*[This is akin to a living review, which will hopefully improve from time to time. Last edited 2020-4-8.]*

This post will collect some models of decoherence and branching. We don’t have a rigorous definition of branches yet but I crudely define models of branching to be models of decoherence^{a } which additionally feature some combination of amplification, irreversibility, redundant records, and/or outcomes with an intuitive macroscopic interpretation.

(Note in particular that I am *not* just listing models for which you can mathematically take a classical limit ( or ) and recover the classical equations of motion; Yaffe has a pleasingly general approach to that task [1], but I’ve previously sketched why that’s an incomplete explanation for classicality.)

I have the following desiderata for models, which tend to be in tension with computational tractability:

- physically realistic
- symmetric (e.g., translationally)
- no ad-hoc system-environment distinction
- Ehrenfest evolution along classical phase-space trajectories (at least on Lyapunov timescales)

Regarding that last one: we would like to recover “classical behavior” in the sense of classical Hamiltonian flow, which (presumably) means continuous degrees of freedom.^{b } Branching only becomes unambiguous in some large-*N* limit, so it seems satisfying models are necessarily messy and difficult to numerically simulate. At the minimum, a good model needs time asymmetry (in the initial state, not the dynamics), sensitive dependence on initial conditions, and a large bath. Most branching will (presumably) be continuous both in time and in number of branches, like a decaying atom where neither the direction nor time of decay are discrete.

Below are some models that have one or more of the above features. For many of these, the historical progression was to first analyze decoherence (tracing out the environment) and then the creation of redundant records (by looking at the correlations within the environment). There are too many cites for me to be comprehensive or historically fair in this brief post, so just email me if you want a more comprehensive bibliography for any of these.

A “spin” refers generically to a two-state quantum system, regardless of whether it has an interpretation in terms of particle spin. An “oscillator” refers generically to a quantum system with a continuous degree of freedom (e.g., position of a particle), regardless of whether there is a harmonic confining potential. An “emergent” system just means a classical degree of freedom that is naturally selected by the decoherent dynamics of the model, rather than being inserted by hand.

#### Models without amplification

**Dirac equation in an inhomogeneous magnetic field**. This is the Stern-Gerlach experiment. It’s nice because the interactions are physical and completely analytic. You can start with a spin uncorrelated with the spatial degrees of freedom, and then see how the inhomogeneous field splits the wavepacket into the two parts. This can be done as a first-year graduate QM homework problem. However, once you have the two parts, you’d need to add amplification/irreversibility to get a proper model of branching, e.g., a model of the phosphorescent screen.

**Two-particle scattering in one dimension**. This is a good example of discrete branching of a continuous variable. (But only in 1D; in higher dimensions, the branches are continuous, being indexed by the scattering angles.) If the two particles interact through a potential that’s a function only of their relative distance, then the center of mass coordinate decouples and this is isomorphic to a single particle scattering from a central potential. The two outcomes are either to tunnel through the barrier or reflect off it. Also a good homework problem, but also has no amplification on its own.

#### Models with amplification and fixed System-Environment split

**Spin decohered by multi-spin apparatus**. The simplest possible model that includes amplification is a central spin measured sequentially by set of other spins through ad-hoc interactions. This is unrealistic, but is deployed judiciously by Zurek in his well-known review article [1] with simple CNOT interactions. An artificial spatial degree of freedom is added to get the Coleman-Hepp model [1, 2] wherein an electron moves along a chain (1D lattice) of atoms and the electron spin is measured and recorded by each atom in turn. Rather than add a spatial degree of freedom or ad-hoc time dependence, one can also just assert that the apparatus spins are coupled much more strongly to the central spin than to each other (while still assuming the Hamiltonian is diagonalized in the basis corresponding to the *z*-axis) [3]. Records have been studied more extensively in this scenario [4, 5].

**Spin decohered by multi-spin apparatus in oscillator bath**. Achieves effective measurement dynamics (as in the previous type of model) using slightly more realistic interactions for a purpose-built measuring apparatus. The apparatus (or “readout”) spins are initialized in a metastable state and coupled to a dissipative bath of oscillators. This is sometimes known as the Curie–Weiss model, and is connected to spontaneous symmetry breaking [4, 5].

**Spin decohered by oscillator bath**. The Jaynes-Cummings model generalized to have multiple modes (oscillators), sometimes known as the spin-boson model. As a model for dissipative reduced dynamics it is well studied [1], but I’m not aware of any work on amplification or records.

**Oscillator decohered by oscillator bath**. Generally a chosen to be harmonic oscillators coupled linearly to each other. Most people just use the reduced dynamics of the central oscillator to study decoherence and diffusion [1], e.g.., Caldiera-Leggett [2]. For sufficiently fast monitoring by a large bath of (individually weakly coupled) oscillators, the reduced dynamics limit to Markovian quantum Brownian motion, which is characterized by a Lindblad equation. One can also show how sufficiently different paths (histories) are recorded in the environment [3] and can also be an idealized form of branching if you analyze the environment, and

**Decoherence by scattering bath of particles**. The system is a particle decohered in the position basis, momentum basis, or (most realistically) an over-complete basis of wavepackets.^{c } Often, the particle is taken to be a charged or dielectric particle decohered by scattering radiation. Usually studied in the idealized limit of Markovian quantum Brownian motion, as described by a Lindblad equation, where the scattering interaction time is taken to zero. Lots of different regimes; too many cites to list. For a neutral particle with a dielectric constant, see chapter 3 of Schlosshauer’s textbook and references therein. Records are discussed in [1][2][3]. One can generalize from a single monitored particle to a large population of them that is monitored by a different (lighter) species [4]. For a charged accelerating particle decohering through bremsstrahlung see the textbooks by Joos et al. and Breuer & Petruccione, and [5]. For a charged non-accelerating particle decohering in the momentum basis, see [6] and, for records, see [7].^{d }

**Decoherence of an electronic junction**. Jiang and Zhang [1] analyze decoherence of electron states in a junction that interacts with virtual Buttiker’s probes [2, 3] or phonons. I haven’t looked at this model carefully.

**Decoherence of a field**. Anglin & Zurek treat an electromagnetic field decohered in the basis of coherent states by a dielectric medium [1].

#### Ad-hoc Models with Amplification and Emergent System

**Brun-Halliwell model**. A 1D interacting spin chain. This model is important because the emergent classical variables — local average hydrodynamic variables — are very general/universal. [1,2,3,4]

#### Physical Models with Amplification and Emergent System?

**Avalanche photodiode**. I’ve never seen a simple tractable model for this, but I know that a good amount of theory does exist somewhere. It would be nice because it’s very physical and common in labs.

**General hydrodynamics**. The holy grail. The quasi-classical degrees of freedom (i.e., the emergent systems) are the approximate eigenstates of local averages of locally conserved quantities like energy and species density.

*[I thank Daniel Ranard and interstice for conversation that prompted this post, and Curt von Keyserlingk for suggesting the paper by Gaveau & Schulman.]*

### Footnotes

(↵ returns to text)

- I take
*decoherence*to mean a model with dynamics taking the form for some tensor decomposition , where is an (approximately) stable orthonormal basis independent of initial state, and where for times and , where is the initial state of and is some characteristic time scale.↵ - In principle you could have discrete degrees of freedom that limit, as , to some sort of discrete classical systems, but most people find this unsatisfying.↵
- Wavepackets are approximate eigenstates of both position and momentum. For a discussion of decoherence with respect to an overcomplete basis of wavepackets, see the introduction of this and references therein.↵
- Charged particles, especially accelerating ones, are the most difficult to handle because one must work with their “dressed” states.↵

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