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Physical Emulation of Nonlinear Spin System Hamiltonians via Closed Loop Feedforward Control of a Collective Atomic Spin

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A cold atomic spin, measured continuously and steered by real-time feedback, can be made to follow the dynamics of target nonlinear spin Hamiltonians such as the Lipkin–Meshkov–Glick model and the kicked top.

desk verdict First experimental QMF emulation of LMG and kicked-top dynamics, with a real latency problem that leaves the demonstration qualitatively right but quantitatively unproven. read the letter →

arxiv 2507.22132 v1 pith:V2IX7QZO submitted 2025-07-29 quant-ph physics.atom-ph

Ian Pannemarsh This is my paper · ORCID
classification quant-phphysics.atom-ph
keywords quantummeasurementandfeedbackcollectiveatomicspinLipkin-Meshkov-GlickmodelkickedtopFaradayrotationQNDchaosFloquettimecrystalmean-fieldemulation
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

This dissertation reports an experimental method for making an ensemble of roughly a million cold cesium atoms evolve as if it were governed by a chosen nonlinear spin Hamiltonian. The method repeatedly measures the collective spin projection $J_z$ without destroying the state and feeds each measurement outcome forward to set the strength of a magnetic rotation, replacing the nonlinear $J_z^2$ term with the product of the measured value and $J_z$. The author demonstrates the approach on two benchmark models: the Lipkin–Meshkov–Glick (LMG) Hamiltonian, where the data show spontaneous symmetry breaking and a dynamical phase transition at the predicted parameter values, and the Quantum Kicked Top, where the data show chaotic trajectories with Lyapunov exponents of the expected order of magnitude and a Floquet time-crystal phase with partial rigidity. If the method is sound, a single apparatus can be reprogrammed to study phase transitions, chaos, and out-of-equilibrium phases in collective spin systems, with atom number acting as a dial between the quantum and classical regimes. The paper is candid about its main shortcoming: roughly $6\,\mu\mathrm{s}$ of control-loop latency spoils the continuous-time LMG emulation, quantified by $\alpha\tau_{\mathrm{delay}} \approx 0.23$, producing a dissipative decay toward the fixed points that a Hamiltonian evolution would not have.

What carries the argument

The load-bearing object is the mean-field linearization of the nonlinear term, $J_z^2 \to \langle J_z\rangle J_z$, which turns a two-body interaction into a single-body rotation whose strength is updated by each measurement. The physical realization is a three-part loop: the Faraday interaction, in which the collective spin projection rotates the polarization of a probe laser by an angle proportional to $\langle J_z\rangle$ (a quantum non-demolition measurement); a field-programmable gate array (FPGA) controller sampling at 500 kHz that digitizes the polarimetry signal and computes the control voltage $V_{\mathrm{ctl}} = g_{\mathrm{ctl}}^{-1}\left(s\Lambda \langle J_z\rangle/\langle J\rangle\right)$; and magnetic coils that execute the resulting rotation about $\hat{z}$ while a constant rotation about $\hat{x}$ runs simultaneously. Supporting machinery keeps the loop faithful within fixed-point hardware: an exponential-decay model of the mean spin length, a Padé approximant for the decay factor, and, for the kicked top, a binary bit-mask operation that implements the kick-angle modulo $2\pi$ so that arbitrarily large nonlinear strengths can be reached.

What would settle it

Repeat the LMG emulation at a tenfold smaller linear rotation rate (so that $\alpha\tau_{\mathrm{delay}} \approx 0.02$) with $s$ fixed; if the spin still decays toward the fixed points with an unchanged settling-time scaling, the claim that loop latency is the primary driver of the dissipative decay is falsified. As a positive control, add artificial delays of roughly 3, 6, and 12 $\mu$s to the loop and check that the signed midpoint decay time $t_{DD}$ shortens monotonically with delay, as the paper's own latency simulations predict.

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Extended reading notes

Core claim

The central claim is that quantum measurement and feedback (QMF) with a mean-field control law makes a collective atomic spin follow the classical (mean-field) dynamics of a target Hamiltonian. Concretely, a QND Faraday measurement yields an outcome $m \propto \langle J_z\rangle$, and the feedback law sets the $z$-rotation rate proportional to $m$, realizing the linearized Hamiltonian $H_{\mathrm{MF}} = -\gamma(1-s)J_x - \gamma \frac{s}{2}\frac{m}{\langle J\rangle}J_z$. For the LMG model this produces the expected bifurcation of the stable fixed point for $s>0.5$: initialized at the unstable point $+\hat{x}$, the spin falls randomly into one of the two symmetry-broken wells, with the choice correlated with the first measurement outcome, and the time-averaged magnetization places the dynamical phase transition at the expected $s=2/3$. For the kicked top, the same protocol run stroboscopically yields trajectory spreads consistent with positive Lyapunov exponents in the chaotic regime and, near $\alpha=\pi$, subharmonic oscillations with a rigidity window that is real but about three times narrower than simulation predicts. The paper argues that the deviations from ideal Hamiltonian evolution—chiefly a non-Hamiltonian decay of the spin toward the model's fixed points in the LMG case and a narrowed rigidity window in the kicked top—are dominated by control-loop latency and measurement transients rather than by quantum projection noise or classical control noise.

Load-bearing premise

The load-bearing premise is that at roughly $10^6$ atoms the measured spin projection can stand in for the operator in the nonlinear term ($J_z^2 \to \langle J_z\rangle J_z$) and that the feedback loop applies the corrected rotation before the state changes appreciably—a condition the paper's own estimate $\alpha\tau_{\mathrm{delay}} \approx 0.23$ shows is only marginally met.

Editorial extensions

If this is right

  • A single apparatus can be reprogrammed to emulate any Hamiltonian whose nonlinear terms are polynomial in $J_z$, simply by changing the feedback control law, without hardware changes.
  • The LMG experiments give an experimental handle on spontaneous symmetry breaking and dynamical phase transitions: the order parameter $Z_\infty$ crosses at the expected $s=2/3$, and the shot-to-shot symmetry-breaking choice is correlated with the first measurement outcome.
  • The kicked-top emulation provides a route to studying quantum chaos in a controlled mesoscopic system; the measured Lyapunov exponents, estimated from the growth of the spread in the elevation angle, sit below the classical values as expected for a lower bound from a single measured projection.
  • The Floquet time-crystal data show subharmonic (period-$2T$) oscillations with finite rigidity around $\alpha=\pi$, indicating the emulator can realize out-of-equilibrium phases of matter on demand.
  • Scaling the atom number from $10^6$ downward should slide the same experiment from the classical to the quantum regime, since the signal scales as $N$ while quantum projection noise scales as $\sqrt{N}$.

Reading between the lines

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

  • The latency-induced decay the paper identifies could be turned into a feature: by deliberately choosing the loop delay, the same apparatus could emulate controlled dissipative or open-system dynamics (for instance Lindblad-type evolution toward fixed points), which the paper does not pursue.
  • Because the symmetry-breaking direction is a function of the first measurement outcome, the setup offers a quantitative probe of the quantum-to-classical transition: at smaller atom number the initial spread in $Z(0)$ grows, so the distribution of decay times $t_{DD}$ should broaden and the correlation between $Z(0)$ and the final well should weaken—a directly testable prediction.
  • The single-projection readout caps the Lyapunov estimate; adding a second probe direction or a stroboscopic tomography step after each kick would remove the lower-bound ambiguity and should bring the experimental Lyapunov exponents closer to the classical spectrum.
  • The discrepancy between the simulated and observed time-crystal rigidity windows (about $0.8\pi$ versus $0.93\pi$) provides a single-number benchmark for the combined effect of latency, transients, and the small bias–probe misalignment, so future implementations can use the measured rigidity width to diagnose control fidelity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper reports an experimental implementation of measurement-based feedback emulation of nonlinear collective-spin Hamiltonians in a cold atomic ensemble of cesium atoms. The protocol uses Faraday QND measurements of Jz and an FPGA feedback loop whose control voltage is proportional to the measured Jz, replacing the quadratic Jz^2 terms of the target Hamiltonians by their mean-field form. The reported results are: LMG symmetry breaking at an unstable fixed point for s > 0.5 with final trajectories near predicted fixed points; LMG order parameters Z_infinity and C_zz^infinity showing a transition near s = 2/3; kicked-top chaos characterized by Lyapunov-exponent estimates from only the first five stroboscopic steps; and a Floquet time-crystal phase whose rigidity window is roughly one third of the simulated width. The manuscript is candid that the energy-conserving LMG evolution is dissipative in the data, that this decay is attributed to control-loop latency with alpha*tau_delay ≈ 0.23, and that a later-discovered probe/bias misalignment corrupts the Jz measurement.

Significance. Demonstrating programmable emulation of nonlinear spin Hamiltonians in a mesoscopic atomic spin would be a useful addition to quantum-simulation toolkits, especially because the atom number can in principle be varied into the quantum regime. The manuscript's strengths are its new experimental data, its use of externally well-studied target models (LMG, kicked top), the consistency of fixed-point locations and the s=2/3 crossing with mean-field theory, and its unusually frank treatment of failure modes such as latency, measurement transients, and probe/bias misalignment. The semiclassical simulations and detailed apparatus documentation are also valuable. The claim is not yet quantitatively established: the operating point violates the paper's own small-latency criterion by an order-one margin, no quantitative fit of the latency/noise model to the data is presented, and the Lyapunov and time-crystal analyses rest on very short or indirectly related observables. As it stands, the paper is a promising experimental demonstration with qualitative agreement, not a closed verification that the emulated dynamics are those of the target Hamiltonians.

major comments (3)
  1. [§6.1.3, Eq. (6.5)] The paper's own latency criterion is violated by the operating point. Equation (6.5) requires alpha*tau_delay ≪ 1, but the reported alpha = 2π × 6.25 kHz and tau_delay ≈ 6 µs give alpha*tau_delay ≈ 0.23. Section 6.1.3 identifies this same latency as the primary driver of the dissipative decay toward fixed points seen in Figs. 6.1 and 6.5, and Fig. 6.8 provides only a qualitative semiclassical simulation. Because the control error is a first-order effect, the symmetry-breaking and phase-transition signatures in Figs. 6.1, 6.5, and 6.6–6.7 could, for all the paper currently shows, be properties of a delayed, dissipative feedback loop rather than of the LMG Hamiltonian. The manuscript should fit the latency/noise model to the experimental time series, report residuals and uncertainties, and either test the predicted dependence on alpha or explicitly show that the observed decay is quantitatively reproduced for multiple operating points.
  2. [§6.2.1, Eq. (6.9)] The Lyapunov-exponent estimator is not validated and is too indirect to support the chaos claim as it stands. Equation (6.9) equates the standard deviation of the measured elevation angle across experimental shots with an exponentially growing separation of neighboring trajectories, then fits this quantity using only the first five stroboscopic steps. Since only Jz is measured and the azimuthal degree of freedom is unobserved, the estimator is at best a heuristic lower bound. No synthetic-data test is given for the same step count, measurement axis, and noise level, so the reported values cannot be distinguished from projection effects or finite-ensemble fluctuations. The text itself notes that standard time-series methods require several hundred steps, which further indicates that this estimator needs separate numerical validation before the agreement shown in Fig. 6.11(b) can be assessed.
  3. [§6.2.2 and Chapter 7] The time-crystal rigidity result is a factor of three narrower than the simulation and is not modeled quantitatively. The experiment shows subharmonic rigidity only out to approximately (1 ± 0.07)π in alpha, whereas Fig. 6.13(a) shows the simulated phase extending to roughly (1 ± 0.2)π; Chapter 7 attributes the discrepancy to latency combined with signal transients and also admits that a tenth-of-a-degree bias/probe misalignment corrupts Jz and therefore the feedback strength. Because the time-crystal phase is one of the headline results of the abstract, the manuscript should either include a quantitative error model that reproduces the observed narrowing, or explicitly present this as a qualitative observation and weaken the corresponding claim. A quantitative statement of how the misalignment biases Jz and the feedback gain is also needed, since it affects all of the LMG and QKT results.
minor comments (5)
  1. [Fig. 6.3] The x-axis is labeled 'Time [s]' although the data span about 1.5 ms; the units appear to be microseconds, not seconds.
  2. [Chapter 7] The summary says the Floquet time-crystal state was observed with 'α = π/2', but Section 6.2.2 states that the FTC phase occurs for α nominally equal to π; one of these statements is a typo.
  3. [§2.1, §2.4] The collective spin is denoted Fz in Eqs. (2.4)–(2.5) and Jz elsewhere; the relationship between these notations should be defined once.
  4. [Throughout] There are several typographical errors (e.g., 'feedforw ard', 'Pad` e', 'inital') that should be corrected in a journal version.
  5. [Chapter 6] A reproducibility statement with raw data and analysis code would strengthen the experimental claims substantially, since the LabVIEW block diagrams alone do not allow quantitative reanalysis.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the feedback control law is an explicit mean-field linearization of the target Hamiltonians, and the experimental comparisons are unfitted, independently measured data.

full rationale

The derivation chain is self-contained. The LMG (Eq. 2.13) and Kicked Top (Eq. 2.17) Hamiltonians are external, well-known models. The feedback unitary (Eq. 2.26) is obtained by an explicit mean-field replacement Jz^2 to <Jz>Jz (Eq. 2.25), and the control voltage (Eq. 5.3) is that unitary translated into calibrated hardware gains. The predicted fixed-point locations (Eq. 2.16), the critical point s=2/3, the Lyapunov exponents, and the time-crystal rigidity are all computed from the target Hamiltonians, not extracted from the data by fitting. The experimental results are new measurements (300 shots per run, unfitted order parameters, fits to raw trajectory spreads for Lyapunov exponents, power spectra), and the paper reports deviations from theory (latency-induced dynamical decay, a roughly three-times narrower time-crystal rigidity window) rather than adjusting theory to match. References [8], [13], and [59] are same-group theoretical papers used as background and comparison, but they are published, parameter-free or independently simulated results with stated assumptions, and the current experiment provides independent falsifiable evidence; the citations are not used to forbid alternatives or to define the target result into existence. The central claim is therefore not reduced to its inputs by construction. A score of 2 reflects the presence of same-group citations in the background theory, not a circular reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters are used to define the central claims; control parameters s, k, and alpha are set from desired rotation strengths. The main assumptions are the mean-field linearization, the QND validity of the Faraday measurement, and the latency/noise model. No new entities are postulated.

assumptions (4)
  • domain assumption Mean-field linearization Jz^2 is replaced by <Jz> Jz, and the measurement outcome m is used as <Jz>.
    Eqs. 2.25-2.26. Valid only for large N and weak measurement backaction; the paper operates at N_eff of about 1e6.
  • domain assumption The Faraday probe provides a QND measurement of Jz with negligible degradation from light shifts.
    Sections 2.5 and 3.5. Relies on low photon scattering and the two-color probe cancellation of the tensor light shift.
  • domain assumption Feedback loop latency and sample rate are small enough that piecewise-constant controls approximate continuous Hamiltonian evolution.
    Sections 5.4 and 6.1.3. The paper finds alpha*tau_delay = 0.23, which is not much smaller than unity, and this is the stated cause of dissipative decay.
  • domain assumption The numerical simulations in Figure 6.8 correctly model the noise and latency budget of the experiment.
    Section 6.1.3. These simulations are used to attribute dynamical decay to latency rather than to an experimental or theoretical error.

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Cite this review

Pith. "Pith review of Physical Emulation of Nonlinear Spin System Hamiltonians via Closed Loop Feedforward Control of a Collective Atomic Spin." pith.science (2026). https://pith.science/paper/V2IX7QZO

@misc{pith2026250722132,
  author       = {Pith},
  title        = {Pith review of: Physical Emulation of Nonlinear Spin System Hamiltonians via Closed Loop Feedforward Control of a Collective Atomic Spin},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V2IX7QZO}},
  note         = {Machine review of arXiv:2507.22132}
}
read the original abstract

In recent decades the field of quantum computation has seen remarkable development. While much progress has been made toward the realization of a fully digital, scalable, and fault tolerant quantum computer, there are still many essential challenges to overcome. In the interim, direct emulation of quantum systems of interest can fill an important gap not only for exploring fundamental questions about many-body physics and the quantum to classical transition, but also for potentially providing alternative methods to verify results from quantum simulations. In this work we will demonstrate a method utilizing closed loop control of the collective magnetic moment of an ensemble of cold neutral atoms via non-destructive measurements to emulate various spin system Hamiltonians. By modifying the feedback control law appropriately we are able to generate nonlinear dynamical behavior in the ensemble, allowing us to explore the physics of collective spin systems at mesoscopic scales. Moreover, controlling the number of atoms in the collective spin can potentially allow us to investigate these dynamics in the transition from fully quantum to the classical limit. In particular, we emulate two models: the Lipkin-Meshkov-Glick (LMG) Hamiltonian, and a closely related model, the Kicked Top. In the former case, we show that our system undergoes a symmetry-breaking phase transition in the expected parameter regime. In the latter, we explore two interesting aspects: the formation of chaos, and a dynamically driven time crystal phase. We will then discuss the advantages and limits of this approach.

Figures

Figures reproduced from arXiv: 2507.22132 by the authors.

Figure 2.1
Figure 2.1. (a) The hyperfine structure of the ground and 1st two excited states of [PITH_FULL_IMAGE:figures/full_fig_p017_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Phase space portraits of the LMG model for several values of [PITH_FULL_IMAGE:figures/full_fig_p023_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Classical energy for the LMG model along ( [PITH_FULL_IMAGE:figures/full_fig_p025_2_3.png] view at source ↗
Figures from the paper (31 more)
Figure 2.4
Figure 2.4. Figure 2.4: Phase space portraits of the KT model with [PITH_FULL_IMAGE:figures/full_fig_p027_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: Schematic of polarimeter used to measure the [PITH_FULL_IMAGE:figures/full_fig_p033_2_5.png]
Figure 3.1
Figure 3.1. Figure 3.1: Relative sizes of the dipole trap, probe, and trapped atom cloud (blue). [PITH_FULL_IMAGE:figures/full_fig_p042_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: Examples of TOF measurements of atom cloud after being dropped [PITH_FULL_IMAGE:figures/full_fig_p045_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Average of 20 images of the atom cloud held by the dipole trap. The [PITH_FULL_IMAGE:figures/full_fig_p047_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: (a) Measurement of the lifetime of atoms held in the dipole trap. By [PITH_FULL_IMAGE:figures/full_fig_p048_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: Example of a polarimetry signal when the collective spin is up along [PITH_FULL_IMAGE:figures/full_fig_p051_3_5.png]
Figure 3.6
Figure 3.6. Figure 3.6: Relative to the baseline decay of the spin when it is up along ˆz [PITH_FULL_IMAGE:figures/full_fig_p056_3_6.png]
Figure 4.1
Figure 4.1. Figure 4.1: µW drive generation connection diagram. A Valon 5015 frequency syn￾thesizer supplies the 9.2 GHz carrier wave, which is modulated by an arbitrary wave￾form generator. In configuration (a), used for spectroscopy, The DS345 generates the modulation frequency, which is …
Figure 4.2
Figure 4.2. Figure 4.2: Diagram of the physical arrangement of the magnetic field coils used [PITH_FULL_IMAGE:figures/full_fig_p062_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: (a) Measured and expected attenuation from the mu-metal and alu [PITH_FULL_IMAGE:figures/full_fig_p066_4_3.png]
Figure 4.4
Figure 4.4. Figure 4.4: Polarimetry measurement variance scaling with averaging window du [PITH_FULL_IMAGE:figures/full_fig_p067_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Variation of total noise in polarimetry measurement with the second [PITH_FULL_IMAGE:figures/full_fig_p069_4_5.png]
Figure 4.6
Figure 4.6. Figure 4.6: Polarimetry measurement variance scaling with [PITH_FULL_IMAGE:figures/full_fig_p071_4_6.png]
Figure 5.1
Figure 5.1. Figure 5.1: General diagram of a feedback loop. The system state, [PITH_FULL_IMAGE:figures/full_fig_p075_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: Detailed diagram of the control loop for the feedforward emulation ex [PITH_FULL_IMAGE:figures/full_fig_p077_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: Waveform timing for feedforward emulation procedure. At t = -0.5 ms, [PITH_FULL_IMAGE:figures/full_fig_p080_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: Rotation pulse timing for QKT emulation program. Two internal digital [PITH_FULL_IMAGE:figures/full_fig_p086_5_4.png]
Figure 6.1
Figure 6.1. Figure 6.1: LMG emulation experimental data with initial state spin up along ˆx [PITH_FULL_IMAGE:figures/full_fig_p091_6_1.png]
Figure 6.2
Figure 6.2. Figure 6.2: (a) Final value of Jz on a shot-to-shot basis for s = 0.7 in LMG emulation experiment with inital state up along ˆx. (b) Correlation plot between the initial (x-axis) and final (y-axis) values of Z. The choice of the upper or lower well is clearly random, and the cor…
Figure 6.3
Figure 6.3. Figure 6.3: Some example trajectories (blue) for s = 0.7 with the midpoints (tDD , Z(tDD)) marked (red points). Also shown are the upper and lower bounds corresponding to the expected mean spin J(t) (black) and the estimated location of the stable fixed points (red lines). -0.02…
Figure 6.4
Figure 6.4. Figure 6.4: Correlation plot between the initial values of [PITH_FULL_IMAGE:figures/full_fig_p093_6_4.png]
Figure 6.5
Figure 6.5. Figure 6.5: LMG emulation experimental data with initial state spin down along [PITH_FULL_IMAGE:figures/full_fig_p095_6_5.png]
Figure 6.6
Figure 6.6. Figure 6.6: Dynamical phase transition comparison between experimental data [PITH_FULL_IMAGE:figures/full_fig_p097_6_6.png]
Figure 6.7
Figure 6.7. Figure 6.7: Experimental data for LMG dynamical phase transition experiment. [PITH_FULL_IMAGE:figures/full_fig_p098_6_7.png]
Figure 6.8
Figure 6.8. Figure 6.8: Numerical simulations comparing the effects of latency and noise in [PITH_FULL_IMAGE:figures/full_fig_p101_6_8.png]
Figure 6.9
Figure 6.9. Figure 6.9: Kicked Top phase space portraits for α = π/2 and k = 2.5. (Right) Regular trajectories are colored red and chaotic motion is blue, with the threshold being λmax = 0.01. (Left) Trajectories are colored by their local maximal Lyapunov exponent. The largest of these eig…
Figure 6.10
Figure 6.10. Figure 6.10: Experimental data for KT emulation with α = π/2. Phase space portraits show the x > 0 hemisphere and are colored according the the local mLCE using the same color scale as in [PITH_FULL_IMAGE:figures/full_fig_p107_6_10.png]
Figure 6.11
Figure 6.11. Figure 6.11: Statistical estimation of the KT mLCE for [PITH_FULL_IMAGE:figures/full_fig_p109_6_11.png]
Figure 6.12
Figure 6.12. Figure 6.12: Power spectral density of Jz as a function of the linear rotation strength α and nonlinear strength k for a state initially up along ˆz in the Kicked Top model. The lighter regions correspond to a Floquet time crystal phase. The expansion of the FTC phase region as …
Figure 6.13
Figure 6.13. Figure 6.13: KT time crystal phase power spectral density as a function of the [PITH_FULL_IMAGE:figures/full_fig_p113_6_13.png]

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