RotorQuant: Quantized Rotor Systems

• RotorQuant refers to quantized rotor systems defined on S¹, characterized by discrete angular momentum and specialized state control for quantum mechanics studies.
• These systems are implemented in platforms like trapped-ion setups, Josephson circuits, and spinor BECs, advancing experimental quantum thermodynamics and metrology.
• RotorQuant methods also extend to CFD applications, enabling efficient analysis of Flettner rotors and benchmarking performance in quantum engines and hybrid devices.

RotorQuant refers collectively to quantized rotor systems—planar or otherwise—which serve as model platforms for foundational studies in quantum mechanics, quantum thermodynamics, and practical rotor-based quantum technologies. “RotorQuant” encompasses experimental, theoretical, and numerical approaches to realizing, probing, controlling, and benchmarking quantum rotors in isolation, as working media in heat engines and refrigerators, as components in hybrid devices, and as objects of precision metrology, leveraging angular momentum quantization and associated phenomena.

1. Quantized Planar Rotors: Hamiltonian, States, and Physical Realizations

The quantum planar rotor is defined on the configuration space $S^1$, with canonical conjugate variables: the angular coordinate $\theta \in [0,2\pi)$ and the angular-momentum operator $L_z = -i\hbar\,\partial_\theta$, whose spectrum is discrete with integer quantum number $m$ ($L_z|m\rangle = \hbar m |m\rangle$). The free-rotor Hamiltonian is $H_0 = L_z^2/(2I)$, with moment of inertia $I$. Its eigenstates are $|m\rangle$ (or equivalently $e^{im\theta}/\sqrt{2\pi}$), with energies $E_m = \hbar^2 m^2 / (2I)$ (Gaida et al., 2024).

Several physical platforms realize RotorQuant systems:

• Trapped-ion planar rotors: Two Coulomb-repelling ions (e.g., $\theta \in [0,2\pi)$0Ca$\theta \in [0,2\pi)$1) in a planar harmonic trap form a rigid rotor of radius $\theta \in [0,2\pi)$2, enabling preparation, superposition, and rotation in a highly isolated environment (Glikin et al., 2023).
• Solid-state implementations: Josephson circuits (phase–number duality), molecular/nanomechanical rotors, and optomechanical setups instantiate the quantum rotor Hamiltonian with tunable potentials (Jr. et al., 2024).
• Spinor BECs: The collective spin state of a spin-1 Bose–Einstein condensate under quadratic Zeeman shift maps to a quantum rotor with angular variable representing the population imbalance (Buchmann et al., 2013).

The critical difference from classical rotors is the discreteness of angular momentum, the $\theta \in [0,2\pi)$3 topology, and the resulting nontrivial phase-space structure (e.g., quantization condition $\theta \in [0,2\pi)$4 for Wigner functions) (Grigorescu, 2018).

2. Quantum Thermodynamics: Rotor Engines and Otto Cycles

RotorQuant systems play a central role in quantum thermodynamics, both as autonomous engines and as working media in externally driven cycles:

• Autonomous quantum rotor engines: A quantum rotor, interacting with a thermalized working fluid (qubits/modes) and dissipative load, accumulates angular momentum as net useful work (flywheel function). Engine performance is characterized via kinetic energy transfer, torque integrals, and ergotropy. Notably, all quantum work metrics (kinetic energy change, time-integrated torque, and ergotropy) are consistent apart from quantum modulations when $\theta \in [0,2\pi)$5 crosses integer multiples of $\theta \in [0,2\pi)$6 (Seah et al., 2018, Seah et al., 2018).
• Quantum Otto cycles: The planar rotor acts as a working medium controlled by external fields. In the case of a magnetic-dipole (flux) rotor, genuine quantum advantages appear: classically, engine/refrigerator operation is forbidden by continuous spectrum properties, but the quantum rotor enables both modes via level discretization and parameter-induced degeneracies in the partition function (Jacobi $\theta \in [0,2\pi)$7-functions) (Gaida et al., 2024).
• Hybrid rotor–qubit devices: Three-body interaction Hamiltonians (rotor plus two qubits, each coupled to a reservoir) permit heat-engine, refrigerator, and accelerator operation in the same framework. Figures of merit include power, efficiency, coefficient of performance (COP), and novel quantifiers such as angular-momentum rectification coefficients which benchmark directionality of work extraction (Leitch et al., 2023).

The table below summarizes performance regimes for hybrid rotor engines (Leitch et al., 2023):

Regime	Optimal $\theta \in [0,2\pi)$8	Asymmetry	Moment of Inertia $\theta \in [0,2\pi)$9
Engine Power	$L_z = -i\hbar\,\partial_\theta$0	$L_z = -i\hbar\,\partial_\theta$1	Moderate
Efficiency	$L_z = -i\hbar\,\partial_\theta$2–$L_z = -i\hbar\,\partial_\theta$3	$L_z = -i\hbar\,\partial_\theta$4	Large (low back-action)
Refrigerator COP	$L_z = -i\hbar\,\partial_\theta$5	$L_z = -i\hbar\,\partial_\theta$6	Small
Heat Rectification	Low–moderate	$L_z = -i\hbar\,\partial_\theta$7	Moderate
Angular-mom. Rect.	Moderate–high	$L_z = -i\hbar\,\partial_\theta$8–$L_z = -i\hbar\,\partial_\theta$9	Moderate–large

Quantum-classical comparison reveals that i) directionality and quantization introduce discrete steps and sawtooth structures in dynamical observables, ii) quantum backaction and uncertainty broaden performance regimes.

3. Decoherence, Uncertainty, and Quantum Control

Understanding orientational decoherence is pivotal for RotorQuant systems:

• Orientational decoherence: Decoherence between angular superpositions scales as $m$0, where $m$1 is the angular-momentum diffusion coefficient. This $m$2-law, verified in two-ion planar rotor experiments, is a general scaling for rotor decoherence (Glikin et al., 2023).
• Decoherence-free subspaces and the design of bosonic rotation codes are enabled by precise knowledge of decoherence rates, particularly for quantum error correction and macroscopic superposition tests (Glikin et al., 2023).
• Uncertainty quantification: The angular-momentum variance $m$3 and various shift-operator-based measures (dispersion $m$4, sine/cosine covariance) define a strict hierarchy of angular uncertainty. The optimal states (von Mises states) saturate these inequalities and have a clear mechanical interpretation—uncertainty corresponds to moments of inertia about axes in $m$5 space (Jr. et al., 2024).
• Coherent control and squeezing: Optical cavity-mediated control of rotor states (spinor BECs in a Fabry–Perot resonator) enables rapid switching between effective potentials and realizes squeezing protocols—variance suppression in angular or angular-momentum quadratures detectable via photon correlation measurements (Buchmann et al., 2013).

4. RotorQuant Phase Space, Distributions, and Entropy

The quantum rotor’s phase space naturally admits a Wigner-function formulation adapted to the $m$6 structure:

• Quantum Wigner distributions: For the planar rotor, the Wigner function $m$7 displays mixed discrete-continuous structure, reflecting the duality between angle and quantized angular momentum. Consistency of marginals enforces quantization conditions $m$8 (Grigorescu, 2018).
• Thermal decoherence to classicality: At finite temperature, loss of off-diagonal coherence in the Wigner kernel induces the emergence of classical sound waves propagating on the ring; the variance $m$9 sets the sound velocity (Grigorescu, 2018).
• Non-thermal entropy: Localization in angle basis (e.g., for coherent rotor states) is measured by non-thermal (Poisson) entropy, reflecting occupation spread over angular-momentum sublevels; for well-localized orbits ($L_z|m\rangle = \hbar m |m\rangle$0), this entropy grows logarithmically (Grigorescu, 2018).

5. Quantum Gear Trains and Interacting Rotor Systems

Coupled quantum rotors (quantum gears) realize synthetic models of rotational information transfer and nanomechanical quantum machinery:

• Hamiltonian and transmission: Two planar rotors with teeth $L_z|m\rangle = \hbar m |m\rangle$1 interact via a $L_z|m\rangle = \hbar m |m\rangle$2-periodic “gear” potential. The long-time-averaged angular momentum transmission ratio $L_z|m\rangle = \hbar m |m\rangle$3 reaches classical benchmarks $L_z|m\rangle = \hbar m |m\rangle$4 for kicks below the quantum break-away threshold; above threshold, quantum interference restores classical transmission at magic kick strengths (Liu et al., 2018).
• Quantum signatures: State revivals in center-of-mass motion and interference-enhanced transmission are direct markers of quantum coherence and Hilbert-space topology; the Bloch structure of the relative pendulum coordinate enables robust angular-momentum transfer even far above classical failure points.
• Work extraction: Useful work outflow (ergotropy) from receiver gears is nearly maximal, signaling minimal loss to non-directional fluctuations (Liu et al., 2018).

6. RotorQuant in Computational Fluid Dynamics: Flettner Rotor Installations

Outside quantum thermodynamics, RotorQuant methodology is implemented in inviscid CFD for aerodynamic evaluation of Flettner rotors:

• Numerical framework: RotorQuant uses finite-volume discretization of the inviscid, incompressible Euler equations, imposing a dynamic circulation-matching body force in cells adjacent to the rotor to enforce prescribed circumferential velocity and circulation (Kühl, 8 May 2025).
• Convection schemes and computational speed: High-order TVD-κ or UDS blending control numerical viscosity, impacting accuracy of lift (within O(10%) error) versus drag (parasitic drag not captured in Eulerian framework). RotorQuant achieves $L_z|m\rangle = \hbar m |m\rangle$5–$L_z|m\rangle = \hbar m |m\rangle$6 savings in CPU time over viscous CFD for early-phase rotor design, making it suitable for design space exploration but not for high-fidelity drag breakdown (Kühl, 8 May 2025).
• Validation: Comparison with potential flow and viscous IDDES/URANS references validates RotorQuant for lift-induced loads and induced-drag scaling, with best agreement for fine circumferential mesh ($L_z|m\rangle = \hbar m |m\rangle$7 cells) and appropriate convection parameter choice.

7. Applications, Interdisciplinary Extensibility, and Benchmarking Protocols

RotorQuant serves as a versatile tool in several cutting-edge quantum and classical research directions:

• Quantum information and simulation: Rotor-based bosonic codes, synthetic dimensions in rotational levels, and ultra-sensitive torque sensors rely on precise decoherence management (Glikin et al., 2023).
• Experimental quantum thermodynamics: Benchmarked protocols (RotorStar plots) enable direct comparison of power, efficiency, COP, rectification, and quantum work quality for any rotor-based quantum machine (Leitch et al., 2023).
• Singular optics, superconducting circuits, and pulse shaping: The rotor algebra and associated uncertainty/entropy structure underpin quantum optics (OAM states), Josephson phase–number duality, and time–frequency coherent control (Jr. et al., 2024).
• Macroscopic quantum experiments: RotorQuant platforms advance foundational tests—decoherence-free subspaces, interference of rotational superpositions, studies of quantum gravity and angular-momentum exchange effects (Glikin et al., 2023).

RotorQuant thus encompasses rigorous quantum theoretical descriptions, experimental realization and control strategies, uncertainty and entropy measures, computational/numerical methodology, and application-agnostic benchmarking frameworks spanning quantum mechanics, quantum thermodynamics, optomechanics, and beyond.