- Open Access
Expanding the neutral-atom gate set: Native iSWAP and exchange gates from dipolar Rydberg interactions
Phys. Rev. Research 8, 033269 – Published 3 September, 2026
DOI: https://doi.org/10.1103/z46h-mgh4
Abstract
We present a native realization of iSWAP and parametrized exchange gates for neutral-atom quantum processing units. Our approach leverages strong dipole-dipole interactions between two different dipole-coupled Rydberg states, employing optimal control techniques to design high-fidelity, time-efficient gate pulses. To minimize experimental complexity, we utilize global driving fields acting identically on all atoms and apply pulse smoothing techniques. While detrimental van der Waals interactions pose a significant challenge, we demonstrate that for both , as a representative alkali atom, and , an alkaline-earth species, high-fidelity pulses can nevertheless be obtained over a broad range of parameters. We identify candidate protocols with reduced susceptibility to noise and analyze their performance under realistic conditions, accounting for atomic motion, Rydberg decay, and experimentally motivated laser frequency and intensity noise. Crucially, we demonstrate that in both Alkali and alkaline-earth-based systems, we can obtain fast iSWAP gates with fidelities of under realistic experimental conditions. These results pave the way for expanding the neutral-atom gate set beyond conventional Rydberg-blockade-based entangling gates.
Physics Subject Headings (PhySH)
Article Text
References (98)
- F. Gyger, M. Ammenwerth, R. Tao, H. Timme, S. Snigirev, I. Bloch, and J. Zeiher, Continuous operation of large-scale atom arrays in optical lattices, Phys. Rev. Res. 6, 033104 (2024).
- M. A. Norcia, H. Kim, W. B. Cairncross, M. Stone, A. Ryou, M. Jaffe, M. O. Brown, K. Barnes, P. Battaglino, T. C. Bohdanowicz, A. Brown, K. Cassella, C.-A. Chen, R. Coxe, D. Crow, J. Epstein, C. Griger, E. Halperin, F. Hummel, A. M. W. Jones, et al., Iterative assembly of atom arrays with cavity-enhanced optical lattices, PRX Quantum 5, 030316 (2024).
- H. J. Manetsch, G. Nomura, E. Bataille, K. H. Leung, X. Lv, and M. Endres, A tweezer array with 6100 highly coherent atomic qubits, Nature (London) 647, 60 (2025).
- N.-C. Chiu, E. C. Trapp, J. Guo, M. H. Abobeih, L. M. Stewart, S. Hollerith, P. L. Stroganov, M. Kalinowski, A. A. Geim, S. J. Evered, S. H. Li, X. Lyu, L. M. Peters, D. Bluvstein, T. T. Wang, M. Greiner, V. Vuletić, and M. D. Lukin, Continuous operation of a coherent 3,000-qubit system, Nature (London) 646, 1075 (2025).
- D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, V. Vuletic, and M. D. Lukin, A quantum processor based on coherent transport of entangled atom arrays, Nature (London) 604, 451 (2022).
- S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletić, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature (London) 622, 268 (2023).
- J. A. Muniz, M. Stone, D. T. Stack, M. Jaffe, J. M. Kindem, L. Wadleigh, E. Zalys-Geller, X. Zhang, C.-A. Chen, M. A. Norcia, J. Epstein, E. Halperin, F. Hummel, T. Wilkason, M. Li, K. Barnes, P. Battaglino, T. C. Bohdanowicz, G. Booth, A. Brown, et al., High-fidelity universal gates in the ground state nuclear spin qubit, PRX Quantum 6, 020334 (2025).
- R. Finkelstein, R. B.-S. Tsai, X. Sun, P. Scholl, S. Direkci, T. Gefen, J. Choi, A. L. Shaw, and M. Endres, Universal quantum operations and ancilla-based readout for tweezer clocks, Nature (London) 634, 321 (2024).
- R. Tao, O. Lib, F. Gyger, H. Timme, M. Ammenwerth, I. Bloch, and J. Zeiher, Universal global gates for a fine-structure qubit in strontium-88, Phys. Rev. Lett. 136, 153602 (2026).
- D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett. 85, 2208 (2000).
- L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, Demonstration of a neutral atom controlled-not quantum gate, Phys. Rev. Lett. 104, 010503 (2010).
- M. H. Goerz, E. J. Halperin, J. M. Aytac, C. P. Koch, and K. B. Whaley, Robustness of high-fidelity Rydberg gates with single-site addressability, Phys. Rev. A 90, 032329 (2014).
- H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuletić, H. Pichler, and M. D. Lukin, Parallel implementation of high-fidelity multiqubit gates with neutral atoms, Phys. Rev. Lett. 123, 170503 (2019).
- S. Jandura and G. Pupillo, Time-optimal two- and three-qubit gates for Rydberg atoms, Quantum 6, 712 (2022).
- C. Fromonteil, D. Bluvstein, and H. Pichler, Protocols for Rydberg entangling gates featuring robustness against quasistatic errors, PRX Quantum 4, 020335 (2023).
- J. Kazemi, M. Schuler, C. Ertler, and W. Lechner, Multiqubit parity gates for Rydberg atoms in various configurations, Phys. Rev. Res. 7, 033269 (2025).
- F. Ginzel, J. Kazemi, V. Torggler, and W. Lechner, Replacement-type quantum gates, Phys. Rev. A 113, 022621 (2026).
- D. Barredo, S. de Léséleuc, V. Lienhard, T. Lahaye, and A. Browaeys, An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays, Science 354, 1021 (2016).
- D. Barredo, V. Lienhard, S. de Léséleuc, T. Lahaye, and A. Browaeys, Synthetic three-dimensional atomic structures assembled atom by atom, Nature (London) 561, 79 (2018).
- A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
- 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. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, et al., Logical quantum processor based on reconfigurable atom arrays, Nature (London) 626, 58 (2024).
- B. W. Reichardt, A. Paetznick, D. Aasen, I. Basov, J. M. Bello-Rivas, P. Bonderson, R. Chao, W. v. Dam, M. B. Hastings, A. Paz, M. P. d. Silva, A. Sundaram, K. M. Svore, A. Vaschillo, Z. Wang, M. Zanner, W. B. Cairncross, C.-A. Chen, D. Crow, H. Kim, et al., Fault-tolerant quantum computation with a neutral atom processor, arXiv:2411.11822.
- D. Bluvstein, A. A. Geim, S. H. Li, S. J. Evered, J. P. Bonilla Ataides, G. Baranes, A. Gu, T. Manovitz, M. Xu, M. Kalinowski, S. Majidy, C. Kokail, N. Maskara, E. C. Trapp, L. M. Stewart, S. Hollerith, H. Zhou, M. J. Gullans, S. F. Yelin, M. Greiner, et al., A fault-tolerant neutral-atom architecture for universal quantum computation, Nature (London) 649, 39 (2026).
- M. Cain, C. Zhao, H. Zhou, N. Meister, J. P. B. Ataides, A. Jaffe, D. Bluvstein, and M. D. Lukin, Correlated decoding of logical algorithms with transversal gates, Phys. Rev. Lett. 133, 240602 (2024).
- P. S. Rodriguez, J. M. Robinson, P. N. Jepsen, Z. He, C. Duckering, C. Zhao, K.-H. Wu, J. Campo, K. Bagnall, M. Kwon, T. Karolyshyn, P. Weinberg, M. Cain, S. J. Evered, A. A. Geim, M. Kalinowski, S. H. Li, T. Manovitz, J. Amato-Grill, J. I. Basham, et al., Experimental demonstration of logical magic state distillation, Nature (London) 645, 620 (2025).
- M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
- G Pelegrí, A. J. Daley, and J. D. Pritchard, High-fidelity multiqubit Rydberg gates via two-photon adiabatic rapid passage, Quantum Sci. Technol. 7, 045020 (2022).
- D. F. Locher, J. Old, K. Brechtelsbauer, J. Holschbach, H. P. Büchler, S. Weber, and M. Müller, Multiqubit Rydberg gates for quantum error correction, PRX Quantum 7, 020354 (2026).
- D. Barredo, H. Labuhn, S. Ravets, T. Lahaye, A. Browaeys, and C. S. Adams, Coherent excitation transfer in a spin chain of three Rydberg atoms, Phys. Rev. Lett. 114, 113002 (2015).
- C. Chen, G. Bornet, M. Bintz, G. Emperauger, L. Leclerc, V. S. Liu, P. Scholl, D. Barredo, J. Hauschild, S. Chatterjee, M. Schuler, A. M. Läuchli, M. P. Zaletel, T. Lahaye, N. Y. Yao, and A. Browaeys, Continuous symmetry breaking in a two-dimensional Rydberg array, Nature (London) 616, 691 (2023).
- B. Sbierski, M. Bintz, S. Chatterjee, M. Schuler, N. Y. Yao, and L. Pollet, Magnetism in the two-dimensional dipolar XY model, Phys. Rev. B 109, 144411 (2024).
- S. de Léséleuc, V. Lienhard, P. Scholl, D. Barredo, S. Weber, N. Lang, H. P. Büchler, T. Lahaye, and A. Browaeys, Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms, Science 365, 775 (2019).
- Z. Yue, Y.-F. Mao, X. Liang, Z.-X. Hua, P. Ge, Y.-X. Chao, K. Li, C. Jia, M. K. Tey, Y. Xu, and L. You, Average topological phase in a disordered Rydberg atom array, Nat. Phys. 22, 844 (2026).
- Y. Chew, T. Tomita, T. P. Mahesh, S. Sugawa, S. de Léséleuc, and K. Ohmori, Ultrafast energy exchange between two single Rydberg atoms on a nanosecond timescale, Nat. Photon. 16, 724 (2022).
- M. H. Goerz, T. Calarco, and C. P. Koch, The quantum speed limit of optimal controlled phasegates for trapped neutral atoms, J. Phys. B: At. Mol. Opt. Phys. 44, 154011 (2011).
- G. Giudici, S. Veroni, G. Giudice, H. Pichler, and J. Zeiher, Fast entangling gates for Rydberg atoms via resonant dipole-dipole interaction, PRX Quantum 6, 030308 (2025).
- Y. Bao, S. S. Yu, 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).
- C. M. Holland, Y. Lu, and L. W. Cheuk, On-demand entanglement of molecules in a reconfigurable optical tweezer array, Science 382, 1143 (2023).
- M. Bergonzoni, S. Jandura, and G. Pupillo, iSWAP gate with polar molecules: Robustness criteria for entangling operations, Phys. Rev. A 112, 032621 (2025).
- M. Bergonzoni, R. R. Riso, and G. Pupillo, Fast quantum gates for neutral atoms separated by a few tens of micrometers, Phys. Rev. Res. (2026), doi:10.1103/hjrh-4327.
- D. M. Abrams, N. Didier, B. R. Johnson, M. P. da Silva, and C. A. Ryan, Implementation of the XY interaction family with calibration of a single pulse, Nat. Electron. 3, 744 (2020).
- E. C. Peterson, G. E. Crooks, and R. S. Smith, Fixed-depth two-qubit circuits and the monodromy polytope, Quantum 4, 247 (2020).
- C. Križan, J. Biznárová, L. Chen, E. Hogedal, A. Osman, C. W. Warren, S. Kosen, H.-X. Li, T. Abad, A. Aggarwal, M. Caputo, J. Fernández-Pendás, A. Gaikwad, L. Grönberg, A. Nylander, R. Rehammar, M. Rommel, O. I. Yuzephovich, A. F. Kockum, J. Govenius, et al., Quantum SWAP gate realized with CZ and iSWAP gates in a superconducting architecture, New J. Phys. 27, 074507 (2025).
- Z. Wang, N. C. Rubin, J. M. Dominy, and E. G. Rieffel, mixers: Analytical and numerical results for the quantum alternating operator ansatz, Phys. Rev. A 101, 012320 (2020).
- M. Drieb-Schön, K. Ender, Y. Javanmard, and W. Lechner, Parity quantum optimization: Encoding constraints, Quantum 7, 951 (2023).
- B. Klaver, S. M. A. Rombouts, M. Fellner, A. Messinger, K. Ender, K. Ludwig, and W. Lechner, swap-less implementation of quantum algorithms, Phys. Rev. A 113, 012443 (2026).
- F. Dreier, C. Fleckenstein, G. Aigner, M. Fellner, R. Stahn, M. Lanthaler, and W. Lechner, Connectivity-aware synthesis of quantum algorithms, arXiv:2501.14020.
- P. Aumann, M. Fellner, D. Alber, M. Cykiert, C. Fleckenstein, R. ter Hoeven, L. Stenzel, R. J. Valencia-Tortora, and W. Lechner, Demonstrating record fidelity for the quantum Fourier transform, arXiv:2604.12465.
- A. Eickbusch, M. McEwen, V. Sivak, A. Bourassa, J. Atalaya, J. Claes, D. Kafri, C. Gidney, C. W. Warren, J. Gross, A. Opremcak, N. Zobrist, K. C. Miao, G. Roberts, K. J. Satzinger, A. Bengtsson, M. Neeley, W. P. Livingston, A. Greene, R. Acharya, et al., Demonstration of dynamic surface codes, Nat. Phys. 21, 1994 (2025).
- G. P. Gehér, D. Byfield, and A. Ruban, Directional codes: A new family of quantum LDPC codes on hexagonal- and square-grid connectivity hardware, arXiv:2507.19430.
- B. Gu, T. Noszko, V. Steffan, J. N. Eberhardt, J. Roffe, J. Eisert, and S. Koutsioumpas, Nearest-neighbour gates are all you need: High-rate quantum low-density parity-check codes on a planar grid, arXiv:2606.19482.
- G. M. Nixon, C. K. McLauchlan, and C. C. van Rest, Vine codes: Low-overhead quantum LDPC codes on a planar square grid, arXiv:2606.20263.
- A. Browaeys, D. Barredo, and T. Lahaye, Experimental investigations of dipole–dipole interactions between a few Rydberg atoms, J. Phys. B: At. Mol. Opt. Phys. 49, 152001 (2016).
- T. G. Walker and M. Saffman, Zeros of Rydberg–Rydberg Föster interactions, J. Phys. B: At. Mol. Opt. Phys. 38, S309 (2005).
- S. Ravets, H. Labuhn, D. Barredo, L. Béguin, T. Lahaye, and A. Browaeys, Coherent dipole–dipole coupling between two single Rydberg atoms at an electrically-tuned Förster resonance, Nat. Phys. 10, 914 (2014).
- I. N. Ashkarin, S. Lepoutre, P. Pillet, I. I. Beterov, I. I. Ryabtsev, and P. Cheinet, Long-range gates via radio-frequency-induced Förster resonances, Phys. Rev. Res. 7, 013034 (2025).
- D. Kurdak, P. R. Banner, Y. Li, S. R. Muleady, A. V. Gorshkov, S. L. Rolston, and J. V. Porto, Enhancement of Rydberg blockade via microwave dressing, Phys. Rev. Lett. 134, 123404 (2025).
- A. L. Shaw, R. Finkelstein, R. B.-S. Tsai, P. Scholl, T. H. Yoon, J. Choi, and M. Endres, Multi-ensemble metrology by programming local rotations with atom movements, Nat. Phys. 20, 195 (2024).
- N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172, 296 (2005).
- J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: Composable transformations of Python + NumPy programs, 2018, https://github.com/jax-ml/jax.
- N. Maskara, S. Ostermann, J. Shee, M. Kalinowski, A. McClain Gomez, R. Araiza Bravo, D. S. Wang, A. I. Krylov, N. Y. Yao, M. Head-Gordon, M. D. Lukin, and S. F. Yelin, Programmable simulations of molecules and materials with reconfigurable quantum processors, Nat. Phys. 21, 289 (2025).
- S. Anand, C. E. Bradley, R. White, V. Ramesh, K. Singh, and H. Bernien, A dual-species Rydberg array, Nat. Phys. 20, 1744 (2024).
- T. M. Graham, Y. Song, J. Scott, C. Poole, L. Phuttitarn, K. Jooya, P. Eichler, X. Jiang, A. Marra, B. Grinkemeyer, M. Kwon, M. Ebert, J. Cherek, M. T. Lichtman, M. Gillette, J. Gilbert, D. Bowman, T. Ballance, C. Campbell, E. D. Dahl, et al., Multi-qubit entanglement and algorithms on a neutral-atom quantum computer, Nature (London) 604, 457 (2022).
- T. M. Graham, L. Phuttitarn, R. Chinnarasu, Y. Song, C. Poole, K. Jooya, J. Scott, A. Scott, P. Eichler, and M. Saffman, Midcircuit measurements on a single-species neutral alkali atom quantum processor, Phys. Rev. X 13, 041051 (2023).
- P. Scholl, A. L. Shaw, R. B.-S. Tsai, R. Finkelstein, J. Choi, and M. Endres, Erasure conversion in a high-fidelity Rydberg quantum simulator, Nature (London) 622, 273 (2023).
- 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).
- H. Katori, M. Takamoto, V. G. Pal'chikov, and V. D. Ovsiannikov, Ultrastable optical clock with neutral atoms in an engineered light shift trap, Phys. Rev. Lett. 91, 173005 (2003).
- A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. L. Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, Sr lattice clock at 1 × fractional uncertainty by remote optical evaluation with a Ca clock, Science 319, 1805 (2008).
- M. D. Swallows, G. K. Campbell, A. D. Ludlow, M. M. Boyd, J. W. Thomsen, M. J. Martin, S. Blatt, T. L. Nicholson, and J. Ye, Precision measurement of fermionic collisions using an optical lattice clock with 1 × inaccuracy, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57, 574 (2010).
- N. D. Lemke, A. D. Ludlow, Z. W. Barber, T. M. Fortier, S. A. Diddams, Y. Jiang, S. R. Jefferts, T. P. Heavner, T. E. Parker, and C. W. Oates, Spin- optical lattice clock, Phys. Rev. Lett. 103, 063001 (2009).
- P. G. Westergaard, J. Lodewyck, L. Lorini, A. Lecallier, E. A. Burt, M. Zawada, J. Millo, and P. Lemonde, Lattice-induced frequency shifts in Sr optical lattice clocks at the level, Phys. Rev. Lett. 106, 210801 (2011).
- M. Bishof, M. J. Martin, M. D. Swallows, C. Benko, Y. Lin, G. Quéméner, A. M. Rey, and J. Ye, Inelastic collisions and density-dependent excitation suppression in a optical lattice clock, Phys. Rev. A 84, 052716 (2011).
- A. Heinz, A. J. Park, N. Šantić, J. Trautmann, S. G. Porsev, M. S. Safronova, I. Bloch, and S. Blatt, State-dependent optical lattices for the strontium optical qubit, Phys. Rev. Lett. 124, 203201 (2020).
- F. Meinert, T. Pfau, and C. Hölzl, Quantum computing device, use, and method, EU Patent Application No. EP20214187.5 (2021).
- S. Pucher, V. Klüsener, F. Spriestersbach, J. Geiger, A. Schindewolf, I. Bloch, and S. Blatt, Fine-structure qubit encoded in metastable strontium trapped in an optical lattice, Phys. Rev. Lett. 132, 150605 (2024).
- G. Unnikrishnan, P. Ilzhöfer, A. Scholz, C. Hölzl, A. Götzelmann, R. K. Gupta, J. Zhao, J. Krauter, S. Weber, N. Makki, H. P. Büchler, T. Pfau, and F. Meinert, Coherent control of the fine-structure qubit in a single alkaline-earth atom, Phys. Rev. Lett. 132, 150606 (2024).
- J. Mögerle, F. Hummel, A. Keil, T. Legrand, E. J. Braun, H. Menke, J. King, B. Olmos, S. Hofferberth, H. P. Büchler, and S. Weber, Accurate modeling of Rydberg atoms and their interactions: Theory and implementation in pairinteraction, arXiv:2605.14993.
- We compute the and coefficients independently for each via perturbation theory of the effective Hamiltonian in the subspace of and with approximately 100 000 basis pair states.
- S. de Léséleuc, D. Barredo, V. Lienhard, A. Browaeys, and T. Lahaye, Analysis of imperfections in the coherent optical excitation of single atoms to Rydberg states, Phys. Rev. A 97, 053803 (2018).
- A. Pagano, S. Weber, D. Jaschke, T. Pfau, F. Meinert, S. Montangero, and H. P. Büchler, Error budgeting for a controlled-phase gate with strontium-88 Rydberg atoms, Phys. Rev. Res. 4, 033019 (2022).
- K.-N. Schymik, S. Pancaldi, F. Nogrette, D. Barredo, J. Paris, A. Browaeys, and T. Lahaye, Single atoms with 6000-second trapping lifetimes in optical-tweezer arrays at cryogenic temperatures, Phys. Rev. Appl. 16, 034013 (2021).
- G. Pichard, D. Lim, E. Bloch, J. Vaneecloo, L. Bourachot, G.-J. Both, G. Mériaux, S. Dutartre, R. Hostein, J. Paris, B. Ximenez, A. Signoles, A. Browaeys, T. Lahaye, and D. Dreon, Rearrangement of individual atoms in a 2000-site optical-tweezer array at cryogenic temperatures, Phys. Rev. Appl. 22, 024073 (2024).
- V. A. Kostelecký and M. M. Nieto, Analytical wave functions for atomic quantum-defect theory, Phys. Rev. A 32, 3243 (1985).
- X. Jiang, J. Scott, M. Friesen, and M. Saffman, Sensitivity of quantum gate fidelity to laser phase and intensity noise, Phys. Rev. A 107, 042611 (2023).
- C. Poole, T. M. Graham, M. A. Perlin, M. Otten, and M. Saffman, Architecture for fast implementation of qLDPC codes with optimized Rydberg gates, Phys. Rev. A 111, 022433 (2025).
- S. R. Cohen and J. D. Thompson, Quantum computing with circular Rydberg atoms, PRX Quantum 2, 030322 (2021).
- S. Ma, G. Liu, P. Peng, B. Zhang, S. Jandura, J. Claes, A. P. Burgers, G. Pupillo, S. Puri, and J. D. Thompson, High-fidelity gates and mid-circuit erasure conversion in an atomic qubit, Nature (London) 622, 279 (2023).
- R. de Keijzer, J. Snijders, A. Carvalho, and S. Kokkelmans, Pulse family optimization for parametrized quantum gates using spectral clustering, Acad. Quantum 1, 7374 (2024).
- H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer Science & Business Media, New York, 2013).
- C. Froese-Fischer, T. Brage, and P. Jonsson, Computational Atomic Structure: An MCHF Approach (Routledge, New York, 2022).
- M. Y. Amusia, Atomic Photoeffect (Springer Science & Business Media, New York, 2013).
- L. Couturier, I. Nosske, F. Hu, C. Tan, C. Qiao, Y. H. Jiang, P. Chen, and M. Weidemüller, Measurement of the strontium triplet Rydberg series by depletion spectroscopy of ultracold atoms, Phys. Rev. A 99, 022503 (2019).
- J. A. Armstrong, J. J. Wynne, and P. Esherick, Bound, odd-parity 1 spectra of the alkaline Earths: Ca, Sr, and Ba, J. Opt. Soc. Am. 69, 211 (1979).
- S. Kunze, R. Hohmann, H.-J. Kluge, J. Lantzsch, L. Monz, J. Stenner, K. Stratmann, K. Wendt, and K. Zimmer, Lifetime measurements of highly excited Rydberg states of strontium I, Z. Phys. D 27, 111 (1993).
- F. Camargo, J. D. Whalen, R. Ding, H. R. Sadeghpour, S. Yoshida, J. Burgdörfer, F. B. Dunning, and T. C. Killian, Lifetimes of ultra-long-range strontium Rydberg molecules, Phys. Rev. A 93, 022702 (2016).
- J. R. Rubbmark and S. A. Borgström, Rydberg series in strontium found in absorption by selectively laser-excited atoms, Phys. Scr. 18, 196 (1978).
- J. A. Muniz, D. J. Young, J. R. K. Cline, and J. K. Thompson, Cavity-QED measurements of the millihertz optical clock transition and determination of its natural linewidth, Phys. Rev. Res. 3, 023152 (2021).
- A. Derevianko, Feasibility of cooling and trapping metastable alkaline-earth atoms, Phys. Rev. Lett. 87, 023002 (2001).