Resonant gain suppression and superluminal group velocity in a multilevel system

doi:10.1364/OE.20.010712

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Resonant gain suppression and superluminal group velocity in a multilevel system

Cui-Li Cui, Chang-Bao Fu, Hong Yang, Qian-Qian Bao, Huai-Liang Xu, and Jin-Hui Wu »View Author Affiliations

Optics Express, Vol. 20, Issue 10, pp. 10712-10720 (2012)
http://dx.doi.org/10.1364/OE.20.010712

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▸ Article Outline
– Abstract
1. Introduction
2. Model and equations
3. Results and discussion
4. Conclusions
– References and links

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Abstract

We investigate the interaction of an open (N + 1)-level extended V-type atomic system (i.e. a closed (N + 2)-level atomic system) with N coherent laser fields and one incoherent pumping field through both analytical and numerical calculations. Our results show that the system can exhibit multiple resonant gain suppressions via perfect quantum destructive interference, which is usually believed to be absent in closed three-level V system and its extended versions involving more atomic levels, with at most N – 1 transparency windows associated with very steep anomalous dispersions occurring in the system. The superluminal group velocity of the probe-laser pulse with at most N – 1 negative values can also be generated and controlled with little gain or absorption.

1. Introduction

During the past decade the group velocity manipulation (either slowing down or speeding up) of weak light pulses has attracted great attention due to its scientific significance (see the Reviews in [1

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

–5

L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008). [CrossRef]

]). Controlling the traveling time of light pulses through certain devices may also lead to important applications, e.g., in optical communications, optical networks, opto-electronic devices, and quantum information processing. In particular, the superluminal light propagation has been attained in a number of different media including atomic gasses [2

A. M. Akulshin and R. J. McLean, “Fast light in atomic media,” J. Opt. 12, 104001 (2010). [CrossRef]

], semiconductor materials [6

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

], room-temperature solids [7

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]

], and optical fibers [8

M. Gonzalez-Herraez, K.-Y. Song, and L. Thevenaz, “Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering,” Appl. Phys. Lett. 87, 081113 (2005). [CrossRef]

, 9

K. Y. Song, K. S. Abedin, and K. Hotate, “Gain-assisted superluminal propagation in tellurite glass fiber based on stimulated Brillouin scattering,” Opt. Express 16, 225–230 (2008). [CrossRef] [PubMed]

]. The underlying physics could be stimulated Brillouin scattering [8

, 9

], coherent gain assisting [10

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000). [CrossRef] [PubMed]

], active Raman gain [11

C. Zhu and G. Huang, “High-order nonlinear Schrodinger equation and weak-light superluminal solitons in active Raman gain media with two control fields,” Opt. Express 19, 1963–1974 (2011). [CrossRef] [PubMed]

], coherent population oscillation [7

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]

, 12

F. Arrieta-Yanez, O. G. Calderon, and S. Melle, “Slow and fast light based on coherent population oscillations in erbium-doped fibres,” J. Opt. 12, 104002 (2010). [CrossRef]

], electromagnetically induced transparency (EIT) [13

C.-L. Cui, J.-K. Jia, J.-W. Gao, Y. Xue, G. Wang, and J.-H. Wu, “Ultraslow and superluminal light propagation in a four-level atomic system,” Phys. Rev. A 76, 033815 (2007). [CrossRef]

], electromagnetically induced absorption (EIA) [14

A. M. Akulshin, S. Barreiro, and A. Lezama, “Steep anomalous dispersion in coherently prepared Rb vapor,” Phys. Rev. Lett. 83, 4277–4280 (1999). [CrossRef]

, 15

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003). [CrossRef]

] (the counterpart of EIT), and resonant gain suppression (RGS) [16

C.-L. C., J.-K. Jia, Y. Zhang, Y. Xue, H.-L. Xu, and J.-H. Wu, “Resonant gain suppression and quantum destructive interference in a three-level open V system,” J. Phys. B: At. Mol. Opt. Phys. 44, 215504 (2011). [CrossRef]

] (the revised version of EIT). Note that the information carried by a light pulse (i.e. the pulse frontier) cannot travel with a velocity exceeding the speed of light in vacuum c as required by the causality, although the pulse center may attain a group velocity much larger than c in an anomalous dispersive medium [17

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

–19

M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A 4, 2104–2108 (1971). [CrossRef]

In this paper, inspired by Ref. [16

] and Ref. [20

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002). [CrossRef]

], we investigate the steady optical response of a (N + 2)-level atomic system and then the resulted superluminal light propagation. The system we considered may be regarded as a (N + 1)-level open system, the extended version of a three-level open V system [16

], because one level is coherently decoupled from the other levels, driven by a weak coherent field (probe) and N – 1 strong coherent fields (couplings). Similar as in Ref. [16

], to attain perfect quantum destructive interference, which has not been proved to exist in the (N +1)-level extended V-type system, the lower level in the open (N +1)-level system should have a spontaneous decay rate much larger than those of the N upper levels. This specific situation may be realized when all N upper levels in the open (N +1)-level system are chosen to be highly excited Rydberg states of radiative lifetimes up to tens of microseconds [21

T. F. Gallagher, Rydberg Atoms (Cambridge University Press Cambridge, England, 1984).

–23

D. Yan, J.-W. Gao, Q.-Q. Bao, H. Yang, H. Wang, and J.-H. Wu, “Electromagnetically induced transparency in a five-level Λ system dominated by two-photon resonant transitions,” Phys. Rev. A 83, 033830 (2011). [CrossRef]

]. With optical Bloch equations, we first obtain a general analytical expression for the probe linear susceptibility, and then an analytical expression for the probe group velocity. These expressions show that at most N – 1 narrow and deep transparency windows, which are in fact the signatures of perfect quantum destructive interference, may be obtained in the open (N + 1)-level system, and that the probe field is superluminal at the transparency frequencies with at most N – 1 different group velocities. Then we consider a few examples with realistic parameters for cold ⁸⁷Rb atoms. Full numerical calculations based on the coupled Maxwell-Bloch equations well confirm our analytical conclusions.

2. Model and equations

We consider a (N + 2)-level atomic system as illustrated in Fig. 1, in which level |1〉, level |2〉, ..., and level |N〉 may refer to N Rydberg states with very high principal quantum numbers while level |g〉 and level |0〉 belong, respectively, to the ground state and the first excited state with the same principal quantum number. In this situation, spontaneous decay rate (Γ_i) of level |i〉 (i = 1, 2,...,N) is expected to be much smaller than that (Γ₀) of level |0〉. As far as cold ⁸⁷Rb atoms are concerned, Γ_i (i = 1, 2,..., N) is about 10 kHz for a highly excited Rydberg state with principal quantum number n ≈ 70 while Γ₀ is equal to 6.0 MHz for the 5P_3/2 state. A monochromatic probe field coherently drives the atomic transition between level |1〉 and level |0〉 with the complex Rabi frequency Ω_p = E⃗_p · d⃗₁₀/2h̄ and the real frequency detuning Δ_p = ω_p − ω₁₀. The nth monochromatic coupling field coherently drives the atomic transition between level |n + 1〉 and level |0〉 with the complex Rabi frequency Ω_n = E⃗_n · d⃗_n_+1,0/2h̄ and the real frequency detuning Δ_n = ω_n − ω_n_+1,0 (n = 1, 2,..., N − 1). A broadband laser [24

M. Mahmoudi, M. Sahrai, and H. Tajalli, “Subluminal and superluminal light propagation via interference of incoherent pumpfields,” Phys. Lett. A 357, 66–71 (2006). [CrossRef]

–26

D. Bullock, J. Evers, and C. H. Keitel, “Modifying spontaneous emission via interferences from incoherent pump fields,” Phys. Lett. A 307, 8–12 (2003). [CrossRef]

] is used as the incoherent pump field to selectively excite some atoms from level |g〉 into level |1〉 at the rate Λ without introducing atomic coherence between level |g〉 and other levels. It is clear that level |g〉 is coherently decoupled from the other N + 1 levels and therefore we can envision here an open (N + 1)-level system consisting of only levels |1〉, |2〉, ..., |N〉, and |0〉, the extended version of a three-level open V system [16

Fig. 1 Schematic diagram of a (N + 2)-level atomic system. Levels |0〉, |1〉, |2〉, ..., and |N〉 make up a (N + 1)-level open system in that level |g〉 is coherently decoupled from them.

Under the rotating-wave and electric-dipole approximations, the interaction Hamiltonian for the open (N + 1)-level system can be written as

H_{I} = - \bar{h} Δ_{p} | 1 〉 〈 1 | - \sum_{n = 1}^{N - 1} \bar{h} Δ_{n} | n + 1 〉 〈 n + 1 | - \bar{h} [Ω_{p} | 1 〉 〈 0 | + \sum_{n = 1}^{N - 1} Ω_{n} | n + 1 〉 〈 0 | + h . c .]

(1)

which allows us to attain the following Bloch equations for density matrix elements

\begin{array}{l} {\dot{ρ}}_{11} & = & - (Γ_{1} + Λ) ρ_{11} + Λ ρ_{g g} + i Ω_{p} ρ_{10}^{*} - i Ω_{p}^{*} ρ_{10} \\ {\dot{ρ}}_{n + 1, n + 1} & = & - Γ_{n + 1} ρ_{n + 1, n + 1} + i Ω_{n} ρ_{n + 1, 0}^{*} - i Ω_{n}^{*} ρ_{n + 1, 0}, n = 1 \sim N - 1 \\ {\dot{ρ}}_{00} & = & - Γ_{0} ρ_{00} + \sum_{n = 1}^{N} Γ_{n} ρ_{n m} - i Ω_{p} ρ_{10}^{*} + i Ω_{p}^{*} ρ_{10} \\ - i \sum_{n = 1}^{N - 1} (Ω_{n} ρ_{n + 1, 0}^{*} - Ω_{n}^{*} ρ_{n + 1, 0}) \\ {\dot{ρ}}_{1, n + 1} & = & [i (Δ_{p} - Δ_{n}) - γ_{1, n + 1}] ρ_{1, n + 1} + i Ω_{p} ρ_{n + 1, 0}^{*} - i Ω_{n}^{*} ρ_{10}, \\ n = 1 \sim N - 1 \end{array}

(2)

\begin{array}{l} {\dot{ρ}}_{m + 1, n + 1} & = & [i (Δ_{m} - Δ_{n}) - γ_{m + 1, n + 1}] ρ_{m + 1, n + 1} + i Ω_{m} ρ_{n + 1, 0}^{*} - i Ω_{n}^{*} ρ_{m + 1, 0}, \\ m \neq n, m, n = 1 \sim N - 1 \\ {\dot{ρ}}_{10} & = & (i Δ_{p} - γ_{10}) ρ_{10} - \sum_{n = 1}^{N - 1} i Ω_{n} ρ_{1, n + 1} - i Ω_{p} (ρ_{11} - ρ_{00}) \\ {\dot{ρ}}_{n + 1, 0} & = & (i Δ_{n} - γ_{n + 1, 0}) ρ_{n + 1, 0} - i \sum_{m \neq n, m = 1}^{N - 1} (Ω_{p} ρ_{n + 1, 1} + Ω_{m} ρ_{n + 1, m + 1}) \\ - i Ω_{n} (ρ_{n + 1, n + 1} - ρ_{00}), n = 1 \sim N - 1 \end{array}

constrained by

ρ_{n m} = ρ_{m n}^{*}

and

\sum_{n = 1}^{N} ρ_{n n} = 1 - ρ_{g g}

. In Eqs. (2), γ₁_n = (Γ₁ + Γ_n + Λ)/2, γ₁₀ = (Γ₁ + Γ₀ + Λ)/2, γ_mn = (Γ_m + Γ_n) /2 and γ_n₀ = (Γ_n + Γ₀) /2 are defined as the decay rates of atomic coherence ρ₁_n, ρ₁₀, ρ_mn, and ρ_n₀ respectively (m ≠ n, m,n = 2 ∼ N).

In the weak probe (Ω_p << Γ₀) and weak pump (Λ << Γ₀) limits, we can analytically solve Eqs. (2) in the steady state to attain

ρ_{10}^{(1)} (\infty)

in the first order of Ω_p but in all order of Ω_n (n = 1 ∼ N − 1), which is proportional to the linear probe susceptibility

\begin{array}{l} χ (Δ_{p}) & = & \frac{N {| d_{10} |}^{2}}{2 \bar{h} ε_{0}} \frac{ρ_{10}^{(1)} (\infty)}{Ω_{p}} \\ = & \frac{N {| d_{10} |}^{2}}{2 \bar{h} ε_{0}} \frac{ρ_{11}^{(0)} (\infty)}{(Δ_{p} + i γ_{10}) - \sum_{n = 1}^{N - 1} Ω_{n}^{2} / (δ_{n} + i γ_{1, n + 1})} \end{array}

(3)

with N being the atomic volume density, d₁₀ the dipole moment on transition |1〉 ↔ |0〉, δ_n = Δ_p − Δ_n the two-photon Raman detuning between the probe and the nth coupling field, and

ρ_{11}^{(0)} (\infty) = Λ / (2 Λ + Γ_{1})

the steady population at level |1〉 in the absence of probe field ω_p. Calculating imaginary and real parts of the linear probe susceptibility with realistic parameters, it is straightforward to examine the absorption and dispersion spectra on transition |1〉 ↔ |0〉 in the next section. Considering Λ ≈ Γ_i << Γ₀ (i = 1, 2,..., N), the probe field is amplified if Im(χ) < 0 in a certain spectral region. In addition, the susceptibility goes to zero when δ_n = 0 with n = 1 ∼ N − 1. Therefore, if all the coupling detunings are different then this open (N +1)-level system will become transparent at N − 1 different frequencies of the probe field.

In the case that M out of the N − 1 coupling detunings Δ_n are equal to Δ and the remaining N − M − 1 are different than Δ (for simplification, we take Δ₁ = Δ₂ = ... = Δ_M = Δ), the susceptibility then approximates

χ (Δ_{p}) = \frac{N {| d_{10} |}^{2}}{2 \bar{h} ε_{0}} \frac{ρ_{11}^{(0)} (\infty)}{(Δ_{p} + i γ_{10}) - \sum_{n = 1}^{M} Ω_{n}^{2} / (Δ_{p} - Δ) - \sum_{n = M + 1}^{N - 1} Ω_{n}^{2} / (Δ_{p} - Δ_{n})}

(4)

which means that N − M transparency windows will appear in the probe gain spectrum. Finally, if all the coupling detunings Δ_n are equal to Δ then the susceptibility reduces to

χ (Δ_{p}) = \frac{N {| d_{10} |}^{2}}{2 \bar{h} ε_{0}} \frac{(Δ_{p} - Δ) ρ_{11}^{(0)} (\infty)}{(Δ_{p} + i γ_{10}) (Δ_{p} - Δ) - \sum_{n = 1}^{N - 1} Ω_{n}^{2}} .

(5)

Thus, all these N − 1 transparency windows will degenerate into a single one.

As far as the propagation dynamics of a pulsed field with central frequency ω_p is concerned, the following Maxwell wave equation in the slowly-varying-envelope approximation is also required:

\frac{\partial}{\partial z} f (z, t) + \frac{1}{c} \frac{\partial}{\partial t} f (z, t) = \frac{i ω_{p}}{2 c} χ (z, t) f (z, t)

(6)

where f(z,t) is the dimensionless pulse envelope (i.e., E⃗_p = 𝒠⃗f (z,t)). In particular, we have f (z,t) ≡ 1 and E⃗_p ≡ 𝒠⃗ in the limit of a cw field. For the convenience of both quantitative calculation and qualitative analysis, we further transform Eq. (6) into the retarded local frame where τ = t − z/c and ξ = z,

\frac{\partial}{\partial α ξ} f (ξ, τ) = \frac{i Γ_{0}}{4 Ω_{p}^{0}} ρ_{10} (ξ, τ)

(7)

with α = N|d₁₀|² ω_p/ε₀h̄cΓ₀ being the propagation constant and

Ω_{p}^{0} = \vec{ℰ} \cdot {\vec{d}}_{10} / 2 \bar{h}

the maximal Rabi frequency.

The group velocity of the probe pulse can be expressed as

υ_{g} = c / [1 + \frac{1}{2} Re (χ) + \frac{ω_{p}}{2} \frac{\partial Re (χ)}{\partial Δ_{p}}]

(8)

which turns for the open (N + 1)-level system into

υ_{g} \approx \frac{c}{1 - Ω_{0}^{2} / Ω_{n}^{2}}

(9)

with Eq. (3) taken into account under the two-photon resonant condition Δ_p = Δ_n (Δ_n ≠ Δ_m, m,n = 1, 2,...,N − 1). Therefore, if none of the coupling detunings are the same, the probe field can propagate with N − 1 different group velocities in the medium. The group velocity υ_g at the nth transparency window center can be controlled via the intensity of the nth coupling laser field. Obviously, it is always larger than the light speed in vacuum c and a critical Rabi frequency

Ω_{0} = {(\frac{N {| d_{10} |}^{2} ω_{p}}{4 \bar{h} ε_{0}} \frac{Λ}{2 Λ + Γ_{1}})}^{1 / 2}

exists for the coupling field. It is also clear that Ω₀ represents a specific value of the coupling Rabi frequency Ω_n (υ_g is negative when Ω_n < Ω₀ whereas positive when Ω_n > Ω₀). Note that one may control Ω₀ by changing the atomic volume density N or the incoherent pumping rate Λ. However, the definition of Ω₀ is valid only for a nonzero Λ although it could be very small. If we set Λ = 0, all atoms under consideration will be located at the coherently decoupled level |g〉 so that the probe and coupling fields interact with nothing, i.e., propagate as in vacuum.

When M out of the N − 1 coupling detunings Δ_n are equal (for simplification, we take Δ₁ = Δ₂ = ... = Δ_M = Δ), the group velocity of the probe pulse then becomes the same as Eq. (9) at the nth transparency window center with n = M + 1,..., N − 1, and

υ_{g} \approx \frac{c}{1 - Ω_{0}^{2} / \sum_{n = 1}^{M} Ω_{n}^{2}}

(10)

around detuning Δ. Finally, if all N fields are on two-photon resonance, the group velocity then becomes

υ_{g} \approx \frac{c}{1 - Ω_{0}^{2} / \sum_{n = 1}^{N - 1} Ω_{n}^{2}}

(11)

at the single transparency window center. To deduce Eq. (10) and Eq. (11), we have assumed that γ₁_n = γ₁_m (n, m = 2 ∼ N) without the loss of generality.

3. Results and discussion

We will now give a few examples of the steady optical response that could occur in the open (N + 1)-level system with realistic parameters for cold ⁸⁷Rb atoms. We plot in Fig. 2 the imaginary (solid curves) and real (dashed curves) parts of the probe susceptibility χ as a function of the probe detuning Δ_p for an open four-level system (N = 3). It is clear that the weak probe field is always amplified around its resonant frequencies and two narrow transparency windows will arise between three gain lines in the case of Δ₁ ≠ Δ₂. In addition, the transparency windows are accompanied by very steep anomalous dispersions as determined by the Kramers-Kronig relation, which is essential for attaining the superluminal light propagation with υ_g > c or even υ_g < 0. The narrow and deep transparency windows are in fact the signatures of perfect quantum destructive interference and can be observed only when the lower level has a spontaneous decay rate much larger than those of all the upper levels Γ_i << Γ₀ (i = 1^∼N). In a closed (N + 1)-level extended V-type system, however, one will find quantum constructive interference instead. These remarks can be verified by the same method as in Ref. [16

] and will not be shown here repeatedly.

Fig. 2 Probe gain Im(χ) (black-solid) and dispersion Re(χ) (red-dashed) as a function of probe detuning Δ_p for an open four-level system (N = 3) with Λ = 0.01 MHz, Γ₀ = 6.0 MHz, Γ₁ = Γ₂ = Γ₃ = 0.01 MHz, Δ₁ = −Δ₂ = −2.0 MHz, N = 5.0 × 10¹⁰ cm⁻³, d₁₀ = 1.0 × 10⁻³⁰ C·m, and (a) Ω₁ = Ω₂ = 2.0 MHz, (b) Ω₁ = 2.0 MHz, Ω₂ = 4.0 MHz.

Note, in particular, that the transparency windows and anomalous dispersions can be either symmetric [see Fig. 2(a)] or asymmetric [see Fig. 2(b)] depending on the field parameters such as Rabi frequencies and frequency detunings. We also plot in Fig. 3 the dynamic evolution of atomic populations of all levels for the case considered in Fig. 2. In this case, only level |g〉 and level |1〉 has non-vanishing populations because the probe field is very weak, level |0〉 has a much larger decay rate than level |1〉, and the coupling field Ω_n will not excite atoms into level |n〉 in the presence of perfect quantum destructive interference. In addition, as we can see from Fig. 4, the middle gain line between Δ_p = Δ₁ and Δ_p = Δ₂ can become very narrow if we increase the coupling Rabi frequencies Ω₁ and Ω₂ [see Fig. 4(a)] or decrease the coupling detuning difference |Δ₁ − Δ₂| [see Fig. 4(b)], while the outboard one near Δ_p = Δ₁ (Δ_p = Δ₂) can become very narrow if we choose a large detuning Δ₁ (Δ₂) of the respective coupling field Ω₁ (Ω₂) [see Fig. 4(c)]. Such dynamically controlled narrow gain lines may have potential applications in the accurate spectroscopic measurement.

Fig. 3 Dynamic evolution of atomic populations in the open four-level system (N = 3) with Ω_p = 0.01 MHz and ρ_gg(0) = 1. Other parameters are the same as in Fig. 2. ρ₁₁ (black-solid); ρ₂₂ (red-dashed); ρ₃₃ (green-dotted); ρ₀₀ (blue-dash-dotted); ρ_gg (magentadash-dot-dotted).

Fig. 4 Probe gain Im(χ) (black-solid) and dispersion Re(χ) (red-dashed) as a function of probe detuning Δ_p for an open four-level system (N = 3) with the same parameters as in Fig. 2(a) except (a) Ω₁ = Ω₂ = 5.0 MHz, (b) Δ₁ = −Δ₂ = −0.5 MHz and (c) Δ₁ = −10.0 MHz.

We also plot a few spectra for the open five- (N = 4) (see Fig. 5) and open six- (N = 5) (see Fig. 6) level systems. As we can see, at most three and four transparency windows appear between four and five gain lines, respectively. Accordingly, we can attain the superluminal light signals when their frequencies fall into these transparency windows accompanied by the anomalous dispersion.

Fig. 5 Probe gain Im(χ) (black-solid) and dispersion Re(χ) (red-dashed) as a function of probe detuning Δ_p for an open five-level system (N = 4) with the same parameters as in Fig. 2 except Γ₁ = Γ₂ = Γ₃ = Γ₄ = 0.01 MHz, Δ₁ = −3.0 MHz, Δ₂ = 0.0, Δ₃ = 3.0 MHz, and (a) Ω₁ = Ω₂ = Ω₃ = 2.0 MHz, (b) Ω₁ = 2.0 MHz, Ω₂ = 3.0 MHz, Ω₃ = 4.0 MHz.

Fig. 6 Probe gain Im(χ) (black-solid) and dispersion Re(χ) (red-dashed) as a function of probe detuning Δ_p for an open six-level system (N = 5) with the same parameters as in Fig. 2 except Γ₁ = Γ₂ = Γ₃ = Γ₄ = Γ₅ = 0.01 MHz, Δ₁ = −3.0 MHz, Δ₂ = −1.0 MHz, Δ₃ = 1.0 MHz, Δ₄ = 3.0 MHz, and (a) Ω₁ = Ω₂ = Ω₃ = Ω₄ = 2.0 MHz, (b) Ω₁ = 2.0 MHz, Ω₂ = 3.0 MHZ, Ω₃ = 4.0 MHz, Ω₄ = 5.0 MHz.

As mentioned above, a narrow and deep transparency window between a gain doublet is the key to attain a superluminal group velocity accompanied by little gain or absorption. We now examine the propagation dynamics of a probe pulse in the open (N + 1)-leve system with realistic parameters for cold ⁸⁷Rb atoms. As an example, we consider here an open four-level system (N = 3). We suppose that the probe pulse is bichromatic

E_{p} = \frac{1}{2} [E_{p 1} f_{1} (z, t) e^{i Δ_{1} t} + E_{p 2} f_{2} (z, t) e^{i Δ_{2} t}] e^{i ω_{10} t} + c . c .

, and the first (second) component E_p₁ (E_p₂) is on Raman resonance with the monochromatic coupling field Ω₁ (Ω₂). Both envelopes of the two components are supposed to be in the Gaussian shape. In Fig. 7 we show the magnitude squared of two-color pulse envelopes at different penetration positions in the medium as a function of the time delay with Δ₁ ≠ Δ₂. It is clear that, as predicted by Eq. (9) with N = 3, both pulse components could be much more advanced than their counterparts propagating in the vacuum, and their group velocities can be controlled by manipulating intensities of the respective coupling fields on Raman resonance. In particular, the group time delay and the group velocity of the first pulse component in Fig. 7(a) are Δτ ≈ −3.24 μs and υ_g ≈ −1.85 × 10⁴m/s, while those of the second pulse component in Fig. 7(b) are Δτ ≈ −1.24 μs and υ_g ≈ −4.84×10⁴m/s. Similar results will also be obtained in the open five- (N = 4), six- (N = 5) level systems and so on, which are not shown here.

Fig. 7 Magnitude squared of (a) the first and (b) the second pulse component envelopes at ξ = 0.0 (black-solid) and 60 mm (red-dashed) as a function of the time delay with Δ₁ = −2.0 MHz, Δ₂ = 2.0 MHz, Ω₁ = 0.8 MHz, and Ω₂ = 1.5 MHz. Other parameters are the same as in Fig. 2 except λ_p₁ ≈ λ_p₂ = 480 nm.

Using the figure of merit F = −c/υ_g to denote how fast a superluminal light signal is, we have quite promising results: F = 1.62 × 10⁴ in Fig. 7(a) and F = 0.62 × 10⁴ in Fig. 7(b). Note that the figure of merit in [10

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000). [CrossRef] [PubMed]

] is only about F = 310 and cannot be easily improved because the two gain lines should be well separated to generate a wide and deep transparency window in the absence of quantum destructive interference. Note also that the figure of merit in [15

] is as high as F = 1.44 × 10⁴. But the superluminal light signal experiences remarkable absorptive loss because the underlying physics of a steep anomalous dispersion is electromagnetically induced absorption (EIA) [27

A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, “Electromagnetically induced absorption in a four-state system,” Phys. Rev. A 61, 011802 (1999). [CrossRef]

We note finally from Eq. (8) that the slow light is attained with ∂Re(χ)/∂Δ_p > 0 while the fast light is attained with ∂Re(χ)/∂Δ_p < 0. So there is no fundamental difference between the slow light and the fast light as far as the underlying physics is concerned. They both originates from the Kramers-Kronig relation between real and imaginary parts of the probe susceptibility. Thus the group velocity of a light pulse contributed by all carrier frequencies can take a value either smaller than c due to the normal dispersion or larger than c due to the anomalous dispersion. Note, however, that the information carried by a light pulse can never propagate with a velocity exceeding c and, according to many models, always propagates with the vacuum light speed c [1

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

]. To conclude, the superluminal light propagation just refers to the group velocity of pulse centers but not to the information velocity of pulse frontiers and therefore dose not contradict the causality.

4. Conclusions

In summary, we have studied the steady optical response of an open (N + 1)-level extended V-type atomic system and the superluminal propagation dynamics of a weak probe pulse. The open (N + 1)-level system is driven by N coherent fields (the probe and the N − 1 couplings) to generate quantum interference and simultaneously interacts with an incoherent field (the pump) to accumulate necessary population from the external ground state. All N upper levels in the open (N + 1)-level system are chosen to be highly excited Rydberg states of radiative lifetimes up to tens of microseconds, so that they could have spontaneous decay rates much smaller than that of the lower level. In this situation, the quantum interference is both destructive and perfect, which is however absent in the closed (N + 1)-level extended V-type system. Our analytical and numerical results show that, due to the perfect quantum destructive interference, at most N − 1 narrow and deep transparency windows accompanied by very steep anomalous dispersions can be observed between N gain lines. And the destructive and perfect quantum interference is therefore essential to attain superluminal light propagation with at most N − 1 different negative group velocities with high figures of merit, which can be controlled by varying the Rabi frequencies of the coupling laser fields on Raman resonance.

Acknowledgments

This work is supported by the National Natural Science Foundation of China ( 11104112 and 11174110), the 49th China Postdoctoral Science Foundation ( 20110491316), and the Basic Scientific Research Foundation of Jilin University.

References and links

1.	R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009). [CrossRef] [PubMed]
2.	A. M. Akulshin and R. J. McLean, “Fast light in atomic media,” J. Opt. 12, 104001 (2010). [CrossRef]
3.	J. Mork, F. Ohman, M. Van Der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev. 3, 30–44 (2009). [CrossRef]
4.	J. Mork, P. Lunnemann, W. Xue, Y. Chen, P. Kaer, and T. R. Nielsen, “Slow and fast light in semiconductor waveguides,” Semicond. Sci. Technol. 25, 083002 (2010). [CrossRef]
5.	L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008). [CrossRef]
6.	S. Chu and S. Wang, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982). [CrossRef]
7.	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]
8.	M. Gonzalez-Herraez, K.-Y. Song, and L. Thevenaz, “Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering,” Appl. Phys. Lett. 87, 081113 (2005). [CrossRef]
9.	K. Y. Song, K. S. Abedin, and K. Hotate, “Gain-assisted superluminal propagation in tellurite glass fiber based on stimulated Brillouin scattering,” Opt. Express 16, 225–230 (2008). [CrossRef] [PubMed]
10.	L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000). [CrossRef] [PubMed]
11.	C. Zhu and G. Huang, “High-order nonlinear Schrodinger equation and weak-light superluminal solitons in active Raman gain media with two control fields,” Opt. Express 19, 1963–1974 (2011). [CrossRef] [PubMed]
12.	F. Arrieta-Yanez, O. G. Calderon, and S. Melle, “Slow and fast light based on coherent population oscillations in erbium-doped fibres,” J. Opt. 12, 104002 (2010). [CrossRef]
13.	C.-L. Cui, J.-K. Jia, J.-W. Gao, Y. Xue, G. Wang, and J.-H. Wu, “Ultraslow and superluminal light propagation in a four-level atomic system,” Phys. Rev. A 76, 033815 (2007). [CrossRef]
14.	A. M. Akulshin, S. Barreiro, and A. Lezama, “Steep anomalous dispersion in coherently prepared Rb vapor,” Phys. Rev. Lett. 83, 4277–4280 (1999). [CrossRef]
15.	K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003). [CrossRef]
16.	C.-L. C., J.-K. Jia, Y. Zhang, Y. Xue, H.-L. Xu, and J.-H. Wu, “Resonant gain suppression and quantum destructive interference in a three-level open V system,” J. Phys. B: At. Mol. Opt. Phys. 44, 215504 (2011). [CrossRef]
17.	L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
18.	C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970). [CrossRef]
19.	M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A 4, 2104–2108 (1971). [CrossRef]
20.	E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002). [CrossRef]
21.	T. F. Gallagher, Rydberg Atoms (Cambridge University Press Cambridge, England, 1984).
22.	D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93, 063001 (2004). [CrossRef] [PubMed]
23.	D. Yan, J.-W. Gao, Q.-Q. Bao, H. Yang, H. Wang, and J.-H. Wu, “Electromagnetically induced transparency in a five-level Λ system dominated by two-photon resonant transitions,” Phys. Rev. A 83, 033830 (2011). [CrossRef]
24.	M. Mahmoudi, M. Sahrai, and H. Tajalli, “Subluminal and superluminal light propagation via interference of incoherent pumpfields,” Phys. Lett. A 357, 66–71 (2006). [CrossRef]
25.	M. Fleischhauer, C. H. Keitel, M. O. Scully, and C. Su, “Lasing without inversion and enhancement of the index of refraction via interference of incoherent pump processes,” Opt. Commun. 87, 109–114 (1992). [CrossRef]
26.	D. Bullock, J. Evers, and C. H. Keitel, “Modifying spontaneous emission via interferences from incoherent pump fields,” Phys. Lett. A 307, 8–12 (2003). [CrossRef]
27.	A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, “Electromagnetically induced absorption in a four-state system,” Phys. Rev. A 61, 011802 (1999). [CrossRef]

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.1670) Quantum optics : Coherent optical effects

ToC Category:
Quantum Optics

History
Original Manuscript: February 16, 2012
Revised Manuscript: April 4, 2012
Manuscript Accepted: April 4, 2012
Published: April 24, 2012

Citation
Cui-Li Cui, Chang-Bao Fu, Hong Yang, Qian-Qian Bao, Huai-Liang Xu, and Jin-Hui Wu, "Resonant gain suppression and superluminal group velocity in a multilevel system," Opt. Express 20, 10712-10720 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-10712

Sort: Author | Year | Journal | Reset

References

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science326, 1074–1077 (2009). [CrossRef] [PubMed]
A. M. Akulshin and R. J. McLean, “Fast light in atomic media,” J. Opt.12, 104001 (2010). [CrossRef]
J. Mork, F. Ohman, M. Van Der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev.3, 30–44 (2009). [CrossRef]
J. Mork, P. Lunnemann, W. Xue, Y. Chen, P. Kaer, and T. R. Nielsen, “Slow and fast light in semiconductor waveguides,” Semicond. Sci. Technol.25, 083002 (2010). [CrossRef]
L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics2, 474–481 (2008). [CrossRef]
S. Chu and S. Wang, “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, “Superluminal and slow light propagation in a room-temperature solid,” Science301, 200–202 (2003). [CrossRef] [PubMed]
M. Gonzalez-Herraez, K.-Y. Song, and L. Thevenaz, “Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering,” Appl. Phys. Lett.87, 081113 (2005). [CrossRef]
K. Y. Song, K. S. Abedin, and K. Hotate, “Gain-assisted superluminal propagation in tellurite glass fiber based on stimulated Brillouin scattering,” Opt. Express16, 225–230 (2008). [CrossRef] [PubMed]
L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature406, 277–279 (2000). [CrossRef] [PubMed]
C. Zhu and G. Huang, “High-order nonlinear Schrodinger equation and weak-light superluminal solitons in active Raman gain media with two control fields,” Opt. Express19, 1963–1974 (2011). [CrossRef] [PubMed]
F. Arrieta-Yanez, O. G. Calderon, and S. Melle, “Slow and fast light based on coherent population oscillations in erbium-doped fibres,” J. Opt.12, 104002 (2010). [CrossRef]
C.-L. Cui, J.-K. Jia, J.-W. Gao, Y. Xue, G. Wang, and J.-H. Wu, “Ultraslow and superluminal light propagation in a four-level atomic system,” Phys. Rev. A76, 033815 (2007). [CrossRef]
A. M. Akulshin, S. Barreiro, and A. Lezama, “Steep anomalous dispersion in coherently prepared Rb vapor,” Phys. Rev. Lett.83, 4277–4280 (1999). [CrossRef]
K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A68, 013810 (2003). [CrossRef]
C.-L. C., J.-K. Jia, Y. Zhang, Y. Xue, H.-L. Xu, and J.-H. Wu, “Resonant gain suppression and quantum destructive interference in a three-level open V system,” J. Phys. B: At. Mol. Opt. Phys.44, 215504 (2011). [CrossRef]
L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A1, 305–313 (1970). [CrossRef]
M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A4, 2104–2108 (1971). [CrossRef]
E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A66, 015802 (2002). [CrossRef]
T. F. Gallagher, Rydberg Atoms (Cambridge University PressCambridge, England, 1984).
D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett.93, 063001 (2004). [CrossRef] [PubMed]
D. Yan, J.-W. Gao, Q.-Q. Bao, H. Yang, H. Wang, and J.-H. Wu, “Electromagnetically induced transparency in a five-level Λ system dominated by two-photon resonant transitions,” Phys. Rev. A83, 033830 (2011). [CrossRef]
M. Mahmoudi, M. Sahrai, and H. Tajalli, “Subluminal and superluminal light propagation via interference of incoherent pumpfields,” Phys. Lett. A357, 66–71 (2006). [CrossRef]
M. Fleischhauer, C. H. Keitel, M. O. Scully, and C. Su, “Lasing without inversion and enhancement of the index of refraction via interference of incoherent pump processes,” Opt. Commun.87, 109–114 (1992). [CrossRef]
D. Bullock, J. Evers, and C. H. Keitel, “Modifying spontaneous emission via interferences from incoherent pump fields,” Phys. Lett. A307, 8–12 (2003). [CrossRef]
A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, “Electromagnetically induced absorption in a four-state system,” Phys. Rev. A61, 011802 (1999). [CrossRef]

Abedin, K. S.
Akulshin, A. M.
Arrieta-Yanez, F.
Bao, Q.-Q.
Barreiro, S.
Bigelow, M. S.
Boyd, R. W.
Brillouin, L.
Bullock, D.
C., C.-L.
Calderon, O. G.
Chen, Y.
Chu, S.
Cote, R.
Crisp, M. D.
Cui, C.-L.
Dogariu, A.
Evers, J.
Eyler, E. E.
Farooqi, S. M.
Fleischhauer, M.
Gallagher, T. F.
Gao, J.-W.
Garrett, C. G. B.
Gauthier, D. J.
Gonzalez-Herraez, M.
Gould, P. L.
Hotate, K.
Huang, G.
Jia, J.-K.
Kaer, P.
Keitel, C. H.
Kim, J. B.
Kim, K.
Kim, S. K.
Knight, P. L.
Krishnan, S.
Kuzmich, A.
Lee, C.
Lepeshkin, N. N.
Lezama, A.
Lunnemann, P.
Mahmoudi, M.
McCumber, D. E.
McLean, R. J.
Melle, S.
Moon, H. S.
Mork, J.
Nielsen, T. R.
Ohman, F.
Paspalakis, E.
Sahrai, M.
Scully, M. O.
Song, K. Y.
Song, K.-Y.
Stanojevic, J.
Su, C.
Taichenachev, A. V.
Tajalli, H.
Thevenaz, L.
Tong, D.
Tumaikin, A. M.
Van Der Poel, M.
Wang, G.
Wang, H.
Wang, L. J.
Wang, S.
Wu, J.-H.
Xu, H.-L.
Xue, W.
Xue, Y.
Yan, D.
Yang, H.
Yudin, V. I.
Yvind, K.
Zhang, Y.
Zhang, Y. P.
Zhu, C.

Appl. Phys. Lett.

M. Gonzalez-Herraez, K.-Y. Song, and L. Thevenaz, “Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering,” Appl. Phys. Lett.87, 081113 (2005). [CrossRef]

J. Opt.

A. M. Akulshin and R. J. McLean, “Fast light in atomic media,” J. Opt.12, 104001 (2010). [CrossRef]
F. Arrieta-Yanez, O. G. Calderon, and S. Melle, “Slow and fast light based on coherent population oscillations in erbium-doped fibres,” J. Opt.12, 104002 (2010). [CrossRef]

J. Phys. B: At. Mol. Opt. Phys.

C.-L. C., J.-K. Jia, Y. Zhang, Y. Xue, H.-L. Xu, and J.-H. Wu, “Resonant gain suppression and quantum destructive interference in a three-level open V system,” J. Phys. B: At. Mol. Opt. Phys.44, 215504 (2011). [CrossRef]

Laser Photon. Rev.

J. Mork, F. Ohman, M. Van Der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photon. Rev.3, 30–44 (2009). [CrossRef]

Nat. Photonics

L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics2, 474–481 (2008). [CrossRef]

Nature

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature406, 277–279 (2000). [CrossRef] [PubMed]

Opt. Commun.

M. Fleischhauer, C. H. Keitel, M. O. Scully, and C. Su, “Lasing without inversion and enhancement of the index of refraction via interference of incoherent pump processes,” Opt. Commun.87, 109–114 (1992). [CrossRef]

Opt. Express

Phys. Lett. A

D. Bullock, J. Evers, and C. H. Keitel, “Modifying spontaneous emission via interferences from incoherent pump fields,” Phys. Lett. A307, 8–12 (2003). [CrossRef]
M. Mahmoudi, M. Sahrai, and H. Tajalli, “Subluminal and superluminal light propagation via interference of incoherent pumpfields,” Phys. Lett. A357, 66–71 (2006). [CrossRef]

Phys. Rev. A

A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, “Electromagnetically induced absorption in a four-state system,” Phys. Rev. A61, 011802 (1999). [CrossRef]
D. Yan, J.-W. Gao, Q.-Q. Bao, H. Yang, H. Wang, and J.-H. Wu, “Electromagnetically induced transparency in a five-level Λ system dominated by two-photon resonant transitions,” Phys. Rev. A83, 033830 (2011). [CrossRef]
C.-L. Cui, J.-K. Jia, J.-W. Gao, Y. Xue, G. Wang, and J.-H. Wu, “Ultraslow and superluminal light propagation in a four-level atomic system,” Phys. Rev. A76, 033815 (2007). [CrossRef]
K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A68, 013810 (2003). [CrossRef]
C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A1, 305–313 (1970). [CrossRef]
M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A4, 2104–2108 (1971). [CrossRef]
E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A66, 015802 (2002). [CrossRef]

Phys. Rev. Lett.

A. M. Akulshin, S. Barreiro, and A. Lezama, “Steep anomalous dispersion in coherently prepared Rb vapor,” Phys. Rev. Lett.83, 4277–4280 (1999). [CrossRef]
S. Chu and S. Wang, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett.48, 738–741 (1982). [CrossRef]
D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett.93, 063001 (2004). [CrossRef] [PubMed]

Science

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science301, 200–202 (2003). [CrossRef] [PubMed]
R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science326, 1074–1077 (2009). [CrossRef] [PubMed]

Semicond. Sci. Technol.

J. Mork, P. Lunnemann, W. Xue, Y. Chen, P. Kaer, and T. R. Nielsen, “Slow and fast light in semiconductor waveguides,” Semicond. Sci. Technol.25, 083002 (2010). [CrossRef]

Other

T. F. Gallagher, Rydberg Atoms (Cambridge University PressCambridge, England, 1984).
L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

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