Preparing quantum states by measurement-feedback control with Bayesian optimization
Yadong Wu1,2,3,4, Juan Yao5,6,7, Pengfei Zhang1,8()
1. Department of Physics, Fudan University, Shanghai 200438, China 2. State Key Laboratory of Surface Physics, Key Laboratory of Micro and Nano Photonic Structures (MOE), Fudan University, Shanghai 200438, China 3. Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China 4. Shanghai Qi Zhi Institute, AI Tower, Xuhui District, Shanghai 200232, China 5. Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China 6. International Quantum Academy, Shenzhen 518048, China 7. Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China 8. Walter Burke Institute for Theoretical Physics & Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
The preparation of quantum states is crucial for enabling quantum computations and simulations. In this work, we present a general framework for preparing ground states of many-body systems by combining the measurement-feedback control process (MFCP) with machine learning techniques. Specifically, we employ Bayesian optimization (BO) to enhance the efficiency of determining the measurement and feedback operators within the MFCP. As an illustration, we study the ground state preparation of the one-dimensional Bose−Hubbard model. Through BO, we are able to identify optimal parameters that can effectively drive the system towards low-energy states with a high probability across various quantum trajectories. Our results open up new directions for further exploration and development of advanced control strategies for quantum computations and simulations.
. [J]. Frontiers of Physics, 2023, 18(6): 61301.
Yadong Wu, Juan Yao, Pengfei Zhang. Preparing quantum states by measurement-feedback control with Bayesian optimization. Front. Phys. , 2023, 18(6): 61301.
M. Geremia J. , K. Stockton J. , Mabuchi H. . Real-time quantum feedback control of atomic spin-squeezing. Science, 2004, 304(5668): 270 https://doi.org/10.1126/science.1095374
2
M. Geremia J. . Deterministic and nondestructively verifiable preparation of photon number states. Phys. Rev. Lett., 2006, 97(7): 073601 https://doi.org/10.1103/PhysRevLett.97.073601
Negretti A. , V. Poulsen U. , Mølmer K. . Quantum superposition state production by continuous observations and feedback. Phys. Rev. Lett., 2007, 99(22): 223601 https://doi.org/10.1103/PhysRevLett.99.223601
5
Sayrin C. , Dotsenko I. , Zhou X. , Peaudecerf B. , Rybarczyk T. , Gleyzes S. , Rouchon P. , Mirrahimi M. , Amini H. , Brune M. , M. Raimond J. , Haroche S. . Real-time quantum feedback prepares and stabilizes photon number states. Nature, 2011, 477(7362): 73 https://doi.org/10.1038/nature10376
6
Zhou X. , Dotsenko I. , Peaudecerf B. , Rybarczyk T. , Sayrin C. , Gleyzes S. , M. Raimond J. , Brune M. , Haroche S. . Field locked to a Fock state by quantum feedback with single photon corrections. Phys. Rev. Lett., 2012, 108(24): 243602 https://doi.org/10.1103/PhysRevLett.108.243602
7
Ristè D. , Dukalski M. , A. Watson C. , de Lange G. , J. Tiggelman M. , M. Blanter Ya. , W. Lehnert K. , N. Schouten R. , DiCarlo L. . Deterministic entanglement of superconducting qubits by parity measurement and feedback. Nature, 2013, 502(7471): 350 https://doi.org/10.1038/nature12513
8
Inoue R. , I. R. Tanaka S. , Namiki R. , Sagawa T. , Takahashi Y. . Unconditional quantumnoise suppression via measurement-based quantum feedback. Phys. Rev. Lett., 2013, 110(16): 163602 https://doi.org/10.1103/PhysRevLett.110.163602
9
C. J. Wade A. , F. Sherson J. , Mølmer K. . Squeezing and entanglement of density oscillations in a Bose-Einstein condensate. Phys. Rev. Lett., 2015, 115(6): 060401 https://doi.org/10.1103/PhysRevLett.115.060401
10
C. Cox K. , P. Greve G. , M. Weiner J. , K. Thompson J. . Deterministic squeezed states with collective measurements and feedback. Phys. Rev. Lett., 2016, 116(9): 093602 https://doi.org/10.1103/PhysRevLett.116.093602
11
Gajdacz M. , J. Hilliard A. , A. Kristensen M. , L. Pedersen P. , Klempt C. , J. Arlt J. , F. Sherson J. . Preparation of ultracold atom clouds at the shot noise level. Phys. Rev. Lett., 2016, 117(7): 073604 https://doi.org/10.1103/PhysRevLett.117.073604
12
Lammers J. , Weimer H. , Hammerer K. . Open-system many-body dynamics through interferometric measurements and feedback. Phys. Rev. A, 2016, 94(5): 052120 https://doi.org/10.1103/PhysRevA.94.052120
13
Sudhir V. , J. Wilson D. , Schilling R. , Schütz H. , A. Fedorov S. , H. Ghadimi A. , Nunnenkamp A. , J. Kippenberg T. . Appearance and disappearance of quantum correlations in measurement-based feedback control of a mechanical oscillator. Phys. Rev. X, 2017, 7(1): 011001 https://doi.org/10.1103/PhysRevX.7.011001
14
J. Briegel H. , Raussendorf R. . Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett., 2001, 86(5): 910 https://doi.org/10.1103/PhysRevLett.86.910
15
Raussendorf R. , Bravyi S. , Harrington J. . Long-range quantum entanglement in noisy cluster states. Phys. Rev. A, 2005, 71(6): 062313 https://doi.org/10.1103/PhysRevA.71.062313
16
Verresen R.Tantivasadakarn N.Vishwanath A., Efficiently preparing Schrödinger’s cat, fractons and non-Abelian topological order in quantum devices, arXiv: 2112.03061 (2021)
17
Tantivasadakarn N.Thorngren R.Vishwanath A.Verresen R., Long-range entanglement from measuring symmetry-protected topological phases, arXiv: 2112.01519 (2021)
18
Y. Zhu G.Tantivasadakarn N.Vishwanath A.Trebst S.Verresen R., Nishimori’s cat: Stable long-range entanglement from finite-depth unitaries and weak measurements, arXiv: 2208.11136 (2022)
19
Y. Lee J.Ji W.Bi Z.Fisher M., Measurement-prepared quantum criticality: From Ising model to gauge theory, and beyond, arXiv: 2208.11699 (2022)
20
Tantivasadakarn N.Vishwanath A.Verresen R., A hierarchy of topological order from finite depth unitaries, measurement and feedforward, arXiv: 2209.06202 (2022)
21
Bravyi S.Kim I.Kliesch A.Koenig R., Adaptive constant-depth circuits for manipulating non-Abelian anyons, arXiv: 2205.01933 (2022)
22
C. Lu T.A. Lessa L.H. Kim I.H. Hsieh T., Measurement as a shortcut to long-range entangled quantum matter, arXiv: 2206.13527 (2022)
Zhang J. , Liu Y. , B. Wu R. , Jacobs K. , Nori F. . Quantum feedback: Theory, experiments, and applications. Phys. Rep., 2017, 679: 1 https://doi.org/10.1016/j.physrep.2017.02.003
A. Muschik C. , Hammerer K. , S. Polzik E. , J. Cirac I. . Quantum teleportation of dynamics and effective interactions between remote systems. Phys. Rev. Lett., 2013, 111(2): 020501 https://doi.org/10.1103/PhysRevLett.111.020501
28
L. Grimsmo A. , S. Parkins A. , S. Skagerstam B. . Rapid steady-state convergence for quantum systems using time-delayed feedback control. New J. Phys., 2014, 16(6): 065004 https://doi.org/10.1088/1367-2630/16/6/065004
29
C. Emary, E. Schöll , T. Brandes. W. Kopylov . Time-delayed feedback control of the Dicke–Hepp–Lieb superradiant quantum phase transition. New J. Phys., 2015, 17(1): 013040 https://doi.org/10.1088/1367-2630/17/1/013040
30
Mazzucchi G. , F. Caballero-Benitez S. , A. Ivanov D. , B. Mekhov I. . Quantum optical feedback control for creating strong correlations in many-body systems. Optica, 2016, 3(11): 1213 https://doi.org/10.1364/OPTICA.3.001213
31
Shankar A. , P. Greve G. , Wu B. , K. Thompson J. , Holland M. . Continuous real-time tracking of a quantum phase below the standard quantum limit. Phys. Rev. Lett., 2019, 122(23): 233602 https://doi.org/10.1103/PhysRevLett.122.233602
32
A. Ivanov D. , Y. Ivanova T. , F. Caballero-Benitez S. , B. Mekhov I. . Cavityless self-organization of ultracold atoms due to the feedback-induced phase transition. Sci. Rep., 2020, 10(1): 10550 https://doi.org/10.1038/s41598-020-67280-3
33
A. Ivanov D. , Yu. Ivanova T. , F. Caballero-Benitez S. , B. Mekhov I. . Feedback-induced quantum phase transitions using weak measurements. Phys. Rev. Lett., 2020, 124(1): 010603 https://doi.org/10.1103/PhysRevLett.124.010603
34
Kroeger K. , Dogra N. , Rosa-Medina R. , Paluch M. , Ferri F. , Donner T. , Esslinger T. . Continuous feedback on a quantum gas coupled to an optical cavity. New J. Phys., 2020, 22(3): 033020 https://doi.org/10.1088/1367-2630/ab73cc
35
H. Muñoz-Arias M. , M. Poggi P. , S. Jessen P. , H. Deutsch I. . Simulating nonlinear dynamics of collective spins via quantum measurement and feedback. Phys. Rev. Lett., 2020, 124(11): 110503 https://doi.org/10.1103/PhysRevLett.124.110503
36
H. Muñoz-Arias M. , H. Deutsch I. , S. Jessen P. , M. Poggi P. . Simulation of the complex dynamics of mean-field p-spin models using measurement-based quantum feedback control. Phys. Rev. A, 2020, 102(2): 022610 https://doi.org/10.1103/PhysRevA.102.022610
37
M. Hurst H. , Guo S. , B. Spielman I. . Feedback induced magnetic phases in binary Bose-Einstein condensates. Phys. Rev. Res., 2020, 2(4): 043325 https://doi.org/10.1103/PhysRevResearch.2.043325
38
A. Ivanov D. , Yu. Ivanova T. , F. Caballero-Benitez S. , B. Mekhov I. . Tuning the universality class of phase transitions by feedback: Open quantum systems beyond dissipation. Phys. Rev. A, 2021, 104(3): 033719 https://doi.org/10.1103/PhysRevA.104.033719
39
M. Hurst H. , B. Spielman I. . Measurement-induced dynamics and stabilization of spinor-condensate domain walls. Phys. Rev. A, 2019, 99(5): 053612 https://doi.org/10.1103/PhysRevA.99.053612
40
T. Young J. , V. Gorshkov A. , B. Spielman I. . Feedback-stabilized dynamical steady states in the bosehubbard model. Phys. Rev. Res., 2021, 3(4): 043075 https://doi.org/10.1103/PhysRevResearch.3.043075
41
M. Wiseman H.J. Milburn G., Quantum Measurement and Control, Cambridge University Press, 2009
Wang C. , Zhai H. . Machine learning of frustrated classical spin models (i): Principal component analysis. Phys. Rev. B, 2017, 96(14): 144432 https://doi.org/10.1103/PhysRevB.96.144432
Wang C. , Zhai H. . Machine learning of frustrated classical spin models (ii): Kernel principal component analysis. Front. Phys., 2018, 13(5): 130507 https://doi.org/10.1007/s11467-018-0798-7
46
Sun N. , Yi J. , Zhang P. , Shen H. , Zhai H. . Deep learning topological invariants of band insulators. Phys. Rev. B, 2018, 98(8): 085402 https://doi.org/10.1103/PhysRevB.98.085402
47
Song T. , Lee H. . Accelerated continuous time quantum Monte Carlo method with machine learning. Phys. Rev. B, 2019, 100(4): 045153 https://doi.org/10.1103/PhysRevB.100.045153
48
Wang C. , Zhai H. , Z. You Y. . Emergent Schrödinger equation in an introspective machine learning architecture. Sci. Bull. (Beijing), 2019, 64(17): 1228 https://doi.org/10.1016/j.scib.2019.07.014
49
Zhang Y. , Mesaros A. , Fujita K. , D. Edkins S. , H. Hamidian M. , Ch’ng K. , Eisaki H. , Uchida S. , C. S. Davis J. , Khatami E. , A. Kim E. . Machine learning in electronic-quantum-matter imaging experiments. Nature, 2019, 570(7762): 484 https://doi.org/10.1038/s41586-019-1319-8
50
S. Rem B. , Käming N. , Tarnowski M. , Asteria L. , Fläschner N. , Becker C. , Sengstock K. , Weitenberg C. . Identifying quantum phase transitions using artificial neural networks on experimental data. Nat. Phys., 2019, 15(9): 917 https://doi.org/10.1038/s41567-019-0554-0
51
Bohrdt A. , S. Chiu C. , Ji G. , Xu M. , Greif D. , Greiner M. , Demler E. , Grusdt F. , Knap M. . Classifying snapshots of the doped Hubbard model with machine learning. Nat. Phys., 2019, 15(9): 921 https://doi.org/10.1038/s41567-019-0565-x
Torlai G. , Timar B. , P. L. van Nieuwenburg E. , Levine H. , Omran A. , Keesling A. , Bernien H. , Greiner M. , Vuletić V. , D. Lukin M. , G. Melko R. , Endres M. . Integrating neural networks with a quantum simulator for state reconstruction. Phys. Rev. Lett., 2019, 123(23): 230504 https://doi.org/10.1103/PhysRevLett.123.230504
54
M. Palmieri A. , Kovlakov E. , Bianchi F. , Yudin D. , Straupe S. , D. Biamonte J. , Kulik S. . Experimental neural network enhanced quantum tomography. npj Quantum Inf., 2020, 6(1): 20 https://doi.org/10.1038/s41534-020-0248-6
55
Wu Y. , Meng Z. , Wen K. , Mi C. , Zhang J. , Zhai H. . Active learning approach to optimization of experimental control. Chin. Phys. Lett., 2020, 37(10): 103201 https://doi.org/10.1088/0256-307X/37/10/103201
56
Saggio V. , E. Asenbeck B. , Hamann A. , Strömberg T. , Schiansky P. , Dunjko V. , Friis N. , C. Harris N. , Hochberg M. , Englund D. , Wölk S. , J. Briegel H. , Walther P. . Experimental quantum speed-up in reinforcement learning agents. Nature, 2021, 591(7849): 229 https://doi.org/10.1038/s41586-021-03242-7
57
Baum Y. , Amico M. , Howell S. , Hush M. , Liuzzi M. , Mundada P. , Merkh T. , R. R. Carvalho A. , J. Biercuk M. . Seantal deep reinforcement learning for error-robust gate-set design on a superconducting quantum computer. PRX Quantum, 2021, 2(4): 040324 https://doi.org/10.1103/PRXQuantum.2.040324
58
Castaldo D. , Rosa M. , Corni S. . Quantum optimal control with quantum computers: A hybrid algorithm featuring machine learning optimization. Phys. Rev. A, 2021, 103(2): 022613 https://doi.org/10.1103/PhysRevA.103.022613
59
V. Sivak V. , Eickbusch A. , Liu H. , Royer B. , Tsioutsios I. , H. Devoret M. . Model-free quantum control with reinforcement learning. Phys. Rev. X, 2022, 12(1): 011059 https://doi.org/10.1103/PhysRevX.12.011059
60
A. Erdman P. , Noé F. . Identifying optimal cycles in quantum thermal machines with reinforcement-learning. npj Quantum Inf., 2022, 8(1): 1 https://doi.org/10.1038/s41534-021-00512-0
61
F. Langer M. , Goeßmann A. , Rupp M. . Representations of molecules and materials for interpolation of quantum-mechanical simulations via machine learning. npj Comput. Mater., 2022, 8(1): 41 https://doi.org/10.1038/s41524-022-00721-x
62
A. Luchnikov I. , O. Kiktenko E. , A. Gavreev M. , Ouerdane H. , N. Filippov S. , K. Fedorov A. . Probing non-Markovian quantum dynamics with data-driven analysis: Beyond “blackbox” machine-learning models. Phys. Rev. Res., 2022, 4(4): 043002 https://doi.org/10.1103/PhysRevResearch.4.043002
P. L. van Nieuwenburg E. , H. Liu Y. , D. Huber S. . Learning phase transitions by confusion. Nat. Phys., 2017, 13(5): 435 https://doi.org/10.1038/nphys4037
H. Liu Y. , P. L. van Nieuwenburg E. . Discriminative cooperative networks for detecting phase transitions. Phys. Rev. Lett., 2018, 120(17): 176401 https://doi.org/10.1103/PhysRevLett.120.176401
Carleo G. , Troyer M. . Solving the quantum many-body problem with artificial neural networks. Science, 2017, 355(6325): 602 https://doi.org/10.1126/science.aag2302
71
Gao X. , M. Duan L. . Efficient representation of quantum many-body states with deep neural networks. Nat. Commun., 2017, 8(1): 662 https://doi.org/10.1038/s41467-017-00705-2
72
Cai Z. , Liu J. . Approximating quantum many-body wave functions using artificial neural networks. Phys. Rev. B, 2018, 97(3): 035116 https://doi.org/10.1103/PhysRevB.97.035116
73
Saito H. . Method to solve quantum few-body problems with artificial neural networks. J. Phys. Soc. Jpn., 2018, 87(7): 074002 https://doi.org/10.7566/JPSJ.87.074002
74
Torlai G. , Mazzola G. , Carrasquilla J. , Troyer M. , Melko R. , Carleo G. . Neuralnetwork quantum state tomography. Nat. Phys., 2018, 14(5): 447 https://doi.org/10.1038/s41567-018-0048-5
75
Wu Y. , Zhang P. , Shen H. , Zhai H. . Visualizing a neural network that develops quantum perturbation theory. Phys. Rev. A, 2018, 98(1): 010701 https://doi.org/10.1103/PhysRevA.98.010701
76
Wang C. , Li H. , Hao Z. , Li X. , Zou C. , Cai P. , Wang Y. , Z. You Y. , Zhai H. . Machine learning identification of impurities in the STM images. Chin. Phys. B, 2020, 29(11): 116805 https://doi.org/10.1088/1674-1056/abc0d5
77
Shahriari B. , Swersky K. , Wang Z. , P. Adams R. , de Freitas N. . Taking the human out of the loop: A review of Bayesian optimization. Proc. IEEE, 2016, 104(1): 148 https://doi.org/10.1109/JPROC.2015.2494218
78
A. Vargas-Hernández R. , Guan Y. , H. Zhang D. , V. Krems R. . Bayesian optimization for the inverse scattering problem in quantum reaction dynamics. New J. Phys., 2019, 21(2): 022001 https://doi.org/10.1088/1367-2630/ab0099
79
Mukherjee R. , Sauvage F. , Xie H. , Löw R. , Mintert F. . Preparation of ordered states in ultra-cold gases using Bayesian optimization. New J. Phys., 2020, 22(7): 075001 https://doi.org/10.1088/1367-2630/ab8677
80
Kuroś A. , Mukherjee R. , Golletz W. , Sauvage F. , Giergiel K. , Mintert F. , Sacha K. . Phase diagram and optimal control for n-tupling discrete time crystal. New J. Phys., 2020, 22(9): 095001 https://doi.org/10.1088/1367-2630/abb03e
T. Belmiro Chu C. , L. Sheu Y. , I. Chu S. . Bayesian optimal control of the ultrashort circularly polarized attosecond pulse generation by two-color polarization gating. Opt. Express, 2021, 29(21): 32900 https://doi.org/10.1364/OE.438212
83
Kuroś A. , Mukherjee R. , Mintert F. , Sacha K. . Controlled preparation of phases in two-dimensional time crystals. Phys. Rev. Res., 2021, 3(4): 043203 https://doi.org/10.1103/PhysRevResearch.3.043203
84
J. Xie Y. , N. Dai H. , S. Yuan Z. , Deng Y. , Li X. , A. Chen Y. , W. Pan J. . Bayesian learning for optimal control of quantum many-body states in optical lattices. Phys. Rev. A, 2022, 106(1): 013316 https://doi.org/10.1103/PhysRevA.106.013316
85
L. Cortes C. , Lefebvre P. , Lauk N. , J. Davis M. , Sinclair N. , K. Gray S. , Oblak D. . Sample-efficient adaptive calibration of quantum networks using Bayesian optimization. Phys. Rev. Appl., 2022, 17(3): 034067 https://doi.org/10.1103/PhysRevApplied.17.034067
86
Jacobs K. , A. Steck D. . A straightforward introduction to continuous quantum measurement. Contemp. Phys., 2006, 47(5): 279 https://doi.org/10.1080/00107510601101934
Chertkov E. , K. Clark B. . Computational inverse method for constructing spaces of quantum models from wave functions. Phys. Rev. X, 2018, 8(3): 031029 https://doi.org/10.1103/PhysRevX.8.031029
Yao Z. , Pan L. , Liu S. , Zhang P. . Bounding entanglement entropy using zeros of local correlation matrices. Phys. Rev. Res., 2022, 4(4): L042037 https://doi.org/10.1103/PhysRevResearch.4.L042037
Ritsch H. , Domokos P. , Brennecke F. , Esslinger T. . Cold atoms in cavity-generated dynamical optical potentials. Rev. Mod. Phys., 2013, 85(2): 553 https://doi.org/10.1103/RevModPhys.85.553
94
J. Elliott T. , Kozlowski W. , F. Caballero-Benitez S. , B. Mekhov I. . Multipartite entangled spatial modes of ultracold atoms generated and controlled by quantum measurement. Phys. Rev. Lett., 2015, 114(11): 113604 https://doi.org/10.1103/PhysRevLett.114.113604
95
Ashida Y. , Ueda M. . Diffraction-unlimited position measurement of ultracold atoms in an optical lattice. Phys. Rev. Lett., 2015, 115(9): 095301 https://doi.org/10.1103/PhysRevLett.115.095301
96
W. Shor P., Fault-tolerant quantum computation, in: Proceedings of 37th Conference on Foundations of Computer Science, pp 56–65, IEEE, 1996
97
Aharonov D.Ben-Or M., Fault-tolerant quantum computation with constant error, in: Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, pp 176–188, 1997
98
J. Preskill, Fault-tolerant quantum computation, in: Introduction to Quantum Computation and Information, pp 213–269, World Scientific, 1998
Gyongyosi L. . Quantum state optimization and computational pathway evaluation for gate-model quantum computers. Sci. Rep., 2020, 10(1): 4543 https://doi.org/10.1038/s41598-020-61316-4
G. D. Torre E. , Diehl S. , D. Lukin M. , Sachdev S. , Strack P. . Keldysh approach for nonequilibrium phase transitions in quantum optics: Beyond the Dicke model in optical cavities. Phys. Rev. A, 2013, 87(2): 023831 https://doi.org/10.1103/PhysRevA.87.023831
105
F. Maghrebi M. , V. Gorshkov A. . Nonequilibrium many-body steady states via Keldysh formalism. Phys. Rev. B, 2016, 93(1): 014307 https://doi.org/10.1103/PhysRevB.93.014307
106
Foss-Feig M. , Niroula P. , T. Young J. , Hafezi M. , V. Gorshkov A. , M. Wilson R. , F. Maghrebi M. . Emergent equilibrium in many-body optical bistability. Phys. Rev. A, 2017, 95(4): 043826 https://doi.org/10.1103/PhysRevA.95.043826
107
Skinner B. , Ruhman J. , Nahum A. . Measurement-induced phase transitions in the dynamics of entanglement. Phys. Rev. X, 2019, 9(3): 031009 https://doi.org/10.1103/PhysRevX.9.031009
108
Choi S. , Bao Y. , L. Qi X. , Altman E. . Quantum error correction in scrambling dynamics and measurement-induced phase transition. Phys. Rev. Lett., 2020, 125(3): 030505 https://doi.org/10.1103/PhysRevLett.125.030505
109
Tang Q. , Zhu W. . Measurement-induced phase transition: A case study in the nonintegrable model by density matrix renormalization group calculations. Phys. Rev. Res., 2020, 2(1): 013022 https://doi.org/10.1103/PhysRevResearch.2.013022
110
Fan R. , Vijay S. , Vishwanath A. , Z. You Y. . Self-organized error correction in random unitary circuits with measurement. Phys. Rev. B, 2021, 103(17): 174309 https://doi.org/10.1103/PhysRevB.103.174309
