Lemos et al. have a relatively recent lettera in Nature where they describe a method of imaging with undetected photons. (An experiment with the same essential quantum features was performed by Zou et al.b way back in 1991, but Lemos et al. have emphasized its implications for imaging.) The idea is conceptually related to decoherence detection, and I want to map one onto the other to flesh out the connection. Their figure 1 gives a schematic of the experiment, and is copied below.
Figure 1 from Lemos et al.: ''Schematic of the experiment. Laser light (green) splits at beam splitter BS1 into modes a and b. Beam a pumps nonlinear crystal NL1, where collinear down-conversion may produce a pair of photons of different wavelengths called signal (yellow) and idler (red). After passing through the object O, the idler reflects at dichroic mirror D2 to align with the idler produced in NL2, such that the final emerging idler f does not contain any information about which crystal produced the photon pair. Therefore, signals c and e combined at beam splitter BS2 interfere. Consequently, signal beams g and h reveal idler transmission properties of object O.''
The first two paragraphs of the letter contain all the meat, encrypted and condensed into an opaque nugget of the kind that Nature loves; it stands as a good example of the lamentable way many quantum experimental articles are written.… [continue reading]
In this post I’m going to give a clean definition of idealized quantum Brownian motion and give a few entry points into the literature surrounding its abstract formulation. A follow-up post will give an interpretation to the components in the corresponding dynamical equation, and some discussion of how the model can be generalized to take into account the ways the idealization may break down in the real world.
I needed to learn this background for a paper I am working on, and I was motivated to compile it here because the idiosyncratic results returned by Google searches, and especially this MathOverflow question (which I’ve answered), made it clear that a bird’s eye view is not easy to find. All of the material below is available in the work of other authors, but not logically developed in the way I would prefer.
Quantum Brownian motion (QBM) is a prototypical and idealized case of a quantum system , consisting of a continuous degree of freedom, that is interacting with a large multi-partite environment , in general leading to varying degrees of dissipation, dispersion, and decoherence of the system. Intuitively, the distinguishing characteristics of QBM is Markovian dynamics induced by the cumulative effect of an environment with many independent, individually weak, and (crucially) “phase-space local” components. We will defined QBM as a particular class of ways that a density matrix may evolve, which may be realized (or approximately realized) by many possible system-environment models. There is a more-or-less precise sense in which QBM is the simplest quantum model capable of reproducing classical Brownian motion in a limit.
In words to be explained: QBM is a class of possible dynamics for an open, quantum, continuous degree of freedom in which the evolution is specified by a quadratic Hamiltonian and linear Lindblad operators.… [continue reading]
The Planck Collaboration has released a paper describing the dust polarization in the CMB for the patch of sky used recently by BICEP2 to announce evidence for primordial gravitational waves. Things look bleak for BICEP2’s claims. See Peter Woit, Sean Carroll, Quanta, Nature, and the New York Times.
In the comments, Peter Woit criticizes the asymmetric way the whole story is likely to be reported:
I think it’s completely accurate at this point to say that BICEP2 has provided zero evidence for primordial gravitational waves, instead is seeing pretty much exactly the expected dust signal.
This may change in the future, based on Planck data, new BICEP2 data, and a joint analysis of the two data sets (although seeing a significant signal this way doesn’t appear very likely), but that’s a separate issue. I don’t think it’s fair to use this possibility to try and evade the implications of the bad science that BICEP2 has done, promoted by press conference, and gotten on the front pages of prominent newspapers and magazines.
This is a perfectly good example of normal science: a group makes claims, they are checked and found to be incorrect. What’s not normal is a massive publicity campaign for an incorrect result, and the open question is what those responsible will now do to inform the public of what has happened. “Science communicators” often are very interested in communicating over-hyped news of a supposed great advance in science, much less interested in explaining that this was a mistake. Some questions about what happens next:
1. Will the New York Times match their front page story “Space Ripples Reveal Big Bang’s Smoking Gun” with a new front page story “Sorry, these guys had it completely wrong?”
… [continue reading]
[Other parts in this series: 1,2,3,4,5,6,7,8.]
A common mistake made by folks newly exposed to the concept of decoherence is to conflate the Schmidt basis with the pointer basis induced by decoherence.
[Show refresher on Schmidt decompsition]
Given any two quantum systems
and a pure joint state
, there always exists a Schmidt decomposition of the form
where and are local orthonormal Schmidt bases on and , respectively.
Now, any state in such a joint Hilbert space can be expressed as for arbitrary fixed orthonormal bases and . What makes the Schmidt decomposition non-trivial is that it has only a single index rather than two indices and . (In particular, this means that the Schmidt decomposition constains at most non-vanishing terms, even if .) The price paid is that the Schmidt bases, and , depend on the state .
When the values in the Schmidt decomposition are non-degenerate, the local bases are unique up to a phase. As evolves in time, this decomposition is defined for each time . The bases and evolve along with it, and can be considered to be a property of the state . In fact, they correspond to the eigenvectors of the respective reduced density matrices of and .
In the ideal case of so-called pure decoherence, the environment begins in an initial state and is coupled to the system through a unitary of the form
with as , where is a conditional unitary on and . The elements of the density matrix of the system evolve as , i.e.… [continue reading]