Abstract

We theoretically study the transport of time-bin entangled photon pairs in a two-dimensional topological photonic system of coupled ring resonators. This system implements the integer quantum Hall model using a synthetic gauge field and exhibits topologically robust edge states. We show that the transport through edge states preserves temporal correlations of entangled photons whereas bulk transport does not preserve these correlations and can lead to significant unwanted temporal bunching or anti-bunching of photons. We study the effect of disorder on the quantum transport properties; while the edge transport remains robust, bulk transport is very susceptible, and in the limit of strong disorder, bulk states become localized. We show that this localization is manifested as an enhanced bunching/anti-bunching of photons. This topologically robust transport of correlations through edge states could enable robust on-chip quantum communication channels and delay lines for information encoded in temporal correlations of photons.

© 2016 Optical Society of America

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2016 (4)

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15, 542–548 (2016).
[Crossref] [PubMed]

S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photon. 10, 180–183 (2016).
[Crossref]

T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, “Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics,” Phys. Rev. A 93, 043827 (2016).
[Crossref]

L. Yuan, Y. Shi, and S. Fan, “Photonic gauge potential in a system with a synthetic frequency dimension,” Opt. Lett. 41, 741–744 (2016).
[Crossref] [PubMed]

2015 (5)

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

T. Ma, A. B. Khanikaev, S. H. Mousavi, and G. Shvets, “Guiding electromagnetic waves around sharp corners: Topologically protected photonic transport in metawaveguides,” Phys. Rev. Lett. 114, 127401 (2015).
[Crossref] [PubMed]

J. Ningyuan, C. Owens, A. Sommer, D. Schuster, and J. Simon, “Time- and site-resolved dynamics in a topological circuit,” Phys. Rev. X 5, 021031 (2015).

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).

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] [PubMed]

2014 (8)

M. Hafezi, “Measuring topological invariants in photonic systems,” Phys. Rev. Lett. 112, 210405 (2014).
[Crossref]

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photon. 8, 701–705 (2014).
[Crossref]

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

M. Hafezi and J. M. Taylor, “Topological physics with light,” Physics Today 67(5), 68 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
[Crossref]

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112, 143604 (2014).
[Crossref] [PubMed]

S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and M. Hafezi, “Topologically robust transport of photons in a synthetic gauge field,” Phys. Rev. Lett. 113, 087403 (2014).
[Crossref] [PubMed]

H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, 4251 (2014).
[Crossref] [PubMed]

2013 (6)

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photon. 7, 322–328 (2013).
[Crossref]

G. Di Giuseppe, L. Martin, A. Perez-Leija, R. Keil, F. Dreisow, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Einstein-Podolsky-Rosen spatial entanglement in ordered and Anderson photonic lattices,” Phys. Rev. Lett. 110, 150503 (2013).
[Crossref] [PubMed]

H. Takesue, N. Matsuda, E. Kuramochi, W. J. Munro, and M. Notomi, “An on-chip coupled resonator optical waveguide single-photon buffer,” Nat. Commun. 4, 2725 (2013).
[Crossref] [PubMed]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7, 1001–1005 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[Crossref]

2012 (4)

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological states and adiabatic pumping in quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
[Crossref] [PubMed]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photon. 6, 782–787 (2012).
[Crossref]

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonic-fermionic quantum walk via integrated photonics,” Phys. Rev. Lett. 108, 010502 (2012).
[Crossref] [PubMed]

A. Schreiber, A. Gábris, P. P. Rohde, and K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref] [PubMed]

2011 (2)

R. O. Umucal ılar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

2010 (3)

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18, 26505–26516 (2010).
[Crossref] [PubMed]

Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, “Quantum correlations in two-particle Anderson localization,” Phys. Rev. Lett. 105, 163905 (2010).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[Crossref] [PubMed]

2009 (2)

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

2008 (1)

S. Mookherjea, J. S. Park, S.-H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photon. 2, 90–93 (2008).
[Crossref]

2002 (1)

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, “Time-bin entangled qubits for quantum communication created by femtosecond pulses,” Phys. Rev. A 66, 062308 (2002).
[Crossref]

1999 (1)

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

1993 (1)

Y. Hatsugai, “Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function,” Phys. Rev. B 48, 11851–11862 (1993).
[Crossref]

Abbarchi, M.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Abouraddy, A. F.

G. Di Giuseppe, L. Martin, A. Perez-Leija, R. Keil, F. Dreisow, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Einstein-Podolsky-Rosen spatial entanglement in ordered and Anderson photonic lattices,” Phys. Rev. Lett. 110, 150503 (2013).
[Crossref] [PubMed]

Amo, A.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Anderson, B. M.

B. M. Anderson, R. Ma, C. Owens, D. I. Schuster, and J. Simon, “Engineering topological materials in microwave cavity arrays,” arXiv1605.03177.

Assefa, S.

Bandaru, P. R.

S. Mookherjea, J. S. Park, S.-H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photon. 2, 90–93 (2008).
[Crossref]

Bloch, J.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

Brendel, J.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

Bromberg, Y.

Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, “Quantum correlations in two-particle Anderson localization,” Phys. Rev. Lett. 105, 163905 (2010).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[Crossref] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[Crossref] [PubMed]

Carusotto, I.

T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, “Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics,” Phys. Rev. A 93, 043827 (2016).
[Crossref]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaître, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

R. O. Umucal ılar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

Chan, C. T.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chen, W.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chen, X.-D.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Cheng, X.

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15, 542–548 (2016).
[Crossref] [PubMed]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Chong, Y. D.

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).

Christodoulides, D. N.

G. Di Giuseppe, L. Martin, A. Perez-Leija, R. Keil, F. Dreisow, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Einstein-Podolsky-Rosen spatial entanglement in ordered and Anderson photonic lattices,” Phys. Rev. Lett. 110, 150503 (2013).
[Crossref] [PubMed]

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Nat. Commun. (3)

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M. C. Rechtsman, Y. Lumer, Y. Plotnik, A. Perez-Leija, A. Szameit, and M. Segev, “Topological protection of photonic path entanglement,” arXiv1605.02053

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic of a 2D lattice of coupled ring resonators implementing the integer quantum-Hall model. Site resonators (black) are coupled using link resonators (grey). The lattice is coupled to input and output waveguides. Edge states transport is confined along the lattice boundary whereas the bulk states follow different paths through the bulk of the lattice. A time-bin entangled photon pair is coupled to the lattice at input and the output temporal correlations are examined. An example single photon temporal wavefunction and the two-photon correlation function is shown at the input and the output. (b) A vertical shift of link resonator introduces direction dependent hopping phase and hence synthetic magnetic field for photons. Photons hopping along right experience a longer path and hence an extra phase compared to photons hopping along left. (c) Single-photon transmission spectrum (solid red line) for a pure 8 × 8 lattice. CW, CCW Edge and bulk bands are shaded in green, red and blue, respectively. In this paper, we use the input/output coupling rate to be same as the coupling rate J between site resonators.
Fig. 2
Fig. 2 (a) Time-correlation Γ(t1, t2) for Ψ+ input state, with σ = 10 T0 and delay τ = 30 T0, where T0 = 1/J. (b,c) Simulated correlation function at the output port of a 8 × 8 lattice for CCW and CW edge states, respectively. The delay incurred in the edge states shifts the correlation function diagonally but correlation of the input state is preserved. The centres of the two time-bins are marked with dashed yellow lines. (d-f) Results for the input state Φ+. Insets show the transmission spectrum and the path followed by edge states. Γ(t1,t2) is normalized such that the maximum is unity.
Fig. 3
Fig. 3 (a–d) Time-correlation function at the input and the output of the lattice for Ψ+ state and three different input frequencies in the bulk band, ω = (−0.52, 0. − 4,0.52)J. The profile is dictated largely by the input excitation frequency and the two photons can bunch at the output even when they are well separated at the input. (e–h) Correlation for the separable state corresponding to the input frequencies in (a–d). For the separable state, the bunching is much less than that for the entangled state. (i–p) Simulation results for Φ+ and the corresponding separable state, where the photons are bunched at the input and can anti-bunch at the output after propagating through bulk states. These results show that the quantum state of two entangled photons is more fragile than the separable state.
Fig. 4
Fig. 4 Correlation function at the output of a disordered system, for Ψ+ input excitation in the (a–d) CCW edge band, (e–h) CW edge band and (i–l) for bulk band; for four different disorder strengths. As disorder increases, the bunching increases and is more significant for bulk band than the edge bands.
Fig. 5
Fig. 5 (a) Calculated probability of bunching for Ψ+ input state, with excitation in the edge and the bulk bands, as a function of disorder. Bunching is more prominent for bulk states and increases with disorder strength. (b) Similarity between the input and the output correlation. Even in the presence of strong disorder, the output correlation for edge states is very similar to that of the input.

Equations (14)

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H = x , y ω 0 a ^ x , y a ^ x , y J ( a ^ x + 1 , y a ^ x , y e i y ϕ + a ^ x , y a ^ x + 1 , y e i y ϕ + a ^ x , y + 1 a ^ x , y + a ^ x , y a ^ x , y + 1 ) ,
| ψ = d t 1 d t 2 ψ ( t 1 , t 2 ; t e , t l ) a ^ ( t 1 ) a ^ ( t 2 ) | 0 ,
| Ψ + = 1 2 ( | e 1 | l 2 + | l 1 | e 2 ) ,
| Φ + = 1 2 ( | e 1 | e 2 + | l 1 | l 2 )
| Φ = 1 2 ( | e 1 | e 2 | l 1 | l 2 ) .
Ψ + ( t 1 , t 2 ; t e , t l ) = A [ exp ( ( t 1 t e ) 2 2 σ 2 ) exp ( ( t 2 t l ) 2 2 σ 2 ) + exp ( ( t 1 t l ) 2 2 σ 2 ) exp ( ( t 2 t e ) 2 2 σ 2 ) ] ,
P B = d t 1 d t 2 δ ( t 2 t 1 ± ε ) Γ ( t 1 , t 2 ) d t 1 d t 2 Γ ( t 1 , t 2 ) ,
S = ( d t 1 d t 2 Γ out ( t 1 , t 2 ) Γ in ( t 1 , t 2 ) ¯ ) 2 d t 1 d t 2 Γ out ( t 1 , t 2 ) d t 1 d t 2 Γ in ( t 1 , t 2 ) ,
| ψ in ( t e , t l ) = A d t 1 d t 2 ψ in ( t 1 , t 2 ; t e , t l ) a ^ ( t 1 ) a ^ ( t 2 ) | 0 .
ψ in ( t 1 , t 2 ; t e , t l ) = 1 2 ( ϕ in , 1 ( t 1 t e ) ϕ in , 2 ( t 2 t l ) + ϕ in , 1 ( t 1 t l ) ϕ in , 2 ( t 2 t e ) ) ,
ψ in ( t 1 , t 2 ; t e , t l ) = 1 2 π d ω 1 d ω 2 ψ ˜ in ( ω 1 , ω 2 ) e i ω 1 t 1 e i ω 2 t 2 ,
ψ ˜ in ( ω 1 , ω 2 ) = ϕ ˜ in , 1 ( ω 1 ) ϕ ˜ in , 2 ( ω 2 ) [ exp ( i ω 1 t e + i ω 2 t l ) + exp ( i ω 1 t l + i ω 2 t e ) ] ,
ψ ˜ out ( ω 1 , ω 2 ) = S ( ω 1 ) S ( ω 2 ) ψ ˜ in ( ω 1 , ω 2 ) = ϕ ˜ out , 1 ( ω 1 ) ϕ ˜ out , 2 ( ω 2 ) [ exp ( i ω 1 t e + i ω 2 t l ) + exp ( i ω 1 t l + i ω 2 t e ) ] ,
ψ out ( t 1 , t 2 ; t e , t l ) = 1 2 π d ω 1 d ω 2 ψ ˜ out ( ω 1 , ω 2 ) e i ω 1 t 1 e i ω 2 t 2 = 1 2 ( ϕ out , 1 ( t 1 t e ) ϕ out , 2 ( t 2 t l ) + ϕ out , 1 ( t 1 t l ) ϕ out , 2 ( t 2 t e ) ) .

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