General phase spaces: from discrete variables to rotor and continuum limits. (English) Zbl 1383.81119
Summary: 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.
MSC:
| 81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
| 70G10 | Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics |
| 81V80 | Quantum optics |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 82D80 | Statistical mechanics of nanostructures and nanoparticles |
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