*[This is akin to a living review, which will hopefully improve from time to time. Last edited 2017-10-11.]*

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. I have the following desiderata for models, which tend to be in tension:

- 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.

Here are some models that have one or more of the above features.

**A single qubit measured by other qubits**. Coupling through CNOT interactions. Used judiciously by Zurek in his well known review article [1], but obviously not super satisfying.

**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.

**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.

**The Coleman-Hepp model**. An electron moving along a line whose spin is measured and recorded by a lattice of fixed spins. Sort of has amplification, but it’s very artificial. [1,2]

**Quantum Brownian Motion**. This comes either in the exact Markov limit obeying a Lindblad equation (which is idealized, but tractable) or in a concrete model with finite microscopic degrees of freedom (like Caldiera-Leggett). Mostly people just use this for decoherence and diffusion, but you can see how sufficiently different paths (histories) decohere using things like the influence functional. If you just consider the reduced dynamics of the system, it’s only a model of decoherence; it can be an idealized form of branching if you analyze the environment. [1,2,3,4]

**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]

**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.

*[I thank Daniel Ranard and interstice for conversation that prompted this post.]*

### 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.↵

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