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arXiv:2505.08241 · Journal of Mathematical Physics

Weak coupling limit for quantum systems with unbounded weakly commuting system operators

Ilya A. Lopatin, Alexander Nikolaevich Pechen

Published Apr 1, 2025• 1 citation
arXivHTMLPDF

Abstract

This work is devoted to a rigorous analysis of the weak coupling limit (WCL) for the reduced dynamics of an open infinite-dimensional quantum system interacting with electromagnetic field or a reservoir formed by Fermi or Bose particles in the dipole approximation. The free system Hamiltonian and the system part of the Hamiltonian describing interaction with the reservoir are considered as unbounded operators with continuous spectrum which are commuting in a weak sense. We derive in the weak coupling limit the reservoir statistics, which is determined by whose terms in the multi-point correlation functions of the reservoir which are non-zero in the WCL. Then we prove that the resulting reduced system dynamics converges to unitary dynamics (such behavior sometimes called as Quantum Cheshire Cat effect) with a modified Hamiltonian which can be interpreted as a Lamb shift to the original Hamiltonian. We obtain exact form of the modified Hamiltonian and estimate the rate of convergence to the limiting dynamics. For Fermi reservoir, we prove the convergence of the full Dyson series. For Bose case the convergence is understood term by term.

DOI
10.1063/5.0239525

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This work is devoted to a rigorous analysis of the weak coupling limit (WCL) for the reduced dynamics of an open infinite-dimensional quantum system interacting with electromagnetic field or a reservoir formed by Fermi or Bose particles in the dipole approximation. The free system Hamiltonian and the system part of the Hamiltonian describing interaction with the reservoir are considered as unbounded operators with continuous spectrum which are commuting in a weak sense. We derive in the weak coupling limit the reservoir statistics, which is determined by whose terms in the multi-point correlation functions of the reservoir which are non-zero in the WCL. Then we prove that the resulting reduced system dynamics converges to unitary dynamics (such behavior sometimes called as Quantum Cheshire Cat effect) with a modified Hamiltonian which can be interpreted as a Lamb shift to the original Hamiltonian. We obtain exact form of the modified Hamiltonian and estimate the rate of convergence to the limiting dynamics. For Fermi reservoir, we prove the convergence of the full Dyson series. For Bose case the convergence is understood term by term.
We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a “small subsystem” and an infinite dimensional one called a “reservoir.” The operator, which we call a “Friedrichs Hamiltonian,” has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before (Accardi et al., 1990) in more complicated models under the name of “stochastic limit.”
We analyse the discrete-time dynamics of a model of non-interacting fermions coupled to an infinite reservoir formed by a bosonic quantum walk on Z{\mathbb Z}Z. This dynamics consists of consecutive applications of free evolutions of the fermions and bosons followed by a local coupling between them. The unitary operator implementing this coupling accounts for energy exchanges between the system and reservoir while it preserves the number of fermions. The free fermion evolution is given by a second-quantized single-particle unitary operator satisfying some genericity assumptions. The free boson evolution is given by the second-quantized shift operator on Z{\mathbb Z}Z. We derive explicitly the Heisenberg dynamics of fermionic observables and obtain a systematic expansion in the large-coupling regime, which we control by using spectral methods. We also prove that the reduced state of the fermions converges in the large-time limit to a mixture of infinite-temperature Gibbs states in each particle sector.
The rigorous derivation of the Uehling-Uhlenbeck equation from morefundamental quantum many-particle systems is a challenging open problem inmathematics. In this paper, we exam the weak coupling limit of quantum NNN-particle dynamics. We assume the integral of the microscopic interaction iszero and we assume W4,1W^{4,1}W4,1 per-particle regularity on the coresspondingBBGKY sequence so that we can rigorously commute limits and integrals. Weprove that, if the BBGKY sequence does converge in some weak sense, thenthis weak-coupling limit must satisfy the infinite quantum Maxwell-Boltzmannhierarchy instead of the expected infinite Uehling-Uhlenbeck hierarchy,regardless of the statistics the particles obey. Our result indicates that,in order to derive the Uehling-Uhlenbeck equation, one must work withper-particle regularity bound below W4,1W^{4,1}W4,1.
We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a “small subsystem” and an infinite dimensional one – a “reservoir”. The operator, which we call a “Friedrichs Hamiltonian”, has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before [AFL] in more complicated models and they are usually referred to as ”stochastic Limit”.
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One of the main methods for describing the dynamics of open quantum systems is the method of quantum master equations. These equations describe the dynamics of the reduced density operator of a system interacting with a reservoir. In this case, averaging is performed over the degrees of freedom of the reservoir, which does not allow one to describe the dynamics of reservoir observables. In this paper we show that applying the Zwanzig projection operator method, which is used in deriving quantum master equations, one can also derive dynamic equations for reservoir observables. As an example, we derive dynamic equations for the average number of quanta (photons, phonons) of a bosonic reservoir in the approximation of its weak coupling to the system in the case of the dipole interaction Hamiltonian.