時間晶體
時間晶體(英語:time crystal)乃一開放系統,其與周圍環境保持非平衡態,呈現時間平移對稱破缺(英語:time translation symmetry breaking)的特性。2017年3月的科學報導指出,此一理論概念已在實驗上獲得證實;隨着時間演進,時間晶體仍無法與環境達到熱平衡。[1][2]
時間晶體的概念首先由諾貝爾物理學獎得主弗朗克·韋爾切克於2012年提出。相對於尋常晶體在空間上呈週期性重複,時間晶體則在時間上呈週期性重複而呈現永動狀態。時間晶體在時間平移對稱上具有自發對稱破缺現象。時間晶體也與零點能量和動態卡西米爾效應有關。
2016年,姚穎(英語:Norman Y. Yao)與加州大學柏克萊分校物理系的同僚提出在實驗室建構時間晶體的藍圖[3];隨後此藍圖經兩組人馬採用,包括馬里蘭大學的Christopher Monroe以及哈佛大學的Mikhail Lukin,兩團隊皆成功創造出時間晶體,實驗成果於2017年3月發表在《自然》期刊。[4][5]
常規晶體是一個三維物體,它們的內部原子按照有規則的順序重複排列而構成。時間晶體是一種四維以上晶體,在時空中擁有一種周期性結構。 一個時間晶體能自發破壞時間平移的對稱性,做空間的非平移運動,時間晶體的構成以『空間』非定域的粒子交叉存在做相互關聯運動,是能效粒子的『額外維』超出『定域空間』的能動量,時間晶體的存在同樣揭示了『超額外維度』的存在意義。
它可以隨着時間改變,但是會持續回到它開始時的相同形態,就如一個鐘的移動的指針周期性的回到它的原始位置。與普通的鐘或者其他周期性的過程不同的是,時間晶體和空間晶體一樣會是最低限度的能量的一種狀態。可以將它看作是一隻可以永遠保持走時精確無誤的鐘,即便是在宇宙達到熱寂之後也是如此。
主要特點
設計方法
2012年7月,來自美國加州大學伯克利分校的李統藏博士以及他來自密歇根大學和清華大學的同事們提出了一種新的方案,有可能實現時間晶體的設想。
首先需要一個離子阱,這是一種利用電場來將某一帶電粒子固定在某一位置上的裝置。這樣做將可以讓這些離子形成一個環狀的晶體,這是因為當離子在極低溫度條件下被捕獲時,它們會相互排斥。隨後科學家施加一個微弱的靜磁場,它將驅動電子自旋。
量子力學指出,離子的自旋能量必須大於0,即便是在這個電子環已經被冷凍至最低能級的情況下也是如此。在這種狀態下,已經不需要電場和磁場來幫助維持這一晶體的形狀以及組成它的各個離子的自旋。這樣做的結果就是獲得一個時間晶體,或者更準確的說是一個時空晶體,因為這個離子環不但在時間上,在空間上也是不斷重複着自身。
研究人員從理論上推理認為,這種時間晶體可以被用作計算機,它可以用不同的自旋狀態當做傳統計算法中的0和1。利用該系統方案,這一設想將是可能的[6]
該方案是基於電場離子阱和粒子之間的庫倫斥力構建的。離子阱的電場將帶電粒子固定住,而庫倫斥力讓它們自發地形成一個空間環狀晶體。在一個微弱的靜態磁場作用下,這一環狀離子晶體將開始永無止境的轉動。由於這一時空晶體已經位於最低量子能態,其時間序列,從理論上說將會永遠持續,即便是當宇宙達到熵的極大值,也就是達到「熱寂」狀態時,情況也是一樣。
製造方法:將10個鐿原子排成一列,然後用兩束激光交替轟擊它們,使得這些原子進入一種穩定且重複的自旋翻轉模式,符合「時間晶體」的定義。另一個來自哈佛大學的研究團隊則通過向鑽石中密集充入氮氣的方式,也製造出了「時間晶體」。
意義
構建一個時空晶體,存在着實際和重要的科學理由:有了這種4維晶體,科學家們將擁有一種全新的,更加有效的手段對複雜的物理屬性和大量粒子的複雜相互作用行為進行研究,或者是研究物理學中所謂的「多體問題」。這種時空晶體同樣可以被用來對量子世界進行研究,如量子糾纏現象,在這種狀態中,當對其中一個粒子進行操作時,另外一個粒子也會相應地發生變化,即便這兩個粒子之間隔開着巨大的距離。
參考文獻
- ^ Ghose, Tia. Time Crystals Created, Suspending Laws of Physics. Live Science (Purch). 2017-03-08 [2017-03-16]. (原始內容存檔於2021-03-18).
- ^ Researcher unveils time crystal as new form of matter. Xinhua English news. [2021-11-15]. (原始內容存檔於2021-05-24).
- ^ "时间晶体"不再是科幻_ 中国青年网. [2021-11-15]. (原始內容存檔於2021-11-17).
- ^ “时间晶体”不再是科幻 人民网. [2021-11-15]. (原始內容存檔於2021-11-15).
- ^ Choi, Soonwon; Choi, Joonhee; Landig, Renate; Kucsko, Georg; Zhou, Hengyun; Isoya, Junichi; Jelezko, Fedor; Onoda, Shinobu; Sumiya, Hitoshi; Khemani, Vedika; von Keyserlingk, Curt; Yao, Norman Y.; Demler, Eugene; Lukin, Mikhail D. Observation of discrete time-crystalline order in a disordered dipolar many-body system (PDF). Nature. 2017, 543 (7644): 221–225 [2017-03-23]. Bibcode:2016arXiv161008057C. ISSN 0028-0836. arXiv:1610.08057v1 . doi:10.1038/nature21426. (原始內容存檔 (PDF)於2019-05-10).
- ^ 科学家打造超宇宙寿命“时空晶体”. 科學探索_科技時代:新浪網. [2019-04-20]. (原始內容存檔於2019-06-04).
延伸閱讀
學術論文
- Beck, Christian; Mackey, Michael C. Could dark energy be measured in the lab? (PDF). Physics Letters B. 2005, 605 (3–4): 295–300 [2021-03-28]. Bibcode:2005PhLB..605..295B. ISSN 0370-2693. arXiv:astro-ph/0406504v2 . doi:10.1016/j.physletb.2004.11.060. (原始內容存檔 (PDF)於2019-10-15).
- Boyle, Latham; Khoo, Jun Yong; Smith, Kendrick. Symmetric Satellite Swarms and Choreographic Crystals (PDF). Physical Review Letters. 2016, 116 (1): 015503 [2021-03-28]. Bibcode:2016PhRvL.116a5503B. ISSN 0031-9007. PMID 26799028. arXiv:1407.5876v2 . doi:10.1103/PhysRevLett.116.015503. (原始內容存檔 (PDF)於2017-02-11).
- Bruno, Patrick. Comment on "Quantum Time Crystals" (PDF). Physical Review Letters. 2013a, 110 (11): 118901 [2021-03-28]. Bibcode:2013PhRvL.110k8901B. ISSN 0031-9007. PMID 25166585. arXiv:1210.4128v1 . doi:10.1103/PhysRevLett.110.118901. (原始內容存檔 (PDF)於2017-02-11).
- Bruno, Patrick. Comment on "Space-Time Crystals of Trapped Ions" (PDF). Physical Review Letters. 2013b, 111 (2) [2021-03-28]. Bibcode:2013PhRvL.111b9301B. ISSN 0031-9007. arXiv:1211.4792v1 . doi:10.1103/PhysRevLett.111.029301. (原始內容存檔 (PDF)於2017-02-11).
- Campisi, Michele; Hänggi, Peter; Talkner, Peter. Colloquium: Quantum fluctuation relations: Foundations and applications (PDF). Reviews of Modern Physics. 2011, 83 (3): 771–791 [2021-03-28]. Bibcode:2011RvMP...83..771C. ISSN 0034-6861. arXiv:1012.2268v5 . doi:10.1103/RevModPhys.83.771. (原始內容存檔 (PDF)於2019-01-27).
- Choi, Soonwon; Choi, Joonhee; Landig, Renate; Kucsko, Georg; Zhou, Hengyun; Isoya, Junichi; Jelezko, Fedor; Onoda, Shinobu; Sumiya, Hitoshi; Khemani, Vedika; von Keyserlingk, Curt; Yao, Norman Y.; Demler, Eugene; Lukin, Mikhail D. Observation of discrete time-crystalline order in a disordered dipolar many-body system (PDF). Nature. 2017, 543 (7644): 221–225 [2017-03-23]. Bibcode:2017Natur.543..221C. ISSN 0028-0836. arXiv:1610.08057v1 . doi:10.1038/nature21426. (原始內容存檔 (PDF)於2019-05-10).
- Chernodub, M. N. Permanently rotating devices: extracting rotation from quantum vacuum fluctuations? (PDF). 2012 [2021-03-28]. Bibcode:2012arXiv1203.6588C. arXiv:1203.6588v1 . (原始內容存檔 (PDF)於2019-10-24).
- Chernodub, M. N. Zero-point fluctuations in rotation: Perpetuum mobile of the fourth kind without energy transfer (PDF). Nuovo Cimento C. 2013a, 5 (36): 53–63 [2021-03-28]. Bibcode:2013arXiv1302.0462C. arXiv:1302.0462v1 . doi:10.1393/ncc/i2013-11523-5. (原始內容存檔 (PDF)於2019-08-14).
- Chernodub, M. N. Rotating Casimir systems: Magnetic-field-enhanced perpetual motion, possible realization in doped nanotubes, and laws of thermodynamics (PDF). Physical Review D. 2013b, 87 (2) [2021-03-28]. Bibcode:2013PhRvD..87b5021C. ISSN 1550-7998. arXiv:1207.3052v2 . doi:10.1103/PhysRevD.87.025021. (原始內容存檔 (PDF)於2019-10-23).
- Copeland, Edmund J.; Sami, M.; Tsujikawa, Shinji. Dynamics of dark energy (PDF). International Journal of Modern Physics D. 2006, 15 (11): 1753–1935 [2021-03-28]. Bibcode:2006IJMPD..15.1753C. ISSN 0218-2718. arXiv:hep-th/0603057 . doi:10.1142/S021827180600942X. (原始內容存檔 (PDF)於2017-02-11).
- Dillenschneider, R.; Lutz, E. Energetics of quantum correlations (PDF). EPL. 2009, 88 (5): 50003 [2021-03-28]. Bibcode:2009EL.....8850003D. ISSN 0295-5075. arXiv:0803.4067 . doi:10.1209/0295-5075/88/50003. (原始內容存檔 (PDF)於2019-10-01).
- Else, Dominic V.; Bauer, Bela; Nayak, Chetan. Floquet Time Crystals (PDF). Physical Review Letters. 2016, 117 (9): 090402 [2021-03-28]. Bibcode:2016PhRvL.117i0402E. ISSN 0031-9007. PMID 27610834. arXiv:1603.08001v4 . doi:10.1103/PhysRevLett.117.090402. (原始內容存檔 (PDF)於2019-10-16).
- Esposito, Massimiliano; Harbola, Upendra; Mukamel, Shaul. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems (PDF). Reviews of Modern Physics. 2009, 81 (4): 1665–1702 [2021-03-28]. Bibcode:2009RvMP...81.1665E. ISSN 0034-6861. arXiv:0811.3717v2 . doi:10.1103/RevModPhys.81.1665. (原始內容存檔 (PDF)於2017-02-11).
- Grifoni, Milena; Hänggi, Peter. Driven quantum tunneling (PDF). Physics Reports. 1998, 304 (5–6): 229–354 [2021-03-28]. Bibcode:1998PhR...304..229G. ISSN 0370-1573. doi:10.1016/S0370-1573(98)00022-2. (原始內容 (PDF)存檔於2017-02-11).
- Guo, Lingzhen; Marthaler, Michael; Schön, Gerd. Phase Space Crystals: A New Way to Create a Quasienergy Band Structure (PDF). Physical Review Letters. 2013, 111 (20) [2021-03-28]. Bibcode:2013PhRvL.111t5303G. ISSN 0031-9007. arXiv:1305.1800v3 . doi:10.1103/PhysRevLett.111.205303. (原始內容存檔 (PDF)於2017-02-11).
- Hasan, M. Z.; Kane, C. L. Colloquium: Topological insulators (PDF). Reviews of Modern Physics. 2010, 82 (4): 3045–3067 [2021-03-28]. Bibcode:2010RvMP...82.3045H. ISSN 0034-6861. arXiv:1002.3895v2 . doi:10.1103/RevModPhys.82.3045. (原始內容存檔 (PDF)於2017-02-11).
- Horodecki, Ryszard; Horodecki, Paweł; Horodecki, Michał; Horodecki, Karol. Quantum entanglement (PDF). Reviews of Modern Physics. 2009, 81 (2): 865–942 [2021-03-28]. Bibcode:2009RvMP...81..865H. ISSN 0034-6861. arXiv:quant-ph/0702225v2 . doi:10.1103/RevModPhys.81.865. (原始內容存檔 (PDF)於2017-02-11).
- Jaffe, R. L. Casimir effect and the quantum vacuum (PDF). Physical Review D. 2005, 72 (2): 021301 [2021-03-28]. Bibcode:2005PhRvD..72b1301J. arXiv:hep-th/0503158 . doi:10.1103/PhysRevD.72.021301. (原始內容存檔 (PDF)於2017-11-05).
- Jarzynski, Christopher. Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale (PDF). Annual Review of Condensed Matter Physics. 2011, 2 (1): 329–351 [2021-03-28]. Bibcode:2011ARCMP...2..329J. ISSN 1947-5454. doi:10.1146/annurev-conmatphys-062910-140506. (原始內容存檔 (PDF)於2017-02-11).
- Jetzer, Philippe; Straumann, Norbert. Josephson junctions and dark energy (PDF). Physics Letters B. 2006, 639 (2): 57–58 [2021-03-28]. Bibcode:2006PhLB..639...57J. ISSN 0370-2693. arXiv:astro-ph/0604522 . doi:10.1016/j.physletb.2006.06.020. (原始內容存檔 (PDF)於2019-01-27).
- Khemani, Vedika; Lazarides, Achilleas; Moessner, Roderich; Sondhi, S. L. Phase Structure of Driven Quantum Systems (PDF). Physical Review Letters. 2016, 116 (25) [2021-03-28]. Bibcode:2016PhRvL.116y0401K. ISSN 0031-9007. arXiv:1508.03344v3 . doi:10.1103/PhysRevLett.116.250401. (原始內容存檔 (PDF)於2019-10-22).
- Lees, J. P. Observation of Time-Reversal Violation in the B0 Meson System (PDF). Physical Review Letters. 2012, 109 (21) [2021-03-28]. Bibcode:2012PhRvL.109u1801L. ISSN 0031-9007. arXiv:1207.5832v4 . doi:10.1103/PhysRevLett.109.211801. (原始內容存檔 (PDF)於2019-01-27).
- Li, Tongcang; Gong, Zhe-Xuan; Yin, Zhang-Qi; Quan, H. T.; Yin, Xiaobo; Zhang, Peng; Duan, L.-M.; Zhang, Xiang. Space-Time Crystals of Trapped Ions (PDF). Physical Review Letters. 2012a, 109 (16) [2021-03-28]. Bibcode:2012PhRvL.109p3001L. ISSN 0031-9007. arXiv:1206.4772v2 . doi:10.1103/PhysRevLett.109.163001. (原始內容存檔 (PDF)於2017-02-11).
- Li, Tongcang; Gong, Zhe-Xuan; Yin, Zhang-Qi; Quan, H. T.; Yin, Xiaobo; Zhang, Peng; Duan, L.-M.; Zhang, Xiang. Reply to Comment on "Space-Time Crystals of Trapped Ions" (PDF). 2012b [2021-03-28]. Bibcode:2012arXiv1212.6959L. arXiv:1212.6959v2 . (原始內容存檔 (PDF)於2017-02-11).
- Lindner, Netanel H.; Refael, Gil; Galitski, Victor. Floquet topological insulator in semiconductor quantum wells (PDF). Nature Physics. 2011, 7 (6): 490–495 [2021-03-28]. Bibcode:2011NatPh...7..490L. ISSN 1745-2473. arXiv:1008.1792v2 . doi:10.1038/nphys1926. (原始內容存檔 (PDF)於2019-10-13).
- Nadj-Perge, S.; Drozdov, I. K.; Li, J.; Chen, H.; Jeon, S.; Seo, J.; MacDonald, A. H.; Bernevig, B. A.; Yazdani, A. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor (PDF). Science. 2014, 346 (6209): 602–607 [2021-03-28]. Bibcode:2014Sci...346..602N. ISSN 0036-8075. PMID 25278507. arXiv:1410.0682v1 . doi:10.1126/science.1259327. (原始內容存檔 (PDF)於2017-02-11).
- Nozières, Philippe. Time crystals: Can diamagnetic currents drive a charge density wave into rotation? (PDF). EPL. 2013, 103 (5): 57008 [2021-03-28]. Bibcode:2013EL....10357008N. ISSN 0295-5075. arXiv:1306.6229v1 . doi:10.1209/0295-5075/103/57008. (原始內容存檔 (PDF)於2019-10-21).
- Sacha, Krzysztof. Modeling spontaneous breaking of time-translation symmetry (PDF). Physical Review A. 2015, 91 (3) [2021-03-28]. Bibcode:2015PhRvA..91c3617S. ISSN 1050-2947. arXiv:1410.3638v3 . doi:10.1103/PhysRevA.91.033617. (原始內容存檔 (PDF)於2017-02-11).
- Schwinger, Julian. Casimir effect in source theory. Letters in Mathematical Physics. 1975, 1 (1): 43–47. Bibcode:1975LMaPh...1...43S. doi:10.1007/BF00405585.
- Schwinger, Julian; DeRaad, Lester L.; Milton, Kimball A. Casimir effect in dielectrics. Annals of Physics. 1978, 115 (1): 1–23. Bibcode:1978AnPhy.115....1S. doi:10.1016/0003-4916(78)90172-0.
- Scully, Marlan O. Extracting Work from a Single Thermal Bath via Quantum Negentropy. Physical Review Letters. 2001, 87 (22): 220601. Bibcode:2001PhRvL..87v0601S. ISSN 0031-9007. PMID 11736390. doi:10.1103/PhysRevLett.87.220601.
- Scully, Marlan O.; Zubairy, M. Suhail; Agarwal, Girish S.; Walther, Herbert. Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence. Science. 2003, 299 (5608): 862–864. Bibcode:2003Sci...299..862S. ISSN 0036-8075. PMID 12511655. doi:10.1126/science.1078955.
- Seifert, Udo. Stochastic thermodynamics, fluctuation theorems and molecular machines (PDF). Reports on Progress in Physics. 2012, 75 (12): 126001 [2021-03-28]. Bibcode:2012RPPh...75l6001S. ISSN 0034-4885. arXiv:1205.4176v1 . doi:10.1088/0034-4885/75/12/126001. (原始內容存檔 (PDF)於2017-02-11).
- Senitzky, I. R. Dissipation in Quantum Mechanics. The Harmonic Oscillator. Physical Review. 1960, 119 (2): 670–679. Bibcode:1960PhRv..119..670S. ISSN 0031-899X. doi:10.1103/PhysRev.119.670.
- Shapere, Alfred; Wilczek, Frank. Classical Time Crystals (PDF). Physical Review Letters. 2012, 109 (16) [2021-03-28]. Bibcode:2012PhRvL.109p0402S. ISSN 0031-9007. arXiv:1202.2537v2 . doi:10.1103/PhysRevLett.109.160402. (原始內容存檔 (PDF)於2017-02-11).
- Shirley, Jon H. Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time. Physical Review. 1965, 138 (4B): B979–B987. Bibcode:1965PhRv..138..979S. ISSN 0031-899X. doi:10.1103/PhysRev.138.B979.
- Smith, J.; Lee, A.; Richerme, P.; Neyenhuis, B.; Hess, P. W.; Hauke, P.; Heyl, M.; Huse, D. A.; Monroe, C. Many-body localization in a quantum simulator with programmable random disorder (PDF). Nature Physics. 2016, 12 (10): 907–911 [2021-03-28]. Bibcode:2016NatPh..12..907S. ISSN 1745-2473. arXiv:1508.07026v1 . doi:10.1038/nphys3783. (原始內容存檔 (PDF)於2017-02-11).
- Maruyama, Koji; Nori, Franco; Vedral, Vlatko. Colloquium* The physics of Maxwell’s demon and information (PDF). Reviews of Modern Physics. 2009, 81 (1): 1–23 [2021-03-28]. Bibcode:2009RvMP...81....1M. ISSN 0034-6861. arXiv:0707.3400 . doi:10.1103/RevModPhys.81.1. (原始內容存檔 (PDF)於2017-02-11).
- Mendonça, J. T.; Dodonov, V. V. Time Crystals in Ultracold Matter (PDF). Journal of Russian Laser Research. 2014, 35 (1): 93–100 [2021-03-28]. ISSN 1071-2836. doi:10.1007/s10946-014-9404-9. (原始內容存檔 (PDF)於2018-06-23).
- Modi, Kavan; Brodutch, Aharon; Cable, Hugo; Paterek, Tomasz; Vedral, Vlatko. The classical-quantum boundary for correlations* Discord and related measures (PDF). Reviews of Modern Physics. 2012, 84 (4): 1655–1707 [2021-03-28]. Bibcode:2012RvMP...84.1655M. ISSN 0034-6861. arXiv:1112.6238 . doi:10.1103/RevModPhys.84.1655. (原始內容存檔 (PDF)於2017-02-11).
- Ray, M. W.; Ruokokoski, E.; Kandel, S.; Möttönen, M.; Hall, D. S. Observation of Dirac monopoles in a synthetic magnetic field (PDF). Nature. 2014, 505 (7485): 657–660 [2021-03-28]. Bibcode:2014Natur.505..657R. ISSN 0028-0836. PMID 24476889. arXiv:1408.3133v1 . doi:10.1038/nature12954. (原始內容存檔 (PDF)於2019-10-20).
- Ray, M. W.; Ruokokoski, E.; Tiurev, K.; Mottonen, M.; Hall, D. S. Observation of isolated monopoles in a quantum field (PDF). Science. 2015, 348 (6234): 544–547 [2021-03-28]. Bibcode:2015Sci...348..544R. ISSN 0036-8075. doi:10.1126/science.1258289. (原始內容存檔 (PDF)於2017-08-08).
- Reimann, Peter; Grifoni, Milena; Hänggi, Peter. Quantum Ratchets (PDF). Physical Review Letters. 1997, 79 (1): 10–13 [2021-03-28]. Bibcode:1997PhRvL..79...10R. ISSN 0031-9007. doi:10.1103/PhysRevLett.79.10. (原始內容 (PDF)存檔於2017-02-11).
- Roßnagel, J.; Abah, O.; Schmidt-Kaler, F.; Singer, K.; Lutz, E. Nanoscale Heat Engine Beyond the Carnot Limit (PDF). Physical Review Letters. 2014, 112 (3): 030602 [2021-03-28]. Bibcode:2014PhRvL.112c0602R. ISSN 0031-9007. PMID 24484127. arXiv:1308.5935 . doi:10.1103/PhysRevLett.112.030602. (原始內容存檔 (PDF)於2016-06-16).
- Roßnagell, J.; Dawkins, S. T.; Tolazzi, K. N.; Abah, O.; Lutz, E.; Schmidt-Kaler, F.; Singer, K. A single-atom heat engine (PDF). Science. 2016, 352 (6283): 325–329 [2021-03-28]. Bibcode:2016Sci...352..325R. ISSN 0036-8075. arXiv:1510.03681 . doi:10.1126/science.aad6320. (原始內容存檔 (PDF)於2017-02-11).
- Tatara, Gen; Kikuchi, Makoto; Yukawa, Satoshi; Matsukawa, Hiroshi. Dissipation Enhanced Asymmetric Transport in Quantum Ratchets (PDF). Journal of the Physical Society of Japan. 1998, 67 (4): 1090–1093 [2021-03-28]. Bibcode:1998JPSJ...67.1090T. ISSN 0031-9015. arXiv:cond-mat/9711045 . doi:10.1143/JPSJ.67.1090. (原始內容存檔 (PDF)於2019-10-01).
- Volovik, G. E. On the broken time translation symmetry in macroscopic systems* Precessing states and off-diagonal long-range order (PDF). JETP Letters. 2013, 98 (8): 491–495 [2021-03-28]. Bibcode:2013JETPL..98..491V. ISSN 0021-3640. arXiv:1309.1845v2 . doi:10.1134/S0021364013210133. (原始內容存檔 (PDF)於2019-01-27).
- von Keyserlingk, C. W.; Khemani, Vedika; Sondhi, S. L. Absolute stability and spatiotemporal long-range order in Floquet systems (PDF). Physical Review B. 2016, 94 (8) [2021-03-28]. Bibcode:2016PhRvB..94h5112V. ISSN 2469-9950. arXiv:1605.00639v3 . doi:10.1103/PhysRevB.94.085112. (原始內容存檔 (PDF)於2019-10-21).
- Wang, Y. H.; Steinberg, H.; Jarillo-Herrero, P.; Gedik, N. Observation of Floquet-Bloch States on the Surface of a Topological Insulator (PDF). Science. 2013, 342 (6157): 453–457 [2021-03-28]. Bibcode:2013Sci...342..453W. ISSN 0036-8075. arXiv:1310.7563v1 . doi:10.1126/science.1239834. (原始內容存檔 (PDF)於2019-10-19).
- Watanabe, Haruki; Oshikawa, Masaki. Absence of Quantum Time Crystals (PDF). Physical Review Letters. 2015, 114 (25): 251603 [2021-03-28]. Bibcode:2015PhRvL.114y1603W. ISSN 0031-9007. PMID 26197119. arXiv:1410.2143v3 . doi:10.1103/PhysRevLett.114.251603. (原始內容存檔 (PDF)於2019-10-12).
- Wilczek, Frank. Quantum Time Crystals (PDF). Physical Review Letters. 2012, 109 (16) [2021-03-28]. Bibcode:2012PhRvL.109p0401W. ISSN 0031-9007. arXiv:1202.2539v2 . doi:10.1103/PhysRevLett.109.160401. (原始內容存檔 (PDF)於2019-10-16).
- Wilczek, Frank. Wilczek Reply* (PDF). Physical Review Letters. 2013a, 110 (11) [2021-03-28]. Bibcode:2013PhRvL.110k8902W. ISSN 0031-9007. doi:10.1103/PhysRevLett.110.118902. (原始內容存檔 (PDF)於2017-02-11).
- Wilczek, Frank. Superfluidity and Space-Time Translation Symmetry Breaking (PDF). Physical Review Letters. 2013b, 111 (25) [2021-03-28]. Bibcode:2013PhRvL.111y0402W. ISSN 0031-9007. arXiv:1308.5949v1 . doi:10.1103/PhysRevLett.111.250402. (原始內容存檔 (PDF)於2019-10-18).
- Willett, R. L.; Nayak, C.; Shtengel, K.; Pfeiffer, L. N.; West, K. W. Magnetic-Field-Tuned Aharonov-Bohm Oscillations and Evidence for Non-Abelian Anyons atν=5/2 (PDF). Physical Review Letters. 2013, 111 (18) [2021-03-28]. Bibcode:2013PhRvL.111r6401W. ISSN 0031-9007. arXiv:1301.2639v1 . doi:10.1103/PhysRevLett.111.186401. (原始內容存檔 (PDF)於2016-06-02).
- Yao, N. Y.; Potter, A. C.; Potirniche, I.-D.; Vishwanath, A. Discrete Time Crystals* Rigidity, Criticality, and Realizations (PDF). Physical Review Letters. 2017, 118 (3) [2021-03-28]. Bibcode:2017PhRvL.118c0401Y. ISSN 0031-9007. arXiv:1608.02589v2 . doi:10.1103/PhysRevLett.118.030401. (原始內容存檔 (PDF)於2019-01-27).
- Yoshii, Ryosuke; Takada, Satoshi; Tsuchiya, Shunji; Marmorini, Giacomo; Hayakawa, Hisao; Nitta, Muneto. Fulde-Ferrell-Larkin-Ovchinnikov states in a superconducting ring with magnetic fields* Phase diagram and the first-order phase transitions (PDF). Physical Review B. 2015, 92 (22) [2021-03-28]. Bibcode:2015PhRvB..92v4512Y. ISSN 1098-0121. arXiv:1404.3519v2 . doi:10.1103/PhysRevB.92.224512. (原始內容存檔 (PDF)於2017-02-11).
- Yukawa, Satoshi; Kikuchi, Macoto; Tatara, Gen; Matsukawa, Hiroshi. Quantum Ratchets (PDF). Journal of the Physical Society of Japan. 1997, 66 (10): 2953–2956 [2021-03-28]. Bibcode:1997JPSJ...66.2953Y. ISSN 0031-9015. arXiv:cond-mat/9706222 . doi:10.1143/JPSJ.66.2953. (原始內容存檔 (PDF)於2017-02-11).
- Yukawa, Satoshi. A Quantum Analogue of the Jarzynski Equality (PDF). Journal of the Physical Society of Japan. 2000, 69 (8): 2367–2370 [2021-03-28]. Bibcode:2000JPSJ...69.2367Y. ISSN 0031-9015. arXiv:cond-mat/0007456 . doi:10.1143/JPSJ.69.2367. (原始內容存檔 (PDF)於2019-01-27).
- Zel'Dovich, Y. B. The quasienergy of a quantum-mechanical system subjected to a periodic action (PDF). Soviet Physics JETP. 1967, 24 (5): 1006–1008 [2021-03-28]. Bibcode:1967JETP...24.1006Z. (原始內容存檔 (PDF)於2017-02-11).
- 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. Observation of a Discrete Time Crystal (PDF). Nature. 2017, 543 (7644): 217–220 [2021-03-28]. Bibcode:2017Natur.543..217Z. ISSN 0028-0836. arXiv:1609.08684v1 . doi:10.1038/nature21413. (原始內容存檔 (PDF)於2019-10-20).
書籍
- Bordag, M.; Mohideen, U.; Mostepanenko, V.M. New developments in the Casimir effect (PDF). Physics Reports. 2001, 353 (1–3): 1–205 [2021-03-28]. Bibcode:2001PhR...353....1B. ISSN 0370-1573. arXiv:quant-ph/0106045 . doi:10.1016/S0370-1573(01)00015-1. (原始內容存檔 (PDF)於2017-02-11).
- Bordag, M.; Mohideen, U.; Mostepanenko, V.M.; Klimchitskaya, G. L. Advances in the Casimir Effect. Oxford: Oxford University Press. 28 May 2009 [2021-03-28]. ISBN 978-0-19-157988-2. (原始內容存檔於2019-03-31).
- Cao, Tian Yu. Conceptual Foundations of Quantum Field Theory. Cambridge: Cambridge University Press. 25 March 2004 [2021-03-28]. ISBN 978-0-521-60272-3. (原始內容存檔於2019-12-15).
- Enz, Charles P. Is the Zero-Point Energy Real?. Enz, C. P.; Mehra, J. (編). Physical Reality and Mathematical Description. Dordrecht: D. Reidel Publishing Company. 1974: 124–132. ISBN 978-94-010-2274-3. doi:10.1007/978-94-010-2274-3_8.
- Greiner, Walter; Müller, B.; Rafelski, J. Quantum Electrodynamics of Strong Fields* With an Introduction into Modern Relativistic Quantum Mechanics. Springer. 2012 [2021-03-28]. ISBN 978-3-642-82274-2. doi:10.1007/978-3-642-82272-8. (原始內容存檔於2020-08-20).
- Lee, T. D. Particle Physics. CRC Press. 15 August 1981 [2021-03-28]. ISBN 978-3-7186-0033-5. (原始內容存檔於2020-08-19).
- Feng, Duan; Jin, Guojun. Introduction to Condensed Matter Physics. singapore: World Scientific. 2005 [2021-03-28]. ISBN 978-981-238-711-0. (原始內容存檔於2019-03-31).
- Milonni, Peter W. The Quantum Vacuum* An Introduction to Quantum Electrodynamics. London: Academic Press. 1994 [2021-03-28]. ISBN 978-0-124-98080-8. (原始內容存檔於2020-08-20).
- Pade, Jochen. Quantum Mechanics for Pedestrians 2* Applications and Extensions. Dordrecht: Springer. 2014 [2021-03-28]. ISBN 978-3-319-00813-4. ISSN 2192-4791. doi:10.1007/978-3-319-00813-4. (原始內容存檔於2020-08-19).
- Schwinger, Julian. Particles, Sources, And Fields, Volume 1* v. 1 (Advanced Books Classics). Perseus. 1998a. ISBN 978-0-738-20053-8.
- Schwinger, Julian. Particles, Sources, And Fields, Volume 2* v. 2 (Advanced Books Classics). Perseus. 1998b. ISBN 978-0-738-20054-5.
- Schwinger, Julian. Particles, Sources, And Fields, Volume 3* v. 3 (Advanced Books Classics). Perseus. 1998c. ISBN 978-0-738-20055-2.
- Sólyom, Jenö. Fundamentals of the Physics of Solids* Volume 1* Structure and Dynamics. Springer. 19 September 2007 [2021-03-28]. ISBN 978-3-540-72600-5. (原始內容存檔於2019-03-31).
- Wilczek, Frank. A Beautiful Question* Finding Nature's Deep Design. Penguin Books Limited. 16 July 2015 [2021-03-28]. ISBN 978-1-84614-702-9. (原始內容存檔於2019-12-15).
出版物
- Aalto University. Physicists discover quantum-mechanical monopoles. phys.org. Science X. 30 April 2015 [2019-01-14]. (原始內容存檔於2015-04-30).
- Aitchison, Ian. Observing the Unobservable. New Scientist. 19 November 1981, 92 (1280): 540–541 [2021-03-28]. ISSN 0262-4079. (原始內容存檔於2020-08-19).
- Amherst College. Physicists create synthetic magnetic monopole predicted more than 80 years ago. phys.org. Science X. 29 January 2014 [2019-01-14]. (原始內容存檔於2014-01-29).
- Aron, Jacob. Computer that could outlive the universe a step closer. newscientist.com. New Scientist. 6 July 2012 [2019-01-14]. (原始內容存檔於2017-02-02).
- Ball, Philip. Focus* New Crystal Type is Always in Motion. physics.aps.org. APS Physics. 8 January 2016 [2019-01-14]. (原始內容存檔於2017-02-03).
- Ball, Philip. Scepticism greets pitch to detect dark energy in the lab. Nature. 8 July 2004, 430 (6996): 126–126 [2021-03-28]. Bibcode:2004Natur.430..126B. ISSN 0028-0836. doi:10.1038/430126b. (原始內容存檔於2019-04-22).
- Cartlidge, Edwin. Scientists build heat engine from a single atom. sciencemag.org. Science Magazine. 21 October 2015 [2019-01-14]. (原始內容存檔於2017-02-01).
- Chandler, David. Topological insulators* Persuading light to mix it up with matter. phys.org. Science X. 24 October 2014 [2019-01-14]. (原始內容存檔於2017-02-08).
- Coleman, Piers. Quantum physics* Time crystals. Nature. 9 January 2013, 493 (7431): 166–167. Bibcode:2013Natur.493..166C. ISSN 0028-0836. doi:10.1038/493166a.
- Cowen, Ron. "Time Crystals" Could Be a Legitimate Form of Perpetual Motion. scientificamerican.com. Scientific American. 27 February 2012 [2019-01-14]. (原始內容存檔於2017-02-02).
- Daghofer, Maria. Viewpoint* Toward Fractional Quantum Hall Physics with Cold Atoms. physics.aps.org. APS Physics. 29 April 2013 [2019-01-14]. (原始內容存檔於2017-02-07).
- Gibney, Elizabeth. The quest to crystallize time. Nature. 2017, 543 (7644): 164–166 [2021-03-28]. Bibcode:2017Natur.543..164G. ISSN 0028-0836. doi:10.1038/543164a. (原始內容存檔於2017-03-13).
- Grossman, Lisa. Death-defying time crystal could outlast the universe. newscientist.com. New Scientist. 18 January 2012 [2019-01-14]. (原始內容存檔於2017-02-02).
- Hackett, Jennifer. Curious Crystal Dances for Its Symmetry. scientificamerican.com. Scientific American. 22 February 2016 [2019-01-14]. (原始內容存檔於2017-02-03).
- Hewitt, John. Creating time crystals with a rotating ion ring. phys.org. Science X. 3 May 2013 [2019-01-14]. (原始內容存檔於2013-07-04).
- Johnston, Hamish. 'Choreographic crystals' have all the right moves. physicsworld.com. Institute of Physics. 18 January 2016 [2019-01-14]. (原始內容存檔於2017-02-03).
- Johannes Gutenberg Universitaet Mainz. Prototype of single ion heat engine created. sciencedaily.com. ScienceDaily. 3 February 2014 [2019-01-14]. (原始內容存檔於2017-02-01).
- Joint Quantum Institute. Floquet Topological Insulators. jqi.umd.edu. Joint Quantum Institute. 22 March 2011 [2019-01-14]. (原始內容存檔於2019-04-03).
- Morgan, James. Elusive magnetic 'monopole' seen in quantum system. bbc.co.uk. BBC. 30 January 2014 [2019-01-14]. (原始內容存檔於2014-01-30).
- Moskowitz, Clara. New Particle Is Both Matter and Antimatter. scientificamerican.com. Scientific American. 2 October 2014 [2019-01-14]. (原始內容存檔於2014-10-09).
- Ouellette, Jennifer. World’s first time crystals cooked up using new recipe. newscientist.com. New Scientist. 31 January 2017 [2019-01-14]. (原始內容存檔於2017-02-01).
- Pilkington, Mark. Zero point energy. theguardian.com (The Guardian). 17 July 2003 [2019-01-14]. (原始內容存檔於2017-02-07).
- Powell, Devin. Can matter cycle through shapes eternally?. Nature. 2013 [2021-03-28]. ISSN 1476-4687. doi:10.1038/nature.2013.13657. (原始內容存檔於2017-02-03).
- Rao, Achintya. BaBar makes first direct measurement of time-reversal violation. physicsworld.com. Institute of Physics. 21 November 2012 [2019-01-14]. (原始內容存檔於2015-03-24).
- Richerme, Phil. Viewpoint* How to Create a Time Crystal. physics.aps.org. APS Physics. 18 January 2017 [2019-01-14]. (原始內容存檔於2017-02-02).
- Thomas, Jessica. Notes from the Editors* The Aftermath of a Controversial Idea. physics.aps.org. APS Physics. 15 March 2013 [2019-01-14]. (原始內容存檔於2017-02-02).
- Qi, Xiao-Liang; Zhang, Shou-Cheng. The quantum spin Hall effect and topological insulators (PDF). Physics Today. 2010, 63 (1): 33–38 [2021-03-28]. Bibcode:2010PhT....63a..33Q. ISSN 0031-9228. arXiv:1001.1602 . doi:10.1063/1.3293411. (原始內容存檔 (PDF)於2017-08-12).
- University of California, Berkeley. Physicists unveil new form of matter—time crystals. phys.org. Science X. 26 January 2017 [2019-01-14]. (原始內容存檔於2017-01-28).
- Weiner, Sophie. Scientists Create A New Kind Of Matter* Time Crystals. popularmechanics.com. Popular mechanics. 28 January 2017 [2019-01-14]. (原始內容存檔於2017-02-03).
- Wolchover, Natalie. Perpetual Motion Test Could Amend Theory of Time. quantamagazine.org. Simons Foundation. 25 April 2013 [2019-01-14]. (原始內容存檔於2017-02-02).
- Wolchover, Natalie. Forging a Qubit to Rule Them All. quantamagazine.org. Simmons Foundation. 15 May 2014 [2019-01-14]. (原始內容存檔於2016-03-15).
- Wood, Charlie. Time crystals realize new order of space-time. csmonitor.com. Christian Science Monitor. 31 January 2017 [2019-01-14]. (原始內容存檔於2017-02-02).
- Yirka, Bob. Physics team proposes a way to create an actual space-time crystal. phys.org. Science X. 9 July 2012 [2019-01-14]. (原始內容存檔於2013-04-15).
- Zakrzewski, Jakub. Viewpoint* Crystals of Time. physics.aps.org. APS Physics. 15 October 2012 [2019-01-14]. (原始內容存檔於2017-02-02).
- Zeller, Michael. Viewpoint* Particle Decays Point to an Arrow of Time. physics.aps.org. APS Physics. 19 November 2012 [2019-01-14]. (原始內容存檔於2017-02-04).
- Zyga, Lisa. Time crystals could behave almost like perpetual motion machines. phys.org. Science X. 20 February 2012 [2019-01-14]. (原始內容存檔於2017-02-03).
- Zyga, Lisa. Physicist proves impossibility of quantum time crystals. phys.org. Space X. 22 August 2013 [2019-01-14]. (原始內容存檔於2017-02-03).
- Zyga, Lisa. Nanoscale heat engine exceeds standard efficiency limit. phys.org. Science X. 27 January 2014 [2019-01-14]. (原始內容存檔於2015-04-04).
- Zyga, Lisa. Physicists propose new definition of time crystals—then prove such things don't exist. phys.org. Science X. 9 July 2015 [2019-01-14]. (原始內容存檔於2015-07-09).
- Zyga, Lisa. Time crystals might exist after all (Update). phys.org. Science X. 9 September 2016 [2019-01-14]. (原始內容存檔於2016-09-11).