Abstract

In contrast to the optomechanically induced transparency (OMIT) defined conventionally, the inverse OMIT behaves as coherent absorption of the input lights in the optomechanical systems. We characterize a feasible inverse OMIT in a multi-channel fashion with a double-sided optomechanical cavity system coupled to a nearby charged nanomechanical resonator via Coulomb interaction, where two counter-propagating probe lights can be absorbed via one of the channels or even via three channels simultaneously with the assistance of a strong pump light. Under realistic conditions, we demonstrate the experimental feasibility of our model by considering two slightly different nanomechanical resonators and the possibility of detecting the energy dissipation of the system. In particular, we find that our model turns to be a unilateral inverse OMIT once the two probe lights are different with a relative phase, and in this case the relative phase can be detected precisely.

© 2015 Optical Society of America

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References

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2015 (4)

F. C. Lei, M. Gao, C. G. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23, 11508–11517 (2015).
[Crossref] [PubMed]

H. Jing, Sahin 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, 9663 (2015).

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91, 063827 (2015).
[Crossref]

R. X. Chen, L. T. Shen, and S. B. Zheng, “Dissipation-induced optomechanical entanglement with the assistance of Coulomb interaction,” Phys. Rev. A 91, 022326 (2015).
[Crossref]

2014 (9)

H. Fu, T. H. Mao, Y. Li, J. F. Ding, J. D. Li, and G. Y. Cao, “Optically mediated spatial localization of collective modes of two coupled cantilevers for high sensitivity optomechanical transducer,” Appl. Phys. Lett. 105, 014108 (2014).
[Crossref]

S. Huang, “Double electromagnetically induced transparency and narrowing of probe absorption in a ring cavity with nanomechanical mirrors,” J. Phys. B: At. Mol. Opt. Phys. 47, 055504 (2014).
[Crossref]

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New. J. Phys. 16, 033023 (2014).
[Crossref]

X. B. Yan, C. L. Cui, K. H. Gu, X. D. Tian, C. B. Fu, and J. H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
[Crossref] [PubMed]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90, 023817 (2014).
[Crossref]

C. H. Dong, J. T. Zhang, V. Fiore, and H. L. Wang, “Optomechanically induced transparency and self-induced oscillations with Bogoliubov mechanical modes,” Optica 1, 425 (2014).
[Crossref]

J. Y. Ma, C. You, L. G. Si, H. Xiong, X. X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, W. J. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nature Communications 5, 5082 (2014).
[Crossref] [PubMed]

2013 (7)

T. Oishi and M. Tomita, “Inverted coupled-resonator-induced transparency,” Phys. Rev. A 88, 013813 (2013).
[Crossref]

Y. H. Ma and L. Zhou, “Electromagnetically induced transparency and quadripartite macroscopic entanglement generated in a ring cavity,” Chin. Phys. B 22, 024204 (2013).
[Crossref]

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. DiGiuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

R. B. Li, L. Deng, and E. W. Hagley, “Fast, all-optical, zero to π continuously controllable kerr phase gate,” Phys. Rev. Lett. 110, 113902 (2013).
[Crossref]

X. G. Zhan, L. G. Si, A. S. Zheng, and X. X. Yang, “Tunable slow light in a quadratically coupled optomechanical system,” J. Phys. B: At. Mol. Opt. Phys. 46(2), 025501 (2013).
[Crossref]

D. Tarhan, S. Huang, and Ö. E. Müstecaplioğlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Phys. Rev. A 87, 013824 (2013).
[Crossref]

J. Q. Zhang, Y. Li, and M. Feng, “Cooling a charged mechanical resonator with time-dependent bias gate voltages,” J. Phys. C 25, 142201 (2013).

2012 (5)

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801 (2012).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86, 053806 (2012).
[Crossref]

C. H. Dong, V. Fiore, Mark C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609 (2012).
[Crossref] [PubMed]

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(01), 013815 (2012).
[Crossref]

2011 (4)

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
[Crossref]

Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” J. Phys. B: At. Mol. Opt. Phys. 44, 165505 (2011).
[Crossref]

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

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit Cavity Electromechanics in the strong-coupling regime,” Nature (London) 471,204–208 (2011).
[Crossref]

2010 (6)

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

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

S. Huang and G. S. Agarwal, “Normal-mode splitting and antibunching in Stokes and anti-Stokes processes in cavity optomechanics: radiation-pressure-induced four-wave-mixing cavity optomechanics,” Phys. Rev. A 81, 033830 (2010).
[Crossref]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[Crossref]

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature (London) 465, 755–758 (2010).
[Crossref]

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

2009 (1)

S. GrÖblacher, K. Hammerer, M. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London) 460, 724–727 (2009).
[Crossref]

2008 (3)

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

2005 (2)

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005).
[Crossref]

W. K. Hensinger, D. W. Utami, H. S. Goan, K. Schwab, C. Monroe, and G. J. Milburn, “Ion trap transducers for quantum electromechanical oscillators,” Phys. Rev. A 72, 041405 (2005).
[Crossref]

2004 (2)

M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74 (2004).
[Crossref] [PubMed]

B. Julsgaard, J. Sherson, and J. I. Cirac, “Experimental demonstration of quantum memory for light,” Nature(London) 432, 482–486 (2004).
[Crossref]

1999 (2)

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[Crossref]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[Crossref]

1996 (2)

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
[Crossref]

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[Crossref] [PubMed]

1992 (1)

S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A. 46, R29–R32 (1992).
[Crossref] [PubMed]

1991 (2)

K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref] [PubMed]

J. E. Field, K. H. Hahn, and S. E. Harris, “Observation of electromagnetically induced transparency in collisionally broadened lead vapor,” Phys. Rev. Lett. 67, 3062–3065 (1991).
[Crossref] [PubMed]

1990 (2)

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[Crossref] [PubMed]

P. L. Knight, M. A. Lauder, and B. J. Dalton, “Laser-induced continuum structure,” Phys. Rep. 190, 1–61 (1990).
[Crossref]

1987 (2)

S. Ravi and G. S. Agarwal, “Absorption spectroscopy of strongly perturbed bound-continuum transitions,” Phys. Rev. A 35,3354–3367 (1987).
[Crossref] [PubMed]

S. L. Haan and G. S. Agarwal, “Stability of dressed states against radiative decay in strongly coupled bound-continuum transitions,” Phys. Rev. A 35, 4592–4604 (1987).
[Crossref] [PubMed]

1986 (1)

H. Bachau, P. Lambropoulos, and R. Shakeshaft, “Theory of laser-induced transitions between autoionizing states of He,” Phys. Rev. A 34, 4785–4792 (1986).
[Crossref] [PubMed]

1984 (1)

1981 (2)

K. Rzaznewski and J. H. Eberly, “Confluence of bound-free coherences in laser-induced autoionization,” Phys. Rev. Lett. 47, 408–412 (1981).
[Crossref]

P. Lambropoulos and P. Zoller, “Autoionizing states in strong laser fields,” Phys. Rev. A 24, 379–397 (1981).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New. J. Phys. 16, 033023 (2014).
[Crossref]

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801 (2012).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
[Crossref]

S. Huang and G. S. Agarwal, “Normal-mode splitting and antibunching in Stokes and anti-Stokes processes in cavity optomechanics: radiation-pressure-induced four-wave-mixing cavity optomechanics,” Phys. Rev. A 81, 033830 (2010).
[Crossref]

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

S. Ravi and G. S. Agarwal, “Absorption spectroscopy of strongly perturbed bound-continuum transitions,” Phys. Rev. A 35,3354–3367 (1987).
[Crossref] [PubMed]

S. L. Haan and G. S. Agarwal, “Stability of dressed states against radiative decay in strongly coupled bound-continuum transitions,” Phys. Rev. A 35, 4592–4604 (1987).
[Crossref] [PubMed]

Alegre, T. P. M.

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

Allman, M. S.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit Cavity Electromechanics in the strong-coupling regime,” Nature (London) 471,204–208 (2011).
[Crossref]

Arcizet, O.

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

Arimondo, E.

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
[Crossref]

Aspelmeyer, M.

S. GrÖblacher, K. Hammerer, M. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London) 460, 724–727 (2009).
[Crossref]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Bachau, H.

H. Bachau, P. Lambropoulos, and R. Shakeshaft, “Theory of laser-induced transitions between autoionizing states of He,” Phys. Rev. A 34, 4785–4792 (1986).
[Crossref] [PubMed]

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. DiGiuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[Crossref]

Bhattacharya, M.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).
[Crossref]

Biancofiore, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. DiGiuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Bochmann, J.

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature (London) 465, 755–758 (2010).
[Crossref]

Boller, K. J.

K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref] [PubMed]

Buu, O.

M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74 (2004).
[Crossref] [PubMed]

Camacho, R.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[Crossref]

Camarota, B.

M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74 (2004).
[Crossref] [PubMed]

Cao, G. Y.

H. Fu, T. H. Mao, Y. Li, J. F. Ding, J. D. Li, and G. Y. Cao, “Optically mediated spatial localization of collective modes of two coupled cantilevers for high sensitivity optomechanical transducer,” Appl. Phys. Lett. 105, 014108 (2014).
[Crossref]

Chan, J.

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

Chang, D.

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

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[Crossref]

Chen, L. B.

C. N. Ren, J. Q. Zhang, L. B. Chen, and Y. J. Gu, “Optomechanical steady-state entanglement induced by electrical interaction,” arxiv:1402.6434 (2014).

Chen, R. X.

R. X. Chen, L. T. Shen, and S. B. Zheng, “Dissipation-induced optomechanical entanglement with the assistance of Coulomb interaction,” Phys. Rev. A 91, 022326 (2015).
[Crossref]

Chen, W. J.

B. Peng, S. K. Özdemir, W. J. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nature Communications 5, 5082 (2014).
[Crossref] [PubMed]

Cheng, J.

Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” J. Phys. B: At. Mol. Opt. Phys. 44, 165505 (2011).
[Crossref]

Cicak, K.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit Cavity Electromechanics in the strong-coupling regime,” Nature (London) 471,204–208 (2011).
[Crossref]

Cirac, J. I.

B. Julsgaard, J. Sherson, and J. I. Cirac, “Experimental demonstration of quantum memory for light,” Nature(London) 432, 482–486 (2004).
[Crossref]

Cui, C. L.

Dalton, B. J.

P. L. Knight, M. A. Lauder, and B. J. Dalton, “Laser-induced continuum structure,” Phys. Rep. 190, 1–61 (1990).
[Crossref]

Deng, L.

R. B. Li, L. Deng, and E. W. Hagley, “Fast, all-optical, zero to π continuously controllable kerr phase gate,” Phys. Rev. Lett. 110, 113902 (2013).
[Crossref]

Deng, Z.

DiGiuseppe, G.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. DiGiuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Ding, J. F.

H. Fu, T. H. Mao, Y. Li, J. F. Ding, J. D. Li, and G. Y. Cao, “Optically mediated spatial localization of collective modes of two coupled cantilevers for high sensitivity optomechanical transducer,” Appl. Phys. Lett. 105, 014108 (2014).
[Crossref]

Dong, C. H.

Du, C. G.

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[Crossref]

Eberly, J. H.

Z. Deng and J. H. Eberly, “Double-resonance effects in strong-field autoionization,” J. Opt. Soc. Am. B 1, 102–107 (1984).
[Crossref]

K. Rzaznewski and J. H. Eberly, “Confluence of bound-free coherences in laser-induced autoionization,” Phys. Rev. Lett. 47, 408–412 (1981).
[Crossref]

Eichenfield, M.

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

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[Crossref]

Feng, M.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91, 063827 (2015).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

J. Q. Zhang, Y. Li, and M. Feng, “Cooling a charged mechanical resonator with time-dependent bias gate voltages,” J. Phys. C 25, 142201 (2013).

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86, 053806 (2012).
[Crossref]

Field, J. E.

S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A. 46, R29–R32 (1992).
[Crossref] [PubMed]

J. E. Field, K. H. Hahn, and S. E. Harris, “Observation of electromagnetically induced transparency in collisionally broadened lead vapor,” Phys. Rev. Lett. 67, 3062–3065 (1991).
[Crossref] [PubMed]

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[Crossref] [PubMed]

Figueroa, E.

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature (London) 465, 755–758 (2010).
[Crossref]

Fiore, V.

Fu, C. B.

Fu, H.

H. Fu, T. H. Mao, Y. Li, J. F. Ding, J. D. Li, and G. Y. Cao, “Optically mediated spatial localization of collective modes of two coupled cantilevers for high sensitivity optomechanical transducer,” Appl. Phys. Lett. 105, 014108 (2014).
[Crossref]

Galassi, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. DiGiuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Gao, M.

Gavartin, E.

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

Genes, C.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Geng, Z.

H. Jing, Sahin 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, 9663 (2015).

Genov, D. A.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

Gigan, S.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Goan, H. S.

W. K. Hensinger, D. W. Utami, H. S. Goan, K. Schwab, C. Monroe, and G. J. Milburn, “Ion trap transducers for quantum electromechanical oscillators,” Phys. Rev. A 72, 041405 (2005).
[Crossref]

GrÖblacher, S.

S. GrÖblacher, K. Hammerer, M. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London) 460, 724–727 (2009).
[Crossref]

Gross, R.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Gu, K. H.

Gu, X.

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90, 023817 (2014).
[Crossref]

Gu, Y. J.

C. N. Ren, J. Q. Zhang, L. B. Chen, and Y. J. Gu, “Optomechanical steady-state entanglement induced by electrical interaction,” arxiv:1402.6434 (2014).

Haan, S. L.

S. L. Haan and G. S. Agarwal, “Stability of dressed states against radiative decay in strongly coupled bound-continuum transitions,” Phys. Rev. A 35, 4592–4604 (1987).
[Crossref] [PubMed]

Hagley, E. W.

R. B. Li, L. Deng, and E. W. Hagley, “Fast, all-optical, zero to π continuously controllable kerr phase gate,” Phys. Rev. Lett. 110, 113902 (2013).
[Crossref]

Hahn, C.

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature (London) 465, 755–758 (2010).
[Crossref]

Hahn, K. H.

J. E. Field, K. H. Hahn, and S. E. Harris, “Observation of electromagnetically induced transparency in collisionally broadened lead vapor,” Phys. Rev. Lett. 67, 3062–3065 (1991).
[Crossref] [PubMed]

Hammerer, K.

S. GrÖblacher, K. Hammerer, M. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London) 460, 724–727 (2009).
[Crossref]

Han, Y.

Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” J. Phys. B: At. Mol. Opt. Phys. 44, 165505 (2011).
[Crossref]

Harris, J. G. E.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

Harris, S. E.

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[Crossref]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[Crossref]

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[Crossref] [PubMed]

S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A. 46, R29–R32 (1992).
[Crossref] [PubMed]

J. E. Field, K. H. Hahn, and S. E. Harris, “Observation of electromagnetically induced transparency in collisionally broadened lead vapor,” Phys. Rev. Lett. 67, 3062–3065 (1991).
[Crossref] [PubMed]

K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref] [PubMed]

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[Crossref] [PubMed]

Hau, L. V.

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[Crossref]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[Crossref]

Hensinger, W. K.

W. K. Hensinger, D. W. Utami, H. S. Goan, K. Schwab, C. Monroe, and G. J. Milburn, “Ion trap transducers for quantum electromechanical oscillators,” Phys. Rev. A 72, 041405 (2005).
[Crossref]

Hill, J. T.

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

Hocke, F.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Huang, S.

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New. J. Phys. 16, 033023 (2014).
[Crossref]

S. Huang, “Double electromagnetically induced transparency and narrowing of probe absorption in a ring cavity with nanomechanical mirrors,” J. Phys. B: At. Mol. Opt. Phys. 47, 055504 (2014).
[Crossref]

D. Tarhan, S. Huang, and Ö. E. Müstecaplioğlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Phys. Rev. A 87, 013824 (2013).
[Crossref]

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801 (2012).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
[Crossref]

S. Huang and G. S. Agarwal, “Normal-mode splitting and antibunching in Stokes and anti-Stokes processes in cavity optomechanics: radiation-pressure-induced four-wave-mixing cavity optomechanics,” Phys. Rev. A 81, 033830 (2010).
[Crossref]

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

S. Huang and M. Tsang, “Electromagnetically induced transparency and optical memories in an optomechanical system with Nmembranes,” arXiv:1403.1340 (2014).

Huebl, H.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Imamoglu, A.

K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref] [PubMed]

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[Crossref] [PubMed]

Jain, M.

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[Crossref] [PubMed]

Jayich, A. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[Crossref]

Jing, H.

H. Jing, Sahin 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, 9663 (2015).

Jing, Q. L.

Julsgaard, B.

B. Julsgaard, J. Sherson, and J. I. Cirac, “Experimental demonstration of quantum memory for light,” Nature(London) 432, 482–486 (2004).
[Crossref]

Karuza, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. DiGiuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Kasapi, A.

S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A. 46, R29–R32 (1992).
[Crossref] [PubMed]

Khurgin, J. B.

Kippenberg, T. J.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

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

Knight, P. L.

P. L. Knight, M. A. Lauder, and B. J. Dalton, “Laser-induced continuum structure,” Phys. Rep. 190, 1–61 (1990).
[Crossref]

Kuzyk, Mark C.

C. H. Dong, V. Fiore, Mark C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609 (2012).
[Crossref] [PubMed]

LaHaye, M.

M. LaHaye, O. Buu, B. Camarota, and K. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74 (2004).
[Crossref] [PubMed]

Lambropoulos, P.

H. Bachau, P. Lambropoulos, and R. Shakeshaft, “Theory of laser-induced transitions between autoionizing states of He,” Phys. Rev. A 34, 4785–4792 (1986).
[Crossref] [PubMed]

P. Lambropoulos and P. Zoller, “Autoionizing states in strong laser fields,” Phys. Rev. A 24, 379–397 (1981).
[Crossref]

Lauder, M. A.

P. L. Knight, M. A. Lauder, and B. J. Dalton, “Laser-induced continuum structure,” Phys. Rep. 190, 1–61 (1990).
[Crossref]

Lei, F. C.

Li, D.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit Cavity Electromechanics in the strong-coupling regime,” Nature (London) 471,204–208 (2011).
[Crossref]

Li, J. D.

H. Fu, T. H. Mao, Y. Li, J. F. Ding, J. D. Li, and G. Y. Cao, “Optically mediated spatial localization of collective modes of two coupled cantilevers for high sensitivity optomechanical transducer,” Appl. Phys. Lett. 105, 014108 (2014).
[Crossref]

Li, R. B.

R. B. Li, L. Deng, and E. W. Hagley, “Fast, all-optical, zero to π continuously controllable kerr phase gate,” Phys. Rev. Lett. 110, 113902 (2013).
[Crossref]

Li, Y.

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram for a double-sided cavity with a nanomechanical resonator NR1 located at the node of the cavity mode and a nanomechanical resonator NR2 outside. NR1 is charged by the bias gate voltage V 1 and subject to the Coulomb force due to another charged NR2 with the bias gate voltage V 2. The optomechanical cavity of length L is driven by three light fields, one of which is the pump field εc with frequency ωc and the other of which are the probe fields εL ( R ) with frequency ωp . The output field is represented by εoutL ( R ). q 1 and q 2 represent the small displacements of NR1 and NR2 from their equilibrium positions, and r 0 is the equilibrium distance between the two charged NRs.
Fig. 2
Fig. 2 The normalized output probe photon number | ε o u t R + ε L | 2 ( | ε o u t L + ε L | 2 ) as functions of the probe detuning D/κ and the effective radiation coupling |G|κ, where D = ωp ωc ωm .
Fig. 3
Fig. 3 The normalized output probe photon number | ε o u t R + ε R | 2 ( | ε o u t L + ε L | 2 ) as a function of the probe detuning D/κ, where D = ωp ωc ωm for identical NRs and D = ωp ωc ω 1 for non-identical case. The red solid, black dashed-dotted, and blue dashed curves are for ω 2 = ω 1, ω 2 = 0.8ω 1 and ω 2 = 1.2ω 1, respectively, if |G| = 6κ.
Fig. 4
Fig. 4 Comparison of identical and non-identical NRs in the variation of photon and phonon numbers with respect to the probe detuning D/κ. The panels: (a) and (d) for the normalized probe photon number 4 κ 2 | ε L | 2 + | ε R | 2 | δ c + | 2 ; (b) and (e) for the normalized mechanical excitation 4 κ 2 | ε L | 2 + | ε R | 2 | δ b 1 + | 2 ; (c) and (d) for 4 κ 2 | ε L | 2 + | ε R | 2 | δ b 2 + | 2 . (a), (b) and (c) plot identical NRs with D = ωp ωc ωm , while (d), (e) and (f) present non-identical NRs (ω 2 = 1.2ω 1) with D = ωp ωc ω 1. The black solid, red dashed, and blue dashed-dotted curves are for pumping rates |G| = 2κ, 4κ, 6κ, respectively.
Fig. 5
Fig. 5 (a) and (c) The strength of the output light from the left-hand side of the optomechanical cavity with different probe light detunings, which remains unchanged for different parameter values; (b) and (d) The phase θ of the unilateral inverse OMIT for different probe light detunings. The left-hand side panels are for identical NRs (ω 2 = ω 1) with D = ωp ωc ωm , while the right-hand side panels for non-identical NRs (ω 2 = 1.2ω 1) with D = ωp ωc ω 1.
Fig. 6
Fig. 6 (a) The decay of the NR versus the relative phase between two probe lights for different effective radiation couplings; (b) The detuning of the probe field from the cavity resonance frequency verses the effective radiation coupling, where Δ = ω 1 and D = δω 1 = δ − Δ.

Tables (1)

Tables Icon

Table 1 The relationship among the normalized output probe photon numbers (εoutR + and εoutL +), the intracavity probe photon numbers ( 4 κ 2 | ε L | 2 + | ε R | 2 | δ c + | 2 ) , and the mechanical excitations ( 4 κ 2 | ε L | 2 + | ε R | 2 | δ b 1 + | 2 , 4 κ 2 | ε L | 2 + | ε R | 2 | δ b 2 + | 2 ) and the summation) for different effective radiation |G| and Coulomb coupling strengths λ in the inverse OMIT. Part I presents the middle channel D = D 0 and part II for the side channels D = D ±. We consider two non-identical NRs with ω 1 = 1.2κ and ω 2 = κ and the identical case with ω 1 = ω 2 = κ. The values in parentheses are for the identical case.

Equations (20)

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H = h ¯ ( ω 0 ω c ) c c + ( p 1 2 2 m 1 + 1 2 m 1 ω 1 2 q 1 2 ) + ( p 2 2 2 m 2 + 1 2 m 2 ω 2 2 q 2 2 ) + i h ¯ ε c ( c c ) + i h ¯ ( c ε L e i δ t h . c . ) + i h ¯ ( c ε R e i δ t h . c . ) + h ¯ g 0 c c q 1 + h ¯ λ 0 q 1 q 2 ,
H = h ¯ Δ c c c + h ¯ ω 1 b 1 b 1 + h ¯ ω 2 b 2 b 2 + h ¯ g c c ( b 1 + b 1 ) + i h ¯ ε c ( c c ) + h ¯ λ ( b 1 + b 1 ) ( b 2 + b 2 ) + i h ¯ ( c ε L e i δ t + h . c . ) + i h ¯ ( c ε R e i δ t h . c . ) ,
b ˙ 1 = ( γ 1 2 + i ω 1 ) b 1 i g c c i λ ( b 2 + b 2 ) + γ 1 b 1 i n , b ˙ = ( γ 2 2 + i ω 2 ) b 2 i λ ( b 1 + b 1 ) + γ 2 b 2 i n , c ˙ = ( 2 κ + i Δ c ) c i g ( b 1 + b 1 ) c + ε c + ( ε L + ε R ) e i δ t + 2 κ ( c i n + d i n ) ,
b 1 s = i g | c s | 2 γ 1 2 + i ω 1 + 8 λ 2 ω 1 ω 2 ( ω 1 + i γ 1 2 ) ( ω 2 + i γ 2 2 ) ( γ 2 2 + i ω 2 ) i g | c s | 2 γ 1 2 + i ω 1 + 8 λ 2 γ 2 2 + i ω 2 , b 2 s = 2 λ ω 1 ( ω 1 + i γ 1 2 ) ( ω 2 i γ 2 2 ) i 2 λ γ 2 2 + i ω 2 b 1 s , c s = ε c 2 κ + i Δ
δ ˙ b 1 = ( γ 1 2 + i ω 1 ) δ b 1 i ( G * δ c + G δ c ) i λ ( δ b 2 + δ b 2 ) + γ 1 b 1 i n , δ ˙ b 2 = ( γ 2 2 + i ω 2 ) δ b 2 i λ ( δ b 1 + δ b 1 ) + γ 2 b 2 i n , δ ˙ c = ( 2 κ + i Δ ) δ c i G ( δ b 1 + δ b 1 ) + ( ε L + ε R ) e i δ t + 2 κ ( c i n + d i n ) ,
δ ˙ b 1 = i ( G * δ c + G δ c ) ( i ω 1 + γ 1 2 ) δ b 1 i λ ( δ b 2 + δ b 2 ) , δ ˙ b 2 = ( i ω 2 + γ 2 2 ) δ b 2 i λ ( δ b 1 + δ b 1 ) , δ ˙ c = ( 2 κ + i Δ ) δ c i G ( δ b 1 + δ b 1 ) + ( ε L + ε R ) e i δ t .
δ o = δ o + e i δ t + δ o e i δ t ,
δ b 1 + = i G * γ 1 2 i ( δ ω 1 ) + λ 2 γ 2 2 i ( δ ω 2 ) δ c + , δ b 2 + = i λ γ 2 2 + i ω i δ δ b 1 + , δ c + = ε L + ε R 2 κ + i ( Δ δ ) + | G | 2 A ,
ε o u t α + ε α e i D t = 2 κ δ c , α = R , L ,
ε o u t α = ε o u t α + e i D t + ε o u t α e i D t , α = R , L ,
ε o u t α + = 2 κ δ c + ε α = ε L + ε R 2 κ + i D + | G | 2 γ 1 2 i ( δ ω 1 ) + λ 2 γ 2 2 i D ε α
ε L = ε R , γ 1 = γ 2 = 2 κ , λ 2 = 1 2 | G | 2 κ 2 .
D 0 = 0 , D ± = ± | G | 2 + λ 2 3 κ 2 = ± 3 2 | G | 2 4 κ 2 ,
4 κ 2 | ε L | 2 + | ε R | 2 | δ c + | 2 = 0.5 ,
4 κ 2 | ε L | 2 + | ε R | 2 | δ b 1 + | 2 = { 2 κ 2 | G | 2 , D 0 = 0 0.75 , D ± = ± 3 2 | G | 2 4 κ 2 ,
4 κ 2 | ε L | 2 + | ε R | 2 | δ b 2 + | 2 = { 1 2 κ 2 | G | 2 , D 0 = 0 0.25 , D ± = ± 3 2 | G | 2 4 κ 2 .
ε o u t L + = 2 κ ( ε L + ε R e i θ ) 2 κ + i ( Δ δ ) + | G | 2 γ 1 2 i ( δ ω 1 ) + λ 2 γ 2 2 i ( δ ω 2 ) ε L = 0.
| G | 2 γ 1 2 i ( δ ω 1 ) + λ 2 γ 2 2 i ( δ ω 2 ) i ( δ Δ ) = 2 κ e i θ .
| G | 2 γ 1 2 i D i D = 2 κ e i θ ,
D ± = ± 1 8 ( 8 | G | 2 + ( 16 κ 2 γ 1 2 ) + 16 | G | 2 ( 16 κ 2 γ 1 2 ) + ( 16 κ 2 + γ 1 2 ) .

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