Physics:Quantum scar

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Perturbation-induced quantum skipping scar in a disordered quantum well with an external magnetic field (for further information, see below and Ref.[1]).

Quantum scarring refers to a phenomenon where the eigenstates of a classically chaotic quantum system have enhanced probability density around the paths of unstable classical periodic orbits.[2][3] The instability of the periodic orbit is a decisive point that differentiates quantum scars from the more trivial observation that the probability density is enhanced in the neighborhood of stable periodic orbits. The latter can be understood as a purely classical phenomenon, a manifestation of the Bohr correspondence principle, whereas in the former, quantum interference is essential. As such, scarring is both a visual example of quantum-classical correspondence, and simultaneously an example of a (local) quantum suppression of chaos.

A classically chaotic system is also ergodic, and therefore (almost) all of its trajectories eventually explore evenly the entire accessible phase space. Thus, it would be natural to expect that the eigenstates of the quantum counterpart would fill the quantum phase space in the uniform manner up to random fluctuations in the semiclassical limit. However, scars are a significant correction to this assumption. Scars can therefore be considered as an eigenstate counterpart of how short periodic orbits provide corrections to the universal spectral statistics of the random matrix theory. There are rigorous mathematical theorems on quantum nature of ergodicity,[4][5][6] proving that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. Nonetheless, the quantum ergodicity theorems do not exclude scarring if the quantum phase space volume of the scars gradually vanishes in the semiclassical limit.

On the classical side, there is no direct analogue of scars. On the quantum side, they can be interpreted as an eigenstate analogy to how short periodic orbits correct the universal random matrix theory eigenvalue statistics. Scars correspond to nonergodic states which are permitted by the quantum ergodicity theorems. In particular, scarred states provide a striking visual counterexample to the assumption that the eigenstates of a classically chaotic system would be without structure. In addition to conventional quantum scars, the field of quantum scarring has undergone its renaissance period, sparked by the discoveries of perturbation-induced scars and many-body scars (see below).

Scar theory

Typical scarred eigenstates of the (Bunimovich) stadium. The figure shows the probability density for three different eigenstates. The scars, referring the regions of concentrated probability density, are generated by (unstable) periodic orbits, two of which are illustrated.

The existence of scarred states is rather unexpected based on the Gutzwiller trace formula,[7][8] which connects the quantum mechanical density of states to the periodic orbits in the corresponding classical system. According to the trace formula, a quantum spectrum is not a result of a trace over all the positions, but it is determined by a trace over all the periodic orbits only. Furthermore, every periodic orbit contributes to an eigenvalue, although not exactly equally. It is even more unlikely that a particular periodic orbit would stand out in contributing to a particular eigenstate in a fully chaotic system, since altogether periodic orbits occupy a zero-volume portion of the total phase space volume. Hence, nothing seems to imply that any particular periodic orbit for a given eigenvalue could have a significant role compared to other periodic orbits. Nonetheless, quantum scarring proves this assumption to be wrong. The scarring was first seen in 1983 by S. W. McDonald in his thesis on the stadium billiard as an interesting numerical observation.[9] They did not show up well in his figure because they were fairly crude "waterfall" plots. This finding was not thoroughly reported in the article discussion about the wave functions and nearest-neighbor level spacing spectra for the stadium billiard.[10] A year later, E. J. Heller published the first examples of scarred eigenfunctions together with a theoretical explanation for their existence.[2] The results revealed large footprints of individual periodic orbits influencing some eigenstates of the classically chaotic Bunimovich stadium, named as scars by E. J. Heller.

A wave packet analysis was a key in proving the existence of the scars, and it is still a valuable tool to understand them. In the original work of E. J. Heller,[2] the quantum spectrum is extracted by propagating a Gaussian wave packet along a periodic orbit. Nowadays, this seminal idea is known as the linear theory of scarring.[2][3][11][12] Scars stand out to the eye in some eigenstates of classically chaotic systems, but are quantified by projection of the eigenstates onto certain test states, often Gaussians, having both average position and average momentum along the periodic orbit. These test states give a provably structured spectrum that reveals the necessity of scars.[13] However, there is no universal measure on scarring; the exact relationship of the stability exponent [math]\displaystyle{ \chi }[/math] to the scarring strength is a matter of definition. Nonetheless, there is a rule of thumb:[3][7] quantum scarring is significant when [math]\displaystyle{ \chi \lt 2\pi }[/math], and the strength scales as [math]\displaystyle{ \chi^{-1} }[/math]. Thus, strong quantum scars are, in general, associated with periodic orbits that are moderately unstable and relative short. The theory predicts the scar enhancement along a classical periodic orbit, but it cannot precisely pinpoint which particular states are scarred and how much. Rather, it can be only stated that some states are scarred within certain energy zones, and by at least by a certain degree.

The linear scarring theory outlined above has been later extended to include nonlinear effects taking place after the wave packet departs the linear dynamics domain around the periodic orbit.[12] At long times, the nonlinear effect can assist the scarring. This stems from nonlinear recurrences associated with homoclinic orbits. A further insight on scarring was acquired with a real-space approach by E. B. Bogomolny[14] and a phase-space alternative by M. V. Berry[15] complementing the wave-packet and Hussimi space methods utilized by E. J. Heller and L. Kaplan.[2][3][12]

The first experimental confirmations of scars were obtained in microwave billiards in the early 1990s.[16][17] Further experimental evidence for scarring has later been delivered by observations in, e.g., quantum wells,[18][19][20] optical cavities[21][22] and the hydrogen atom.[23] In the early 2000s, the first observations were achieved in an elliptical billiard.[24] Many classical trajectories converge in this system and lead to pronounced scarring at the foci, commonly called as quantum mirages.[25] In addition, recent numerical results indicated the existence of quantum scars in ultracold atomic gases.[26]

As well as there being no universal measure for the level of scarring, there is also no generally accepted definition of it. Originally, it was stated[2] that certain unstable periodic orbits are shown to permanently scar some quantum eigenfunctions as [math]\displaystyle{ \hbar \rightarrow 0 }[/math], in the sense that extra density surrounds the region of the periodic orbit. However, a more formal definition for scarring would be the following:[3] A quantum eigenstate of a classically chaotic system is scarred by a periodic if its density on the classical invariant manifolds near and all along that periodic is systematically enhanced above the classical, statistically expected density along that orbit. An fascinating consequence of this enhancement is antiscarring.[3][27] Since there may be strongly scarred states among the eigenstates, the necessity for a uniform average over a large number of states requires the existence of antiscars with low probability in the region of ''regular'' scars. Furthermore, it has been realized[27] that some decay processes have antiscarred states with anomalously long escape times.

Most of the research on quantum scars has been restricted to non-relativistic quantum systems described by the Schrödinger equation, where the dependence of the particle energy on momentum is quadratic. However, scarring can occur in a relativistic quantum systems described by the Dirac equation, where the energy-momentum relation is linear instead.[28][29][30] Heuristically, these relativistic scars are a consequence of the fact that both spinor components satisfy the Helmholtz equation, in analogue to the time-independent Schrödinger equation. Therefore, relativistic scars have the same origin as the conventional scarring[2] introduced by E. J. Heller. Nevertheless, there is a difference in terms of the recurrence with respect to energy variation. Furthermore, it was shown that the scarred states can lead to strong conductance fluctuations in the corresponding open quantum dots via the mechanism of resonant transmission.[28]

In addition to scarring described above, there are several similar phenomena, either connected by theory or appearance. First of all, when scars are visually identified, some of the states may remind of classical ''bouncing-ball'' motion, excluded from quantum scars into its own category. For example, a stadium billiard supports these highly nonergodic eigenstates, which reflect trapped bouncing motion between the straight walls.[3] It has been shown that the bouncing states persist at the limit [math]\displaystyle{ \hbar \rightarrow 0 }[/math], but at the same time this result suggests a diminishing percentage of all the states in the agreement with the quantum ergodicity theorems of Schnirelman, Colin de Verdiere, and Zelditch.[4][5][6] Secondly, scarring should not be confused with statistical fluctuations. Similar structures of an enhanced probability density occur even as random superpositions of plane waves,[31] in the sense of the Berry conjecture.[32][33] Furthermore, there is a genre of scars, not caused by actual periodic orbits, but their remnants, known as ghosts. They refer to periodic orbits that are found in a nearby system in the sense of some tunable, external system parameter.[34][35] Scarring of this kind has been associated to almost-periodic orbits.[36] Another subclass of ghosts stems from complex periodic orbits which exist in the vicinity of bifurcation points.[37][38]

Perturbation-induced quantum scars

Example of scarring in disordered quantum dots. The unperturbed potential has the shape of [math]\displaystyle{ V(r) \propto r^5 }[/math] , and it is perturbed with randomly scattered Gaussian bumps (red markers denote the locations and size of the bumps). The figure shows one of the eigenstates of the perturbed quantum well that is strongly scarred by a periodic orbit of the unperturbed system (solid blue line).

A new class of quantum scars was discovered in disordered two-dimensional nanostructures.[39][40][1][41][42] Even though similar in appearance to ordinary quantum scars described above, these scars have a fundamentally different origin. In the case, the disorder arising from small perturbations (see red dots in the figure) is sufficient to destroy classical long-time stability. Hence, there is no moderately unstable periodic in the classical counterpart to which a scar would corresponds in the ordinary scar theory. Instead, scars are formed around periodic orbits of the corresponding unperturbed system. Ordinary scar theory is further excluded by the behavior of the scars as a function of the disorder strength. When the potential bumps are made stronger while keeping they otherwise unchanged, the scars grow stronger and then fade away without changing their orientation. In contrary, a scar caused by conventional theory should become rapidly weaker due to the increase of the stability exponent of a periodic orbit with increasing disorder. Furthermore, comparing scars at different energies reveals that they occur in only a few distinct orientations This too contradicts predictions of ordinary scar theory.

Many-body quantum scarring

The area of quantum many-body scars is a subject of active research.[43][44]

Scars have occurred in investigations for potential applications of Rydberg states to quantum computing, specifically acting as qubits for quantum simulation.[45][46] The particles of the system in an alternating ground state-Rydberg state configuration continually entangled and disentangled rather than remaining entangled and undergoing thermalization.[45][46][47] Systems of the same atoms prepared with other initial states did thermalize as expected.[46][47] The researchers dubbed the phenomenon "quantum many-body scarring".[48][49]

The causes of quantum scarring are not well understood.[45] One possible proposed explanation is that quantum scars represent integrable systems, or nearly do so, and this could prevent thermalization from ever occurring.[50] This has drawn criticisms arguing that a non-integrable Hamiltonian underlies the theory.[51] Recently, a series of works[52][53] has related the existence of quantum scarring to an algebraic structure known as dynamical symmetries.[54][55]

Fault-tolerant quantum computers are desired, as any perturbations to qubit states can cause the states to thermalize, leading to loss of quantum information.[45] Scarring of qubit states is seen as a potential way to protect qubit states from outside disturbances leading to decoherence and information loss.

See also

References

  1. ↑ 1.0 1.1 Keski-Rahkonen, J.; Luukko, P. J. J.; Kaplan, L.; Heller, E. J.; Räsänen, E. (2017-09-20). "Controllable quantum scars in semiconductor quantum dots". Physical Review B 96 (9): 094204. doi:10.1103/PhysRevB.96.094204. https://link.aps.org/doi/10.1103/PhysRevB.96.094204. 
  2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 Heller, Eric J. (1984-10-15). "Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits". Physical Review Letters 53 (16): 1515–1518. doi:10.1103/PhysRevLett.53.1515. https://link.aps.org/doi/10.1103/PhysRevLett.53.1515. 
  3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 3.6 Heller, Eric Johnson (2018). The semiclassical way to dynamics and spectroscopy. Princeton: Princeton University Press. ISBN 978-1-4008-9029-3. OCLC 1034625177. https://www.worldcat.org/oclc/1034625177. 
  4. ↑ 4.0 4.1 Zelditch, Steven (1987-12-01). "Uniform distribution of eigenfunctions on compact hyperbolic surfaces". Duke Mathematical Journal 55 (4). doi:10.1215/S0012-7094-87-05546-3. ISSN 0012-7094. https://projecteuclid.org/journals/duke-mathematical-journal/volume-55/issue-4/Uniform-distribution-of-eigenfunctions-on-compact-hyperbolic/10.1215/S0012-7094-87-05546-3.full. 
  5. ↑ 5.0 5.1 Shnirelman, Alexander (1974). "Ergodic properties of eigenfunctions". Uspekhi Matematicheskikh Nauk 29: 181–182. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=4463&option_lang=eng. 
  6. ↑ 6.0 6.1 Colin de Verdière, Yves (1985). "Ergodicité et fonctions propres du laplacien". Communications in Mathematical Physics 102 (3): 497–502. doi:10.1007/BF01209296. https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-102/issue-3/Ergodicit%c3%a9-et-fonctions-propres-du-laplacien/cmp/1104114465.full?tab=ArticleLink. 
  7. ↑ 7.0 7.1 Gutzwiller, M. C. (1990). Chaos in classical and quantum mechanics. New York: Springer-Verlag. ISBN 0-387-97173-4. OCLC 22754223. https://www.worldcat.org/oclc/22754223. 
  8. ↑ Gutzwiller, Martin C. (1971-03-01). "Periodic Orbits and Classical Quantization Conditions". Journal of Mathematical Physics 12 (3): 343–358. doi:10.1063/1.1665596. ISSN 0022-2488. Bibcode: 1971JMP....12..343G. https://aip.scitation.org/doi/10.1063/1.1665596. 
  9. ↑ McDonald, S.W. (1983-09-01) (in en). Wave dynamics of regular and chaotic rays. pp. LBL–14837, 5373229. doi:10.2172/5373229. http://www.osti.gov/servlets/purl/5373229-gClYUs/. 
  10. ↑ McDonald, Steven W.; Kaufman, Allan N. (1979-04-30). "Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories". Physical Review Letters 42 (18): 1189–1191. doi:10.1103/PhysRevLett.42.1189. https://link.aps.org/doi/10.1103/PhysRevLett.42.1189. 
  11. ↑ Kaplan, L (1999-01-01). "Scars in quantum chaotic wavefunctions" (in en). Nonlinearity 12 (2): R1–R40. doi:10.1088/0951-7715/12/2/009. ISSN 0951-7715. https://doi.org/10.1088/0951-7715/12/2/009. 
  12. ↑ 12.0 12.1 12.2 Kaplan, L.; Heller, E. J. (1998-04-10). "Linear and Nonlinear Theory of Eigenfunction Scars" (in en). Annals of Physics 264 (2): 171–206. doi:10.1006/aphy.1997.5773. ISSN 0003-4916. https://www.sciencedirect.com/science/article/pii/S0003491697957730. 
  13. ↑ Antonsen, T. M.; Ott, E.; Chen, Q.; Oerter, R. N. (1 January 1995). "Statistics of wave-function scars". Physical Review E 51 (1): 111–121. doi:10.1103/PhysRevE.51.111. PMID 9962623. Bibcode: 1995PhRvE..51..111A. 
  14. ↑ Bogomolny, E. B. (1988-06-01). "Smoothed wave functions of chaotic quantum systems" (in en). Physica D: Nonlinear Phenomena 31 (2): 169–189. doi:10.1016/0167-2789(88)90075-9. ISSN 0167-2789. https://dx.doi.org/10.1016/0167-2789%2888%2990075-9. 
  15. ↑ Berry, Michael Victor (1989-05-08). "Quantum scars of classical closed orbits in phase space". Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 423 (1864): 219–231. doi:10.1098/rspa.1989.0052. https://royalsocietypublishing.org/doi/10.1098/rspa.1989.0052. 
  16. ↑ Sridhar, S. (1991-08-12). "Experimental observation of scarred eigenfunctions of chaotic microwave cavities". Physical Review Letters 67 (7): 785–788. doi:10.1103/PhysRevLett.67.785. PMID 10044988. https://link.aps.org/doi/10.1103/PhysRevLett.67.785. 
  17. ↑ Stein, J.; Stöckmann, H.-J. (1992-05-11). "Experimental determination of billiard wave functions". Physical Review Letters 68 (19): 2867–2870. doi:10.1103/PhysRevLett.68.2867. PMID 10045515. https://link.aps.org/doi/10.1103/PhysRevLett.68.2867. 
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  19. ↑ Wilkinson, P. B.; Fromhold, T. M.; Eaves, L.; Sheard, F. W.; Miura, N.; Takamasu, T. (April 1996). "Observation of 'scarred' wavefunctions in a quantum well with chaotic electron dynamics" (in en). Nature 380 (6575): 608–610. doi:10.1038/380608a0. ISSN 1476-4687. https://www.nature.com/articles/380608a0. 
  20. ↑ Narimanov, E. E.; Stone, A. Douglas (1998-01-05). "Origin of Strong Scarring of Wave Functions in Quantum Wells in a Tilted Magnetic Field". Physical Review Letters 80 (1): 49–52. doi:10.1103/PhysRevLett.80.49. https://link.aps.org/doi/10.1103/PhysRevLett.80.49. 
  21. ↑ Lee, Sang-Bum; Lee, Jai-Hyung; Chang, Joon-Sung; Moon, Hee-Jong; Kim, Sang Wook; An, Kyungwon (2002-01-02). "Observation of Scarred Modes in Asymmetrically Deformed Microcylinder Lasers". Physical Review Letters 88 (3): 033903. doi:10.1103/PhysRevLett.88.033903. PMID 11801060. https://link.aps.org/doi/10.1103/PhysRevLett.88.033903. 
  22. ↑ Harayama, Takahisa; Fukushima, Tekehiro; Davis, Peter; Vaccaro, Pablo O.; Miyasaka, Tomohiro; Nishimura, Takehiro; Aida, Tahito (2003-01-31). "Lasing on scar modes in fully chaotic microcavities". Physical Review E 67 (1): 015207. doi:10.1103/PhysRevE.67.015207. PMID 12636553. https://link.aps.org/doi/10.1103/PhysRevE.67.015207. 
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  28. ↑ 28.0 28.1 Huang, Liang; Lai, Ying-Cheng; Ferry, David K.; Goodnick, Stephen M.; Akis, Richard (2009-07-27). "Relativistic Quantum Scars". Physical Review Letters 103 (5): 054101. doi:10.1103/PhysRevLett.103.054101. PMID 19792502. https://link.aps.org/doi/10.1103/PhysRevLett.103.054101. 
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  30. ↑ Song, Min-Yue; Li, Zi-Yuan; Xu, Hong-Ya; Huang, Liang; Lai, Ying-Cheng (2019-10-03). "Quantization of massive Dirac billiards and unification of nonrelativistic and relativistic chiral quantum scars". Physical Review Research 1 (3): 033008. doi:10.1103/PhysRevResearch.1.033008. 
  31. ↑ O’Connor, P.; Gehlen, J.; Heller, E. J. (1987-03-30). "Properties of random superpositions of plane waves". Physical Review Letters 58 (13): 1296–1299. doi:10.1103/PhysRevLett.58.1296. PMID 10034395. https://link.aps.org/doi/10.1103/PhysRevLett.58.1296. 
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  34. ↑ Bellomo, Paolo; Uzer, T. (1994-09-01). "State scarring by ghosts of periodic orbits". Physical Review E 50 (3): 1886–1893. doi:10.1103/PhysRevE.50.1886. PMID 9962190. https://link.aps.org/doi/10.1103/PhysRevE.50.1886. 
  35. ↑ Bellomo, Paolo; Uzer, T. (1995-02-01). "Quantum scars and classical ghosts". Physical Review A 51 (2): 1669–1672. doi:10.1103/PhysRevA.51.1669. PMID 9911757. https://link.aps.org/doi/10.1103/PhysRevA.51.1669. 
  36. ↑ Biswas, Debabrata (2000-05-01). "Closed almost-periodic orbits in semiclassical quantization of generic polygons". Physical Review E 61 (5): 5129–5133. doi:10.1103/PhysRevE.61.5129. PMID 11031557. https://link.aps.org/doi/10.1103/PhysRevE.61.5129. 
  37. ↑ Kuś, Marek; Haake, Fritz; Delande, Dominique (1993-10-04). "Prebifurcation periodic ghost orbits in semiclassical quantization". Physical Review Letters 71 (14): 2167–2171. doi:10.1103/PhysRevLett.71.2167. PMID 10054605. https://link.aps.org/doi/10.1103/PhysRevLett.71.2167. 
  38. ↑ Sundaram, Bala; Scharf, Rainer (1995-05-15). "A standard perspective on ghosts" (in en). Physica D: Nonlinear Phenomena. Quantum Complexity in Mesoscopic Systems 83 (1): 257–270. doi:10.1016/0167-2789(94)00267-T. ISSN 0167-2789. https://dx.doi.org/10.1016/0167-2789%2894%2900267-T. 
  39. ↑ Keski-Rahkonen, J.; Ruhanen, A.; Heller, E. J.; Räsänen, E. (2019-11-21). "Quantum Lissajous Scars". Physical Review Letters 123 (21): 214101. doi:10.1103/PhysRevLett.123.214101. PMID 31809168. https://link.aps.org/doi/10.1103/PhysRevLett.123.214101. 
  40. ↑ Luukko, Perttu J. J.; Drury, Byron; Klales, Anna; Kaplan, Lev; Heller, Eric J.; Räsänen, Esa (2016-11-28). "Strong quantum scarring by local impurities" (in en). Scientific Reports 6 (1): 37656. doi:10.1038/srep37656. ISSN 2045-2322. PMID 27892510. 
  41. ↑ Keski-Rahkonen, J; Luukko, P J J; Åberg, S; Räsänen, E (2019-01-21). "Effects of scarring on quantum chaos in disordered quantum wells" (in en). Journal of Physics: Condensed Matter 31 (10): 105301. doi:10.1088/1361-648x/aaf9fb. ISSN 0953-8984. PMID 30566927. https://doi.org/10.1088/1361-648X/aaf9fb. 
  42. ↑ Keski-Rahkonen, Joonas (2020) (in en). Quantum Chaos in Disordered Two-Dimensional Nanostructures. Tampere University. ISBN 978-952-03-1699-0. https://trepo.tuni.fi/handle/10024/123296. 
  43. ↑ Lin, Cheng-Ju; Motrunich, Olexei I. (2019). "Exact Quantum Many-body Scar States in the Rydberg-blockaded Atom Chain". Physical Review Letters 122 (17): 173401. doi:10.1103/PhysRevLett.122.173401. PMID 31107057. 
  44. ↑ Moudgalya, Sanjay; Regnault, Nicolas; Bernevig, B. Andrei (2018-12-27). "Entanglement of Exact Excited States of AKLT Models: Exact Results, Many-Body Scars and the Violation of Strong ETH". Physical Review B 98 (23): 235156. doi:10.1103/PhysRevB.98.235156. ISSN 2469-9950. 
  45. ↑ 45.0 45.1 45.2 45.3 "Quantum Scarring Appears to Defy Universe's Push for Disorder". March 20, 2019. https://www.quantamagazine.org/quantum-scarring-appears-to-defy-universes-push-for-disorder-20190320/. 
  46. ↑ 46.0 46.1 46.2 Bernien, Hannes; Schwartz, Sylvain; Keesling, Alexander; Levine, Harry; Omran, Ahmed; Pichler, Hannes; Choi, Soonwon; Zibrov, Alexander S. et al. (November 30, 2017). "Probing many-body dynamics on a 51-atom quantum simulator" (in en). Nature 551 (7682): 579–584. doi:10.1038/nature24622. ISSN 1476-4687. PMID 29189778. Bibcode: 2017Natur.551..579B. 
  47. ↑ 47.0 47.1 Turner, C. J.; Michailidis, A. A.; Abanin, D. A.; Serbyn, M.; Papić, Z. (October 22, 2018). "Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations". Physical Review B 98 (15): 155134. doi:10.1103/PhysRevB.98.155134. Bibcode: 2018PhRvB..98o5134T. 
  48. ↑ Papić, Z.; Serbyn, M.; Abanin, D. A.; Michailidis, A. A.; Turner, C. J. (May 14, 2018). "Weak ergodicity breaking from quantum many-body scars" (in en). Nature Physics 14 (7): 745–749. doi:10.1038/s41567-018-0137-5. ISSN 1745-2481. Bibcode: 2018NatPh..14..745T. http://eprints.whiterose.ac.uk/130860/1/1711.03528v2.pdf. 
  49. ↑ Ho, Wen Wei; Choi, Soonwon; Pichler, Hannes; Lukin, Mikhail D. (January 29, 2019). "Periodic Orbits, Entanglement, and Quantum Many-Body Scars in Constrained Models: Matrix Product State Approach". Physical Review Letters 122 (4): 040603. doi:10.1103/PhysRevLett.122.040603. PMID 30768339. Bibcode: 2019PhRvL.122d0603H. 
  50. ↑ Khemani, Vedika; Laumann, Chris R.; Chandran, Anushya (2019). "Signatures of integrability in the dynamics of Rydberg-blockaded chains". Physical Review B 99 (16): 161101. doi:10.1103/PhysRevB.99.161101. Bibcode: 2019PhRvB..99p1101K. 
  51. ↑ Choi, Soonwon; Turner, Christopher J.; Pichler, Hannes; Ho, Wen Wei; Michailidis, Alexios A.; Papić, Zlatko; Serbyn, Maksym; Lukin, Mikhail D. et al. (2019). "Emergent SU(2) dynamics and perfect quantum many-body scars". Physical Review Letters 122 (22): 220603. doi:10.1103/PhysRevLett.122.220603. PMID 31283292. 
  52. ↑ Moudgalya, Sanjay; Regnault, Nicolas; Bernevig, B. Andrei (2020-08-20). "[math]\displaystyle{ \eta }[/math]-pairing in Hubbard models: From spectrum generating algebras to quantum many-body scars". Physical Review B 102 (8): 085140. doi:10.1103/PhysRevB.102.085140. https://link.aps.org/doi/10.1103/PhysRevB.102.085140. 
  53. ↑ Bull, Kieran; Desaules, Jean-Yves; Papić, Zlatko (2020-04-27). "Quantum scars as embeddings of weakly broken Lie algebra representations". Physical Review B 101 (16): 165139. doi:10.1103/PhysRevB.101.165139. https://link.aps.org/doi/10.1103/PhysRevB.101.165139. 
  54. ↑ Buča, Berislav; Tindall, Joseph; Jaksch, Dieter (2019-04-15). "Non-stationary coherent quantum many-body dynamics through dissipation" (in en). Nature Communications 10 (1): 1730. doi:10.1038/s41467-019-09757-y. ISSN 2041-1723. PMID 30988312. Bibcode: 2019NatCo..10.1730B. 
  55. ↑ Medenjak, Marko; Buča, Berislav; Jaksch, Dieter (2020-07-20). "Isolated Heisenberg magnet as a quantum time crystal". Physical Review B 102 (4): 041117. doi:10.1103/PhysRevB.102.041117. Bibcode: 2020PhRvB.102d1117M. https://link.aps.org/doi/10.1103/PhysRevB.102.041117. 



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