Vortex NOON states for rotation sensing

Nathan Dupont; Nathan Goldman; Peter Schlagheck; Simon Dengis

Vortex NOON states for rotation sensing

T0 review · 0 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read Vortex NOON states of bosonic atoms detect external rotations at the Heisenberg limit.

desk verdict This paper gives a concrete, workable proposal for making vortex NOON states on usable timescales and turning them into a rotation sensor at the Heisenberg limit. read the letter →

arxiv 2606.29509 v1 pith:VQCNIPG7 submitted 2026-06-28 cond-mat.quant-gas

Simon Dengis , Nathan Dupont , Peter Schlagheck , Nathan Goldman This is my paper · ORCID

classification cond-mat.quant-gas

keywords NOONstatesvortexmodesrotationsensingHeisenberglimitBose-Hubbardmodelcoldatomsquantummetrologycounterdiabaticdriving

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a scheme to generate NOON states from few-body bosonic vortices in a weakly anisotropic two-dimensional harmonic trap for cold atoms. These states arise because the single-particle p orbitals create an effective two-mode Bose-Hubbard model whose vortex modes carry opposite circulation and become entangled through collective tunneling once the system enters the self-trapping regime. Two acceleration methods—geodesic counterdiabatic driving for small particle numbers and resonance- or chaos-assisted tunneling for larger numbers—bring preparation times down to experimentally accessible scales while keeping fidelities near unity. The resulting states then enter an interferometric protocol whose intrinsic sensitivity to rotation yields detection of infinitesimal external rotations at the Heisenberg limit. A reader would care because the construction supplies a concrete, realizable route to quantum-enhanced rotation sensors inside standard bosonic Josephson junctions.

What carries the argument

Effective two-mode Bose-Hubbard model defined by vortex modes (p_x ± i p_y) carrying opposite circulation in the p orbitals of the trap.

What would settle it

An experiment that fails to produce a spectrally isolated NOON manifold or that measures rotation sensitivity no better than the standard quantum limit would falsify the central claim.

Watch

Extended reading notes

Core claim

Vortex NOON states of few-body bosonic vortices can be generated in a weakly anisotropic two-dimensional harmonic trap where single-particle p orbitals define an effective two-mode Bose-Hubbard model with modes (p_x ± i p_y) carrying opposite circulation. In the self-trapping regime the NOON manifold is spectrally isolated and collective tunneling produces highly entangled vortex NOON states; geodesic counterdiabatic driving and resonance-chaos-assisted tunneling accelerate their creation on relevant timescales with near-unit fidelity. An interferometric protocol that exploits the states' intrinsic sensitivity to rotation then detects infinitesimal external rotations at the Heisenberg limit.

Load-bearing premise

The single-particle p orbitals in the weakly anisotropic trap define an effective two-mode Bose-Hubbard model whose vortex modes carry opposite circulation.

Editorial extensions

If this is right

The NOON manifold becomes spectrally isolated in the self-trapping regime.
Geodesic counterdiabatic driving produces the states for small particle numbers on short timescales.
Resonance- and chaos-assisted tunneling produces the states for larger particle numbers.
The interferometric protocol reaches Heisenberg-limited rotation detection.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same vortex-orbital construction could be attempted in other trap anisotropies or with higher angular-momentum orbitals.
Combining the two acceleration strategies might further shorten preparation times beyond what either achieves alone.
The metrological protocol may generalize to other many-body systems that host circulating modes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main thing to know is that the authors map p orbitals in a weakly anisotropic 2D trap onto an effective two-mode Bose-Hubbard model, isolate the NOON manifold in the self-trapping regime, and then supply two acceleration routes—geodesic counterdiabatic driving for small N and resonance/chaos-assisted tunneling for larger N—to reach high-fidelity states before the natural tunneling time becomes prohibitive. They close with an interferometric sequence that converts the ±2N angular-momentum difference into a rotation-induced phase.

What is actually new is the specific choice of vortex combinations (p_x ± i p_y) together with the tailored acceleration protocols and the explicit sensing sequence. The paper does a reasonable job spelling out the parameter windows where the effective model holds, where the NOON states are spectrally isolated, and where the accelerated protocols achieve near-unit fidelity.

The soft spots are modest and standard for this style of proposal. Everything rests on the two-mode truncation remaining accurate once anisotropy and interactions are turned on; that assumption is plausible but would benefit from a direct check against the full single-particle spectrum. The chaos-assisted route may also be more sensitive to trap imperfections than the counterdiabatic one. The metrological gain itself is the usual NOON scaling once the states exist, so the advance is in the generation method rather than a new fundamental limit.

This is for people working on quantum metrology with ultracold atoms or on bosonic Josephson junctions that involve higher orbitals. A reader who already knows the two-mode literature will see the concrete protocols and parameter choices as the useful part.

I would send it to peer review. The construction is internally consistent, builds on established tools, and supplies enough detail to be checked.

Referee Report

0 major / 2 minor

Summary. The paper introduces a scheme to generate NOON states of few-body bosonic vortices in a weakly anisotropic 2D harmonic trap, where single-particle p orbitals define an effective two-mode Bose-Hubbard model with vortex modes (p_x ± i p_y) carrying opposite circulation. In the self-trapping regime the NOON manifold is spectrally isolated; collective tunneling produces highly entangled vortex NOON states, but on long timescales. Two acceleration protocols (geodesic counterdiabatic driving for small N and resonance/chaos-assisted tunneling for larger N) are developed to reach near-unit fidelity on usable timescales. An interferometric protocol is then presented that maps the ±2N angular-momentum difference onto a rotation-induced phase, recovering Heisenberg-limited rotation sensing.

Significance. If the effective two-mode modeling and fidelity claims hold, the work supplies a concrete, experimentally relevant route to rotation metrology with entangled vortex states in cold atoms. The use of standard Josephson-junction constructions together with explicit parameter regimes, fidelity benchmarks, and a closed interferometric sequence is a strength; the absence of ad-hoc free parameters in the core derivation further supports the proposal.

minor comments (2)

The abstract states that the NOON manifold 'becomes spectrally isolated' in the self-trapping regime; a brief quantitative statement of the gap size relative to other manifolds (e.g., in terms of the interaction strength U/J) would help readers assess robustness without consulting the figures.
The two acceleration protocols are presented as complementary; a short table or paragraph comparing the accessible particle-number ranges, required control precision, and residual fidelity loss for each method would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed and positive summary of our manuscript on vortex NOON states for rotation sensing. The recommendation for minor revision is noted. No specific major comments were provided in the report, so we have no point-by-point responses at this time. We are happy to incorporate any minor changes or clarifications if the editor or referee identifies them.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper's derivation begins from the standard effective two-mode Bose-Hubbard model for p_x ± i p_y vortex orbitals in a weakly anisotropic trap, a well-established approximation in the literature. Spectral isolation of the NOON manifold, collective tunneling, acceleration protocols (geodesic counterdiabatic driving and resonance/chaos-assisted tunneling), and the interferometric protocol mapping ±2N angular momentum to rotation phase are all constructed using conventional quantum mechanics, Josephson-junction techniques, and metrology methods without any reduction to fitted parameters defined by the target claim, self-definitional loops, or load-bearing self-citations. The Heisenberg-limit sensitivity follows directly from the intrinsic properties of the NOON states within the model, rendering the chain self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the validity of the effective two-mode model and the isolation of the NOON manifold in the self-trapping regime; no free parameters or invented entities are identifiable from the abstract.

assumptions (1)

domain assumption The single-particle p orbitals define an effective two-mode Bose-Hubbard model with vortex modes carrying opposite circulation.
This modeling choice is the foundational premise stated in the abstract on which the entire scheme depends.

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Cite this review

Pith. "Pith review of Vortex NOON states for rotation sensing." pith.science (2026). https://pith.science/paper/VQCNIPG7

@misc{pith2026260629509,
  author       = {Pith},
  title        = {Pith review of: Vortex NOON states for rotation sensing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQCNIPG7}},
  note         = {Machine review of arXiv:2606.29509}
}

abstract

We introduce a scheme to generate NOON states of few-body bosonic vortices and demonstrate their application as high-precision rotation sensors. Our approach is based on cold atoms in a weakly anisotropic two-dimensional harmonic trap, where the single-particle p orbitals define an effective two-mode Bose-Hubbard model with vortex modes $(\mathrm{p}_x\pm\mathrm{i}\mathrm{p}_y)$ carrying opposite circulation. In the self-trapping regime, we show that the NOON manifold becomes spectrally isolated, and collective tunneling processes give rise to highly entangled vortex NOON states. However, these states emerge on prohibitively long timescales. To overcome this limitation, we develop two complementary acceleration strategies: geodesic counterdiabatic driving for small particle numbers, and resonance- and chaos-assisted tunneling in the semiclassical regime at larger particle numbers. Both approaches enable the generation of NOON states on experimentally relevant timescales while preserving near-unit fidelities. Finally, we quantify the metrological advantage of vortex NOON states by introducing an interferometric protocol that exploits their intrinsic sensitivity to rotation, enabling the detection of infinitesimal external rotations at the Heisenberg limit. Our work opens the door to rotation sensors based on atomic NOON states, generically realizable in bosonic Josephson junctions with vortex-type orbitals.

Figures

Figures reproduced from arXiv: 2606.29509 by the authors.

**Figure 1.** Vortex NOON states for rotation sensing. Schematic representation of an ensemble of bosons confined in a two-dimensional harmonic trap (gray parabolic area). The goal is to create within this system a NOON state of vortices, |NOON⟩ = (|N, 0⟩ + |0, N⟩)/ √ 2, a superposition of one component where all bosons circulate in one direction (red) and another where they all circulate in the opposite direction (blue). Here, … view at source ↗

**Figure 2.** One-body subspace. Top left: First six levels Enx,ny of the 2D quantum harmonic oscillator Eq. (1) for Ω = 0 with anisotropy ∆/ℏω¯ (exaggerated for clarity compared to the weakly anisotropic regime considered; see Eqs. (4) and (3)) and subspace of interest {|px ⟩, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Interferometry scheme with NOON states. From left to right: The NOON operator Uˆ is applied to the fully-polarized initial Fock state |N, 0⟩, creating the NOON state. In the absence of trap anisotropy (∆ = 0), the NOON components |N, 0⟩ and |0, N⟩ accumulate respectively a phase ±NΩT during the interrogation time T, amounting to the relative phase φ = 2NΩT (proportional to the hatched area of the interferometer in … view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Many-body spectrum, Bloch sphere and NOON state preparation in self-trapping regime. (a) Sketch of the two-mode system described by the Hamiltonian (9) in the vortex basis {|⟲⟩ ≡ |ψ+⟩, |⟳⟩ ≡ |ψ−⟩} (5), with 2-body interaction energy −2U/3, intermode coupling ∆ (i.e. a…

**Figure 5.** Figure 5: Geodesic counterdiabatic driving for fast NOON-state preparation. (a) Spectrum of the BoseHubbard Hamiltonian (22) under geodesic driving (G, dashed gray) and geodesic counterdiabatic driving (GCD, solid black) as a function of the bias ℏΩ/∆, for N = 5 particles, U = …

**Figure 6.** Figure 6: Performance of the GCD with protocol duration and particle number. (a,b) Magnitude of the effective coupling |∆eff | (32) as a function of (a) the total protocol duration T (in units of ℏ/∆) for different particle numbers N, and (b) the particle number N for differen…

**Figure 7.** Figure 7: Floquet-Magnus emulation of the complex hopping. Time evolution of the populations in |N, 0⟩ (red) and |0, N⟩ (blue) for geodesic counterdiabatic driving (Eq. (33)) (dark colors) and for its emulation by a Hamiltonian with oscillating terms (Eq. (34)) (light colors),…

**Figure 8.** Figure 8: Accelerated NOON-state preparation via resonance- and chaos-assisted tunneling. (a) Left: Stroboscopic Poincar´e section in the (φ, z)-plane (black dots), with the partial barriers (red dashed) delimiting the regular islands near z = ±1 for ℏeff = 2/N = 0.2. Right: Foc…

**Figure 9.** Figure 9: summarizes the phenomenology of the RATCAT acceleration as the modulation angular frequency ω is varied, at a fixed amplitude of modulation ℏΩ = 3.8 ∆. As established in Sec. III B 1, this amplitude sets the degree of non-integrability: it must be large enough to de…

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quantum Many-Body Metrology of Rotation Sensing with Strong Interactions
quant-ph 2026-08 conditional novelty 6.0 of 10

For a few bosons in a ring with two weak links, strong interactions make the many-body quantum Fisher information at zero rotation nonzero and large, improving slow-rotation sensing beyond the Bose-Hubbard prediction.
Time-independent counterdiabatic driving for emergent two-level subspaces in many-body systems
quant-ph 2026-07 conditional novelty 6.0 of 10

Geodesic parameter trajectories make the counterdiabatic Hamiltonian time-independent for effective two-level systems, reducing shortcuts to adiabaticity to fixed-amplitude controls.

Reference graph

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139 extracted references · 7 canonical work pages · cited by 2 Pith papers

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Shortcut to adiabaticity The possibility of adiabatically generating a NOON state is opened by the separation of the energy lev- els corresponding to the states of interest. The adia- batic theorem states that when the Hamiltonian varies slowly compared to the intrinsic timescale set by the level splitting between the energy eigenstates, the system re- ma...
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However, impor- tant connections have been established between geodesic and counterdiabatic driving [50, 51, 100, 101]

Geodesic counterdiabatic driving A priori, the CDH is defined independently of the choice of the driving function Ω. However, impor- tant connections have been established between geodesic and counterdiabatic driving [50, 51, 100, 101]. The combination of these two driving strategies, known as geodesic counterdiabatic driving (GCD), makes it possi- ble to...
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Driven two-mode system Assuming, as before, control over the energy difference between the vortex modes±, we consider the following periodic drive of the Hamiltonian (22): ˆH(t) =− U 3 ˆb†2 + ˆb2 + + ˆb†2 − ˆb2 − − ∆ 2 ˆb† +ˆb− + ˆb† −ˆb+ −ℏΩ cos(ωt)(ˆn+ −ˆn−),(40) withℏΩ andωthe amplitude and angular frequency of the modulation. By analogy with Bloch’s t...
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Resonance- and chaos-assisted tunneling In the vicinity of a nonlinearr:s-resonance chain inside the regular island ( ˜I < ˜Ic), the driven Hamiltonian (43) admits the effective description [45, 46] ˜H(r:s) eff (˜I, ˜θ) = (˜I− ˜Ir:s)2 2 ˜mr:s + 2˜Vr:scos(r˜θ),(48) parametrized by the resonance action ˜Ir:s, the effec- tive mass ˜m r:s and the resonance co...
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Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.

[1] [1]

Shortcut to adiabaticity The possibility of adiabatically generating a NOON state is opened by the separation of the energy lev- els corresponding to the states of interest. The adia- batic theorem states that when the Hamiltonian varies slowly compared to the intrinsic timescale set by the level splitting between the energy eigenstates, the system re- ma...

[2] [2]

However, impor- tant connections have been established between geodesic and counterdiabatic driving [50, 51, 100, 101]

Geodesic counterdiabatic driving A priori, the CDH is defined independently of the choice of the driving function Ω. However, impor- tant connections have been established between geodesic and counterdiabatic driving [50, 51, 100, 101]. The combination of these two driving strategies, known as geodesic counterdiabatic driving (GCD), makes it possi- ble to...

[3] [3]

Floquet-Magnus emulation of the complex coupling The need to introduce a complex coupling ∆ eff may, however, pose practical difficulties. A viable solution to implement ∆eff is through Floquet engineering, by adding an oscillating term to the Bose-Hubbard Hamiltonian, emulating the additional complex phase [98, 99, 104, 105]. We show that such an alterna...

[4] [4]

By analogy with Bloch’s theorem in space, time-periodic Hamiltonians are conveniently treated within the Floquet formalism [110]

Driven two-mode system Assuming, as before, control over the energy difference between the vortex modes±, we consider the following periodic drive of the Hamiltonian (22): ˆH(t) =− U 3 ˆb†2 + ˆb2 + + ˆb†2 − ˆb2 − − ∆ 2 ˆb† +ˆb− + ˆb† −ˆb+ −ℏΩ cos(ωt)(ˆn+ −ˆn−),(40) withℏΩ andωthe amplitude and angular frequency of the modulation. By analogy with Bloch’s t...

[5] [5]

Quantizing Eq

Resonance- and chaos-assisted tunneling In the vicinity of a nonlinearr:s-resonance chain inside the regular island ( ˜I < ˜Ic), the driven Hamiltonian (43) admits the effective description [45, 46] ˜H(r:s) eff (˜I, ˜θ) = (˜I− ˜Ir:s)2 2 ˜mr:s + 2˜Vr:scos(r˜θ),(48) parametrized by the resonance action ˜Ir:s, the effec- tive mass ˜m r:s and the resonance co...

[6] [6]

As established in Sec

NOON state preparation from RAT and CAT Figure 9 summarizes the phenomenology of the RAT- CAT acceleration as the modulation angular frequencyω is varied, at a fixed amplitude of modulationℏΩ = 3.8 ∆. As established in Sec. III B 1, this amplitude sets the de- gree of non-integrability: it must be large enough to de- velop the connected chaotic sea and th...

[7] [7]

Pro- jecting ˆH1b (2) onto this manifold directly yields Eq. (6), with (E 10 +E 01)/2 = 2ℏ¯ω,E10 −E 01 =−∆, ˆnx −ˆny = ˆb† +ˆb− + ˆb† −ˆb+ and ˆLz = iRℏ(ˆb† yˆbx − ˆb† xˆby) =Rℏ(ˆn+ −ˆn−).(B5) The interaction Hamiltonian (7) is derived from the two-mode bosonic field (B2) as ˆHint = g2D 2 ZZ ∞ −∞ dxdy ˆψ†2(x, y) ˆψ2(x, y).(B6) Expanding the fields in the ...

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