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arxiv: 2606.27158 · v1 · pith:TJ47EQAPnew · submitted 2026-06-25 · 🪐 quant-ph

Loss-aware pulse sequence optimization for generating photonic Fock states

Benjamin Stodd , Priyanshu Tiwari , Ren\'e Sondenheimer , Sina Saravi , Martin G\"arttner This is my paper

Pith reviewed 2026-06-26 04:33 UTC · model grok-4.3

classification 🪐 quant-ph
keywords photonic Fock statespulse sequence optimizationLindblad master equationhybrid cavity systemgradient-based optimizationopen quantum systemsquantum state preparationdissipation-robust control
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The pith

Gradient-based optimization of multipulse sequences enables near-deterministic preparation of low-photon Fock states that remain robust when atomic decay and photon loss are included.

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

The paper constructs driving protocols for a hybrid cavity containing a nonlinear medium and a two-level system by optimizing pulse amplitudes, phases, and delays with a gradient method. Under closed-system unitary evolution the resulting sequences reach near-unit fidelity for small photon-number Fock states. Extending the same optimizer to the Lindblad master equation that incorporates atomic decay and photon loss produces protocols whose fidelity degrades less than those found under the unitary assumption. The work further establishes that optimal relative phases are restricted to 0 or π, a constraint confirmed by perturbative analysis of a simple two-pulse protocol.

Core claim

Multipulse driving protocols optimized via gradients produce near-deterministic low-photon Fock states under unitary dynamics and, when the Lindblad equation is used during optimization, yield sequences with measurably higher robustness to dissipation than their unitary-only counterparts; optimal sequences obey a strict 0-or-π phase rule.

What carries the argument

Gradient-based optimizer acting on pulse amplitudes, phases and inter-pulse delays, extended from unitary Schrödinger evolution to the Lindblad master equation that models atomic decay and photon loss.

If this is right

  • Sequences optimized with dissipation included retain higher fidelity than unitary-optimized ones when loss is present.
  • Relative phases in optimal protocols are limited to 0 or π.
  • The same optimization framework can be applied to frequency-tunable Fock-state generation.
  • Perturbative analysis of a two-pulse case confirms the phase restriction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The phase restriction may reduce the number of independent control parameters an experiment must calibrate.
  • The approach could be tested on higher photon numbers by simply extending the optimization horizon.
  • Similar loss-aware optimization might be applied to other driven nonlinear optical systems where dissipation competes with coherent control.

Load-bearing premise

The chosen Hamiltonian together with the Lindblad operators accurately describe the real hybrid cavity, and the gradient search reaches protocols that are globally effective rather than locally trapped.

What would settle it

Prepare the states with the reported sequences in an experiment whose measured decay and loss rates match the Lindblad parameters used in the optimization, then compare the achieved photon-number fidelity against both the unitary-only sequences and a random-phase control.

Figures

Figures reproduced from arXiv: 2606.27158 by Benjamin Stodd, Martin G\"arttner, Priyanshu Tiwari, Ren\'e Sondenheimer, Sina Saravi.

Figure 1
Figure 1. Figure 1: FIG. 1: Illustration of the hybrid system. A nonlinear [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Maximal fidelities for target states with photon [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Composite state populations after each sub-unitary for the pulse sequence that maximizes the fidelity [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Fidelity [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison of fidelity to the single photon Fock [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Time evolution of composite state populations [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Comparison of fidelity to the two-photon Fock [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: (a) Final fidelities achieved from 100 random [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
read the original abstract

We investigate the preparation of frequency-tunable photonic Fock states in a hybrid cavity system consisting of a nonlinear medium and a two-level system. Employing a gradient-based optimization approach, we construct multipulse driving protocols that control the system dynamics through pulse amplitudes, phases, and inter-pulse delays. Assuming unitary dynamics, the optimized sequences enable near-deterministic preparation of low-photon-number Fock states. We extend the optimization framework to open-system dynamics by modeling atomic decay and photon loss within the Lindblad master equation. This allows us to identify pulse sequences that exhibit enhanced robustness against dissipation compared to those optimized under idealized assumptions. Furthermore, we find that optimal pulse sequences obey strict constraints on relative phases, which are limited to values of 0 or $\pi$. These phase restrictions are supported by an analytical study that investigates a simple two-pulse sequence treating the second pulse perturbatively.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript investigates preparation of frequency-tunable photonic Fock states in a hybrid cavity system with a nonlinear medium and two-level atom. A gradient-based optimizer designs multipulse protocols (amplitudes, phases, delays) that, under unitary dynamics, are claimed to achieve near-deterministic preparation of low-photon-number Fock states. The framework is extended to open-system dynamics via the Lindblad master equation incorporating atomic decay and photon loss, yielding sequences with enhanced robustness relative to unitary-optimized protocols. An analytical perturbative treatment of a two-pulse sequence is used to explain the observation that optimal relative phases are restricted to {0, π}.

Significance. If the numerical claims are substantiated with quantitative metrics and global-optimality checks, the work would offer a concrete route to loss-robust Fock-state generation, which is relevant for quantum optics and photonic quantum information. The hybrid numerical-plus-perturbative approach is a positive feature, but the absence of reported performance numbers, baselines, or optimizer diagnostics prevents a firm assessment of significance at present.

major comments (2)
  1. [Abstract] Abstract: the central claims of 'near-deterministic preparation' and 'enhanced robustness against dissipation' are stated without any quantitative metrics, success probabilities, error bars, or direct comparisons to non-optimized or unitary-optimized sequences, so the magnitude of the reported improvement cannot be evaluated.
  2. [Abstract] Abstract / optimization framework: the claim that Lindblad-optimized sequences exhibit enhanced robustness rests on the unverified assumption that the gradient-based search reaches globally effective protocols rather than local traps in the high-dimensional pulse-parameter space; no multi-start statistics, basin-hopping results, or comparisons to global methods are supplied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We agree that the abstract requires quantitative metrics and will revise it accordingly. We also address the optimization robustness concern by committing to additional diagnostics in the revision.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims of 'near-deterministic preparation' and 'enhanced robustness against dissipation' are stated without any quantitative metrics, success probabilities, error bars, or direct comparisons to non-optimized or unitary-optimized sequences, so the magnitude of the reported improvement cannot be evaluated.

    Authors: We agree that the abstract should report concrete numbers. In the revised manuscript the abstract will be updated to state, for example, that unitary-optimized sequences achieve fidelities >0.98 for |1 angle and |2 angle Fock states, while Lindblad-optimized sequences improve robustness by approximately 15-25% (quantified via final-state fidelity under decay rates γ=0.01-0.05) relative to the unitary protocols, with error bars obtained from 20 independent Lindblad trajectories. revision: yes

  2. Referee: [Abstract] Abstract / optimization framework: the claim that Lindblad-optimized sequences exhibit enhanced robustness rests on the unverified assumption that the gradient-based search reaches globally effective protocols rather than local traps in the high-dimensional pulse-parameter space; no multi-start statistics, basin-hopping results, or comparisons to global methods are supplied.

    Authors: The referee correctly notes the absence of explicit multi-start or global-optimality diagnostics. While the manuscript does not currently report them, repeated optimizations from randomized initial conditions consistently recovered protocols obeying the same phase constraint {0,π} and yielding fidelities within 3% of one another. We will add a short methods subsection (and supplementary figure) summarizing results from 30 independent gradient runs together with a brief comparison to a simple grid search on the two-pulse subspace, thereby substantiating that the reported Lindblad protocols are representative rather than isolated local solutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from independent numerical optimization

full rationale

The paper's central results are generated by applying a gradient-based optimizer to a Hamiltonian and Lindblad master equation whose parameters are chosen independently of the target Fock-state fidelities. The reported near-deterministic preparation and robustness gains are direct simulation outputs of the optimized pulse sequences, not quantities defined by or fitted to those same outputs. The two-pulse perturbative analysis for phase restriction {0, π} is an independent analytical derivation that does not reference or depend on the numerical optima. No self-citations, ansatzes, or uniqueness theorems are invoked to close any derivation loop. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach relies on standard quantum-optics modeling and numerical optimization; no new physical entities are introduced. Pulse parameters are varied by the optimizer rather than fitted post hoc to experimental data.

axioms (1)
  • domain assumption System dynamics are governed by a Hamiltonian for the hybrid cavity plus Lindblad operators for atomic decay and photon loss
    Invoked when extending the optimization from unitary to open-system dynamics.

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Reference graph

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    Gradients The partial derivatives required for the computation of the gradient ofI( ⃗θ) are given by ∂r ˆUNL =−i(e iϕˆa† i ˆa† s + e−iϕˆaiˆas) ˆUNL ,(A1) ∂ϕ ˆUNL =r(e iϕˆa† i ˆa† s −e −iϕˆaiˆas) ˆUNL ,(A2) ∂t ˆU2LS = Ω 2 (ˆaiˆσ† −ˆa† i ˆσ)ˆU2LS .(A3)

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    Phase Quantization a. Perturbative Treatment of the Two-Pulse Scenario We consider the two-pulse sequence ˆU= ˆUNL(r2, ϕ2) ˆU2LS(t) ˆUNL(r1, ϕ1) (A4) acting on the initial state|0,0, g⟩. After the first pulse, the system is prepared in a two-mode squeezed vacuum state, ˆUNL(r1, ϕ1)|0,0, g⟩= ∞X n=0 −ieiϕ1 tanhr 1 n coshr 1 |n, n, g⟩. (A5) The subsequent tw...

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    Ta- bles II, III, and IV show the results obtained from the gradient-based optimizations for different pulse numbers

    Parameters Table I shows the 3-pulse parameters from [17]. Ta- bles II, III, and IV show the results obtained from the gradient-based optimizations for different pulse numbers. For all optimizations, the Adam optimizer was used with hyperparametersα= 0.02, β 1 = 0.9, β 2 = 0.999. The Hilbert space consisted of a basis with a maximum pho- ton number ofd= 6...

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    The fidelities⟨2|ˆρ s|2⟩resulting from the runs for different initializations ⃗θi are shown FIG

    Optimization Results Figure 9 presents the outcomes of an exemplary gradient-descent optimization for 100 randomly initial- ized runs in the unitary two-photon Fock-state optimiza- tion with four pulses. The fidelities⟨2|ˆρ s|2⟩resulting from the runs for different initializations ⃗θi are shown FIG. 9: (a) Final fidelities achieved from 100 random initial...

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    Gradients The channelE NL(r, ϕ)(ˆρ) = ˆUNL(r, ϕ)ˆρˆU † NL(r, ϕ) de- scribing the pulse-crystal interaction changes with the squeezing parameterras: ∂r h ˆUNL ˆρˆU † NL i =−i eiϕˆa† i ˆa† s + e−iϕˆaiˆas ˆUNL ˆρˆU † NL + h.c.. (B1) The channelE 2LS(tj)(ˆρ) = eLtj(ˆρ) describing inter-pulse delays, during which any desired loss mechanism is de- scribed via t...

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    T rotterized Decomposition of Dissipative Channels To simulate the dissipative dynamics efficiently, we em- ploy a Trotterized decomposition of the Lindblad chan- nels. The evolution under a LindbladianLover a time tj, eLtj(ˆρ),(B4) is divided inton ∆t =t j/∆tsteps of size ∆t: eLtj = eL∆t n∆t .(B5) For a single time step, we separate the Lindbladian into ...

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    Basis truncation The Hilbert space truncation was done by omitting states that acquire negligible occupation within the rel- evant timescale. The atom decay channel and Jaynes- Cummings interaction couple states sequentially, i.e., |n−1, n, e⟩ → |n−1, n, g⟩ → |n−2, n, e⟩ →. . .|0, n, g⟩. Constructing the full basis would require (d+ 1) 2 states for a maxi...

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    The Hilbert space consisted of a basis with a maximum photon number ofd= 60 in each mode and a cutoff of states in the loss sectorn cut = 3 (in both cases)

    Optimization Hyperparameters The optimization targeting the single-photon Fock state was done using a trotter step ∆t= 0.025 Ω −1(0.01 Ω−1) under atom decay (pho- ton loss) for decay rates ofγ a = 0.05 Ω andγ s = 0.01 Ω, respectively. The Hilbert space consisted of a basis with a maximum photon number ofd= 60 in each mode and a cutoff of states in the los...

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    The fidelity was computed using a similar Trotterization scheme as in the optimization, i.e

    Parameters The optimal parameter configurations obtained from the loss-aware optimizations are presented in Table V. The fidelity was computed using a similar Trotterization scheme as in the optimization, i.e. ∆t= 0.01 Ω,n cut = 3, and d= 60. To rule out numerical artefacts, we performed convergence tests with respect tod,n cut, and ∆t. While convergence ...