Abstract

Temporal modes (TMs) are field-orthogonal broadband wave-packet states of light occupying a common frequency band, and they can encode information in a higher-dimensional alphabet compared to, say, photon polarization. The key—yet still missing—ingredient for the full implementation of a system deploying TMs is a highly selective quantum pulse gate—a multiplexing device that can route photonic packets according to their temporal shape with high temporal-mode discrimination and high efficiency, figures of merit that together we call high selectivity. Here, we present the first implementation of a highly selective quantum pulse gate. The method is a generalization of all-optical Ramsey interferometry, so far demonstrated only for continuous-wave light fields [Phys. Rev. Lett. 117, 223601 (2016) [CrossRef]  ]. As applied to temporal modes, we refer to the method as temporal-mode interferometry.

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

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References

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    [Crossref]
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2018 (1)

S. R. Nambiar and S. K. Selvaraja, “High-efficiency broad-bandwidth subwavelength grating-based fiber-chip coupler in silicon-on-insulator,” Opt. Eng. 57, 017115 (2018).
[Crossref]

2017 (2)

2016 (3)

2015 (2)

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
[Crossref]

D. V. Reddy, M. G. Raymer, and C. J. McKinstrie, “Sorting photon wave packets using temporal-mode interferometry based on multiple-stage quantum frequency conversion,” Phys. Rev. A 91, 012323 (2015).
[Crossref]

2014 (2)

2013 (1)

2012 (3)

2011 (2)

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
[Crossref]

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011).
[Crossref]

2010 (1)

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref]

2007 (2)

E. Frumker and Y. Silberberg, “Phase and amplitude pulse shaping with two-dimensional phase-only spatial light modulators,” J. Opt. Soc. Am. B 24, 2940–2947 (2007).
[Crossref]

J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang, and D. Jaksch, “Mapping broadband single-photon wave packets into an atomic memory,” Phys. Rev. A 75, 011401 (2007).
[Crossref]

2002 (1)

2001 (1)

Z. Zheng and A. M. Weiner, “Coherent control of second harmonic generation using spectrally phase coded femtosecond waveforms,” Chem. Phys. 267, 161–171 (2001).
[Crossref]

1992 (1)

J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68, 2153–2156 (1992).
[Crossref]

1966 (1)

U. M. Titulaer and R. J. Glauber, “Density operators for coherent fields,” Phys. Rev. 145, 1041–1050 (1966).
[Crossref]

Allgaier, M.

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Ansari, V.

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Berroth, M.

Brecht, B.

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
[Crossref]

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011).
[Crossref]

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
[Crossref]

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Butschke, J.

Cargill, D. S.

Chou, M.-H.

Christ, A.

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011).
[Crossref]

Cirac, J. I.

J. I. Cirac, L. M. Duan, and P. Zöller, “Quantum optical implementation of quantum information processing,” arXiv: quant-ph/0405030 (2004).

Clemmen, S.

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

Donohue, J. M.

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Duan, L. M.

J. I. Cirac, L. M. Duan, and P. Zöller, “Quantum optical implementation of quantum information processing,” arXiv: quant-ph/0405030 (2004).

Eckstein, A.

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
[Crossref]

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011).
[Crossref]

Farsi, A.

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

Fejer, M. M.

Frumker, E.

Gaeta, A. L.

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

Glauber, R. J.

U. M. Titulaer and R. J. Glauber, “Density operators for coherent fields,” Phys. Rev. 145, 1041–1050 (1966).
[Crossref]

Harder, G.

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Huang, J.

J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68, 2153–2156 (1992).
[Crossref]

Huang, Y.-P.

Ikuta, R.

Imoto, N.

Jain, N.

Jaksch, D.

J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang, and D. Jaksch, “Mapping broadband single-photon wave packets into an atomic memory,” Phys. Rev. A 75, 011401 (2007).
[Crossref]

Kanter, G. S.

Koashi, M.

Kobayashi, T.

Kumar, P.

Langrock, C.

Letzkus, F.

Manurkar, P.

Matsuki, K.

McGuinness, H. J.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref]

McKinstrie, C. J.

D. V. Reddy, M. G. Raymer, and C. J. McKinstrie, “Sorting photon wave packets using temporal-mode interferometry based on multiple-stage quantum frequency conversion,” Phys. Rev. A 91, 012323 (2015).
[Crossref]

D. V. Reddy, M. G. Raymer, and C. J. McKinstrie, “Efficient sorting of quantum-optical wave packets by temporal-mode interferometry,” Opt. Lett. 39, 2924–2927 (2014).
[Crossref]

D. V. Reddy, M. G. Raymer, C. J. McKinstrie, L. Mejling, and K. Rottwitt, “Temporal mode selectivity by frequency conversion in second-order nonlinear optical waveguides,” Opt. Express 21, 13840–13863 (2013).
[Crossref]

L. Mejling, D. S. Cargill, C. J. McKinstrie, K. Rottwitt, and R. O. Moore, “Effects of nonlinear phase modulation on Bragg scattering in the low-conversion regime,” Opt. Express 20, 27454–27475 (2012).
[Crossref]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref]

Mejling, L.

Miki, S.

Moore, R. O.

Nambiar, S. R.

S. R. Nambiar and S. K. Selvaraja, “High-efficiency broad-bandwidth subwavelength grating-based fiber-chip coupler in silicon-on-insulator,” Opt. Eng. 57, 017115 (2018).
[Crossref]

Nunn, J.

J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang, and D. Jaksch, “Mapping broadband single-photon wave packets into an atomic memory,” Phys. Rev. A 75, 011401 (2007).
[Crossref]

Parameswaran, K. R.

Quesada, N.

N. Quesada and J. E. Sipe, “High efficiency in mode-selective frequency conversion,” Opt. Lett. 41, 364–367 (2016).
[Crossref]

N. Quesada and J. E. Sipe, “Effects of time ordering in quantum nonlinear optics,” Phys. Rev. A 90, 063840 (2014).
[Crossref]

Radic, S.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref]

Ramelow, S.

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

Raymer, M. G.

D. V. Reddy and M. G. Raymer, “Engineering temporal-mode-selective frequency conversion in nonlinear optical waveguides: from theory to experiment,” Opt. Express 25, 12952–12966 (2017).
[Crossref]

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
[Crossref]

D. V. Reddy, M. G. Raymer, and C. J. McKinstrie, “Sorting photon wave packets using temporal-mode interferometry based on multiple-stage quantum frequency conversion,” Phys. Rev. A 91, 012323 (2015).
[Crossref]

D. V. Reddy, M. G. Raymer, and C. J. McKinstrie, “Efficient sorting of quantum-optical wave packets by temporal-mode interferometry,” Opt. Lett. 39, 2924–2927 (2014).
[Crossref]

D. V. Reddy, M. G. Raymer, C. J. McKinstrie, L. Mejling, and K. Rottwitt, “Temporal mode selectivity by frequency conversion in second-order nonlinear optical waveguides,” Opt. Express 21, 13840–13863 (2013).
[Crossref]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref]

J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang, and D. Jaksch, “Mapping broadband single-photon wave packets into an atomic memory,” Phys. Rev. A 75, 011401 (2007).
[Crossref]

Reddy, D. V.

Rosa, M. F.

Roslund, J.

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Rottwitt, K.

Sansoni, L.

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Selvaraja, S. K.

S. R. Nambiar and S. K. Selvaraja, “High-efficiency broad-bandwidth subwavelength grating-based fiber-chip coupler in silicon-on-insulator,” Opt. Eng. 57, 017115 (2018).
[Crossref]

Silberberg, Y.

Silberhorn, C.

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
[Crossref]

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011).
[Crossref]

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
[Crossref]

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Silver, M.

Sipe, J. E.

N. Quesada and J. E. Sipe, “High efficiency in mode-selective frequency conversion,” Opt. Lett. 41, 364–367 (2016).
[Crossref]

N. Quesada and J. E. Sipe, “Effects of time ordering in quantum nonlinear optics,” Phys. Rev. A 90, 063840 (2014).
[Crossref]

Suche, H.

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011).
[Crossref]

Surmacz, K.

J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang, and D. Jaksch, “Mapping broadband single-photon wave packets into an atomic memory,” Phys. Rev. A 75, 011401 (2007).
[Crossref]

Terai, H.

Titulaer, U. M.

U. M. Titulaer and R. J. Glauber, “Density operators for coherent fields,” Phys. Rev. 145, 1041–1050 (1966).
[Crossref]

Treps, N.

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Vogel, W.

Waldermann, F. C.

J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang, and D. Jaksch, “Mapping broadband single-photon wave packets into an atomic memory,” Phys. Rev. A 75, 011401 (2007).
[Crossref]

Walmsley, I. A.

J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang, and D. Jaksch, “Mapping broadband single-photon wave packets into an atomic memory,” Phys. Rev. A 75, 011401 (2007).
[Crossref]

Wang, Z.

J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang, and D. Jaksch, “Mapping broadband single-photon wave packets into an atomic memory,” Phys. Rev. A 75, 011401 (2007).
[Crossref]

Weiner, A. M.

Yamamoto, T.

Yamashita, T.

Yamazaki, D.

Zaoui, W. S.

Zheng, Z.

Zöller, P.

J. I. Cirac, L. M. Duan, and P. Zöller, “Quantum optical implementation of quantum information processing,” arXiv: quant-ph/0405030 (2004).

Chem. Phys. (1)

Z. Zheng and A. M. Weiner, “Coherent control of second harmonic generation using spectrally phase coded femtosecond waveforms,” Chem. Phys. 267, 161–171 (2001).
[Crossref]

J. Opt. Soc. Am. B (2)

New J. Phys. (1)

B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, “From quantum pulse gate to quantum pulse shaper—engineered frequency conversion in nonlinear optical waveguides,” New J. Phys. 13, 065029 (2011).
[Crossref]

Opt. Eng. (1)

S. R. Nambiar and S. K. Selvaraja, “High-efficiency broad-bandwidth subwavelength grating-based fiber-chip coupler in silicon-on-insulator,” Opt. Eng. 57, 017115 (2018).
[Crossref]

Opt. Express (6)

Opt. Lett. (2)

Optica (1)

Phys. Rev. (1)

U. M. Titulaer and R. J. Glauber, “Density operators for coherent fields,” Phys. Rev. 145, 1041–1050 (1966).
[Crossref]

Phys. Rev. A (4)

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

N. Quesada and J. E. Sipe, “Effects of time ordering in quantum nonlinear optics,” Phys. Rev. A 90, 063840 (2014).
[Crossref]

D. V. Reddy, M. G. Raymer, and C. J. McKinstrie, “Sorting photon wave packets using temporal-mode interferometry based on multiple-stage quantum frequency conversion,” Phys. Rev. A 91, 012323 (2015).
[Crossref]

J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang, and D. Jaksch, “Mapping broadband single-photon wave packets into an atomic memory,” Phys. Rev. A 75, 011401 (2007).
[Crossref]

Phys. Rev. Lett. (3)

J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68, 2153–2156 (1992).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref]

S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “Ramsey interference with single photons,” Phys. Rev. Lett. 117, 223601 (2016).
[Crossref]

Phys. Rev. X (1)

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
[Crossref]

Other (2)

J. I. Cirac, L. M. Duan, and P. Zöller, “Quantum optical implementation of quantum information processing,” arXiv: quant-ph/0405030 (2004).

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv: 1607.03001 (2016).

Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. Schematic for a two-stage two-color all-optical Ramsey interferometer using three-wave mixing. The net interferometric phase is a combination of the phase imparted to the three distinct beams (Δϕp+ΔϕsΔϕr). BS stands for beamsplitter. Also shown are the temporal-mode shapes used in the experiment, as defined in Eqs. (3)–(5).
Fig. 2.
Fig. 2. Two-stage, TM-selective all-optical Ramsey interferometer setup. Constructed by doublepassing optical pulses through the same nonlinear MgO:PPLN waveguide. The pump pulse from the pulse shaper is split into two beams at the polarizing beamsplitter (PBS). Pump-1 travels through the waveguide from left to right and loops back to the PBS in the counterclockwise direction, and is eventually rejected by the broadband Faraday isolator. Pump-2 loops back to the PBS in the clockwise direction, thus going through the waveguide from right to left. The two pump pulses are within the waveguide at different times, forming the two stages. The signal-in pulse copropagates with pump-1 through the waveguide, then gets separated from the pump through a dichroic mirror (DM2), and is then backreflected into the “second-stage” waveguide from an end mirror labeled “s-mirror.” The signal-out pulse is measured at the reject port of the isolator. The register-mid pulse generated from partial conversion of signal-in in the first-stage is backreflected into the second stage from the “r-mirror.” The “s-mirror” and “r-mirror” elements are mounted on high-precision linear translation stages for effective interferometric phase control via subwavelength displacements (ΔLj=λjΔϕj/4π for j{s,r}). HWP stands for half-wave plate. DM1 and DM3 are dichroic mirrors. Signal (register)-band beam paths are colored green (blue), while the pump-band beams are red.
Fig. 3.
Fig. 3. Two-stage, TM-selective optical Ramsey interferometric fringes visible in CE versus s-mirror displacement. For pump-pulse shapes (a) p0, (b) p1, and (c) p2, with various signal TM shapes (s0, s1, s2). Shape functions defined in Eqs. (3)–(5). The relative phase shifts among the fringe patterns both across and within the subplots are due to system drifts in between data runs.
Fig. 4.
Fig. 4. (a) Conversion efficiency versus combined-mirror-displacement for pump and signal shapes p0 and s0. r-mirror always moved in +10  nm steps. The legend items stand for s-mirror step sizes of 0 nm (A¯), 20  nm (B¯), and +20  nm (C¯). (b) Peak CE for various pump-signal shape combinations, and single-stage optimum levels for theoretically predicted exact Schmidt modes (dashed). The numbers in the insets are the three-mode separabilities for the two-stage scheme and the single-stage scheme (parenthesized). (c) Cross correlation from Stage-2 only CE versus s (r)-mirror displacement with register (signal)-mid beam blocked, demonstrating relative temporal widths. Pump and signal-in shapes were p0 and s0.
Fig. 5.
Fig. 5. Stage-2-only CE versus s-mirror displacement with register-mid beam blocked. For pump shape p0 and signal shape (a) s0 and (b) s1 at various pump-1 powers and low pump-2 power. The average pump-1 beam powers listed in the legend are for pulse rates of 80 MHz. Positive displacement is towards the waveguide.

Equations (5)

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H^I(t)=γ0LdzAp(tβpz)A^s(z,t)A^r(z,t)+h.a.,
U^I=Texp[idtH^I(t)]=exp(Ω^1+Ω^2+Ω^3+),
HG0(ω)=1N0exp((ωω0)22Δω2),
HG1(ω)=1N1(ω0.8Δω)exp((ωω0)22(0.8Δω)2),
HG2(ω)=1N2[2(ω0.89Δω)21]×exp((ωω0)22(0.8078Δω)2).