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\(\mathcal{PT}\)-Symmetric Periodic Optical Potentials

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Abstract

In quantum theory, any Hamiltonian describing a physical system is mathematically represented by a self-adjoint linear operator to ensure the reality of the associated observables. In an attempt to extend quantum mechanics into the complex domain, it was realized few years ago that certain non-Hermitian parity-time (\(\mathcal{PT}\)) symmetric Hamiltonians can exhibit an entirely real spectrum. Much of the reported progress has been remained theoretical, and therefore hasn’t led to a viable experimental proposal for which non Hermitian quantum effects could be observed in laboratory experiments. Quite recently however, it was suggested that the concept of \(\mathcal{PT}\)-symmetry could be physically realized within the framework of classical optics. This proposal has, in turn, stimulated extensive investigations and research studies related to \(\mathcal{PT}\)-symmetric Optics and paved the way for the first experimental observation of \(\mathcal{PT}\)-symmetry breaking in any physical system. In this paper, we present recent results regarding \(\mathcal{PT}\)-symmetric Optics.

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References

  1. Ablowitz, M.J., Musslimani, Z.H.: Spectral renormalization method for computing self-localized solutions to nonlinear systems. Opt. Lett. 30, 2140–2142 (2005)

    Article  ADS  Google Scholar 

  2. Ahmed, Z.: Energy band structure due to a complex, periodic, PT invariant potential. Phys. Lett. A 286, 231–235 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Bagchi, B., Quesne, C., Znojil, M.: Generalized continuity equation and modified normalization in PT-symmetric quantum mechanics. Mod. Phys. Lett. A 16, 2047 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Bender, C.M.: Introduction to PT symmetric quantum theory. Contemp. Phys. 46, 277–292 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  5. Bender, C.M.: Four easy pieces. J. Phys. A, Math. Gen. 39, 9993–10012 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Bender, C.M.: Making sense of non Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  7. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Bender, C.M., Brody, D.C., Jones, H.F.: Must a Hamiltonian be Hermitian? Am. J. Phys. 71, 1095–1102 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Bender, C.M., Brody, D.C., Jones, H.F., Meister, B.K.: Faster than Hermitian quantum mechanics. Phys. Rev. Lett. 98, 040403 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  10. Bender, C.M., Hook, D.W., Meisinger, P.N., Wang, Q.H.: Complex correspondence principle. Phys. Rev. Lett. 104, 061601 (2010)

    Article  ADS  Google Scholar 

  11. Bender, C.M., Hook, D.W., Meisinger, P.N., Wang, Q.H.: Probability density in the complex plane. Ann. Phys. (2010)

  12. Bendix, O., Fleischmann, R., Kottos, T., Shapiro, B.: Exponentially fragile PT symmetry in lattices with localized eigenmodes. Phys. Rev. Lett. 103, 030402 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  13. Berry, M.V.: Physics of nonHermitian degeneracies. Czech J. Phys. 54, 1039–1047 (2004)

    Article  ADS  Google Scholar 

  14. Berry, M.V.: Optical lattices with PT symmetry are not transparent. J. Phys. A, Math. Theor. 41, 244007 (2008)

    Article  ADS  Google Scholar 

  15. Boyd, J.K.: One-dimensional crystal with a complex periodic potential. J. Math. Phys. 42, 15–29 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Cervero, J.M.: PT symmetry in one-dimensional quantum periodic potentials. Phys. Lett. A 317, 26–31 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Christodoulides, D.N., Joseph, R.J.: Discrete self-focusing in nonlinear arrays of coupled wave-guides. Opt. Lett. 13, 794–796 (1988)

    Article  ADS  Google Scholar 

  18. Christodoulides, D.N., Lederer, F., Silberberg, Y.: Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003)

    Article  ADS  Google Scholar 

  19. Cottey, A.A.: Floquet’s theorem and band theory in one dimension. Am. J. Phys. 39, 1235–1244 (1971)

    Article  ADS  Google Scholar 

  20. Eastham, M.: The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, Edinburgh (1973)

    MATH  Google Scholar 

  21. El-Ganainy, R., Makris, K.G., Christodoulides, D.N., Musslimani, Z.H.: Theory of coupled optical PT symmetric structures. Opt. Lett. 32, 2632–2634 (2007)

    Article  ADS  Google Scholar 

  22. Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.N.: Observation of PT symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)

    Article  ADS  Google Scholar 

  23. Khare, A., Sukhatme, U.: Analytically solvable PT-invariant periodic potentials. Phys. Lett. A 324, 406–414 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Khare, A., Sukhatme, U.: Periodic potentials and PT symmetry. J. Phys. A, Math. Gen. 39, 10133–10142 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Klaiman, S., Günther, U., Moiseyev, N.: Visualization of branch points in PT symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  26. Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, 809–821 (1959)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Kotani, S.: Generalized Floquet theory for stationary Schrödinger operators in one dimension. Chaos Solitons Fractals 8, 1817–1854 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Kottos, T.: Broken symmetry makes light work. Nat. Phys. 6, 166–167 (2010)

    Article  Google Scholar 

  29. Longhi, S.: Bloch oscillations in complex crystals with PT symmetry. Phys. Rev. Lett. 103, 123601 (2009)

    Article  ADS  Google Scholar 

  30. Longhi, S.: Dynamic localization and transport in complex crystals. Phys. Rev. B 80, 235102 (2009)

    Article  ADS  Google Scholar 

  31. Longhi, S.: Spectral singularities in a non-Hermitian Friedrichs-Fano-Anderson model. Phys. Rev. B 80, 165125 (2009)

    Article  ADS  Google Scholar 

  32. Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008)

    Article  ADS  Google Scholar 

  33. Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: PT symmetric optical lattices. Phys. Rev. A 81, 063807 (2010)

    Article  ADS  Google Scholar 

  34. Meiman, N.N.: The theory of one-dimensional Schrödinger operators with a periodic potential. J. Math. Phys. 18, 834–848 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  35. Midya, B., Roy, B., Roychoudhury, R.: A note on the PT invariant periodic potential V(x)=4cos 2 x+4iV 0sin 2x. Phys. Lett. A 374, 2605–2607 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  36. Mostafazadeh, A.: Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102, 220402 (2009)

    Article  ADS  Google Scholar 

  37. Mostafazadeh, A.: Resonance phenomenon related to spectral singularities, complex barrier potential, and resonating waveguides. Phys. Rev. A 80, 032711 (2009)

    Article  ADS  Google Scholar 

  38. Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)

    Article  ADS  Google Scholar 

  39. Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Analytical solutions to a class of nonlinear Schrödinger equations with PT like potentials. J. Phys. A, Math. Theor. 41, 244019 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  40. Odeh, F., Keller, J.B.: Partial differential equations with periodic coefficients and Bloch waves in crystals. J. Math. Phys. 5, 1499 (1964)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Rüter, C.E., Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of parity–time symmetry in optics. Nat. Phys. 6, 192–195 (2010)

    Article  Google Scholar 

  42. Staliunas, K., Herrero, R., Vilaseca, R.: Subdiffraction and spatial filtering due to periodic spatial modulation of the gain-loss profile. Phys. Rev. A 80, 013821 (2009)

    Article  ADS  Google Scholar 

  43. Ultanir, E.A., Stegeman, G.I., Michaelis, D., Lange, C.H., Lederer, F.: Stable dissipative solitons in semiconductor optical amplifiers. Phys. Rev. Lett. 90, 253903 (2003)

    Article  ADS  Google Scholar 

  44. West, C.T., Kottos, T., Prosen, T.: PT symmetric wave chaos. Phys. Rev. Lett. 104, 054102 (2010)

    Article  ADS  Google Scholar 

  45. Znojil, M.: PT symmetric square well. Phys. Lett. A 285, 7–10 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. Znojil, M.: A return to observability near exceptional points in a schematic PT symmetric model. Phys. Lett. B 647, 225–230 (2007)

    Article  MathSciNet  ADS  Google Scholar 

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Makris, K.G., El-Ganainy, R., Christodoulides, D.N. et al. \(\mathcal{PT}\)-Symmetric Periodic Optical Potentials. Int J Theor Phys 50, 1019–1041 (2011). https://doi.org/10.1007/s10773-010-0625-6

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  • DOI: https://doi.org/10.1007/s10773-010-0625-6

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