Skip to main content
SpringerLink
Log in

Analysis of Adiabatic Approximation Using Stable Hamiltonian Method

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In this paper, we deal with the adiabatic approximation of general Hamiltonians by splitting it into two parts, with one part a Hamiltonian that has at least one time-independent eigenstate up to a phase factor. We first develop the method of finding this kind of Hamiltonians. Then the relationship between adiabatic approximation and these Hamiltonians is discussed. Applying this to a general case, we give both a necessary condition and a sufficient condition for adiabatic approximation, followed by a spin-half example to illustrate.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ehrenfest, P.: On adiabatic changes of a system in connection with the quantum theory. Proc. Amsterdam Acad. 19, 576–597 (1916)

    Google Scholar 

  2. Born, M., Fock, V.: Beweis des adiabatensatzes. Z. Phys. 51(3–4), 165–180 (1928)

    Article  ADS  MATH  Google Scholar 

  3. Schwinger, J.: On nonadiabatic processes in inhomogeneous fields. Phys. Rev. 51(8), 648–651 (1937)

    Article  ADS  Google Scholar 

  4. Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn. 5(6), 435–439 (1950)

    Article  ADS  Google Scholar 

  5. Landau, L.D.: On the theory of transfer of energy at collisions II. Phys. Z. Sowjetunion 2(46) (1932)

  6. Zener, C.: Dissociation of excited diatomic molecules by external perturbations. Proc. R. Soc. Lond. Ser. A 140(842), 660–668 (1933)

    Article  ADS  MATH  Google Scholar 

  7. Gell-Mann, M., Low, F.: Bound states in quantum field theory. Phys. Rev. 84(2), 350–354 (1951)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 392(1802), 45–57 (1984)

    Article  ADS  MATH  Google Scholar 

  9. Farhi, E., Goldstone, J., Gutmann, S., et al.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292(5516), 472–475 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Joshi, C., Larson, J., Jonson, M., et al.: Entanglement of distant optomechanical systems. Phys. Rev. A 85(3), 033805 (2012)

    Article  ADS  Google Scholar 

  11. Mohammady, M.H., Morley, G.W., Nazir, A., et al.: Analysis of quantum coherence in bismuth-doped silicon: a system of strongly coupled spin qubits. Phys. Rev. B 85(5), 094404 (2012)

    Article  ADS  Google Scholar 

  12. Rigolin, G., Ortiz, G.: Adiabatic theorem for quantum systems with spectral degeneracy. Phys. Rev. A 85(6), 062111 (2012)

    Article  ADS  Google Scholar 

  13. Cullimore, M., Everitt, M.J., Ormerod, M.A., et al.: Relationship between minimum gap and success probability in adiabatic quantum computing. J. Phys. A, Math. Theor. 45(50), 505305 (2012)

    Article  MathSciNet  Google Scholar 

  14. Marzlin, K.P., Sanders, B.C.: Inconsistency in the application of the adiabatic theorem. Phys. Rev. Lett. 93(16), 160408 (2004)

    Article  ADS  Google Scholar 

  15. Tong, D.M., Singh, K., Kwek, L.C., et al.: Quantitative conditions do not guarantee the validity of the adiabatic approximation. Preprint (2005). arXiv:quant-ph/0509073

  16. Duki, S., Mathur, H., Narayan, O.: Preceding comment. Phys. Rev. Lett. 97, 128901 (2006)

    Article  ADS  Google Scholar 

  17. Ma, J., Zhang, Y., Wang, E., et al.: Comment II on inconsistency in the application of the adiabatic theorem. Phys. Rev. Lett. 97(12), 128902 (2006)

    Article  ADS  Google Scholar 

  18. Du, J., Hu, L., Wang, Y., et al.: Experimental study of the validity of quantitative conditions in the quantum adiabatic theorem. Phys. Rev. Lett. 101(6), 060403 (2008)

    Article  ADS  Google Scholar 

  19. Amin, M.H.S.: Consistency of the adiabatic theorem. Phys. Rev. Lett. 102, 220401 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  20. Tong, D.M., Singh, K., Kwek, L.C., et al.: Sufficiency criterion for the validity of the adiabatic approximation. Phys. Rev. Lett. 98(15), 150402 (2007)

    Article  ADS  Google Scholar 

  21. Maamache, M., Saadi, Y.: Adiabatic theorem and generalized geometrical phase in the case of continuous spectra. Phys. Rev. Lett. 101(15), 150407 (2008)

    Article  ADS  Google Scholar 

  22. Lidar, D.A., Rezakhani, A.T., Hamma, A.: Adiabatic approximation with exponential accuracy for many-body systems and quantum computation. J. Math. Phys. 50, 102106 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  23. Duan, Q.H., Chen P, X., Wu, W.: Adiabatic conditions and the uncertainty relation. Preprint (2011). arXiv:1102.0128

  24. Cao, H.X., Guo, Z.H., Chen, Z.L., et al.: Quantitative sufficient conditions for adiabatic approximation. Sci. China, Ser. G, Phys. Astron. 1(215) (2013)

  25. Larson, J., Stenholm, S.: Validity of adiabaticity in cavity QED. Phys. Rev. A 73(3), 033805 (2006)

    Article  ADS  Google Scholar 

  26. Tong, D.M., Yi, X.X., Fan, X.J., et al.: Examination of the adiabatic approximation in open systems. Phys. Lett. A 372(14), 2364–2367 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. O’Hara, M.J., O’Leary, D.P.: Adiabatic theorem in the presence of noise. Phys. Rev. A 77(4), 042319 (2008)

    Article  ADS  Google Scholar 

  28. Barthel, T., Kasztelan, C., McCulloch, I.P., et al.: Magnetism, coherent many-particle dynamics, and relaxation with ultracold bosons in optical superlattices. Phys. Rev. A 79(5), 053627 (2009)

    Article  ADS  Google Scholar 

  29. Gu, S.J.: Fidelity susceptibility and quantum adiabatic condition in thermodynamic limits. Phys. Rev. E 79(6), 061125 (2009)

    Article  ADS  Google Scholar 

  30. Duan, Q.H., Chen, P.X.: Realization of universal adiabatic quantum computation with fewer physical resources. Phys. Rev. A 84(4), 042332 (2011)

    Article  ADS  Google Scholar 

  31. Guo, Z., Cao, H.: Time evolution and adiabatic approximation in PT-symmetric quantum mechanics. Preprint (2012). arXiv:1212.4615

  32. Freedman, H.I., Lawson, J.D.: Systems with constant eigenvectors with applications to exact and numerical solutions of ordinary differential equations. Linear Algebra Appl. 8(4), 369–374 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  33. Freedman, H.I.: Functionally commutative matrices and matrices with constant eigenvectors. Linear Multilinear Algebra 4(2), 107–113 (1976)

    Article  Google Scholar 

  34. Tong, D.M.: Quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation. Phys. Rev. Lett. 104(12), 120401 (2010)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The work was supported by Basic Sciences Training Funds of China NO. J1103212.

Author information

Authors and Affiliations

  1. Department of Physics, Shandong University, Jinan, 250100, China

    Yi-Tian Ding

Authors
  1. Yi-Tian Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi-Tian Ding.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ding, YT. Analysis of Adiabatic Approximation Using Stable Hamiltonian Method. Int J Theor Phys 53, 1628–1636 (2014). https://doi.org/10.1007/s10773-013-1960-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-013-1960-1

Keywords

Search

Navigation