## 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.

© 2012 OSA

## 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 [1–5]). 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], semiconductor materials [6], room-temperature solids [7], and optical fibers [8, 9]. The underlying physics could be stimulated Brillouin scattering [8, 9], coherent gain assisting [10], active Raman gain [11], coherent population oscillation [7, 12], electromagnetically induced transparency (EIT) [13], electromagnetically induced absorption (EIA) [14, 15] (the counterpart of EIT), and resonant gain suppression (RGS) [16] (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 [17–19].

In this paper, inspired by Ref. [16] and Ref. [20], 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 [21–23]. 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 ^{87}Rb 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 (Γ

_{0}) of level |0〉. As far as cold

^{87}Rb 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 Γ

_{0}is equal to 6.0 MHz for the 5

*P*

_{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}*d⃗*

_{10}/2

*h̄*and the real frequency detuning Δ

*=*

_{p}*ω*−

_{p}*ω*

_{10}. The

*n*th 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}/2

*h̄*and the real frequency detuning Δ

*=*

_{n}*ω*−

_{n}*ω*

_{n}_{+1,0}(

*n*= 1, 2,...,

*N*− 1). A broadband laser [24–26] 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].

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

*γ*

_{1}

*= (Γ*

_{n}_{1}+ Γ

*+ Λ)/2,*

_{n}*γ*

_{10}= (Γ

_{1}+ Γ

_{0}+ Λ)/2,

*γ*= (Γ

_{mn}*+ Γ*

_{m}*) /2 and*

_{n}*γ*

_{n}_{0}= (Γ

*+ Γ*

_{n}_{0}) /2 are defined as the decay rates of atomic coherence

*ρ*

_{1}

*,*

_{n}*ρ*

_{10},

*ρ*, and

_{mn}*ρ*

_{n}_{0}respectively (

*m*≠

*n*,

*m,n*= 2 ∼

*N*).

In the weak probe (Ω* _{p}* << Γ

_{0}) and weak pump (Λ << Γ

_{0}) limits, we can analytically solve Eqs. (2) in the steady state to attain ${\rho}_{10}^{(1)}(\infty )$ in the first order of Ω

*but in all order of Ω*

_{p}*(*

_{n}*n*= 1 ∼

*N*− 1), which is proportional to the linear probe susceptibility

*N*being the atomic volume density,

*d*

_{10}the dipole moment on transition |1〉 ↔ |0〉,

*δ*= Δ

_{n}*− Δ*

_{p}*the two-photon Raman detuning between the probe and the*

_{n}*n*th coupling field, and ${\rho}_{11}^{(0)}(\infty )=\Lambda /(2\Lambda +{\Gamma}_{1})$ the steady population at level |1〉 in the absence of probe field

*ω*. 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

_{p}*<< Γ*

_{i}_{0}(

*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*= 0 with

_{n}*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 Δ

_{1}= Δ

_{2}= ... = Δ

*= Δ), the susceptibility then approximates*

_{M}*N*−

*M*transparency windows will appear in the probe gain spectrum. Finally, if all the coupling detunings Δ

*are equal to Δ then the susceptibility reduces to*

_{n}*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:

*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*,

*α*=

*N*|

*d*

_{10}|

^{2}

*ω*/

_{p}*ε*

_{0}

*h̄c*Γ

_{0}being the propagation constant and ${\Omega}_{p}^{0}=\overrightarrow{\mathcal{E}}\cdot {\overrightarrow{d}}_{10}/2\overline{h}$ the maximal Rabi frequency.

The group velocity of the probe pulse can be expressed as

*N*+ 1)-level system into with Eq. (3) taken into account under the two-photon resonant condition Δ

*= Δ*

_{p}*(Δ*

_{n}*≠ Δ*

_{n}*,*

_{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*at the

_{g}*n*th transparency window center can be controlled via the intensity of the

*n*th coupling laser field. Obviously, it is always larger than the light speed in vacuum

*c*and a critical Rabi frequency ${\Omega}_{0}={\left(\genfrac{}{}{0.1ex}{}{N{\left|{d}_{10}\right|}^{2}{\omega}_{p}}{4\overline{h}{\epsilon}_{0}}\genfrac{}{}{0.1ex}{}{\Lambda}{2\Lambda +{\Gamma}_{1}}\right)}^{1/2}$ exists for the coupling field. It is also clear that Ω

_{0}represents a specific value of the coupling Rabi frequency Ω

*(*

_{n}*υ*is negative when Ω

_{g}*< Ω*

_{n}_{0}whereas positive when Ω

*> Ω*

_{n}_{0}). Note that one may control Ω

_{0}by changing the atomic volume density

*N*or the incoherent pumping rate Λ. However, the definition of Ω

_{0}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 Δ

_{1}= Δ

_{2}= ... = Δ

*= Δ), the group velocity of the probe pulse then becomes the same as Eq. (9) at the*

_{M}*n*th transparency window center with

*n*=

*M*+ 1,...,

*N*− 1, and

*N*fields are on two-photon resonance, the group velocity then becomes

*γ*

_{1}

*=*

_{n}*γ*

_{1}

*(*

_{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 ^{87}Rb 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 Δ

_{1}≠ Δ

_{2}. 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*< 0. The narrow and deep transparency windows are in fact the signatures of

_{g}*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

*<< Γ*

_{i}_{0}(

*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.

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

*= Δ*

_{p}_{1}and Δ

*= Δ*

_{p}_{2}can become very narrow if we increase the coupling Rabi frequencies Ω

_{1}and Ω

_{2}[see Fig. 4(a)] or decrease the coupling detuning difference |Δ

_{1}− Δ

_{2}| [see Fig. 4(b)], while the outboard one near Δ

*= Δ*

_{p}_{1}(Δ

*= Δ*

_{p}_{2}) can become very narrow if we choose a large detuning Δ

_{1}(Δ

_{2}) of the respective coupling field Ω

_{1}(Ω

_{2}) [see Fig. 4(c)]. Such dynamically controlled narrow gain lines may have potential applications in the accurate spectroscopic measurement.

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.

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 ^{87}Rb atoms. As an example, we consider here an open four-level system (*N* = 3). We suppose that the probe pulse is bichromatic
${E}_{p}=\genfrac{}{}{0.1ex}{}{1}{2}\left[{E}_{p1}{f}_{1}(z,t){e}^{i{\Delta}_{1}t}+{E}_{p2}{f}_{2}(z,t){e}^{i{\Delta}_{2}t}\right]{e}^{i{\omega}_{10}t}+c.c.$, and the first (second) component *E _{p}*

_{1}(

*E*

_{p}_{2}) is on Raman resonance with the monochromatic coupling field Ω

_{1}(Ω

_{2}). 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 Δ

_{1}≠ Δ

_{2}. 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 Δ

*τ*≈ −3.24

*μ*s and

*υ*≈ −1.85 × 10

_{g}^{4}m/s, while those of the second pulse component in Fig. 7(b) are Δ

*τ*≈ −1.24

*μ*s and

*υ*≈ −4.84×10

_{g}^{4}m/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.

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 × 10

^{4}in Fig. 7(a) and

*F*= 0.62 × 10

^{4}in Fig. 7(b). Note that the figure of merit in [10] 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 × 10

^{4}. But the superluminal light signal experiences remarkable absorptive loss because the underlying physics of a steep anomalous dispersion is electromagnetically induced absorption (EIA) [27].

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

*< 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*

_{p}*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]. 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.

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