Probing excited-state quantum phase transitions with trapped cold ions

Marek Kucha\v{r}; Michal Macek

arxiv: 2603.28509 · v2 · submitted 2026-03-30 · 🪐 quant-ph · cond-mat.stat-mech· nucl-th

Probing excited-state quantum phase transitions with trapped cold ions

Marek Kucha\v{r} , Michal Macek This is my paper

Pith reviewed 2026-05-14 21:39 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechnucl-th

keywords excited-state quantum phase transitionstrapped ionsExtended Rabi Modelquantum criticalitysuperradiant phasewitness observablesPaul trapqubit-phonon coupling

0 comments

The pith

Concrete protocols allow a single trapped ion to probe excited-state quantum phase transitions by varying its qubit-phonon coupling.

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

The paper proposes using a single ion in a radio-frequency Paul trap to experimentally realize quantum criticality from excited-state quantum phase transitions in the Extended Rabi Model. It identifies a class of excited states between two critical ESQPT energies in the superradiant phase and defines witness observables based on their properties. These observables are studied for critical scaling and state evolution when the coupling strength is changed linearly over time at different rates. The theoretical model parameters are mapped to experimental trapped-ion settings, with simulations including open-system effects for realistic setups.

Core claim

A specific class of excited states in the Extended Rabi Model occurring between two critical ESQPT energies in its (anti)Jaynes-Cummings superradiant phase can be used to define ESQPT witness observables whose critical scaling behaviors can be observed by driving the qubit-phonon coupling strength across the critical points in a trapped ion experiment.

What carries the argument

The Extended Rabi Model Hamiltonian and its excited states in the superradiant phase, which motivate the definition of ESQPT witness observables that exhibit critical scaling when the system is driven across quantum critical points.

If this is right

The witness observables display distinct critical scaling near the ESQPT points.
Driving the coupling strength linearly at finite rates produces different state evolutions across the critical energies.
The ERM control parameters map directly to experimental parameters in trapped ion setups.
Unitary evolutions and open-system corrections can both be simulated for state-of-the-art ion traps.

Where Pith is reading between the lines

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

Implementing these protocols could enable studies of quantum criticality in minimal systems without requiring large ensembles of particles.
Similar driving techniques might apply to other quantum simulators for exploring excited-state transitions.
The resilience of the single-ion setup could make ESQPTs more accessible for experimental verification than in many-body systems.

Load-bearing premise

The identified excited states in the Extended Rabi Model can be prepared, maintained, and driven across critical points in a trapped ion with enough fidelity that the witness observables stay measurable before decoherence takes over.

What would settle it

Measuring whether the proposed witness observables exhibit the predicted critical scaling behaviors when the qubit-phonon coupling is varied across the critical values in a laboratory trapped-ion experiment.

Figures

Figures reproduced from arXiv: 2603.28509 by Marek Kucha\v{r}, Michal Macek.

**Figure 1.** Figure 1: FIG. 1: (a) Different qubit-phonon interaction phases of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Level dynamics as in Fig. 2 (b), but color-coded [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Expectation values of the phonon-number operator [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Strength functions, Eq. (16), in instantaneous quench [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Same as in Fig. 6 but with the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Rabi oscillations numerically obtained by evolv [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: Vacuum populations of the down-projected states [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10: Same as in Fig. 9, but for fixed [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11: Color-coded map of the inner and outer phase-space [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗

read the original abstract

We propose concrete protocols to realize quantum criticality due to excited-state quantum phase transitions (ESQPTs) experimentally in presumably the simplest and most resilient system involving a single trapped ion oscillating in a radio-frequency Paul trap. We identify a specific class of excited states of the Extended Rabi Model (ERM) Hamiltonian, which occur between two critical ESQPT energies of the model in its (anti)Jaynes-Cummings superradiant phase. Properties of these states motivate the definition of several ESQPT witness observables. We study their critical scaling behaviors as well as various distinct state evolutions by driving the system across the quantum criticalities by changing the qubit-phonon coupling strength linearly in time at different finite rates. A mapping of the theoretical control parameters of the ERM to the experimental parameters of a trapped ion setup is provided, and simulations are performed for values referencing existing state-of-the-art setups, addressing both unitary state evolutions as well as relevant open-system corrections.

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.

Desk Editor's Note private letter to a colleague

The paper gives a workable experimental proposal for ESQPTs in a single trapped ion with parameter mapping and open-system simulations, though the survival of critical signatures under decoherence is the key thing to check.

read the letter

The main point of this paper is a proposal to observe excited-state quantum phase transitions using a single ion in a Paul trap. They focus on the extended Rabi model, pick out a class of excited states between the critical energies in the superradiant phase, and suggest driving the system linearly across those points while tracking witness observables. What is new here is the detailed mapping of the model parameters to actual trapped-ion controls like the qubit-phonon coupling strength, and the specific protocols for finite-rate ramps at different speeds. They also define witnesses based on the state properties and run simulations that include open-system effects for realistic parameters from current setups. This turns the ESQPT concept into something that could be tested without needing a large many-body system. The paper does a reasonable job making the case for this minimal platform. The choice of a single ion keeps things simple and resilient, and addressing decoherence channels like motional heating and qubit relaxation shows they are thinking about real experiments. The unitary evolutions and the open-system corrections are both considered, which is better than many pure theory papers. The soft spot is around whether the critical signatures actually survive in practice. The stress-test concern about decoherence during the driving is relevant because if the Lindblad terms wash out the non-analytic features or the scaling of the witnesses on the timescales needed to cross the points, then the protocols won't work as hoped. The abstract says they perform simulations for state-of-the-art values, but I would need to see the specific results on how much the witnesses deviate and whether the scaling remains detectable. If the gaps are small and ramps not slow enough, that could be a problem. Also, preparing and maintaining those specific excited states with high fidelity might be trickier than assumed. This kind of work is for experimentalists in ion trapping who want to explore quantum criticality in controllable systems, and for theorists who study ESQPTs and need benchmarks. A reader interested in quantum simulation proposals would get concrete ideas from the parameter mapping and the driving protocols. It deserves a serious referee. The idea is coherent, the model is established, and the simulations add credibility even if some aspects need more detail in revision.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes concrete protocols to realize and probe excited-state quantum phase transitions (ESQPTs) in a single trapped ion in a radio-frequency Paul trap by mapping the Extended Rabi Model (ERM) to the experimental system. It identifies a specific class of excited states between two critical ESQPT energies in the (anti)Jaynes-Cummings superradiant phase, defines several witness observables, examines their critical scaling and state evolution under linear-in-time ramps of the qubit-phonon coupling at varying finite rates, provides an explicit mapping of ERM parameters to trapped-ion controls, and reports both unitary and open-system simulations using state-of-the-art Paul-trap parameters.

Significance. If the simulations confirm that the proposed witness observables retain measurable critical scaling signatures under realistic decoherence, the work would establish a minimal and resilient platform for experimental ESQPT studies, enabling direct tests of non-analytic features in driven open quantum systems with current trapped-ion technology and potentially informing protocols for other criticality phenomena.

major comments (2)

[Simulations section (open-system analysis)] Simulations section (open-system analysis): The central claim that ESQPT witness observables remain measurable requires that decoherence (motional heating, qubit relaxation, dephasing) does not wash out non-analytic scaling on the timescales of the linear ramps; the abstract references such simulations but provides no explicit Lindblad operators, quantitative ratios of ramp rate to decoherence rates, or scaling plots demonstrating survival of the signatures, leaving the load-bearing assumption unverified.
[Mapping and state-preparation paragraph] Mapping and state-preparation paragraph: The identification of the specific class of excited states between the two critical ESQPT energies and the protocol for their preparation and maintenance must be shown to be compatible with the finite ramp rates; without error analysis or fidelity estimates for state preparation in the presence of the trap parameters, the feasibility of crossing the critical points with sufficient coherence cannot be assessed.

minor comments (1)

[Abstract] Abstract: The phrase 'presumably the simplest and most resilient system' would benefit from a one-sentence comparison to other platforms (e.g., cavity QED or superconducting circuits) to clarify the advantage.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive comments that will help improve the clarity and completeness of our work. We address the major comments below.

read point-by-point responses

Referee: Simulations section (open-system analysis): The central claim that ESQPT witness observables remain measurable requires that decoherence (motional heating, qubit relaxation, dephasing) does not wash out non-analytic scaling on the timescales of the linear ramps; the abstract references such simulations but provides no explicit Lindblad operators, quantitative ratios of ramp rate to decoherence rates, or scaling plots demonstrating survival of the signatures, leaving the load-bearing assumption unverified.

Authors: We appreciate the referee highlighting the need for more explicit details in the open-system analysis. Our simulations do incorporate open-system effects using a Lindblad master equation with operators corresponding to motional heating, qubit relaxation, and dephasing, employing parameters from state-of-the-art trapped-ion experiments. To address this comment, we will revise the simulations section to explicitly list the Lindblad operators, provide quantitative ratios between the ramp rates and decoherence rates, and include additional scaling plots that demonstrate the persistence of the non-analytic signatures under realistic decoherence levels. This will strengthen the verification of our central claim. revision: yes
Referee: Mapping and state-preparation paragraph: The identification of the specific class of excited states between the two critical ESQPT energies and the protocol for their preparation and maintenance must be shown to be compatible with the finite ramp rates; without error analysis or fidelity estimates for state preparation in the presence of the trap parameters, the feasibility of crossing the critical points with sufficient coherence cannot be assessed.

Authors: We agree that a more detailed analysis of state preparation is warranted to fully assess feasibility. In the revised manuscript, we will expand the relevant paragraph to include error analysis and fidelity estimates for preparing the identified class of excited states, taking into account the finite ramp rates and the specific trap parameters. This will provide a clearer demonstration of the compatibility with experimental constraints and the maintenance of coherence during the crossing of critical points. revision: yes

Circularity Check

0 steps flagged

No circularity: proposal maps ERM to ion trap via independent definitions and simulations

full rationale

The paper defines witness observables from properties of identified excited states in the Extended Rabi Model, studies their scaling under linear driving, and maps parameters to Paul-trap experiments with open-system Lindblad simulations using standard decoherence rates. None of these steps reduce by construction to fitted inputs or self-citations; the central claims rest on explicit Hamiltonian evolution and numerical integration rather than renaming or re-deriving quantities from the same data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the Extended Rabi Model faithfully captures the trapped-ion physics and that the identified excited states remain accessible under realistic driving and decoherence.

axioms (1)

domain assumption The Extended Rabi Model Hamiltonian accurately describes the qubit-phonon dynamics of a single trapped ion in a Paul trap.
Invoked when mapping theoretical parameters to experimental controls and when claiming the states can be realized.

pith-pipeline@v0.9.0 · 5470 in / 1306 out tokens · 51169 ms · 2026-05-14T21:39:40.958134+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We identify a specific class of excited states of the Extended Rabi Model (ERM) Hamiltonian... Properties of these states motivate the definition of several ESQPT witness observables... linear-interaction-ramp driving protocols
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

semiclassical Hamiltonian h(x',p',m') = x'^2 + p'^2/2 + m' p^2 λ^2 (x'^2 + δ^2 p'^2) + 1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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By choosing the non-interacting ground state, we do not imply that it is optimal for application of all the discussed critical phenomena in quantum state manipulation, how- ever, it is definitely the best state for acquiring a proof of concept experimentally due to its enhanced dephasing 15 resilience, e.g. compared to higher Fock states. Detailed optimiz...

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P. Cejnar, P. Str´ ansk´ y, M. Macek and M. Kloc, J. Phys. A: Math. Theor. 54, 133001 (2021)

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Cejnar, M

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J., Puebla, R., & Plenio, M

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Peres, Phys

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I. Pietik¨ ainen, et al., PRX Quantum5, 040307, (2024)

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Mølmer, Y

K. Mølmer, Y. Castin, and J. Dalibard, JOSA B,10, 524-538, (1993)

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Macek and A

M. Macek and A. Leviatan, Ann. Phys.351, 302 (2014)

work page 2014

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Ch´ avez-Carlos et al., npj Quantum Information9, 76 (2023)

J. Ch´ avez-Carlos et al., npj Quantum Information9, 76 (2023)

work page 2023

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Iachello et al.,J

F. Iachello et al.,J. Phys. A: Math. Theor.56495305 (2023)

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Iachello et al., J

F. Iachello et al., J. Phys. A: Math. Theor.57415302 (2024)

work page 2024

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Obˇ sil, et al., Rev

P. Obˇ sil, et al., Rev. Sci. Instrum.90, 083201 (2019)

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The S2 and S2’ excited phases for|δ| ̸= 1 in fact display remnants of U(1) symmetry of the (a)JC Hamiltonians, endowed with the unusual kinetic termH∝ −p 2 locally aroundx=p= 0 related to the (locally) Mexican-hat- form of the semiclassical (a)JC Hamiltonians of Eq. (11). A more rigorous interpretation possibly in terms of quasi- and partial dynamical sym...

work page

[36] [36]

compared to higher Fock states

By choosing the non-interacting ground state, we do not imply that it is optimal for application of all the discussed critical phenomena in quantum state manipulation, how- ever, it is definitely the best state for acquiring a proof of concept experimentally due to its enhanced dephasing 15 resilience, e.g. compared to higher Fock states. Detailed optimiz...

work page