Abstract
The dynamics of a nonstationary quantum system whose Hamiltonian explicitly depends on time is called adiabatic if a system state that is an eigenstate of the Hamiltonian at the initial instant of time remains close to this eigenstate throughout the evolution. The degree of such closeness depends on the smallness of the parameter that determines the rate of change of the Hamiltonian. It is usually believed that one of the factors playing a decisive role for the stability of the adiabatic dynamics is the structure of the spectrum of the Hamiltonian. As the quantum adiabatic theorem states in its usual formulation, deviations from the adiabatic evolution can be estimated from above by the ratio of the rate of change of the Hamiltonian to the minimum distance between the energy of the state that approximates the adiabatic dynamics and the rest of the spectrum of the Hamiltonian. We analyze this dependence and prove theorems showing that the efficiency of the adiabatic approximation is more influenced by the rate of change of the Hamiltonian eigenvectors than by the dynamics of the spectrum. In a vast majority of physically meaningful cases, it turns out that controlling the dynamics of eigenvectors is sufficient for ensuring the adiabaticity, regardless of the dynamics of the spectrum as such.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This condition is necessary for applying the convergence theorem, as we see below. If it is violated, the first integral in (12) also diverges, and the whole estimate becomes trivial.
The above conditions are also satisfied by the representation of the Hamiltonian for \(s=s^*\) in the form \(\mathrm H_{s^*}(\lambda)=\mu\mathbb{I}+\lambda A+o(\lambda)\), but the identity matrix \(\mathbb{I}\) can be discarded without loss of generality.
See relation (25) and the preceding paragraph.
References
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 1: Mechanics, Elsevier (1982); Vol. 5: Statistical Physics, Elsevier (2013).
M. Born, “Das Adiabatenprinzip in der Quantenmechanik,” Z. Phys., 40, 167–192 (1926).
M. Born and V. Fock, “Beweis des adiabatensatzes,” Z. Phys., 51, 165–180 (1928).
T. Kato, “On the adiabatic theorem of quantum mechanics,” J. Phys. Soc. Japan, 5, 435–439 (1950).
A. Messiah, Quantum Mechanics (Dover Books on Physics), Dover, Mineola, NY (2014).
J. E. Avron and A. Elgart, “Adiabatic theorem without a gap condition,” Commun. Math. Phys., 203, 445–463 (1999); arXiv: math-ph/9805022.
S. Teufel, “A Note on the adiabatic theorem without gap condition,” Lett. Math. Phys., 58, 261–266 (2001).
O. Lychkovskiy, O. Gamayun, and V. Cheianov, “Time scale for adiabaticity breakdown in driven many-body systems and orthogonality catastrophe,” Phys. Rev. Lett., 119, 200401, 6 pp. (2017); arXiv: 1611.00663.
N. Il’in, A. Aristova, and O. Lychkovskiy, “Adiabatic theorem for closed quantum systems initialized at finite temperature,” Phys. Rev. A, 104, L030202, 6 pp. (2021).
O. Lychkovskiy, O. Gamayun, and V. Cheianov, “Necessary and sufficient condition for quantum adiabaticity in a driven one-dimensional impurity-fluid system,” Phys. Rev. B, 98, 024307, 9 pp. (2018); arXiv: 1804.03726.
R. Schützhold and G. Schaller, “Adiabatic quantum algorithms as quantum phase transitions: First versus second order,” Phys. Rev. A, 74, 060304, 4 pp. (2006); arXiv: quant-ph/0608017.
J. Latorre and R. Orús, “Adiabatic quantum computation and quantum phase transitions,” Phys. Rev. A, 69, 062302, 5 pp. (2004); arXiv: quant-ph/0308042.
J. M. Bowman, “Reduced dimensionality theory of quantum reactive scattering,” J. Phys. Chem., 95, 4960–4968 (1991).
U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. A new concept and experimental results,” J. Chem. Phys., 92, 5363–5376 (1990).
K. Bergmann, N. V. Vitanov, and B. W. Shore, “Perspective: Stimulated Raman adiabatic passage: The status after 25 years,” J. Phys. Chem., 142, 170901, 21 pp. (2015).
J. C. Budich and B. Trauzettel, “From the adiabatic theorem of quantum mechanics to topological states of matter,” Phys. Status Solidi RRL, 7, 109–129 (2013).
A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (Dover Books on Physics), Dover, Mineola, NY (2003).
D. J. Thouless, “Quantization of particle transport,” Phys. Rev. B, 27, 6083–6087 (1983).
M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, “A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice,” Nature Phys., 12, 350–354 (2016); arXiv: 1507.02225.
E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, “Quantum computation by adiabatic evolution,” arXiv: quant-ph/0001106.
E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, “A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem,” Science, 292, 472–475 (2001); arXiv: quant-ph/0104129.
D. A. Lidar, A. T. Rezakhani, and A. Hamma, “Adiabatic approximation with exponential accuracy for many-body systems and quantum computation,” J. Math. Phys., 50, 102106, 26 pp. (2009).
M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: Dark-state polaritons,” Phys. Rev. A, 65, 022314, 12 pp. (2002); arXiv: quant-ph/0106066.
M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, “Geometry and non-adiabatic response in quantum and classical systems,” Phys. Rep., 697, 1–87 (2017).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover, Mineola, NY (1999).
M. M. Wilde, Quantum Information Theory, Cambridge Univ. Press, New York (2013).
D. Markham, J. A. Miszczak, Z. Puchała, and K. Życzkowski, “Quantum state discrimination: A geometric approach,” Phys. Rev. A, 77, 042111, 9 pp. (2008); arXiv: 0711.4286.
L. D. Landau, “Zur theorie der energieubertragung. II,” Phys. Z. Sowjetunion, 2, 46–51 (1932).
C. Zener, “Non-adiabatic crossing of energy levels,” Proc. R. Soc. London Ser. A, 137, 696–702 (1932).
Acknowledgments
The author is grateful to O. V. Lychkovskiy for the useful discussions.
Funding
This paper was supported by the Russian Science Foundation grant No. 17-71-20158.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares no conflicts of interest.
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 211, pp. 121–135 https://doi.org/10.4213/tmf10195.
Rights and permissions
About this article
Cite this article
Ilyin, N.B. Quantum adiabatic theorem with energy gap regularization. Theor Math Phys 211, 545–557 (2022). https://doi.org/10.1134/S0040577922040080
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577922040080