Quantum simulation of Hofstadter butterfly with synthetic gauge fields on two-dimensional superconducting-qubit lattices

doi:10.1007/s11467-023-1319-x

Frontiers of Physics

2023, Vol. 18

Issue (6): 61302 https://doi.org/10.1007/s11467-023-1319-x

本期目录

Quantum simulation of Hofstadter butterfly with synthetic gauge fields on two-dimensional superconducting-qubit lattices

Wei Feng¹, Dexi Shao¹, Guo-Qiang Zhang¹, Qi-Ping Su¹, Jun-Xiang Zhang²(

), Chui-Ping Yang¹(

)

¹. School of Physics, Hangzhou Normal University, Hangzhou 311121, China
². School of Physics, Zhejiang University, Hangzhou 310027, China

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Abstract：

Motivated by recent realizations of two-dimensional (2D) superconducting-qubit lattices, we propose a protocol to simulate Hofstadter butterfly with synthetic gauge fields in superconducting circuits. Based on the existing 2D superconducting-qubit lattices, we construct a generalized Hofstadter model on zigzag lattices, which has a fractal energy spectrum similar to the original Hofstadter butterfly. By periodically modulating the resonant frequencies of qubits, we engineer a synthetic gauge field to mimic the generalized Hofstadter Hamiltonian. A spectroscopic method is used to demonstrate the Hofstadter butterfly from the time evolutions of experimental observables. We numerically simulate the dynamics of the system with realistic parameters, and the results show a butterfly spectrum clearly. Our proposal provides a promising way to realize the Hofstadter butterfly on the latest 2D superconducting-qubit lattices and will stimulate the quantum simulation of novel properties induced by magnetic fields in superconducting circuits.

Key words： quantum simulation superconducting circuits superconducting qubit quantum computation

收稿日期: 2023-03-22 出版日期: 2023-06-30

Corresponding Author(s): Jun-Xiang Zhang,Chui-Ping Yang

引用本文:

. [J]. Frontiers of Physics, 2023, 18(6): 61302.
Wei Feng, Dexi Shao, Guo-Qiang Zhang, Qi-Ping Su, Jun-Xiang Zhang, Chui-Ping Yang. Quantum simulation of Hofstadter butterfly with synthetic gauge fields on two-dimensional superconducting-qubit lattices. Front. Phys. , 2023, 18(6): 61302.

链接本文:

https://academic.hep.com.cn/fop/CN/10.1007/s11467-023-1319-x
https://academic.hep.com.cn/fop/CN/Y2023/V18/I6/61302

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1	P. Feynman R., Simulating physics with computers, Int. J. Theor. Phys. 21(6−7), 467 (1982)
2	M. Georgescu I. , Ashhab S. , Nori F. . Quantum simulation. Rev. Mod. Phys., 2014, 86(1): 153 https://doi.org/10.1103/RevModPhys.86.153
3	Buluta I. , Nori F. . Quantum simulators. Science, 2009, 326(5949): 108 https://doi.org/10.1126/science.1177838
4	Somaroo S. , H. Tseng C. , F. Havel T. , Laflamme R. , G. Cory D. . Quantum simulations on a quantum computer. Phys. Rev. Lett., 1999, 82(26): 5381 https://doi.org/10.1103/PhysRevLett.82.5381
5	Simon J. , S. Bakr W. , Ma R. , E. Tai M. , M. Preiss P. , Greiner M. . Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature, 2011, 472(7343): 307 https://doi.org/10.1038/nature09994
6	Bloch I. , Dalibard J. , Nascimbene S. . Quantum simulations with ultracold quantum gases. Nat. Phys., 2012, 8(4): 267 https://doi.org/10.1038/nphys2259
7	Kim K. , S. Chang M. , Korenblit S. , Islam R. , E. Edwards E. , K. Freericks J. , D. Lin G. , M. Duan L. , Monroe C. . Quantum simulation of frustrated Ising spins with trapped ions. Nature, 2010, 465(7298): 590 https://doi.org/10.1038/nature09071
8	R. Hofstadter D. . Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B, 1976, 14(6): 2239 https://doi.org/10.1103/PhysRevB.14.2239
9	R. Dean C. , Wang L. , Maher P. , Forsythe C. , Ghahari F. , Gao Y. , Katoch J. , Ishigami M. , Moon P. , Koshino M. , Taniguchi T. , Watanabe K. , L. Shepard K. , Hone J. , Kim P. . Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature, 2013, 497(7451): 598 https://doi.org/10.1038/nature12186
10	A. Ponomarenko L. , V. Gorbachev R. , L. Yu G. , C. Elias D. , Jalil R. , A. Patel A. , Mishchenko A. , S. Mayorov A. , R. Woods C. , R. Wallbank J. , Mucha-Kruczynski M. , A. Piot B. , Potemski M. , V. Grigorieva I. , S. Novoselov K. , Guinea F. , I. Fal’ko V. , K. Geim A. . Cloning of Dirac fermions in graphene superlattices. Nature, 2013, 497(7451): 594 https://doi.org/10.1038/nature12187
11	Hunt B. , D. Sanchez-Yamagishi J. , F. Young A. , Yankowitz M. , J. LeRoy B. , Watanabe K. , Taniguchi T. , Moon P. , Koshino M. , Jarillo-Herrero P. , C. Ashoori R. . Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science, 2013, 340(6139): 1427 https://doi.org/10.1126/science.1237240
12	S. Wu Q. , Liu J. , Guan Y. , V. Yazyev O. . Landau levels as a probe for band topology in graphene moiré superlattices. Phys. Rev. Lett., 2021, 126(5): 056401 https://doi.org/10.1103/PhysRevLett.126.056401
13	V. Rozhkov A. , O. Sboychakov A. , L. Rakhmanov A. , Nori F. . Electronic properties of graphene-based bilayer systems. Phys. Rep., 2016, 648: 1 https://doi.org/10.1016/j.physrep.2016.07.003
14	Dalibard J. , Gerbier F. , Juzeliunas G. , Öhberg P. . Colloquium: Artificial gauge potentials for neutral atoms. Rev. Mod. Phys., 2011, 83(4): 1523 https://doi.org/10.1103/RevModPhys.83.1523
15	Galitski V. , B. Spielman I. . Spin‒orbit coupling in quantum gases. Nature, 2013, 494(7435): 49 https://doi.org/10.1038/nature11841
16	Gerbier F. , Dalibard J. . Gauge fields for ultracold atoms in optical superlattices. New J. Phys., 2010, 12(3): 033007 https://doi.org/10.1088/1367-2630/12/3/033007
17	Cho J. , G. Angelakis D. , Bose S. . Fractional quantum Hall state in coupled cavities. Phys. Rev. Lett., 2008, 101(24): 246809 https://doi.org/10.1103/PhysRevLett.101.246809
18	O. Umucalılar R. , Carusotto I. . Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A, 2011, 84(4): 043804 https://doi.org/10.1103/PhysRevA.84.043804
19	Roushan P. , Neill C. , Megrant A. , Chen Y. , Babbush R. , Barends R. , Campbell B. , Chen Z. , Chiaro B. , Dunsworth A. , Fowler A. , Jeffrey E. , Kelly J. , Lucero E. , Mutus J. , J. J. O’Malley P. , Neeley M. , Quintana C. , Sank D. , Vainsencher A. , Wenner J. , White T. , Kapit E. , Neven H. , Martinis J. . Chiral ground-state currents of interacting photons in a synthetic magnetic field. Nat. Phys., 2017, 13(2): 146 https://doi.org/10.1038/nphys3930
20	Koch J. , A. Houck A. , L. Hur K. , M. Girvin S. . Time-reversal-symmetry breaking in circuit-QED-based photon lattices. Phys. Rev. A, 2010, 82(4): 043811 https://doi.org/10.1103/PhysRevA.82.043811
21	Nunnenkamp A. , Koch J. , M. Girvin S. . Synthetic gauge fields and homodyne transmission in Jaynes‒Cummings lattices. New J. Phys., 2011, 13(9): 095008 https://doi.org/10.1088/1367-2630/13/9/095008
22	Marcos D. , Rabl P. , Rico E. , Zoller P. . Superconducting circuits for quantum simulation of dynamical gauge fields. Phys. Rev. Lett., 2013, 111(11): 110504 https://doi.org/10.1103/PhysRevLett.111.110504
23	P. Wang Y. , L. Yang W. , Hu Y. , Y. Xue Z. , Wu Y. . Detecting topological phases of microwave photons in a circuit quantum electrodynamics lattice. npj Quantum Inf., 2016, 2(1): 16015 https://doi.org/10.1038/npjqi.2016.15
24	H. Yang Z. , P. Wang Y. , Y. Xue Z. , L. Yang W. , Hu Y. , H. Gao J. , Wu Y. . Circuit quantum electrodynamics simulator of flat band physics in a Lieb lattice. Phys. Rev. A, 2016, 93(6): 062319 https://doi.org/10.1103/PhysRevA.93.062319
25	Alaeian H. , W. S. Chang C. , V. Moghaddam M. , M. Wilson C. , Solano E. , Rico E. . Creating lattice gauge potentials in circuit QED: The bosonic Creutz ladder. Phys. Rev. A, 2019, 99(5): 053834 https://doi.org/10.1103/PhysRevA.99.053834
26	J. Zhao Y. , W. Xu X. , Wang H. , X. Liu Y. , M. Liu W. . Vortex‒Meissner phase transition induced by a two-tone-drive-engineered artificial gauge potential in the fermionic ladder constructed by superconducting qubit circuits. Phys. Rev. A, 2020, 102(5): 053722 https://doi.org/10.1103/PhysRevA.102.053722
27	Guan X. , L. Feng Y. , Y. Xue Z. , Chen G. , T. Jia S. . Synthetic gauge field and chiral physics on two-leg superconducting circuits. Phys. Rev. A, 2020, 102(3): 032610 https://doi.org/10.1103/PhysRevA.102.032610
28	Jaksch D. , Zoller P. . Creation of effective magnetic fields in optical lattices: The Hofstadter butterfly for cold neutral atoms. New J. Phys., 2003, 5: 56 https://doi.org/10.1088/1367-2630/5/1/356
29	Graß T. , Muschik C. , Celi A. , W. Chhajlany R. , Lewenstein M. . Synthetic magnetic fluxes and topological order in one-dimensional spin systems. Phys. Rev. A, 2015, 91(6): 063612 https://doi.org/10.1103/PhysRevA.91.063612
30	Banerjee R. , C. H. Liew T. , Kyriienko O. . Realization of Hofstadter’s butterfly and a one-way edge mode in a polaritonic system. Phys. Rev. B, 2018, 98(7): 075412 https://doi.org/10.1103/PhysRevB.98.075412
31	Aidelsburger M. , Atala M. , Lohse M. , T. Barreiro J. , Paredes B. , Bloch I. . Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett., 2013, 111(18): 185301 https://doi.org/10.1103/PhysRevLett.111.185301
32	Miyake H. , A. Siviloglou G. , J. Kennedy C. , C. Burton W. , Ketterle W. . Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett., 2013, 111(18): 185302 https://doi.org/10.1103/PhysRevLett.111.185302
33	Q. You J. , Nori F. . Atomic physics and quantum optics using superconducting circuits. Nature, 2011, 474(7353): 589 https://doi.org/10.1038/nature10122
34	Gu X.F. Kockum A.Miranowicz A.Liu Y.Nori F., Microwave photonics with superconducting quantum circuits, Phys. Rep. 718–719, 1 (2017)
35	Clarke J. , K. Wilhelm F. . Superconducting quantum bits. Nature, 2008, 453(7198): 1031 https://doi.org/10.1038/nature07128
36	J. J. O’Malley P. , Babbush R. , D. Kivlichan I. , Romero J. , R. McClean J. . et al.. Scalable quantum simulation of molecular energies. Phys. Rev. X, 2016, 6(3): 031007 https://doi.org/10.1103/PhysRevX.6.031007
37	Xu K. , J. Chen J. , Zeng Y. , R. Zhang Y. , Song C. , Liu W. , Guo Q. , Zhang P. , Xu D. , Deng H. , Huang K. , Wang H. , Zhu X. , Zheng D. , Fan H. . Emulating many-body localization with a superconducting quantum processor. Phys. Rev. Lett., 2018, 120(5): 050507 https://doi.org/10.1103/PhysRevLett.120.050507
38	Feng W. , Q. Zhang G. , P. Su Q. , X. Zhang J. , P. Yang C. . Generation of Greenberger‒Horne‒Zeilinger states on two-dimensional superconducting-qubit lattices via parallel multiqubit-gate operations. Phys. Rev. Appl., 2022, 18(6): 064036 https://doi.org/10.1103/PhysRevApplied.18.064036
39	P. Su Q. , Zhang Y. , Bin L. , P. Yang C. . Efficient scheme for realizing a multiplex-controlled phase gate with photonic qubits in circuit quantum electrodynamics. Front. Phys., 2022, 17(5): 53505 https://doi.org/10.1007/s11467-022-1163-4
40	Liu T. , Q. Guo B. , H. Zhou Y. , L. Zhao J. , L. Fang Y. , C. Wu Q. , P. Yang C. . Transfer of quantum entangled states between superconducting qubits and microwave field qubits. Front. Phys., 2022, 17(6): 61502 https://doi.org/10.1007/s11467-022-1166-1
41	P. Su Q. , Zhang H. , P. Yang C. . Transferring quantum entangled states between multiple single-photonstate qubits and coherent-state qubits in circuit QED. Front. Phys., 2021, 16(6): 61501 https://doi.org/10.1007/s11467-021-1098-1
42	Liu T. , F. Zheng Z. , Zhang Y. , L. Fang Y. , P. Yang C. . Transferring entangled states of photonic cat-state qubits in circuit QED. Front. Phys., 2020, 15(2): 21603 https://doi.org/10.1007/s11467-019-0949-5
43	Arute F. , Arya K. , Babbush R. , Bacon D. , C. Bardin J. . et al.. Quantum supremacy using a programmable superconducting processor. Nature, 2019, 574(7779): 505 https://doi.org/10.1038/s41586-019-1666-5
44	Gong M. , Wang S. , Zha C. , C. Chen M. , L. Huang H. . et al.. Quantum walks on a programmable two-dimensional 62-qubit superconducting processor. Science, 2021, 372(6545): 948 https://doi.org/10.1126/science.abg7812
45	Wu Y. , S. Bao W. , Cao S. , Chen F. , C. Chen M. . et al.. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett., 2021, 127(18): 180501 https://doi.org/10.1103/PhysRevLett.127.180501
46	Zhang X. , Jiang W. , Deng J. , Wang K. , Chen J. , Zhang P. , Ren W. , Dong H. , Xu S. , Gao Y. , Jin F. , Zhu X. , Guo Q. , Li H. , Song C. , V. Gorshkov A. , Iadecola T. , Liu F. , X. Gong Z. , Wang Z. , L. Deng D. , Wang H. . Digital quantum simulation of Floquet symmetry protected topological phases. Nature, 2022, 607(7919): 468 https://doi.org/10.1038/s41586-022-04854-3
47	Ren W. , Li W. , Xu S. , Wang K. , Jiang W. , Jin F. , Zhu X. , Chen J. , Song Z. , Zhang P. , Dong H. , Zhang X. , Deng J. , Gao Y. , Zhang C. , Wu Y. , Zhang B. , Guo Q. , Li H. , Wang Z. , Biamonte J. , Song C. , L. Deng D. , Wang H. . Experimental quantum adversarial learning with programmable superconducting qubits. Nat. Comput. Sci., 2022, 2(11): 711 https://doi.org/10.1038/s43588-022-00351-9
48	Liu W. , Feng W. , Ren W. , W. Wang D. , Wang H. . Synthesizing three-body interaction of spin chirality with superconducting qubits. Appl. Phys. Lett., 2020, 116(11): 114001 https://doi.org/10.1063/1.5140884
49	Ren W. , Liu W. , Song C. , Li H. , Guo Q. , Wang Z. , Zheng D. , S. Agarwal G. , O. Scully M. , Y. Zhu S. , Wang H. , W. Wang D. . Simultaneous excitation of two noninteracting atoms with time-frequency correlated photon pairs in a superconducting circuit. Phys. Rev. Lett., 2020, 125(13): 133601 https://doi.org/10.1103/PhysRevLett.125.133601
50	Guo Q. , Cheng C. , Li H. , Xu S. , Zhang P. , Wang Z. , Song C. , Liu W. , Ren W. , Dong H. , Mondaini R. , Wang H. . Stark many-body localization on a superconducting quantum processor. Phys. Rev. Lett., 2021, 127(24): 240502 https://doi.org/10.1103/PhysRevLett.127.240502
51	Ye Y. , Y. Ge Z. , Wu Y. , Wang S. , Gong M. , R. Zhang Y. , Zhu Q. , Yang R. , Li S. , Liang F. , Lin J. , Xu Y. , Guo C. , Sun L. , Cheng C. , Ma N. , Y. Meng Z. , Deng H. , Rong H. , Y. Lu C. , Z. Peng C. , Fan H. , Zhu X. , W. Pan J. . Propagation and localization of collective excitations on a 24-qubit superconducting processor. Phys. Rev. Lett., 2019, 123(5): 050502 https://doi.org/10.1103/PhysRevLett.123.050502
52	J. Kollár A. , Fitzpatrick M. , A. Houck A. . Hyperbolic lattices in circuit quantum electrodynamics. Nature, 2019, 571(7763): 45 https://doi.org/10.1038/s41586-019-1348-3
53	G. Harper P. . Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A, 1955, 68(10): 874 https://doi.org/10.1088/0370-1298/68/10/304
54	K. Das K. , Christ J. . Realizing the Harper model with ultracold atoms in a ring lattice. Phys. Rev. A, 2019, 99(1): 013604 https://doi.org/10.1103/PhysRevA.99.013604
55	Roushan P. , Neill C. , Tangpanitanon J. , M. Bastidas V. , Megrant A. , Barends R. , Chen Y. , Chen Z. , Chiaro B. , Dunsworth A. , Fowler A. , Foxen B. , Giustina M. , Jeffrey E. , Kelly J. , Lucero E. , Mutus J. , Neeley M. , Quintana C. , Sank D. , Vainsencher A. , Wenner J. , White T. , Neven H. , G. Angelakis D. , Martinis J. . Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science, 2017, 358(6367): 1175 https://doi.org/10.1126/science.aao1401
56	Beaudoin F. , P. da Silva M. , Dutton Z. , Blais A. . First-order sidebands in circuit QED using qubit frequency modulation. Phys. Rev. A, 2012, 86(2): 022305 https://doi.org/10.1103/PhysRevA.86.022305
57	D. Strand J.Ware M.Beaudoin F.A. Ohki T.R. Johnson B.Blais A.L. T. Plourde B., First-order sideband transitions with flux-driven asymmetric transmon qubits, Phys. Rev. B 87, 220505(R) (2013)
58	Li X. , Ma Y. , Han J. , Chen T. , Xu Y. , Cai W. , Wang H. , P. Song Y. , Y. Xue Z. , Q. Yin Z. , Sun L. . Perfect quantum state transfer in a superconducting qubit chain with parametrically tunable couplings. Phys. Rev. Appl., 2018, 10(5): 054009 https://doi.org/10.1103/PhysRevApplied.10.054009
59	Senko C. , Smith J. , Richerme P. , Lee A. , C. Campbell W. , Monroe C. . Coherent imaging spectroscopy of a quantum many-body spin system. Science, 2014, 345(6195): 430 https://doi.org/10.1126/science.1251422
60	Jurcevic P. , Hauke P. , Maier C. , Hempel C. , P. Lanyon B. , Blatt R. , F. Roos C. . Spectroscopy of interacting quasiparticles in trapped ions. Phys. Rev. Lett., 2015, 115(10): 100501 https://doi.org/10.1103/PhysRevLett.115.100501
61	R. Johansson J. , D. Nation P. , Nori F. . QuTiP: An open-source Python framework for the dynamics of open quantum systems. Comput. Phys. Commun., 2012, 183(8): 1760 https://doi.org/10.1016/j.cpc.2012.02.021
62	R. Johansson J. , D. Nation P. , Nori F. . QuTiP2: A Python framework for the dynamics of open quantum systems. Comput. Phys. Commun., 2013, 184(4): 1234 https://doi.org/10.1016/j.cpc.2012.11.019
63	W. Wang D. , Song C. , Feng W. , Cai H. , Xu D. , Deng H. , Li H. , Zheng D. , Zhu X. , Wang H. , Y. Zhu S. , O. Scully M. . Synthesis of antisymmetric spin exchange interaction and chiral spin clusters in superconducting circuits. Nat. Phys., 2019, 15(4): 382 https://doi.org/10.1038/s41567-018-0400-9

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