Abstract

Activating transitions between internal states of physical systems has emerged as an appealing approach to create lattices and complex networks. In such a scheme, the internal states or modes of a physical system are regarded as lattice sites or network nodes in an abstract space whose dimensionality may exceed the systems’ apparent (geometric) dimensionality. This introduces the notion of synthetic dimensions, thus providing entirely novel pathways for fundamental research and applications. Here, we analytically show that the propagation of multiphoton states through multiport waveguide arrays gives rise to synthetic dimensions where a single waveguide system generates a multitude of synthetic lattices. Since these synthetic lattices exist in photon-number space, we introduce the concept of pseudo-energy and demonstrate its utility for studying multiphoton interference processes. Specifically, the spectrum of the associated pseudo-energy operator generates a unique ordering of the relevant states. Together with generalized pseudo-energy ladder operators, this allows for representing the dynamics of multiphoton states by way of pseudo-energy term diagrams that are associated with a synthetic atom. As a result, the pseudo-energy representation leads to concise analytical expressions for the eigensystem of N photons propagating through M nearest-neighbor coupled waveguides. In the regime where N2 and M3, nonlocal coupling in Fock space gives rise to hitherto unknown all-optical dark states that display intriguing nontrivial dynamics.

© 2020 Chinese Laser Press

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2019 (7)

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[Crossref]

T. Ozawa and H. M. Price, “Topological quantum matter in synthetic dimensions,” Nat. Rev. Phys. 1, 349–357 (2019).
[Crossref]

H. Cai, J. Liu, J. Wu, Y. He, S.-Y. Zhu, J.-X. Zhang, and D.-W. Wang, “Experimental observation of momentum-space chiral edge currents in room-temperature atoms,” Phys. Rev. Lett. 122, 023601 (2019).
[Crossref]

M. A. Quiroz-Juarez, A. Perez-Leija, K. Tschernig, B. M. Rodriguez-Lara, O. S. Magana-Loaiza, K. Busch, Y. N. Joglekar, and R. de J. Leon-Montiel, “Exceptional points of any order in a single, lossy waveguide beam splitter by photon-number-resolved detection,” Photon. Res. 7, 862–867 (2019).
[Crossref]

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

F. J. Furch, W. D. Engel, T. Witting, A. Perez-Leija, M. J. J. Vrakking, and A. Mermillod-Blondin, “Single-step fabrication of surface waveguides in fused silica with few-cycle laser pulses,” Opt. Lett. 44, 4267–4270 (2019).
[Crossref]

O. S. Magana-Loaiza, R. de J. Leon-Montiel, A. Perez-Leija, A. B. URen, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5, 80 (2019).
[Crossref]

2018 (2)

2017 (2)

H. M. Price, T. Ozawa, and N. Goldman, “Synthetic dimensions for cold atoms from shaking a harmonic trap,” Phys. Rev. A 95, 023607 (2017).
[Crossref]

H. M. Price, T. Ozawa, and N. Goldman, “Synthetic dimensions for cold atoms from shaking a harmonic trap,” Phys. Rev. A 95, 023607 (2017).
[Crossref]

2016 (9)

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]

T. Bilitewski and N. R. Cooper, “Synthetic dimensions in the strong-coupling limit: supersolids and pair superfluids,” Phys. Rev. A 94, 023630 (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]

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]

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gräfe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional Fourier transforms,” Nat. Commun. 7, 11027 (2016).
[Crossref]

K. Wang, S. Weimann, S. Nolte, A. Perez-Leija, and A. Szameit, “Measuring the Aharonov-Anandan phase in multiport photonic systems,” Opt. Lett. 41, 1889–1892 (2016).
[Crossref]

D.-W. Wang, H. Cai, R.-B. Liu, and M. O. Scully, “Mesoscopic superposition states generated by synthetic spin-orbit interaction in Fock-state lattices,” Phys. Rev. Lett. 116, 220502 (2016).
[Crossref]

M. Gräfe, R. Heilmann, M. Lebugle, D. Guzman-Silva, A. Perez-Leija, and A. Szameit, “Integrated photonic quantum walks,” J. Opt. 18, 103002 (2016).
[Crossref]

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser Photon. Rev. 9, 363–384 (2016).
[Crossref]

2015 (2)

M. Lebugle, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6, 8273 (2015).
[Crossref]

D.-W. Wang, R.-B. Liu, S.-Y. Zhu, and M. O. Scully, “Superradiance lattice,” Phys. Rev. Lett. 114, 043602 (2015).
[Crossref]

2013 (2)

D. Jukić and H. Buljan, “Four-dimensional photonic lattices and discrete tesseract solitons,” Phys. Rev. A 87, 013814 (2013).
[Crossref]

M. K. Nezhad, A. R. Bahrampour, M. Golshani, S. M. Mahdavi, and A. Langari, “Phase transition to spatial Bloch-like oscillation in squeezed photonic lattices,” Phys. Rev. A 88, 023801 (2013).
[Crossref]

2012 (4)

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock oscillator lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801–3803 (2012).
[Crossref]

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

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

2011 (2)

A. Regensburger, C. Bersch, B. Hinrichs, G. Onishchukov, A. Schreiber, C. Silberhorn, and U. Peschel, “Photon propagation in a discrete fiber network: an interplay of coherence and losses,” Phys. Rev. Lett. 107, 233902 (2011).
[Crossref]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref]

2010 (2)

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref]

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

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]

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

2007 (1)

P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of coupled-mode theory in waveguiding structures,” Phys. Rev. E 75, 016608 (2007).
[Crossref]

1999 (1)

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[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]

1990 (1)

D. Gill and E. Tadmor, “An O(N2) method for computing the eigensystem of N × N symmetric tridiagonal matrices by the divide and conquer approach,” SIAM J. Sci. Comput. 11, 161–173 (1990).
[Crossref]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref]

Abouraddy, A. F.

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock oscillator lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

Aitchison, J. S.

P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of coupled-mode theory in waveguiding structures,” Phys. Rev. E 75, 016608 (2007).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Aleahmad, P.

Auzinsh, M.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Bahrampour, A. R.

M. K. Nezhad, A. R. Bahrampour, M. Golshani, S. M. Mahdavi, and A. Langari, “Phase transition to spatial Bloch-like oscillation in squeezed photonic lattices,” Phys. Rev. A 88, 023801 (2013).
[Crossref]

Bandres, M. A.

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[Crossref]

Barnum, T. J.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[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]

Bergmann, K.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Bersch, C.

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

A. Regensburger, C. Bersch, B. Hinrichs, G. Onishchukov, A. Schreiber, C. Silberhorn, and U. Peschel, “Photon propagation in a discrete fiber network: an interplay of coherence and losses,” Phys. Rev. Lett. 107, 233902 (2011).
[Crossref]

Bilitewski, T.

T. Bilitewski and N. R. Cooper, “Synthetic dimensions in the strong-coupling limit: supersolids and pair superfluids,” Phys. Rev. A 94, 023630 (2016).
[Crossref]

Bromberg, Y.

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

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Seyfang, G.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Shadbolt, P. J.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Shi, Y.

Silberberg, Y.

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

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[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]

A. Regensburger, C. Bersch, B. Hinrichs, G. Onishchukov, A. Schreiber, C. Silberhorn, and U. Peschel, “Photon propagation in a discrete fiber network: an interplay of coherence and losses,” Phys. Rev. Lett. 107, 233902 (2011).
[Crossref]

Sipe, J. E.

P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of coupled-mode theory in waveguiding structures,” Phys. Rev. E 75, 016608 (2007).
[Crossref]

Steel, M. J.

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser Photon. Rev. 9, 363–384 (2016).
[Crossref]

Szameit, A.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[Crossref]

K. Tschernig, R. de J. León-Montiel, O. S. Magana-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]

K. Wang, S. Weimann, S. Nolte, A. Perez-Leija, and A. Szameit, “Measuring the Aharonov-Anandan phase in multiport photonic systems,” Opt. Lett. 41, 1889–1892 (2016).
[Crossref]

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser Photon. Rev. 9, 363–384 (2016).
[Crossref]

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gräfe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional Fourier transforms,” Nat. Commun. 7, 11027 (2016).
[Crossref]

M. Gräfe, R. Heilmann, M. Lebugle, D. Guzman-Silva, A. Perez-Leija, and A. Szameit, “Integrated photonic quantum walks,” J. Opt. 18, 103002 (2016).
[Crossref]

M. Lebugle, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6, 8273 (2015).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801–3803 (2012).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock oscillator lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref]

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref]

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

Tadmor, E.

D. Gill and E. Tadmor, “An O(N2) method for computing the eigensystem of N × N symmetric tridiagonal matrices by the divide and conquer approach,” SIAM J. Sci. Comput. 11, 161–173 (1990).
[Crossref]

Thompson, M. G.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Tichy, M.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gräfe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional Fourier transforms,” Nat. Commun. 7, 11027 (2016).
[Crossref]

Tschernig, K.

URen, A. B.

O. S. Magana-Loaiza, R. de J. Leon-Montiel, A. Perez-Leija, A. B. URen, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5, 80 (2019).
[Crossref]

Verde, M. R.

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Vitanov, N. V.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Vrakking, M. J. J.

Wang, D.-W.

H. Cai, J. Liu, J. Wu, Y. He, S.-Y. Zhu, J.-X. Zhang, and D.-W. Wang, “Experimental observation of momentum-space chiral edge currents in room-temperature atoms,” Phys. Rev. Lett. 122, 023601 (2019).
[Crossref]

D.-W. Wang, H. Cai, R.-B. Liu, and M. O. Scully, “Mesoscopic superposition states generated by synthetic spin-orbit interaction in Fock-state lattices,” Phys. Rev. Lett. 116, 220502 (2016).
[Crossref]

D.-W. Wang, R.-B. Liu, S.-Y. Zhu, and M. O. Scully, “Superradiance lattice,” Phys. Rev. Lett. 114, 043602 (2015).
[Crossref]

Wang, H.-L.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Wang, K.

Weidt, S.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Weihs, G.

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gräfe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional Fourier transforms,” Nat. Commun. 7, 11027 (2016).
[Crossref]

Weimann, S.

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[Crossref]

K. Wang, S. Weimann, S. Nolte, A. Perez-Leija, and A. Szameit, “Measuring the Aharonov-Anandan phase in multiport photonic systems,” Opt. Lett. 41, 1889–1892 (2016).
[Crossref]

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gräfe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional Fourier transforms,” Nat. Commun. 7, 11027 (2016).
[Crossref]

Wichmann, G.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Withford, M. J.

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser Photon. Rev. 9, 363–384 (2016).
[Crossref]

Witting, T.

Wu, J.

H. Cai, J. Liu, J. Wu, Y. He, S.-Y. Zhu, J.-X. Zhang, and D.-W. Wang, “Experimental observation of momentum-space chiral edge currents in room-temperature atoms,” Phys. Rev. Lett. 122, 023601 (2019).
[Crossref]

Xiao, M.

You, C.

O. S. Magana-Loaiza, R. de J. Leon-Montiel, A. Perez-Leija, A. B. URen, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5, 80 (2019).
[Crossref]

Yuan, L.

Zhang, J.-X.

H. Cai, J. Liu, J. Wu, Y. He, S.-Y. Zhu, J.-X. Zhang, and D.-W. Wang, “Experimental observation of momentum-space chiral edge currents in room-temperature atoms,” Phys. Rev. Lett. 122, 023601 (2019).
[Crossref]

Zhu, J.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Zhu, S.-Y.

H. Cai, J. Liu, J. Wu, Y. He, S.-Y. Zhu, J.-X. Zhang, and D.-W. Wang, “Experimental observation of momentum-space chiral edge currents in room-temperature atoms,” Phys. Rev. Lett. 122, 023601 (2019).
[Crossref]

D.-W. Wang, R.-B. Liu, S.-Y. Zhu, and M. O. Scully, “Superradiance lattice,” Phys. Rev. Lett. 114, 043602 (2015).
[Crossref]

Zhu, X.-F.

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Zilberberg, O.

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]

J. Opt. (1)

M. Gräfe, R. Heilmann, M. Lebugle, D. Guzman-Silva, A. Perez-Leija, and A. Szameit, “Integrated photonic quantum walks,” J. Opt. 18, 103002 (2016).
[Crossref]

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

J. Phys. B (2)

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

K. Bergmann, H.-C. Nägerl, C. Panda, G. Gabrielse, E. Miloglyadov, M. Quack, G. Seyfang, G. Wichmann, S. Ospelkaus, A. Kuhn, S. Longhi, A. Szameit, P. Pirro, B. Hillebrands, X.-F. Zhu, J. Zhu, M. Drewsen, W. K. Hensinger, S. Weidt, T. Halfmann, H.-L. Wang, G. S. Paraoanu, N. V. Vitanov, J. Mompart, T. Busch, T. J. Barnum, D. D. Grimes, R. W. Field, M. G. Raizen, E. Narevicius, M. Auzinsh, D. Budker, A. Pálffy, and C. H. Keitel, “Roadmap on STIRAP applications,” J. Phys. B 52, 202001 (2019).
[Crossref]

Laser Photon. Rev. (2)

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser Photon. Rev. 9, 363–384 (2016).
[Crossref]

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

Nat. Commun. (2)

S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gräfe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, “Implementation of quantum and classical discrete fractional Fourier transforms,” Nat. Commun. 7, 11027 (2016).
[Crossref]

M. Lebugle, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6, 8273 (2015).
[Crossref]

Nat. Photonics (1)

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics 6, 45–49 (2012).
[Crossref]

Nat. Rev. Phys. (1)

T. Ozawa and H. M. Price, “Topological quantum matter in synthetic dimensions,” Nat. Rev. Phys. 1, 349–357 (2019).
[Crossref]

Nature (2)

E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature 567, 356–360 (2019).
[Crossref]

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

npj Quantum Inf. (1)

O. S. Magana-Loaiza, R. de J. Leon-Montiel, A. Perez-Leija, A. B. URen, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5, 80 (2019).
[Crossref]

Opt. Lett. (5)

Optica (1)

Photon. Res. (1)

Phys. Rev. A (8)

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock oscillator lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

M. K. Nezhad, A. R. Bahrampour, M. Golshani, S. M. Mahdavi, and A. Langari, “Phase transition to spatial Bloch-like oscillation in squeezed photonic lattices,” Phys. Rev. A 88, 023801 (2013).
[Crossref]

D. Jukić and H. Buljan, “Four-dimensional photonic lattices and discrete tesseract solitons,” Phys. Rev. A 87, 013814 (2013).
[Crossref]

H. M. Price, T. Ozawa, and N. Goldman, “Synthetic dimensions for cold atoms from shaking a harmonic trap,” Phys. Rev. A 95, 023607 (2017).
[Crossref]

H. M. Price, T. Ozawa, and N. Goldman, “Synthetic dimensions for cold atoms from shaking a harmonic trap,” Phys. Rev. A 95, 023607 (2017).
[Crossref]

T. Bilitewski and N. R. Cooper, “Synthetic dimensions in the strong-coupling limit: supersolids and pair superfluids,” Phys. Rev. A 94, 023630 (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]

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

Phys. Rev. E (1)

P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of coupled-mode theory in waveguiding structures,” Phys. Rev. E 75, 016608 (2007).
[Crossref]

Phys. Rev. Lett. (9)

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

D.-W. Wang, R.-B. Liu, S.-Y. Zhu, and M. O. Scully, “Superradiance lattice,” Phys. Rev. Lett. 114, 043602 (2015).
[Crossref]

D.-W. Wang, H. Cai, R.-B. Liu, and M. O. Scully, “Mesoscopic superposition states generated by synthetic spin-orbit interaction in Fock-state lattices,” Phys. Rev. Lett. 116, 220502 (2016).
[Crossref]

H. Cai, J. Liu, J. Wu, Y. He, S.-Y. Zhu, J.-X. Zhang, and D.-W. Wang, “Experimental observation of momentum-space chiral edge currents in room-temperature atoms,” Phys. Rev. Lett. 122, 023601 (2019).
[Crossref]

A. Regensburger, C. Bersch, B. Hinrichs, G. Onishchukov, A. Schreiber, C. Silberhorn, and U. Peschel, “Photon propagation in a discrete fiber network: an interplay of coherence and losses,” Phys. Rev. Lett. 107, 233902 (2011).
[Crossref]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[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]

SIAM J. Sci. Comput. (1)

D. Gill and E. Tadmor, “An O(N2) method for computing the eigensystem of N × N symmetric tridiagonal matrices by the divide and conquer approach,” SIAM J. Sci. Comput. 11, 161–173 (1990).
[Crossref]

Other (4)

P. Lambropoulos and D. Petrosyan, Fundamentals of Quantum Optics and Quantum Information, Vol. 23 (Springer, 2007).

K. Irwin and G. Hilton, Transition-Edge Sensors (Springer, 2005), pp. 63–150.

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

S. Fan, “Photonic gauge potential and synthetic dimension with integrated photonics platforms,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2017), paper SM3O.1.

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Figures (8)

Fig. 1.
Fig. 1. 1D array of M identical nearest-neighbour evanescently coupled waveguides with coupling coefficients κm,m+1.
Fig. 2.
Fig. 2. Probability distribution |m,Nm|U^(z)|ψ(0)|2 for the initial state |ψ(0)=|5,5 propagating through a waveguide beam splitter with (a) β1=β2=1 (discrete “diffraction” in state space) and (b) β1=0 and β2=4 (“Bloch oscillations” in state space).
Fig. 3.
Fig. 3. Probability distribution |n1,n2,n3|U^(z)|ψ(0)|2 for the initial state |ψ(0)=|1,0,1 propagating through a balanced three-waveguide beam splitter (κ1=κ2=1) with (a) β1=β2=β3=0 and (b) β1=β3=0 and β2=2. At the dotted horizontal line, the state has evolved almost exactly into a two-photon NOON state in state space.
Fig. 4.
Fig. 4. Pseudo-energy term diagrams for (a) N=1 photon in M=3 coupled waveguides, (b) N=2 photons in M=2 coupled waveguides, and (c) N=2 photons in M=3 waveguides. Horizontal lines symbolize the different Fock states; vertical arrows indicate allowed transitions along with the corresponding pseudo-exchange energy.
Fig. 5.
Fig. 5. Matrix components of the effective Hamiltonian Hμν for N=2 photons propagating in M=3 identical, nearest-neighbor-coupled waveguides (β1=β2 and κ1=κ2=1).
Fig. 6.
Fig. 6. (a) 2D Fock graph for M=3 waveguides excited by N=2 indistinguishable photons. The corresponding adjacency matrix is induced by the effective Hamiltonian in Fig. 5 according to Eq. (31). (b) Sample trial implementation of the (M=3,N=2) Fock graph for a single photon and six waveguides arranged in 2D. Dotted lines indicate additional crosstalk between the waveguides, which is topologically unavoidable in this and any other real-space configuration that we have considered. Therefore, to the best of our knowledge, the synthetic coupled structure in (a) cannot be implemented in the single-photon regime.
Fig. 7.
Fig. 7. (a) Overview of several 2D and 3D embeddings of Fock graphs Aμ,ν(N,M) for M=2,,6 waveguides excited by N=1,,5 indistinguishable photons. Different node colors indicate layer-like structures that emerge for N3,M4 (all nodes in the same layer feature the same color). For readability, we have omitted the node labels as well as the graphs for M5,N4. (b) Smallest example of an isomorphic pair of planar Fock graphs with N=2,M=4 and N=3,M=3, respectively.
Fig. 8.
Fig. 8. Evolution of the probabilities |Kν|U^(z)|ψ|2 of the state |ψ as defined in Eq. (37).

Tables (1)

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Table 1. Possible Lattice Configurations for States Arising in a Waveguide Trimer Excited by Two Photons

Equations (44)

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idc0dt=f(t)c1(t),
idcmdt=ωmcm(t)+f(t)(mcm1(t)+m+1cm+1(t)).
H^=j=1M(βja^ja^j+κj,j1a^j1a^j+κj,j+1a^j+1a^j).
ida^mdz=βma^m+κm,m1a^m1+κm,m+1a^m+1,
|Ψ(0)=(a^1(0))n1(a^M(0))nMn1!nM!|0z(m=1MU1,m(z)a^m(z))n1(m=1MUM,m(z)a^m(z))nMn1!nM!|0.
|1m=|0,,1mthwaveguide,,0,
iddz|1m=βm|1m+κm,m1|1m1+κm,m+1|1m+1,
H^=β1a^1a^1+β2a^2a^2+κa^1a^2+κa^1a^2.
id|m,ndz=(β1m+β2n)|m,n+Cm|m1,n+1+Cm+1|m+1,n1,
H^=β1a^1a^1+β2a^2a^2+β3a^3a^3+κ1(a^1a^2+a^2a^1)+κ2(a^2a^3+a^3a^2).
iddz|200=2β1|200+2κ1|110,
iddz|110=(β1+β2)|110+κ2|101+2κ1(|200+|020),
iddz|020=2β2|020+2κ1|110+2κ2|011,
iddz|101=(β1+β3)|101+κ1|011+κ2|110,
iddz|011=(β2+β3)|011+κ1|101+2κ2(|002+|020),
iddz|002=2β3|002+2κ2|011.
|n1,,nM[n1..nM]N+1=n1×(N+1)0++nM×(N+1)M1.
K^(N,M)=m=1M(N+1)m1n^m,
K^(N,M)|n1,,nM=K(n1,,nM)|n1,,nM,
{[2.0.0]3,[1.1.0]3,[0.2.0]3,[1.0.1]3,[0.1.1]3,[0.0.2]3}={2,4,6,10,12,18}.
|2,0,0=|[2.0.0]3=2=|K1,|1,1,0=|[1.1.0]3=4=|K2,|0,2,0=|[0.2.0]3=6=|K3,|1,0,1=|[1.0.1]3=10=|K4,|0,1,1=|[0.1.1]3=12=|K5,|0,0,2=|[0.0.2]3=18=|K6.
|Kν=|[n1(ν),,nM(ν)]N+1=|n1(ν),,nM(ν),
nm(ν)=(Kν/(N+1)m1)#(N+1),
a^ia^j|n1,,nM=(ni+1)nj|n1,,ni+1,,nj1,,nM.
Kμ=[n1,,ni+1,,nj1,,nM]N+1=Kν+(N+1)i1(N+1)j1.
ΔKij=(N+1)i1(N+1)j1=ΔKji,
Kμ|κija^ia^j|Kν=κij(ni(ν)+1)nj(ν)δKμ,Kν+ΔKij.
|ΔKij|=|KμKν|.
iddz|Kμ=m=1Mβmnm(μ)|Kμ+ν=1NFi,j=1Mκij(ni(ν)+1)nj(ν)δKμ,Kν+ΔKij|Kν.
iddz|Kμ=Nβ|Kμ+ν=1NFi=1M1κi((ni(ν)+1)ni+1(ν)δKμ,KνΔKi+ni(ν)(ni+1(ν)+1)δKμ,Kν+ΔKi)|Kν.
Aμν(N,M)=Θ(Hμν),
Aμν(N,M)=Aμν(M1,N+1)N,M,
Hμν=(0100001011200010010012001200011201000010).
(λ1,,λ6)=(2,1,0,0,1,2),
|ϕ3=(120012012)and|ϕ5=12(110011)
|ψ=12(|ϕ3+|ϕ5)=12(112012120).
|ψ=12(|200+12|11012|10112|011).
|ϕn=m=1Mum(n)a^m|0=m=1Mum(n)|Km,
H^|ϕn=λn|ϕn,
ϕ^n=m=1Mum(n)a^m.
|n˜1,,n˜M=m=1Mϕ^mn˜m|0,
|K˜ν=m=1M(k=1Muk(m)a^k)n˜m(ν)|0.
cμ(ν)=Kμ|m=1M(k=1Muk(m)a^k)n˜m(ν)|0.
λ˜ν=m=1Mn˜m(ν)λm.