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

Peng-Cheng Ma; Yin Xiao; Ya-Fei Yu; Zhi-Ming Zhang

doi:10.1364/OE.22.003621

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

Peng-Cheng Ma, Yin Xiao, Ya-Fei Yu, and Zhi-Ming Zhang

Optics Express
Vol. 22,
Issue 3,
pp. 3621-3628
(2014)
•https://doi.org/10.1364/OE.22.003621

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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)

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Table of Contents Category
- Slow and Fast Light
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Optical resonators (140.4780)
- All-optical devices (230.1150)
- Micro-optical devices (230.3990)

About this Article
History
- Original Manuscript: December 30, 2013
- Revised Manuscript: January 27, 2014
- Manuscript Accepted: January 28, 2014
- Published: February 6, 2014

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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.

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 (10³ − 10⁶) 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_pr, amplitude ɛ_pu and ε_pr) 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 I_b 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],

H_{total} = H_{N R} + H_{C P B} + H_{int},

H_{N R} = \bar{h} ω_{n} a^{†} a,

H_{C P B} = \frac{1}{2} \bar{h} ω_{q} σ_{z} - \frac{1}{2} E_{J} \cos [\frac{π Φ_{x} (t)}{ϕ_{0}}] σ_{x},

H_{int} = \bar{h} λ (a^{†} + a) σ_{z},

where H_NR is the Hamiltonian of the nanomechanical resonator, a^† and a are the phonon creation and annihilation operators of the NR. H_CPB is the Hamiltonian of CPB qubit which can be characterized by the pseudospin operators σ_z and σ_x = σ₊ + σ₋. E_J is the maximum Josephson energy. h̄ω_q = 4E_c(2n_c − 1) is the electrostatic energy,

E_{c} = \frac{e^{2}}{2 C_{Σ}}

is the charging energy with C_Σ = C_b + C_g + 2C_J is the total CPB capacitance. C_b, C_g, C_J 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. n_c = (C_bV_b + C_gV_g)/(2e) is the dimensionless gate charge, where V_b is the voltage between the NR and the CPB island, and V_g is the gate voltage of the CPB qubit. Then n_c can be precisely tuned to give proper qubit performance by adjusting the V_b and V_g. Displacement x of the NR gives rise to linear modulation of the capacitance between NR and CPB island, C_b(x) ≃ C_b(0) + (∂C_b/∂x), which modulates the electrostatic energy of CPB and then lead to modlulate the capacitive coupling constant

λ = \frac{2 C_{g} V_{g} E_{c}}{e \bar{h} C_{b}} \frac{\partial C_{b}}{\partial x} Δ x_{z p}

with Δx_zp 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_x(t) =Φ_q(t) + Φ_b through the CPB qubit loop of an effective area S with Φ₀ = h̄/(2e) being the flux quantum [31]. Here, Φ_q(t) = μ₀SI(t)/(2πl), l is the distance between the MW line and the qubit and μ₀ is the vacuum permeability. Φ_q(t) and Φ_b can be controlled by the MW current I(t) = ε_pucos(ω_put) + ε_prcos(ω_prt + θ) and the direct current I_b 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 I_b and the MW current I(t) satisfy Φ_b ≫ Φ_q(t) and

Φ_{b} / Φ_{0} = \frac{1}{2}

, we get

E_{J} \cos [\frac{π Φ_{x} (t)}{ϕ_{0}}] ≃ - E_{J} \frac{π Φ_{q} (t)}{ϕ_{0}}

, Applying a frame rotating at the frequency ω_pu of pump current, the Hamiltonian of the total system becomes

H_{p u} = \frac{1}{2} \bar{h} Δ σ_{z} + \bar{h} ω_{n} a^{†} a + \bar{h} λ (a^{†} + a) σ_{z} + \bar{h} Ω (σ_{+} + σ_{-}) + μ ε_{p r} (σ_{+} e^{- i δ t} + σ_{-} e^{i δ t}),

where δ = ω_pr − ω_pu is the detuning of probe current and the pump current, Δ= ω_q − ω_pu 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 = \frac{μ ε_{p u}}{\bar{h}}

is the effective“ Rabi frequency” of the pump current, and μ = (μ₀Sh̄E_J)/(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 ω_pu and ω_pr and a direct current I_b are applied to the microwave line beside the CPB to control the flux Φ_x through the CPB loop.

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We set q = a^† + a. By using the Heisenberg equation $i \bar{h} \frac{d O}{d t} = [O, H]$ 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_z 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:

\frac{d σ_{-}}{d t} = - (\frac{1}{T_{2}} + i Δ + i q) σ_{-} + i Ω σ_{z} + \frac{i μ ε_{p r}}{\bar{h}} e^{- i δ t} σ_{z},

\frac{d σ_{z}}{d t} = - (σ_{z} + 1) \frac{1}{T_{1}} - 2 i Ω (σ_{+} - σ_{-}) - \frac{2 i μ}{\bar{h}} (σ_{+} ε_{p r} e^{- i δ t} - σ_{-} ε_{p r}^{*} e^{i δ t}),

\frac{d^{2} q}{d t^{2}} + γ_{n} \frac{d q}{d t} + ω_{n}^{2} q = - 4 ω_{n} λ^{2} σ_{z}

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, T₁ is the CPB qubit relaxation time, T₂ is the CPB qubit dephasing time,

γ_{n} = \frac{ω_{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

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},

where each solution contains three item O₀, O₊₁, O₋₁ (with O = σ₋, σ_z, q), corresponding to the responses at the frequencies ω_pu, ω_pr, and 2ω_pu − ω_pr, respectively [35]. Supposing O₀ ≫ 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

q_{0} = - 4 λ_{0} ω_{n} k_{0}, σ_{0} = \frac{i μ Ω k_{0}}{1 + i (Δ - 4 λ_{0} ω_{0} k_{0})},

the population inversion

k_{0} = σ_{0}^{z}

of the CPB is determined by the following equation:

(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.

and the solution of σ₊₁ is

σ_{+ 1} = \frac{T_{2} μ ε_{p r}}{\bar{h}} \frac{\frac{- 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)} + \frac{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}},

which corresponds to the effective linear susceptibility as follow [13]:

χ_{eff}^{(1)} (ω_{p r}) = \frac{μ σ_{+ 1}}{ε_{p r}} = \frac{T_{2} μ^{2}}{\bar{h}} χ^{(1)} (ω_{p r}),

where χ⁽¹⁾(ω_pr) is the dimensionless linear susceptibility. In all above equations

λ_{0} = \frac{λ^{2}}{ω_{n}^{2}}

, γ₀ = γ_nT₂, Ω₀ =ΩT₂, δ₀ = δT₂, Δ₀ =ΔT₂, and

η = \frac{ω_{0}^{2}}{ω_{0}^{2} - i γ_{0} δ_{0} - δ_{0}^{2}},

β = \frac{\frac{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}} + \frac{\frac{- 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 = \frac{1}{1 + i Δ_{0} - 4 i λ_{0} ω_{0} k_{0} - i δ_{0}} + \frac{1}{1 - i Δ_{0} + 4 i λ_{0} ω_{0} k_{0}} .

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

v_{g} = \frac{c}{n + ω_{p r} (\frac{d n}{d ω_{p r}})},

where

n ≃ 1 + 2 π χ_{eff}^{(1)}

, and we can get

\frac{c}{v_{g}} = 1 + 2 π Re χ_{eff}^{(1)} {(ω_{p r})}_{ω_{p r} = ω_{q}} + 2 π ω_{p r} Re {(\frac{d χ_{eff}^{(1)}}{d ω_{p r}})}_{ω_{p r} = ω_{q}} .

When

Re χ_{eff}^{(1)} {(ω_{p r})}_{ω_{p r} = ω_{q}} = 0

, the above equation reduces to

\frac{c}{v_{g}} - 1 = 2 π ω_{p r} \frac{μ^{2} T_{2}}{\bar{h}} Re {(\frac{d χ^{(1)} (ω_{p r})}{d ω_{p r}})}_{ω_{p r} = ω_{q}} = \frac{1}{T_{2}} Σ Re {(\frac{d χ^{(1)} (ω_{p r})}{d ω_{p r}})}_{ω_{p r} = ω_{q}},

where Eq. (15) has been used and

Σ = 2 π ω_{p r} \frac{μ^{2} T_{2}^{2}}{\bar{h}}

. 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 E_c/h̄ = 2π × 40GHz and E_J/h̄ = 2π × 4GHz such that E_c ≫ E_J [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 ω_n = 2π×133MHz, the quality factor Q = 5000 [15], the coupling constant λ = 0.1ω_n = 2π × 13.3MHz [22]. For S = 1μm², l = 10μm, and ε_pu = 200μA we have $\frac{μ}{\bar{h}} = \frac{μ_{0} S E_{J}}{8 \bar{h} l ϕ_{0}} ≃ 30 G H z A^{- 1}$ and Ω₀ =ΩT₂ = (με_puT₂)/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_pr for Δ_pu = ω_n with parameters λ₀ = 0.01, Q=5000, ω_n = 2π × 133MHz, T₁ = 0.25μs, T₂ = 0.05μs, Ω₀ = 3. We find obviously that at Δ_pr = 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 Δ_pr = ω_q − ω_pr for Δ_pu = ω_n with parameters λ₀ = 0.01, Q=5000, ω_n = 2π × 133MHz, T₁ = 0.25μs, T₂ = 0.05μs, Ω₀ = 3.

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Figure 3 shows the group velocity n_g(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 ω_n. The condition Δ_pu = ω_n 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 n_g(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 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_pr when the detuning Δ_pu = 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 Δ_pr = 0. Figure 5 shows the group velocity index n_g(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 Δ_pr while detuning Δ_pu = 0. The parameters are λ₀ = 0.01, Q=5000, ω_n = 2π × 133MHz, T₁ = 0.25μs, T₂ = 0.05μs, Ω₀ = 3.

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Fig. 5 The group velocity index $n_{g} = \frac{c}{v_{g}}$ (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.

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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 ω_pr = ω_q and scan the pump frequency from Δ_pu = ω_n to Δ_pu = 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).

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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]

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]

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]

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]

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]

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]

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]

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]

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

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

References

View by:
Article Order
|
Year
|
Author
|
Publication

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[Crossref] [PubMed]
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]
S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[Crossref]
L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 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, 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]
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]
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]
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]
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]
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]
K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Phys. Today 58, 36–42 (2005).
[Crossref]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
O. Astafiev, 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]
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]
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]
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]
G. S. Agarwal, “Electromagnetic-field-induced transparency in high-density exciton systems,” Phys. Rev. A 51, R2711–R2714 (1995).
[Crossref] [PubMed]
G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]
S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83, 023823 (2011).
[Crossref]
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]
S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A 46, R29–R32 (1992).
[Crossref] [PubMed]
P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering, ” Phys. Rev. B 70, 205304 (2004).
[Crossref]
J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature 453, 1031–1042 (2008).
[Crossref] [PubMed]
M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

2013 (1)

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]

2011 (3)

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

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]

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]

2010 (3)

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]

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]

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

2009 (5)

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]

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]

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]

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]

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

2008 (2)

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]

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

2006 (3)

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]

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]

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]

2005 (3)

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

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]

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

2004 (4)

P. Rabl, A. Shnirman, and 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, and J. S. Tsai, “Quantum noise in the Josephson charge qubit,” Phys. Rev. Lett. 93, 267007 (2004).
[Crossref]

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]

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]

2003 (4)

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]

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]

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]

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]

2002 (1)

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]

2001 (2)

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]

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]

1999 (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]

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]

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]

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]

1996 (1)

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]

1995 (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]

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

1992 (1)

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

1982 (1)

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

Agarwal, G. S.

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

G. S. Agarwal and 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.

Armour, A. D.

Astafiev, O.

O. Astafiev, 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]

Behroozi, C. H.

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

Bennink, R. S.

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]

Bialczak, R. C.

Bigelow, M. S.

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]

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]

Bjarklev, A.

Blencow, M. P.

Boyd, R. W.

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

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]

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]

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]

Budker, D.

Callegari, C.

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]

Capmany, J.

Chan, J.

Chang, D. E.

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]

Chang, S. W.

Chang-Hasnain, C. J.

Chen, Y.

Chiorescu, I.

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]

Chu, S.

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

Chuang, S. L.

Clarke, J.

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

Cleland, A. N.

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]

Dutton, Z.

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

Eardley, M.

Echternach, P. M.

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]

Eichenfield, M.

Ekinci, K. L.

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]

Feng, X. L.

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]

Field, J. E.

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

Fleischhauer, M.

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

Fry, E. S.

Gauthier, D. J.

R. W. Boyd and 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, and Y. W. Jiang, “Nanomechanical-resonator-assisted induced transparency in a Cooper-pair box system,” New J. Phys. 10, 095016 (2008).
[Crossref]

Gross, R.

Hafezi, M.

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]

Ham, B. S.

Harmansand, C. J. P. M.

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]

Harris, S. E.

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]

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]

S. E. Harris, J. E. Field, and 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, and 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.

Hill, J. T.

Hocke, F.

Hofheinz, M.

Hollberg, L.

Huang, S.

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

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

Huebl, H.

Imamoglu, A.

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

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).
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Jain, M.

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]

Jiang, Y. W.

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]

Kasapi, A.

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]

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

Kash, M. M.

Kimball, D. F.

Kippenberg, T. J.

Kitching, J.

Knappe, S.

Ku, P. C.

LaHaye, M. D.

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]

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Lepeshkin, N. N.

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]

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]

Li, J. J.

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]

Li, T.

Lin, C. H.

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]

Lin, Q.

Liu, Y. X.

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).
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Lucero, E.

Lukin, M. D.

Marangos, J. P.

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

Marx, A.

Mayer Alegre, T. P.

Mooij, J. E.

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]

Moreland, J.

Mork, J.

Musser, J. A.

Nakamura, Y.

O. Astafiev, 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]

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]

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]

Neeley, M.

Nori, F.

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).
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O’Connell, A. D.

Painter, O.

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]

Palinginis, P.

Pashkin, Y. A.

O. Astafiev, 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]

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).
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Rabl, P.

P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering, ” Phys. Rev. B 70, 205304 (2004).
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Rochester, S. M.

Rostovtsev, Y.

Roukes, M. L.

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).
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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]

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

Safavi-Naeini, A. H.

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]

Sales, S.

Sank, D.

Sautenkov, V. A.

Schliesser, A.

Schwab, K. C.

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]

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

Scully, M. O.

Sedgwick, F.

Shahriar, M. S.

Shnirman, A.

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

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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]

Sudarshanam, V. S.

Suh, J.

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]

Sun, C. P.

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).
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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]

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]

Tsai, J. S.

O. Astafiev, 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]

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]

Turukhin, A. V.

Wang, H.

Wang, Y. D.

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]

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]

Wang, Y. J.

Wei, L.

Wei, L. F.

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]

Weides, M.

Welch, G. R.

Wenner, J.

Wilhelm, F. K.

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

Wilson-Rae, I.

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]

Winger, M.

Wong, S.

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

Wong, V.

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]

Xue, W.

Yamamoto, T.

O. Astafiev, 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]

Yang, Y. T.

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]

Yashchuk, V. V.

Yin, G. Y.

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]

Yuan, X. Z.

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]

Zhang, P.

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]

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]

Zhou, X.

Zhu, K. D.

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]

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]

Zibrov, A. S.

Zoller, P.

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]

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

Appl. Phys. Lett. (1)

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]

Nano Lett. (2)

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]

Nat. Phys. (1)

Nature (6)

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]

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]

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

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]

New J. Phys. (2)

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]

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]

Opt. Express (1)

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]

Opt. Lett. (3)

Phys. Rev. A (6)

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[Crossref] [PubMed]

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

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

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]

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

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]

Phys. Rev. B (2)

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

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]

Phys. Rev. Lett. (11)

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]

O. Astafiev, 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]

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]

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

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]

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]

Phys. Today (1)

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

Rev. Mod. Phys. (1)

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

Science (3)

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]

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

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]

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Fig. 1 Schematic diagram of the coupled NR-CPB system. The microwave currents with frequencies ω_pu and ω_pr and a direct current I_b are applied to the microwave line beside the CPB to control the flux Φ_x through the CPB loop.

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Fig. 2 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_pr for Δ_pu = ω_n with parameters λ₀ = 0.01, Q=5000, ω_n = 2π × 133MHz, T₁ = 0.25μs, T₂ = 0.05μs, Ω₀ = 3.

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Fig. 3 The group velocity index n_g(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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Fig. 4 The dimensionless imaginary part (Imχ⁽¹⁾) and real part (Reχ⁽¹⁾) of linear optical susceptibility as a function of detuning Δ_pr while detuning Δ_pu = 0. The parameters are λ₀ = 0.01, Q=5000, ω_n = 2π × 133MHz, T₁ = 0.25μs, T₂ = 0.05μs, Ω₀ = 3.

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Fig. 5 The group velocity index

n_{g} = \frac{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.

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Equations (21)

Equations on this page are rendered with MathJax. Learn more.

H_{total} = H_{N R} + H_{C P B} + H_{int},

H_{N R} = \bar{h} ω_{n} a^{†} a,

H_{C P B} = \frac{1}{2} \bar{h} ω_{q} σ_{z} - \frac{1}{2} E_{J} \cos [\frac{π Φ_{x} (t)}{ϕ_{0}}] σ_{x},

H_{int} = \bar{h} λ (a^{†} + a) σ_{z},

H_{p u} = \frac{1}{2} \bar{h} Δ σ_{z} + \bar{h} ω_{n} a^{†} a + \bar{h} λ (a^{†} + a) σ_{z} + \bar{h} Ω (σ_{+} + σ_{-}) + μ ε_{p r} (σ_{+} e^{- i δ t} + σ_{-} e^{i δ t}),

\frac{d σ_{-}}{d t} = - (\frac{1}{T_{2}} + i Δ + i q) σ_{-} + i Ω σ_{z} + \frac{i μ ε_{p r}}{\bar{h}} e^{- i δ t} σ_{z},

\frac{d σ_{z}}{d t} = - (σ_{z} + 1) \frac{1}{T_{1}} - 2 i Ω (σ_{+} - σ_{-}) - \frac{2 i μ}{\bar{h}} (σ_{+} ε_{p r} e^{- i δ t} - σ_{-} ε_{p r}^{*} e^{i δ t}),

\frac{d^{2} q}{d t^{2}} + γ_{n} \frac{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} = \frac{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} = \frac{T_{2} μ ε_{p r}}{\bar{h}} \frac{\frac{- 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)} + \frac{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}) = \frac{μ σ_{+ 1}}{ε_{p r}} = \frac{T_{2} μ^{2}}{\bar{h}} χ^{(1)} (ω_{p r}),

η = \frac{ω_{0}^{2}}{ω_{0}^{2} - i γ_{0} δ_{0} - δ_{0}^{2}},

β = \frac{\frac{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}} + \frac{\frac{- 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 = \frac{1}{1 + i Δ_{0} - 4 i λ_{0} ω_{0} k_{0} - i δ_{0}} + \frac{1}{1 - i Δ_{0} + 4 i λ_{0} ω_{0} k_{0}} .

v_{g} = \frac{c}{n + ω_{p r} (\frac{d n}{d ω_{p r}})},

\frac{c}{v_{g}} = 1 + 2 π Re χ_{eff}^{(1)} {(ω_{p r})}_{ω_{p r} = ω_{q}} + 2 π ω_{p r} Re {(\frac{d χ_{eff}^{(1)}}{d ω_{p r}})}_{ω_{p r} = ω_{q}} .

\frac{c}{v_{g}} - 1 = 2 π ω_{p r} \frac{μ^{2} T_{2}}{\bar{h}} Re {(\frac{d χ^{(1)} (ω_{p r})}{d ω_{p r}})}_{ω_{p r} = ω_{q}} = \frac{1}{T_{2}} Σ Re {(\frac{d χ^{(1)} (ω_{p r})}{d ω_{p r}})}_{ω_{p r} = ω_{q}},