pith. machine review for the scientific record. sign in

arxiv: 2604.09341 · v1 · submitted 2026-04-10 · 🪐 quant-ph

Recognition: unknown

Algebraic structure of Fock-state lattices

Piergiorgio Ferraro , Caio B. Naves , Jonas Larson
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:28 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Fock-state latticesLie algebrasphase-space geometryintegrable systemsquantum transportcurved spacesLie superalgebrasbosonic systems
0
0 comments X

The pith

Fock-state lattices arise from Lie algebra generators, linking their connectivity and dynamics to phase-space geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Fock-state lattices can be systematically built by applying the generators of a Lie algebra to Fock states. Cartan generators label the individual sites while root generators set the allowed hops between them. This algebraic origin directly determines the lattice dimension, its bond structure, and the symmetries that govern transport or revival. The same construction yields a corresponding Lie phase space, often with curvature, so that lattice dynamics inherit geometric features from that phase space. The approach does not cover every integrable Hamiltonian, especially those nonlinear in the generators, and mixed bosonic-fermionic systems may instead require a Lie superalgebra.

Core claim

Starting from a Lie algebra, a Fock-state lattice is defined by letting its Cartan generators label the sites and its root generators label the bonds. For several standard Lie algebras this produces both an explicit lattice and its associated Lie phase space, establishing a direct map between FSL dynamics and phase-space geometry. Both the lattice and the phase space can carry nontrivial curvature. The construction fails for many integrable Hamiltonians that are nonlinear in the generators and, for systems with mixed degrees of freedom, the appropriate underlying structure is often a Lie superalgebra rather than an ordinary Lie algebra.

What carries the argument

The mapping from a Lie algebra to a Fock-state lattice in which Cartan generators define sites and root generators define bonds, simultaneously producing a Lie phase space whose geometry governs the lattice dynamics.

If this is right

  • Dimensionality, bond connectivity, and symmetry constraints of any FSL are fixed once the underlying algebra is identified.
  • Transport and revival phenomena are restricted by the root-system structure of the algebra.
  • Dynamics on the lattice inherit the geometry, including curvature, of the associated Lie phase space.
  • Systems with nonlinear Hamiltonians or mixed bosonic-fermionic degrees of freedom generally require structures beyond ordinary Lie algebras.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same algebraic construction could be used to engineer synthetic lattices whose curvature is chosen in advance by selecting the Lie algebra.
  • Experimental signatures of curved phase-space geometry might appear as modified revival times or anomalous diffusion in multi-mode bosonic systems.
  • The framework suggests a classification of all FSLs that admit an algebraic origin, leaving a remainder class that must be treated by other means.

Load-bearing premise

The action of any Lie algebra's generators on Fock states always yields a lattice whose connectivity and geometry remain physically meaningful and extend to the dynamics of integrable Hamiltonians.

What would settle it

An explicit integrable Hamiltonian whose Fock-state connectivity and allowed transitions cannot be reproduced by the generators of any Lie algebra or Lie superalgebra.

Figures

Figures reproduced from arXiv: 2604.09341 by Caio B. Naves, Jonas Larson, Piergiorgio Ferraro.

Figure 1
Figure 1. Figure 1: FIG. 1. Demonstration of the Lie-algebra phase space (LPS, [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Demonstration of the time evolution in the FSL (d) [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Two snapshots of the distribution [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Snapshots of the fourth root of the distribution [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

We analyze Fock-state lattices (FSLs) from an algebraic viewpoint. Starting from a Lie algebra, we associate a FSL constructed from the action of its generators: diagonal (Cartan) generators define the lattice sites, while off-diagonal (root) generators determine the lattice bonds. This construction reveals that identifying an underlying algebraic structure provides direct physical insight into FSLs, including their dimensionality, connectivity, symmetry constraints, and possible transport and revival phenomena. By examining several common Lie algebras, we identify not only their associated FSLs but also the corresponding Lie phase spaces, thereby establishing a systematic connection between FSL dynamics and phase-space geometry. In many cases, both the phase space and the FSL exhibit nontrivial curvature, opening possibilities for exploring quantum dynamics in curved synthetic spaces. We further address whether every integrable Hamiltonian admits an underlying Lie algebra that reproduces the same FSL structure. We show that this is not generally the case, particularly for Hamiltonians that are nonlinear in the generators, and that for systems combining different types of degrees of freedom the appropriate underlying structure may instead be a Lie superalgebra.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that Fock-state lattices (FSLs) admit a systematic algebraic construction from Lie algebras, with Cartan generators labeling sites and root generators labeling bonds. This yields direct insight into FSL dimensionality, connectivity, symmetry constraints, transport, and revival phenomena, while linking FSL dynamics to the geometry of the corresponding Lie phase spaces (often curved). The authors illustrate the construction for several common Lie algebras and show that the correspondence does not hold in general for nonlinear Hamiltonians or for mixed bosonic-fermionic systems (where superalgebras are required instead).

Significance. If the generator-to-lattice mapping is free of post-hoc choices, the framework supplies a useful organizing principle that connects abstract algebraic data to concrete geometric features of FSLs and their dynamics. The explicit treatment of scope limitations (nonlinear Hamiltonians, superalgebras) is a strength, as is the emphasis on phase-space curvature as a route to synthetic curved-space quantum mechanics.

minor comments (3)
  1. The abstract refers to 'several common Lie algebras' without naming them; an explicit list (e.g., su(2), su(1,1), etc.) together with the corresponding FSL dimensions and curvatures would improve readability.
  2. A compact table or diagram summarizing the site/bond assignments for each algebra examined would make the central construction easier to verify at a glance.
  3. The discussion of why nonlinear Hamiltonians fall outside the construction would benefit from one fully worked counter-example showing that no Lie algebra reproduces the same connectivity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The recognition that the generator-to-lattice mapping supplies an organizing principle linking algebraic data to geometric features of FSLs, together with the explicit discussion of scope limitations, is appreciated. As the report contains no enumerated major comments, we have no specific points to address point-by-point. We will incorporate any minor editorial or presentational suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines an explicit mapping from any Lie algebra's Cartan and root generators to FSL sites and bonds, then applies this construction to several standard algebras to recover their associated lattices and phase spaces. This is a direct, non-predictive association rather than a derivation that reduces to fitted inputs or prior self-citations. The subsequent claim that the mapping supplies geometric insight follows immediately from the algebraic structure itself. The negative result—that not every integrable Hamiltonian admits such a Lie algebra—is established by counterexamples (nonlinear generators and bosonic-fermionic mixtures requiring superalgebras), which are independent of the main construction and do not rely on self-referential theorems. No step matches the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard Lie-algebra theory (Cartan decomposition, root systems) and the assumption that Fock-space operators realize these algebras; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The generators of a Lie algebra act on Fock states such that diagonal (Cartan) elements label distinct sites and off-diagonal (root) elements define bonds.
    Invoked in the opening construction of the FSL from the algebra.
  • domain assumption The resulting lattice geometry and connectivity capture the essential symmetry and transport properties of the physical system.
    Used to claim direct physical insight into dimensionality, connectivity, and revival phenomena.

pith-pipeline@v0.9.0 · 5491 in / 1420 out tokens · 56485 ms · 2026-05-10T17:28:09.401096+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

75 extracted references · 6 canonical work pages · 1 internal anchor

  1. [1]

    I. M. Georgescu, S. Ashhab, and F. Nori, Quantum sim- ulation, Reviews of Modern Physics86, 153 (2014)

    2014

  2. [2]

    Altman, K

    E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Dem- ler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriks- son, K.-M. C. Fu,et al., Quantum simulators: Architec- tures and opportunities, PRX quantum2, 017003 (2021)

    2021

  3. [3]

    Bloch, J

    I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Reviews of modern physics 80, 885 (2008)

    2008

  4. [4]

    Lewenstein, A

    M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, Ultracold atomic gases in optical lat- tices: mimicking condensed matter physics and beyond, Advances in Physics56, 243 (2007)

    2007

  5. [5]

    Blatt and C

    R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nature Physics8, 277 (2012)

    2012

  6. [6]

    J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. 16 Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Engineered two-dimensional ising interactions in a trapped-ion quantum simulator with hundreds of spins, Nature484, 489 (2012)

    2012

  7. [7]

    Schneider, D

    C. Schneider, D. Porras, and T. Schaetz, Experimental quantum simulations of many-body physics with trapped ions, Reports on Progress in Physics75, 024401 (2012)

    2012

  8. [8]

    Aspuru-Guzik and P

    A. Aspuru-Guzik and P. Walther, Photonic quantum sim- ulators, Nature physics8, 285 (2012)

    2012

  9. [9]

    M. J. Hartmann, Quantum simulation with interacting photons, Journal of Optics18, 104005 (2016)

    2016

  10. [10]

    Grass, D

    T. Grass, D. Bercioux, U. Bhattacharya, M. Lewenstein, H. S. Nguyen, and C. Weitenberg, Colloquium: Synthetic quantum matter in nonstandard geometries, Reviews of Modern Physics97, 011001 (2025)

    2025

  11. [11]

    A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeli¯ unas, and M. Lewenstein, Synthetic gauge fields in synthetic dimensions, Physical review let- ters112, 043001 (2014)

    2014

  12. [12]

    Ozawa, H

    T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, Synthetic dimensions in integrated photon- ics: From optical isolation to four-dimensional quantum hall physics, Physical Review A93, 043827 (2016)

    2016

  13. [13]

    Arg¨ uello-Luengo, U

    J. Arg¨ uello-Luengo, U. Bhattacharya, A. Celi, R. W. Chhajlany, T. Grass, M. P lodzie´ n, D. Rakshit, T. Sala- mon, P. Stornati, L. Tarruell,et al., Synthetic dimen- sions for topological and quantum phases, Communica- tions Physics7, 143 (2024)

    2024

  14. [14]

    D. Yu, W. Song, L. Wang, R. Srikanth, S. Kaushik Srid- har, T. Chen, C. Huang, G. Li, X. Qiao, X. Wu,et al., Comprehensive review on developments of synthetic di- mensions, Photonics Insights4, R06 (2025)

    2025

  15. [15]

    H. M. Price, T. Ozawa, and N. Goldman, Synthetic di- mensions for cold atoms from shaking a harmonic trap, Physical Review A95, 023607 (2017)

    2017

  16. [16]

    T. Chen, C. Huang, I. Velkovsky, K. R. Hazzard, J. P. Covey, and B. Gadway, Strongly interacting rydberg atoms in synthetic dimensions with a magnetic flux, Na- ture communications15, 2675 (2024)

    2024

  17. [17]

    Boada, A

    O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, Dirac equation for cold atoms in artificial curved space- times, New Journal of Physics13, 035002 (2011)

    2011

  18. [18]

    Bermudez, T

    A. Bermudez, T. Schaetz, and D. Porras, Synthetic gauge fields for vibrational excitations of trapped ions, Physical review letters107, 150501 (2011)

    2011

  19. [19]

    Manovitz, Y

    T. Manovitz, Y. Shapira, N. Akerman, A. Stern, and R. Ozeri, Quantum simulations with complex geometries and synthetic gauge fields in a trapped ion chain, PRX quantum1, 020303 (2020)

    2020

  20. [20]

    D.-W. Wang, H. Cai, R.-B. Liu, and M. O. Scully, Meso- scopic superposition states generated by synthetic spin- orbit interaction in fock-state lattices, Physical review letters116, 220502 (2016)

    2016

  21. [21]

    Saugmann and J

    P. Saugmann and J. Larson, Fock-state-lattice approach to quantum optics, Physical Review A108, 033721 (2023)

    2023

  22. [22]

    Larson and T

    J. Larson and T. Mavrogordatos,The Jaynes–Cummings model and its descendants: modern research directions (IoP Publishing, 2021)

    2021

  23. [23]

    J. Yuan, H. Cai, and D.-W. Wang, Quantum simula- tion in Fock-state lattices, Advances in Physics: X9, 2325611 (2024), publisher: Taylor & Francis eprint: https://doi.org/10.1080/23746149.2024.2325611

    work page doi:10.1080/23746149.2024.2325611 2024

  24. [24]

    Cai and D.-W

    H. Cai and D.-W. Wang, Topological phases of quantized light, National Science Review8, nwaa196 (2021)

    2021

  25. [25]

    Moessner and A

    R. Moessner and A. P. Ramirez, Geometrical frustration, Physics Today59, 24 (2006)

    2006

  26. [26]

    J. Deng, H. Dong, C. Zhang, Y. Wu, J. Yuan, X. Zhu, F. Jin, H. Li, Z. Wang, H. Cai, C. Song, H. Wang, J. Q. You, and D.-W. Wang, Observing the quantum topology of light, Science378, 966 (2022), publisher: American Association for the Advancement of Science

    2022

  27. [27]

    C. Wu, W. Liu, Y. Jia, G. Chen, and F. Chen, Obser- vation of topological pumping of a defect state in a fock photonic lattice, Physical Review A107, 033501 (2023)

    2023

  28. [28]

    Yang, J.-S

    M. Yang, J.-S. Xu, C.-F. Li, and G.-C. Guo, Simulating topological materials with photonic synthetic dimensions in cavities, Quantum Frontiers1, 10 (2022)

    2022

  29. [29]

    Sriram, S

    S. Sriram, S. K. Sridhar, and A. Dutt, Quantized topo- logical phases beyond square lattices in floquet synthetic dimensions, Optical Materials Express15, 272 (2025)

    2025

  30. [30]

    J. Yang, Y. Li, Y. Yang, X. Xie, Z. Zhang, J. Yuan, H. Cai, D.-W. Wang, and F. Gao, Realization of all-band- flat photonic lattices, Nature Communications15, 1484 (2024)

    2024

  31. [31]

    Zhang, W

    J. Zhang, W. Huang, J. Chu, J. Qiu, X. Sun, Z. Tao, J. Zhang, L. Zhang, Y. Zhou, Y. Chen,et al., Synthetic multidimensional aharonov-bohm cages in fock state lat- tices, Physical Review Letters134, 070601 (2025)

    2025

  32. [32]

    Y. Yao, L. Xiang, Z. Guo, Z. Bao, Y.-F. Yang, Z. Song, H. Shi, X. Zhu, F. Jin, J. Chen,et al., Observation of many-body fock space dynamics in two dimensions, Na- ture Physics19, 1459 (2023)

    2023

  33. [33]

    Larson, Floquet analysis perspective of driven light– matter interaction models, Journal of the Optical Society of America B41, C235 (2024)

    J. Larson, Floquet analysis perspective of driven light– matter interaction models, Journal of the Optical Society of America B41, C235 (2024)

    2024

  34. [34]

    X. Zhao, Y. Xu, L.-M. Kuang, and J.-Q. Liao, Dark- state engineering in fock-state lattices, Physical Review Research7, 033070 (2025)

    2025

  35. [35]

    M. Li, W. Cai, Z. Hua, Y. Xu, Y. Zhou, Z.-J. Chen, X.-B. Zou, G.-C. Guo, L. Sun, and C.-L. Zou, Scalable generation of macroscopic fock states exceeding 10,000 photons, arXiv preprint arXiv:2601.05118 (2026)

    work page arXiv 2026

  36. [36]

    Ben-Ami, M

    T. Ben-Ami, M. Heyl, and R. Moessner, Many-body cages: Disorder-free glassiness from flat bands in fock space, and many-body rabi oscillations, arXiv preprint arXiv:2504.13086 (2025)

    work page arXiv 2025

  37. [37]

    Localized Fock Space Cages in Kinetically Constrained Models

    C. Jonay and F. Pollmann, Localized fock space cages in kinetically constrained models, arXiv preprint arXiv:2504.20987 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  38. [38]

    W. Yao, S. Li, Z. Liu, Y. Li, Z. Xie, X. Zhao, X. Cheng, Y. Li, Z.-Y. Xue, and Y. Lin, Non-abelian aharonov- bohm caging in synthetic dimensions with a trapped ion, arXiv preprint arXiv:2602.13796 (2026)

    work page arXiv 2026

  39. [39]

    C. B. Naves and J. Larson, Liouville-fock-state lat- tices: From environment-induced frustration to synthetic lattice dynamics, Physical Review Research7, 043233 (2025)

    2025

  40. [40]

    D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Metal– insulator transition in a weakly interacting many-electron system with localized single-particle states, Annals of physics321, 1126 (2006)

    2006

  41. [41]

    J. Yuan, H. Cai, and D.-W. Wang, Quantum simulation in fock-state lattices, Advances in Physics: X9, 2325611 (2024)

    2024

  42. [42]

    J. J. Sakurai and J. Napolitano,Modern quantum me- chanics(Cambridge University Press, 2020)

    2020

  43. [43]

    Auerbach,Interacting electrons and quantum mag- netism(Springer Science & Business Media, 2012)

    A. Auerbach,Interacting electrons and quantum mag- netism(Springer Science & Business Media, 2012)

    2012

  44. [44]

    Anishetty, M

    R. Anishetty, M. Mathur, and I. Raychowdhury, Irre- 17 ducible su (3) schwinger bosons, Journal of mathematical physics50(2009)

    2009

  45. [45]

    Georgi,Lie algebras in particle physics: from isospin to unified theories(Taylor & Francis, 2000)

    H. Georgi,Lie algebras in particle physics: from isospin to unified theories(Taylor & Francis, 2000)

    2000

  46. [46]

    Gilmore,Lie groups, Lie algebras, and some of their applications(Courier Corporation, 2006)

    R. Gilmore,Lie groups, Lie algebras, and some of their applications(Courier Corporation, 2006)

    2006

  47. [47]

    A. M. Perelomov, Generalized coherent states and some of their applications, Soviet Physics Uspekhi20, 703 (1977)

    1977

  48. [48]

    A. M. Perelomov,Generalized Coherent States and Their Applications, Theoretical and Mathematical Physics (Springer, Berlin, Heidelberg, 1986)

    1986

  49. [49]

    M. O. Scully and M. S. Zubairy,Quantum optics(Cam- bridge university press, 1997)

    1997

  50. [50]

    H. J. Carmichael,Statistical methods in quantum op- tics 1: master equations and Fokker-Planck equations (Springer Science & Business Media, 2013)

    2013

  51. [51]

    W. P. Schleich,Quantum optics in phase space(John Wiley & Sons, 2015)

    2015

  52. [52]

    G. S. Agarwal, Relation between atomic coherent-state representation, state multipoles, and generalized phase- space distributions, Physical Review A24, 2889 (1981)

    1981

  53. [53]

    Brif and A

    C. Brif and A. Mann, Phase-space formulation of quantum mechanics and quantum state reconstruction for physical systems with lie-group symmetries, arXiv preprint quant-ph/9809052 (1998)

    work page arXiv 1998

  54. [54]

    A. B. Klimov and H. De Guise, General approach to quasi-distribution functions, Journal of Physics A: Math- ematical and Theoretical43, 402001 (2010)

    2010

  55. [55]

    Tilma, M

    T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro, and K. Nemoto, Wigner functions for arbitrary quantum systems, Physical review letters117, 180401 (2016)

    2016

  56. [56]

    A. B. Klimov and S. M. Chumakov,A group-theoretical approach to quantum optics: models of atom-field inter- actions, Vol. 10 (Wiley Online Library, 2009)

    2009

  57. [57]

    Hartmann, F

    T. Hartmann, F. Keck, H. Korsch, and S. Mossmann, Dynamics of bloch oscillations, New Journal of Physics 6, 2 (2004)

    2004

  58. [58]

    Susskind and J

    L. Susskind and J. Glogower, Quantum mechanical phase and time operator, Physics Physique Fizika1, 49 (1964)

    1964

  59. [59]

    Pegg and S

    D. Pegg and S. Barnett, Phase properties of the quantized single-mode electromagnetic field, Physical Review A39, 1665 (1989)

    1989

  60. [60]

    Zhang, R

    W.-M. Zhang, R. Gilmore,et al., Coherent states: The- ory and some applications, Reviews of Modern Physics 62, 867 (1990)

    1990

  61. [61]

    Christandl, N

    M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Perfect state transfer in quantum spin networks, Physical review letters92, 187902 (2004)

    2004

  62. [62]

    Mathur and D

    M. Mathur and D. Sen, Coherent states for su (3), Jour- nal of Mathematical Physics42, 4181 (2001)

    2001

  63. [63]

    Buˇ zek, Su (1, 1) squeezing of su (1, 1) generalized coherent states, Journal of Modern Optics37, 303 (1990)

    V. Buˇ zek, Su (1, 1) squeezing of su (1, 1) generalized coherent states, Journal of Modern Optics37, 303 (1990)

    1990

  64. [64]

    Duan, Quantum walk for su (1, 1), Physical Review A 107, 022224 (2023)

    L. Duan, Quantum walk for su (1, 1), Physical Review A 107, 022224 (2023)

    2023

  65. [65]

    Mandel and E

    L. Mandel and E. Wolf,Optical coherence and quantum optics(Cambridge university press, 1995)

    1995

  66. [66]

    R. Y. Chiao and T. F. Jordan, Lorentz-group berry phases in squeezed light, Physics Letters A132, 77 (1988)

    1988

  67. [67]

    Hong-Yi and H

    F. Hong-Yi and H. Zaidi, Berry phase for the su (1, 1) co- herent state, Canadian Journal of Physics66, 978 (1988)

    1988

  68. [68]

    Buck and C

    B. Buck and C. Sukumar, Exactly soluble model of atom- phonon coupling showing periodic decay and revival, Physics Letters A81, 132 (1981)

    1981

  69. [69]

    Buˇ zek, Jaynes-cummings model with intensity- dependent coupling interacting with holstein-primakoff su (1, 1) coherent state, Physical review A39, 3196 (1989)

    V. Buˇ zek, Jaynes-cummings model with intensity- dependent coupling interacting with holstein-primakoff su (1, 1) coherent state, Physical review A39, 3196 (1989)

    1989

  70. [70]

    Holstein and H

    T. Holstein and H. Primakoff, Field dependence of the in- trinsic domain magnetization of a ferromagnet, Physical Review58, 1098 (1940)

    1940

  71. [71]

    C. C. Gerry, Correlated two-mode su (1, 1) coherent states: nonclassical properties, Journal of the Optical So- ciety of America B8, 685 (1991)

    1991

  72. [72]

    Gilles and P

    L. Gilles and P. Knight, Non-classical properties of two- mode su (1, 1) coherent states, Journal of Modern Optics 39, 1411 (1992)

    1992

  73. [73]

    Braak, Integrability of the rabi model, Physical Re- view Letters107, 100401 (2011)

    D. Braak, Integrability of the rabi model, Physical Re- view Letters107, 100401 (2011)

    2011

  74. [74]

    Gazeau,Coherent states in quantum physics(Wiley Online Library, 2009)

    J.-P. Gazeau,Coherent states in quantum physics(Wiley Online Library, 2009)

    2009

  75. [75]

    V. I. Arnol’d,Mathematical methods of classical mechan- ics, Vol. 60 (Springer Science & Business Media, 2013)

    2013