## Adapted Caldeira-Leggett

Several new papers have been posted since May 2021. Contextual updates to these pages reflecting the new work will appear eventually. In the mean time, here are links and abstracts:

arXiv 2202.12353 **Equilibration and "Thermalization'' in the Adapted Caldeira-Leggett model**

Abstract: I explore the processes of equilibration exhibited by the Adapted Caldeira-Leggett (ACL) model, a small unitary "toy model" developed for numerical studies of quantum decoherence between an SHO and an environment. I demonstrate how dephasing allows equilibration to occur in a wide variety of situations. While the finite model size and other "unphysical" aspects prevent the notions of temperature and thermalization from being generally applicable, certain primitive aspects of thermalization can be realized for particular parameter values. I link the observed behaviors to intrinsic properties of the global energy eigenstates, and argue that the phenomena I observe contain elements which might be key ingredients that lead to ergodic behavior in larger more realistic systems. The motivations for this work range from curiosity about phenomena observed in earlier calculations with the ACL model, to much larger questions related to the nature of equilibrium, thermalization and the emergence of physical laws.

arXiv:2105.14017 **Einselection, Equilibrium and Cosmology**

Abstract: Our observed Universe has a very strong arrow of time rooted in its low entropy starting point. This low entropy start can be related to various "tuning puzzles" about the early state of the Universe. Here we explore the relationship between the arrow of time and the emergence of classical from quantum in the hopes of ultimately gaining insights into cosmological initial conditions. Our focus is on einselection, the process whereby interactions with an environment select preferred states for a quantum system. This process plays an essential role in the emergence of classical from quantum. Studies of einselection have so far been limited to cases that exhibit an arrow of time. Here we study the ability of equilibrium systems to exhibit einselection -- and investigate whether detailed balance prevents this -- motivated by the question of whether classicality requires an arrow of time. We present calculations in the adapted Caldeira-Leggett model which demonstrate that einselection can indeed take place in equilibrium systems, and show how this phenomenon is tied to histories which express an arrow of time, despite the global equilibrium. We discuss some interesting implications of our results for cosmology and cosmological initial conditions. We are intrigued and a bit surprised by the role the consistent histories formalism has ended up playing in our analysis.

arXiv:2105.14032 **Copycat process in the early stages of einselection**

Abstract: 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.

arXiv:2105.14040 **Adapted Caldeira-Leggett Model**

Abstract: We preset a variant of the Caldeira-Leggett (CL) model of a harmonic oscillator coupled to an environment. The CL model is a standard tool for studying the physics of decoherence. Our "adapted Caldeira-Leggett" (ACL) model is built in a finite Hilbert space which makes it suitable for numerical studies. Taking a numerical approach allows us to avoid the limitations of standard approximation schemes used with the CL model. We are able to evolve the ACL model in a fully reversible unitary manner, without the built-in time asymmetry and other assumptions that come with the master equation methods typically used. We have used the ACL model to study new topics in the field of decoherence and einselection where the full unitary evolution is essential to our work. Those results (reported in companion papers) include an examination of the relationship between einselection and the arrow of time, and studies of the very earliest stages of einselection. This paper provides details about the ACL model and our numerical methods. Our numerical approach makes it straightforward to explore and plot any property of the physical system. Thus we believe the examples and illustrations we present here may provide a helpful resource for those wishing to improve their familiarity with standard decoherence results, as well as those looking to probe the underpinnings of our companion papers. We expect the ACL model will be a useful tool for exploring additional phenomena that cannot be studied using traditional approximation schemes.

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Andrew Arrasmith and I have developed an "Adapted Caldeira-Leggett" (ACL) model. Like the original Caldeira-Leggett (CL) model it is useful for exploring the quantum aspects of a system interacting with an environment. Unlike the CL model, the ACL occupies a finite Hilbert space and is amenable to numerical calculations. We have performed various numerical explorations which in the first instance show decoherence and einselection phenomena similar to those seen with the CL model. Our next exploration has been to study these phenomena under equilibrium conditions, where there is no arrow of time. This is a very different regime vs the usual CL studies, which use master equation techniques that presuppose an arrow of time throughout. As of April 2019 publications on these topics are in preparation. Here are some of my recent talks on the subject.

- April 4, 2019 University of Nottingham Seminar (Slides here)
- See also the Copenhagen talks on this page from April 2018 (under "Reviews of cosmic inflation"). The work was not as full developed then, but those talks give more of the motivations and background.
- Also, for general background information on decoherence and einselection, here are some excellent review articles by Zurek and by Schlosshauer.

The **"copycat state"**: We observe an interesting two step process toward einselection. If one starts with an initial "Schrödinger Cat" superposition of wavepackets, the first step involves the density matrix acquiring a 2nd (small) nonzero eigenvalue with a corresponding eigenstate that has a "copycat" form. This 2nd eigenstate looks something like a mirror image of the original Schrödinger cat state. It remains stable for a while, and then on the full decoherence time the two eigenstates "collapse" into the single wavepacket (pointer basis) form. These slides illustrate the whole process.