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arXiv:2207.01598 · Journal of Functional Analysis

Norm approximation for the Fröhlich dynamics in the mean-field regime

Nikolai Leopold

Published Apr 24, 2023• 1 citation
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Abstract

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.

DOI
10.1016/j.jfa.2023.109979

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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.
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.
This thesis proves certain results concerning an important question in non-equilibrium quantum statistical mechanics which is the derivation of effective evolution equations approximating the dynamics of a system of large number of bosons initially at equilibrium (ground state at very low temperatures). The dynamics of such systems are governed by the time-dependent linear many-body Schroedinger equation from which it is typically difficult to extract useful information due to the number of particles being large. We will study quantitatively (i.e. with explicit bounds on the error) how a suitable one particle non-linear Schroedinger equation arises in the mean field limit as number of particles N → ∞ and how the appropriate corrections to the mean field will provide better approximations of the exact dynamics. In the first part of this thesis we consider the evolution of N bosons, where N is large, with two-body interactions of the form N³ᵝv(Nᵝ⋅), 0≤β≤1. The parameter β measures the strength and the range of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [18,19] by Grillakis-Machedon-Margetis. We extend the results for 0 ≤ β < 1/3 in [19, 20] to the case of β < 1/2 and obtain an error bound of the form p(t)/Nᵅ, where α>0 and p(t) is a polynomial, which implies a specific rate of convergence as N → ∞. In the second part, utilizing estimates of the type discussed in the first part, we compare the exact evolution with the mean field approximation in the sense of marginals. We prove that the exact evolution is close to the approximate in trace norm for times of the order o(1)√N compared to log(o(1)N) as obtained in Chen-Lee-Schlein [6] for the Hartree evolution. Estimates of similar type are obtained for stronger interactions as well.
We consider the time evolution of the renormalized Nelson model, which describes N bosons linearly coupled to a quantized scalar field, in the mean-field limit of many particles N\gg 1 with coupling constant proportional to N^{-1/2} . First, we show that initial states exhibiting Bose–Einstein condensation for the particles and approximating a coherent state for the quantum field retain their structure under the many-body time evolution. Concretely, the dynamics of the reduced densities are approximated by solutions of two coupled PDEs, the Schrödinger–Klein–Gordon equations. Second, we construct a renormalized Bogoliubov evolution that describes the quantum fluctuations around the Schrödinger–Klein–Gordon equations. This evolution is used to extend the approximation of the evolved many-body state to the full norm topology. In summary, we provide a comprehensive analysis of the Nelson model that reveals the role of renormalization in the mean-field Bogoliubov theory.
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.
Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N\mathit{N}N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as N−1/2\mathit{N}^{-1/2}N−1/2 in the limit of large N\mathit{N}N.
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.