General phase spaces: from discrete variables to rotor and continuum limits

TL;DR

This work develops a cohesive framework to relate discrete-variable, rotor, and continuous-variable quantum phase spaces via explicit limit-taking procedures, and then applies it to six foundational models to generate dv, rotor, andcv generalizations. It demonstrates how dv Hamiltonians can be systematically mapped to rotor and cv counterparts, preserving key symmetries and exposing connections to the quantum Hall effect, optomechanics, and lattice gauge theories. The resulting rotor and cv generalizations include novel constructions (e.g., rotor versions of the almost Mathieu operator and CV toric/cubic codes) and provide a platform to study low-energy limits, spectral properties, and potential experimental realizations. The techniques offer a versatile route to explore topological, many-body, and gauge-theoretic physics across phase-space formalisms with implications for quantum simulation and quantum information processing.

Abstract

We provide a basic introduction to discrete-variable, rotor, and continuous-variable quantum phase spaces, explaining how the latter two can be understood as limiting cases of the first. We extend the limit-taking procedures used to travel between phase spaces to a general class of Hamiltonians (including many local stabilizer codes) and provide six examples: the Harper equation, the Baxter parafermionic spin chain, the Rabi model, the Kitaev toric code, the Haah cubic code (which we generalize to qudits), and the Kitaev honeycomb model. We obtain continuous-variable generalizations of all models, some of which are novel. The Baxter model is mapped to a chain of coupled oscillators and the Rabi model to the optomechanical radiation pressure Hamiltonian. The procedures also yield rotor versions of all models, five of which are novel many-body extensions of the almost Mathieu equation. The toric and cubic codes are mapped to lattice models of rotors, with the toric code case related to U(1) lattice gauge theory.

Figures (3)

• Figure 1: In phase space, four elementary translations along position and momentum define a closed circuit, here oriented clockwise, and its corresponding enclosed area, represented here in grey. Quantum physics associates a phase factor to an area in phase space, measured in units of Planck's constant.
• Figure 2: (a) The $\textnormal{dv}$ phase space can be thought of as consisting of a degree of freedom hopping from site to site. The set of sites forms a ring. The position variable $s$ is a site index and is thus an integer modulo $\mathcal{N}$, the number of sites along the ring. (b) If hopping between two sites takes the universal same amount of time, phase space is fully discrete, with the momentum $m$ belonging also to the set of integers modulo $\mathcal{N}$. Both position and momentum have periodic boundary conditions, as indicated by the set of black and white arrows, and thus phase space has the topology of a torus.
• Figure 3: Eigenvalues of $\frac{{\cal N}}{2\pi}(H_{\textnormal{dv}}^{\text{sho}}-2)$ (\ref{['eq:hdft']}) vs. ${\cal N}$ for $\mathcal{L}=\mathcal{M}=1$. One can see that they approach true harmonic oscillator eigenvalues of an integer plus a half (horizontal lines) as ${\cal N}\rightarrow\infty$. This behavior persists for higher values of $\mathcal{L}$.