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

Research on slow and fast light systems has increased from both theoretical and experimental aspects in physics [1,2]. The first superluminal light propagation was observed in a resonant system [3], where the laser propagates without appreciable shape distortion but experiences very strong resonant absorption. Various techniques have been developed to realized slow and fast light in atomic vapors [46] and solid materials [7,8]. To reduce absorption, most of those works [47] are based on the electromagnetically induced transparency (EIT) or coherent population oscillation (CPO) [9]. However, EIT-based slow light in general has limitations in potential applications of ultrahigh speed information processing due to its narrow transparency spectrum. Coherent population oscillation (CPO) was introduced as a robust physical mechanism to overcome the defect of EIT and had less limitations to achieve ultraslow light in solids [8, 10].

On the other hand, nanomechanical resonators are currently under rapid development owing to their combination of large quality factors (103 − 106) and high natural frequencies (MHzGHz) together with important applications [1113], such as high precision measurement [14], zeptogram-scale nanomechanical mass sensing [15], quantum measurement [16] and laser cooling of a NR mode to its quantum ground state [17]. Recently, slow light with an optomechanical crystal array has been researched both theoretically and experimentally [18,19]. More recently, slowing and advancing of light has been experimentally realized in microwave regime using circuit nanoelectromechanics system [20]. In the present paper, we theoretically investigate the slow and fast light effects in microwave regime with the NR and CPB coupled system. Much attention has been paid to this coupled system, such as to probe the decay of the NR [21], to prepare the NR in a Fock state [22], and to cool the NR to its ground state [23]. Further more, this coupled NR-CPB system has been realized in experiments [24,25]. Based on the above-mentioned achievements, in this paper, we study the slow and superluminal light in microwave regime with this NR-CPB coupled system by changing the detuning of pump current from CPB qubit frequency. The slow and fast light can be switched by simply adjusting a proper pump-CPB detuning from the NR frequency to zero. Our results indicate some potential applications in microwave photonics [26,27], nonlinear optics, and optical communication.

The organization of this paper is as follows: In Section 2 we introduce the theoretical mode and derive the analytical expressions. In Section 3 we discuss the results according to numerical calculations. Section 4 presents the conclusions.

2. Theoretical mode and analytical expressions

We begin with the Hamiltonian approximating the NR-CPB coupled system. As shown schematically in Fig. 1, the NR is capacitively coupled to a CPB qubit consisting of two Josephson junctions forming a SQUID loop [28,29], which allows us to control its effective Josephson energy with a small external magnetic field or magnetic flux. Two microwave currents (pump and probe current with frequency ωpu and ωpr, amplitude ɛpu and εpr) are simultaneously applied to a microwave (WM) line [30] beside the CPB to induce the oscillating magnetic fields in the Josephson junction SQUID loop of the CPB qubit. Also, a direct current Ib 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, 31, 32],

Htotal=HNR+HCPB+Hint,
HNR=h¯ωnaa,
HCPB=12h¯ωqσz12EJcos[πΦx(t)ϕ0]σx,
Hint=h¯λ(a+a)σz,
where HNR is the Hamiltonian of the nanomechanical resonator, a and a are the phonon creation and annihilation operators of the NR. HCPB is the Hamiltonian of CPB qubit which can be characterized by the pseudospin operators σz and σx = σ+ + σ. EJ is the maximum Josephson energy. h̄ωq = 4Ec(2nc − 1) is the electrostatic energy, Ec=e22CΣ is the charging energy with CΣ = Cb + Cg + 2CJ is the total CPB capacitance. Cb, Cg, CJ 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. nc = (CbVb + CgVg)/(2e) is the dimensionless gate charge, where Vb is the voltage between the NR and the CPB island, and Vg is the gate voltage of the CPB qubit. Then nc can be precisely tuned to give proper qubit performance by adjusting the Vb and Vg. Displacement x of the NR gives rise to linear modulation of the capacitance between NR and CPB island, Cb(x) ≃ Cb(0) + (∂Cb/∂x), which modulates the electrostatic energy of CPB and then lead to modlulate the capacitive coupling constant λ=2CgVgEceh¯CbCbxΔxzp with Δxzp 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 Φx(t) =Φq(t) + Φb through the CPB qubit loop of an effective area S with Φ0 = /(2e) being the flux quantum [31]. Here, Φq(t) = μ0SI(t)/(2πl), l is the distance between the MW line and the qubit and μ0 is the vacuum permeability. Φq(t) and Φb can be controlled by the MW current I(t) = εpucos(ωput) + εprcos(ωprt + θ) and the direct current Ib in the MW line, respectively. For simplicity, we suppose the phase factor 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 Ib and the MW current I(t) satisfy Φb ≫ Φq(t) and Φb/Φ0=12, we get EJcos[πΦx(t)ϕ0]EJπΦq(t)ϕ0, Applying a frame rotating at the frequency ωpu of pump current, the Hamiltonian of the total system becomes
Hpu=12h¯Δσz+h¯ωnaa+h¯λ(a+a)σz+h¯Ω(σ++σ)+μεpr(σ+eiδt+σeiδt),
where δ = ωprωpu is the detuning of probe current and the pump current, Δ= ωqωpu 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, Ω=μεpuh¯ is the effective“ Rabi frequency” of the pump current, and μ = (μ0Sh̄EJ)/(8lΦ0) is the effective “electric dipole moment” of the qubit.

 

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.

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We set q = a + a. By using the Heisenberg equation ih¯dOdt=[O,H] and the commutation relation [σz, σ±] = ±σ±, [σ+, σ] = σz, [a, a] = 1, we can obtain the equations of motion for σ, σz and q [33,34]. Then we add, phenomenally, the decay rates to these equations and take the average of these equations. For convenience, in the following we denote the average value 〈O(t)〉 with variable O(t). The resulting equations of motion are as follows:

dσdt=(1T2+iΔ+iq)σ+iΩσz+iμεprh¯eiδtσz,
dσzdt=(σz+1)1T12iΩ(σ+σ)2iμh¯(σ+εpreiδtσεpr*eiδt),
d2qdt2+γndqdt+ωn2q=4ωnλ2σz
In obtaining above equations we have taken the semiclassical approach by factorizing the NR and CPB qubit degrees, i.e.〉 = 〈q〉〈σ〉, which ignores correlation between these systems. In above equations, T1 is the CPB qubit relaxation time, T2 is the CPB qubit dephasing time, γn=ωnQ is the decay of the NR due to the coupling to a reservoir of “background” modes and other intrinsic processes [17], with Q is the quality factor. To solve above equations we make following assumptions
σ(t)=σ0+σ+1eiδt+σ1eiδt,
σz(t)=σ0z+σ+1zeiδt+σ1zeiδt,
q(t)=q0+q+1eiδt+q1eiδt,
where each solution contains three item O0, O+1, O−1 (with O = σ, σz, q), corresponding to the responses at the frequencies ωpu, ωpr, and 2ωpuωpr, respectively [35]. Supposing O0O±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
q0=4λ0ωnk0,σ0=iμΩk01+i(Δ4λ0ω0k0),
the population inversion k0=σ0z of the CPB is determined by the following equation:
(k0+1)[(Δ04λ0ω0k0)2+1]+4T1/T2Ωc2k0=0.
and the solution of σ+1 is
σ+1=T2μεprh¯8λ0ω0ηT1/T2Ω02k02θ(1+iΔ04iλ0ω0k0)(T1/T2iδ012T1/T2β)+2iT1/T2Ω02k0θT1/T2iδ012T1/T2β+ik01+iΔ04iλ0ω0k0iδ0,
which corresponds to the effective linear susceptibility as follow [13]:
χeff(1)(ωpr)=μσ+1εpr=T2μ2h¯χ(1)(ωpr),
where χ(1)(ωpr) is the dimensionless linear susceptibility. In all above equations λ0=λ2ωn2, γ0 = γnT2, Ω0T2, δ0 = δT2, Δ0T2, and
η=ω02ω02iγ0δ0δ02,
β=4iλ0ω0k0ηΩ021+iΔ04iλ0ω0k0+ω021+iΔ04iλ0ω0k0iδ0+4iλ0ω0k0ηΩ021iΔ0+4iλ0ω0k0+ω021iΔ0+4iλ0ω0k0iδ0,
θ=11+iΔ04iλ0ω0k0iδ0+11iΔ0+4iλ0ω0k0.

In terms of this model, we can determine the light group velocity as [36,37],

vg=cn+ωpr(dndωpr),
where n1+2πχeff(1), and we can get
cvg=1+2πReχeff(1)(ωpr)ωpr=ωq+2πωprRe(dχeff(1)dωpr)ωpr=ωq.
When Reχeff(1)(ωpr)ωpr=ωq=0, the above equation reduces to
cvg1=2πωprμ2T2h¯Re(dχ(1)(ωpr)dωpr)ωpr=ωq=1T2ΣRe(dχ(1)(ωpr)dωpr)ωpr=ωq,
where Eq. (15) has been used and Σ=2πωprμ2T22h¯. We can find from this expression that when the dispersion is steeply positive or negative, the group velocity can be significantly reduced or increased. In the following section we will present some numerical results.

3. Numerical results and discussion

For illustration of the numerical results, we choose the realistically reasonable parameters to demonstrate the slow and fast light effect based on the coupled NR-CPB system. All the parameters used here are accessible in experiment. Typical parameters of the CPB charge qubit are designed Ec/ = 2π × 40GHz and EJ/ = 2π × 4GHz such that EcEJ [38]. Experiments by many researchers have demonstrated that the CPB’ s excited state has a lifetime of up to 2μs and the coherence time of a superposition state is as long as 0.5μs, i.e. T1 = 2μs and T2 = 0.5μs [11, 39]. The NR resonance frequency ωn = 2π×133MHz, the quality factor Q = 5000 [15], the coupling constant λ = 0.1ωn = 2π × 13.3MHz [22]. For S = 1μm2, l = 10μm, and εpu = 200μA we have μh¯=μ0SEJ8h¯lϕ030GHzA1 and Ω0T2 = (μεpuT2)/ = 3.

Figure 2 illustrates the behavior of 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. We find obviously that at Δpr = 0, there is a steep positive slope related to zero absorption. This large dispersive characteristics can lead to the possibility of implementation of slow light effect.

 

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.

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Figure 3 shows the group velocity ng(in units of Σ) as a function of the effective Rabi frequency Ω2 and the parameters used are the same as in Fig. 2. It is clear that near Ω2 = 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 ωn. The condition Δpu = ωn 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 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]. Therefore, in our structure one can obtain the slow output light without absorption by only adjusting the pump-CPB detuning to the frequency of nanomechanical resonator.

 

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.

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Similarly, the NR-CPB system can also implement the superluminal effect without absorption when the pump current detuning Δpu = 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 (Imχ(1)) and (Reχ(1)) as a function of detuning Δpr when the detuning Δpu = 0. We can find that Fig. 4(a) is similar to Yuan et.al. [32] which also describes the absorbtion in this coupled system. From Fig. 4 we can find that the large dispersion relates to a very steep negative slope. It means the superluminal effect without absorption at Δpr = 0. Figure 5 shows the group velocity index ng(in units of Σ) of fast light as a function of efficient Rabi frequency Ω2.

 

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.

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Fig. 5 The group velocity index ng=cvg (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.

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According to above discussions, it can be found clearly that the NR-CPB coupled system provides us an efficient way to switch between slow light and fast light by simply adjusting the pump detuning in terms of the mechanically induced coherent population oscillation. In experiments, one can fix the probe field with frequency ωpr = ωq and scan the pump frequency from Δpu = ωn to Δpu = 0, then one can efficiently switch the probe field from slow to fast.

4. Conclusion

In conclusion, we have investigated the tunable superluminal and slow light effects in microwave regime with a coupled NR-CPB system. It can provide us an efficient and convenient way to switch between slow and fast light. The greatest advantage of our system is that we can efficiently switch from slow to fast light by only adjusting the pump-CPB deturning from the NR frequency to zero. Our scheme may have potential applications in various applications such as optical communication, microwave photonics and nonlinear optics.

Finally, we hope that the results of this paper can be tested by experiments in the near future. Recently, LaHaye et al.[24] and Suh et al.[25] have reported experimental results on nanomechanical measurements of a superconducting qubit, maybe one can use a similar experimental setup to test our predicted effects.

Acknowledgments

This work was supported by the Major Research Plan of the NSFC (Grant No. 91121023), the NSFC (Grant Nos. 61378012 and 60978009), the SRFDPHEC (Grant No. 20124407110009), the “973” Program (Grant Nos. 2011CBA00200 and 2013CB921804), the PCSIRT (Grant No. IRT1243), the SRFGS of SCNU (Grant No. 2012kyjj119), the NSF of JHEI (Grant No. 12KJD140002), and the PFET of HNU (Grant No. 11HSQNZ07).

References and links

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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. 82, 5229–5232 (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]  

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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]  

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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]  

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

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

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References

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  1. R. W. Boyd, D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
    [CrossRef] [PubMed]
  2. A. Kasapi, M. Jain, G. Y. Yin, S. E. Harris, “Electromagnetically induced transparency: propagation dynamics,” Phys. Rev. Lett. 74, 2447–2450 (1995).
    [CrossRef] [PubMed]
  3. S. Chu, 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, C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
    [CrossRef]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. K. C. Schwab, M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58, 36–42 (2005).
    [CrossRef]
  12. 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).
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  15. Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, M. L. Roukes, “Zeptogram-scalenanomechanical mass sensing,” Nano Lett. 6, 583–586 (2006).
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  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, A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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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]

Goan, H. S.

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]

Gross, R.

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]

Hafezi, M.

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]

Ham, B. S.

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]

Harmansand, C. J. P. M.

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]

Harris, S. E.

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]

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

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

Hau, L. V.

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]

Hemmer, P. R.

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]

Hill, J. T.

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).
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Hocke, F.

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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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