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arXiv:2005.02098 · Pure and Applied Analysis

Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron

Nikolai Leopold, David Mitrouskas, Simone Rademacher, Benjamin Schlein, Robert Seiringer

Published Dec 31, 2021• 15 citations
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Abstract

We consider the Fröhlich Hamiltonian with large coupling constant $α$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order $α^2$. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order $α^2$.

DOI
10.2140/paa.2021.3.653

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We consider the Fröhlich Hamiltonian with large coupling constant ααα. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order α2α^2α2. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order α2α^2α2.
We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state ϕ0 ⊗ξα, where ϕ0 is the electron ground state of the Pekar energy functional and ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2. In the present note we prove that a similar approximation holds for t = O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α-2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2, while the phonon fluctuations around the coherent state ξα can be described by a time-dependent Bogoliubov transformation.
We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau-Pekar equations. These describe a Bose-Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.
Abstract We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state φ0⊗ξα\varphi _0 \otimes \xi _\alpha φ0​⊗ξα​ φ0⊗ξα , where φ0\varphi _0φ0​ φ0 is the electron ground state of the Pekar energy functional and ξα\xi _\alpha ξα​ ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2\alpha ^2α2 α2 . In the present note we prove that a similar approximation holds for t=O(α2)t=O(\alpha ^2)t=O(α2) t=O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α−2\alpha ^{-2}α−2 α-2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2\alpha ^2α2 α2 , while the phonon fluctuations around the coherent state ξα\xi _\alpha ξα​ ξα can be described by a time-dependent Bogoliubov transformation.
We study the time evolution of the Fröhlich Hamiltonian in a mean-field limit in which many particles weakly couple to the quantized phonon field. Assuming that the particles are initially in a Bose-Einstein condensate and that the excitations of the phonon field are initially in a coherent state we provide an effective dynamics which approximates the time evolved many-body state in norm, provided that the number of particles is large. The approximation is given by a product state which evolves according to the Landau–Pekar equations and which is corrected by a Bogoliubov dynamics. In addition, we extend the results from Leopold et al. (2021) [27] about the approximation of the time evolved many-body state in trace-norm topology to a larger class of many-body initial states with an improved rate of convergence.
Starting from recent advances in the first-principles modeling of polarons, variational polaron equations in the strong-coupling adiabatic approximation are formulated in Bloch space. In this framework, polaron formation energy as well as individual electron, phonon, and electron-phonon contributions are obtained. We suggest an efficient gradient-based optimization algorithm and apply these equations to the generalized Fr\"ohlich model with anisotropic nondegenerate electronic bands, both in two- and three-dimensional cases. The effect of the divergence of Fr\"ohlich electron-phonon matrix elements at $\mathrm{\ensuremath{\Gamma}}$ point is treated analytically, improving the convergence with respect to the sampling in reciprocal space. The whole methodology is validated by obtaining the known asymptotic solution of the standard Fr\"ohlich model in isotropic scenario and also by comparing our results with the Gaussian ansatz approach, showing the difference between the numerically exact and Gaussian trial wavefunctions. Additionally, decomposition of the energy into individual terms allows one to recover the Pekar's 1:2:3:4 theorem, which is shown to be valid even in the anisotropic case. We expect that the improvements in the formalism and numerical implementation will be applicable beyond the large polaron hypothesis inherent to Fr\"ohlich model.