rigorously from first principles, i.e., from a full microscopic quantum-mechanical many-body problem involving an impurity in an otherwise homogeneous system. We accomplish this for the N + 1 Bose gas in the mean-field regime by showing that a suitable polaron-type Hamiltonian arises at weak interactions as a low-energy effective theory for this problem. In the second part, we investigate rigorously the ground state of the model at fixed momentum and for large values of the coupling constant. Qualitatively, the system is expected to display a transition from the quasi-particle behavior at small momenta, where the dispersion relation is parabolic and the particle moves through the medium dragging along a cloud of phonons, to the radiative behavior at larger momenta where the polaron decelerates and emits free phonons. At the same time, in the strong coupling regime, the bosonic field is expected to behave purely classically. Accordingly, the effective mass of the polaron at strong coupling is conjectured to be asymptotically equal to the one obtained from the semiclassical counterpart of the problem, first studied by Landau and Pekar in the 1940s. For polaron models with regularized form factors and phonon dispersion relations of superfluid type, i.e., bounded below by a linear function of the wavenumbers for all phonon momenta as in the interacting Bose gas, we prove that for a large window of momenta below the radiation threshold, the energy-momentum relation at strong coupling is indeed essentially a parabola with semi-latus rectum equal to the Landau–Pekar effective mass, as expected. For the Fröhlich polaron describing electrons in polar crystals where the dispersion relation is of the optical type and the form factor is formally UV–singular due to the nature of the point charge-dipole interaction, we are able to give the corresponding upper bound. In contrast to the regular case, this requires the inclusion of the quantum fluctuations of the phonon field, which makes the problem considerably more difficult. The results are supplemented by studies on the absolute ground-state energy at strong coupling, a proof of the divergence of the effective mass with the coupling constant for a wide class of polaron models, as well as the discussion of the apparent UV singularity of the Fröhlich model and the application of the techniques used for its removal for the energy estimates.
A method is presented for deriving a set of kinetic equations for a system of electrons and phonons in a simplified model of a metal. By employing the second quantization representation for the creation and annihilation operators of the electrons and the phonons, an hierarchy of equations for the distribution functions and correlation functions is introduced. This hierarchy is studied, using an approach originally developed by Bogoliubov, where both truncation of the hierarchy and irreversibility are achieved under general assumptions. A set of kinetic equations is obtained for an homogeneous system, where the electrons dynamically shield both each other and the phonons.
The inhomogeneous master equation obtained in a previous paper is employed to derive a hierarchy of moment equations (quantum mechanical) analogous to the BBGKY hierarchy; the method used is the generating function according to Laplace. In this master equation, and in all these moment equations as well, the effect of the interaction between the particles of the system is represented by relaxation terms; further, there are streaming terms due to the external field. The interactions considered are of a binary nature, fermion–boson or boson–boson. The resulting equations have a simple form (especially for fermion–boson interaction) since the terms representing the interaction can be cast in a form in which only the Boltzmann operator (appropriate to the one-body description) and the second order Fokker–Planck moment appear. The Boltzmann equations obtained by the new method are the same as before (K. M. van Vliet. J/Math. Phys.).