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arXiv:2401.07690 · Physical Review A

Encoding position by spins: Objectivity in the boson-spin model

Tae-Hun Lee, Jarosław K. Korbicz

Published May 3, 2024• 4 citations
arXivHTMLPDF

Abstract

We investigate quantum objectivity in the boson-spin model, where a central harmonic oscillator interacts with a thermal bath of spin-$\frac{1}{2}$ systems. We analyze how information about a continuous position variable can be encoded into discrete, finite-dimensional environments. More precisely, we study conditions under which the so-called spectrum broadcast structures can be formed in the model. These are multipartite quantum-state structures, representing a mode-refined form of decoherence. Working in the recoil-less limit, we use the Floquet theory to show that despite its apparent simplicity, the model has a rich structure with different regimes, depending on the motion of the central system. In one of them, the faithful encoding of the position and hence objectivity are impossible irrespectively of the resources used. In another, large enough collections of spins will faithfully encode the position information. We derive the characteristic length scales, corresponding to decoherence and precision of the encoding.

DOI
10.1103/physreva.109.052204

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We investigate quantum objectivity in the boson-spin model, where a central harmonic oscillator interacts with a thermal bath of spin-12\frac{1}{2}21​ systems. We analyze how information about a continuous position variable can be encoded into discrete, finite-dimensional environments. More precisely, we study conditions under which the so-called spectrum broadcast structures can be formed in the model. These are multipartite quantum-state structures, representing a mode-refined form of decoherence. Working in the recoil-less limit, we use the Floquet theory to show that despite its apparent simplicity, the model has a rich structure with different regimes, depending on the motion of the central system. In one of them, the faithful encoding of the position and hence objectivity are impossible irrespectively of the resources used. In another, large enough collections of spins will faithfully encode the position information. We derive the characteristic length scales, corresponding to decoherence and precision of the encoding.
We analyze decoherence and objectivization processes in spin-spin models for arbitrary spins without interactions between different spins in the environment. We first derive the most general analytic form of the decoherence factor in the measurement limit, where the interaction Hamiltonian dominates the rest. We then analyze thermal environments and derive exact, analytic formulas for both the decoherence factor and the state fidelity of postinteraction environment states, conditioned on different states of the system. This allows us to analyze the objectivization process of the state of the central spin during the interaction. We do so using so-called spectrum broadcast structures (SBS), which are specific multipartite quantum states encoding a certain operation notion of objectivity. We analyze analytically (for short times) and numerically how higher spin influences the efficiency of decoherence and objectivization processes. As expected, the higher the spin of the environment, the more efficient decoherence and objectivization become. This work is a generalization of previous studies, limited to spin-1/21/21/2 systems only, and we hope will be useful in future objectivity experiments.
Objectivity constitutes one of the main features of the macroscopic classical world. An important aspect of the quantum-to-classical transition issue is to explain how such a property arises from the microscopic quantum theory. Recently, within the framework of open quantum systems, there has been proposed such a mechanism in terms of the so-called spectrum broadcast structures. These are multipartite quantum states of the system of interest and a part of its environment, assumed to be under an observation. This approach requires a departure from the standard open quantum systems methods, as the environment cannot be completely neglected. In the present paper we study the emergence of such a state structure in one of the canonical models of the condensed-matter theory: the spin-boson model, describing the dynamics of a two-level system coupled to an environment made up by a large number of harmonic oscillators. We pay much attention to the behavior of the model in the non-Markovian regime, in order to provide a testbed to analyze how the non-Markovian nature of the evolution affects the surfacing of a spectrum broadcast structure.
In this paper, we contribute to the mathematical foundations of the recently established theory of Spectrum Broadcast Structures (SBS). These are multipartite quantum states, encoding an operational notion of objectivity and exhibiting a more advanced form of decoherence. We study SBS and asymptotic convergence to SBS in the case of a central system interacting with N environments via the von Neumann-type measurement interactions, ubiquitous in the theory of open quantum systems. We will be focusing on the case where the system is modeled by an infinite-dimensional Hilbert space and the operators associated with the system in the Hamiltonian have purely continuous spectrum. Such a setup yields mathematical complications that have hitherto not been addressed in the theory of SBS.
Quantum systems interacting with environments tend to have their information lost from transmission through the correlations generated with the degrees of freedom of the environment. For situations where we have non- Markovian environments, the information contained in the environment may return to the system and such effect can be captured by non-Markovian witnesses. In the present work, we use a central qubit coupled to a spin chain with Ising interactions subject to a magnetic field, i.e., a central spin model, and solve the exact dynamics of a system subject to dephasing dynamics via Kraus operators. For such we use two witnesses to analyze the presence of non-Markovianity: the BLP trace-based measure and the conditional past-future correlator (CPF). Furthermore, we see how such behavior suppresses the classic plateau in Partial Information Plot (PIP) from the paradigm of quantum Darwinism, as well as objective information from Spectrum Broadcast Structure (SBS). In addition to the system point of view, we show the impossibility of encoding accessible information for measurement in the environment for any model limit.
The birth of objective properties from the subjective quantum world has been one of the key questions in the quantum-to-classical transition. Based on recent results in the field, we study it in a quantum mechanical model of a boson-boson interaction—quantum Brownian motion. Using various simplifications, we prove a formation for thermal environments of, so called, spectrum broadcast structures, responsible for perceived objectivity. In the quantum measurement limit we prove that this structure is always formed, providing the characteristic timescales. Including self-Hamiltonians of the environment, we show the exponential scaling of the effect with the size of the environment. Finally, in the full model we numerically study the influence of squeezing in the initial state of the environment, showing broader regions of formation than for non-squeezed thermal states.
Emergence of objective, classical properties in quantum systems can be described in the modern language of quantum information theory. In this work, we present an example of such an analysis. We apply the quantum channel theory to a boson-spin model of open quantum systems and calculate, using recoilless approximation and the Floquet theory, the Holevo quantity, which bounds the capacity of the channel, broadcasting information about the central system into its environment. We analyze both the short-time regime, showing quadratic in time initial growth of the capacity, and the asymptotic regime. Complicated dependence on the model parameters, such as temperature, tunneling energy for the environment, etc., is also analyzed, showing, e.g., regimes where the Holevo bound reaches its maximum.