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arxiv: 2605.03468 · v1 · submitted 2026-05-05 · 🪐 quant-ph

Universal qutrit control in asymmetric-top molecules

Qian-Qian Hong , Zhi-Jian Zheng , Zhe-Jun Zhang , Xin-Xia Jian , Chuan-Cun Shu This is my paper

Pith reviewed 2026-05-07 17:22 UTC · model grok-4.3

classification 🪐 quant-ph
keywords qutrit controlasymmetric-top moleculespulse-area theoremmicrowave controlrotational statessingle-qutrit gatesquantum information processingphase operations
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The pith

Asymmetric-top molecules support universal single-qutrit control via analytic microwave pulse sequences.

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

The paper establishes a method to implement any single-qutrit gate on three rotational eigenstates of an asymmetric-top molecule. It separates the gate into directly addressable rotations and auxiliary-state phase shifts, then derives an exact mapping from gate parameters to control fields. This analytic route replaces numerical optimization and is verified in simulations for a concrete molecule with low leakage. A sympathetic reader would care because molecular rotors offer long coherence times and the method gives a systematic design tool for multilevel quantum control.

Core claim

We present a theoretical framework for universal single-qutrit control in asymmetric-top molecules. The qutrit is encoded in three rotational eigenstates with an auxiliary state providing independent phase control. An analytic protocol combines directly addressable SU(2) rotations with auxiliary-state-mediated phase operations. A multilevel pulse-area theorem supplies an explicit analytic mapping between gate parameters and control fields, enabling systematic design of high-fidelity microwave pulse sequences. Numerical simulations confirm robustness for Walsh-Hadamard gates with minimal leakage, and four SU(2) decomposition strategies are compared for error sensitivity.

What carries the argument

The multilevel pulse-area theorem, which supplies an explicit analytic mapping between desired gate parameters and the required microwave control fields.

If this is right

  • Arbitrary single-qutrit gates become implementable with high fidelity using microwave pulses in molecules such as 1,2-propanediol.
  • Phase-error sensitivity of the gate depends on the chosen sequence for decomposing the SU(2) rotations.
  • Amplitude errors propagate along specific coherence pathways that can be traced analytically.
  • The approach minimizes leakage out of the three-state computational subspace in the simulated cases.

Where Pith is reading between the lines

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

  • The same pulse-area mapping could be applied to other multilevel systems whose level structure resembles the rotational ladder of asymmetric tops.
  • If the auxiliary state remains stable under real decoherence, sequences of many gates become feasible without repeated numerical re-optimization.
  • Experimental tests in trapped or cooled asymmetric-top molecules would directly check whether the analytic predictions survive laboratory noise sources.

Load-bearing premise

The auxiliary state can be addressed independently to apply phase shifts inside the computational manifold without causing leakage or decoherence that the pulse-area theorem fails to capture.

What would settle it

A simulation or experiment that applies the derived pulse sequences for an arbitrary qutrit gate and measures fidelity well below the predicted high value due to unaccounted population loss from the auxiliary state.

Figures

Figures reproduced from arXiv: 2605.03468 by Chuan-Cun Shu, Qian-Qian Hong, Xin-Xia Jian, Zhe-Jun Zhang, Zhi-Jian Zheng.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of universal single-qutrit control in an a view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Average gate fidelities for elementary operations an view at source ↗
Figure 4
Figure 4. Figure 4: shows the analytical error coefficients C gate ξ and C ψin ξ , together with the corresponding fidelities obtained from molecular simulations as functions of the error parameter ξ. Figures 4(a) and (b) indicate that all four decomposition se￾quences have identical analytical gate-error coefficients. The average gate fidelities exhibit the same dependence on ampli￾tude error, confirming that target gate sen… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Error dynamics in the generalized Bloch-vector repr view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The same simulations as in Fig view at source ↗
read the original abstract

We present a theoretical framework for universal single-qutrit control in asymmetric-top molecules, advancing molecular quantum information processing. In this approach, the qutrit is encoded in three rotational eigenstates, with an auxiliary state providing independent phase control within the computational manifold. We explore an analytic protocol for arbitrary single-qutrit gates, combining directly addressable SU(2) rotations with auxiliary-state-mediated phase operations. To support this, we derive a multilevel pulse-area theorem that provides an explicit analytic mapping between gate parameters and control fields, enabling systematic design of high-fidelity microwave pulse sequences. Numerical simulations with 1,2-propanediol confirm the robustness of our approach, achieving Walsh-Hadamard gates with minimal leakage from the computational subspace. We further examine four SU(2) decomposition strategies and find that phase-error sensitivity depends on the decomposition sequence, while amplitude errors propagate along specific coherence pathways. Our results establish asymmetric-top molecules as a viable platform for qutrit-based quantum operations and offer an analytical method for precise quantum control of complex multilevel systems.

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 / 1 minor

Summary. The manuscript presents a theoretical framework for universal single-qutrit control in asymmetric-top molecules. The qutrit is encoded in three rotational eigenstates with an auxiliary state providing independent phase control. An analytic protocol combines directly addressable SU(2) rotations with auxiliary-mediated phase operations, supported by a derived multilevel pulse-area theorem that maps gate parameters explicitly to microwave control fields. Numerical simulations on 1,2-propanediol are reported to achieve Walsh-Hadamard gates with minimal leakage from the computational subspace, and four SU(2) decomposition strategies are compared for sensitivity to phase and amplitude errors.

Significance. If the multilevel pulse-area theorem and auxiliary-state isolation hold, the work supplies an analytic tool for high-fidelity qutrit gate design in molecular systems, which would be a useful advance for molecular quantum information processing. The explicit parameter-to-field mapping and the comparison of decomposition strategies are strengths; the numerical checks on a concrete molecule add concreteness. Significance is tempered by the need to confirm that the four-level truncation remains valid under realistic pulse bandwidths.

major comments (2)
  1. The multilevel pulse-area theorem is derived under an implicit four-level truncation (three computational states plus one auxiliary). In asymmetric-top molecules the rotational spectrum is dense; weak dipole couplings from the auxiliary level to states outside this manifold can produce leakage or uncontrolled phase shifts not captured by the theorem. This assumption is load-bearing for the claimed analytic mapping and universal control, yet the manuscript provides no quantitative bound on such couplings for the pulse parameters used.
  2. Numerical simulations are reported to show minimal leakage only for Walsh-Hadamard gates on 1,2-propanediol. No fidelity values, error bars, or explicit verification that the pulse-area theorem holds for other gates or pulse sequences are supplied. Without these metrics the robustness claim for arbitrary single-qutrit gates cannot be assessed.
minor comments (1)
  1. The abstract states 'minimal leakage' without reporting quantitative fidelity or leakage probabilities; the main text should supply these numbers together with the simulation parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: The multilevel pulse-area theorem is derived under an implicit four-level truncation (three computational states plus one auxiliary). In asymmetric-top molecules the rotational spectrum is dense; weak dipole couplings from the auxiliary level to states outside this manifold can produce leakage or uncontrolled phase shifts not captured by the theorem. This assumption is load-bearing for the claimed analytic mapping and universal control, yet the manuscript provides no quantitative bound on such couplings for the pulse parameters used.

    Authors: We agree that the four-level truncation is central to the analytic mapping and that the dense rotational spectrum of asymmetric-top molecules requires careful consideration of off-manifold couplings. The auxiliary state was selected for its relative isolation in the 1,2-propanediol spectrum, but the original manuscript does not supply explicit bounds. We will add a quantitative estimate of leakage and phase shifts from external states, derived from the known dipole moments, selection rules, and the specific pulse bandwidths and durations used in the protocol. revision: yes

  2. Referee: Numerical simulations are reported to show minimal leakage only for Walsh-Hadamard gates on 1,2-propanediol. No fidelity values, error bars, or explicit verification that the pulse-area theorem holds for other gates or pulse sequences are supplied. Without these metrics the robustness claim for arbitrary single-qutrit gates cannot be assessed.

    Authors: The numerical section focuses on the Walsh-Hadamard gate as a concrete demonstration, with leakage described qualitatively. We concur that explicit fidelity metrics and broader verification are needed to support universality. In the revised manuscript we will report average gate fidelities with error bars for the Walsh-Hadamard gate and additional representative single-qutrit gates, together with direct checks that the pulse-area theorem parameters produce the expected evolution for those sequences. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from Hamiltonian

full rationale

The paper derives its multilevel pulse-area theorem via standard time-dependent perturbation theory applied to the four-level truncation of the asymmetric-top molecular Hamiltonian, yielding an explicit analytic mapping from gate parameters to control fields. This follows directly from the Schrödinger equation and SU(2) decomposition without presupposing the target gate in the inputs, without fitting parameters to the desired outcome, and without load-bearing self-citations or imported uniqueness theorems. The auxiliary-state phase control is introduced as an independent addressable transition within the model, and numerical validation on 1,2-propanediol serves as external check rather than definitional closure. The central claims remain independent of the results they produce.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard quantum-mechanical assumptions about rotational eigenstates and coherent microwave driving; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Three rotational eigenstates plus one auxiliary state can be isolated sufficiently to form a closed computational manifold under microwave driving.
    Invoked when the qutrit is encoded and the auxiliary state is used for phase control.
  • standard math The multilevel pulse-area theorem derived from the time-dependent Schrödinger equation provides an exact analytic mapping for the chosen pulse shapes.
    Central to the claim of systematic gate design.

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Reference graph

Works this paper leans on

73 extracted references · 73 canonical work pages

  1. [1]

    = ˆU(η,χ) ˆUb ˆUc ˆUa π/4 3 π/2 π/4 0 arcsin(1 / √

    7 π/6 π/4 π/6 π/4 4 π/3 2 π/3 5 π/6 ˆU (2) gen. = ˆU(η,χ) ˆUb ˆUc ˆUa π/4 3 π/2 π/4 0 arcsin(1 / √

    work page

  2. [2]

    = ˆU(η,χ) ˆUb ˆUa ˆUc arcsin(1/ √

    3 π/2 5 π/6 2 π/3 ˆU (3) gen. = ˆU(η,χ) ˆUb ˆUa ˆUc arcsin(1/ √

    work page

  3. [3]

    = ˆU(η,χ) ˆUa ˆUc ˆUb π/4 4 π/3 π/4 11 π/6 arcsin(1 / √

    3 π/2 π/4 0 π/4 3 π/2 3 π/2 5 π/6 ˆU (4) gen. = ˆU(η,χ) ˆUa ˆUc ˆUb π/4 4 π/3 π/4 11 π/6 arcsin(1 / √

    work page

  4. [4]

    7 π/6 5 π/6 3 π/2 D. Analytical design of the control pulse sequence The implementation of a single-qutrit gate relies on decom- posing the target unitary into a sequence of elementary SU(2 ) operations acting on selected two-level subspaces, follow ed by a diagonal phase operation. In practice, each elementary operation is realized by a resonant control ...

    work page

  5. [5]

    This analyt ic mapping enables systematic and explicit design of pulse se- quences for arbitrary single-qutrit gates

    are deter- mined directly from the target gate parameters. This analyt ic mapping enables systematic and explicit design of pulse se- quences for arbitrary single-qutrit gates. E. Average gate fidelity and state fidelity To evaluate the analytically designed pulse sequences, we numerically compute the exact rotational dynamics using th e evolution operator ...

    work page

  6. [6]

    Within the computational subspace, the qutrit state is described by the density matri x ˆρ(t) = |ψ(t)⟩⟨ψ(t)|

    and ( 18). Within the computational subspace, the qutrit state is described by the density matri x ˆρ(t) = |ψ(t)⟩⟨ψ(t)|. Any qutrit density matrix admits an expan- sion in the Gell-Mann basis {ˆλk}8 k=1 (see Appendix C) [46]: ˆρ(t) = 1 3 ˆI + 1 2 8∑ k=1 sk(t) ˆλk, (19) where ˆI is the 3 ×3 identity, and sk(t) = Tr[ ˆρ(t) ˆλk] = ⟨ˆλk⟩ are the generalized B...

    work page

  7. [7]

    In the molecular qutrit system, we simulate ampli- tude errors by applying relative errors to the rotation angl es, θa, b, c → (1 +ξ)θa, b, c

    and ( 13) indi- cate that reductions in average gate fidelity and target-sta te fidelity are determined by analytical error coe fficients; larger coefficients reflect greater sensitivity to parameter perturba- tions. In the molecular qutrit system, we simulate ampli- tude errors by applying relative errors to the rotation angl es, θa, b, c → (1 +ξ)θa, b, c. Figu...

    work page

  8. [8]

    Next, we consider the ordering ˆU p ˆUa ˆUb ˆUc

    Therefore, the decomposition ˆU p ˆUc ˆUb ˆUa is inconsis- tent and does not yield a valid solution. Next, we consider the ordering ˆU p ˆUa ˆUb ˆUc. In this case, the relevant matrix elements are M12 = ieiφa sinθa cosθb, M22 = eiηcosθa cosθb, M32 = iei(χ−φb) sinθb, M31 = iei(χ−φc) sinθc cosθb. (D9) 11 Matching their moduli to those of ˆUFT again gives θa...

    work page

  9. [9]

    (D11) Substituting Eqs

    (D10) The phases are determined by M12 = 1/ √ 3, M22 = ei2π/3/ √ 3, M32 = ei4π/3/ √ 3, M31 = 1/ √ 3, and M33 = ei2π/3/ √ 3, result- ing in φa = 3π 2 , η = 2π 3 , χ = 2π 3 , φ b = 11π 6 , φ c = 7π 6 . (D11) Substituting Eqs. ( D10) and ( D11) into the matrix element M11 yields M11 = 1 2 + 1 4 √ 3 −i 4, (D12) which di ffers from the target value ( ˆUFT)11 = 1/ √

    work page

  10. [10]

    This analysis confirms that only four of the six possible decompo- sition sequences yield valid real solutions for the qutrit Walsh- Hadamard gate

    There- fore, the decomposition ˆU p ˆUa ˆUb ˆUc is also inconsistent. This analysis confirms that only four of the six possible decompo- sition sequences yield valid real solutions for the qutrit Walsh- Hadamard gate

    work page

  11. [11]

    B. P . Lanyon, M. Barbieri, M. P . Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, Simplifying quantum logic using higher- dimensional Hilbert spaces, Nat. Phys. 5, 134 (2009)

    work page 2009

  12. [12]

    Cozzolino, B

    D. Cozzolino, B. Da Lio, D. Bacco, and L. K. Oxenløwe, High - dimensional quantum communication: benefits, progress, an d future challenges, Adv. Quantum Technol. 2, 1900038 (2019)

    work page 2019

  13. [13]

    Y . Wang, Z. Hu, B. C. Sanders, and S. Kais, Qudits and high-dimensional quantum computing, Front. Phys. 8, 589504 (2020)

    work page 2020

  14. [14]

    Sawant, J

    R. Sawant, J. A. Blackmore, P . D. Gregory, J. Mur-Petit, D. Jaksch, J. Aldegunde, J. M. Hutson, M. R. Tarbutt, and S. L. Cornish, Ultracold polar molecules as qudits, New J. Phys. 22, 013027 (2020)

    work page 2020

  15. [15]

    E. O. Kiktenko, A. S. Nikolaeva, and A. K. Fedorov, Collo- quium: Qudits for decomposing multiqubit gates and realizi ng quantum algorithms, Rev. Mod. Phys. 97, 021003 (2025)

    work page 2025

  16. [16]

    Morvan, V

    A. Morvan, V . V . Ramasesh, M. S. Blok, J. M. Kreikebaum, K. O’Brien, L. Chen, B. K. Mitchell, R. K. Naik, D. I. Santiago, and I. Siddiqi, Qutrit randomized benchmarking, Phys. Rev. Lett. 126, 210504 (2021)

    work page 2021

  17. [17]

    M. S. Blok, V . V . Ramasesh, T. Schuster, K. O’Brien, J. M. Kreikebaum, D. Dahlen, A. Morvan, B. Y oshida, N. Y . Yao, and I. Siddiqi, Quantum information scrambling on a superco n- ducting qutrit processor, Phys. Rev. X 11, 021010 (2021)

    work page 2021

  18. [18]

    N. Goss, A. Morvan, B. Marinelli, B. K. Mitchell, L. B. Nguyen, R. K. Naik, L. Chen, C. J¨ unger, J. M. Kreikebaum, D. I. Santiago, et al. , High-fidelity qutrit entangling gates for superconducting circuits, Nat. Commun. 13, 7481 (2022)

    work page 2022

  19. [19]

    K. Luo, W. Huang, Z. Tao, L. Zhang, Y . Zhou, J. Chu, W. Liu, B. Wang, J. Cui, S. Liu, F. Yan, M.-H. Y ung, Y . Chen, T. Yan, and D. Y u, Experimental realization of two qutrits gate with tunable coupling in superconducting circ uits, Phys. Rev. Lett. 130, 030603 (2023)

    work page 2023

  20. [20]

    N. Goss, S. Ferracin, A. Hashim, A. Carignan-Dugas, J. M . Kreikebaum, R. K. Naik, D. I. Santiago, and I. Siddiqi, Exten d- ing the computational reach of a superconducting qutrit pro ces- sor, npj Quantum Inf. , 101 (2024)

    work page 2024

  21. [21]

    Ammenwerth, H

    M. Ammenwerth, H. Timme, F. Gyger, R. Tao, I. Bloch, and J. Zeiher, Realization of a fast triple-magic all-optical q utrit in 88Sr, Phys. Rev. Lett. 135, 143401 (2025)

    work page 2025

  22. [22]

    N. V . Vitanov, Synthesis of arbitrary SU(3) transforma tions of atomic qutrits, Phys. Rev. A 85, 032331 (2012)

    work page 2012

  23. [23]

    Dogra, Arvind, and K

    S. Dogra, Arvind, and K. Dorai, Determining the par- ity of a permutation using an experimental NMR qutrit, Phys. Lett. A 378, 3452 (2014)

    work page 2014

  24. [24]

    Lindon, A

    J. Lindon, A. Tashchilina, L. W. Cooke, and L. J. LeBlanc, Complete unitary qutrit control in ultracold atom s, Phys. Rev. Appl. 19, 034089 (2023)

    work page 2023

  25. [25]

    L. E. Fischer, A. Chiesa, F. Tacchino, D. J. Egger, S. Car retta, and I. Tavernelli, Universal qudit gate synthesis for trans mons, PRX Quantum 4, 030327 (2023)

    work page 2023

  26. [26]

    M. A. Y urtalan, J. Shi, M. Kononenko, A. Lupascu, and S. A sh- hab, Implementation of a walsh-hadamard gate in a supercon- ducting qutrit, Phys. Rev. Lett. 125, 180504 (2020)

    work page 2020

  27. [27]

    Kononenko, M

    M. Kononenko, M. A. Y urtalan, S. Ren, J. Shi, S. Ash- hab, and A. Lupascu, Characterization of control in a superconducting qutrit using randomized benchmarking, Phys. Rev. Res. 3, L042007 (2021)

    work page 2021

  28. [28]

    Y . Wang, K. Snizhko, A. Romito, Y . Gefen, and K. Murch, Dissipative preparation and stabilization of many- body quantum states in a superconducting qutrit array, Phys. Rev. A 108, 013712 (2023)

    work page 2023

  29. [29]

    Ringbauer, M

    M. Ringbauer, M. Meth, L. Postler, R. Stricker, R. Blatt , P . Schindler, and T. Monz, A universal qudit quantum processor with trapped ions, Nat. Phys. 18, 1053 (2022)

    work page 2022

  30. [30]

    P . Hrmo, B. Wilhelm, L. Gerster, M. W. van Mourik, M. Hu- ber, R. Blatt, P . Schindler, T. Monz, and M. Ringbauer, Nativ e qudit entanglement in a trapped ion quantum processor, Nat. Commun. 14, 2242 (2023)

    work page 2023

  31. [31]

    Burshtein, S

    A. Burshtein, S. Fraenkel, M. Goldstein, and R. Finkels tein, Ro- bust control and entanglement of qudits in neutral atom arra ys, Phys. Rev. Res. 8, 013055 (2026)

    work page 2026

  32. [32]

    H. Xu, C. Song, W. Liu, G. Xue, F. Su, H. Deng, Y . Tian, D. Zheng, S. Han, Y .-P . Zhong, et al. , Coherent population transfer between uncoupled or weakly coupled states in ladd er- type superconducting qutrits, Nat. Commun. 7, 11018 (2016)

    work page 2016

  33. [33]

    T. Roy, Z. Li, E. Kapit, and D. Schuster, Two-qutrit quan - tum algorithms on a programmable superconducting processo r, Phys. Rev. Appl. 19, 064024 (2023)

    work page 2023

  34. [34]

    J.-L. Wu, Y . Wang, S.-L. Su, Y . Xia, Y . Jiang, and J. Song,Dis- crimination of enantiomers through quantum interference a nd quantum zeno effect, Opt. Express 28, 33475 (2020)

    work page 2020

  35. [35]

    J.-L. Wu, Y . Wang, J.-X. Han, C. Wang, S.-L. Su, Y . Xia, Y . Jiang, and J. Song, Two-path interference for enantiomer-selective state transfer of chiral molecul es, Phys. Rev. Appl. 13, 044021 (2020)

    work page 2020

  36. [36]

    M.-R. Cai, C. Ye, H. Dong, and Y . Li, Enantiodetec- tion of chiral molecules via two-dimensional spectroscopy , Phys. Rev. Lett. 129, 103201 (2022)

    work page 2022

  37. [37]

    Y . Guo, X. Gong, S. Ma, and C.-C. Shu, Cyclic three-level - pulse-area theorem for enantioselective state transfer of chiral 12 molecules, Phys. Rev. A 105, 013102 (2022)

    work page 2022

  38. [38]

    J. Lee, J. Bischo ff, A. O. Hernandez-Castillo, B. Sar- takov, G. Meijer, and S. Eibenberger-Arias, Quan- titative study of enantiomer-specific state transfer, Phys. Rev. Lett. 128, 173001 (2022)

    work page 2022

  39. [39]

    Cheng, Y .-Y

    J.-J. Cheng, Y .-Y . Chen, Y . Li, and L. Zhang, Enantiomer-specific state transfer of chiral molecules in cyclic three-level systems with SU(2) structures, Phys. Rev. A 107, 013718 (2023)

    work page 2023

  40. [40]

    Hong, X.-J

    Q.-Q. Hong, X.-J. Song, L. Xu, and C.-C. Shu, Robust control of enantioselective state transfer in chiral molec ules, ChemPhysChem 27, e202500705 (2026)

    work page 2026

  41. [41]

    C. P . Koch, M. Lemeshko, and D. Sugny, Quantum control of molecular rotation, Rev. Mod. Phys. 91, 035005 (2019)

    work page 2019

  42. [42]

    Leibscher, E

    M. Leibscher, E. Pozzoli, C. P´ erez, M. Schnell, M. Siga - lotti, U. Boscain, and C. P . Koch, Full quantum control of enantiomer-selective state transfer in chiral molecules d espite degeneracy, Commun. Phys. 5, 110 (2022)

    work page 2022

  43. [43]

    J. Lee, E. Abdiha, B. G. Sartakov, G. Meijer, and S. Eibenberger-Arias, Near-complete chiral selection in r ota- tional quantum states, Nat. Commun. 15, 7441 (2024)

    work page 2024

  44. [44]

    F. E. Bergg¨ otz, M. Leibscher, W. Sun, C. P . Koch, and M. Schnell, Di fferentiating between enantiomers with nuclear quadrupole coupling using microwave three-wave mixing, J. Phys. Chem. Lett. 16, 12087 (2025)

    work page 2025

  45. [45]

    R. N. Zare and W. G. Harter, Angular momentum: understan d- ing spatial aspects in chemistry and physics, New Y ork 120 (1988)

    work page 1988

  46. [46]

    Shchedrin, C

    G. Shchedrin, C. O’Brien, Y . Rostovtsev, and M. O. Scull y, An- alytic solution and pulse area theorem for three-level atom s, Phys. Rev. A 92, 063815 (2015)

    work page 2015

  47. [47]

    Hong, L.-B

    Q.-Q. Hong, L.-B. Fan, C.-C. Shu, and N. E. Henriksen, Ge n- eration of maximal three-state field-free molecular orient ation with terahertz pulses, Phys. Rev. A 104, 013108 (2021)

    work page 2021

  48. [48]

    Fan, C.-C

    L.-B. Fan, C.-C. Shu, D. Dong, J. He, N. E. Henriksen, and F. Nori, Quantum coherent control of a single molecular- polariton rotation, Phys. Rev. Lett. 130, 043604 (2023)

    work page 2023

  49. [49]

    Q.-Q. Hong, D. Dong, N. E. Henriksen, F. Nori, J. He, and C .- C. Shu, Precise quantum control of molecular rotation towar d a desired orientation, Phys. Rev. Res. 7, L012049 (2025)

    work page 2025

  50. [50]

    Hong, Z.-J

    Q.-Q. Hong, Z.-J. Zhang, C.-C. Shu, J. He, D. Dong, and D. Ding, Precise quantum control of unidirectional field-fr ee molecular orientation, Phys. Rev. A 113, 013118 (2026)

    work page 2026

  51. [51]

    Jian, Z.-J

    X.-X. Jian, Z.-J. Zheng, J.-J. Jiang, L. Zhou, C.-C. Shu, and J. He, All-optical raman control of ultracold atomic hyperfine states using the pulsed jump protocol, Phys. Rev. A 112, 013108 (2025)

    work page 2025

  52. [52]

    Hong, Z.-Z

    Q.-Q. Hong, Z.-Z. Lian, C.-C. Shu, and N. E. Henriksen, Per- spectives: Quantum control of field-free molecular orientation, Phys. Chem. Chem. Phys. 25, 32763 (2023)

    work page 2023

  53. [53]

    Fan, H.-J

    L.-B. Fan, H.-J. Li, Q. Chen, H. Zhou, H. Liu, and C.-C. Shu, Maximizing orientation of a three-state molecule in a cavity with analytically designed pulses, Phys. Rev. A 111, 033119 (2025)

    work page 2025

  54. [54]

    Yang, Q.-Q

    J. Yang, Q.-Q. Hong, S. Ma, S.-S. Ma, and C.-C. Shu, Multi - level pulse-area theorem for precise quantum control of mol ec- ular vibration and rotation, Phys. Rev. A 113, 033106 (2026)

    work page 2026

  55. [55]

    M. A. Nielsen, A simple formula for the average gate fidel ity of a quantum dynamical operation, Phys. Lett. A 303, 249 (2002)

    work page 2002

  56. [56]

    C. M. Caves and G. J. Milburn, Qutrit entanglement, Opt. Com- mun. 179, 439 (2000)

    work page 2000

  57. [57]

    A. B. Klimov, R. Guzm´ an, J. C. Retamal, and C. Saave- dra, Qutrit quantum computer with trapped ions, Phys. Rev. A 67, 062313 (2003)

    work page 2003

  58. [58]

    Patterson, M

    D. Patterson, M. Schnell, and J. M. Doyle, Enantiomer-s pecific detection of chiral molecules via microwave spectroscopy, Na- ture 497, 475 (2013)

    work page 2013

  59. [59]

    Mitra, N

    D. Mitra, N. B. Vilas, C. Hallas, L. Anderegg, B. L. Augen - braun, L. Baum, C. Miller, S. Raval, and J. M. Doyle, Direct laser cooling of a symmetric top molecule, Science 369, 1366 (2020)

    work page 2020

  60. [60]

    L. Baum, N. B. Vilas, C. Hallas, B. L. Augenbraun, S. Rava l, D. Mitra, and J. M. Doyle, 1D magneto-optical trap of poly- atomic molecules, Phys. Rev. Lett. 124, 133201 (2020)

    work page 2020

  61. [61]

    N. B. Vilas, C. Hallas, L. Anderegg, P . Robichaud, A. Win - nicki, D. Mitra, and J. M. Doyle, Magneto-optical trapping a nd sub-doppler cooling of a polyatomic molecule, Nature 606, 70 (2022)

    work page 2022

  62. [62]

    Z. Zeng, S. Deng, S. Yang, and B. Yan, Three-dimensional magneto-optical trapping of barium monofluoride, Phys. Rev. Lett. 133, 143404 (2024)

    work page 2024

  63. [63]

    Langen, G

    T. Langen, G. V altolina, D. Wang, and J. Ye, Quantum stat e manipulation and cooling of ultracold molecules, Nat. Phys. 20, 702 (2024)

    work page 2024

  64. [64]

    S. L. Cornish, M. R. Tarbutt, and K. R. Hazzard, Quantum c om- putation and quantum simulation with ultracold molecules, Nat. Phys. 20, 730 (2024)

    work page 2024

  65. [65]

    Anderegg, L

    L. Anderegg, L. W. Cheuk, Y . Bao, S. Burchesky, W. Ketter le, K.-K. Ni, and J. M. Doyle, An optical tweezer array of ultracold molecules, Science 365, 1156 (2019)

    work page 2019

  66. [66]

    Y . Bao, S. S. Y u, L. Anderegg, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, Dipolar spin-exchange and entanglement between molecules in an optical tweezer array, Science 382, 1138 (2023)

    work page 2023

  67. [67]

    C. M. Holland, Y . Lu, and L. W. Cheuk, On-demand entan- glement of molecules in a reconfigurable optical tweezer arr ay, Science 382, 1143 (2023)

    work page 2023

  68. [68]

    D. K. Ruttley, T. R. Hepworth, A. Guttridge, and S. L. Cor nish, Long-lived entanglement of molecules in magic-wavelength optical tweezers, Nature 637, 827 (2025)

    work page 2025

  69. [69]

    Yao, Photonic generation of microwave arbitrary wav eforms, Opt

    J. Yao, Photonic generation of microwave arbitrary wav eforms, Opt. Commun. 284, 3723 (2011)

    work page 2011

  70. [70]

    Merrill and K

    J. Merrill and K. Brown, Progress in compensating pulse se- quences for quantum computation, Adv. Chem. Phys. 154, 241 (2014)

    work page 2014

  71. [71]

    Z. Zeng, L. Zhang, Y . Zhang, H. Tian, Z. Zhang, S. Zhang, H. Li, and Y . Liu, Microwave pulse generation via employing an electric signal modulator to achieve time-domain mode lock ing in an optoelectronic oscillator, Opt. Lett. 46, 2107 (2021)

    work page 2021

  72. [72]

    Z. Bao, Y . Li, Z. Wang, J. Wang, J. Yang, H. Xiong, Y . Song, Y . Wu, H. Zhang, and L. Duan, A cryogenic on-chip microwave pulse generator for large-scale superconducting quantum c om- puting, Nat. Commun. 15, 5958 (2024)

    work page 2024

  73. [73]

    T. Xia, M. Lichtman, K. Maller, A. W. Carr, M. J. Piotrow- icz, L. Isenhower, and M. Sa ffman, Randomized benchmark- ing of single-qubit gates in a 2D array of neutral-atom qubit s, Phys. Rev. Lett. 114, 100503 (2015)

    work page 2015