Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 126, 10 pages      arXiv:0712.3910      https://doi.org/10.3842/SIGMA.2007.126
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Faster than Hermitian Time Evolution

Carl M. Bender
Physics Department, Washington University, St. Louis, MO 63130, USA

Received October 22, 2007, in final form December 22, 2007; Published online December 26, 2007

Abstract
For any pair of quantum states, an initial state |Iñ and a final quantum state |Fñ, in a Hilbert space, there are many Hamiltonians H under which |Iñ evolves into |Fñ. Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, Emax and Emin, is held fixed. We can then determine the Hamiltonian H that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time τ. For Hermitian Hamiltonians, τ has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of τ can be made arbitrarily small because for PT-symmetric Hamiltonians the path from the vector |Iñ to the vector |Fñ, as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.

Key words: brachistochrone; PT quantum mechanics; parity; time reversal; time evolution; unitarity.

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References

  1. Bender C.M., Boettcher S., Meisinger P.N., PT-symmetric quantum mechanics, J. Math. Phys. 40 (1999), 2201-2209, quant-ph/9809072.
  2. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
  3. Bender C.M., Introduction to PT-symmetric quantum theory, Contemp. Phys. 46 (2005), 277-292, quant-ph/0501052.
  4. Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70 (2007), 947-1018, hep-th/0703096.
  5. Dorey P., Dunning C., Tateo R., The ODE/IM correspondence, J. Phys. A: Math. Theor. 40 (2007), R205-R283, hep-th/0703066.
  6. Dorey P., Dunning C., Tateo R., Supersymmetry and the spontaneous breakdown of PT symmetry, J. Phys. A: Math. Gen. 34 (2001), L391-L400, hep-th/0104119.
    Dorey P., Dunning C., Tateo R., Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics, J. Phys. A: Math. Gen. 34 (2001), 5679-5704, hep-th/0103051.
  7. Bender C.M., Brody D.C., Jones H.F., Complex extension of quantum mechanics, Phys. Rev. Lett. 89 (2002), 270401, 4 pages, quant-ph/0208076.
    Bender C.M., Brody D.C., Jones H.F., Must a Hamiltonian be Hermitian?, Amer. J. Phys. 71 (2003), 1095-1102, hep-th/0303005.
  8. Bender C.M., Brody D.C., Jones H.F., Scalar quantum field theory with a complex cubic interaction, Phys. Rev. Lett. 93 (2004), 251601, 4 pages, hep-th/0402011.
  9. Bender C.M., Brandt S.F., Chen J.-H., Wang Q., The C operator in PT-symmetric quantum field theory transforms as a Lorentz scalar, Phys. Rev. D 71 (2005), 065010, 7 pages, hep-th/0412316.
  10. Wu T.T., Ground state of a Bose system of hard spheres, Phys. Rev. 115 (1959), 1390-1404.
  11. Hollowood T., Quantum solitons in affine Toda field theories, Nuclear Phys. B 384 (1992), 523-540, hep-th/9110010.
  12. Fisher M.E., Yang-Lee edge singularity and j3 field theory, Phys. Rev. Lett. 40 (1978), 1610-1613.
    Cardy J.L., Conformal invariance and the Yang-Lee edge singularity in two dimensions, Phys. Rev. Lett. 54 (1985), 1354-1356.
    Cardy J.L., Mussardo G., S-matrix of the Yang-Lee edge singularity in two dimensions, Phys. Lett. B 225 (1989), 275-278.
    Zamolodchikov A.B., Two-point correlation function in scaling Lee-Yang model, Nuclear Phys. B 348 (1991), 619-641.
  13. Brower R.C., Furman M.A., Moshe M., Critical exponents for the Reggeon quantum spin model, Phys. Lett. B 76 (1978), 213-219.
    Harms B.C., Jones S.T., Tan C.-I., Complex energy spectra in reggeon quantum mechanics with quartic interactions, Nuclear Phys. 171 (1980), 392-412.
    Harms B.C., Jones S.T., Tan C.-I., New structure in the energy spectrum of reggeon quantum mechanics with quartic couplings, Phys. Lett. B 91 (1980), 291-295.
  14. Günther U., Stefani F., Znojil M., MHD a2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator, J. Math. Phys. 46 (2005), 063504, 22 pages, math-ph/0501069.
    Günther U., Samsonov B.F., Stefani F., A globally diagonalizable a2-dynamo operator, SUSY QM and the Dirac equation, J. Phys. A: Math. Theor. 40 (2007), F169-F176, math-ph/0611036.
  15. de Morisson Faria C.F., Fring A., Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation, Laser Phys. 17 (2007), 424-437, quant-ph/0609096.
  16. Lee T.D., Some special examples in renormalizable field theory, Phys. Rev. 95 (1954), 1329-1334.
  17. Källén G., Pauli W., On the mathematical structure of T.D. Lee's model of a renormalizable field theory, Mat.-Fys. Medd. 30 (1955), no. 7.
  18. Bender C.M., Brandt S.F., Chen J.-H., Wang Q., Ghost busting: PT-symmetric interpretation of the Lee model, Phys. Rev. D 71 (2005), 025014, 11 pages, hep-th/0411064.
  19. Barton G., Introduction to advanced field theory, John Wiley & Sons, New York, 1963, Chap. 12.
  20. Bender C.M., Mannheim P.D., No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model, Phys. Rev. Lett., to appear, arXiv:0706.0207.
  21. Mostafazadeh A., Exact PT-symmetry is equivalent to Hermiticity, J. Phys. A: Math. Gen. 36 (2003), 7081-7091, quant-ph/0304080.
  22. Buslaev V., Grecchi V., Equivalence of unstable anharmonic oscillators and double wells, J. Phys. A: Math. Gen. 26 (1993), 5541-5549.
    Jones H.F., Mateo J., An equivalent Hermitian Hamiltonian for the non-Hermitian -x4 potential, Phys. Rev. D 73 (2006), 085002, 4 pages, quant-ph/0601188.
    Bender C.M., Brody D.C., Chen J.-H., Jones H.F., Milton K.A., Ogilvie M.C., Equivalence of a complex PT-symmetric quartic Hamiltonian and a Hermitian quartic Hamiltonian with an anomaly, Phys. Rev. D 74 (2006), 025016, 10 pages, hep-th/0605066.
  23. Carlini A., Hosoya A., Koike T., Okudaira Y., Quantum brachistochrone, Phys. Rev. Lett. 96 (2006), 060503, 4 pages, quant-ph/0511039.
  24. Brody D.C., Hook D.W., On optimum Hamiltonians for state transformations, J. Phys. A: Math. Gen. 39 (2006), L167-L170, quant-ph/0601109.
  25. Bender C.M., Brody D.C., Jones H.F., Meister B.K., Faster than Hermitian quantum mechanics, Phys. Rev. Lett. 98 (2007), 040403, 4 pages, quant-ph/0609032.
  26. Assis P.E.G., Fring A., The quantum brachistochrone problem for non-Hermitian Hamiltonians, quant-ph/0703254.
    Günther U., Rotter I., Samsonov B.F., Projective Hilbert space structures at exceptional points, J. Phys. A: Math. Theor. 40 (2007), 8815-8833, arXiv:0704.1291.
    Martin D., Is PT-symmetric quantum mechanics just quantum mechanics in a non-orthogonal basis?, quant-ph/0701223.
    Mostafazadeh A., Quantum brachistochrone problem and the geometry of the state space in pseudo-Hermitian quantum mechanics, Phys. Rev. Lett. 99 (2007), 130502, 4 pages, arXiv:0706.3844.
    Günther U., Samsonov B.F., Non-unitary operator equivalence classes, the PT-symmetric brachistochrone problem and Lorentz boosts, arXiv:0709.0483.
    Rotter I., The brachistochrone problem in open quantum systems, arXiv:0708.3891.
    Mostafazadeh A., Physical meaning of Hermiticity and shortcomings of the composite (Hermitian + non-Hermitian) quantum theory of Günther and Samsonov, arXiv:0709.1756.

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