View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Okina PHYSICAL REVIEW A 99, 053846 (2019) Half-cycle soliton and periodic waves of arbitrarily high amplitude in a two-level system Hervé Leblond Laboratoire de Photonique d’Angers, EA 4464, Université d’Angers, 2 Boulevard Lavoisier, 49000 Angers, France (Received 8 February 2019; published 28 May 2019) We show that the electromagnetic wave propagation problem based on a two-level atomic model admits exact traveling wave solutions, of both soliton and cnoidal wave types. These solutions extend the already known solutions of the same type of both the mKdV and sG equations, which have been derived from the same initial model, respectively, in a long-wave and in a short-wave approximation. The continuation is such that no modification of the wave profile is required, but that the wave velocity only has to be corrected. DOI: 10.1103/PhysRevA.99.053846 I. INTRODUCTION A more complicated model, assuming at least two transi- tions and containing both mKdV and sG-type contributions, Since the experimental achievement of optical pulses has been shown to be very promising in the description of whose duration is close to a single optical cycle in 1999 [1–3], few-cycle-pulses (FCPs) propagation [19]. It has been shown considerable effort has been made to propose an adequately that it is the most general model of this type [20]. Both the renewed theoretical description of the wave propagation. mKdV and the sG models can be generalized to many more In the highest intensity range, the medium can be consid- realistic situations, taking into account the vector character of ered as fully ionized [4,5]. This is a fortiori true in the rel- the field [21], the transverse dimensions of the space [22,23], ativistic nonlinear optics regime [6], where the wave electric and the more complex atomic structure of the material [24]. field may largely exceed the atomic field (the highest peak In the first section of this paper, we present the two-level power yet achieved is close to 10 PW [7], which focused model and derive the exact solution. In the second section, on a micrometer square would produce a intensities up to we compare this result to that of the mKdV model, which 28 / 2 10 W m ). was derived in the long-wave approximation from the same We will restrict ourselves here to the situation where the starting point. In the third section, an analogous comparison atomic structure is not destroyed by the light pulse. Within is performed with the short-wave asymptotic, which is the sG this limit, the main theoretical approaches that have been model. developed are the quantum one [8], the envelope approach, which uses generalizations of the nonlinear Schrödinger equation [9,10], and models proposed to completely avoid II. EXACT WAVE SOLUTIONS the use of the slowly varying envelope approximation A. The two-level model (SVEA) [11]. Let us consider the Schrödinger–von Neumann equation Several non-SVEA models have been described, based on various descriptions of the medium. Some are macroscopic, ∂ρ ih¯ = [H,ρ], (1) such as the so-called short-pulse equation [12], and more ∂t complicated models [13,14]. Other models are based on a ρ = − μ · microscopic model of the atoms, which may be a classical where is the density matrix operator, and H H0 E, with one [15] or use the density matrix formalism [16,17]. However, all these studies use some low-amplitude, or ωa 0 H0 = h¯ , (2) “weakly nonlinear,” approximation, while the formation of 0 ωb few-cycle optical solitons requires in principle electric fields whose amplitude becomes comparable to that of the atoms. the two-level Hamiltonian, and Our goal in this paper is to propose a nonperturbative 0 μ approach to optical wave propagation. We considered the μ = ∗ e , (3) μ 0 x simplest model of a material which can be written in the frame of the quantum mechanics: the two-level model in the density the atomic electric dipole moment operator, assumed to be matrix formalism. Two asymptotic models have been derived directed along the direction defined by the unitary vector ex. from the two-level one beyond the SVEA, one being based The evolution of the electric field E is governed by the on the modified Korteweg–de Vries (mKdV) equation, and wave equation the other on the sine-Gordon (sG) equation, depending if the central pulse frequency is well above or well below the −1 ∂2 P ∇(∇ · E) −∇2E = E + , (4) 2 2 frequency of the transition [18]. c ∂t ε0 2469-9926/2019/99(5)/053846(8) 053846-1 ©2019 American Physical Society HERVÉ LEBLOND PHYSICAL REVIEW A 99, 053846 (2019) ∗ in which c is the speed of light in vacuum, ε0 the dielectric so that 2ρt μ = d + iq, with d and q real. = (ωb − ωa )is permittivity in vacuum, and the polarization density P is the transition angular frequency, related to ρ by w = (ρb − ρa ), (18) P = NTr(ρμ). (5) is the population difference, negative in the considered situa- We are looking for an exact progressive wave solution of tion, and ρ0 = (ρa + ρb)/2. these equations. We restrict ourselves to a linearly polarized Then it is seen that dρ0/dθ = 0, in accordance with the wave and assume a planar wave propagation along z. Hence, property of the density matrix Tr(ρ) = 1, and the system we seek for a solution of the form reduces to = − z E E t ex (6) dw v =− γ dq, θ 2 (19) of Eqs. (1)–(5). d dd Obviously, we must have =−q, θ (20) z d P = P t − ex, (7) dq v = ( + 2γ |μ|2w)d, (21) dθ and (4) reduces to where we have set −1 d2E −1 d2 P = E + , (8) N 2 2 2 2 v dθ c dθ ε0 γ = , (22) ε0h¯χ where θ = t − z/v is the retarded time. We are looking for either localized solutions E or periodic for brevity. Equation (20) is solved straightforwardly as solutions with mean value zero. Hence P must have the same −1 dd q = . (23) property, the integration constants must vanish, and Eq. (8) dθ reduces to Using (23) into Eq. (19) allows us to integrate it as = ε χ , P 0 E (9) γ w = d2 − l, (24) where we have defined the “susceptibility” χ by c2 where l is some constant. If E is localized, then d is as well, χ = − 1. (10) − v2 and ( l)isthevalueofw in the absence of a wave. In this case, ρb = ρth is the thermal excitation, and ρa = 1 − ρth,so We write the components of the density matrix as = − ρ < that l 1 2 th, and 0 l 1, where the equality occurs ρ ρ if all atoms are initially in the fundamental state. Since a ρ = a t , ρ∗ ρ (11) periodic wave is a mathematical limit of a very long wave t b packet, the same interpretation still holds in this case. and the expression of the polarization density (5) can be Using the expression (23)ofq into Eq. (21) yields an written as equation for d: ∗ ∗ = ρ μ + ρ μ . 2 P N( t t ) (12) d d = Fd − 2Gd3, (25) θ 2 Replacing E by this expression of P divided by ε0χ in Eq. (1) d we get the following set of equations: where we have set ρ = γ |μ|2 − d a = N ρ μ∗ + ρ∗μ ρ μ∗ − ρ∗μ , F (2l ) (26) ih¯ ( t t )( t t ) (13) dθ ε0χ ρ and d t =− ω − ω ρ − N ρ μ∗ + ρ∗μ ρ − ρ μ, ih¯ h¯( b a ) t ( t t )( b a ) = γ 2|μ|2. dθ ε0χ G (27) (14) Multiplying Eq. (25)bydd/dθ and integrating yield dρb N ∗ ∗ ∗ ∗ 2 ih =− ρ μ + ρ μ ρ μ − ρ μ , dd 2 4 ¯ θ ε χ ( t t )( t t ) (15) = Fd − Gd + C, (28) d 0 dθ the fourth equation being the complex conjugate of (14). where C is a constant, and finally We define ∗ ∗ dd d = (ρt μ + ρ μ), (16) θ =± √ + θ0, (29) t C + Fd2 − Gd4 which is the atomic transition dipole moment, so that θ P = Nd, and 0 being another constant. It corresponds to translation invari- ance and the ± sign to invariance by symmetry. We can restrict = ρ μ∗ − ρ∗μ , + q ( t t ) (17) ourselves to the sign without loss of generality. 053846-2 HALF-CYCLE SOLITON AND PERIODIC WAVES OF … PHYSICAL REVIEW A 99, 053846 (2019) We write the soliton parameter as p = 2 acosh(2)/ t, where t is the full width at half maximum of the pulse. Notice that the wave profile depends neither on the atomic density nor on the resonance frequency; these quantities affect only the velocity v. C. The periodic solution 1. The cnoidal wave We set for convenience Q = C + Fd2 − Gd4. (36) When C < 0, the dipole moment d never can take the value zero in (29), which excludes that the corresponding solution can represent a wave oscillating around zero. However, for > FIG. 1. Example of soliton profiles for several values of the C 0, the two roots of Q have opposite signs. Let us denote 2 soliton parameter, corresponding to pulse durations of t = 1fs by dm the positive root. It is clear that (dotted, black line), 2 fs (dashed, magenta), 3 fs (solid, green), 4 fs √ F + F 2 + 4GC (one dash, two dots, cyan), and 5 fs (one dash, 10 dots, red).