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Bosonization of Noise Effects in Nonlocal Quantum Dynamics

Michele Fantechi, Marco Merkli

TL;DR

The paper shows that quantum systems S nonlocally coupled to an environment R with 1/√M scaling exhibit universal bosonization of the environmental influence in the thermodynamic limit M→∞. By a Dyson-series analysis and a matching of two-point correlations, the authors construct a Gaussian fluctuation reservoir F of harmonic oscillators whose vacuum dynamics yields the exact reduced dynamics of S, thereby replacing R with F in H_SF=H_S+H_F+∑_q(G_q⊗X_q+G_q^†⊗X_q^†). Central to the result is the quantum central limit theorem, which enforces Wick factorization of multi-time correlations and enables an explicit, physically transparent mapping to a finite set of oscillators. The work establishes a universal framework for treating nonlocal noise and fluctuations, with implications for analyzing decoherence, entanglement, and transport in large quantum baths, and clarifies when ground-state versus thermal representations of F are appropriate.∎

Abstract

Quantum systems that interact non-locally with an environment are paradigms for exploring collective phenomena. They naturally emerge in various physical contexts involving long-range, many-body interactions. We consider a general class of such open systems characterized by a coupling to the environment which is inversely proportional to the square root of the environment size. We show that the induced system dynamics has a universal bosonic nature: the same evolution arises from coupling the system to a collection of noninteracting bosonic modes, independently of the microscopic structure of the original environment. This emergent "bosonization" of the environment's influence results from the scaling of the coupling in the thermodynamic limit and is a manifestation of the quantum central limit theorem. While the effect has been observed in specific models before, we show that it is, in fact, a universal feature.

Bosonization of Noise Effects in Nonlocal Quantum Dynamics

TL;DR

The paper shows that quantum systems S nonlocally coupled to an environment R with 1/√M scaling exhibit universal bosonization of the environmental influence in the thermodynamic limit M→∞. By a Dyson-series analysis and a matching of two-point correlations, the authors construct a Gaussian fluctuation reservoir F of harmonic oscillators whose vacuum dynamics yields the exact reduced dynamics of S, thereby replacing R with F in H_SF=H_S+H_F+∑_q(G_q⊗X_q+G_q^†⊗X_q^†). Central to the result is the quantum central limit theorem, which enforces Wick factorization of multi-time correlations and enables an explicit, physically transparent mapping to a finite set of oscillators. The work establishes a universal framework for treating nonlocal noise and fluctuations, with implications for analyzing decoherence, entanglement, and transport in large quantum baths, and clarifies when ground-state versus thermal representations of F are appropriate.∎

Abstract

Quantum systems that interact non-locally with an environment are paradigms for exploring collective phenomena. They naturally emerge in various physical contexts involving long-range, many-body interactions. We consider a general class of such open systems characterized by a coupling to the environment which is inversely proportional to the square root of the environment size. We show that the induced system dynamics has a universal bosonic nature: the same evolution arises from coupling the system to a collection of noninteracting bosonic modes, independently of the microscopic structure of the original environment. This emergent "bosonization" of the environment's influence results from the scaling of the coupling in the thermodynamic limit and is a manifestation of the quantum central limit theorem. While the effect has been observed in specific models before, we show that it is, in fact, a universal feature.
Paper Structure (5 sections, 3 theorems, 93 equations, 1 figure)

This paper contains 5 sections, 3 theorems, 93 equations, 1 figure.

Table of Contents

  1. Proof of Theorem
  2. Dynamics as $M\rightarrow\infty$ by the Dyson series
  3. Equivalent Gaussian reservoir: Proof of the theorem
  4. Emergence of the ground state for ${\rm F}$
  5. Wick's theorem

Key Result

Theorem 1

The system dynamics $\rho_{\rm S}(t)$ defined in 7-1, is equivalently given, for all $t\in{\mathbb R}$, by where $H_{{\rm S}{\rm F}}$ is the interacting ${\rm S}{\rm F}$ Hamiltonian MX and $\rho_{\rm F}$ is the ground state rhovac.

Figures (1)

  • Figure 1: Illustration of the main result. A system ${\rm S}$ with Hamiltonian $H_{\rm S}$ interacting with two different reservoirs. Left sketch: ${\rm S}$ interacts equally with the $M$ elements of a reservoir, each a $d_{\rm R}$-level system with Hamiltonian $h_{\rm R}$. The interaction operator is $\propto 1/\sqrt M$, called a mesoscopic scaling. Right sketch: ${\rm S}$ interacts with a reservoir of (maximally) $d_{\rm R}^2$ independent bosonic modes indexed by $(k,l)$, all in their ground state. Each oscillator corresponds to a transition $E_k\rightarrow E_l$ between energy levels of $h_{\rm R}$ allowed by the interaction ${\rm S}{\rm R}$ operator $v$. The interaction operator $X$ of ${\rm S}$ with the bosonic modes is linear in the creation and annihilation operators. The left-right arrow $\leftrightarrow$ indicates our main result: The reduced dynamics of ${\rm S}$ obtained from the left model (as $M\rightarrow\infty$) and the right model is the same. The result holds for arbitrary ${\rm S}$ and regardless of the fine details of the components of ${\rm R}$.

Theorems & Definitions (3)

  • Theorem
  • Proposition 1
  • Proposition 2