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Motional states of two interacting atoms in an optical tweezer can form a qubit-oscillator module, with a universal bosonic gate set driven by stroboscopic trap modulation.

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2026-08-03 18:08 UTC pith:W2XLNCXW

load-bearing objection Genuinely new architecture, but the abstract outruns the simulations and the 1D model is pushed into a regime the paper itself rules out. the 3 major comments →

arxiv 2512.06429 v2 pith:W2XLNCXW submitted 2025-12-06 quant-ph physics.atom-ph

Hybrid qubit-oscillator module from motional states of two interacting atoms

Jaeyong Hwang , Tianrui Xu , Sean R. Muleady , Steven K. Pampel , Gur Lubin , Dawson P. Hewatt , Cindy A. Regal , Ana Maria Rey This is my paper
classification quant-ph physics.atom-ph
keywords optical tweezersmotional statesqubit-oscillator modulecontact interactionbosonic gatessqueezingquantum sensinghybrid quantum information
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proposes building a hybrid qubit-oscillator module entirely from the motion of two atoms trapped in one optical tweezer. The idea is to use the center-of-mass motion of the pair as a bosonic oscillator and the relative motion, made anharmonic by atom-atom contact interactions, as the qubit. By stroboscopically shifting and modulating the tweezer, the authors show that a universal set of bosonic gates (displacement, rotation, squeezing) and their qubit-controlled versions can be generated with high simulated fidelity, without using internal atomic states for the gates. If correct, this gives atomic tweezer arrays a route to spin-boson simulation and precision sensing that avoids spin-dependent noise.

Core claim

The central claim is that the intrinsic s-wave contact interaction between two atoms in a harmonic trap supplies the nonlinearity needed to turn the pair's relative motion into a qubit, while the center-of-mass motion acts as a harmonic oscillator, and that stroboscopic modulation of a single optical tweezer can implement displacement, rotation, squeezing, and their controlled versions. The paper derives effective gate generators from the fifth-order expansion of the time-averaged potential, numerically checks fidelities (e.g., infidelities around 10^-5 for displacement and squeezing, with all gates at about 99% or better for representative parameters), and gives state-preparation, readout,

What carries the argument

The central object is the two-body Hamiltonian split into center-of-mass and relative coordinates, with the relative coordinate subject to a contact delta-potential of strength u. That delta potential shifts the even harmonic levels of the relative motion, creating an anharmonicity that isolates the two lowest even states as the qubit. The control mechanism is stroboscopic potential painting: rapidly cycling the tweezer center through multiple beam positions produces a time-averaged potential whose low-order expansion coefficients can be set independently, and sinusoidal modulation at combinations of the oscillator frequency and the qubit splitting resonantly selects each desired gate genera

Load-bearing premise

The load-bearing premise is that the 1D delta-function contact-interaction model remains accurate at the interaction strengths used in the simulations (u/ℏω_x = 0.86 for the displacement and controlled-displacement gates), even though the paper states the 1D approximation is valid only for u much smaller than ℏω_x,y,z; no 3D benchmark is provided.

What would settle it

Run a full 3D two-atom simulation, or measure the relative-motion spectrum and controlled-displacement gate fidelity in an experiment, at the End Matter parameters (u/ℏω_x = 0.86, ω_x = 2π × 140 kHz, with the listed transverse frequencies). If the anharmonicity or gate fidelities deviate from the 1D prediction by more than the quoted infidelities, the central claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A universal set of bosonic operations (displacement, rotation, squeezing) and qubit-controlled counterparts can be generated with roughly 99% or higher simulated fidelity using only atomic motion.
  • Because no internal states are used for the gates, the scheme avoids spin-dependent noise sources such as magnetic-field fluctuations and differential light shifts.
  • The module is estimated to detect magnetic dipolar interactions with about 10 Hz sensitivity in one second and sub-Hz resolution within minutes in a 20x20 tweezer array.
  • The qubit-oscillator pair is a natural building block for spin-boson quantum simulation and hybrid discrete-continuous variable processing, with scalability envisioned through dipolar or Rydberg-dressed coupling between arrays.
  • Estimated gate times (microseconds for displacement, up to tens of milliseconds for controlled gates) are compatible with existing tweezer-array cooling and control.

Where Pith is reading between the lines

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

  • Editorial inference: if a full 3D benchmark confirms the 1D contact model at the strong interactions used in the simulations, the same stroboscopic scheme could likely be extended to even stronger interactions, yielding faster controlled gates than the estimated 0.1-10 ms.
  • Editorial inference: the readout protocol reintroduces internal atomic states via Raman transitions, so the practical immunity to spin noise applies to the gate operations themselves, not necessarily to the measurement step.
  • Editorial inference: a minimal experimental test would be to implement only the displacement gate and verify the resulting Schrodinger-cat state in the center-of-mass motion, validating the stroboscopic painting technique before attempting the more complex controlled gates.
  • Editorial inference: the same trap-modulation response could serve as a local probe of other two-body parameters, such as the s-wave scattering length or trap anharmonicity, not only magnetic dipolar interactions.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The manuscript proposes a hybrid qubit-oscillator platform in which two interacting atoms in a stroboscopically engineered optical tweezer encode a qubit in the relative motional states and a bosonic mode in the center-of-mass motion. The authors derive an effective 1D Hamiltonian, show that time-modulated trap amplitudes can generate displacement, rotation, squeezing, and qubit-controlled versions of these gates, and report numerical fidelities from time-dependent simulations. They also outline state preparation and readout and claim applicability to sensing of dipolar interactions.

Significance. If the central dynamical model is valid, this is a conceptually new architecture: an all-motional qubit-oscillator module in neutral-atom tweezer arrays, with no internal atomic states used for the gate operations. The target unitaries are defined independently of the simulation, and the gate parameters are derived from modulation amplitudes; the optimization of lambda is implementation tuning rather than fitting the answer. The paper provides explicit trap parameters, analytically derived matrix elements, and numerically optimized fidelities, which makes the protocol concrete and falsifiable. These strengths justify publication if the model-validity and overclaim issues identified below are resolved.

major comments (3)
  1. [End Matter; after Eq. (5); Eq. (S14)] The 1D contact-interaction model is used at u/hbar*omega_x = 0.86 (D, CD) and 0.36 (S, CS), while the paper states after Eq. (5) that the regime of validity is u << hbar*omega_{x,y,z}. With the End Matter trap frequencies, 0.86*hbar*omega_x = 0.62*hbar*omega_z and 0.17*hbar*omega_y, a large violation. The qubit splitting, anharmonicity, the coefficients c1-c3 in Table S1, and the simulated fidelities all derive from the 1D delta-potential spectrum. No 3D two-body benchmark or convergence check in the transverse confinement is presented. The reported >99% fidelities are therefore not yet established for the actual 3D tweezer trap; a 3D benchmark or a restriction to interaction strengths where the 1D reduction is controlled is needed.
  2. [Abstract; Fig. 3 discussion; End Matter] The abstract promises detection of magnetic dipolar interactions with ~10 Hz sensitivity in one second and sub-Hz resolution within a few minutes in a 20x20 array 'under realistic experimental imperfections'. No sensitivity analysis appears anywhere in the body or the Supplemental Material. The only quantitative performance results are gate fidelities from closed-system evolution, and the text explicitly states that external decoherence is not considered. Since the quoted gate times extend to 13 ms, where motional decoherence is non-negligible for the proposed system, the abstract's quantitative claims are unsupported. Substantiate them with a decoherence model and parameter estimates, or remove them.
  3. [Fig. 3; Quantum gate generation] The text says 'All displacement, rotation, and squeezing gates, and their controlled counterparts, show >=99% gate fidelities up to some given amplitudes', but for |alpha|=3 the CD infidelity is 1.7e-1, and the text later restricts the CD gate to |alpha| <= 1. The abstract and introduction present a universal gate set as a main result; the restricted amplitude range for the controlled displacement should be stated directly with the fidelity data, and the cost of concatenating small-CD operations for larger displacements should be discussed.
minor comments (3)
  1. [Eq. (8)] The fidelity is defined for a single input state |up>|0>, and Fig. S1 uses displaced inputs for R, SR, and CR. Please state explicitly whether the quoted numbers are state-specific fidelities or process-level fidelities; as written, 'gate fidelity' is ambiguous.
  2. [Fig. 3 table] The infidelity entry '<= 10^{-0.8}' for the CD gate is awkward; use a decimal value such as approximately 1.6e-1 for readability.
  3. [End Matter, around Eq. (8)] The term 'full 3D time-dependent Hamiltonian' in the text describing Eq. (8) is misleading, since the simulations use the 1D effective potential after transverse ground-state projection. Please reword to avoid confusion.

Circularity Check

0 steps flagged

No significant circularity: derivation is self-contained, and the λ optimization is implementation tuning rather than fitting the claimed result.

full rationale

The derivation chain is self-contained. The target unitaries (D, R, S, CD, CR, CS) are standard bosonic and controlled operations defined independently of the numerical simulation (Eq. (7), Table S1). The effective gate Hamiltonians are obtained by explicit polynomial expansion of the stroboscopically averaged Gaussian tweezer potential (Eqs. (2), (4), (S14)) and by analytic/perturbative relative-motional matrix elements from the Busch-type solution for a harmonic trap with a contact δ-interaction [36,37]—not from the gate-fidelity data. The reported fidelities are computed by full time-dependent evolution of the wave equation including all higher-order terms, with the modulation strength λ optimized as a control parameter; choosing a pulse amplitude to realize an independently specified unitary is not fitting the target into the claim. The self-citations [12,16] provide experimental context, tomography background, and prior control ideas, but none is load-bearing for the central gate construction, and no uniqueness theorem or ansatz is imported from the authors' prior work. The End-Matter use of u/ℏω_x = 0.86 despite the stated 1D validity condition u ≪ ℏω_{x,y,z} is a legitimate model-validity and correctness concern, but it does not make the derivation circular.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The ledger reflects that the central protocol depends on standard low-energy scattering theory, a time-averaged stroboscopic potential, freezing of y/z motion, and truncation of the potential expansion at fifth order. The most fragile inputs are the interaction-strength choice u/hbar*omega_x = 0.86 and the unmodeled stroboscopic/decoherence effects, neither of which is benchmarked against a full 3D simulation or experimental data.

free parameters (3)
  • interaction strength ratio u/hbar*omega_x = 0.86 for D/CD gates, 0.36 for S/CS gates
    Chosen by adjusting omega_y and omega_z to maximize anharmonicity and reduce leakage; the 0.86 value lies outside the stated u << hbar*omega_{x,y,z} validity regime.
  • modulation strength lambda = Optimized per gate and amplitude; optimal values shown in Fig. 3 inset
    Free control parameter numerically optimized to maximize gate fidelity; gate parameters alpha, gamma, xi scale with lambda*T.
  • beam center positions zeta_j and trap depths U_j(0) = zeta = {-1.14,-0.56,0.04,0.38,0.90} for jmax=5; zeta = {-0.6775,0,0.6775} for jmax=3; U(0)/V0 listed in End Matter
    Hand-chosen to minimize uncontrolled higher-order potential terms; reported gate fidelities depend on these design choices.
axioms (5)
  • domain assumption Stroboscopic time-averaged potential approximation
    The potential is assumed to equal a sum of jmax Gaussian profiles because switching is much faster than the trap frequencies; Floquet heating and finite-switching-rate corrections are neglected. Invoked in the Setup section.
  • domain assumption y and z motional degrees of freedom remain in their ground states
    The 3D dynamics are reduced to 1D by integrating out y,z ground states. If radial heating or non-adiabatic modulation populates y/z, the qubit-oscillator Hamiltonian is invalid. Invoked before Eq. (2).
  • domain assumption Truncation of the potential expansion at kmax=5 with negligible higher-order terms
    The potential is expanded to fifth order and O((x')^6) terms are neglected because the wavefunction is assumed to remain within the beam waist; the controlled-displacement infidelity at |alpha|=3 shows these terms are not fully negligible.
  • standard math Contact pseudopotential representation of s-wave scattering
    The delta-potential with derivative regularization is the standard low-energy model for ultracold s-wave scattering and is used throughout the relative-mode Hamiltonian and gate derivations.
  • standard math Rotating-wave approximation in gate synthesis
    Rapidly rotating terms are neglected in the interaction picture; the supplemental partially accounts for higher-order corrections for the R, SR, and CR gates.

pith-pipeline@v1.3.0-alltime-deepseek · 18036 in / 14045 out tokens · 94015 ms · 2026-08-03T18:08:24.984761+00:00 · methodology

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read the original abstract

We propose a qubit-oscillator platform based on the motional states of two interacting atoms in an optical tweezer. By stroboscopically modulating an engineered trap with tunable anharmonicity, we implement a complete set of bosonic operations and their qubit-controlled counterparts with high fidelity. This motional control enables accurate detection of magnetic dipolar interactions with $\sim10$ Hz sensitivity in one second, reaching sub-Hz resolution within a few minutes in a $20\times20$ tweezer array under realistic experimental imperfections. Our approach establishes a versatile platform for motional quantum control of two atoms, with applications to spin-boson physics and precision sensing of interaction potentials and trapping environments.

Figures

Figures reproduced from arXiv: 2512.06429 by Ana Maria Rey, Cindy A. Regal, Dawson P. Hewatt, Gur Lubin, Jaeyong Hwang, Sean R. Muleady, Steven K. Pampel, Tianrui Xu.

Figure 1
Figure 1. Figure 1: Hybrid qubit–oscillator module from the motion of [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Implementing controlled displacement (CD) and controlled squeezing (CS) gates. (a) Starting from [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A list of native gates: displacement (D), rotation (R), spin rotation (SR), squeezing (S), controlled displacement [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

55 extracted references · 2 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Bloch, J

    I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys.80, 885 (2008)

    2008

  2. [2]

    ˆSx,2, followed by population measurements such as⟨ ˆSz,1 ˆSz,2⟩ [35]. Conclusion.—We have introduced protocols, using na- tive contact interactions between two atoms in a time- averaged optical tweezer potential, to realize a con- trollable hybrid qubit-oscillator module using only the atomic motion, without involving any internal atomic states for gate ...

  3. [3]

    Bloch, J

    I. Bloch, J. Dalibard, and S. Nascimbène, Quantum sim- ulations with ultracold quantum gases, Nat. Phys.8, 267 (2012)

    2012

  4. [4]

    Schäfer, T

    F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultra- cold atoms in optical lattices, Nat. Rev. Phys. 2, 411 (2020)

    2020

  5. [5]

    Gross and W

    C. Gross and W. S. Bakr, Quantum gas microscopy for single atom and spin detection, Nat. Phys. 17, 1316 (2021)

    2021

  6. [6]

    Ashkin, Optical trapping and manipulation of neutral particles usinglasers, Proc

    A. Ashkin, Optical trapping and manipulation of neutral particles usinglasers, Proc. Natl. Acad. Sci. U.S.A. 94, 4853 (1997)

    1997

  7. [7]

    A. M. Kaufman and K.-K. Ni, Quantum science with optical tweezer arrays of ultracold atoms and molecules, Nat. Phys. 17, 1324 (2021)

    2021

  8. [8]

    Saffman, T

    M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys.82, 2313 (2010)

    2010

  9. [9]

    Browaeys and T

    A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020)

    2020

  10. [10]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semegh- ini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. Lukin, Logical quantum processor based on reconfig- urable atom arrays, Nature626, 58 (2024)

    2024

  11. [11]

    Saffman, X

    M. Saffman, X. L. Zhang, A. T. Gill, L. Isenhower, and T. G. Walker, Rydberg state mediated quantum gates and entanglement of pairs of neutral atoms, J. Phys. Conf. Ser. 264, 012023 (2011)

    2011

  12. [12]

    R. B.-S. Tsai, X. Sun, A. L. Shaw, R. Finkelstein, and M. Endres, Benchmarking and fidelity response theory of high-fidelity rydberg entangling gates, PRX Quantum 6, 010331 (2025)

    2025

  13. [13]

    M. O. Brown, S. R. Muleady, W. J. Dworschack, R. J. Lewis-Swan, A. M. Rey, O. Romero-Isart, and C. A. Re- gal, Time-of-flight quantum tomography of an atom in an optical tweezer, Nat. Phys.19, 569 (2023)

    2023

  14. [14]

    Lienhard, R

    V. Lienhard, R. Martin, Y. T. Chew, T. Tomita, K. Ohmori, and S. de Léséleuc, arXiv:2505.10092

    arXiv

  15. [15]

    P. Lunt, P. Hill, J. Reiter, P. M. Preiss, M. Gałka, and S. Jochim, Realization of a Laughlin State of Two RapidlyRotatingFermions,Phys.Rev.Lett. 133,253401 (2024)

    2024

  16. [16]

    A. L. Shaw, P. Scholl, R. Finkelstein, R. B.-S. Tsai, J. Choi, and M. Endres, Erasure cooling, control, and hyperentanglement of motion in optical tweezers, Science 388, 845 (2025)

    2025

  17. [17]

    P. T. Grochowski, H. Pichler, C. A. Regal, and O. Romero-Isart, Quantum control of continuous systems via nonharmonic potential modulation, Quantum9, 1824 (2025)

    2025

  18. [18]

    W. Huie, C. Conefrey-Shinozaki, Z. Jia, P. Draper, and J. P. Covey, arXiv:2507.18426

    arXiv

  19. [19]

    Lloyd and S

    S. Lloyd and S. L. Braunstein, Quantum computation 6 over continuous variables, Phys. Rev. Lett. 82, 1784 (1999)

    1999

  20. [20]

    U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa, Hybrid discrete- and continuous- variablequantuminformation,Nat.Phys. 11,713(2015)

    2015

  21. [21]

    A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quantum Computing with Rotation-Symmetric Bosonic Codes, Phys. Rev. X10, 011058 (2020)

    2020

  22. [22]

    Y. Liu, S. Singh, K. C. Smith, E. Crane, J. M. Martyn, A. Eickbusch, A. Schuckert, R. D. Li, J. Sinanan-Singh, M. B. Soley, T. Tsunoda, I. L. Chuang, N. Wiebe, and S. M. Girvin, arXiv:2407.10381

    arXiv

  23. [23]

    Crane, K

    E. Crane, K. C. Smith, T. Tomesh, A. Eickbusch, J. M. Martyn, S. Kühn, L. Funcke, M. A. DeMarco, I. L. Chuang, N. Wiebe, A. Schuckert, and S. M. Girvin, arXiv:2409.03747

    Pith/arXiv arXiv

  24. [24]

    Blais, R.-S

    A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Phys. Rev. A69, 062320 (2004)

    2004

  25. [25]

    Clarke and F

    J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature 453, 1031 (2008)

    2008

  26. [26]

    Xiang, S

    Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems, Rev. Mod. Phys.85, 623 (2013)

    2013

  27. [27]

    N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Ex- tending the lifetime of a quantum bit with error correc- tion in superconducting circuits, Nature536, 441 (2016)

    2016

  28. [28]

    Blais, A

    A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys.93, 025005 (2021)

    2021

  29. [29]

    Porras and J

    D. Porras and J. I. Cirac, Bose-Einstein Condensa- tion and Strong-Correlation Behavior of Phonons in Ion Traps, Phys. Rev. Lett.93, 263602 (2004)

    2004

  30. [30]

    Blatt and C

    R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys.8, 277 (2012)

    2012

  31. [31]

    Schneider, D

    C. Schneider, D. Porras, and T. Schaetz, Experimental quantum simulations of many-body physics with trapped ions, Rep. Prog. Phys.75, 024401 (2012)

    2012

  32. [32]

    P.-Y. Hou, J. J. Wu, S. D. Erickson, D. C. Cole, G.Zarantonello,A.D.Brandt,S.Geller,A.Kwiatkowski, S. Glancy, E. Knill, A. C. Wilson, D. H. Slichter, and D. Leibfried, Coherent coupling and non-destructive measurement of trapped-ion mechanical oscillators, Nat. Phys. 20, 1636 (2024)

    2024

  33. [33]

    Hartke, B

    T. Hartke, B. Oreg, N. Jia, and M. Zwierlein, Quantum register of fermion pairs, Nature601, 537 (2022)

    2022

  34. [34]

    Lacki, P

    M. Lacki, P. Zoller, and M. A. Baranov, Stroboscopic painting of optical potentials for atoms with subwave- length resolution, Phys. Rev. A100, 033610 (2019)

    2019

  35. [35]

    Z. Z. Yan, B. M. Spar, M. L. Prichard, S. Chi, H.-T. Wei, E. Ibarra-García-Padilla, K. R. A. Hazzard, and W. S. Bakr, Two-Dimensional Programmable Tweezer Arrays of Fermions, Phys. Rev. Lett.129, 123201 (2022)

    2022

  36. [36]

    See Supplemental Material

  37. [37]

    Busch, B.-G

    T. Busch, B.-G. Englert, K. Rzażewski, and M. Wilkens, Two Cold Atoms in a Harmonic Trap, Found. Phys.28, 549 (1998)

    1998

  38. [38]

    Idziaszek and T

    Z. Idziaszek and T. Calarco, Two atoms in an anisotropic harmonic trap, Phys. Rev. A71, 050701 (2005)

    2005

  39. [39]

    S. K. Pampel, M. Marinelli, M. O. Brown, J. P. D’Incao, and C. A. Regal, Quantifying Light-Assisted Collisions in Optical Tweezers across the Hyperfine Spectrum, Phys. Rev. Lett. 134, 013202 (2025)

    2025

  40. [40]

    Urvoy, Z

    A. Urvoy, Z. Vendeiro, J. Ramette, A. Adiyatullin, and V. Vuletić, Direct Laser Cooling to Bose-Einstein Con- densation in a Dipole Trap, Phys. Rev. Lett.122, 203202 (2019)

    2019

  41. [41]

    Flühmann and J

    C. Flühmann and J. P. Home, Direct Characteristic- Function Tomography of Quantum States of the Trapped-Ion Motional Oscillator, Phys. Rev. Lett.125, 043602 (2020)

    2020

  42. [42]

    Gottesman, A

    D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A64, 012310 (2001)

    2001

  43. [43]

    Campagne-Ibarcq, A

    P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys- Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Quantum error correction of a qubit encoded in grid states of an oscillator, Nature584, 368 (2020)

    2020

  44. [44]

    Flühmann, T

    C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevit- sky, K. Mehta, and J. P. Home, Encoding a qubit in a trapped-ion mechanical oscillator, Nature 566, 513 (2019)

    2019

  45. [45]

    Singh, B

    S. Singh, B. Royer, and S. M. Girvin, arXiv:2504.19992

    Pith/arXiv arXiv

  46. [46]

    W. H. Zurek, Sub-Planck structure in phase space and its relevance for quantum decoherence, Nature412, 712 (2001)

    2001

  47. [47]

    G. M. Tino, Testing gravity with cold atom interferom- etry: Results and prospects, Quantum Sci. Technol. 6, 024014 (2021)

    2021

  48. [48]

    P.T.GrochowskiandR.Filip,OptimalPhase-Insensitive Force Sensing with Non-Gaussian States, Phys. Rev. Lett. 135, 230802 (2025)

    2025

  49. [49]

    M. Kang, H. Nuomin, S. N. Chowdhury, J. L. Yuly, K. Sun, J. Whitlow, J. Valdiviezo, Z. Zhang, P. Zhang, D. N. Beratan, and K. R. Brown, Seeking a quan- tum advantage with trapped-ion quantum simulations of condensed-phase chemical dynamics, Nat. Rev. Chem.8, 340 (2024)

    2024

  50. [50]

    Navickas, R

    T. Navickas, R. J. MacDonell, C. H. Valahu, V. C. Olaya- Agudelo, F. Scuccimarra, M. J. Millican, V. G. Matsos, H. L. Nourse, A. D. Rao, M. J. Biercuk, C. Hempel, I. Kassal, and T. R. Tan, Experimental Quantum Sim- ulation of Chemical Dynamics, J. Am. Chem. Soc.147, 23566 (2025)

    2025

  51. [51]

    Hybrid qubit-oscillator module with motional states of two trapped interacting atoms

    D. F. James and J. Jerke, Effective Hamiltonian the- ory and its applications in quantum information, Can. J. Phys. 85, 625 (2007). End Matter Inoursimulations,weuseparametersrelevantfor 87Rb atoms with a tweezer beam waistW = 700nm, trap fre- quency ωx = 2π × 140kHz (V0 = 1mK), and a perturba- tion parameter ϵx = 0 .041. To reduce qubit leakage by increa...

    2007

  52. [52]

    The matrix element of the interaction between harmonic basis|2m⟩ and |2n⟩ can be calculated as Vmn = Z |ψ2m(r)|2√πuδ(r)|ψ2n(r)|2dr = u − 1 2 m+n s 2m m 2n n (S7) where ψ2n(r) = π−1/4 √ 22n(2n)! exp −r2/2 H2n(r) is dimensionless wavefunction of the harmonic oscillator state|2n⟩, and Hn(r) is the n-th Hermite polynomial. The Hamiltonian in the even-number h...

  53. [53]

    + β ˆSx,2(ˆa2 + ˆa† 2)), and assuming the contact interaction is small and can be neglected during this measurement protocol, the state after the rotation and spin-dependent force becomes: Y j=1,2 ˆDj(2β ˆSx,j)e−i ˆSx,j θ |g⟩1 |g⟩2 |ψ⟩ = 1 2 Y j=1,2 e−iθ/2 |+⟩j ⊗ ˆDj(β) + eiθ/2 |−⟩j ⊗ ˆDj(−β) |ψ⟩ = 1 2 e−iθ |+⟩1 |+⟩2 ˆDR(β √

  54. [54]

    + eiθ |−⟩1 |−⟩2 ˆDR(−β √ 2) |ψ⟩ + 1 2 |+⟩1 |−⟩2 ˆDr(β √

  55. [55]

    + |−⟩1 |+⟩2 ˆDr(− √ 2β) |ψ⟩ . (S22) A joint measurement of both spin operators yields 4⟨ ˆSz,1 ˆSz,2⟩θ = 1 2 cos(2θ)Re[χR(β′)] + 1 2 sin(2θ)Im[χR(β′)] + 1 2 Re[χr(β′)] (S23) where β′ = 2 √ 2β, χR(β′) = ⟨ψ| ˆDR(β′) |ψ⟩, and χr(β′) = ⟨ψ| ˆDr(β′) |ψ⟩. Note that due to the indistinguishability of the two atoms, the relative coordinate can only occupy even-num...