Abstract

Exceptional points (EPs) are degeneracies of non-Hermitian operators where, in addition to the eigenvalues, the corresponding eigenmodes become degenerate. Classical and quantum photonic systems with EPs have attracted tremendous attention due to their unusual properties, topological features, and an enhanced sensitivity that depends on the order of the EP, i.e., the number of degenerate eigenmodes. Yet, experimentally engineering higher-order EPs in classical or quantum domains remain an open challenge due to the stringent symmetry constraints that are required for the coalescence of multiple eigenmodes. Here, we analytically show that the number-resolved dynamics of a single, lossy waveguide beam splitter, excited by N indistinguishable photons and post-selected to the N-photon subspace, will exhibit an EP of order N+1. By using the well-established mapping between a beam splitter Hamiltonian and the perfect state transfer model in the photon-number space, we analytically obtain the time evolution of a general N-photon state and numerically simulate the system’s evolution in the post-selected manifold. Our results pave the way toward realizing robust, arbitrary-order EPs on demand in a single device.

© 2019 Chinese Laser Press

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References

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  1. L. Landau and E. Lifshitz, Quantum Mechanics: Non Relativistic Theory (Pergamon, 1977), Vol. 3.
  2. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  3. Y. N. Joglekar, C. Thompson, D. D. Scott, and G. Gautam, “Optical waveguide arrays: quantum effects and PT symmetry breaking,” Eur. Phys. J. 63, 30001 (2013).
    [Crossref]
  4. L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).
    [Crossref]
  5. 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]
  6. M. Mueller and I. Rotter, “Exceptional points in open quantum systems,” J. Phys. A 41, 244018 (2008).
    [Crossref]
  7. W. D. Heiss, “The physics of exceptional points,” J. Phys. A 45, 444016 (2012).
    [Crossref]
  8. T. Kato, Perturbation Theory for Linear Operators (Springer Science & Business Media, 2013), Vol. 132.
  9. 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]
  10. K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
    [Crossref]
  11. M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. Murch, “Quantum state tomography across the exceptional point in a single dissipative qubit,” arXiv:1901.07968 (2019).
  12. J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. 10, 855 (2019).
    [Crossref]
  13. Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).
  14. J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
    [Crossref]
  15. H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
    [Crossref]
  16. S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
    [Crossref]
  17. 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]
  18. J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
    [Crossref]
  19. 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]
  20. W. Chen, S. Kaya Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
    [Crossref]
  21. H. Zhao, Z. Chen, R. Zhao, and L. Feng, “Exceptional point engineered glass slide for microscopic thermal mapping,” Nat. Commun. 9, 1764 (2018).
    [Crossref]
  22. Q. Zhong, D. N. Christodoulides, M. Khajavikhan, K. G. Makris, and R. El-Ganainy, “Power-law scaling of extreme dynamics near higher-order exceptional points,” Phys. Rev. A 97, 020105 (2018).
    [Crossref]
  23. S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
    [Crossref]
  24. H.-K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9, 4320 (2018).
    [Crossref]
  25. G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
    [Crossref]
  26. L. Banchi, W. S. Kolthammer, and M. S. Kim, “Multiphoton tomography with linear optics and photon counting,” Phys. Rev. Lett. 121, 250402 (2018).
    [Crossref]
  27. O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).
  28. W. K. Lai, V. Buek, and P. L. Knight, “Nonclassical fields in a linear directional coupler,” Phys. Rev. A 43, 6323–6336 (1991).
    [Crossref]
  29. K. Tschernig, R. de J. León-Montiel, O. S. Magaña-Loaiza, A. Szameit, K. Busch, and A. Perez-Leija, “Multiphoton discrete fractional Fourier dynamics in waveguide beam splitters,” J. Opt. Soc. Am. B 35, 1985–1989 (2018).
    [Crossref]
  30. E. M. Graefe, U. Gnther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
    [Crossref]
  31. M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92, 187902 (2004).
    [Crossref]
  32. A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
    [Crossref]
  33. A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
    [Crossref]
  34. Y. N. Joglekar and A. Saxena, “Robust PT-symmetric chain and properties of its Hermitian counterpart,” Phys. Rev. A 83, 050101 (2011).
    [Crossref]
  35. Y. N. Joglekar, C. Thompson, and G. Vemuri, “Tunable waveguide lattices with nonuniform parity-symmetric tunneling,” Phys. Rev. A 83, 063817 (2011).
    [Crossref]
  36. R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
    [Crossref]
  37. R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
    [Crossref]
  38. N. A. Mortensen, P. A. D. Gonçalves, M. Khajavikhan, D. N. Christodoulides, C. Tserkezis, and C. Wolff, “Fluctuations and noise-limited sensing near the exceptional point of parity-time-symmetric resonator systems,” Optica 5, 1342–1346 (2018).
    [Crossref]
  39. S. Longhi, “Loschmidt echo and fidelity decay near an exceptional point,” arXiv:1905.03553 (2019).
  40. J. Wei and E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963).
    [Crossref]
  41. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973), Vol. 7.
  42. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
    [Crossref]
  43. J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, “NOON states of nine quantized vibrations in two radial modes of a trapped ion,” Phys. Rev. Lett. 121, 160502 (2018).
    [Crossref]
  44. M. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, “Higher order exceptional points in discrete photonics platforms,” in Parity-Time Symmetry and Its Applications (Springer, 2018), pp. 261–275.
  45. A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
    [Crossref]

2019 (2)

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. 10, 855 (2019).
[Crossref]

S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
[Crossref]

2018 (9)

H.-K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9, 4320 (2018).
[Crossref]

L. Banchi, W. S. Kolthammer, and M. S. Kim, “Multiphoton tomography with linear optics and photon counting,” Phys. Rev. Lett. 121, 250402 (2018).
[Crossref]

R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
[Crossref]

J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, “NOON states of nine quantized vibrations in two radial modes of a trapped ion,” Phys. Rev. Lett. 121, 160502 (2018).
[Crossref]

H. Zhao, Z. Chen, R. Zhao, and L. Feng, “Exceptional point engineered glass slide for microscopic thermal mapping,” Nat. Commun. 9, 1764 (2018).
[Crossref]

Q. Zhong, D. N. Christodoulides, M. Khajavikhan, K. G. Makris, and R. El-Ganainy, “Power-law scaling of extreme dynamics near higher-order exceptional points,” Phys. Rev. A 97, 020105 (2018).
[Crossref]

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]

K. Tschernig, R. de J. León-Montiel, O. S. Magaña-Loaiza, A. Szameit, K. Busch, and A. Perez-Leija, “Multiphoton discrete fractional Fourier dynamics in waveguide beam splitters,” J. Opt. Soc. Am. B 35, 1985–1989 (2018).
[Crossref]

N. A. Mortensen, P. A. D. Gonçalves, M. Khajavikhan, D. N. Christodoulides, C. Tserkezis, and C. Wolff, “Fluctuations and noise-limited sensing near the exceptional point of parity-time-symmetric resonator systems,” Optica 5, 1342–1346 (2018).
[Crossref]

2017 (4)

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (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]

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

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
[Crossref]

2016 (6)

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
[Crossref]

2014 (1)

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]

2013 (3)

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

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

2012 (1)

W. D. Heiss, “The physics of exceptional points,” J. Phys. A 45, 444016 (2012).
[Crossref]

2011 (2)

Y. N. Joglekar and A. Saxena, “Robust PT-symmetric chain and properties of its Hermitian counterpart,” Phys. Rev. A 83, 050101 (2011).
[Crossref]

Y. N. Joglekar, C. Thompson, and G. Vemuri, “Tunable waveguide lattices with nonuniform parity-symmetric tunneling,” Phys. Rev. A 83, 063817 (2011).
[Crossref]

2010 (1)

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

2009 (1)

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 (3)

M. Mueller and I. Rotter, “Exceptional points in open quantum systems,” J. Phys. A 41, 244018 (2008).
[Crossref]

E. M. Graefe, U. Gnther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
[Crossref]

2004 (1)

M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92, 187902 (2004).
[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]

1991 (1)

W. K. Lai, V. Buek, and P. L. Knight, “Nonclassical fields in a linear directional coupler,” Phys. Rev. A 43, 6323–6336 (1991).
[Crossref]

1963 (1)

J. Wei and E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963).
[Crossref]

Abbasi, M.

M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. Murch, “Quantum state tomography across the exceptional point in a single dissipative qubit,” arXiv:1901.07968 (2019).

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[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]

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

Aragón, J. L.

R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
[Crossref]

Assawaworrarit, S.

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
[Crossref]

Banchi, L.

L. Banchi, W. S. Kolthammer, and M. S. Kim, “Multiphoton tomography with linear optics and photon counting,” Phys. Rev. Lett. 121, 250402 (2018).
[Crossref]

Bartley, T. J.

G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
[Crossref]

Bender, C. M.

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

Bian, Z.

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

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]

Bohm, J.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Buek, V.

W. K. Lai, V. Buek, and P. L. Knight, “Nonclassical fields in a linear directional coupler,” Phys. Rev. A 43, 6323–6336 (1991).
[Crossref]

Busch, K.

K. Tschernig, R. de J. León-Montiel, O. S. Magaña-Loaiza, A. Szameit, K. Busch, and A. Perez-Leija, “Multiphoton discrete fractional Fourier dynamics in waveguide beam splitters,” J. Opt. Soc. Am. B 35, 1985–1989 (2018).
[Crossref]

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Chan, C. T.

S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
[Crossref]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Chapman, R. J.

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

Chen, W.

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

Chen, Y.

S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
[Crossref]

Chen, Z.

H. Zhao, Z. Chen, R. Zhao, and L. Feng, “Exceptional point engineered glass slide for microscopic thermal mapping,” Nat. Commun. 9, 1764 (2018).
[Crossref]

Christandl, M.

M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92, 187902 (2004).
[Crossref]

Christodoulides, D. N.

Q. Zhong, D. N. Christodoulides, M. Khajavikhan, K. G. Makris, and R. El-Ganainy, “Power-law scaling of extreme dynamics near higher-order exceptional points,” Phys. Rev. A 97, 020105 (2018).
[Crossref]

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]

N. A. Mortensen, P. A. D. Gonçalves, M. Khajavikhan, D. N. Christodoulides, C. Tserkezis, and C. Wolff, “Fluctuations and noise-limited sensing near the exceptional point of parity-time-symmetric resonator systems,” Optica 5, 1342–1346 (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]

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[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]

Clerk, A. A.

H.-K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9, 4320 (2018).
[Crossref]

Corrielli, G.

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

Crespi, A.

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

Datta, N.

M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92, 187902 (2004).
[Crossref]

de Melo, L.

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. 10, 855 (2019).
[Crossref]

Ding, K.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Domínguez-Juárez, J. L.

R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
[Crossref]

Doppler, J.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Duan, L.-M.

J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, “NOON states of nine quantized vibrations in two radial modes of a trapped ion,” Phys. Rev. Lett. 121, 160502 (2018).
[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]

Ekert, A.

M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92, 187902 (2004).
[Crossref]

El-Ganainy, R.

Q. Zhong, D. N. Christodoulides, M. Khajavikhan, K. G. Makris, and R. El-Ganainy, “Power-law scaling of extreme dynamics near higher-order exceptional points,” Phys. Rev. A 97, 020105 (2018).
[Crossref]

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]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (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]

M. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, “Higher order exceptional points in discrete photonics platforms,” in Parity-Time Symmetry and Its Applications (Springer, 2018), pp. 261–275.

Fan, S.

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
[Crossref]

Feng, L.

H. Zhao, Z. Chen, R. Zhao, and L. Feng, “Exceptional point engineered glass slide for microscopic thermal mapping,” Nat. Commun. 9, 1764 (2018).
[Crossref]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).
[Crossref]

Garcia-Gracia, H.

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]

Gautam, G.

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

Ge, L.

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).
[Crossref]

Gerrits, T.

G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
[Crossref]

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Girschik, A.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Gnther, U.

E. M. Graefe, U. Gnther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

Gonçalves, P. A. D.

Graefe, E. M.

E. M. Graefe, U. Gnther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

Guo, A.

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]

Harder, G.

G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
[Crossref]

Harris, J. G. E.

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref]

Harter, A. K.

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. 10, 855 (2019).
[Crossref]

R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
[Crossref]

Hassan, A. U.

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]

Heiss, W. D.

W. D. Heiss, “The physics of exceptional points,” J. Phys. A 45, 444016 (2012).
[Crossref]

Hodaei, H.

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]

Hou, B.

S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
[Crossref]

Huang, Z.

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

Jiang, L.

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref]

Joglekar, Y. N.

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. 10, 855 (2019).
[Crossref]

R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
[Crossref]

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

Y. N. Joglekar and A. Saxena, “Robust PT-symmetric chain and properties of its Hermitian counterpart,” Phys. Rev. A 83, 050101 (2011).
[Crossref]

Y. N. Joglekar, C. Thompson, and G. Vemuri, “Tunable waveguide lattices with nonuniform parity-symmetric tunneling,” Phys. Rev. A 83, 063817 (2011).
[Crossref]

M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. Murch, “Quantum state tomography across the exceptional point in a single dissipative qubit,” arXiv:1901.07968 (2019).

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

Kato, T.

T. Kato, Perturbation Theory for Linear Operators (Springer Science & Business Media, 2013), Vol. 132.

Kay, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Kaya Ozdemir, S.

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

Keil, R.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

Khajavikhan, M.

Q. Zhong, D. N. Christodoulides, M. Khajavikhan, K. G. Makris, and R. El-Ganainy, “Power-law scaling of extreme dynamics near higher-order exceptional points,” Phys. Rev. A 97, 020105 (2018).
[Crossref]

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]

N. A. Mortensen, P. A. D. Gonçalves, M. Khajavikhan, D. N. Christodoulides, C. Tserkezis, and C. Wolff, “Fluctuations and noise-limited sensing near the exceptional point of parity-time-symmetric resonator systems,” Optica 5, 1342–1346 (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]

M. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, “Higher order exceptional points in discrete photonics platforms,” in Parity-Time Symmetry and Its Applications (Springer, 2018), pp. 261–275.

Kim, K.

J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, “NOON states of nine quantized vibrations in two radial modes of a trapped ion,” Phys. Rev. Lett. 121, 160502 (2018).
[Crossref]

Kim, M. S.

L. Banchi, W. S. Kolthammer, and M. S. Kim, “Multiphoton tomography with linear optics and photon counting,” Phys. Rev. Lett. 121, 250402 (2018).
[Crossref]

Knight, P. L.

W. K. Lai, V. Buek, and P. L. Knight, “Nonclassical fields in a linear directional coupler,” Phys. Rev. A 43, 6323–6336 (1991).
[Crossref]

Kolthammer, W. S.

L. Banchi, W. S. Kolthammer, and M. S. Kim, “Multiphoton tomography with linear optics and photon counting,” Phys. Rev. Lett. 121, 250402 (2018).
[Crossref]

Korsch, H. J.

E. M. Graefe, U. Gnther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

Kuhl, U.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Kwek, L.-C.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Lai, W. K.

W. K. Lai, V. Buek, and P. L. Knight, “Nonclassical fields in a linear directional coupler,” Phys. Rev. A 43, 6323–6336 (1991).
[Crossref]

Landahl, A. J.

M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92, 187902 (2004).
[Crossref]

Landau, L.

L. Landau and E. Lifshitz, Quantum Mechanics: Non Relativistic Theory (Pergamon, 1977), Vol. 3.

Lau, H.-K.

H.-K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9, 4320 (2018).
[Crossref]

León-Montiel, R. de J.

R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
[Crossref]

K. Tschernig, R. de J. León-Montiel, O. S. Magaña-Loaiza, A. Szameit, K. Busch, and A. Perez-Leija, “Multiphoton discrete fractional Fourier dynamics in waveguide beam splitters,” J. Opt. Soc. Am. B 35, 1985–1989 (2018).
[Crossref]

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Li, J.

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. 10, 855 (2019).
[Crossref]

Libisch, F.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Lifshitz, E.

L. Landau and E. Lifshitz, Quantum Mechanics: Non Relativistic Theory (Pergamon, 1977), Vol. 3.

Lita, A. E.

G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
[Crossref]

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
[Crossref]

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Liu, J.

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. 10, 855 (2019).
[Crossref]

Longhi, S.

S. Longhi, “Loschmidt echo and fidelity decay near an exceptional point,” arXiv:1905.03553 (2019).

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973), Vol. 7.

Lu, W.

S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
[Crossref]

Luo, L.

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. 10, 855 (2019).
[Crossref]

Lv, D.

J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, “NOON states of nine quantized vibrations in two radial modes of a trapped ion,” Phys. Rev. Lett. 121, 160502 (2018).
[Crossref]

Ma, G.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Magaña-Loaiza, O. S.

K. Tschernig, R. de J. León-Montiel, O. S. Magaña-Loaiza, A. Szameit, K. Busch, and A. Perez-Leija, “Multiphoton discrete fractional Fourier dynamics in waveguide beam splitters,” J. Opt. Soc. Am. B 35, 1985–1989 (2018).
[Crossref]

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Mailybaev, A. A.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Makris, K. G.

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]

Q. Zhong, D. N. Christodoulides, M. Khajavikhan, K. G. Makris, and R. El-Ganainy, “Power-law scaling of extreme dynamics near higher-order exceptional points,” Phys. Rev. A 97, 020105 (2018).
[Crossref]

Mason, D.

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref]

Milburn, T. J.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Miller, A. J.

Mirin, R. P.

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Moiseyev, N.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Morandotti, R.

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]

Mortensen, N. A.

Moya-Cessa, H.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

Mueller, M.

M. Mueller and I. Rotter, “Exceptional points in open quantum systems,” J. Phys. A 41, 244018 (2008).
[Crossref]

Murch, K.

M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. Murch, “Quantum state tomography across the exceptional point in a single dissipative qubit,” arXiv:1901.07968 (2019).

Musslimani, Z. H.

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]

Naghiloo, M.

M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. Murch, “Quantum state tomography across the exceptional point in a single dissipative qubit,” arXiv:1901.07968 (2019).

Nam, S. W.

G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
[Crossref]

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
[Crossref]

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Niederle, A. E.

E. M. Graefe, U. Gnther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

Nolte, S.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Norman, E.

J. Wei and E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963).
[Crossref]

Onanga, F. A.

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

Osellame, R.

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

Perez-Leija, A.

K. Tschernig, R. de J. León-Montiel, O. S. Magaña-Loaiza, A. Szameit, K. Busch, and A. Perez-Leija, “Multiphoton discrete fractional Fourier dynamics in waveguide beam splitters,” J. Opt. Soc. Am. B 35, 1985–1989 (2018).
[Crossref]

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Peruzzo, A.

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

Quintero-Torres, R.

R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
[Crossref]

Quiroz-Juárez, M. A.

R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
[Crossref]

Rabl, P.

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Rodriguez-Lara, B. M.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Rotter, I.

M. Mueller and I. Rotter, “Exceptional points in open quantum systems,” J. Phys. A 41, 244018 (2008).
[Crossref]

Rotter, S.

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]

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

Ruzicka, F.

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

Salamo, G. J.

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]

Santandrea, M.

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

Saxena, A.

Y. N. Joglekar and A. Saxena, “Robust PT-symmetric chain and properties of its Hermitian counterpart,” Phys. Rev. A 83, 050101 (2011).
[Crossref]

Scott, D. D.

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

Silberberg, Y.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

Silberhorn, C.

G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
[Crossref]

Siviloglou, G. A.

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]

Szameit, A.

K. Tschernig, R. de J. León-Montiel, O. S. Magaña-Loaiza, A. Szameit, K. Busch, and A. Perez-Leija, “Multiphoton discrete fractional Fourier dynamics in waveguide beam splitters,” J. Opt. Soc. Am. B 35, 1985–1989 (2018).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Teimourpour, M.

M. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, “Higher order exceptional points in discrete photonics platforms,” in Parity-Time Symmetry and Its Applications (Springer, 2018), pp. 261–275.

Thompson, C.

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

Y. N. Joglekar, C. Thompson, and G. Vemuri, “Tunable waveguide lattices with nonuniform parity-symmetric tunneling,” Phys. Rev. A 83, 063817 (2011).
[Crossref]

Tschernig, K.

Tserkezis, C.

U’Ren, A. B.

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Um, M.

J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, “NOON states of nine quantized vibrations in two radial modes of a trapped ion,” Phys. Rev. Lett. 121, 160502 (2018).
[Crossref]

Vemuri, G.

Y. N. Joglekar, C. Thompson, and G. Vemuri, “Tunable waveguide lattices with nonuniform parity-symmetric tunneling,” Phys. Rev. A 83, 063817 (2011).
[Crossref]

Volatier-Ravat, M.

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]

Wang, K.

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

Wang, S.

S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
[Crossref]

Wei, J.

J. Wei and E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963).
[Crossref]

Wiersig, J.

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

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[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]

Wittek, S.

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]

Wolff, C.

Xiao, L.

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

Xiao, M.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Xu, H.

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref]

Xue, P.

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

Yang, L.

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

Yi, W.

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

You, C.

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

Yu, X.

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
[Crossref]

Yung, M.-H.

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

Zhan, X.

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

Zhang, J.

J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, “NOON states of nine quantized vibrations in two radial modes of a trapped ion,” Phys. Rev. Lett. 121, 160502 (2018).
[Crossref]

Zhang, J.-N.

J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, “NOON states of nine quantized vibrations in two radial modes of a trapped ion,” Phys. Rev. Lett. 121, 160502 (2018).
[Crossref]

Zhang, Z. Q.

S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
[Crossref]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Zhao, G.

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

Zhao, H.

H. Zhao, Z. Chen, R. Zhao, and L. Feng, “Exceptional point engineered glass slide for microscopic thermal mapping,” Nat. Commun. 9, 1764 (2018).
[Crossref]

Zhao, R.

H. Zhao, Z. Chen, R. Zhao, and L. Feng, “Exceptional point engineered glass slide for microscopic thermal mapping,” Nat. Commun. 9, 1764 (2018).
[Crossref]

Zhong, Q.

Q. Zhong, D. N. Christodoulides, M. Khajavikhan, K. G. Makris, and R. El-Ganainy, “Power-law scaling of extreme dynamics near higher-order exceptional points,” Phys. Rev. A 97, 020105 (2018).
[Crossref]

M. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, “Higher order exceptional points in discrete photonics platforms,” in Parity-Time Symmetry and Its Applications (Springer, 2018), pp. 261–275.

Commun. Phys. (1)

R. de J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits,” Commun. Phys. 1, 88 (2018).
[Crossref]

Eur. Phys. J. (1)

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

J. Math. Phys. (1)

J. Wei and E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. A (3)

M. Mueller and I. Rotter, “Exceptional points in open quantum systems,” J. Phys. A 41, 244018 (2008).
[Crossref]

W. D. Heiss, “The physics of exceptional points,” J. Phys. A 45, 444016 (2012).
[Crossref]

E. M. Graefe, U. Gnther, H. J. Korsch, and A. E. Niederle, “A non-Hermitian PT symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points,” J. Phys. A 41, 255206 (2008).
[Crossref]

Nat. Commun. (5)

S. Wang, B. Hou, W. Lu, Y. Chen, Z. Q. Zhang, and C. T. Chan, “Arbitrary order exceptional point induced by photonic spin-orbit interaction in coupled resonators,” Nat. Commun. 10, 832 (2019).
[Crossref]

H.-K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. 9, 4320 (2018).
[Crossref]

H. Zhao, Z. Chen, R. Zhao, and L. Feng, “Exceptional point engineered glass slide for microscopic thermal mapping,” Nat. Commun. 9, 1764 (2018).
[Crossref]

J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. 10, 855 (2019).
[Crossref]

R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, “Experimental perfect state transfer of an entangled photonic qubit,” Nat. Commun. 7, 11339 (2016).
[Crossref]

Nat. Photonics (1)

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752–762 (2017).
[Crossref]

Nat. Phys. (1)

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]

Nature (5)

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]

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

J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
[Crossref]

H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref]

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546, 387–390 (2017).
[Crossref]

Opt. Express (1)

Optica (1)

Phys. Rev. A (7)

Q. Zhong, D. N. Christodoulides, M. Khajavikhan, K. G. Makris, and R. El-Ganainy, “Power-law scaling of extreme dynamics near higher-order exceptional points,” Phys. Rev. A 97, 020105 (2018).
[Crossref]

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

W. K. Lai, V. Buek, and P. L. Knight, “Nonclassical fields in a linear directional coupler,” Phys. Rev. A 43, 6323–6336 (1991).
[Crossref]

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

Y. N. Joglekar and A. Saxena, “Robust PT-symmetric chain and properties of its Hermitian counterpart,” Phys. Rev. A 83, 050101 (2011).
[Crossref]

Y. N. Joglekar, C. Thompson, and G. Vemuri, “Tunable waveguide lattices with nonuniform parity-symmetric tunneling,” Phys. Rev. A 83, 063817 (2011).
[Crossref]

Phys. Rev. Lett. (7)

M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92, 187902 (2004).
[Crossref]

G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
[Crossref]

L. Banchi, W. S. Kolthammer, and M. S. Kim, “Multiphoton tomography with linear optics and photon counting,” Phys. Rev. Lett. 121, 250402 (2018).
[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]

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

J. Zhang, M. Um, D. Lv, J.-N. Zhang, L.-M. Duan, and K. Kim, “NOON states of nine quantized vibrations in two radial modes of a trapped ion,” Phys. Rev. Lett. 121, 160502 (2018).
[Crossref]

Phys. Rev. X (1)

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6, 021007 (2016).
[Crossref]

Science (1)

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref]

Other (8)

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973), Vol. 7.

M. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, “Higher order exceptional points in discrete photonics platforms,” in Parity-Time Symmetry and Its Applications (Springer, 2018), pp. 261–275.

M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. Murch, “Quantum state tomography across the exceptional point in a single dissipative qubit,” arXiv:1901.07968 (2019).

T. Kato, Perturbation Theory for Linear Operators (Springer Science & Business Media, 2013), Vol. 132.

L. Landau and E. Lifshitz, Quantum Mechanics: Non Relativistic Theory (Pergamon, 1977), Vol. 3.

Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, “Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit,” arXiv:1903.09806 (2019).

O. S. Magaña-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” arXiv:1901.00122 (2019).

S. Longhi, “Loschmidt echo and fidelity decay near an exceptional point,” arXiv:1905.03553 (2019).

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Figures (3)

Fig. 1.
Fig. 1. (a) Schematic of a single, lossy waveguide beam splitter excited with N indistinguishable photons prepared in the state |m)|Nm,m=|Nma|mb, where a represents the neutral (gray) waveguide and b is the lossy (red) waveguide. (b) Mapping onto the N-photon subspace spanned by (N+1) multiphoton states |m), represented as a tight-binding lattice model. The coupling between adjacent “modes” is given by matrix elements of J^x; the linearly increasing loss is also shown. (c) Flow of eigenvalues of H^N for N=4. R(λr) shows level attraction with an EP of order five at Γ=2κ; I(λr) shows the emergence of slow modes past the transition. (d) Intensity I(z) shows the fraction of trials where the system remains in the N-photon subspace, i.e., the post-selection probability. It reflects the order of the exceptional point. The beam splitter parameters are ω0=κ=1  cm1, and the initial state is |ψ(0)=|0).
Fig. 2.
Fig. 2. Evolution of spin-projections Jr for the N+1 eigenmodes in the post-selected manifold with N=4 (left column) and N=5 (right column) photons, considering different values of the dissipation coefficient: (a), (b) Γ=Γc/2, (c), (d) Γ=0.99Γc, and (e), (f) Γ=1.5Γc. The vector coordinates in the (Jx,Jy,Jz) space are defined by the expectation values of the J^α operators in each eigenstate. When Γ<Γc, the spin projections are in the xy plane; at the EP, they coalesce along the positive y axis; and when Γ>Γc, they are in the xz plane.
Fig. 3.
Fig. 3. Mode occupation dynamics in the post-selected manifold with NOON state input. (a) For N=5 and small loss, the dynamics show asymmetric oscillations. (b) At the EP, P(|m),z) reaches a steady state with most of the weight localized in the low-loss region. (c) After the transition, the steady-state is reached more slowly. (d)–(f) show qualitatively similar results for an N=8 NOON state input. The waveguide beam splitter parameters are set to ω0=κ=1  cm1.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

H^=ω0(a^a^+b^b^)+κ(a^b^+a^b^)iΓb^b^,
H^N=(ω0iΓ/2)N^+2κJ^xiΓJ^z,
J^z=(b^b^a^a^)/2,
J^x=(a^b^+a^b^)/2,
λr=(ω0iΓ/2)N+r4κ2Γ2,
I(z)=ψ(0)|G(z)G(z)|ψ(0),
G(z)=ei(ω0iΓ/2)N^zeif+(z)J^+eifz(z)J^zeif(z)J^,
zf+(z)=κ[1+f+2(z)]Γf+(z),zfz(z)=iΓ+2iκf+(z),zf(z)=κexp[ifz(z)].
f±(z)=Γ2κ+Δλ2κ[tan(zΔλ/2)Γ/Δλ1+(Γ/Δλ)tan(zΔλ/2)],
fz(z)=2iln[cos(zΔλ2)+ΓΔλsin(zΔλ2)].
P(|m);z)=|(m|ϕ(z)|2ϕ(z)|ϕ(z),

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