Abstract

A waveguide loop coupled to two external line waveguides by a 50/50 beam splitter forms a Sagnac interferometer. We consider the situation where two Λ-type three-level emitters are symmetrically coupled to the loop of a Sagnac interferometer and a single photon is input through one end of the line waveguides. Since the incoming photon is always in a superposition of the clockwise and counterclockwise modes of the loop and the two emitters are positioned symmetrically with respect to the input port of photon, the processes of photon scattering at the two emitters are symmetric and coherent. When the separation of the emitters and the coupling strengths of the emitters with the waveguide loop take some special values, due to quantum interference, a frequency down-conversion can certainly happen at one of the two emitters during the photon scattering but one cannot know at which emitter the frequency down-conversion takes place. This indistinguishability of the coherent frequency down-conversion processes can result in the generation of the symmetric or antisymmetric two-qubit maximally entangled states of the emitters. In the present scheme, a single photon comes in and goes out of the waveguide loop, and no photon localization modes exists. The entangled states result from the coherent frequency down-conversion processes of the emitters. Thus, the resulting entangled states are stable if the two lower-lying states of the emitters have no decay. We also investigate the influence of the dissipation of the emitters and the finite bandwidth of an input photon wavepacket on the success probability of entanglement generation, and find that the present scheme is robust to these effects and feasible with current available technologies.

© 2017 Optical Society of America

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2016 (12)

Z. Liao, X. Zeng, H. Nha, and M. S. Zubairy, “Photon transport in a one-dimensional nanophotonic waveguide QED system,” Phys. Scr. 91, 063004 (2016).
[Crossref]

N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. V. Kupriyanov, and J. Laurat, “Large Bragg Reflection from One-Dimensional Chains of Trapped Atoms Near a Nanoscale Waveguide,” Phys. Rev. Lett. 117, 133603 (2016).
[Crossref] [PubMed]

H. L. Sørensen, J. B. Béguin, K. W. Kluge, I. Iakoupov, A. S. Sørensen, J. H. Müller, E. S. Polzik, and J. Appel, “Coherent Backscattering of Light Off One-Dimensional Atomic Strings,” Phys. Rev. Lett. 117, 133604 (2016).
[Crossref] [PubMed]

C. H. Yan and L. F. Wei, “Photonic switches with ideal switching contrasts for waveguide photons,” Phys. Rev. A 94, 053816 (2016).
[Crossref]

Z. Liao, H. Nha, and M. S. Zubairy, “Single-photon frequency-comb generation in a one-dimensional waveguide coupled to two atomic arrays,” Phys. Rev. A 93, 033851 (2016).
[Crossref]

Z. Chen, Y. Zhou, and J. T. Shen, “Photon antibunching and bunching in a ring-resonator waveguide quantum electrodynamics system,” Opt. Lett. 41(14), 3313–3316 (2016).
[Crossref] [PubMed]

M. T. Cheng, X. S Ma, J. Y. Zhang, and B. Wang, “Single photon transport in two waveguides chirally coupled by a quantum emitter,” Opt. Express 24(17), 19988–19993 (2016).
[Crossref] [PubMed]

P. Facchi, M. S. Kim, S. Pascazio, F. V. Pepe, D. Pomarico, and T. Tufarelli, “Bound states and entanglement generation in waveguide quantum electrodynamics,” Phys. Rev. A 94, 043839 (2016).
[Crossref]

V. Paulisch, H. J. Kimble, and A. González-Tudela, “Universal quantum computation in waveguide QED using decoherence free subspaces,” New J. Phys. 18, 043041 (2016).
[Crossref]

I. M. Mirza and J. C. Schotland, “Multiqubit entanglement in bidirectional-chiral-waveguide QED,” Phys. Rev. A 94, 012302 (2016).
[Crossref]

I. M. Mirza and J. C. Schotland, “Two-photon entanglement in multiqubit bidirectional-waveguide QED,” Phys. Rev. A 94, 012309 (2016).
[Crossref]

D. Rieländer, A. Lenhard, M. Mazzera, and H. de Riedmatten, “Cavity enhanced telecom heralded single photons for spin-wave solid state quantum memories,” New J. Phys. 18, 123013 (2016).
[Crossref]

2015 (11)

G. Schunk, U. Vogl, D. V. Strekalov, M. Förtsch, F. Sedlmeir, H. G. L. Schwefel, M. Göbelt, S. Christiansen, G. Leuchs, and C. Marquardt, “Interfacing transitions of different alkali atoms and telecom bands using one narrowband photon pair source,” Optica 2, 773–778 (2015).
[Crossref]

A. González-Tudela, V. Paulisch, D. E. Chang, H. J. Kimble, and J. I. Cirac, “Deterministic generation of arbitrary photonic states assisted by dissipation,” Phys. Rev. Lett. 115, 163603 (2015).
[Crossref] [PubMed]

H. Pichler, T. Ramos, A. J. Daley, and P. Zoller, “Quantum optics of chiral spin networks,” Phys. Rev. A 91, 042116 (2015).
[Crossref]

L. Yuan, S. Xu, and S. Fan, “Achieving nonreciprocal unidirectional single-photon quantum transport using the photonic AharonovĺCBohm effect,” Opt. Lett. 40(22), 5140–5143 (2015).
[Crossref] [PubMed]

J. Lu, Z. H. Wang, and L. Zhou, “T-shaped single-photon router,” Opt. Express 23(18), 22955–22962 (2015).
[Crossref] [PubMed]

C. Gonzalez-Ballestero, A. Gonzalez-Tudela, F. J. Garcia-Vidal, and E. Moreno, “Chiral route to spontaneous entanglement generation,” Phys. Rev. B 92, 155304 (2015).
[Crossref]

S. A. H. Gangaraj, A. Nemilentsau, G. W. Hanson, and S. Hughes, “Transient and steady-state entanglement mediated by three-dimensional plasmonic waveguides,” Opt. Express 23(17), 22330–22346 (2015).
[Crossref] [PubMed]

Z. Liao, X. Zeng, S.-Y. Zhu, and M. S. Zubairy, “Single photon transport through an atomic chain coupled to a one-dimensional nanophotonic waveguide,” Phys. Rev. A 92, 023806 (2015).
[Crossref]

J. Borregaard, P. Kómár, E. M. Kessler, M. D. Lukin, and A. S. Sørensen, “Long-distance entanglement distribution using individual atoms in optical cavities,” Phys. Rev. A 92, 012307 (2015).
[Crossref]

A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115, 063601 (2015).
[Crossref] [PubMed]

C. H. Yan and L. F. Wei, “Single photon transport along a one-dimensional waveguide with a side manipulated cavity QED system,” Opt. Express 23(8), 10374–10384 (2015).
[Crossref] [PubMed]

2014 (6)

A. Goban, C. L. Hung, S. P. Yu, J.D. Hood, J.A. Muniz, J.H. Lee, M.J. Martin, A.C. McClung, K.S. Choi, D.E. Chang, O. Painter, and H.J. Kimble, “AtomĺClight interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

L. T. Shen, R. X. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Preparation of two-qubit steady entanglement through driving a single qubit,” Opt. Lett. 39(20), 6046–6049 (2014).
[Crossref] [PubMed]

W. B. Yan and H. Fan, “Single-photon quantum router with multiple output ports,” Sci. Rep. 4, 4820 (2014).
[Crossref] [PubMed]

T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
[Crossref] [PubMed]

A. Zheng, X. Y. Lü, and J. Liu, “Single-photon frequency conversion for generation of entanglement via constructive interference in Sagnac interferometers,” J. Phys. B: At. Mol. Opt. Phys. 47, 055501 (2014).
[Crossref]

C. Gonzalez-Ballestero, E. Moreno, and F. J. Garcia-Vidal, “Generation, manipulation, and detection of two-qubit entanglement in waveguide QED,” Phys. Rev. A 89, 042328 (2014).
[Crossref]

2013 (9)

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D Matsukevich, and C. Kurtsiefer, “Narrow band source of transform-limited photon pairs via four-wave mixing in a cold atomic ensemble,” Phys. Rev. Lett. 111, 123602 (2013)
[Crossref] [PubMed]

A. González-Tudela and D. Porras, “Mesoscopic entanglement induced by spontaneous emission in solid-state quantum optics,” Phys. Rev. Lett. 110, 080502 (2013).
[Crossref] [PubMed]

H. Zheng and H. U. Baranger, “Persistent quantum beats and long-distance entanglement from waveguide-mediated interactions,” Phys. Rev. Lett. 110, 113601 (2013).
[Crossref] [PubMed]

W. B. Yan, J. F. Huang, and H. Fan, “Tunable single-photon frequency conversion in a Sagnac interferometer,” Sci. Rep. 3, 3555 (2013).
[Crossref] [PubMed]

C. Gonzalez-Ballestero, F. J. García-Vidal, and E. Moreno, “Non-Markovian effects in waveguide-mediated entanglement,” New J. Phys. 15, 073015 (2013).
[Crossref]

J. Yang, G. W. Lin, Y. P. Niu, and S. Q. Gong, “Quantum entangling gates using the strong coupling between two optical emitters and nanowire surface plasmons,” Opt. Express 21(13), 15618–15626 (2013).
[Crossref] [PubMed]

X. R. Jin, L. Sun, X. Yang, and J. Gao, “Quantum entanglement in plasmonic waveguides with near-zero mode indices,” Opt. Lett. 38(20), 4078–4081 (2013).
[Crossref] [PubMed]

I. C. Hoi, C. M. Wilson, G. Johansson, J. Lindkvist, B. Peropadre, T. Palomaki, and P. Delsing, “Microwave quantum optics with an artificial atom in one-dimensional open space,” New J. Phys. 15, 025011 (2013).
[Crossref]

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-Mediated Interactions Between Distant Artificial Atoms,” Science 342, 1494–1496 (2013).
[Crossref] [PubMed]

2012 (5)

I. C. Hoi, T. Palomaki, J. Lindkvist, G. Johansson, P. Delsing, and C. M. Wilson, “Generation of Nonclassical Microwave States Using an Artificial Atom in 1D Open Space,” Phys. Rev. Lett. 108, 263601 (2012).
[Crossref] [PubMed]

N. P. de Leon, B. J. shields, C. L. Yu, D. E. Englund, A. V. Akimove, M. D. Lukin, and H. Park, “Tailoring Light-Matter Interaction with a Nanoscale Plasmon Resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

M. Bradford, K. C. Obi, and J. T. Shen, “Efficient Single-Photon Frequency Conversion Using a Sagnac Interferometer,” Phys. Rev. Lett. 108, 103902 (2012).
[Crossref] [PubMed]

M. Bradford and J.T. Shen, “Single-photon frequency conversion by exploiting quantum interference,” Phys. Rev. A 85, 043814 (2012).
[Crossref]

K. Stannigel, P. Rabl, and P. Zoller, “Driven-dissipative preparation of entangled states in cascaded quantum-optical networks,” New J. Phys. 14, 063014 (2012).
[Crossref]

2011 (3)

J. Li and R. Yu, “Single-plasmon scattering grating using nanowire surface plasmon coupled to nanodiamond nitrogen-vacancy center,” Opt. Express 19(21), 20991–21002 (2011).
[Crossref] [PubMed]

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of Two Qubits Mediated by One-Dimensional Plasmonic Waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[Crossref] [PubMed]

D. Martín-Cano, A. González-Tudela, L. Martín-Moreno, and F. J. García-Vidal, “Dissipation-driven generation of two-qubit entanglement mediated by plasmonic waveguides,” Phys. Rev. B 84, 235306 (2011).
[Crossref]

2010 (4)

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327, 840–843 (2010).
[Crossref] [PubMed]

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of Two Individual Neutral Atoms Using Rydberg Blockade,” Phys. Rev. Lett. 104, 010502 (2010).
[Crossref] [PubMed]

W. Chen, G. Y. Chen, and Y. N. Chen, “Coherent transport of nanowire surface plasmons coupled to quantum dots,” Opt. Express 18(10), 10360–10368 (2010).
[Crossref] [PubMed]

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82, 075427 (2010).
[Crossref]

2009 (1)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

2008 (1)

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008).
[Crossref] [PubMed]

2007 (2)

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450, 402–406 (2007).
[Crossref] [PubMed]

J. T. Shen and S. Fan, “Strongly correlated multiparticle transport in one dimension through a quantum impurity,” Phys. Rev. A 76, 062709 (2007).
[Crossref]

2006 (1)

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, “Single-photon Sagnac interferometer,” J. Phys. B: At. Mol. Opt. Phys. 39, 1011–1016 (2006).
[Crossref]

Abdumalikov, A. A.

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327, 840–843 (2010).
[Crossref] [PubMed]

Akimov, A. V.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450, 402–406 (2007).
[Crossref] [PubMed]

Akimove, A. V.

N. P. de Leon, B. J. shields, C. L. Yu, D. E. Englund, A. V. Akimove, M. D. Lukin, and H. Park, “Tailoring Light-Matter Interaction with a Nanoscale Plasmon Resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

Alibart, O.

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, “Single-photon Sagnac interferometer,” J. Phys. B: At. Mol. Opt. Phys. 39, 1011–1016 (2006).
[Crossref]

Appel, J.

H. L. Sørensen, J. B. Béguin, K. W. Kluge, I. Iakoupov, A. S. Sørensen, J. H. Müller, E. S. Polzik, and J. Appel, “Coherent Backscattering of Light Off One-Dimensional Atomic Strings,” Phys. Rev. Lett. 117, 133604 (2016).
[Crossref] [PubMed]

Astafiev, O.

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327, 840–843 (2010).
[Crossref] [PubMed]

Baldi, P.

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, “Single-photon Sagnac interferometer,” J. Phys. B: At. Mol. Opt. Phys. 39, 1011–1016 (2006).
[Crossref]

Baranger, H. U.

H. Zheng and H. U. Baranger, “Persistent quantum beats and long-distance entanglement from waveguide-mediated interactions,” Phys. Rev. Lett. 110, 113601 (2013).
[Crossref] [PubMed]

Béguin, J. B.

H. L. Sørensen, J. B. Béguin, K. W. Kluge, I. Iakoupov, A. S. Sørensen, J. H. Müller, E. S. Polzik, and J. Appel, “Coherent Backscattering of Light Off One-Dimensional Atomic Strings,” Phys. Rev. Lett. 117, 133604 (2016).
[Crossref] [PubMed]

Bertocchi, G.

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, “Single-photon Sagnac interferometer,” J. Phys. B: At. Mol. Opt. Phys. 39, 1011–1016 (2006).
[Crossref]

Blais, A.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-Mediated Interactions Between Distant Artificial Atoms,” Science 342, 1494–1496 (2013).
[Crossref] [PubMed]

Blatt, R.

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008).
[Crossref] [PubMed]

Borregaard, J.

J. Borregaard, P. Kómár, E. M. Kessler, M. D. Lukin, and A. S. Sørensen, “Long-distance entanglement distribution using individual atoms in optical cavities,” Phys. Rev. A 92, 012307 (2015).
[Crossref]

Bradford, M.

M. Bradford, K. C. Obi, and J. T. Shen, “Efficient Single-Photon Frequency Conversion Using a Sagnac Interferometer,” Phys. Rev. Lett. 108, 103902 (2012).
[Crossref] [PubMed]

M. Bradford and J.T. Shen, “Single-photon frequency conversion by exploiting quantum interference,” Phys. Rev. A 85, 043814 (2012).
[Crossref]

Browaeys, A.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of Two Individual Neutral Atoms Using Rydberg Blockade,” Phys. Rev. Lett. 104, 010502 (2010).
[Crossref] [PubMed]

Chandra, A.

N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. V. Kupriyanov, and J. Laurat, “Large Bragg Reflection from One-Dimensional Chains of Trapped Atoms Near a Nanoscale Waveguide,” Phys. Rev. Lett. 117, 133603 (2016).
[Crossref] [PubMed]

Chang, D. E.

A. González-Tudela, V. Paulisch, D. E. Chang, H. J. Kimble, and J. I. Cirac, “Deterministic generation of arbitrary photonic states assisted by dissipation,” Phys. Rev. Lett. 115, 163603 (2015).
[Crossref] [PubMed]

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450, 402–406 (2007).
[Crossref] [PubMed]

Chang, D.E.

A. Goban, C. L. Hung, S. P. Yu, J.D. Hood, J.A. Muniz, J.H. Lee, M.J. Martin, A.C. McClung, K.S. Choi, D.E. Chang, O. Painter, and H.J. Kimble, “AtomĺClight interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

Chen, G. Y.

Chen, R. X.

Chen, W.

Chen, Y. N.

Chen, Z.

Cheng, M. T.

Chng, B.

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D Matsukevich, and C. Kurtsiefer, “Narrow band source of transform-limited photon pairs via four-wave mixing in a cold atomic ensemble,” Phys. Rev. Lett. 111, 123602 (2013)
[Crossref] [PubMed]

Choi, K.S.

A. Goban, C. L. Hung, S. P. Yu, J.D. Hood, J.A. Muniz, J.H. Lee, M.J. Martin, A.C. McClung, K.S. Choi, D.E. Chang, O. Painter, and H.J. Kimble, “AtomĺClight interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

Christiansen, S.

Cirac, J. I.

A. González-Tudela, V. Paulisch, D. E. Chang, H. J. Kimble, and J. I. Cirac, “Deterministic generation of arbitrary photonic states assisted by dissipation,” Phys. Rev. Lett. 115, 163603 (2015).
[Crossref] [PubMed]

A. González-Tudela, V. Paulisch, H. J. Kimble, and J. I. Cirac, “Reliable multiphoton generation in waveguide QED,” https://arxiv.org/abs/1603.01243 (2016).

Corzo, N. V.

N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. V. Kupriyanov, and J. Laurat, “Large Bragg Reflection from One-Dimensional Chains of Trapped Atoms Near a Nanoscale Waveguide,” Phys. Rev. Lett. 117, 133603 (2016).
[Crossref] [PubMed]

Daley, A. J.

H. Pichler, T. Ramos, A. J. Daley, and P. Zoller, “Quantum optics of chiral spin networks,” Phys. Rev. A 91, 042116 (2015).
[Crossref]

T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
[Crossref] [PubMed]

de Leon, N. P.

N. P. de Leon, B. J. shields, C. L. Yu, D. E. Englund, A. V. Akimove, M. D. Lukin, and H. Park, “Tailoring Light-Matter Interaction with a Nanoscale Plasmon Resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

de Riedmatten, H.

D. Rieländer, A. Lenhard, M. Mazzera, and H. de Riedmatten, “Cavity enhanced telecom heralded single photons for spin-wave solid state quantum memories,” New J. Phys. 18, 123013 (2016).
[Crossref]

Delsing, P.

I. C. Hoi, C. M. Wilson, G. Johansson, J. Lindkvist, B. Peropadre, T. Palomaki, and P. Delsing, “Microwave quantum optics with an artificial atom in one-dimensional open space,” New J. Phys. 15, 025011 (2013).
[Crossref]

I. C. Hoi, T. Palomaki, J. Lindkvist, G. Johansson, P. Delsing, and C. M. Wilson, “Generation of Nonclassical Microwave States Using an Artificial Atom in 1D Open Space,” Phys. Rev. Lett. 108, 263601 (2012).
[Crossref] [PubMed]

Dzsotjan, D.

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82, 075427 (2010).
[Crossref]

Englund, D. E.

N. P. de Leon, B. J. shields, C. L. Yu, D. E. Englund, A. V. Akimove, M. D. Lukin, and H. Park, “Tailoring Light-Matter Interaction with a Nanoscale Plasmon Resonator,” Phys. Rev. Lett. 108, 226803 (2012).
[Crossref] [PubMed]

Evellin, C.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of Two Individual Neutral Atoms Using Rydberg Blockade,” Phys. Rev. Lett. 104, 010502 (2010).
[Crossref] [PubMed]

Facchi, P.

P. Facchi, M. S. Kim, S. Pascazio, F. V. Pepe, D. Pomarico, and T. Tufarelli, “Bound states and entanglement generation in waveguide quantum electrodynamics,” Phys. Rev. A 94, 043839 (2016).
[Crossref]

Fan, H.

W. B. Yan and H. Fan, “Single-photon quantum router with multiple output ports,” Sci. Rep. 4, 4820 (2014).
[Crossref] [PubMed]

W. B. Yan, J. F. Huang, and H. Fan, “Tunable single-photon frequency conversion in a Sagnac interferometer,” Sci. Rep. 3, 3555 (2013).
[Crossref] [PubMed]

Fan, S.

L. Yuan, S. Xu, and S. Fan, “Achieving nonreciprocal unidirectional single-photon quantum transport using the photonic AharonovĺCBohm effect,” Opt. Lett. 40(22), 5140–5143 (2015).
[Crossref] [PubMed]

J. T. Shen and S. Fan, “Strongly correlated multiparticle transport in one dimension through a quantum impurity,” Phys. Rev. A 76, 062709 (2007).
[Crossref]

Fedorov, A.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-Mediated Interactions Between Distant Artificial Atoms,” Science 342, 1494–1496 (2013).
[Crossref] [PubMed]

Fleischhauer, M.

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82, 075427 (2010).
[Crossref]

Förtsch, M.

Gaëtan, A.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of Two Individual Neutral Atoms Using Rydberg Blockade,” Phys. Rev. Lett. 104, 010502 (2010).
[Crossref] [PubMed]

Gangaraj, S. A. H.

Gao, J.

Garcia-Vidal, F. J.

C. Gonzalez-Ballestero, A. Gonzalez-Tudela, F. J. Garcia-Vidal, and E. Moreno, “Chiral route to spontaneous entanglement generation,” Phys. Rev. B 92, 155304 (2015).
[Crossref]

C. Gonzalez-Ballestero, E. Moreno, and F. J. Garcia-Vidal, “Generation, manipulation, and detection of two-qubit entanglement in waveguide QED,” Phys. Rev. A 89, 042328 (2014).
[Crossref]

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of Two Qubits Mediated by One-Dimensional Plasmonic Waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[Crossref] [PubMed]

García-Vidal, F. J.

C. Gonzalez-Ballestero, F. J. García-Vidal, and E. Moreno, “Non-Markovian effects in waveguide-mediated entanglement,” New J. Phys. 15, 073015 (2013).
[Crossref]

D. Martín-Cano, A. González-Tudela, L. Martín-Moreno, and F. J. García-Vidal, “Dissipation-driven generation of two-qubit entanglement mediated by plasmonic waveguides,” Phys. Rev. B 84, 235306 (2011).
[Crossref]

Goban, A.

N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. V. Kupriyanov, and J. Laurat, “Large Bragg Reflection from One-Dimensional Chains of Trapped Atoms Near a Nanoscale Waveguide,” Phys. Rev. Lett. 117, 133603 (2016).
[Crossref] [PubMed]

A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115, 063601 (2015).
[Crossref] [PubMed]

A. Goban, C. L. Hung, S. P. Yu, J.D. Hood, J.A. Muniz, J.H. Lee, M.J. Martin, A.C. McClung, K.S. Choi, D.E. Chang, O. Painter, and H.J. Kimble, “AtomĺClight interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

Göbelt, M.

Gong, S. Q.

Gonzalez-Ballestero, C.

C. Gonzalez-Ballestero, A. Gonzalez-Tudela, F. J. Garcia-Vidal, and E. Moreno, “Chiral route to spontaneous entanglement generation,” Phys. Rev. B 92, 155304 (2015).
[Crossref]

C. Gonzalez-Ballestero, E. Moreno, and F. J. Garcia-Vidal, “Generation, manipulation, and detection of two-qubit entanglement in waveguide QED,” Phys. Rev. A 89, 042328 (2014).
[Crossref]

C. Gonzalez-Ballestero, F. J. García-Vidal, and E. Moreno, “Non-Markovian effects in waveguide-mediated entanglement,” New J. Phys. 15, 073015 (2013).
[Crossref]

Gonzalez-Tudela, A.

C. Gonzalez-Ballestero, A. Gonzalez-Tudela, F. J. Garcia-Vidal, and E. Moreno, “Chiral route to spontaneous entanglement generation,” Phys. Rev. B 92, 155304 (2015).
[Crossref]

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of Two Qubits Mediated by One-Dimensional Plasmonic Waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[Crossref] [PubMed]

González-Tudela, A.

V. Paulisch, H. J. Kimble, and A. González-Tudela, “Universal quantum computation in waveguide QED using decoherence free subspaces,” New J. Phys. 18, 043041 (2016).
[Crossref]

A. González-Tudela, V. Paulisch, D. E. Chang, H. J. Kimble, and J. I. Cirac, “Deterministic generation of arbitrary photonic states assisted by dissipation,” Phys. Rev. Lett. 115, 163603 (2015).
[Crossref] [PubMed]

A. González-Tudela and D. Porras, “Mesoscopic entanglement induced by spontaneous emission in solid-state quantum optics,” Phys. Rev. Lett. 110, 080502 (2013).
[Crossref] [PubMed]

D. Martín-Cano, A. González-Tudela, L. Martín-Moreno, and F. J. García-Vidal, “Dissipation-driven generation of two-qubit entanglement mediated by plasmonic waveguides,” Phys. Rev. B 84, 235306 (2011).
[Crossref]

A. González-Tudela, V. Paulisch, H. J. Kimble, and J. I. Cirac, “Reliable multiphoton generation in waveguide QED,” https://arxiv.org/abs/1603.01243 (2016).

Gouraud, B.

N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. V. Kupriyanov, and J. Laurat, “Large Bragg Reflection from One-Dimensional Chains of Trapped Atoms Near a Nanoscale Waveguide,” Phys. Rev. Lett. 117, 133603 (2016).
[Crossref] [PubMed]

Grangier, P.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of Two Individual Neutral Atoms Using Rydberg Blockade,” Phys. Rev. Lett. 104, 010502 (2010).
[Crossref] [PubMed]

Gulati, G. K.

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D Matsukevich, and C. Kurtsiefer, “Narrow band source of transform-limited photon pairs via four-wave mixing in a cold atomic ensemble,” Phys. Rev. Lett. 111, 123602 (2013)
[Crossref] [PubMed]

Hanson, G. W.

Hemmer, P. R.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450, 402–406 (2007).
[Crossref] [PubMed]

Hoi, I. C.

I. C. Hoi, C. M. Wilson, G. Johansson, J. Lindkvist, B. Peropadre, T. Palomaki, and P. Delsing, “Microwave quantum optics with an artificial atom in one-dimensional open space,” New J. Phys. 15, 025011 (2013).
[Crossref]

I. C. Hoi, T. Palomaki, J. Lindkvist, G. Johansson, P. Delsing, and C. M. Wilson, “Generation of Nonclassical Microwave States Using an Artificial Atom in 1D Open Space,” Phys. Rev. Lett. 108, 263601 (2012).
[Crossref] [PubMed]

Hood, J. D.

A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115, 063601 (2015).
[Crossref] [PubMed]

Hood, J.D.

A. Goban, C. L. Hung, S. P. Yu, J.D. Hood, J.A. Muniz, J.H. Lee, M.J. Martin, A.C. McClung, K.S. Choi, D.E. Chang, O. Painter, and H.J. Kimble, “AtomĺClight interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

Horodecki, K.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

Horodecki, M.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

Horodecki, P.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

Horodecki, R.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

Huang, J. F.

W. B. Yan, J. F. Huang, and H. Fan, “Tunable single-photon frequency conversion in a Sagnac interferometer,” Sci. Rep. 3, 3555 (2013).
[Crossref] [PubMed]

Hughes, S.

Hung, C. L.

A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115, 063601 (2015).
[Crossref] [PubMed]

A. Goban, C. L. Hung, S. P. Yu, J.D. Hood, J.A. Muniz, J.H. Lee, M.J. Martin, A.C. McClung, K.S. Choi, D.E. Chang, O. Painter, and H.J. Kimble, “AtomĺClight interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

Iakoupov, I.

H. L. Sørensen, J. B. Béguin, K. W. Kluge, I. Iakoupov, A. S. Sørensen, J. H. Müller, E. S. Polzik, and J. Appel, “Coherent Backscattering of Light Off One-Dimensional Atomic Strings,” Phys. Rev. Lett. 117, 133604 (2016).
[Crossref] [PubMed]

Inomata, K.

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327, 840–843 (2010).
[Crossref] [PubMed]

Jin, X. R.

Johansson, G.

I. C. Hoi, C. M. Wilson, G. Johansson, J. Lindkvist, B. Peropadre, T. Palomaki, and P. Delsing, “Microwave quantum optics with an artificial atom in one-dimensional open space,” New J. Phys. 15, 025011 (2013).
[Crossref]

I. C. Hoi, T. Palomaki, J. Lindkvist, G. Johansson, P. Delsing, and C. M. Wilson, “Generation of Nonclassical Microwave States Using an Artificial Atom in 1D Open Space,” Phys. Rev. Lett. 108, 263601 (2012).
[Crossref] [PubMed]

Kessler, E. M.

J. Borregaard, P. Kómár, E. M. Kessler, M. D. Lukin, and A. S. Sørensen, “Long-distance entanglement distribution using individual atoms in optical cavities,” Phys. Rev. A 92, 012307 (2015).
[Crossref]

Kim, M. S.

P. Facchi, M. S. Kim, S. Pascazio, F. V. Pepe, D. Pomarico, and T. Tufarelli, “Bound states and entanglement generation in waveguide quantum electrodynamics,” Phys. Rev. A 94, 043839 (2016).
[Crossref]

Kimble, H. J.

V. Paulisch, H. J. Kimble, and A. González-Tudela, “Universal quantum computation in waveguide QED using decoherence free subspaces,” New J. Phys. 18, 043041 (2016).
[Crossref]

A. González-Tudela, V. Paulisch, D. E. Chang, H. J. Kimble, and J. I. Cirac, “Deterministic generation of arbitrary photonic states assisted by dissipation,” Phys. Rev. Lett. 115, 163603 (2015).
[Crossref] [PubMed]

A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115, 063601 (2015).
[Crossref] [PubMed]

A. González-Tudela, V. Paulisch, H. J. Kimble, and J. I. Cirac, “Reliable multiphoton generation in waveguide QED,” https://arxiv.org/abs/1603.01243 (2016).

Kimble, H.J.

A. Goban, C. L. Hung, S. P. Yu, J.D. Hood, J.A. Muniz, J.H. Lee, M.J. Martin, A.C. McClung, K.S. Choi, D.E. Chang, O. Painter, and H.J. Kimble, “AtomĺClight interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

Kluge, K. W.

H. L. Sørensen, J. B. Béguin, K. W. Kluge, I. Iakoupov, A. S. Sørensen, J. H. Müller, E. S. Polzik, and J. Appel, “Coherent Backscattering of Light Off One-Dimensional Atomic Strings,” Phys. Rev. Lett. 117, 133604 (2016).
[Crossref] [PubMed]

Kómár, P.

J. Borregaard, P. Kómár, E. M. Kessler, M. D. Lukin, and A. S. Sørensen, “Long-distance entanglement distribution using individual atoms in optical cavities,” Phys. Rev. A 92, 012307 (2015).
[Crossref]

Kupriyanov, D. V.

N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. V. Kupriyanov, and J. Laurat, “Large Bragg Reflection from One-Dimensional Chains of Trapped Atoms Near a Nanoscale Waveguide,” Phys. Rev. Lett. 117, 133603 (2016).
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H. Pichler, T. Ramos, A. J. Daley, and P. Zoller, “Quantum optics of chiral spin networks,” Phys. Rev. A 91, 042116 (2015).
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Z. Liao, H. Nha, and M. S. Zubairy, “Single-photon frequency-comb generation in a one-dimensional waveguide coupled to two atomic arrays,” Phys. Rev. A 93, 033851 (2016).
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Z. Liao, X. Zeng, S.-Y. Zhu, and M. S. Zubairy, “Single photon transport through an atomic chain coupled to a one-dimensional nanophotonic waveguide,” Phys. Rev. A 92, 023806 (2015).
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Nat. Commun. (1)

A. Goban, C. L. Hung, S. P. Yu, J.D. Hood, J.A. Muniz, J.H. Lee, M.J. Martin, A.C. McClung, K.S. Choi, D.E. Chang, O. Painter, and H.J. Kimble, “AtomĺClight interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
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K. Stannigel, P. Rabl, and P. Zoller, “Driven-dissipative preparation of entangled states in cascaded quantum-optical networks,” New J. Phys. 14, 063014 (2012).
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Opt. Express (7)

Opt. Lett. (4)

Optica (1)

Phys. Rev. A (11)

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I. M. Mirza and J. C. Schotland, “Multiqubit entanglement in bidirectional-chiral-waveguide QED,” Phys. Rev. A 94, 012302 (2016).
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H. Pichler, T. Ramos, A. J. Daley, and P. Zoller, “Quantum optics of chiral spin networks,” Phys. Rev. A 91, 042116 (2015).
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Z. Liao, H. Nha, and M. S. Zubairy, “Single-photon frequency-comb generation in a one-dimensional waveguide coupled to two atomic arrays,” Phys. Rev. A 93, 033851 (2016).
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C. Gonzalez-Ballestero, A. Gonzalez-Tudela, F. J. Garcia-Vidal, and E. Moreno, “Chiral route to spontaneous entanglement generation,” Phys. Rev. B 92, 155304 (2015).
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H. Zheng and H. U. Baranger, “Persistent quantum beats and long-distance entanglement from waveguide-mediated interactions,” Phys. Rev. Lett. 110, 113601 (2013).
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T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
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A. González-Tudela and D. Porras, “Mesoscopic entanglement induced by spontaneous emission in solid-state quantum optics,” Phys. Rev. Lett. 110, 080502 (2013).
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N. P. de Leon, B. J. shields, C. L. Yu, D. E. Englund, A. V. Akimove, M. D. Lukin, and H. Park, “Tailoring Light-Matter Interaction with a Nanoscale Plasmon Resonator,” Phys. Rev. Lett. 108, 226803 (2012).
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A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115, 063601 (2015).
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Figures (8)

Fig. 1
Fig. 1 Schematic of the system. Two emitters A and B are coupled to a Sagmac interferometer. An external routing element is used to distinguish the input and output photons.
Fig. 2
Fig. 2 Energy levels of the Λ-type quantum emitters A and B. The coupling strengths corresponding to the transition paths are V 1 A, V 2 A, V 1 B, and V 2 B, respectively.
Fig. 3
Fig. 3 Generation road of the state |ϕ21s〉 in the even subspace. With the conditions shown in the figure, the system evolves from initial state, through state |ϕ31s〉, to the target state, in which the emitters are in symmetric state |ϕ21s〉.
Fig. 4
Fig. 4 Generation road of the state |ϕ21a〉 in the even subspace. With the conditions shown in the figure, the system evolves from initial state, through state |ϕ31s〉, to the target state, in which the emitters are in antisymmetric state |ϕ21a〉.
Fig. 5
Fig. 5 The evolution to the state |ϕ21s〉 in the odd subspace. With the conditions shown in the figure, the system evolves from initial state, through state |ϕ31a〉, to the target state, in which the emitters are in symmetric state |ϕ21s〉.
Fig. 6
Fig. 6 The evolution to the state |ϕ21a〉 in the odd subspace. With the conditions shown in the figure, the system evolves from initial state, through state |ϕ31a〉, to the target state, in which the emitters are in symmetric state |ϕ21a〉.
Fig. 7
Fig. 7 Success probability p as a function of the parameter Γτ, for different values of α (α = Γ12) with γ = 0.
Fig. 8
Fig. 8 Success probability p as a function of the Purcell factor P, for different values of α. The Intensity FWHM of the input photon is τ = 8 Γ−1.

Equations (76)

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S bs in = 1 2 ( 1 1 1 1 ) .
S loop = ( 1 0 0 1 ) .
S bs out = 1 2 ( 1 1 1 1 ) .
H = H wg + H em + H int .
H wg = d x c cw ( x ) ( i c x ) c cw ( x ) + d x c ccw ( x ) ( i c x ) c ccw ( x ) ,
H em = α = A , B [ ω 1 σ 11 α + ω 2 σ 22 α + ( ω 3 i γ / 2 ) σ 33 α ] ,
H int = d x δ ( x x A ) V 1 A { [ c cw ( x ) + c ccw ( x ) ] σ 13 A + σ 31 A [ c cw ( x ) + c ccw ( x ) ] } + d x δ ( x x A ) V 2 A { [ c cw ( x ) + c ccw ( x ) ] σ 23 A + σ 32 A [ c cw ( x ) + c ccw ( x ) ] } + d x δ ( x x B ) V 1 B { [ c cw ( x ) + c ccw ( x ) ] σ 13 B + σ 31 B [ c cw ( x ) + c ccw ( x ) ] } + d x δ ( x x B ) V 2 B { [ c cw ( x ) + c ccw ( x ) ] σ 23 B + σ 32 B [ c cw ( x ) + c ccw ( x ) ] } ,
c e ( x ) = 1 2 [ c cw ( x ) + c ccw ( x ) ] ,
c o ( x ) = 1 2 [ c cw ( x ) c ccw ( x ) ] .
| g = | 1 A , 1 B ,
| ϕ 31 s = 1 2 ( | 3 A , 1 B + | 1 A , 3 B ) ,
| ϕ 31 a = 1 2 ( | 3 A , 1 B | 1 A , 3 B ) ,
| ϕ 21 s = 1 2 ( | 2 A , 1 B + | 1 A , 2 B ) ,
| ϕ 21 a = 1 2 ( | 2 A , 1 B | 1 A , 2 B ) .
H = H wg e + H wg o + H em + H int ,
H wg e = d x c e ( x ) ( i c x ) c e ( x ) ,
H wg o = d x c o ( x ) ( i c x ) c o ( x ) ,
H em = 2 ω 1 | g g | + ( ω 1 + ω 2 ) | ϕ 21 s ϕ 21 s | + ( ω 1 + ω 2 ) | ϕ 21 a ϕ 21 a | + [ ω 1 + ( ω 3 i γ / 2 ) ] | ϕ 31 s ϕ 31 s | + [ ω 1 + ( ω 3 i γ / 2 ) ] | ϕ 31 a ϕ 31 a | ,
H int = V 1 d x [ δ ( x + d / 2 ) + δ ( x d / 2 ) ] { c e ( x ) σ 13 s + σ 31 s c e ( x ) } + V 1 d x [ δ ( x + d / 2 ) δ ( x d / 2 ) ] { c o ( x ) σ 13 a + σ 31 a c o ( x ) } + V 2 2 d x [ δ ( x + d / 2 ) + δ ( x d / 2 ) ] { c e ( x ) σ 23 s + σ 32 s c e ( x ) } + V 2 2 d x [ δ ( x + d / 2 ) δ ( x d / 2 ) ] { c o ( x ) σ 23 a + σ 32 a c o ( x ) } .
σ 31 s = 1 2 ( σ 31 A + σ 31 B ) = | ϕ 31 s g | ,
σ 31 a = 1 2 ( σ 31 A σ 31 B ) = | ϕ 31 a g | ,
σ 23 s = σ 23 A + σ 23 B = | ϕ 21 s ϕ 31 s | + | ϕ 21 a ϕ 31 a | ,
σ 23 a = σ 23 A σ 23 B = | ϕ 21 s ϕ 31 s | + | ϕ 21 a ϕ 31 a | ,
| Ψ k e = d x f 11 e ( k , x ) c e ( x ) | , g + f 31 s | , ϕ 31 s + d x f 21 s e ( k 2 , x ) c e ( x ) | , ϕ 21 s + d x f 21 a o ( k 2 , x ) c o ( x ) | , ϕ 21 a ,
| Ψ k o = d x f 11 o ( k , x ) c o ( x ) | , g + f 31 a | , ϕ 31 a + d x f 21 s o ( k 2 , x ) c o ( x ) | , ϕ 21 s + d x f 21 a e ( k 2 , x ) c e ( x ) | , ϕ 21 a ,
f 11 e ( k , x ) = 1 2 π e i k x [ θ ( d / 2 x ) + t 11 A B e θ ( x + d / 2 ) θ ( d / 2 x ) + t 11 e θ ( x d / 2 ) ] ,
f 21 s e ( k 2 , x ) = 1 2 π e i k 2 x [ t 21 sAB e θ ( x + d / 2 ) θ ( d / 2 x ) + t 21 s e θ ( x d / 2 ) ] ,
f 21 a o ( k 2 , x ) = 1 2 π e i k 2 x [ t 21 aAB o θ ( x + d / 2 ) θ ( d / 2 x ) + t 21 a o θ ( x d / 2 ) ] .
f 31 s = 2 / 2 π icos ( k d / 2 ) V 1 ( e i k d + 1 ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 11 AB e = Γ 2 i ( Δ + i γ / 2 ) ( e i k d + 1 ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 11 e = ( e i k d + 1 ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ( e i k d + 1 ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 21 sAB e = 2 cos ( k d / 2 ) e i k 2 d / 2 Γ 1 Γ 2 ( e i k d + 1 ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 21 s e = 2 2 cos ( k d / 2 ) cos ( k 2 d / 2 ) Γ 1 Γ 2 ( e i k d + 1 ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 21 aAB o = 2 cos ( k d / 2 ) e i k 2 d / 2 Γ 1 Γ 2 ( e i k d + 1 ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 21 a o = 2 2 icos ( k d / 2 ) sin ( k 2 d / 2 ) Γ 1 Γ 2 ( e i k d + 1 ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) .
f 11 o ( k , x ) = 1 2 π e i k x [ θ ( d / 2 x ) + t 11 AB o θ ( x + d / 2 ) θ ( d / 2 x ) + t 11 o θ ( x d / 2 ) ] ,
f 21 s o ( k 2 , x ) = 1 2 π e i k 2 x [ t 21 sAB o θ ( x + d / 2 ) θ ( d / 2 x ) + t 21 s o θ ( x d / 2 ) ] ,
f 21 a e ( k 2 , x ) = 1 2 π e i k 2 x [ t 21 aAB e θ ( x + d / 2 ) θ ( d / 2 x ) + t 21 a e θ ( x d / 2 ) ] ,
f 31 a = 2 2 π sin ( k d / 2 ) V 1 ( 1 e i k d ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 11 AB o = Γ 2 i ( Δ + i γ / 2 ) ( 1 e i k d ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 11 o = ( 1 e i k d ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ( 1 e i k d ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 21 sAB o = 2 i sin ( k d / 2 ) e i k 2 d / 2 Γ 1 Γ 2 ( 1 e i k d ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 21 s o = 2 2 sin ( k d / 2 ) sin ( k 2 d / 2 ) Γ 1 Γ 2 ( 1 e i k d ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 21 aAB e = 2 i sin ( k d / 2 ) e i k 2 d / 2 Γ 1 Γ 2 ( 1 e i k d ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) ,
t 21 a e = 2 2 i sin ( k d / 2 ) cos ( k 2 d / 2 ) Γ 1 Γ 2 ( 1 e i k d ) Γ 1 + Γ 2 i ( Δ + i γ / 2 ) .
| Ψ k e = | Ψ k e in + 1 E H 0 + i H int | Ψ k e ,
| Ψ k e = | Ψ k e out + 1 E H 0 i H int | Ψ k e .
| Ψ k e in = d x e i k x 2 π c e ( x ) | , g ,
| Ψ k e out = d x t 11 e e i k x 2 π c e ( x ) | , g + d x t 21 s e e i k 2 x 2 π c e ( x ) | , ϕ 21 s + d x t 21 a o e i k 2 x 2 π c o ( x ) | , ϕ 21 a .
| Ψ k o = | Ψ k o in + 1 E H 0 + i H int | Ψ k o ,
| Ψ k o = | Ψ k o out + 1 E H 0 i H int | Ψ k o ,
| Ψ k o in = d x e i k x 2 π c o ( x ) | , g ,
| Ψ k o out = d x t 11 o e i k x 2 π c o ( x ) | , g + d x t 21 s o e i k 2 x 2 π c o ( x ) | , ϕ 21 s + d x t 21 a e e i k 2 x 2 π c e ( x ) | , ϕ 21 a .
ϕ in ( x , t ) = ( 4 ln 2 π c 2 τ 2 ) 1 / 4 exp { [ 2 ln 2 ( c t x ) / ( c τ ) ] 2 } exp { i k 13 ( c t x ) } .
ϕ ˜ in ( k ) = ( c 2 τ 2 4 π ln 2 ) 1 / 4 exp { [ c τ / ( 2 2 ln 2 ) ] 2 ( k k 13 ) 2 } .
p = d k | t 21 s e ( k ) ϕ ˜ in ( k ) | 2 π ln 2 4 α Γ τ ( α + 1 ) 2 ( 2 α + 1 α + 1 + 1 2 P ) exp [ 1 4 ln 2 ( 2 α + 1 α + 1 + 1 2 P ) 2 ( Γ τ ) 2 ] × erfc [ 1 4 ln 2 ( 2 α + 1 α + 1 + 1 2 P ) Γ τ ] ,
p | t 21 s e ( k 13 ) | 2 = 8 α ( α + 1 ) 2 ( 2 α + 1 α + 1 + 1 2 P ) 2 .
p 1 3 4 P .
i c x f 11 e ( k , x ) + 2 ω 1 f 11 e ( k , x ) + V 1 [ δ ( x + d / 2 ) + δ ( x d / 2 ) ] f 31 s = ( 2 ω 1 + c k ) f 11 e ( k , x ) ,
( ω 1 + ω 3 i γ / 2 ) f 31 s + V 1 [ f 11 e ( k , d / 2 ) + f 11 e ( k , d / 2 ) ] + 1 2 V 2 [ f 21 s e ( k 2 , d / 2 ) + f 21 s e ( k 2 , d / 2 ) ] + 1 2 V 2 [ f 21 a o ( k 2 , d / 2 ) f 21 a o ( k 2 , d / 2 ) ] = ( 2 ω 1 + c k ) f 31 s ,
i c x f 21 s e ( k 2 , x ) + ( ω 1 + ω 2 ) f 21 s e ( k 2 , x ) + 1 2 V 2 [ δ ( x + d / 2 ) + δ ( x d / 2 ) ] f 31 s = ( 2 ω 1 + c k ) f 21 s e ( k 2 , x ) ,
i c x f 21 a o ( k 2 , x ) + ( ω 1 + ω 2 ) f 21 a o ( k 2 , x ) + 1 2 V 2 [ δ ( x + d / 2 ) δ ( x d / 2 ) ] f 31 s = ( 2 ω 1 + c k ) f 21 a o ( k 2 , x ) .
i c 2 π e i k d / 2 ( 1 / t 11 AB e ) + V 1 f 31 s = 0 ,
i c 2 π e i k d / 2 ( t 11 AB e + t 11 e ) + V 1 f 31 s = 0 ,
V 1 2 π e i k d / 2 1 + t 11 AB e 2 + V 1 2 π e i k d / 2 t 11 AB e + t 11 e 2 + V 2 4 π e i k 2 d / 2 t 21 sAB e 2 + V 2 4 π e i k 2 d / 2 t 21 s AB e + t 21 s e 2 + V 2 4 π e i k 2 d / 2 t 21 aAB o 2 V 2 4 π e i k 2 d / 2 t 21 a AB o + t 21 a o 2 = ( Δ + i γ 2 ) f 31 s ,
i c 2 π e i k 2 d / 2 t 21 sAB e + V 2 2 f 31 s = 0 ,
i c 2 π e i k 2 d / 2 ( t 21 sAB e t 21 s e ) + V 2 2 f 31 s = 0 ,
i c 2 π e i k 2 d / 2 t 21 aAB o + V 2 2 f 31 s = 0 ,
i c 2 π e i k 2 d / 2 ( t 21 aAB o t 21 a o ) V 2 2 f 31 s = 0 .
, g | c e ( x ) | Ψ k e = , g | c e ( x ) | Ψ k e in + , g | c e ( x ) 1 E H 0 + i H int | Ψ k e .
, g | c e ( x ) | Ψ k e = f 11 e ( k , x ) = e i k x 2 π θ ( x d / 2 ) + t 11 AB e e i k x 2 π θ ( x + d / 2 ) θ ( x + d / 2 ) + t 11 e e i k x 2 π θ ( x d / 2 ) ,
, g | c e ( x ) 1 E H 0 + i | Ψ k e = e i k x 2 π θ ( x d / 2 ) + t 11 AB e e i k x 2 π θ ( x + d / 2 ) θ ( x + d / 2 ) + t 11 e e i k x 2 π θ ( x d / 2 ) ,
, g | c e ( x ) | Ψ k e in = , g | c e ( x ) | Ψ k e , g | c e ( x ) 1 E H 0 + i H int | Ψ k e = e i k x 2 π .
, g | c e ( x ) | Ψ k e out = , g | c e ( x ) | Ψ k e , g | c e ( x ) 1 E H 0 + i H int | Ψ k e = t 11 e e i k x 2 π ,
, ϕ 21 s | c e ( x ) | Ψ k e out = , ϕ 21 s | c e ( x ) | Ψ k e , ϕ 21 s | c e ( x ) 1 E H 0 i H int | Ψ k e = t 21 s e e i k 2 x 2 π ,
, ϕ 21 a | c o ( x ) | Ψ k e out = , ϕ 21 a | c o ( x ) | Ψ k e , ϕ 21 a | c o ( x ) 1 E H 0 i H int | Ψ k e = t 21 a o e i k 2 x 2 π .

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