Abstract
In this chapter we describe how \(\mathbb {PT}\)-symmetric systems can be implemented in a passive fashion, that is, without using gain, by employing modulated waveguide structures. To this end, we present the underlying theoretical ideas as well as the details of the implementation of such passive structures. As an application, we experimentally demonstrate the transition from ballistic to diffusive transport in passive \(\mathbb {PT}\)-symmetric waveguide arrays.
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Notes
- 1.
This operation is quite common in quantum field theory (especially in lattice quantum field theory), where it is used to transform the Minkowski metric to an Euclidean one, allowing methods of statistical mechanics to be used for evaluating path integrals on a lattice [20].
- 2.
The results presented in this section, although specific to the case of a step-index potential, are valid for any arbitrary (but well defined) refractive index distribution, as it will be discussed at the end of this paragraph.
- 3.
Notice, that Eq. (10) differs slightly from the one obtained by applying the Kramers-Henneberger transformation, since the extra term appearing in Eq. (10) should be of the form \(\nu ^2 d\cos {}(\nu Z)\psi \) instead of \(i\nu d\sin {}(\nu z)\partial \psi /\partial X\). Here, we choose this second form, since it offers a better way to calculate the losses of the system analytically. However, it is not difficult to prove that both versions of the transformed equation lead to equivalent results.
- 4.
Notice that, in defining the functions \(\mathbb {T}(\beta _b,\nu )\) and \(\mathbb {U}(\beta _b,\nu )\), we have neglected a term proportional to δ(β a − β b + ν), since it accounts for an off-resonance term.
- 5.
\(\bar {\beta }\) can be anyway eliminated by means of a suitable gauge transformation.
- 6.
The amplitude of the modulation has chosen to be small, in order to avoid the onset of a z-modulated coupling constant.
References
Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)
El-Ganainy, R., Makris, K.G., Christodoulides, D.N., Musslimani, Z.H.: Theory of coupled optical PT-symmetric structures. Opt. Lett. 32(17), 2632–2634 (2007)
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)
Zheng, M.C., Christodoulides, D.N., Fleischmann, R., Kottos, T.: PT optical lattices and universality in beam dynamics. Phys. Rev. A 82, 010103 (2010)
Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)
Dmitriev, S.V., Sukhorukov, A.A., Kivshar, Y.S.: Binary parity-time-symmetric nonlinear lattices with balanced gain and loss. Opt. Lett. 35(17), 2976–2978 (2010)
Longhi, S.: Bloch oscillations in complex crystals with PT symmetry. Phys. Rev. Lett. 103, 123601 (2009)
Longhi, S.: Spectral singularities in a non-Hermitian Friedrichs-Fano-Anderson model. Phys. Rev. B 80, 165125 (2009)
Szameit, A., Rechtsman, M.C., Bahat-Treidel, O., Segev, M.: PT-symmetry in honeycomb photonic lattices. Phys. Rev. A 84, 021806 (2011)
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)
Gräfe, M., Heilmann, R., Keil, R., Eichelkraut, T., Heinrich, M., Nolte, S., Szameit, A.: Correlations of indistinguishable particles in non-Hermitian lattices. New J. Phys. 15(3), 033008 (2013)
Eichelkraut, T., Heilmann, R., Weimann, S., Stützer, S., Dreisow, F., Christodoulides, D.N., Nolte, S., Szameit, A.: Mobility transition from ballistic to diffusive transport in non-Hermitian lattices. Nat. Commun. 4, 2533 (2013)
Ornigotti, M., Szameit, A.: Quasi PT-symmetry in passive photonic lattices. J. Opt. 16(6), 065501 (2014)
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)
Regensburger, A., Bersch, C., Miri, M.-A., Onishchukov, G., Christodoulides, D.N., Peschel, U.: Parity-time synthetic photonic lattices. Nature 488, 167–171 (2012)
El-Ganainy, R., Makris, K.G., Christodoulides, D.N., Musslimani, H.Z.: Theory of coupled optical pt-symmetric structures. Opt. Lett. 32, 2632 (2007)
Bendix, O., Fleischmann, R., Kottos, T., Shapiro, B.: Optical structures with local pt-symmetry. J. Phys. A Math. Theor. 43, 263505 (2010)
Yariv, A.: Optical Electronics, 3rd edn. CBS College Publishing, New York (1985)
Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Perseus Books, Reading, Massachusetts (1995)
Rohe, H.J.: Lattice Gauge Theories: An Introduction. World Scientific Publishing, Singapore (2012)
Chan, J.W., Huser, T., Risbud, S., Krol, D.M.: Structural changes in fused silica after exposure to focused femtosecond laser pulses. Opt. Lett. 26(21), 1726–1728 (2001)
Chan, J.W., Huser, T.R., Risbud, S.H., Krol, D.M.: Modification of the fused silica glass network associated with waveguide fabrication using femtosecond laser pulses. Appl. Phys. A 76(3), 367–372 (2003)
Mansour, I., Caccavale, F.: An improved procedure to calculate the refractive index profile from the measured near-field intensity. J. Lightwave Technol. 14(3), 423–428 (1996)
Szameit, A., Dreisow, F., Heinrich, M., Pertsch, T., Nolte, S., Tünnermann, A., Suran, E., Louradour, F., Barthélémy, A., Longhi, S.: Image reconstruction in segmented femtosecond laser-written waveguide arrays. Appl. Phys. Lett. 93(18), 181109 (2008)
Szameit, A., Garanovich, I.L., Heinrich, M., Sukhorukov, A.A., Dreisow, F., Pertsch, T., Nolte, S., Tünnermann, A., Longhi, S., Kivshar, Y.S.: Observation of two-dimensional dynamic localization of light. Phys. Rev. Lett. 104, 223903 (2010)
Heinrich, M., Szameit, A., Dreisow, F., Keil, R., Minardi, S., Pertsch, T., Nolte, S., Tünnermann, A., Lederer, F.: Observation of three-dimensional discrete-continuous X waves in photonic lattices. Phys. Rev. Lett. 103, 113903 (2009)
Naether, U., Meyer, J.M., Stützer, S., Tünnermann, A., Nolte, S., Molina, M.I., Szameit, A.: Anderson localization in a periodic photonic lattice with a disordered boundary. Opt. Lett. 37(4), 485–487 (2012)
Kramers, H.A.: Collected Scientific Papers. North-Holland, Amsterdam (1956)
Henneberger, W.C.: Perturbation method for atoms in intense light beams. Phys. Rev. Lett. 21, 838 (1968)
Breuer, H.P.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2007)
Byron, F.W., Fuller, R.W.: Mathematics of Classica and Quantum Physics. Dover, New York (1992)
Golshani, M., Weimann, S., Jafari, Kh., Khazaei Nezhad, M., Langari, A., Bahrampour, A.R., Eichelkraut, T., Mahdavi, S.M., Szameit, A.: Impact of loss on the wave dynamics in photonic waveguide lattices. Phys. Rev. Lett. 113, 123903 (2014)
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Eichelkraut, T., Weimann, S., Kremer, M., Ornigotti, M., Szameit, A. (2018). Passive \(\mathbb {PT}\)-Symmetry in Laser-Written Optical Waveguide Structures. In: Christodoulides, D., Yang, J. (eds) Parity-time Symmetry and Its Applications. Springer Tracts in Modern Physics, vol 280. Springer, Singapore. https://doi.org/10.1007/978-981-13-1247-2_5
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