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

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.

22 pages, 3 figures; part of special issue on Rabi model; v2 minor changes

References in corpus (20)

• Local stabilizer codes in three dimensions without string logical operators
• Target space supergeometry of $\eta$ and $\lambda$-deformed strings
• An Integrable Deformation of the AdS5 x S5 Superstring
• Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal
• Quantum phase transition in the Dicke model with critical and non-critical entanglement
• The ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ mirror model as a string
• Discrete Abelian Gauge Theories for Quantum Simulations of QED
• Qudit surface codes and gauge theory with finite cyclic groups
• Supergravity background of the lambda-deformed AdS_3 x S^3 supercoset
• Aharonov-Bohm effect in the tunnelling of a quantum rotor in a linear Paul trap
• Double Wick rotating Green-Schwarz strings
• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects
• Generalized Kitaev Models and Slave Genons
• Anyonic statistics with continuous variables
• Integrability of the eta-deformed Neumann-Rosochatius model
• On discrete structures in finite Hilbert spaces
• Energy spectrum and critical exponents of the free parafermion $Z_N$ spin chain
• $Z_N$ symmetric chiral Rabi model: a new $N$-level system
• $Z_3$ generalization of the Kitaev's spin-1/2 model