Abstract

Over the past decade, parity-time (PT)-symmetric Hamiltonians have been experimentally realized in classical, optical settings with balanced gain and loss, or in quantum systems with localized loss. In both realizations, the PT-symmetry-breaking transition occurs at the exceptional point of the non-Hermitian Hamiltonian, where its eigenvalues and the corresponding eigenvectors both coincide. Here, we show that in lossy systems, the PT transition is a phenomenon that broadly occurs without an attendant exceptional point, and is driven by the potential asymmetry between the neutral and the lossy regions. With experimentally realizable quantum models in mind, we investigate dimer and trimer waveguide configurations with one lossy waveguide. We validate the tight-binding model results by using the beam-propagation-method analysis. Our results pave a robust way toward studying the interplay between passive PT transitions and quantum effects in dissipative photonic configurations.

© 2018 Chinese Laser Press

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References

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2018 (2)

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

A. K. Harter, F. A. Onanga, and Y. N. Joglekar, “Veiled symmetry of disordered Parity-Time lattices: protected PT-threshold and the fate of localization,” Sci. Rep. 8, 44 (2018).
[Crossref]

2017 (7)

Y. Fu, X. Zhang, Y. Xu, and H. Chen, “Design of zero index metamaterials with PT symmetry using epsilon-near-zero media with defects,” J. App. Phys. 121, 054503 (2017).
[Crossref]

Y. Fu and Y. Xu, “Asymmetric effects in waveguide systems using PT symmetry and zero index metamaterials,” Sci. Rep. 7, 12476 (2017).
[Crossref]

M. H. Teimourpour and R. El-Ganainy, “Laser self-termination in trimer photonic molecules,” J. Opt. 19, 075801 (2017).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13, 1117–1123 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Garcia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

J. D. Huerta Morales and B. M. Rodriguez-Lara, “Photon propagation through linearly active dimers,” Appl. Sci. 7, 587 (2017).
[Crossref]

2016 (1)

A. K. Harter, T. E. Lee, and Y. N. Joglekar, “PT-breaking threshold in spatially asymmetric Aubry-Andre-Harper models: hidden symmetry and topological states,” Phys. Rev. A 93, 062101 (2016).
[Crossref]

2015 (1)

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref]

2014 (7)

R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A 90, 013802 (2014).
[Crossref]

M. Ornigotti and A. Szameit, “Quasi PT-symmetry in passive photonic lattices,” J. Opt. 16, 065501 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

2013 (2)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Y. N. Joglekar, C. Thompson, D. D. Scott, and G. Vemuri, “Optical waveguide arrays: quantum effects and PT symmetry breaking,” Eur. Phys. J. Appl. Phys. 63, 30001 (2013).
[Crossref]

2012 (4)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
[Crossref]

H. Eleuch and I. Rotter, “Avoided level crossings in open quantum systems,” Fortschr. Phys. 61, 194–204 (2012).
[Crossref]

M. Lietrzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

2010 (3)

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

A. Mostafazadeh, “Pseudo-Hermitian representation of quantum mechanics,” Int. J. Geom. Methods Mod. Phys. 07, 1191–1306 (2010).
[Crossref]

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B: At. Mol. Opt. Phys. 43, 163001 (2010).
[Crossref]

2009 (2)

M. Znojil, “Three-Hilbert-space formulation of quantum mechanics,” SIGMA 5, 001 (2009).
[Crossref]

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

2008 (2)

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

M. Znojil, “Time-dependent version of crypto-Hermitian quantum theory,” Phys. Rev. D 78, 085003 (2008).
[Crossref]

2007 (2)

2005 (1)

I. Rotter and A. F. Sadreev, “Avoided level crossings, diabolic points, and branch points in the complex plane in an open double quantum dot,” Phys. Rev. E 71, 036227 (2005).
[Crossref]

2004 (1)

A. Mostafazadeh and A. Batal, “Physical aspects of pseudo-Hermitian and PT-symmetric quantum mechanics,” J. Phys. A 37, 11645–11679 (2004).
[Crossref]

2003 (2)

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095–1102 (2003).
[Crossref]

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A 36, 7081–7091 (2003).
[Crossref]

2002 (2)

A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205–214 (2002).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1990 (1)

W. D. Weiss and A. L. Sannino, “Avoided level crossing and exceptional points,” J. Phys. A 23, 1167–1178 (1990).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
[Crossref]

Aimez, V.

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

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Batal, A.

A. Mostafazadeh and A. Batal, “Physical aspects of pseudo-Hermitian and PT-symmetric quantum mechanics,” J. Phys. A 37, 11645–11679 (2004).
[Crossref]

Bender, C. M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095–1102 (2003).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Bian, Z. H.

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13, 1117–1123 (2017).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Brandstetter, M.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095–1102 (2003).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Cerjan, A.

M. Lietrzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

Chen, H.

Y. Fu, X. Zhang, Y. Xu, and H. Chen, “Design of zero index metamaterials with PT symmetry using epsilon-near-zero media with defects,” J. App. Phys. 121, 054503 (2017).
[Crossref]

Chen, W.

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Christodoulides, D. N.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Garcia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

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M. Znojil, “Three-Hilbert-space formulation of quantum mechanics,” SIGMA 5, 001 (2009).
[Crossref]

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J. J. Sakurai, Modern Quantum Mechanics (Addison Wesley, 1996).

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in dissipative Floquet system of ultracold atoms,” arXiv:1608.05061 (2016).

T. Kato, Perturbation Theory for Linear Operators (Springer, 1976).

S. Scheel and A. Szameit, “PT-symmetric photonic quantum systems with gain and loss do not exist,” arXiv:1805.10876 (2018).

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Figures (6)

Fig. 1.
Fig. 1. Imaginary parts of the spectrum Eq. (1) for nonzero perturbations δ/J={0.01,0.1}. The schematic PT dimer with gain (red) and loss (blue) sites is shown. When γ/J1, Iλ± grow linearly with the gain-loss strength, but are nonzero. The divergence of their derivative at the threshold γ=J is smoothed out as δ>0 increases. In this case, the system is always in the PT-broken phase, no matter how small Iλ±(γ)0 are.
Fig. 2.
Fig. 2. Decay rates Γ± of the two eigenmodes of Eq. (5) show the emergence of a slowly decaying mode for γJ, not only at δ=0, the prototypical exceptional point case, but also for a wide range of δ/J0. Inset: the PT transition threshold is determined by dΓ/dγ changing its sign from positive to negative. These results are even in the offset δ and thus remain the same for δ<0.
Fig. 3.
Fig. 3. Net transmission T(γ) shows an upturn with increasing loss strength γ signaling the passive PT transition [29]. It shows minimal change from its δ=0 value [29] when δ/J is increased all the way to unity, i.e., the system is removed far from the exceptional point. These results are even in δ and thus remain unchanged for δ<0. Inset: the lattice-model results (line) are consistent with the BPM results (stars) obtained with sample parameters in Ref. [29]; see Section 5 for details.
Fig. 4.
Fig. 4. PT transition in three coupled waveguides with lossy center waveguide, Eq. (6). The decay rates Γk show the emergence of a slow mode near γPT22J for a wide range of center-site potential. These results are even in δ and thus remain valid for δ<0.
Fig. 5.
Fig. 5. PT transition in three coupled waveguides with first lossy waveguide, Eq. (7). When δ=0, the two modes with equal decay rates undergo a sign change for dΓk/dγ near γ/J1.4. When onsite potential δ>0 is introduced, the two degenerate modes split, and the eigenmode decay-rate diagram hints at the existence of an exceptional point at δ/J=0.75.
Fig. 6.
Fig. 6. BPM results for the intensity I(x,z) of a symmetric initial pulse in a waveguide trimer defined by Hamiltonian Eq. (6) with (a), (b) δ=0 and (c), (d) δ/J=1. (a) When γ/J=2.75<γPT, the pulse decays quickly. (b) At a much larger loss, γ/J=10, the pulse propagates longer, giving rise to increased transmission. (c) For γ/J=2.75, the pulse decays quickly. (d) At γ/J=10, the pulse propagates longer. In both cases, δ=0 and δ/J=1, the emergence of a slowly decaying mode is clear; in the latter case, the Hamiltonian is far removed from an exceptional point.

Equations (9)

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λ±(γ,δ)=±J2γ2+δ22iγδ.
HD(γ)=Jσxiγ(1σz)=Jσx2iγ|22|.
Γ+=γγ2J2γJJ22γ,
dΓdγ=0  at  γ=γPT,
H2(γ,δ)=(0JJ2iγ2δ).
Htc=H0(iγ+δ)|22|.
Hte=H0(iγ+δ)|11|.
Ht(γ,δ,J)=HtcJ(|13|+|31|),
λt±=12(iγ+δ+J)±128J2+[(δJ)+iγ]2.