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Discontinuity, Nonlinearity, and Complexity 13(1) (2024) 49--64 | DOI:10.5890/DNC.2024.03.004

F. Revuelta$^{1}$, E. Vergini$^2$, R. M. Benito$^1$, F. Borondo$^{3}$

$^1$ Grupo de Sistemas Complejos, Escuela T'ecnica Superior de Ingenier'i a Agron'omica, Alimentaria y de Biosistemas, Universidad Polit'ecnica de Madrid, Avda.~Puerta de Hierro 2-4, 28040 Madrid, Spain

$^2$ Departamento de Física, Comisión Nacional de Energía Atómica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina

$^3$ Departamento de Qu'imica, Universidad Aut'onoma de Madrid, Cantoblanco, 28049 Madrid, Spain

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Abstract

In this work, we calculate degenerate eigenstates of a coupled quartic oscillator with a high degree of chaoticity in a narrow energy window. For this purpose, we extend the range of applicability of the scar functions defined by us in F. Revuelta \emph{et al.}, Phys. Rev. E \textbf{102}, 042210 (2020) to this more complicated case. Our method allows the reconstruction of individual eigenstates in terms of the shortest unstable periodic orbits of the system. As in the case of nondegenerate states, the basis size is of the same order of the number of computed eigenfunctions and depends on the ratio between the Heisenberg and the Ehrenfest times.

References

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54. [54]	Bogomolny, E.B. (1988), %Smoothed wave functions of chaotic quantum systems, %Physica D: Nonlinear Phenomena, %31, 169-189. % % % %

55. [55]	dePolavieja, G.G., Borondo, F., and Benito, R.M. (1994), %Scars in groups of eigenstates in a classically chaotic system, %Physical Review Letters, %73, 1613-1616. % %

56. [56]	Berry, M.V. (1989), %Quantum scars of classical closed orbits in phase space, %Proceedings of the Royal Society A, %423, 219–231. % % %

57. [57]	%Arranz, F.J., Borondo, F., and Benito, R.M. (1996), %Distribution of zeros of the Husimi function in a realistic Hamiltonian molecular system, %Physical Review E, %54, 2458-2464. % %

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64. [64]	%Arranz, F.J., Seidel, L., Giralda, C.G., Benito, R.M. and Borondo, F. (2010), %Scars at the edge of the transition from order to chaos in the isomerizing %molecular systems LiNC-LiCN and HCN-HNC, and HO$_2$, %Physical Review E, %82, 026201. % % % % %

65. [65]	Vergini E.G. and Carlo, G.G. (2001), %Semiclassical construction of resonances with hyperbolic structure: % the scar function, %Journal of Physics A: Mathematical and General, %34, 4525-4552. % % %

66. [66]	Carlo, G.G., Vergini, E.G., and Lustemberg, P. (2002), %Scar functions in the Bunimovich stadium billiard, %Journal of Physics A: Mathematical and General, %35, 7965-7982. % % % %

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69. [69]	%Revuelta, F., Vergini, E., Benito, R.M., and Borondo, F. (2016), %Scar functions, barriers for chemical reactivity, and vibrational basis sets, %Journal of Physical Chemistry A, %120, 4928-4938. % % %

70. [70]	%Revuelta, F., Vergini, E.G., Benito, R.M., and Borondo, F. (2020), %Short-periodic-orbit method for excited chaotic eigenfunctions, %Physical Review E, %102, 042210. % % %

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73. [73]	%Revuelta, F., Vergini, E., Benito, R.M. and Borondo, F. (2017), %Semiclassical basis sets for the computation of molecular vibrational states, %The Journal of Chemical Physics 146, 014107. % % %

74. [74]

    Revuelta, F., Vergini, E., %Benito, R.M., and Borondo, F. (2022), %Computing chaotic eigenvectors in narrow energy windows, %Chapter 6 of %Nonlinear Dynamics and Complexity. %Mathematical Modelling of Real-World Problems %(Editor: Pinto, C. M. A.), %Volume 36 of Nonlinear Systems and Complexity %(Editor: Luo, A.C.J.), %Springer-Verlag: Cham. % %

75. [75]	% Davis, M.J. and Heller, E.J. (1979), %Semiclassical gaussian basis set method for molecular vibrational wave functions, %Journal of Chemical Physics, %71, 3383-3395. % % %

76. [76]	%Reimers, J.R. and Heller, E.J. (1985), %The exact eigenfunctions and eigenvalues of a two‐dimensional rigid % rotor obtained using gaussian wave packet dynamics, %Journal of Chemical Physics, %83, 511-515. % % %

77. [77]	%Poirier, B. and Salam, A. (2004), %Quantum dynamics calculations using symmetrized, orthogonal % Weyl-Heisenberg wavelets with a phase space truncation scheme. II. % Construction and optimization, %Journal of Chemical Physics, %121, 1690-1703. % % %

78. [78]	%Poirier, B. and Salam, A. (2004), %Quantum dynamics calculations using symmetrized, orthogonal % Weyl-Heisenberg wavelets with a phase space truncation scheme. III. % Representations and calculations, %Journal of Chemical Physics, %121, 1704-1724. % % %

79. [79]	%Halverson, T. and Poirier, B. (2014), %Calculation of exact vibrational spectra for P$_2$O and CH$_2$NH using a % phase space wavelet basis, %Journal of Chemical Physics, %140, 204112. % % % %

80. [80]	%Halverson, T. and Poirier, B. (2012), %Accurate quantum dynamics calculations using symmetrized gaussians on % a doubly dense von Neumann lattice, %Journal of Chemical Physics, %137, 224101. % % % %

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82. [82]	Shimshovitz, A., B\v{a}cic, Z., and Tannor, D.J. (2014), %The von Neumann basis in non-cartesian coordinates: Application to % floppy triatomic molecules, %Journal of Chemical Physics, %141, 234106. % % % % %

83. [83]	%Bernien, H., Schwartz, S., Keesling, A., Levine, H. Omran, A., %Pichler, H., Choi, S., Zibrov, A. S., Endres, M., Greiner, M., %Vuletic, V., and Lukin, M. D. (2017), %Probing many-body dynamics on a 51-atom quantum simulator, %Nature (London), %551, 579. % %

84. [84]	% Turner, C.J., Michailidis, A.A., Abanin, D.A., Serbyn, M., and Papic, Z. (2018), %Weak ergodicity breaking from quantum many-body scars, %Nature Physics, %14, 745. % % %

85. [85]	%Wei Ho, W., Choi, S., Pichler, H., and Lukin, M.D. (2019), %Periodic orbits, entanglement, and quantum many-body Scars in %constrained models: Matrix product state approach, %Physical Review Letters, %122, 040603. % % %

86. [86]	%Choi, S., Turner, C.J., Pichler, H., Ho, W.W., %Michailidis, A.A., Papić, Z., Serbyn, M., Lukin, M.D., and Abanin, D.A. (2019), %Emergent SU(2) dynamics and perfect %quantum many-body scars, %Physical Review Letters, %122, 220603. % % %

87. [87]	%Turner, C.J., Desaules, J.-Y., Bull, K., and Papić, Z. (2021), %Correspondence principle for many-body scars in ultracold Rydberg atoms, %Physical Review X, %11, 021021. % % %

88. [88]	%Desaules, J.-Y., Hudomal, A., Turner, C.J., and Papić, Z. (2021), %Proposal for realizing quantum scars in the tilted 1D Fermi-Hubbard model, %Physical Review Letters, %126, 210601. % % %

89. [89]	%Dahlqvist, R. and Russberg, G. (1990), %Existence of stable orbits in the $x^2 y^2$ potential, %Physical Review Letters, %65, 2837. % %

90. [90]	%Kapela, T. and Simó, C. (2017), %Rigorous KAM results around arbitrary %periodic orbits for Hamiltonian systems, %Nonlinearity, %30, 965. % % % %

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93. [93]	%Revuelta, F., Benito, R.M., Borondo, F., and Vergini, E. (2013), %Using basis sets of scar functions, %Physical Review E, %87, 042921. % % % %

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96. [96]	Vergini, E.G., Sibert III, E.L., %Revuelta, F., Benito, R.M. and Borondo, F. (2010), %Diagonal matrix elements in a scar function basis set, %Europhysics Letters, %89, 40013. % % % %

97. [97]	%Littlejohn, R.G., Cargo, M., Carrington, T., Mitchell, K.A., and Poirier, B. (2002), %A general framework for discrete variable representation basis sets, %Journal of Chemical Physics, %116, 8691-8703. % % %

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99. [99]	Vergini, E.G. (2012), %Semiclassical approach to long time propagation in quantum chaos: % predicting scars, %Physical Review Letters, %108, 264101. % % %

100. [100]

     Vergini, E.G. (2014), %Semiclassical propagator to evaluate off-diagonal matrix elements of % the evolution operator between quantum states, %Physical Review E, %89, 020901. %