Abstract.
We study stable propagation of multiple shape-preserving optical pulses in an inhomogeneously broadened multi-level atomic medium. By analytically solving the Maxwell-Schrödinger equations governing the evolution of N coupled optical fields and atomic amplitudes we show that N pulsed optical waves coupling to (N+1)-levels can be automatically matched with the same soliton waveform and identical yet very slow propagation velocity. Several sets of coupled soliton solutions for two different (N+1)-level models are given and their stability is studied by using a numerical simulation.
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It is a direct generalization of a three-state V-type system. N upper levels in such system can be Zeeman or hyperfine split of the level of atomic excited state
To make the coherent transient approximation be valid, the typical temporal width of the optical pulses, τ0, is assumed to be short (e.g. 10-9 s). In the case of an external static magnetic field B having the order of magnitude of one Tesla, the frequency band width corresponding two Zeeman sub-levels is around 1.4 ×1010 s-1, which is much larger than the typical band width of the optical pulses, given by τ0 -1=1.0× 109 s-1. If the detuning Δ≤1.0×109 s-1, one can have a fairly exact resonance between each optical pulse and corresponding two upper and lower levels. Thus the rotating-wave approximation in such case can be used safely. In fact, such approximation has been widely used in references [18–22, 28] for the systems with a multi-level configuration
The polarization of the Λ-type (V-type) system reads \({\bf P}={\cal N}_a \sum_{j=1}^N [{\bf p}_{j0}a_0a_j^* \exp[i(k_j z-\omega_j t)]\)+c.c. (\({\bf P}={\cal N}_a \sum_{j+1}^N [{\bf p}_{0j}a_0^* a_j \exp[i(k_j z-\omega_j t)]\)+c.c.). Equation (4c) is obtained from the Maxwell equation under an slowly-varying envelope approximation new
A.C. Newell, J.V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, California, 1992), Chap. 5
The solution ansatz with forms of hyperbolic functions is widely used in nonlinear wave equations that posses solitonlike solutions. The key of this technique is the use of a closure of hyperbolic functions after taking differentiating operations, i.e. the derivative of a hyperbolic function is still a hyperbolic function, see references [10, 11], G. Huang, J. Phys. A 33, 8477 (2000). The exponential factor φ(z) is introduced to account for the effect of detuning rah.
We have set the left boundary of the medium as z=z0=0
A.V. Rybin et al., J. Phys. A 38, L177 (2005), and L357 (2005)
The coupled soliton solution for the case without the inhomogeneous broadening takes the form of equation (5) but with K=κ0τ0, φ(z)=0, and 1/V=1/c+κ0/B1 2. Note that all coupled soliton solutions for the generalized Λ-type and V-type (N+1)-level systems given here and in the following are reduced to the coupled soliton solutions of corresponding (N+1)-level systems in the absence of the inhomogeneous broadening by just taking g(Δ)=δ(Δ)
L. Deng, M.G. Payne, E.W. Hagley, Phys. Rev. A 70, 063813 (2004). The system displays the interaction resulting from self- and cross-phase modulation effects, which provide the nonlinearity supporting the formation of the coupled optical soliton. The waveform matching of different pulses is due to the cross-phase modulation effect.
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Huang, G., Hang, C. & Deng, L. Propagation of shape-preserving optical pulses in inhomogeneously broadened multi-level systems. Eur. Phys. J. D 40, 437–444 (2006). https://doi.org/10.1140/epjd/e2006-00161-8
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DOI: https://doi.org/10.1140/epjd/e2006-00161-8
PACS.
- 42.50.Md Optical transient phenomena: quantum beats, photon echo, free-induction decay, dephasings and revivals, optical nutation, and self-induced transparency
- 42.50.Gy Effects of atomic coherence on propagation, absorption, and amplification of light; electromagnetically induced transparency and absorption