Derivation of an effective evolution equation for a strongly coupled polaron

Abstract

Abstract. Fröhlich's polaron Hamiltonian describes an electron coupled to the quantized phonon field of an ionic crystal. We show that in the strong coupling limit the dynamics of the polaron is approximated by an effective non-linear partial differential equation due to Landau and Pekar, in which the phonon field is treated as a classical field.

Mean-Field Limits of Particles in Interaction with Quantized Radiation Fields

“…These involve fewer degrees of freedom, are less exact but easier to investigate. Effective evolution equations for particles that interact with quantized radiation fields have rigorously been derived for example in [9,6,1,27,7,8,14,10]. The general setting in these works is given by the tensor product of two Hilbert spaces…”

On the dynamics of the mean-field polaron in the high-frequency limit

“…Here x ∈ R 3 , t ∈ R, and ∆ −1 |u| 2 = −(4π| · |) −1 * |u| 2 . The parameter ε > 0 plays the role of the electronphonon coupling strength or the inverse of the phonon frequency in a more fundamental, quantum field theoretic model of the polaron [FG,BNAS00].…”

“…While it appears natural, mathematically, to study the limit ε → 0 of (1)-(2), from a physical point of view the limit ε → ∞ is even more relevant, because the system (1)-(2) is believed, and partly proven, to describe the strong coupling limit, ε → ∞, of the Fröhlich model of large polarons [FS14,FG]. We remark that the Nelson model, which is similar to the Fröhlich model, in a classical limit leads to the Schrödinger-Klein-Gordon system [AF14].…”

Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit

“…In this respect our analysis is more general than the one contained in the works mentioned above [GNV06,FS14,FG17,Gri16], although we do not address any dynamical question: in all those papers indeed the initial state of the field must be of very special type, i.e., a (squeezed) coherent state, which is already semiclassical from a certain point of view. On the opposite, we make very weak restrictions on the possible field configurations, and show explicitly how such a freedom influences the effective Schrödinger operator for the particles.…”

“…More generally, a regime in which there emerges a behavior similar to the quasi-classical limit is the adiabatic decoupling generated by a separation between fast and slow degrees of freedom [PST03,Teu03,PST07], and also the non-relativistic limit of electrons coupled to a quantum field [Ara90,Hir93,Hir98]. In spite of a completely different physical meaning, there are also strong mathematical analogies with the strong coupling limit for the Fröhlich polaron [FS14,FG17,Gri16] (see also below). Finally, we want to mention the works [BCFS07, BCF`13] about the effective mass and dynamics of a quantum particle interacting with the electromagnetic field in QED.…”