Faster than Hermitian Time Evolution

Carl M. Bender

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

For any pair of quantum states, an initial state $|I\rangle$ and a final quantum state $|F\rangle$, in a Hilbert space, there are many Hamiltonians $H$ under which $|I\rangle$ evolves into $|F\rangle$. Let us impose the constraint that the difference between the largest and smallest eigenvalues of $H$, $E_{\max}$ and $E_{\min}$, 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 $\tau$. For Hermitian Hamiltonians, $\tau$ has a nonzero lower bound. However, among non-Hermitian $\mathcal{PT}$-symmetric Hamiltonians satisfying the same energy constraint, $\tau$ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of $\tau$ can be made arbitrarily small because for $\mathcal{PT}$-symmetric Hamiltonians the path from the vector $|I\rangle$ to the vector $|F\rangle$, 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.

Détail

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

Bibliographie

[1] Bender C. M., Boettcher S., Meisinger P. N., “$\mathcal{PT}$-symmetric quantum mechanics”, J. Math. Phys., 40 (1999), 2201–2209 ; quant-ph/9809072 | DOI | MR

[2] Bender C. M., Boettcher S., “Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry”, Phys. Rev. Lett., 80 (1998), 5243–5246 ; physics/9712001 | DOI | MR | Zbl

[3] Bender C. M., “Introduction to $\mathcal{PT}$-symmetric quantum theory”, Contemp. Phys., 46, 2005, 277–292 ; quant-ph/0501052 | MR

[4] Bender C. M., “Making sense of non-Hermitian Hamiltonians”, Rep. Prog. Phys., 70 (2007), 947–1018 ; hep-th/0703096 | DOI | MR

[5] Dorey P., Dunning C., Tateo R., “The ODE/IM correspondence”, J. Phys. A: Math. Theor., 40 (2007), R205–R283 ; hep-th/0703066 | DOI | MR | Zbl

[6] Dorey P., Dunning C., Tateo R., “Supersymmetry and the spontaneous breakdown of $\mathcal{PT}$ symmetry”, J. Phys. A: Math. Gen., 34 (2001), L391–L400 ; ; Dorey P., Dunning C., Tateo R., “Spectral equivalences, Bethe ansatz equations, and reality properties in $\mathcal{PT}$-symmetric quantum mechanics”, J. Phys. A: Math. Gen., 34 (2001), 5679–5704 ; hep-th/0104119hep-th/0103051 | DOI | MR | Zbl | DOI | MR | Zbl

[7] Bender C. M., Brody D. C., Jones H. F., “Complex extension of quantum mechanics”, Phys. Rev. Lett., 89 (2002), 270401, 4 pp., ages ; ; Bender C. M., Brody D. C., Jones H. F., “Must a Hamiltonian be Hermitian?”, Amer. J. Phys., 71 (2003), 1095–1102 ; quant-ph/0208076hep-th/0303005 | DOI | MR | DOI | MR

[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 pp., ages ; hep-th/0402011 | DOI | MR

[9] Bender C. M., Brandt S. F., Chen J.-H., Wang Q., “The $\mathcal C$ operator in $\mathcal{PT}$-symmetric quantum field theory transforms as a Lorentz scalar”, Phys. Rev. D, 71 (2005), 065010, 7 pp., ages ; hep-th/0412316 | DOI

[10] Wu T. T., “Ground state of a Bose system of hard spheres”, Phys. Rev., 115 (1959), 1390–1404 | DOI | MR | Zbl

[11] Hollowood T., “Quantum solitons in affine Toda field theories”, Nuclear Phys. B, 384 (1992), 523–540 ; hep-th/9110010 | DOI | MR

[12] Fisher M. E., “Yang–Lee edge singularity and $\varphi^3$ 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 | DOI | DOI | MR | DOI | MR | DOI | MR

[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 | DOI | DOI | DOI

[14] Günther U., Stefani F., Znojil M., “MHD $\alpha^2$-dynamo, Squire equation and $\mathcal{PT}$-symmetric interpolation between square well and harmonic oscillator”, J. Math. Phys., 46 (2005), 063504, 22 pp., ages ; ; Günther U., Samsonov B. F., Stefani F., “A globally diagonalizable $\alpha^2$-dynamo operator, SUSY QM and the Dirac equation”, J. Phys. A: Math. Theor., 40 (2007), F169–F176 ; math-ph/0501069math-ph/0611036 | DOI | MR | Zbl | DOI | MR | Zbl

[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 | DOI

[16] Lee T. D., “Some special examples in renormalizable field theory”, Phys. Rev., 95 (1954), 1329–1334 | DOI | MR | Zbl

[17] Källén G., Pauli W., “On the mathematical structure of T. D. Lee's model of a renormalizable field theory”, Mat.-Fys. Medd., 30:7 (1955), 23 | MR | Zbl

[18] Bender C. M., Brandt S. F., Chen J.-H., Wang Q., “Ghost busting: $\mathcal{PT}$-symmetric interpretation of the Lee model”, Phys. Rev. D, 71 (2005), 025014, 11 pp., ages ; hep-th/0411064 | DOI

[19] Barton G., Introduction to advanced field theory, Chap. 12, John Wiley Sons, New York, 1963 | MR | Zbl

[20] Bender C. M., Mannheim P. D., “No-ghost theorem for the fourth-order derivative Pais–Uhlenbeck oscillator model”, Phys. Rev. Lett., 100:11 (2008), 110402, 4 pp., ages ; arXiv:0706.0207 | DOI | MR

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

[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 $-x^4$ potential”, Phys. Rev. D, 73 (2006), 085002, 4 pp., ages ; ; Bender C. M., Brody D. C., Chen J.-H., Jones H. F., Milton K. A., Ogilvie M. C., “Equivalence of a complex $\mathcal{PT}$-symmetric quartic Hamiltonian and a Hermitian quartic Hamiltonian with an anomaly”, Phys. Rev. D, 74 (2006), 025016, 10 pp., ages ; quant-ph/0601188hep-th/0605066 | DOI | MR | Zbl | DOI | MR | DOI | MR

[23] Carlini A., Hosoya A., Koike T., Okudaira Y., “Quantum brachistochrone”, Phys. Rev. Lett., 96 (2006), 060503, 4 pp., ages ; quant-ph/0511039 | DOI

[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 | DOI | MR | Zbl

[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 pp., ages ; quant-ph/0609032 | DOI | MR

[26] Assis P. E. G., Fring A., The quantum brachistochrone problem for non-Hermitian Hamiltonians, ; Günther U., Rotter I., Samsonov B. F., “Projective Hilbert space structures at exceptional points”, J. Phys. A: Math. Theor., 40 (2007), 8815–8833 ; ; Martin D., Is $\mathcal{PT}$-symmetric quantum mechanics just quantum mechanics in a non-orthogonal basis?, ; Mostafazadeh A., “Quantum brachistochrone problem and the geometry of the state space in pseudo-Hermitian quantum mechanics 4 pages”, Phys. Rev. Lett., 99 (2007), 130502 ; quant-ph/0703254arXiv:0704.1291quant-ph/0701223arXiv:0706.3844 | MR | DOI | MR | Zbl | DOI