We identify and describe unique early time behavior of a quantum system initially in a superposition, interacting with its environment. This behavior---the copycat process---occurs after the system begins to decohere, but before complete einselection. To illustrate this behavior analytic solutions for the system density matrix, its eigenvalues, and eigenstates a short time after system-environment interactions begin are provided. Features of the solutions and their connection to observables are discussed, including predictions for the continued evolution of the eigenstates towards einselection, time dependence of spin expectation values, and an estimate of the system's decoherence time. In particular we explore which aspects of the early stages of decoherence exhibit quadratic evolution to leading order, and which aspects exhibit more rapid linear behavior. Many features of our early time perturbative solutions are agnostic of the spectrum of the environment. We also extend our work beyond short time perturbation theory to compare with numerical work from a companion paper.
We identify and describe unique early time behavior of a quantum system initially in a superposition, interacting with its environment. This behavior---the copycat process---occurs after the system begins to decohere, but before complete einselection. To illustrate this behavior analytic solutions for the system density matrix, its eigenvalues, and eigenstates a short time after system-environment interactions begin are provided. Features of the solutions and their connection to observables are discussed, including predictions for the continued evolution of the eigenstates towards einselection, time dependence of spin expectation values, and an estimate of the system's decoherence time. In particular we explore which aspects of the early stages of decoherence exhibit quadratic evolution to leading order, and which aspects exhibit more rapid linear behavior. Many features of our early time perturbative solutions are agnostic of the spectrum of the environment. We also extend our work beyond short time perturbation theory to compare with numerical work from a companion paper.
We study dynamics of quantum open systems, paying special attention to these aspects of their evolution which are relevant to the transition from quantum to classical. We begin with a discussion of the conditional dynamics of simple systems. The resulting models are straightforward but suffice to illustrate basic physical ideas behind quantum measurements and decoherence. To discuss decoherence and environment-induced superselection (einselection) in a more general setting, we sketch perturbative as well as exact derivations of several master equations valid for various systems. Using these equations we study einselection employing the general strategy of the predictability sieve. Assumptions that are usually made in the discussion of decoherence are critically reexamined along with the “standard lore” to which they lead. Restoration of quantum-classical correspondence in systems that are classically chaotic is discussed. The dynamical second law - it is shown - can be traced to the same phenomena that allow for the restoration of the correspondence principle in decohering chaotic systems (where it is otherwise lost on a very short time-scale). Quantum error correction is discussed as an example of an anti-decoherence strategy. Implications of decoherence and einselection for the interpretation of quantum theory are briefly pointed out.
The act of measurement on a quantum state is supposed to collapse the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schr\odinger's equation. There are two general approaches to the study of wave-function collapse: one called the consistent or decoherent histories approach and the other, the environmental decoherence approach, which studies the effect of the environment upon the quantum system, to explain wave-function collapse. In the environmental decoherence approach, one usually studies a Markovian-approximated Master equation to study the time-evolution of reduced density matrix and obtains the long-term dependence of the off-diagonal elements of this matrix. We do not make a Markovian assumption and study a particularly simple and calculable example. We find, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential, which is a new result (the long-term behavior is, of course, still exponential). This allows one to connect the Fermi-golden rule quadratic-in-time behavior of a transition probability to the exponential long-time behavior of a collapsing wave-function.
Chapter 4 begins to discuss decoherence, and, thus, to address the overarching question: How does the classical world—classical states that are responsible for the objective reality of our everyday experience—emerge from within the Universe that is, as we know from compelling experimental evidence, made out of quantum stuff. The short answer to this question is that decoherence selects (from the vast number of superpositions that populate Hilbert space in the process of environment-induced superselection (also known as einselection) the few states that are—in contrast to all the other alternatives—stable in spite of their immersion in the environment. Decoherence is illustrated with a detailed discussion of two models. A spin decohered by an environment of spins as well as quantum Brownian motion have become paradigmatic models of decoherence for good reason: They are exactly solvable and yet they capture (albeit in an idealized manner) the emergence of the preferred classical states in settings that are relevant for quantum measurements and for Newtonian dynamics in effectively classical phase space.
The environment – external or internal degrees of freedom coupled to the system – can, in effect, monitor some of its observables. As a result, the eigenstates of these observables decohere and behave like classical states: Continuous destruction of superpositions leads to environment-induced superselection (einselection). Here I investigate it in the context of quantum chaos (i.e., quantum dynamics of systems which are classically chaotic). I show that the evolution of a chaotic macroscopic (but, ultimately, quantum) system is not just difficult to predict (requiring an accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg indeterminacy (which limits the resolution available in the initial conditions). This happens after a time t ℏ which is only logarithmic in the Planck constant. A definitely macroscopic (if somewhat outrageous) example is afforded by various components of the solar system which are chaotic, with the Lyapunov timescales ranging from a bit more than a month (Hyperion) to millions of years (planetary system as a whole). On the timescale t ℏ the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of radii of their orbits into “Schrödinger cat – like” states, and the concept of a trajectory would cease to apply. In reality, such paradoxical states are eliminated by decoherence which helps restore quantum-classical correspondence. The price for the recovery of classicality is the loss of predictability: In the classical limit (associated with effective decoherence, and not just with the smallness of ℏ) the rate of increase of the von Neumann entropy of the decohering system is independent of the strength of the coupling to the environment, and equal to the sum of the positive Lyapunov exponents. Algorithmic aspects of entropy production are briefly explored to illustrate the effect of decoherence from the point of view of the observer. We show that “decoherence strikes twice”, introducing unpredictability into the system and extracting quantum coherence from the observers memory, where it enters as a price for the classicality of his records.
In discrete quantum systems, quantum decoherence is the disappearance of simple phase relations as a result of interactions with an environment. For many applications, the question is not necessarily how to avoid (inevitable) system-environment interactions, but rather how to design environments that optimally preserve a system's phase relations in spite of such interactions. The formation of system-environment entanglement is a major driving mechanism for decoherence, and a detailed understanding of this process could inform strategies for conserving coherence optimally. This requires scalable, flexible, and systematically improvable quantum dynamical methods that retain detailed information about the entanglement properties of the environment, yet very few current methods offer this combination of features. Here we address this need by introducing a theoretical framework wherein we combine the truncated Wigner approximation with standard time-dependent perturbation theory, allowing for computing expectation values of operators in the combined system-environment Hilbert space. We demonstrate the utility of this framework by applying it to the spin-boson model, representative of qubits and simple donor-acceptor systems. For this model, our framework provides an analytical description of perturbative contributions to expectation values. We monitor how quantum decoherence at zero temperature is accompanied by entanglement formation with individual environmental degrees of freedom. Based on this entanglement behavior, we find that the selective suppression of low-frequency environmental modes is particularly effective for mitigating quantum decoherence.
Description: This work explores the conditions under which quantum superposition may emerge or persist at cosmological scales, challenging the common assumption that the early universe was the most favorable environment for quantum coherence. While the universe is widely believed to have originated from a quantum regime, the extreme interaction density, thermal noise, and intense field couplings of the early universe are argued to have caused rapid decoherence, effectively suppressing sustained quantum superposition beyond minimal scales. In this sense, the early universe is interpreted as a decoherence-dominated environment rather than a coherent quantum domain. The paper proposes an alternative perspective in which the late universe, characterized by cosmic expansion, reduced interaction density, and lower environmental noise, may offer more favorable conditions for partial isolation and the re-emergence of quantum superposition under specific circumstances. This conceptual framework reframes the quantum–classical transition as an environmentally driven process rather than a purely temporal or scale-dependent one, contributing to ongoing discussions in quantum cosmology, decoherence theory, and the foundations of physics. 🔹 Keywords: Quantum cosmology; Quantum superposition; Decoherence; Early universe; Late universe; Environmental noise; Quantum–classical transition; Interaction density; Foundations of physics; Conceptual physics