Abstract

We theoretically demonstrate an efficient method to control slow and fast light in microwave regime with a coupled system consisting of a nanomechanical resonator (NR) and a superconducting Cooper-pair box (CPB). Using the pump-probe technique, we find that both slow and fast light effects of the probe field can appear in this coupled system. Furthermore, we show that a tunable switch from slow light to fast light can be achieved by only adjusting the pump-CPB detuning from the NR frequency to zero. Our coupled system may have potential applications, for example, in optical communication, microwave photonics, and nonlinear optics.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  35. S. Huang, G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
    [CrossRef]
  36. R. S. Bennink, R. W. Boyd, C. R. Stroud, V. Wong, “Enhanced self-action effects by electromagnetically induced transparency in the two-level atom,” Phys. Rev. A 63, 033804 (2001).
    [CrossRef]
  37. S. E. Harris, J. E. Field, A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A 46, R29–R32 (1992).
    [CrossRef] [PubMed]
  38. P. Rabl, A. Shnirman, P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering, ” Phys. Rev. B 70, 205304 (2004).
    [CrossRef]
  39. J. Clarke, F. K. Wilhelm, “Superconducting quantum bits,” Nature 453, 1031–1042 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]

2013 (1)

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[CrossRef]

2011 (3)

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13, 023003 (2011).
[CrossRef]

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

S. Huang, G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
[CrossRef]

2010 (3)

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

J. Suh, M. D. LaHaye, P. M. Echternach, K. C. Schwab, M. L. Roukes, “Parametric amplification and back-action noise squeezing by a qubit-coupled nanoresonator,” Nano Lett. 10, 3990–3994 (2010).
[CrossRef] [PubMed]

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

2009 (5)

2008 (2)

X. Z. Yuan, H. S. Goan, C. H. Lin, K. D. Zhu, Y. W. Jiang, “Nanomechanical-resonator-assisted induced transparency in a Cooper-pair box system,” New J. Phys. 10, 095016 (2008).
[CrossRef]

J. Clarke, F. K. Wilhelm, “Superconducting quantum bits,” Nature 453, 1031–1042 (2008).
[CrossRef] [PubMed]

2006 (3)

C. P. Sun, L. F. Wei, Y. X. Liu, F. Nori, “Quantum transducers: Integrating transmission lines and nanomechanical resonators via charge qubits,” Phys. Rev. A 73, 022318 (2006).
[CrossRef]

Y. J. Wang, M. Eardley, S. Knappe, J. Moreland, L. Hollberg, J. Kitching, “Magnetic resonance in an atomic vapor excited by a mechanical resonator,” Phys. Rev. Lett. 97, 227602 (2006).
[CrossRef] [PubMed]

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, M. L. Roukes, “Zeptogram-scalenanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
[CrossRef] [PubMed]

2005 (3)

P. Zhang, Y. D. Wang, C. P. Sun, “Cooling mechanism for a nanomechanical resonator by periodic coupling to a Cooper pair box,” Phys. Rev. Lett. 95, 097204 (2005).
[CrossRef]

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

K. C. Schwab, M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58, 36–42 (2005).
[CrossRef]

2004 (4)

P. Rabl, A. Shnirman, P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering, ” Phys. Rev. B 70, 205304 (2004).
[CrossRef]

O. Astafiev, Y. A. Pashkin, Y. Nakamura, T. Yamamoto, J. S. Tsai, “Quantum noise in the Josephson charge qubit,” Phys. Rev. Lett. 93, 267007 (2004).
[CrossRef]

I. Wilson-Rae, P. Zoller, A. Imamoglu, “Laser cooling of a nanomechanical resonator mode to its quantum ground state,” Phys. Rev. Lett. 92, 075507 (2004).
[CrossRef] [PubMed]

P. C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S. W. Chang, S. L. Chuang, “Slow light in semiconductor quantum wells,” Opt. Lett. 29, 2291–2293 (2004).
[CrossRef] [PubMed]

2003 (4)

P. Zhang, Y. D. Wang, C. P. Sun, “Quantum measurement of a coupled nanomechanical resonator-Cooper-pair box system,” Phys. Rev. B 68, 155311 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef] [PubMed]

I. Chiorescu, Y. Nakamura, C. J. P. M. Harmansand, J. E. Mooij, “Coherent quantum dynamics of a super-conducting flux qubit, ” Science 299, 1869–1871 (2003).
[CrossRef] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef] [PubMed]

2002 (1)

A. D. Armour, M. P. Blencow, K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. 88, 148301 (2002).
[CrossRef] [PubMed]

2001 (2)

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

R. S. Bennink, R. W. Boyd, C. R. Stroud, V. Wong, “Enhanced self-action effects by electromagnetically induced transparency in the two-level atom,” Phys. Rev. A 63, 033804 (2001).
[CrossRef]

1999 (4)

Y. Nakamura, Y. A. Pashkin, J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature 398, 786–788 (1999).
[CrossRef]

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

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767–1770 (1999).
[CrossRef]

1996 (1)

A. N. Cleland, M. L. Roukes, “Fabrication of high frequency nanometer scale mechanical resonators from bulk Si crystals,” Appl. Phys. Lett. 69, 2653–2655 (1996).
[CrossRef]

1995 (2)

G. S. Agarwal, “Electromagnetic-field-induced transparency in high-density exciton systems,” Phys. Rev. A 51, R2711–R2714 (1995).
[CrossRef] [PubMed]

A. Kasapi, M. Jain, G. Y. Yin, S. E. Harris, “Electromagnetically induced transparency: propagation dynamics,” Phys. Rev. Lett. 74, 2447–2450 (1995).
[CrossRef] [PubMed]

1992 (1)

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

1982 (1)

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

Agarwal, G. S.

S. Huang, G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
[CrossRef]

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

G. S. Agarwal, “Electromagnetic-field-induced transparency in high-density exciton systems,” Phys. Rev. A 51, R2711–R2714 (1995).
[CrossRef] [PubMed]

Alkeskjold, T. T.

Ansmann, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Armour, A. D.

A. D. Armour, M. P. Blencow, K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. 88, 148301 (2002).
[CrossRef] [PubMed]

Astafiev, O.

O. Astafiev, Y. A. Pashkin, Y. Nakamura, T. Yamamoto, J. S. Tsai, “Quantum noise in the Josephson charge qubit,” Phys. Rev. Lett. 93, 267007 (2004).
[CrossRef]

Behroozi, C. H.

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

Bennink, R. S.

R. S. Bennink, R. W. Boyd, C. R. Stroud, V. Wong, “Enhanced self-action effects by electromagnetically induced transparency in the two-level atom,” Phys. Rev. A 63, 033804 (2001).
[CrossRef]

Bialczak, R. C.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef] [PubMed]

Bjarklev, A.

Blencow, M. P.

A. D. Armour, M. P. Blencow, K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. 88, 148301 (2002).
[CrossRef] [PubMed]

Boyd, R. W.

R. W. Boyd, D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[CrossRef] [PubMed]

R. S. Bennink, R. W. Boyd, C. R. Stroud, V. Wong, “Enhanced self-action effects by electromagnetically induced transparency in the two-level atom,” Phys. Rev. A 63, 033804 (2001).
[CrossRef]

Budker, D.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767–1770 (1999).
[CrossRef]

Callegari, C.

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, M. L. Roukes, “Zeptogram-scalenanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
[CrossRef] [PubMed]

Capmany, J.

Chan, J.

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

Chang, D. E.

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

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13, 023003 (2011).
[CrossRef]

Chang, S. W.

Chang-Hasnain, C. J.

Chen, Y.

Chiorescu, I.

I. Chiorescu, Y. Nakamura, C. J. P. M. Harmansand, J. E. Mooij, “Coherent quantum dynamics of a super-conducting flux qubit, ” Science 299, 1869–1871 (2003).
[CrossRef] [PubMed]

Chu, S.

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

Chuang, S. L.

Clarke, J.

J. Clarke, F. K. Wilhelm, “Superconducting quantum bits,” Nature 453, 1031–1042 (2008).
[CrossRef] [PubMed]

Cleland, A. N.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

A. N. Cleland, M. L. Roukes, “Fabrication of high frequency nanometer scale mechanical resonators from bulk Si crystals,” Appl. Phys. Lett. 69, 2653–2655 (1996).
[CrossRef]

Dutton, Z.

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

Eardley, M.

Y. J. Wang, M. Eardley, S. Knappe, J. Moreland, L. Hollberg, J. Kitching, “Magnetic resonance in an atomic vapor excited by a mechanical resonator,” Phys. Rev. Lett. 97, 227602 (2006).
[CrossRef] [PubMed]

Echternach, P. M.

J. Suh, M. D. LaHaye, P. M. Echternach, K. C. Schwab, M. L. Roukes, “Parametric amplification and back-action noise squeezing by a qubit-coupled nanoresonator,” Nano Lett. 10, 3990–3994 (2010).
[CrossRef] [PubMed]

M. D. LaHaye, J. Suh, P. M. Echternach, K. C. Schwab, M. L. Roukes, “Nanomechanical measurements of a superconducting qubit,” Nature 459, 960–964 (2009).
[CrossRef] [PubMed]

Eichenfield, M.

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

Ekinci, K. L.

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, M. L. Roukes, “Zeptogram-scalenanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
[CrossRef] [PubMed]

Feng, X. L.

Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, M. L. Roukes, “Zeptogram-scalenanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
[CrossRef] [PubMed]

Field, J. E.

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

Fleischhauer, M.

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

Fry, E. S.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Gauthier, D. J.

R. W. Boyd, D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the coupled NR-CPB system. The microwave currents with frequencies ωpu and ωpr and a direct current Ib are applied to the microwave line beside the CPB to control the flux Φx through the CPB loop.

Fig. 2
Fig. 2

The absorption (Imχ(1)) and dispersion (Reχ(1)) of the probe current as a function of the probe-qubit detuning Δpr = ωqωpr for Δpu = ωn with parameters λ0 = 0.01, Q=5000, ωn = 2π × 133MHz, T1 = 0.25μs, T2 = 0.05μs, Ω0 = 3.

Fig. 3
Fig. 3

The group velocity index ng(in units of Σ) as a function of the effective Rabi frequency Ω2, the parameters (except Ω0) used are the same as in Fig. 2.

Fig. 4
Fig. 4

The dimensionless imaginary part (Imχ(1)) and real part (Reχ(1)) of linear optical susceptibility as a function of detuning Δpr while detuning Δpu = 0. The parameters are λ0 = 0.01, Q=5000, ωn = 2π × 133MHz, T1 = 0.25μs, T2 = 0.05μs, Ω0 = 3.

Fig. 5
Fig. 5

The group velocity index n g = c v g (in units of Σ) of fast light as a function of efficient Rabi frequency Ω2, the parameters (except Ω0) used are the same as in Fig. 4.

Equations (21)

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H total = H N R + H C P B + H int ,
H N R = h ¯ ω n a a ,
H C P B = 1 2 h ¯ ω q σ z 1 2 E J cos [ π Φ x ( t ) ϕ 0 ] σ x ,
H int = h ¯ λ ( a + a ) σ z ,
H p u = 1 2 h ¯ Δ σ z + h ¯ ω n a a + h ¯ λ ( a + a ) σ z + h ¯ Ω ( σ + + σ ) + μ ε p r ( σ + e i δ t + σ e i δ t ) ,
d σ d t = ( 1 T 2 + i Δ + i q ) σ + i Ω σ z + i μ ε p r h ¯ e i δ t σ z ,
d σ z d t = ( σ z + 1 ) 1 T 1 2 i Ω ( σ + σ ) 2 i μ h ¯ ( σ + ε p r e i δ t σ ε p r * e i δ t ) ,
d 2 q d t 2 + γ n d q d t + ω n 2 q = 4 ω n λ 2 σ z
σ ( t ) = σ 0 + σ + 1 e i δ t + σ 1 e i δ t ,
σ z ( t ) = σ 0 z + σ + 1 z e i δ t + σ 1 z e i δ t ,
q ( t ) = q 0 + q + 1 e i δ t + q 1 e i δ t ,
q 0 = 4 λ 0 ω n k 0 , σ 0 = i μ Ω k 0 1 + i ( Δ 4 λ 0 ω 0 k 0 ) ,
( k 0 + 1 ) [ ( Δ 0 4 λ 0 ω 0 k 0 ) 2 + 1 ] + 4 T 1 / T 2 Ω c 2 k 0 = 0.
σ + 1 = T 2 μ ε p r h ¯ 8 λ 0 ω 0 η T 1 / T 2 Ω 0 2 k 0 2 θ ( 1 + i Δ 0 4 i λ 0 ω 0 k 0 ) ( T 1 / T 2 i δ 0 1 2 T 1 / T 2 β ) + 2 i T 1 / T 2 Ω 0 2 k 0 θ T 1 / T 2 i δ 0 1 2 T 1 / T 2 β + i k 0 1 + i Δ 0 4 i λ 0 ω 0 k 0 i δ 0 ,
χ eff ( 1 ) ( ω p r ) = μ σ + 1 ε p r = T 2 μ 2 h ¯ χ ( 1 ) ( ω p r ) ,
η = ω 0 2 ω 0 2 i γ 0 δ 0 δ 0 2 ,
β = 4 i λ 0 ω 0 k 0 η Ω 0 2 1 + i Δ 0 4 i λ 0 ω 0 k 0 + ω 0 2 1 + i Δ 0 4 i λ 0 ω 0 k 0 i δ 0 + 4 i λ 0 ω 0 k 0 η Ω 0 2 1 i Δ 0 + 4 i λ 0 ω 0 k 0 + ω 0 2 1 i Δ 0 + 4 i λ 0 ω 0 k 0 i δ 0 ,
θ = 1 1 + i Δ 0 4 i λ 0 ω 0 k 0 i δ 0 + 1 1 i Δ 0 + 4 i λ 0 ω 0 k 0 .
v g = c n + ω p r ( d n d ω p r ) ,
c v g = 1 + 2 π Re χ eff ( 1 ) ( ω p r ) ω p r = ω q + 2 π ω p r Re ( d χ eff ( 1 ) d ω p r ) ω p r = ω q .
c v g 1 = 2 π ω p r μ 2 T 2 h ¯ Re ( d χ ( 1 ) ( ω p r ) d ω p r ) ω p r = ω q = 1 T 2 Σ Re ( d χ ( 1 ) ( ω p r ) d ω p r ) ω p r = ω q ,

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