Abstract

Exceptional points of parity-time (PT) symmetric systems hold an intriguing potential for highly sensitive sensors. Here, we theoretically explore the role of mesoscopic fluctuations and noise on the spectral and temporal properties of systems of PT-symmetric-coupled gain–loss resonators operating near the exceptional point, where eigenvalues and eigenvectors coalesce. We show that experimentally inevitable detuning in the frequencies of the uncoupled resonators leads to an unavoidable modification of the conditions for reaching the exceptional point, while, as this point is approached in ensembles of resonator pairs, statistical averaging significantly smears the spectral features. We discuss how these fluctuations affect the sensitivity of sensors based on coupled PT-symmetric resonators. Finally, we show that temporal fluctuations in the detuning and gain of these sensors lead to at least a quadratic growth of the optical power in time, implying that maintaining operation at the exceptional point over a long period can be rather challenging. Our theoretical analysis clarifies issues central to the realization of PT-symmetric devices, and should facilitate future experimental work in the field.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (4)

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]

S. Longhi, “Exceptional points and photonic catastrophe,” Opt. Lett. 43, 2929–2932 (2018).
[Crossref]

G. Dolfo and J. Vigué, “Damping of coupled harmonic oscillators,” Eur. J. Phys. 39, 025005 (2018).
[Crossref]

W. Langbein, “No exceptional precision of exceptional-point sensors,” Phys. Rev. A 98, 023805 (2018).
[Crossref]

2017 (7)

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

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

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

Z. Gao, S. Fryslie, B. Thompson, P. Carney, and K. Choquette, “Parity-time symmetry in coherently coupled vertical cavity laser arrays,” Optica 4, 323–329 (2017).
[Crossref]

J. Miller, “Exceptional points make for exceptional sensors,” Phys. Today 70(10), 23–26 (2017).
[Crossref]

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
[Crossref]

S. Longhi, “Parity-time symmetry meets photonics: a new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
[Crossref]

2014 (5)

J. Liu, P. D. Garcia, S. Ek, N. Gregersen, T. Suhr, M. Schubert, J. Mørk, S. Stobbe, and P. Lodahl, “Random nanolasing in the Anderson localized regime,” Nat. Nanotechnol. 9, 285–289 (2014).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, 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. Özdemir, 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]

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)

E. Amooghorban, N. A. Mortensen, and M. Wubs, “Quantum optical effective-medium theory for loss-compensated metamaterials,” Phys. Rev. Lett. 110, 153602 (2013).
[Crossref]

R. Ameling and H. Giessen, “Microcavity plasmonics: strong coupling of photonic cavities and plasmons,” Laser Photon. Rev. 7, 141–169 (2013).
[Crossref]

2012 (2)

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]

J. Grgić, J. R. Ott, F. Wang, O. Sigmund, A.-P. Jauho, J. Mørk, and N. A. Mortensen, “Fundamental limitations to gain enhancement in periodic media and waveguides,” Phys. Rev. Lett. 108, 183903 (2012).
[Crossref]

2011 (1)

J. Grgić, E. Campaioli, S. Raza, P. Bassi, and N. A. Mortensen, “Coupled-resonator optical waveguides: Q-factor and disorder influence,” Opt. Quantum Electron. 42, 511–519 (2011).
[Crossref]

2010 (4)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref]

J. R. Ott, N. A. Mortensen, and P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[Crossref]

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327, 1352–1355 (2010).
[Crossref]

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]

2009 (2)

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]

M. Svedendahl, S. Chen, A. Dmitriev, and M. Käll, “Refractometric sensing using propagating versus localized surface plasmons: a direct comparison,” Nano Lett. 9, 4428–4433 (2009).
[Crossref]

2007 (2)

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

M. C. Goorden and M. Büttiker, “Two-particle scattering matrix of two interacting mesoscopic conductors,” Phys. Rev. Lett. 99, 146801 (2007).
[Crossref]

2006 (1)

A. M. Lunde, K. Flensberg, and L. I. Glazman, “Interaction-induced resonance in conductance and thermopower of quantum wires,” Phys. Rev. Lett. 97, 256802 (2006).
[Crossref]

2003 (1)

2001 (1)

N. A. Mortensen, K. Flensberg, and A.-P. Jauho, “Coulomb drag in coherent mesoscopic systems,” Phys. Rev. Lett. 86, 1841–1844 (2001).
[Crossref]

2000 (1)

B. N. Narozhny and I. L. Aleiner, “Mesoscopic fluctuations of the coulomb drag,” Phys. Rev. Lett. 84, 5383–5386 (2000).
[Crossref]

1999 (1)

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]

1997 (1)

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
[Crossref]

1994 (1)

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994).
[Crossref]

1985 (1)

P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55, 1622–1625 (1985).
[Crossref]

1980 (1)

B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee, “Magnetoresistance and Hall effect in a disordered two-dimensional electron gas,” Phys. Rev. B 22, 5142–5153 (1980).
[Crossref]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[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]

Aleiner, I. L.

B. N. Narozhny and I. L. Aleiner, “Mesoscopic fluctuations of the coulomb drag,” Phys. Rev. Lett. 84, 5383–5386 (2000).
[Crossref]

Altshuler, B. L.

B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee, “Magnetoresistance and Hall effect in a disordered two-dimensional electron gas,” Phys. Rev. B 22, 5142–5153 (1980).
[Crossref]

Ameling, R.

R. Ameling and H. Giessen, “Microcavity plasmonics: strong coupling of photonic cavities and plasmons,” Laser Photon. Rev. 7, 141–169 (2013).
[Crossref]

Amooghorban, E.

E. Amooghorban, N. A. Mortensen, and M. Wubs, “Quantum optical effective-medium theory for loss-compensated metamaterials,” Phys. Rev. Lett. 110, 153602 (2013).
[Crossref]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

Balachandran, R. M.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994).
[Crossref]

Bassi, P.

J. Grgić, E. Campaioli, S. Raza, P. Bassi, and N. A. Mortensen, “Coupled-resonator optical waveguides: Q-factor and disorder influence,” Opt. Quantum Electron. 42, 511–519 (2011).
[Crossref]

Beenakker, C. W. J.

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
[Crossref]

Bender, C. M.

B. Peng, S. K. Özdemir, 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]

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

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. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Büttiker, M.

M. C. Goorden and M. Büttiker, “Two-particle scattering matrix of two interacting mesoscopic conductors,” Phys. Rev. Lett. 99, 146801 (2007).
[Crossref]

Campaioli, E.

J. Grgić, E. Campaioli, S. Raza, P. Bassi, and N. A. Mortensen, “Coupled-resonator optical waveguides: Q-factor and disorder influence,” Opt. Quantum Electron. 42, 511–519 (2011).
[Crossref]

Carney, P.

Chen, S.

M. Svedendahl, S. Chen, A. Dmitriev, and M. Käll, “Refractometric sensing using propagating versus localized surface plasmons: a direct comparison,” Nano Lett. 9, 4428–4433 (2009).
[Crossref]

Chen, W.

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

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref]

Choquette, K.

Chow, W.

Christodoulides, D.

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-Gracia, 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]

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]

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. 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]

Datta, S.

S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University, 1995).

Deutsch, C.

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

Dmitriev, A.

M. Svedendahl, S. Chen, A. Dmitriev, and M. Käll, “Refractometric sensing using propagating versus localized surface plasmons: a direct comparison,” Nano Lett. 9, 4428–4433 (2009).
[Crossref]

Dolfo, G.

G. Dolfo and J. Vigué, “Damping of coupled harmonic oscillators,” Eur. J. Phys. 39, 025005 (2018).
[Crossref]

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref]

Duchesne, D.

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]

Ek, S.

J. Liu, P. D. Garcia, S. Ek, N. Gregersen, T. Suhr, M. Schubert, J. Mørk, S. Stobbe, and P. Lodahl, “Random nanolasing in the Anderson localized regime,” Nat. Nanotechnol. 9, 285–289 (2014).
[Crossref]

El-Ganainy, R.

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]

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Figures (4)

Fig. 1.
Fig. 1. Schematic illustration of a PT-symmetric dimer formed by two identical (no frequency detuning) coupled optical resonators, but with opposite values of the gain–loss parameter G(g/2κ). In the absence of gain–loss (G=0), the two resonators form common hybridized states with splitting by 2κ, while for the exceptional point (EP) at G=1, the system is degenerate, i.e., Ω+=Ω, with coalescing eigenstates.
Fig. 2.
Fig. 2. Plots of the distribution of complex eigenfrequencies Ω=Ω+iΩ for varying G. The upper panel shows P(Ω), while P(Ω) is displayed in the lower panel for an ensemble of coupled resonators with σ=0.1. Clearly, the sample-to-sample fluctuations are pronounced as one approaches the exceptional point. The right-hand panels show eigenfrequency distributions for specific values of G=(0.5, 1, 1.5) corresponding to the dashed vertical lines in the left-hand panels.
Fig. 3.
Fig. 3. Plot of the distribution of P(Ω) versus ΩΩ¯ at the exceptional point (G=1) for ensembles of coupled resonators with σ=0.05,0.1,0.2,0.3, and 0.4. Data points are the results of numerical ensemble averaging of the spectra associated with Eq. (2b), while the filled curve shows the approximate universal result from Eq. (8).
Fig. 4.
Fig. 4. Sensitivity-diminution factor (filled curve) versus Δ0/σ, with σ representing the noise/fluctuations. The integral I(Δ0/σ) in Eq. (10) is also shown (red line), along with the asymptotic behaviors in Eq. (11) indicated by dashed lines.

Equations (17)

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it(a(t)b(t))ψ(t)=(ωaig2κκωb+ig2)H(a(t)b(t)),
(Ω¯+ΔiG11Ω¯Δ+iG)(ab)=Ω(ab),
Ω±=Ω¯±1(G+iΔ)2,
ψ±=(ab)±=(i(G+iΔ)±1(G+iΔ)21).
Ω±Ω¯±(1+12Δ2),
Ω±GΔ.
Ω±Ω¯±(1i)Δ.
P0(Δ)=12πσexp(12Δ2/σ2).
Σ=Ω+Ω2|Δ|.
Σ=2dΔ|Δ|P(Δ)=25/4Γ(34)σπ1.64σ.
P(Ω)1σF(ΩΩ¯σ),
P(Δ)=12πσexp[12(ΔΔ0)2/σ2],
ΣΔ0=2πσ1/2dx|x+Δ0σ|xexp(12x2)I(Δ0/σ).
I(Δ0/σ){Δ0σ,Δ0σπ2σΔ0,Δ0σ,
iτψ=H0ψ;H0=(Ω¯i11Ω¯+i),
ψ(τ)=U0(τ)ψ(0)=exp(iH0τ)ψ(0).
U(τ)=exp(iΩ¯τ)n=0(iτ)nn!An=exp(iΩ¯τ)(IiAτ),

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