Abstract

We study the optical transmission characteristics of coupled spinning optomechanical resonators with pump-probe driven lasers. Under the steady-state conditions, we focus on how changing the optical Sagnac effect due to same or opposite spinning directions of the resonators can give rise to non-reciprocal and delayed probe light transmission. We find that coupled resonators can exhibit distinct transmission features, can generate negative group delays (slow as well as fast light) and offer additional control of the probe light transmission as compared to the case of a single spinning resonator. Our results can be useful in achieving chiral light propagation in quantum communication technologies without using traditional magneto-optical means.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
[Crossref]

2018 (9)

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-hermitian time-floquet systems,” Phys. Rev. Lett. 120(8), 087401 (2018).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref]

Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

J. Li and Y. Wu, “Quality of photon antibunching in two cavity-waveguide arrangements on a chip,” Phys. Rev. A 98(5), 053801 (2018).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
[Crossref]

I. M. Mirza and J. C. Schotland, “Influence of disorder on electromagnetically induced transparency in chiral waveguide quantum electrodynamics,” J. Opt. Soc. Am. B 35(5), 1149–1158 (2018).
[Crossref]

H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424–1430 (2018).
[Crossref]

2017 (8)

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017).
[Crossref]

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

A. Metelmann and A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95(1), 013837 (2017).
[Crossref]

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017).
[Crossref]

W. Zeng, W. Nie, L. Li, and A. Chen, “Ground-state cooling of a mechanical oscillator in a hybrid optomechanical system including an atomic ensemble,” Sci. Rep. 7(1), 17258 (2017).
[Crossref]

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: quantum interferences and correlations,” Phys. Rev. A 95(2), 023832 (2017).
[Crossref]

2016 (6)

D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016).
[Crossref]

F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
[Crossref]

S. Davuluri, “Optomechanics for absolute rotation detection,” Phys. Rev. A 94(1), 013808 (2016).
[Crossref]

Y.-L. Liu, G.-Z. Wang, Y.-x. Liu, and F. Nori, “Mode coupling and photon antibunching in a bimodal cavity containing a dipole quantum emitter,” Phys. Rev. A 93(1), 013856 (2016).
[Crossref]

I. M. Mirza, “Strong coupling optical spectra in dipole–dipole interacting optomechanical tavis- cummings models,” Opt. Lett. 41(11), 2422–2425 (2016).
[Crossref]

2015 (8)

I. M. Mirza, “Real-time emission spectrum of a hybrid atom-optomechanical cavity,” J. Opt. Soc. Am. B 32(8), 1604–1614 (2015).
[Crossref]

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

S. Davuluri and S. Zhu, “Controlling optomechanically induced transparency through rotation,” Europhys. Lett. 112(6), 64002 (2015).
[Crossref]

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).
[Crossref]

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

2014 (3)

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014).
[Crossref]

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in $\mathcal {PT}$PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89(1), 013824 (2014).
[Crossref]

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

2013 (4)

C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
[Crossref]

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref]

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103(16), 161105 (2013).
[Crossref]

2012 (4)

2011 (3)

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

A. Kamal, J. Clarke, and M. H. Devoret, “Noiseless non-reciprocity in a parametric active device,” Nat. Phys. 7(4), 311–315 (2011).
[Crossref]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

2010 (6)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

D. Witthaut and A. S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. 12(4), 043052 (2010).
[Crossref]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010).
[Crossref]

Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18(8), 8367–8382 (2010).
[Crossref]

2009 (3)

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009).
[Crossref]

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref]

2005 (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

2000 (1)

G. B. Malykin, “The sagnac effect: correct and incorrect explanations,” Phys.-Usp. 43(12), 1229–1252 (2000).
[Crossref]

1985 (1)

C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
[Crossref]

Abdumalikov, A.

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

Agarwal, G. S.

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in $\mathcal {PT}$PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89(1), 013824 (2014).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. Scheme of the system consisting of two series-coupled spinning optomechanical resonators.
Fig. 2.
Fig. 2. Probe transmission rate as a function of detuning $\Delta _{p}$ for a single resonator and a double coupled resonator system. (a) Absence of spin in all cases and increasing resonator-resonator coupling $J$ for the two-resonator case. (b) Bottom resonator spinning in the clockwise direction (with rate $\Omega _1=40$kHz) while upper resonator may or may not be spinning in the same direction. We have also incorporated the scenario when the upper resonator is not spinning. Note that in this and later plots, for $N=2$ we have assumed $J/\kappa _{ex}=1$ and $\vert \Omega _{1}\vert =\vert \Omega _{2}\vert =\vert \Omega \vert$ unless stated otherwise.
Fig. 3.
Fig. 3. Probe transmission for larger spinning rate $|\Omega |=100$kHz. We have focused on a frequency region where non-reciprocal light transmission is evident. Rest of the parameters are the same as used in Fig. 2.
Fig. 4.
Fig. 4. The group delay of probe light as a function of spinning rate magnitude for (a) a single and (b) a double spinning resonator. In the double resonator case, we have only plotted the unique situation where for a range of $|\Omega |$ fast light can be achieved. For comparison, we have plotted the single spinning resonator case in (a) where one can only achieve slow light for all $|\Omega |$ value. Parameters are the same as used in Fig. 2.
Fig. 5.
Fig. 5. Off-resonance transmission enhancement factor as a function of spinning rate for the double resonator case in which one can achieve the fast light. For comparison we have plotted the corresponding single resonator enhancement factor as well. Parameters are the same as used in Fig. 2.
Fig. 6.
Fig. 6. Probe transmission rate as a function of detuning $\Delta _{p}$ for a double coupled resonator system. Bottom resonator spinning in the counterclockwise direction (with rate $\Omega _1=40$kHz) while upper resonator may or may not be spinning in the same direction. We have also incorporated the scenario when the upper resonator is not spinning. We have set $J/\kappa _{ex}=1$.
Fig. 7.
Fig. 7. Probe transmission for larger spinning rate $|\Omega |=100$kHz. The yellow highlighted regions show the possibility of optical nonreciprocal transmission with different combinations of spinning directions. Rest of the parameters are the same as used in Fig. 2.
Fig. 8.
Fig. 8. The group delay of probe light as a function of spinning rate magnitude for various spin direction options. Parameters are the same as used in Fig. 2.

Equations (16)

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H ^ = H ^ 0 + H ^ i n t + H ^ d r , H ^ 0 = j = 1 2 ( Δ c j a ^ j a ^ j + p ^ j 2 2 m j + 1 2 m j ω m j 2 x ^ j 2 + p ^ j θ 2 2 m j r j 2 ) , H ^ d r = j = 1 2 i κ e x ( ε l j a ^ j + ε p 1 a ^ 1 e i ( ω p ω l j ) t H . c . ) , H ^ i n t = j = 1 2 ξ j x ^ j a ^ j a ^ j + J 1 , 2 ( a 1 a ^ 2 + a ^ 2 a ^ 1 ) .
ω c j ω c j + Δ s a g j , where Δ s a g j := n j r j Ω j ω c j c ( 1 1 n j 2 λ j n j d n j d λ j ) .
d a ^ 1 ( t ) d t = i ( Δ c 1 ξ 1 x ^ 1 i β 1 ) a ^ 1 + κ e x ( ε l 1 + ε p 1 e i η 1 t ) i J 1 , 2 a ^ 2 , d a ^ 2 ( t ) d t = i ( Δ c 2 ξ 2 x ^ 2 i β 2 ) a ^ 2 + κ e x ε l 2 i J 1 , 2 a ^ 1 , d 2 x ^ 1 ( t ) d t 2 = ξ 1 m 1 a ^ 1 a ^ 1 ω m 1 2 x ^ 1 + p ^ 1 θ 2 m 1 2 r 1 3 γ m 1 d x ^ 1 d t , d 2 x ^ 2 ( t ) d t 2 = ξ 2 m 2 a ^ 2 a ^ 2 ω m 2 2 x ^ 2 + p ^ 2 θ 2 m 2 2 r 2 3 γ m 2 d x ^ 2 d t , d θ ^ 1 ( t ) d t = p ^ 1 θ m 1 r 1 2 , d θ ^ 2 ( t ) d t = p ^ 2 θ m 2 r 2 2 , d p ^ j θ ( t ) d t = 0 , d p ^ 2 θ ( t ) d t = 0.
d a ^ 1 ( t ) d t = i ( Δ c 1 i β 1 ) a ^ 1 + i ξ 1 x ^ 1 a ^ 1 + κ e x ( ε l 1 + ε p 1 e i η 1 t ) i J 1 , 2 a ^ 2 , d a ^ 2 ( t ) d t = i ( Δ c 2 i β 2 ) a ^ 2 + i ξ 2 x ^ 1 a ^ 2 + κ e x ε l 2 i J 1 , 2 a ^ 1 , d 2 x ^ 1 ( t ) d t 2 = ( ω m 1 2 + γ m 1 d d t ) x ^ 1 + ξ 1 m 1 a ^ 1 a ^ 1 + p ^ 1 θ 2 m 1 2 r 1 3 , d 2 x ^ 2 ( t ) d t 2 = ( ω m 2 2 + γ m 2 d d t ) x ^ 2 + ξ 2 m 2 a ^ 2 a ^ 2 + p ^ 2 θ 2 m 2 2 r 2 3 , d θ ^ 1 ( t ) d t = p ^ 1 θ m 1 r 1 2 , d θ ^ 2 ( t ) d t = p ^ 2 θ m 2 r 2 2 , d p ^ 1 θ ( t ) d t = 0 , d p ^ 2 θ ( t ) d t = 0.
a ^ 1 a 1 + δ a 1 e i η 1 t + δ a + 1 e i η 1 t , a ^ 2 a 2 + δ a 2 e i η 2 t + δ a + 2 e i η 2 t , x ^ 1 x 1 + δ x 1 e i η 1 t + δ x 1 e i η 1 t , x ^ 2 x 2 + δ x 2 e i η 2 t + δ x 2 e i η 2 t .
a 1 = ( κ e x ε l i J 1 , 2 a 2 ) β 1 + i Δ c 1 i ξ 1 x 1 , a 2 = ( κ e x ε l i J 2 , 1 a 1 ) β 2 + i Δ c 2 i ξ 2 x 2 , x 1 = ( ξ 1 | a 1 | 2 + m 1 r 1 Ω 1 2 ) m 1 ω m 1 2 , x 2 = ( ξ 2 | a 2 | 2 + m 2 r 2 Ω 2 2 ) m 2 ω m 2 2 .
δ a 1 ( β 1 + i Δ c 1 i ξ 1 x 1 i η 1 ) i ξ 1 a 1 δ x 1 = κ e x ε p 1 i J 1 , 2 δ a 2 , δ a 2 ( β 2 + i Δ c 2 i ξ 2 x 2 i η 2 ) i ξ 2 a 2 δ x 2 = i J 1 , 2 δ a 1 , δ a + 1 ( β 1 i Δ c 1 + i ξ 1 x 1 i η 1 ) + i ξ 1 a 1 δ x 1 = i J 1 , 2 δ a 2 , δ a + 2 ( β 2 i Δ c 2 + i ξ 2 x 2 i η 2 ) + i ξ 2 a 2 δ x 2 = i J 1 , 2 δ a 1 , m 1 ( ω m 1 2 η 1 i η 1 γ m 1 ) δ x 1 = ξ 1 ( a 1 δ a 1 + a 1 δ a + 1 ) , m 2 ( ω m 2 2 η 2 i η 2 γ m 2 ) δ x 2 = ξ 2 ( a 2 δ a 2 + a 2 δ a + 2 ) .
T t p 2 = < a ^ o u t a ^ o u t > < a ^ i n a ^ i n > .
a ^ o u t = a ^ i n κ e x δ a 1 = ε p 1 κ e x δ a 1 .
T = | 1 κ e x ε p 1 δ a 1 | 2 .
δ a 1 = κ e x ε p 1 { i ξ 2 | a | 2 + m ( β ~ ) Γ m } i ξ 2 | a | 2 ( β ~ ) ( β ~ ) { i ξ 2 | a | 2 + m ( β ~ ) Γ m } ,
a = κ e x ε l β + i Δ c i ξ x , x = ( ξ | a | 2 + m r Ω 2 ) m ω m 2 .
τ g = d arg ( t p ) d Δ p
G . D . = τ g ( Ω 1 0 , Ω 2 0 ) τ g ( Ω 1 = 0 , Ω 2 = 0 ) | Δ p 1.
E . F . = T ( Ω 1 0 , Ω 2 0 ) T ( Ω 1 = 0 , Ω 2 = 0 ) | Δ p 1.
E . F . = T ( Ω 0 ) T ( Ω = 0 ) | Δ p 1.

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