A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

Eric J. Pap ; Daniël Boer ; Holger Waalkens

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

Détail

Keywords: adiabatic quantum mechanics, geometric phase, exceptional point, quantum geometric tensor.

Bibliographie

[1] Aharonov Y., Anandan J., “Phase change during a cyclic quantum evolution”, Phys. Rev. Lett., 58 (1987), 1593–1596 | DOI | MR

[2] Anandan J., Aharonov Y., “Geometry of quantum evolution”, Phys. Rev. Lett., 65 (1990), 1697–1700 | DOI | MR | Zbl

[3] Berry M. V., “Quantal phase factors accompanying adiabatic changes”, Proc. Roy. Soc. London Ser. A, 392 (1984), 45–57 | DOI | MR | Zbl

[4] Berry M. V., “The quantum phase, five years after”, Geometric Phases in Physics, Adv. Ser. Math. Phys., 5, eds. A. Shapere, F. Wilczeck, World Sci. Publ., Teaneck, NJ, 1989, 7–28

[5] Berry M. V., Uzdin R., “Slow non-Hermitian cycling: exact solutions and the Stokes phenomenon”, J. Phys. A: Math. Theor., 44 (2011), 435303, 26 pp. | DOI | Zbl

[6] Berry M. V., Wilkinson M., “Diabolical points in the spectra of triangles”, Proc. Roy. Soc. London Ser. A, 392 (1984), 15–43 | DOI | MR | Zbl

[7] Born M., Fock V., “Beweis des Adiabatensatzes”, Z. Phys., 51 (1928), 165–180 | DOI | Zbl

[8] Chruściński D., Jamiołkowski A., Geometric phases in classical and quantum mechanics, Progress in Mathematical Physics, 36, Birkhäuser Boston, Inc., Boston, MA, 2004 | DOI | Zbl

[9] Cohen E., Larocque H., Bouchard F., Nejadsattari F., Gefen Y., Karimi E., “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond”, Nature Rev. Phys., 1 (2019), 437–449, arXiv: 1912.12596 | DOI

[10] Cui X.-D., Zheng Y., “Geometric phases in non-Hermitian quantum mechanics”, Phys. Rev. A, 86 (2012), 064104, 4 pp. | DOI

[11] Cui X.-D., Zheng Y., “Unification of the family of Garrison–Wright's phases”, Sci. Rep., 4 (2014), 5813, 8 pp. | DOI

[12] Garrison J. C., Wright E. M., “Complex geometrical phases for dissipative systems”, Phys. Lett. A, 128 (1988), 177–181 | DOI | MR

[13] Heiss W. D., “Phases of wave functions and level repulsion”, Eur. Phys. J. D, 7 (1999), 1–4, arXiv: quant-ph/9901023 | DOI

[14] Heiss W. D., “The physics of exceptional points”, J. Phys. A: Math. Theor., 45 (2012), 444016, 11 pp., arXiv: 1210.7536 | DOI | Zbl

[15] Höller J., Read N., Harris J. G. E., “Non-Hermitian adiabatic transport in spaces of exceptional points”, Phys. Rev. A, 102 (2020), 032216, 9 pp., arXiv: 1809.07175 | DOI | MR

[16] Kato T., “On the adiabatic theorem of quantum mechanics”, J. Phys. Soc. Japan, 5 (1950), 435–439 | DOI

[17] Kato T., Perturbation theory for linear operators, Grundleheren der Mathematischen Wissenschaften, 132, Springer, Berlin – Heidelberg, 1966 | DOI | Zbl

[18] Lee J. M., Introduction to smooth manifolds, Graduate Texts in Mathematics, 218, Springer, New York, 2012 | DOI

[19] Mandel A., Gonçalves J. Z., “Construction of open sets of free $k$-tuples of matrices”, Proc. Edinburgh Math. Soc., 30 (1987), 121–131 | DOI | MR | Zbl

[20] Manini N., Pistolesi F., “Off-diagonal geometric phases”, Phys. Rev. Lett., 85 (2000), 3067–3071, arXiv: quant-ph/9911083 | DOI

[21] Mehri-Dehnavi H., Mostafazadeh A., “Geometric phase for non-Hermitian Hamiltonians and its holonomy interpretation”, J. Math. Phys., 49 (2008), 082105, 17 pp., arXiv: 0807.3405 | DOI | MR | Zbl

[22] Milburn T. J., Doppler J., Holmes C. A., Portolan S., Rotter S., Rabl P., “General description of quasiadiabatic dynamical phenomena near exceptional points”, Phys. Rev. A, 92 (2015), 052124, 12 pp., arXiv: 1410.1882 | DOI

[23] Miri M.-A., Alù A., “Exceptional points in optics and photonics”, Science, 363 (2019), 42 pp. | DOI | Zbl

[24] Mostafazadeh A., “Exact $PT$-symmetry is equivalent to Hermiticity”, J. Phys. A: Math. Gen., 36 (2003), 7081–7091, arXiv: quant-ph/0304080 | DOI | Zbl

[25] Nenciu G., Rasche G., “On the adiabatic theorem for nonselfadjoint Hamiltonians”, J. Phys. A: Math. Gen., 25 (1992), 5741–5751 | DOI | MR

[26] Pancharatnam S., “Generalized theory of interference, and its applications. I Coherent pencils”, Proc. Indian Acad. Sci. Sect. A, 44 (1956), 247–262 | DOI | MR

[27] Pap E. J., Boer D., Waalkens H., “Non-Abelian nature of systems with multiple exceptional points”, Phys. Rev. A, 98 (2018), 023818, 9 pp., arXiv: 1805.04291 | DOI

[28] Pap E. J., Waalkens H., Frames of group-sets and their application in bundle theory, arXiv: 2010.03913

[29] Pati A. K., “Relation between “phases” and “distance” in quantum evolution”, Phys. Lett. A, 159 (1991), 105–112 | DOI | MR

[30] Provost J. P., Vallee G., “Riemannian structure on manifolds of quantum states”, Comm. Math. Phys., 76 (1980), 289–301 | DOI | MR | Zbl

[31] Shapere A., Wilczek F. (eds.), Geometric phases in physics, Advanced Series in Mathematical Physics, 5, World Sci. Publ. Co., Inc., Teaneck, NJ, 1989 | DOI | MR | Zbl

[32] Simon B., “Holonomy, the quantum adiabatic theorem, and Berry's phase”, Phys. Rev. Lett., 51 (1983), 2167–2170 | DOI | MR

[33] Tanaka A., Cheon T., “Bloch vector, disclination and exotic quantum holonomy”, Phys. Lett. A, 379 (2015), 1693–1698, arXiv: 1409.5211 | DOI | Zbl

[34] Tanaka A., Cheon T., “Path topology dependence of adiabatic time evolution”, Functional Analysis and Operator Theory for Quantum Physics, EMS Ser. Congr. Rep., eds. J. Dittrich, H. Kovařík, A. Laptev, Eur. Math. Soc., Zürich, 2017, 531–542, arXiv: 1512.06983 | DOI | Zbl

[35] Tanaka A., Cheon T., Kim S. W., “Gauge invariants of eigenspace and eigenvalue anholonomies: examples in hierarchical quantum circuits”, J. Phys. A: Math. Theor., 45 (2012), 335305, 20 pp., arXiv: 1203.5412 | DOI | Zbl

[36] Uzdin R., Mailybaev A., Moiseyev N., “On the observability and asymmetry of adiabatic state flips generated by exceptional points”, J. Phys. A: Math. Theor., 44 (2011), 435302, 8 pp. | DOI | MR | Zbl

[37] Wilczek F., Zee A., “Appearance of gauge structure in simple dynamical systems”, Phys. Rev. Lett., 52 (1984), 2111–2114 | DOI | MR

[38] Wojcik C. C., Sun X.-Q., Bzdušek T., Fan S., “Homotopy characterization of non-Hermitian Hamiltonians”, Phys. Rev. B, 101 (2020), 205417, 17 pp., arXiv: 1911.12748 | DOI

[39] Xu H., Mason D., Jiang L., Harris J. G. E., “Topological energy transfer in an optomechanical system with exceptional points”, Nature, 537 (2016), 80–83, arXiv: 1602.06881 | DOI

[40] Zhang D.-J., Wang Q.-H., Gong J., “Quantum geometric tensor in PT-symmetric quantum mechanics”, Phys. Rev. A, 99 (2018), 042104, 13 pp., arXiv: 1811.04638 | DOI