[Paper Review] From repeated to continuous quantum interactions

Stéphane Attal, Yan Pautrat|ArXiv.org|Nov 4, 2003

Quantum Mechanics and Applications12 references4 citations

TL;DR

This paper establishes a rigorous Hamiltonian derivation of quantum Langevin equations as the continuous-time limit of discrete, repeated quantum interactions between a small system and a chain of ancillary systems. By introducing three distinct time scales in the interaction Hamiltonian—order 1, √h, and h—it shows that the limit evolution is governed by a quantum stochastic differential equation, justifying the standard quantum Langevin equations used in quantum optics and continual measurement without ad hoc assumptions.

ABSTRACT

We consider the general physical situation of a quantum system $\H_0$ interacting with a chain of exterior systems $\bigotimes_\N \H$, one after the other, during a small interval of time $h$ and following some Hamiltonian $H$ on $\H_0 \otimes \H$. We discuss the passage to the limit to continuous interactions ($h o 0$) in a setup which allows to compute the limit of this Hamiltonian evolution in a single state space: a continuous field of exterior systems $\otimes_{\R} \H$. Surprisingly, the passage to the limit shows the necessity for 3 different time scales in $H$. The limit evolution equation is shown to spontaneously produce quantum noises terms: we obtain a quantum Langevin equation as limit of the Hamiltonian evolution. For the very first time, these quantum Langevin equations are obtained as the effective limit from repeated to continuous interactions and not only as a model. These results justify the usual quantum Langevin equations considered in continual quantum measurement or in quantum optics. We show that the three time scales correspond to the normal regime, the weak coupling limit and the low density limit. Our approach allows to consider these two physical limits altogether for the first time. Their combination produces an effective Hamiltonian on the small system, which had never been described before. We apply these results to give an Hamiltonian description of the von Neumann measurement. We also consider the approximation of continuous time quantum master equations by discrete time ones. In particular we show how any Lindblad generator is obtained as the limit of completely positive maps.

Motivation & Objective

• To rigorously derive quantum Langevin equations as the continuous limit of discrete, repeated quantum interactions.
• To unify the weak coupling and low-density limits within a single framework by identifying three distinct time scales in the interaction Hamiltonian.
• To provide a Hamiltonian foundation for quantum Langevin equations, rather than treating them as phenomenological models.
• To show how Lindblad generators emerge as the limit of discrete completely positive maps, linking open quantum systems to fundamental dynamics.
• To apply the framework to the von Neumann measurement model, offering a Hamiltonian description of quantum measurement processes.

Proposed method

• Model the interaction of a small system with a discrete chain of ancillary systems over time intervals of length h, using a fixed Hamiltonian H on each pair.
• Embed the discrete chain into a continuous Fock field over ℝ⁺, allowing the limit h → 0 to be taken in a single Hilbert space.
• Introduce three distinct time scales in the Hamiltonian: one of order 1, one of order √h, and one of order h, to ensure a nontrivial limit.
• Use unitary evolution on the extended system to define the discrete dynamics, then prove convergence of the Heisenberg evolution to a solution of a quantum stochastic differential equation.
• Apply quantum stochastic calculus (Hudson-Parthasarathy) to describe the limit evolution as a quantum Langevin equation.
• Establish conditions under which the discrete evolution maps converge to a Lindblad generator, showing that any Lindbladian arises as such a limit.

Experimental results

Research questions

• RQ1 How can quantum Langevin equations be derived from fundamental Hamiltonian dynamics of repeated interactions?
• RQ2 What renormalization conditions on the interaction Hamiltonian are necessary to obtain a nontrivial continuous limit?
• RQ3 How do the weak coupling and low-density limits emerge simultaneously within a unified framework?
• RQ4 Can the standard Lindblad master equation be obtained as the limit of discrete, unitary, completely positive evolutions?
• RQ5 What is the Hamiltonian description of the von Neumann measurement process in this repeated interaction framework?

Key findings

• The passage to the continuous limit requires three distinct time scales in the interaction Hamiltonian: one of order 1, one of order √h, and one of order h.
• The limit evolution is described by a quantum stochastic differential equation (quantum Langevin equation), which emerges naturally from the Hamiltonian dynamics without ad hoc assumptions.
• The quantum Langevin equation obtained is the standard one used in quantum optics and continual measurement, such as the Wigner-Weisskopf model for spontaneous emission.
• Any Lindblad generator arises as the limit of discrete, completely positive maps when the interaction matrices satisfy appropriate scaling conditions.
• The convergence of the discrete evolution to the continuous one holds in the strong operator topology and implies weak convergence of Heisenberg evolutions.
• The framework provides a Hamiltonian derivation of the von Neumann measurement process, showing how continuous observation emerges from repeated interactions.