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时间晶体是一种特殊的物质状态, 它指多体系统在内部自组织的作用下, 自发产生时间周期性振荡的现象. 近期, 无需外部周期性驱动的连续时间晶体已在耗散固体材料中实现, 并呈现出长时间稳定振荡的特性. 然而, 在多体系统中, 连续时间晶体的系统参数, 包括原子间相互作用强度、均匀性、频率失谐以及驱动场强度等, 均呈现高度的复杂性和关联性; 这些参数对连续时间晶体振荡周期形成的物理机制和耦合效应的影响尚不明确. 本文基于掺铒晶体中构建的连续时间晶体, 通过理论分析揭示了时间晶体振荡周期与驱动光场强度、偶极-偶极相互作用、原子间跃迁强度差异以及耗散系数之间的内在关联. 研究表明, 即便在这些参数动态变化引起的扰动下, 时间晶体的振荡周期仍展现出显著的鲁棒性.
Continuous time crystals represent a novel state of many-body systems, self-organizing into time-periodic oscillations without the need for external periodic driving. Recent experiments have demonstrated the realization of such systems in dissipative solid-state materials, where persistent temporal order is autonomously sustained. A defining characteristic of time crystals is their robustness, signifying the ability to maintain rhythmic behavior despite various disturbances, including fluctuations in internal parameters and external noise., which is of both scientific value and potential for technological applications Although prior studies have established the existence of robustness in specific experimental parameters, a systematic framework for quantifying and predicting their resilience to perturbations is lacking, and the underlying physics of this robustness remains inadequately understood. Key unresolved questions include how nonlinear interactions and feedback mechanisms contribute to stability, and what the critical thresholds are for parameter variations beyond which temporal order collapses. This paper addresses these gaps by systematically analyzing how internal parameters and external influences affect the oscillation period and overall stability. Internally, the dynamics are dictated by dipole-dipole interactions and atomic transition strengths, which define the system’s emergent temporal symmetry breaking. Externally, the system’s response is controlled by the intensity of the optical driving field and the rates of energy dissipation. A key finding is the identification of an intrinsic feedback mechanism that dynamically stabilizes the time crystal. This mechanism acts as a restorative force, correcting for deviations caused by minor disturbances and maintaining the coherence of the oscillatory phase. Moreover, the system displays nonlinear dynamical behavior, characterized by two distinct regimes: one where stable oscillations continue under moderate disturbances, and another where stronger disturbances induce a dynamical phase transition, leading to disordered states or a switch between dynamically unstable and stable states. These results provide a thorough understanding of the diverse behaviors observed in continuous time crystals and create a vital theoretical foundation for exploiting their unique properties in advanced applications like quantum information processing and precision metrology. -
Keywords:
- continuous-time crystals /
- many-body systems /
- erbium-doped crystals /
- periodic oscillations /
- robustness
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图 1 系统的理论模型 (a) 多体相互作用示意图. 上图, 所有离子都处于基态(如绿色的自旋所示). 下图, 当部分离子被光学激发到激发态(如粉色的自旋所示), 会引起局域磁场变化∆B. (b) 系统的能级结构. 通过光学跃迁, 能级结构由分立的两对双重态组成. 由于邻近离子诱导的磁场变化, 自旋能级劈裂, 形成了四能级结构. 蓝色和红色的双箭头分别表示不同的耦合强度系数$ t_1 $和$ t_2 $
Fig. 1. Theoretical model of the system. (a) Illustration of the many body interactions. Top, all ions in their ground states (as shown by the green spins). Bottem, when some ions are optically excited (as illustrated by the pink spins), they introduce local magnetic field variance ∆B to their adjacent atoms. (b) The energy structure of the system. The energy structure consists of two pairs of spin doublets separated by an optical transition. Due to the magnetic field induced by neighboring ions, the spin levels are split out, forming a four-level structure. The blue and red bidirectional arrows represent the optical transitions with different coupling strengths coefficients $ t_1 $ and $ t_2 $, respectively.
图 2 时间晶体中能级布居数随时间的演化. 其中$ \Omega = 0.73 $ MHz, $ \Delta_s = 90 $ MHz, $ t_1/t_2 = 2.28:1 $ (a) 布居数$ \rho_{44} $在长时间极限下表现出持续的振荡. (b) 时间晶体状态下$ \rho_{44}(t) $的频谱图. (c) 不同能级布居数之间的线性关系. 深蓝色, $ \rho_{11} $和$ \rho_{33} $的依赖关系; 浅蓝色, $ \rho_{22} $和$ \rho_{44} $的依赖关系. (d) 不同能级布居数之间形成的极限环. 深蓝色, $ \rho_{11} $和$ \rho_{22} $的依赖关系; 浅蓝色, $ \rho_{11} $和$ \rho_{44} $的依赖关系
Fig. 2. Populatioin Evolution of time crystaline phase. Here we used $ \Omega = 0.73 $ MHz, $ \Delta_s = 90 $ MHz, and $ t_1/t_2 = 2.28:1 $. (a) The populations $ \rho_{44} $ as a function of time. (b) Spectrogram of $ \rho_{44}(t) $ in the time crystaline phase. (c) Linear dependence of the populations of differnt energy levels. Dark blue, the dependence between $ \rho_{11} $ and $ \rho_{33} $; light blue, the dependence between $ \rho_{22} $ and $ \rho_{44} $). (d) Limit cycles formed between the populations of different levels. Dark blue, the dependence between $ \rho_{11} $ and $ \rho_{22} $; light blue, the dependence between $ \rho_{11} $ and $ \rho_{44} $).
图 3 时间晶体振荡周期对拉比频率的依赖关系. 其中$ \Delta_s = 90 $ MHz, $ t_1/t_2 = 2.28:1 $ (a) 布居数$ \rho_{44} $的周期随拉比频率$\Omega $的变化情况. (b) 对应拉比频率下, 布居数$ \rho_{44} $随时间的演化. $ \Omega = 0.55 $ MHz时的$ \rho_{44}(t) $, 振荡快速衰减, 表明未形成时间晶体; $ \Omega = 0.74 $ MHz和$ \Omega = 0.87 $ MHz时的$ \rho_{44}(t) $, 持续的周期性振荡, 表明形成了时间晶体, 但形成的时间晶体的周期不同; $ \Omega = 0.88 $ MHz时的$ \rho_{44}(t) $, 周期性振荡消失, 表明时间晶体瓦解
Fig. 3. Dependence of the period of the time crystal on the driving Rabi frequency. Here we used $ \Delta_s = 90 $ MHz, $ t_1/t_2 = 2.28:1 $. (a) Period of the populations $ \rho_{44} $ as a function of Rabi frequency $\Omega $. (b) Evolution of the populations $ \rho_{44} $ with time at the corresponding Rabi frequency. At $ \Omega = 0.55 $ MHz, the oscillation of $ \rho_{44}(t) $ rapidly decays, indicating the absence of a time crystal; The sustained periodic oscillation of $ \rho_{44}(t) $ at $ \Omega = 0.74 $ MHz and $ \Omega = 0.87 $ MHz, indicate the formation of a time crystal, but with different periods; At $ \Omega = 0.88 $ MHz, the disappearance of periodic oscillations of $ \rho_{44}(t) $, indicating the disintegration of the time crystal.
图 4 $ \rho_{44} $的动力学演化与频谱图 (a) 对应噪声强度下, $ \rho_{44} $的时域响应. 左子图: 施加白噪声的时域波形(噪声频谱范围为$ [0, 5] $ kHz). 右子图: $ \rho_{44}(t) $在时间窗t = 4.8—5.3 ms的局部动力学放大. 驱动光场振幅变化的范围为2%—8%. (b) 对应噪声强度下, $ \rho_{44}(t) $的频谱图. 在噪声强度从2%增至8%的变化范围内, 系统的特征频率稳定在无噪声时的11.2 kHz附近
Fig. 4. The dynamic evolution and frequency spectrum of $ \rho_ {44} $. (a) Under the corresponding noise intensity time-domain response of $ \rho_ {44} $. Left subplot: Time-domain waveform with white noise applied (the noise spectrum range is $ [0, 5] $ kHz). Right subplot: The local dynamic amplification of $ \rho_ {44} (t) $ in the time window of t = 4.8–5.3 ms. The noise amplitute of the driving field is varyiing from 2% to 8%. (b) Spectra of $ \rho_{44}(t) $ at the corresponding noise intensity. Although the spectral analysis shows a linewidth broadening effect as the noise intensity increases from 2% to 8%, the eigenfrequency of the system stabilizes around 11.2 kHz.
图 5 时间晶体振荡周期对多体相互作用的依赖关系. 其中$ \Omega = 0.73 $ MHz, $ t_1/t_2 = 2.28:1 $ (a) 布居数$ \rho_{44} $的振荡周期随激发诱导的频移$ \Delta_s $的变化情况. (b) 对应激发诱导频移下, 布居数$ \rho_{44} $随时间的演化. $ \Delta_s = 30 $ MHz时的$ \rho_{44}(t) $, 快速的拉比振荡, 但频率正比于外加光场, 表明未形成时间晶体; $ \Delta_s = 90 $ MHz和$ \Delta_s = 117 $ MHz时的$ \rho_{44}(t) $, 持续的周期性振荡, 表明形成了时间晶体, 且形成的时间晶体的周期非常接近; $ \Delta_s = 130 $ MHz时的$ \rho_{44}(t) $, 复杂的准周期振荡, 表明时间晶体瓦解, 系统进入混沌状态
Fig. 5. Dependence of the period of the time crystal on the many-body interaction. Here we used $ \Omega = 0.73 $ MHz, $ t_1/t_2 = 2.28:1 $. (a) Period of the populations $ \rho_{44} $ as a function of the many-body interaction $ \Delta_s $. (b) Evolution of the populations $ \rho_{44} $ with time at the corresponding many-body interaction strength. At $ \Delta_s = 30 $ MHz, $ \rho_{44}(t) $ exhibits rapid Rabi oscillations, but the frequency is proportional to the applied light field, indicating the absence of a time crystal. The sustained periodic oscillation of $ \rho_{44}(t) $ at $ \Delta_s = 90 $ MHz and $ \Delta_s = 117 $ MHz, indicating the formation of a time crystal, and the period of the formed time crystal is very close; The complex quasi periodic oscillation of $ \rho_{44}(t) $ at $ \Delta_s = 130 $ MHz, indicating the disintegration of the time crystal and the system entering a chaotic state.
图 6 时间晶体振荡周期对跃迁耦合强度系数的依赖关系. 其中$ \Omega = 0.73 $ MHz, $ \Delta_s = 90 $ MHz (a) 布局数$ \rho_{44} $振荡周期随跃迁耦合强度系数的变化情况. (b) 对应跃迁强度系数下, 布居数$ \rho_{44} $随时间的演化. $ t_1/t_2 = 1.52 $时的$ \rho_{44}(t) $, 振荡快速衰减, 表明未形成时间晶体; $ t_1/t_2 = 1.64 $、$ t_1/t_2 = 1.69 $和$ t_1/t_2 = 1.76 $时的$ \rho_{44}(t) $(分别对应图a中的标记), 持续的周期性振荡, 表明形成了时间晶体, 且形成的时间晶体的周期非常接近
Fig. 6. Dependence of the period of the time crystal on the ratio of optical transition strengths. Here we used $ \Omega = 0.73 $ MHz, $ \Delta_s = 90 $ MHz. (a) Period of the populations $ \rho_{44} $ as a function of the ratio the ratio of optical transition strengths. (b) Evolution of the populations $ \rho_{44} $ at the corresponding ratio. At $ t_1/t_2 = 1.52 $, the oscillation of $ \rho_ {44}(t) $ rapidly decays, indicating the absence of a time crystal; The sustained periodic oscillation of $ \rho_ {44}(t) $ at $ t_1/t_2 = 1.64 $, $ t_1/t_2 = 1.69 $, and $ t_1/t_2 = 1.76 $(corresponding to the markings in 6 a respectively), indicating the formation of a time crystal with very close periods.
图 7 时间晶体振荡周期对衰减系数$ \gamma_0 $的依赖关系 (a) 布居数$ \rho_{44} $的振荡周期随衰减系数$ \gamma_0 $的变化情况. (b) 对应衰减系数下, 布居数$ \rho_{44} $随时间的演化
Fig. 7. Dependence of the period of the time crystal on the decay rate $ \gamma_0 $. (a) Period of the populations $ \rho_{44} $ as a function of the decay rate $ \gamma_0 $. (b) Evolution of the populations $ \rho_{44} $ with time at the corresponding decay rate as marked.
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[1] Li T, Gong Z X, Yin Z Q, Quan H, Yin X, Zhang P, Duan L M, Zhang X 2012 Phys. Rev. Lett. 109 163001
Google Scholar
[2] Sacha K 2015 Phys. Rev. A 91 033617
Google Scholar
[3] Zhang J, Hess P W, Kyprianidis A, Becker P, Lee A, Smith J, Pagano G, Potirniche I D, Potter A C, Vishwanath A, Yao N Y, Monroe C 2017 Nature 543 217
Google Scholar
[4] Wilczek F 2012 Phys. Rev. Lett. 109 160401
Google Scholar
[5] Shapere A, Wilczek F 2012 Phys. Rev. Lett. 109 160402
Google Scholar
[6] Else D V, Bauer B, Nayak C 2016 Phys. Rev. Lett. 117 090402
Google Scholar
[7] Sacha K, Zakrzewski J 2017 Rep. Prog. Phys. 81 016401
[8] Yao N Y, Nayak C 2018 Phys. Today 71 40
[9] Cai Z, Huang Y, Liu W V 2020 Chin. Phys. Lett. 37 050503
Google Scholar
[10] Hannaford P, Sacha K 2022 Europhys. Lett. 139 10001
Google Scholar
[11] Tucker K, Zhu B, Lewis-Swan R J, Marino J, Jimenez F, Restrepo J G, Rey A M 2018 New J. Phys. 20 123003
Google Scholar
[12] Keßler H, Cosme J G, Hemmerling M, Mathey L, Hemmerich A 2019 Phys. Rev. A 99 053605
Google Scholar
[13] Liu T, Ou J Y, MacDonald K F, Zheludev N I 2023 Nat. Phys. 19 986
Google Scholar
[14] Choi S, Choi J, Landig R, Kucsko G, Zhou H, Isoya J, Jelezko F, Onoda S, Sumiya H, Khemani V, von Keyserlingk C, Yao N Y, Demler E, Lukin M D 2017 Nature 543 221
Google Scholar
[15] Russomanno A, Iemini F, Dalmonte M, Fazio R 2017 Phys. Rev. B 95 214307
Google Scholar
[16] Smits J, Liao L, Stoof H, van der Straten P 2018 Phys. Rev. Lett. 121 185301
Google Scholar
[17] Autti S, Eltsov V, Volovik G 2018 Phys. Rev. Lett. 120 215301
Google Scholar
[18] Kyprianidis A, Machado F, Morong W, Becker P, Collins K S, Else D V, Feng L, Hess P W, Nayak C, Pagano G, Yao N Y, Monroe C 2021 Science 372 1192
Google Scholar
[19] Randall J, Bradley C, Van Der Gronden F, Galicia A, Abobeih M, Markham M, Twitchen D, Machado F, Yao N, Taminiau T 2021 Science 374 1474
Google Scholar
[20] Mi X, Ippoliti M, Quintana C, Greene A, Chen Z, Gross J, Arute F, Arya K, Atalaya J, Babbush R, Bardin J C, Basso J, Bengtsson A, Bilmes A, Bourassa A, Brill L, Broughton M, Buckley B B, Buell D A, Burkett B, Bushnell N, Chiaro B, Collins R, Courtney W, Debroy D, Demura S, Derk A R, Dunsworth A, Eppens D, Erickson C, Farhi E, Fowler A G, Foxen B, Gidney C, Giustina M, Harrigan M P, Harrington S D, Hilton J, Ho A, Hong S, Huang T, Huff A, Huggins W J, Ioffe L B, Isakov V Sergei, Iveland J, Jeffrey E, Jiang Z, Jones C, Kafri D, Khattar T, Kim S, Kitaev A, Klimov V Paul, Korotkov A N, Kostritsa F, Landhuis D, Laptev P, Lee J, Lee K, Locharla A, Lucero E, Martin O, McClean J R, McCourt T, McEwen M, Miao K C, Mohseni M, Montazeri S, Mruczkiewicz W, Naaman O, Neeley M, Neill C, Newman M, Niu M Y, O’Brien T E, Opremcak A, Ostby E, Pato B, Petukhov A, Rubin N C, Sank D, Satzinger K J, Shvarts V, Su Y, Strain D, Szalay M, Trevithick M D, Villalonga B, White T, Yao Z J, Yeh P, Yoo J, Zalcman A, Neven H, Boixo S, Smelyanskiy V, Megrant A, Kelly J, Chen Y, Sondhi S L, Moessner R, Kechedzhi K, Khemani V, Roushan P 2022 Nature 601 531
Google Scholar
[21] 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 C Hekangand Song, Gorshkov V Alexey, Iadecola T, Liu F, Gong Z X, Wang Z, Deng D L, Wang H 2022 Nature 607 468
Google Scholar
[22] Huang B, Wu Y H, Liu W V 2018 Phys. Rev. Lett. 120 110603
Google Scholar
[23] Pizzi A, Knolle J, Nunnenkamp A 2019 Phys. Rev. Lett. 123 150601
Google Scholar
[24] Rovny J, Blum R L, Barrett S E 2018 Phys. Rev. Lett. 120 180603
Google Scholar
[25] Pal S, Nishad N, Mahesh T, Sreejith G 2018 Phys. Rev. Lett. 120 180602
Google Scholar
[26] O’ Sullivan J, Lunt O, Zollitsch C W, Thewalt M, Morton J J, Pal A 2020 New J. Phys. 22 085001
Google Scholar
[27] Horowicz Y, Katz O, Raz O, Firstenberg O 2021 Proc. Natl. Acad. Sci. 118 e2106400118
Google Scholar
[28] Cabot A, Carollo F, Lesanovsky I 2022 Phys. Rev. B 106 134311
Google Scholar
[29] Khemani V, Lazarides A, Moessner R, Sondhi S L 2016 Phys. Rev. Lett. 116 250401
Google Scholar
[30] Vlasov R, Lemeza A, Gladush M 2013 Laser Phys. Lett. 10 045401
Google Scholar
[31] Gong Z, Hamazaki R, Ueda M 2018 Phys. Rev. Lett. 120 040404
Google Scholar
[32] Iemini F, Russomanno A, Keeling J, Schirò M, Dalmonte M, Fazio R 2018 Phys. Rev. Lett. 121 035301
Google Scholar
[33] Buča B, Tindall J, Jaksch D 2019 Nat. Commun. 10 1730
Google Scholar
[34] Piazza F, Ritsch H 2015 Phys. Rev. Lett. 115 163601
Google Scholar
[35] Booker C, Buča B, Jaksch D 2020 New J. Phys. 22 085007
Google Scholar
[36] Keßler H, Kongkhambut P, Georges C, Mathey L, Cosme J G, Hemmerich A 2021 Phys. Rev. Lett. 127 043602
Google Scholar
[37] Taheri H, Matsko A B, Maleki L, Sacha K 2022 Nat. Commun. 13 848
Google Scholar
[38] Martínez-Romero J S, Halevi P 2018 Phys.Rev.A 98 053852
Google Scholar
[39] Dong R Y, Liu Y M, Sui J Y, Zhang H F 2023 IEEE Trans. Antennas Propag. 72 674
[40] Dong R y, Wang S, Zou J H, Zhang H f 2023 Opt. Lett. 48 2627
Google Scholar
[41] Valdez-García J, Halevi P 2024 Phys. Rev. A 109 063517
Google Scholar
[42] Kongkhambut P, Skulte J, Mathey L, Cosme J G, Hemmerich A, Keßler H 2022 Science 377 670
Google Scholar
[43] Chen Y H, Zhang X 2023 Nat. Commun. 14 6161
Google Scholar
[44] Wadenpfuhl K, Adams C S 2023 Phys. Rev. Lett. 131 143002
Google Scholar
[45] Krishna M, Solanki P, Hajdušek M, Vinjanampathy S 2023 Phys. Rev. Lett. 130 150401
Google Scholar
[46] Wu X, Wang Z, Yang F, Gao R, Liang C, Tey M K, Li X, Pohl T, You L 2024 Nat. Phys. 20 1389
Google Scholar
[47] Ding D, Bai Z, Liu Z, Shi B, Guo G, Li W, Adams C S 2024 Sci. Adv. 10 eadl5893
Google Scholar
[48] Zhou P, Li X X, Xing X Y, Chen Y H, Zhang X D 2022 Acta Phys. Sin. 71 102
[49] Calderón O, Antón M, Carreño F 2003 Eur. Phys. J. D 25 77
Google Scholar
[50] Chen Y H, Horvath S P, Longdell J J, Zhang X 2021 Phys. Rev. Lett. 126 110601
Google Scholar
[51] Greilich A, Kopteva N, Kamenskii A, Sokolov P, Korenev V, Bayer M 2024 Nat. Phys. 20 631
Google Scholar
[52] Chen S, Raha M, Phenicie C M, Ourari S, Thompson J D 2020 Science 370 592
Google Scholar
[53] Lee T E, Häffner H, Cross M 2011 Phys. Rev. A: At., Mol., Opt. Phys. 84 031402
Google Scholar
[54] Kos P, Ljubotina M, Prosen T 2018 Phys. Rev. X 8 021062
[55] Scarlatella O, Clerk A A, Fazio R, Schiró M 2021 Phys. Rev. X 11 031018
[56] Li T, Li H X, Chen Y H, Zhang X 2024 Europhys. Lett. 147 55001
Google Scholar
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