Abstract

The recent emerging field of synthetic dimension in photonics offers a variety of opportunities for manipulating different internal degrees of freedom of photons such as the spectrum of light. While nonlinear optical effects can be incorporated into these photonic systems with synthetic dimensions, these nonlinear effects typically result in long-range interactions along the frequency axis. Thus, it has been difficult to use the synthetic dimension concept to study a large class of Hamiltonians that involves local interactions. Here we show that a Hamiltonian that is locally interacting along the synthetic dimension can be achieved in a dynamically modulated ring resonator incorporating χ(3) nonlinearity, provided that the group velocity dispersion of the waveguide forming the ring is specifically designed. As a demonstration we numerically implement a Bose–Hubbard model and explore photon blockade effect in the synthetic frequency space. Our work opens new possibilities for studying fundamental many-body physics in the synthetic space in photonics, with potential applications in optical quantum communication and quantum computation.

© 2020 Chinese Laser Press

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2020 (3)

2019 (10)

M. Radulaski, J. L. Zhang, Y.-K. Tzeng, K. G. Lagoudakis, H. Ishiwata, C. Dory, K. A. Fischer, Y. A. Kelaita, S. Sun, P. C. Maurer, K. Alassaad, G. Ferro, Z.-X. Shen, N. A. Melosh, S. Chu, and J. Vučković, “Nanodiamond integration with photonic devices,” Laser Photon. Rev. 13, 1800316 (2019).
[Crossref]

M. B. Mia, N. Jaidye, and S. Kim, “Extremely high dispersions in heterogeneously coupled waveguides,” Opt. Express 27, 10426–10437 (2019).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafei, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photon. Res. 7, 630–641 (2019).

A. Dutt, M. Monkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nat. Commun. 10, 3122 (2019).
[Crossref]

D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, “Colloquium: many-body localization, thermalization, and entanglement,” Rev. Mod. Phys. 91, 021001 (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]

L. Yuan, Q. Lin, A. Zhang, M. Xiao, X. Chen, and S. Fan, “Photonic gauge potential in one cavity with synthetic frequency and orbital angular momentum dimensions,” Phys. Rev. Lett. 122, 083903 (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]

2018 (5)

Y. Baum and G. Refael, “Setting boundaries with memory: generation of topological boundary states in Floquet-induced synthetic crystals,” Phys. Rev. Lett. 120, 106402 (2018).
[Crossref]

L. Yuan, Q. Lin, M. Xiao, and S. Fan, “Synthetic dimension in photonics,” Optica 5, 1396–1405 (2018).
[Crossref]

X.-W. Luo, C. Zhang, G.-C. Guo, and Z.-W. Zhou, “Topological photonic orbital-angular-momentum switch,” Phys. Rev. A 97, 043841 (2018).
[Crossref]

L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B 97, 104105 (2018).
[Crossref]

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum control through discrete frequency diffraction in the presence of photonic gauge potentials,” Phys. Rev. Lett. 120, 133901 (2018).
[Crossref]

2017 (13)

B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, “Spectral photonic lattices with complex long-range coupling,” Optica 4, 1433–1436 (2017).
[Crossref]

T. Ozawa and I. Carusotto, “Synthetic dimensions with magnetic fields and local interactions in photonic lattices,” Phys. Rev. Lett. 118, 013601 (2017).
[Crossref]

Y. Zhang and Y. Zhu, “Generation of Weyl points in coupled optical microdisk-resonator arrays via external modulation,” Phys. Rev. A 96, 013811 (2017).
[Crossref]

X.-W. Luo, X. Zhou, J.-S. Xu, C.-F. Li, G.-C. Guo, C. Zhang, and Z.-W. Zhou, “Synthetic-lattice enabled all-optical devices based on orbital angular momentum of light,” Nat. Commun. 8, 16097 (2017).
[Crossref]

X.-F. Zhou, X.-W. Luo, S. Wang, G.-C. Guo, X. Zhou, H. Pu, and Z.-W. Zhou, “Dynamically manipulating topological physics and edge modes in a single degenerate optical cavity,” Phys. Rev. Lett. 118, 083603 (2017).
[Crossref]

S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, and J. Ye, “Spin-Corbit-coupled fermions in an optical lattice clock,” Nature 542, 66–70 (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]

I. Martin, G. Refael, and B. Halperin, “Topological frequency conversion in strongly driven quantum systems,” Phys. Rev. X 7, 041008 (2017).
[Crossref]

K. G. L. Pedersen and M. Pletyukhov, “Few-photon scattering on Bose-Hubbard lattices,” Phys. Rev. A 96, 023815 (2017).
[Crossref]

L. Yuan, M. Xiao, S. Xu, and S. Fan, “Creating anyons from photons using a nonlinear resonator lattice subject to dynamic modulation,” Phys. Rev. A 96, 043864 (2017).
[Crossref]

M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, T. Menke, D. Borgnia, P. M. Preiss, F. Grusdt, A. M. Kaufman, and M. Greiner, “Microscopy of the interacting Harper-CHofstadter model in the two-body limit,” Nature 546, 519–523 (2017).
[Crossref]

D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89, 021001 (2017).
[Crossref]

L. Stern, B. Desiatov, N. Mazurski, and U. Levy, “Strong coupling and high-contrast all-optical modulation in atomic cladding waveguides,” Nat. Commun. 8, 14461 (2017).
[Crossref]

2016 (6)

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

L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Inguscio, and L. Fallani, “Synthetic dimensions and spin-orbit coupling with an optical clock transition,” Phys. Rev. Lett. 117, 220401 (2016).
[Crossref]

L. Yuan and S. Fan, “Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator,” Optica 3, 1014–1018 (2016).
[Crossref]

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7, 13731 (2016).
[Crossref]

D. V. Strekalov, C. Marquardt, A. B. Matsko, H. G. L. Schwefel, and G. Leuchs, “Nonlinear and quantum optics with whispering gallery resonators,” J. Opt. 18, 123002 (2016).
[Crossref]

2015 (8)

X.-W. Luo, X. Zhou, C.-F. Li, J.-S. Xu, G.-C. Guo, and Z.-W. Zhou, “Quantum simulation of 2D topological physics in a 1D array of optical cavities,” Nat. Commun. 6, 7704 (2015).
[Crossref]

M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, “Observation of chiral edge states with neutral fermions in synthetic Hall ribbons,” Science 349, 1510–1513 (2015).
[Crossref]

B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman, “Visualizing edge states with an atomic Bose gas in the quantum Hall regime,” Science 349, 1514–1518 (2015).
[Crossref]

B. Gadway, “Atom-optics approach to studying transport phenomena,” Phys. Rev. A 92, 043606 (2015).
[Crossref]

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

S. Kato and T. Aoki, “Strong coupling between a trapped single atom and an all-fiber cavity,” Phys. Rev. Lett. 115, 093603 (2015).
[Crossref]

R. Ritter, N. Gruhler, W. Pernice, H. Kübler, T. Pfau, and R. Löw, “Atomic vapor spectroscopy in integrated photonic structures,” Appl. Phys. Lett. 107, 041101 (2015).
[Crossref]

A. Biella, L. Mazza, I. Carusotto, D. Rossini, and R. Fazio, “Photon transport in a dissipative chain of nonlinear cavities,” Phys. Rev. A 91, 053815 (2015).
[Crossref]

2014 (3)

D. E. Chang, V. Vuletić, and M. D. Lukin, “Quantum nonlinear optics–photon by photon,” Nat. Photonics 8, 685–694 (2014).
[Crossref]

Y. Zhu, J. A. Greenberg, N. A. Husein, and D. J. Gauthier, “Giant all-optical tunable group velocity dispersion in an optical fiber,” Opt. Express 22, 14382–14391 (2014).
[Crossref]

A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliūnas, and M. Lewenstein, “Synthetic gauge fields in synthetic dimensions,” Phys. Rev. Lett. 112, 043001 (2014).
[Crossref]

2013 (4)

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Figures (4)

Fig. 1.
Fig. 1. (a) A ring resonator, composed by two types of single-mode waveguides A and B, undergoing the dynamic modulation. (b) A ring under the dynamic modulation supports a synthetic lattice along the frequency dimension. (c) The ring is coupled with the through-port and drop-port waveguides.
Fig. 2.
Fig. 2. Four-wave-mixing processes in a ring with a third-order nonlinear susceptibility including the hyper-parametric processes (SPM, XPM, and others) and the THG process.
Fig. 3.
Fig. 3. (a) A general four-wave-mixing process in a waveguide with the third-order susceptibility χ(3). (b) The plot of ΔIsinc2(ΔkL) showing the effect of the phase mismatching.
Fig. 4.
Fig. 4. Normalized distributions of the two-photon correlation probability Pm,n. (a)–(c) The input photon pair is |ϕ(4,4) with g=0, g=2J, and g=10J, respectively. (d)–(f) The input photon pair is |ϕ(15,15) with g=0, g=2J, and g=10J, respectively. Positions inside two dashed lines correspond to Pm,m.

Equations (20)

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HBH=Jm(amam+1+h.c.)g2mamamamam,
T=ei2αcos(Ωt),
2πΩEmτΔEm=iα(Em+1+Em1),
H0=Jm(amam+1+h.c.).
H2=Jm(amam+1+h.c.)g2m,n,p,qamanapaqg3m,n(aman3+h.c.),
A(z,ω)z=iβ(ω)A(z,ω),
β(ω)β(ω0)=dβdω|ω0(ωω0)+12d2βdω2|ω0(ωω0)2+.
A(z=L,ω)=A(z=0,ω)eiβ0Li(ωω0)L/vgiβ2(ωω0)2L/2.
A˜3z=iω022β0c2χ(3)A˜1A˜2A˜4*eiΔkz,
A˜4z=iω022β0c2χ(3)A˜1A˜2A˜3*eiΔkz,
Δk=β22[(ω3ω0)2+(ω4ω0)2(ω1ω0)2(ω2ω0)2].
2πΩA˜mτ=iα(A˜m+1+A˜m1)+iω022β0c2χ(3)|A˜m|2A˜m+inm2ω02β0c2χ(3)|A˜n|2A˜m,
H3=Jm(amam+1+h.c.)g2mamamamamgnmanamaman.
nmanamaman=mamam(nmanan)=mamam(Namam)=N2Nmamamamam.
H4=Jm(amam+1+h.c.)+g2mamamamamg(N2N).
dam(t)dt=i[H3,am]γam(t)+iγcin,m(t),
cout,m(t)=cin,m(t)iγam(t),
dout,m(t)=iγam(t),
|ϕ(p,q)=dt1dt2f(t1+t22)h(t1t2)cin,q(t2)cin,p(t1)|0,
Pm,n=dtGm,n(2)(t,t).

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