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Resonant gain suppression and superluminal group velocity in a multilevel system Cui-Li Cui,1,2 Chang-Bao Fu,1 Hong Yang,1 Qian-Qian Bao,1 Huai-Liang Xu,2 and Jin-Hui Wu1,∗ 2 College

1 College of Physics, Jilin University, Changchun 130012, China of Electronic Science and Engineering, Jilin University, Changchun 130012, China ∗ [email protected]

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 Optical Society of America OCIS codes: (270.0270) Quantum optics; (270.1670) Coherent optical effects.

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Received 16 Feb 2012; revised 4 Apr 2012; accepted 4 Apr 2012; published 24 Apr 2012

7 May 2012 / Vol. 20, No. 10 / OPTICS EXPRESS 10712

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). 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). 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). 19. M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A 4, 2104–2108 (1971). 20. E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002). 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). 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). 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). 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). 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). 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).

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, optoelectronic 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

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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.

(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 MaxwellBloch 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 (Γi ) of level |i (i = 1, 2, ..., N) is expected to be much smaller than that (Γ0 ) of level |0. As far as cold 87 Rb atoms are concerned, Γ (i = 1, 2, ..., N) is about 10 kHz for a highly excited Rydberg i state with principal quantum number n ≈ 70 while Γ0 is equal to 6.0 MHz for the 5P3/2 state. A monochromatic probe field coherently drives the atomic transition between level |1 and − → → − level |0 with the complex Rabi frequency Ω p = E p · d 10 /2¯h 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 Ωn = E n · d n+1,0 /2¯h and the real frequency detuning Δn = ωn − ωn+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 N−1 N−1 HI = −¯hΔ p |1 1| − ∑n=1 h¯ Δn |n + 1 n + 1| − h¯ Ω p |1 0| + ∑n=1 Ωn |n + 1 0| + h.c. (1) which allows us to attain the following Bloch equations for density matrix elements ∗ − (Γ1 + Λ) ρ11 + Λρgg + iΩ p ρ10 − iΩ∗p ρ10

ρ˙ 11

=

ρ˙ n+1,n+1

=

∗ −Γn+1 ρn+1,n+1 + iΩn ρn+1,0 − iΩ∗n ρn+1,0 , n = 1 ∼ N − 1

ρ˙ 00

=

∗ −Γ0 ρ00 + ∑n=1 Γn ρnn − iΩ p ρ10 + iΩ∗p ρ10

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N



ρ˙ 1,n+1 ρ˙ m+1,n+1

N−1 ∗ −i∑n=1 Ωn ρn+1,0 − Ω∗n ρn+1,0

∗ = [i (Δ p − Δn ) − γ1,n+1 ] ρ1,n+1 + iΩ p ρn+1,0 − iΩ∗n ρ10 , n = 1 ∼ N −1

=

(2)

∗ [i (Δm − Δn ) − γm+1,n+1 ] ρm+1,n+1 + iΩm ρn+1,0 − iΩ∗n ρm+1,0 ,

m = n, m, n = 1 ∼ N − 1

ρ˙ 10 ρ˙ n+1,0

= (iΔ p − γ10 ) ρ10 − ∑n=1 iΩn ρ1,n+1 − iΩ p (ρ11 − ρ00 ) N−1

= (iΔn − γn+1,0 ) ρn+1,0 − i∑m=n,m=1 (Ω p ρn+1,1 + Ωm ρn+1,m+1 ) N−1

−iΩn (ρn+1,n+1 − ρ00 ) , n = 1 ∼ N − 1 ∗ ρmn

and ∑Nn=1 ρnn = 1 − ρgg . In Eqs. (2), γ1n = (Γ1 + Γn + Λ) /2, γ10 = constrained by ρnm = (Γ1 + Γ0 + Λ) /2, γmn = (Γm + Γn ) /2 and γn0 = (Γn + Γ0 ) /2 are defined as the decay rates of atomic coherence ρ1n , ρ10 , ρmn , and ρn0 respectively (m = n, m, n = 2 ∼ N). In the weak probe (Ω p