## 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 [4–6] and solid materials [7,8]. To reduce absorption, most of those works [4–7] 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 (10^{3} − 10^{6}) and high natural frequencies (*MHz* − *GHz*) together with important applications [11–13], 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 *ω _{pu}* and

*ω*, amplitude

_{pr}*ɛ*and

_{pu}*ε*) 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

_{pr}*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, 31, 32],

_{b}*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, ${E}_{c}=\frac{{e}^{2}}{2{C}_{\mathrm{\Sigma}}}$ is the charging energy with

_{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 $\lambda =\frac{2{C}_{g}{V}_{g}{E}_{c}}{e\overline{h}{C}_{b}}\frac{\partial {C}_{b}}{\partial x}\mathrm{\Delta}{x}_{zp}$ with Δ

_{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]. Here, Φ

*(*

_{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 ${\mathrm{\Phi}}_{b}/{\mathrm{\Phi}}_{0}=\frac{1}{2}$, we get ${E}_{J}\mathit{cos}\left[\frac{\pi {\mathrm{\Phi}}_{x}(t)}{{\varphi}_{0}}\right]\simeq -{E}_{J}\frac{\pi {\mathrm{\Phi}}_{q}(t)}{{\varphi}_{0}}$, Applying a frame rotating at the frequency

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

_{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, $\mathrm{\Omega}=\frac{\mu {\epsilon}_{pu}}{\overline{h}}$ is the effective“ Rabi frequency” of the pump current, and

_{pu}*μ*= (

*μ*

_{0}

*Sh̄E*)

_{J}*/*(8

*l*Φ

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

We set *q* = *a*^{†} + *a*. By using the Heisenberg equation
$i\overline{h}\frac{dO}{dt}=[O,H]$ and the commutation relation [*σ _{z}*,

*σ*

_{±}] = ±

*σ*

_{±}, [

*σ*

_{+},

*σ*

_{−}] =

*σ*, [

_{z}*a*,

*a*

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

*σ*

_{−},

*σ*and

_{z}*q*[33,34]. 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:

*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, ${\gamma}_{n}=\frac{{\omega}_{n}}{Q}$ 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

*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]. Supposing

_{pr}*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

*σ*

_{+1}is

*χ*

^{(1)}(

*ω*) is the dimensionless linear susceptibility. In all above equations ${\lambda}_{0}=\frac{{\lambda}^{2}}{{\omega}_{n}^{2}}$,

_{pr}*γ*

_{0}=

*γ*

_{n}T_{2}, Ω

_{0}=Ω

*T*

_{2},

*δ*

_{0}=

*δT*

_{2}, Δ

_{0}=Δ

*T*

_{2}, and

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

where $n\simeq 1+2\pi {\chi}_{\mathit{eff}}^{(1)}$, and we can get## 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 *E _{c}*/

*h̄*= 2

*π*× 40

*GHz*and

*E*/

_{J}*h̄*= 2

*π*× 4

*GHz*such that

*E*≫

_{c}*E*[38]. Experiments by many researchers have demonstrated that the CPB’ s excited state has a lifetime of up to 2

_{J}*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, 39]. The NR resonance frequency

*ω*= 2

_{n}*π*×133

*MHz*, the quality factor

*Q*= 5000 [15], the coupling constant

*λ*= 0.1

*ω*= 2

_{n}*π*× 13.3

*MHz*[22]. For

*S*= 1

*μm*

^{2},

*l*= 10

*μm*, and

*ε*= 200

_{pu}*μA*we have $\frac{\mu}{\overline{h}}=\frac{{\mu}_{0}S\hspace{0.17em}{E}_{J}}{8\overline{h}l{\varphi}_{0}}\simeq 30GHz{A}^{-1}$ and Ω

_{0}=Ω

*T*

_{2}= (

*με*

_{pu}T_{2})/

*h̄*= 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 Δ* _{pr}* =

*ω*−

_{q}*ω*for Δ

_{pr}*=*

_{pu}*ω*with parameters

_{n}*λ*

_{0}= 0.01,

*Q*=5000,

*ω*= 2

_{n}*π*× 133

*MHz*,

*T*

_{1}= 0.25

*μs*,

*T*

_{2}= 0.05

*μs*, Ω

_{0}= 3. We find obviously that at Δ

*= 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.*

_{pr}Figure 3 shows the group velocity *n _{g}*(in units of Σ) as a function of the effective Rabi frequency Ω

^{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]. 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.

Similarly, the NR-CPB system can also implement the superluminal effect without absorption when the pump current detuning Δ* _{pu}* = 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 Δ

*when the detuning Δ*

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

_{pu}*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 Δ

*= 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}.

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 *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

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).

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