## Microwave field controlled slow and fast light with a coupled system consisting of a nanomechanical resonator and a Cooper-pair box |

Optics Express, Vol. 22, Issue 3, pp. 3621-3628 (2014)

http://dx.doi.org/10.1364/OE.22.003621

Acrobat PDF (979 KB)

### 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

## 1. Introduction

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

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

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

4. 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 **397**, 594–598 (1999). [CrossRef]

6. D. Budker, D. F. Kimball, S. M. Rochester, and 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]

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

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

4. 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 **397**, 594–598 (1999). [CrossRef]

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

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

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

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

^{3}− 10

^{6}) and high natural frequencies (

*MHz*−

*GHz*) together with important applications [11

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

13. J. J. Li and K. D. Zhu, “An efficient optical knob from slow light to fast in a coupled nanomechanical resonator-quantum dot system,” Opt. Express **17**, 19874–19881 (2009). [CrossRef] [PubMed]

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

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

16. 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, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature **464**, 697–703 (2010). [CrossRef]

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

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

19. A. H. Safavi-Naeini, T. P. Mayer 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**, 69–73 (2011). [CrossRef] [PubMed]

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

21. A. D. Armour, M. P. Blencow, and 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]

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

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

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

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

26. W. Xue, S. Sales, J. Capmany, and J. Mork, “Microwave phase shifter with controllable power response based on slow-and fast-light effects in semiconductor optical amplifiers,” Opt. Lett. **34**, 929–931 (2009). [CrossRef] [PubMed]

27. L. Wei, W. Xue, Y. Chen, T. T. Alkeskjold, and A. Bjarklev, “Optically fed microwave true-time delay based on a compact liquid-crystal photonic-bandgap-fiber device,” Opt. Lett. **34**, 2757–2759 (2009). [CrossRef] [PubMed]

## 2. Theoretical mode and analytical expressions

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

29. O. Astafiev 1, Y. A. Pashkin, Y. Nakamura, T. Yamamoto, and J. S. Tsai, “Quantum noise in the Josephson charge qubit,” Phys. Rev. Lett. **93**, 267007 (2004). [CrossRef]

*ω*and

_{pu}*ω*, amplitude

_{pr}*ɛ*and

_{pu}*ε*) are simultaneously applied to a microwave (WM) line [30

_{pr}30. I. Chiorescu, Y. Nakamura, C. J. P. M. Harmansand, and J. E. Mooij, “Coherent quantum dynamics of a super-conducting flux qubit, ” Science **299**, 1869–1871 (2003). [CrossRef] [PubMed]

*I*is applied to the MW line to control the magnetic flux through the SQUID loop and the effective Josephson coupling of the CPB qubit. The Hamiltonian of the total system can be written as [22

_{b}22. P. Zhang, Y. D. Wang, and C. P. Sun, “Quantum measurement of a coupled nanomechanical resonator-Cooper-pair box system,” Phys. Rev. B **68**, 155311 (2003). [CrossRef]

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

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

*H*is the Hamiltonian of the nanomechanical resonator,

_{NR}*a*

^{†}and

*a*are the phonon creation and annihilation operators of the NR.

*H*is the Hamiltonian of CPB qubit which can be characterized by the pseudospin operators

_{CPB}*σ*and

_{z}*σ*=

_{x}*σ*

_{+}+

*σ*

_{−}.

*E*is the maximum Josephson energy.

_{J}*h̄ω*= 4

_{q}*E*(2

_{c}*n*− 1) is the electrostatic energy,

_{c}*C*

_{Σ}=

*C*+

_{b}*C*+ 2

_{g}*C*is the total CPB capacitance.

_{J}*C*,

_{b}*C*,

_{g}*C*are, respectively, the capacitance between the NR and the CPB island, the gate capacitance of the CPB qubit, and the capacitance of each Josephson junction.

_{J}*n*= (

_{c}*C*+

_{b}V_{b}*C*)/(2

_{g}V_{g}*e*) is the dimensionless gate charge, where

*V*is the voltage between the NR and the CPB island, and

_{b}*V*is the gate voltage of the CPB qubit. Then

_{g}*n*can be precisely tuned to give proper qubit performance by adjusting the

_{c}*V*and

_{b}*V*. Displacement

_{g}*x*of the NR gives rise to linear modulation of the capacitance between NR and CPB island,

*C*(

_{b}*x*) ≃

*C*(0) + (

_{b}*∂C*), which modulates the electrostatic energy of CPB and then lead to modlulate the capacitive coupling constant

_{b}/∂x*x*is the zero-point uncertainty of the NR. The coupling between the MW line and CPB qubit in the second term of Eq. (3) results from the totally applied magnetic flux Φ

_{zp}*(*

_{x}*t*) =Φ

*(*

_{q}*t*) + Φ

*through the CPB qubit loop of an effective area*

_{b}*S*with Φ

_{0}=

*h̄*/(2

*e*) being the flux quantum [31

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

*(*

_{q}*t*) =

*μ*

_{0}

*SI*(

*t*)/(2

*πl*),

*l*is the distance between the MW line and the qubit and

*μ*

_{0}is the vacuum permeability. Φ

*(*

_{q}*t*) and Φ

*can be controlled by the MW current*

_{b}*I*(

*t*) =

*ε*(

_{pu}cos*ω*) +

_{pu}t*ε*(

_{pr}cos*ω*+

_{pr}t*θ*) and the direct current

*I*in the MW line, respectively. For simplicity, we suppose the phase factor

_{b}*θ*= 0 as it is not difficult to find that the results of this paper do not depend on the value of

*θ*. Modulating the current

*I*and the MW current

_{b}*I*(

*t*) satisfy Φ

*≫ Φ*

_{b}*(*

_{q}*t*) and

*ω*of pump current, the Hamiltonian of the total system becomes where

_{pu}*δ*=

*ω*−

_{pr}*ω*is the detuning of probe current and the pump current, Δ=

_{pu}*ω*−

_{q}*ω*is the detuning of the qubit resonance and the pump current. In analogy to the case of a two-level atom driven by bichromatic electromagnetic waves, here,

_{pu}*μ*= (

*μ*

_{0}

*Sh̄E*)

_{J}*/*(8

*l*Φ

_{0}) is the effective “electric dipole moment” of the qubit.

*q*=

*a*

^{†}+

*a*. By using the Heisenberg equation

*σ*,

_{z}*σ*

_{±}] = ±

*σ*

_{±}, [

*σ*

_{+},

*σ*

_{−}] =

*σ*, [

_{z}*a*,

*a*

^{†}] = 1, we can obtain the equations of motion for

*σ*

_{−},

*σ*and

_{z}*q*[33

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

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

*O*(

*t*)〉 with variable

*O*(

*t*). The resulting equations of motion are as follows: In obtaining above equations we have taken the semiclassical approach by factorizing the NR and CPB qubit degrees,

*i.e.*〈

*qσ*

_{−}〉 = 〈

*q*〉〈

*σ*

_{−}〉, which ignores correlation between these systems. In above equations,

*T*

_{1}is the CPB qubit relaxation time,

*T*

_{2}is the CPB qubit dephasing time,

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

*Q*is the quality factor. To solve above equations we make following assumptions where each solution contains three item

*O*

_{0},

*O*

_{+1},

*O*

_{−1}(with

*O*=

*σ*

_{−},

*σ*,

_{z}*q*), corresponding to the responses at the frequencies

*ω*,

_{pu}*ω*, and 2

_{pr}*ω*−

_{pu}*ω*, respectively [35

_{pr}35. S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A **83**, 023823 (2011). [CrossRef]

*O*

_{0}≫

*O*

_{±1}, Eqs. (6), (7) and (8) can be solved by treating

*O*

_{±1}as perturbation. After substituting Eqs. (9), (10) and (11) into Eqs. (6), (7) and (8) and ignoring the second-order small terms, we can obtain the steady-state mean values of the system as the population inversion

*σ*

_{+1}is

13. J. J. Li and K. D. Zhu, “An efficient optical knob from slow light to fast in a coupled nanomechanical resonator-quantum dot system,” Opt. Express **17**, 19874–19881 (2009). [CrossRef] [PubMed]

*χ*

^{(1)}(

*ω*) is the dimensionless linear susceptibility. In all above equations

_{pr}*γ*

_{0}=

*γ*

_{n}T_{2}, Ω

_{0}=Ω

*T*

_{2},

*δ*

_{0}=

*δT*

_{2}, Δ

_{0}=Δ

*T*

_{2}, and

36. R. S. Bennink, R. W. Boyd, C. R. Stroud, and 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, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A **46**, R29–R32 (1992). [CrossRef] [PubMed]

## 3. Numerical results and discussion

*E*/

_{c}*h̄*= 2

*π*× 40

*GHz*and

*E*/

_{J}*h̄*= 2

*π*× 4

*GHz*such that

*E*≫

_{c}*E*[38

_{J}38. P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering, ” Phys. Rev. B **70**, 205304 (2004). [CrossRef]

*μs*and the coherence time of a superposition state is as long as 0.5

*μs*,

*i.e. T*

_{1}= 2

*μs*and

*T*

_{2}= 0.5

*μs*[11

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

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

*ω*= 2

_{n}*π*×133

*MHz*, the quality factor

*Q*= 5000 [15

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

*λ*= 0.1

*ω*= 2

_{n}*π*× 13.3

*MHz*[22

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

*S*= 1

*μm*

^{2},

*l*= 10

*μm*, and

*ε*= 200

_{pu}*μA*we have

_{0}=Ω

*T*

_{2}= (

*με*

_{pu}T_{2})/

*h̄*= 3.

*n*(in units of Σ) as a function of the effective Rabi frequency Ω

_{g}^{2}and the parameters used are the same as in Fig. 2. It is clear that near Ω

^{2}= 0.05(

*MHz*)

^{2}, the slow light index can be obtained as 600. That is, the output will be 600 times slower than the input. The physical origin of this result is due to the so called mechanically induced coherent population oscillation, which induces quantum interference between the resonator and two MW currents (pump and probe field). The simultaneous presence of pump and probe fields generates a radiation force at the NR frequency

*ω*. The condition Δ

_{n}*=*

_{pu}*ω*just corresponds to that the pump field couples to the optical transition via the Stokes process and the system becomes fully transparent to the probe field. On the other hand, the displace

_{n}*x*of NR from equilibrium position alters the capacitance of the CPB qubit and its resonance frequency. In this case, the system is similar to the conventional three-level systems in EIT studies [40

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

*= 0. In order to illustrate it more clearly, we plot Figs. 4 and 5 with the same experimental data as in Fig. 2. In Fig. 4, we also describe the theoretical variation of (*

_{pu}*Imχ*

^{(1)}) and (

*Reχ*

^{(1)}) as a function of detuning Δ

*when the detuning Δ*

_{pr}*= 0. We can find that Fig. 4(a) is similar to Yuan*

_{pu}*et.al.*[32

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

*= 0. Figure 5 shows the group velocity index*

_{pr}*n*(in units of Σ) of fast light as a function of efficient Rabi frequency Ω

_{g}^{2}.

*ω*=

_{pr}*ω*and scan the pump frequency from Δ

_{q}*=*

_{pu}*ω*to Δ

_{n}*= 0, then one can efficiently switch the probe field from slow to fast.*

_{pu}## 4. Conclusion

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

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

## Acknowledgments

## References and links

1. | R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science |

2. | A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency: propagation dynamics,” Phys. Rev. Lett. |

3. | S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. |

4. | 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 |

5. | M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. |

6. | D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. |

7. | A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. |

8. | M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. |

9. | P. C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S. W. Chang, and S. L. Chuang, “Slow light in semiconductor quantum wells,” Opt. Lett. |

10. | M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science |

11. | K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today |

12. | A. N. Cleland and M. L. Roukes, “Fabrication of high frequency nanometer scale mechanical resonators from bulk Si crystals,” Appl. Phys. Lett. |

13. | J. J. Li and K. D. Zhu, “An efficient optical knob from slow light to fast in a coupled nanomechanical resonator-quantum dot system,” Opt. Express |

14. | Y. J. Wang, M. Eardley, S. Knappe, J. Moreland, L. Hollberg, and J. Kitching, “Magnetic resonance in an atomic vapor excited by a mechanical resonator,” Phys. Rev. Lett. |

15. | Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, and M. L. Roukes, “Zeptogram-scalenanomechanical mass sensing,” Nano Lett. |

16. | 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, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature |

17. | I. Wilson-Rae, P. Zoller, and A. Imamoglu, “Laser cooling of a nanomechanical resonator mode to its quantum ground state,” Phys. Rev. Lett. |

18. | D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. |

19. | A. H. Safavi-Naeini, T. P. Mayer 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 |

20. | X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. |

21. | A. D. Armour, M. P. Blencow, and K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-pair box,” Phys. Rev. Lett. |

22. | P. Zhang, Y. D. Wang, and C. P. Sun, “Quantum measurement of a coupled nanomechanical resonator-Cooper-pair box system,” Phys. Rev. B |

23. | P. Zhang, Y. D. Wang, and C. P. Sun, “Cooling mechanism for a nanomechanical resonator by periodic coupling to a Cooper pair box,” Phys. Rev. Lett. |

24. | M. D. LaHaye, J. Suh, P. M. Echternach, K. C. Schwab, and M. L. Roukes, “Nanomechanical measurements of a superconducting qubit,” Nature |

25. | J. Suh, M. D. LaHaye, P. M. Echternach, K. C. Schwab, and M. L. Roukes, “Parametric amplification and back-action noise squeezing by a qubit-coupled nanoresonator,” Nano Lett. |

26. | W. Xue, S. Sales, J. Capmany, and J. Mork, “Microwave phase shifter with controllable power response based on slow-and fast-light effects in semiconductor optical amplifiers,” Opt. Lett. |

27. | L. Wei, W. Xue, Y. Chen, T. T. Alkeskjold, and A. Bjarklev, “Optically fed microwave true-time delay based on a compact liquid-crystal photonic-bandgap-fiber device,” Opt. Lett. |

28. | Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature |

29. | O. Astafiev 1, Y. A. Pashkin, Y. Nakamura, T. Yamamoto, and J. S. Tsai, “Quantum noise in the Josephson charge qubit,” Phys. Rev. Lett. |

30. | I. Chiorescu, Y. Nakamura, C. J. P. M. Harmansand, and J. E. Mooij, “Coherent quantum dynamics of a super-conducting flux qubit, ” Science |

31. | C. P. Sun, L. F. Wei, Y. X. Liu, and F. Nori, “Quantum transducers: Integrating transmission lines and nanomechanical resonators via charge qubits,” Phys. Rev. A |

32. | X. Z. Yuan, H. S. Goan, C. H. Lin, K. D. Zhu, and Y. W. Jiang, “Nanomechanical-resonator-assisted induced transparency in a Cooper-pair box system,” New J. Phys. |

33. | G. S. Agarwal, “Electromagnetic-field-induced transparency in high-density exciton systems,” Phys. Rev. A |

34. | G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A |

35. | S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A |

36. | R. S. Bennink, R. W. Boyd, C. R. Stroud, and V. Wong, “Enhanced self-action effects by electromagnetically induced transparency in the two-level atom,” Phys. Rev. A |

37. | S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A |

38. | P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering, ” Phys. Rev. B |

39. | J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature |

40. | M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. |

**OCIS Codes**

(140.4780) Lasers and laser optics : Optical resonators

(230.1150) Optical devices : All-optical devices

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Slow and Fast Light

**History**

Original Manuscript: December 30, 2013

Revised Manuscript: January 27, 2014

Manuscript Accepted: January 28, 2014

Published: February 6, 2014

**Citation**

Peng-Cheng Ma, Yin Xiao, Ya-Fei Yu, and Zhi-Ming Zhang, "Microwave field controlled slow and fast light with a coupled system consisting of a nanomechanical resonator and a Cooper-pair box," Opt. Express **22**, 3621-3628 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3621

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