Quantum eigenstates from classical Gibbs distributions27 July 2020
Some interesting work Anatoli Polkovnikov is now on arXiv and submitted to SciPost Physics! In our new work on Quantum eigenstates from classical Gibbs distributions we investigate what happens when we try to express classical mechanics in the quantum mechanical language of states and operators and how to get from Gibbs ensembles to the Schrödinger equation?
Classical and quantum mechanics tend to be expressed in different languages, either continuous phase-space variables (position and momentum) or operators/states. Uncertainty leads to phase-space probability distributions (CM) or density matrices (QM). We can always try to express a density matrix as a classical probability distribution, asking the probability that a quantum particle has a given position or momentum. This leads to what's known as a 'Wigner quasiprobability distribution'. This behaves a lot like a probability distribution — but it can be negative! Still, if we treat it as a regular probability distribution to calculate physical observables we generally get sensible answers. Here we do the reverse: starting from a classical probability distribution, we map this to something that looks a lot like a quantum density matrix, where we can express observables such as position and momentum as the usual operators. Interestingly, something similar happens as in Wigner's function: while we can calculate expectation values of observables in the familiar way, this object is not quite a density matrix because its eigenvalues (probabilities) can be negative! It's easy to see why such quasiprobabilities need to be negative: if they weren't, we would have a classical uncertainty principle. For classical distributions with sufficient uncertainty, the negative quasiprobabilities disappear and we end up with an exact density matrix. One class of distributions with built-in uncertainty is that of Gibbs distributions. Given the temperature of a system, we know the classical probability of finding a system with given position and momentum, where higher temperatures lead to more uncertainty. Writing out the eigenvalue equation for their classical eigenstates, a saddle-point approximation returns the stationary Schrödinger equation! The approximation is controlled by the temperature and really works surprisingly well at not-too-low temperatures, and we explore this in the paper through various numerical and analytical examples.
We discuss how the language of wave functions (state vectors) and associated non-commuting Hermitian operators naturally emerges from classical mechanics by applying the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schrödinger equation follows from the Liouville equation, with ℏ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (quantized) energies, where these states directly correspond to quantum eigenstates. We show that this correspondence is particularly pronounced for canonical Gibbs ensembles, where classical eigenstates satisfy an integral eigenvalue equation that reduces to the Schrödinger equation in a saddle-point approximation controlled by the inverse temperature. We illustrate this correspondence by showing that some paradigmatic examples such as tunneling, band structures, and quantum eigenstates in chaotic potentials can be reproduced to a surprising precision from a classical Gibbs ensemble, without any reference to quantum mechanics and with all parameters (including ℏ) on the order of unity. Interestingly, it is now classical mechanics which allows for apparent negative probabilities to occupy eigenstates, dual to the negative probabilities in Wigner's quasiprobability distribution. These negative probabilities are shown to disappear when allowing sufficient uncertainty in the classical distributions.
Maximum Velocity Circuits in Physical Review Research08 July 2020
My work on Maximum Velocity Ciruits with Austen Lamacraft is now published in Physical Review Research!A key image and blurb are also highlighted on the PRResearch homepage.
Quantum Chaos 2020 Seminars08 June 2020
A wonderful idea from Pablo Poggi: starting this week, we are organizing a series of online talks about topics related to Quantum Chaos in its various forms! Topics including (but not limited to)
- Manifestations of chaos in quantum systems
- Quantum information scrambling
- Ergodicity and thermalization in closed many-body quantum systems
- Quantum simulations of complex quantum dynamics
More information (including the schedule of upcoming talks) can be accessed at https://sites.google.com/view/pablopoggi/qchaos2020-seminars