[1] [1] JI A C, SUN Q, XIE X C, et al. Josephson effect for photons in two weakly linked microcavities[J]. Physical Review Letters, 2009, 102(2): 023602. DOI: 10.1103/PhysRevLett.102.023602.

[2] [2] JI A C, XIE X C, LIU W M. Quantum magnetic dynamics of polarized light in arrays of microcavities[J]. Physical Review Letters, 2007, 99(18): 183602. DOI: 10.1103/PhysRevLett.99.183602.

[3] [3] QI R, YU X L, LI Z B, et al. Non-Abelian Josephson effect between two F=2 spinor Bose-Einstein condensates in double optical traps[J]. Physical Review Letters, 2009, 102(18): 185301. DOI: 10.1103/PhysRevLett.102.185301.

[4] [4] MOODIE I R, BALLANTINE E K, KEELING J. Generalized classes of continuous symmetries in two-mode Dicke models[J]. Physical Review A, 2018, 97(3): 033802. DOI: 10.1103/PhysRevA.97.033802.

[5] [5] NAGY D, SZIRMAI G, DOMOKOS P. Critical exponent of a quantum-noise-driven phase transition: The open-system Dicke model[J]. Physical Review A, 2011, 84(4): 043637. DOI: 10.1103/PhysRevA.84.043637.

[6] [6] DOMINIK J, WINTERAUER. Multistable particlefield dynamics in cavity-generated optical lattices[J]. Physical Review A, 2015, 91(5): 53829. DOI: 10.1103/PhysRevA.91.053829.

[7] [7] MUNIZ J A, BARBERENA D. Exploring dynamical phase transitions with cold atoms in an optical cavity[J]. Nature, 2020, 580(7805): 602‒607. DOI: 10.1038/s41586-020-2224-x.

[8] [8] RITSCH H, DOMOKOS P. Cold atoms in cavity-generated dynamical optical potentials[J]. Reviews of Modern Physics, 2013, 85(2): 553‒601. DOI: 10.1103/RevModPhys.85.553.

[9] [9] XU Y J, PU H, FALLAS P D. Multicriticality and quantum fluctuation in a generalized Dicke model[J]. Physical Review A, 2021, 104(4): 043708. DOI: 10.1103/PhysRevA.104.043708.

[10] [10] CABALLERO-BENITEZ S F, MEKHOV I B. Quantum optical lattices for emergent many-body phases of ultracold atoms[J]. Physical Review Letters, 2015, 115(24): 243604. DOI: 10.1103/PhysRevLett.115.243604.

[11] [11] BAKHTIARI M R, RITSCH H, HEMMERICH A, et al. Nonequilibrium phase transition of interacting bosons in an intracavity optical lattice[J]. Physical Review Letters, 2015, 114(12): 123601. DOI: 10.1103/PhysRevLett.114.123601.

[12] [12] DOGRA N, HUBER S D, BRENNECKE F, et al. Phase transitions in a Bose-Hubbard model with cavity-mediated global-range interactions[J]. Physical Review A, 2016, 94(2): 023632. DOI: 10.1103/PhysRevA.94.023632.

[13] [13] HEBIB Y, ZHANG C, YANG J, et al. Quantum phases of lattice dipolar bosons coupled to a high-finesse cavity[J]. Physical Review A, 2023, 107(4): 043318. DOI: 10.1103/PhysRevA.107.043318.

[14] [14] SHARMA S, KRAUS R. Quantum critical behavior of entanglement in lattice bosons with cavity-mediated long-range interactions[J]. Physical Review Letters, 2022, 129(14): 143001. DOI: 10.1103/PhysRevLett.129.143001.

[15] [15] CHEN Y, YU Z, ZHAI H. Quantum phase transitions of the Bose-Hubbard model inside a cavity[J]. Physical Review A, 2016, 93(4): 041601. DOI: 10.1103/PhysRevA.93.041601.

[16] [16] QIN J L, ZHOU L. Supersolid gap soliton in a Bose-Einstein condensate and optical ring cavity coupling system[J]. Physical Review E, 2022, 105(5): 054214. DOI: 10.1103/PhysRevE.105.054214.

[17] [17] BAUMANN K. The Dicke quantum phase transition with a superfluid gas in an optical cavity[J]. Nature, 2010, 464: 1301-1306. DOI: 10.1038/nature09009.

[18] [18] LEONARD J, MORALES A. Supersolid formation in a quantum gas breaking a continuous translational symmetry[J]. Nature, 2017, 543(7643): 87‒90. DOI: 10.1038/nature21067.

[19] [19] LEONARD J, MORALES A, ZUPANCIC P, et al. Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas[J]. Science, 2017, 358(6369): 1415‒1418. DOI: 10.1126/science.aan2608.

[20] [20] DOGRA N, LANDINI M, KROEGER K, et al. Dissipation-induced structural instability and chiral dynamics in a quantum gas[J]. Science, 2019, 366(6472): 1496‒1499. DOI: 10.1126/science.aaw4465.

[21] [21] KROEGER K, DOGRA N. Continuous feedback on a quantum gas coupled to an optical cavity[J]. New Journal of Physics, 2020, 22(3): 033020. DOI: 10.1088/1367-2630/ab73cc.

[22] [22] FAN J T, CHEN G, JIA S T. Atomic self-organization emerging from tunable quadrature coupling[J]. Physical Review A, 2020, 101(6): 063627. DOI: 10.1103/PhysRevA.101.063627.

[23] [23] LI X, DREON D, ZUPANCIC P, et al. Measuring the dynamics of a first order structural phase transition between two con-figurations of a superradiant crystal[J]. Physical Review Research, 2020, 3(1): 012024. DOI: 10.1103/PhysRevResearch.3.L012024.

[24] [24] DREON D, BAUMGRTNER A, LI X, et al. Self-oscillating pump in a topological dissipative atom-cavity system[J]. Nature, 2022, 608(7923): 494‒498. DOI: 10.1038/S41586-022-04970-0.

[25] [25] JAKSCH D, BRUDER C, CIRAC J I, et al. Cold bosonic atoms in optical lattices[J]. Physical Review Letters, 1998, 81(15): 3108. DOI: 10.1103/PhysRevLett.81.3108.

[26] [26] BLOCH I. Quantum gases in optical lattices[J]. Physics, 2013, 17(4): 25‒29. DOI: 10.1088/2058-7058/17/4/32.

[27] [27] HAN J R, ZHANG T, WANG Y Z, et al. Quantum phase transition of two-component Bose-Einstein condensates in optical lattices[J]. Physics Letters A, 2004, 332(1-2): 131‒140. DOI: 10.1016/j.physleta.2004.09.059.

[28] [28] ZWERGER W. Mott-Hubbard transition of cold atoms in optical lattices[J]. Journal of Optics B: Quantum and Semiclassical Optics, 2003, 5(2): S9. DOI: 10.1088/1464-4266/5/2/352.

[29] [29] GREINER M, MANDEL O, ESSLINGER T, et al. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms[J]. Nature, 2002, 415(6867): 39‒44. DOI: 10.1038/415039a.

[30] [30] STOFERLE T, MORITZ H, SCHORI C, et al. Transition from a strongly interacting 1D superfluid to a Mott insulator[J]. Physical Review Letters, 2004, 92(13): 130403. DOI: 10.1103/PhysRevLett.92.130403.

[31] [31] ISACSSON A, CHA M C, SENGUPTA K, et al. Superfluid-insulator transitions of two-species Bosons in an optical lattice[J]. Physical Review B, 2005, 72(18): 184507. DOI: 10.1103/PhysRevB.72.184507.

[32] [32] GUAN X, FAN J T, ZHOU X F, et al. Two-component lattice bosons with cavity-mediated long-range interaction[J]. Physical Review A, 2019, 100(1): 013617. DOI: 10.1103/PhysRevA.100.013617.

[33] [33] JORDENS R, STROHMAIER N, GUNTER K, et al. A Mott insulator of fermionic atoms in an optical lattice[J]. Nature, 2008, 455(7210): 204‒207. DOI: 10.1038/nature07244.

[34] [34] DANSHITA I, WILLIAMS J E, MELO C A R S D, et al. Quantum phases of bosons in double-well optical lattices[J]. Physical Review A, 2007, 76(4): 043606. DOI: 10.1103/PhysRevA.76.043606.

[35] [35] DANSHITA I, CARLOS A R. Reentrant quantum phase transition in double-well optical lattices[J]. Physical Review A, 2008, 77(6): 3195‒3199. DOI: 10.1103/PhysRevA.77.063609.

[36] [36] DENG H, DAI H. Cluster Gutzwiller study of the Bose-Hubbard ladder: Ground-state phase diagram and many-body Landau-Zener dynamics[J]. Physical Review A, 2015, 92(2): 023618. DOI: 10.1103/PhysRevA.92.023618.

[37] [37] QIAO X, ZHANG X B. Quantum phases of interacting bosons on biased two-leg ladders with magnetic flux[J]. Physical Review A, 2021, 104(5): 053323. DOI: 10.1103/PhysRevA.104.053323.

[38] [38] PADHAN A, PARIDA R. Quantum phases of constrained bosons on a two-leg Bose-Hubbard ladder[J]. Physical Review A, 2023, 108(1): 013316. DOI: 10.1103/PhysRevA.108.013316.

[39] [39] KLINDER J, THORWART M. Observation of a Superradiant Mott Insulator in the Dicke-Hubbard Model[J]. Physical Review Letters, 2015, 115(23): 230403. DOI: 10.1103/PhysRevLett.115.230403.

[40] [40] LANDIG R, HRUBY L, DOGRA N, et al. Quantum phases from competing short- and long-range interactions in an optical lattice[J]. Nature, 2016, 532(7600): 476‒479. DOI: 10.1038/nature17409.

[41] [41] DUTTA O, GAJDA M, HAUKE P, et al. Non-standard Hubbard models in optical lattices: a review[J]. Reports on Progress in Physics, 2015, 78(6): 066001. DOI: 10.1088/0034-4885/78/6/066001.

[42] [42] GORAL K, SANTOS L, LEWENSTEIN M. Quantum phases of dipolar bosons in optical lattices[J]. Physical Review Letters, 2002, 88(17): 170406. DOI: 10.1103/PhysRevLett.88.170406.

[43] [43] KOVRIZHIN D L, PAI G V, SINHA S. Density wave and supersolid phases of correlated bosons in an optical lattice[J]. Europhysics Letters, 2005, 72(2): 162. DOI: 10.1209/epl/i2005-10231-y.

[44] [44] SILVER A O, HOHENADLER M, BHASEEN M J, et al. Bose-Hubbard models coupled to cavity light fields[J]. Physical Review A, 2010, 81(2): 023617. DOI: 10.1103/PhysRevA.81.023617.

[45] [45] BHASEEN M J, SILVER A O, HOHENADLER M, et al. Polaritons and pairing phenomena in Bose-Hubbard mixtures[J]. Physical Review Letters, 2009, 102(13): 135301. DOI: 10.1103/PhysRevLett.102.135301.

[46] [46] ZHENG W, COOPER N R. Superradiance induced particle flow via dynamical gauge coupling[J]. Physical Review Letters, 2016, 117(17): 175302. DOI: 10.1103/PhysRevLett.117.175302.

[47] [47] MIYAKE H, SIVILOGLOU G A. Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices[J]. Physical Review Letters, 2013, 111(18): 185302. DOI: 10.1103/PhysRevLett.111.185302.

[48] [48] JAKSCH D, ZOLLER P. Creation of effective magnetic fields in optical lattices: The Hofstadter butterfly for cold neutral atoms[J]. New Journal of Physics, 2003, 5(1): 1‒11. DOI: 10.1088/1367-2630/5/1/356.