Lasing without inversion

H-- Fearn

Lasing without inversion

H-- Fearn

1992, Zeitschrift f�r Physik D Atoms, Molecules and Clusters

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We develop a nonlinear theory of lasing without inversion in a four level atomic system. The appearance of lasing without inversion is the result of atomic coherence induced between two lower levels by a Raman process. We consider both nondegenerate and degenerate configurations and obtain expressions for the linear gain and the laser output intensity. We identify the conditions for ideal lasing without inversion.

Z. Phys. D - Atoms, Moleculesand Clusters 22, 483-493 (1992) Atoms, Molecules Ze,,.ch.,. f~ir Physik D and Clusters © Springer-Verlag 1992 Lasing without inversion II. Raman process created atomic coherence S.Y. Zhu 1, M.O. Scully 2, H. Fearn 1, and L.M. Narducci* 1 Center for Advanced Studies and Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA z Max-Planek Institut ffir Quantenoptik, W-8046 Garching, Federal Republic of Germany Received 1 July 1991 Abstract. We develop a nonlinear theory of lasing without inversion in a four level atomic system. The appearance of lasing without inversion is the result of atomic coherence induced between two lower levels by a Raman process. We consider both nondegenerate and degenerate configurations and obtain expressions for the linear gain and the laser output intensity. We identify the conditions for ideal lasing without inversion. PACS: 42.50.Md; 42.55. - f; 32.80. - t I. Introduction In a previous paper [1] we have studied the process of lasing without inversion in open and closed systems, for both nondegenerate and degenerate configurations (open systems are such that the lasing levels can decay to other lower lying states, while in closed systems the decay processes are confined to the active levels of interest; the adjectives nondegenerate and degenerate refer to the amplification of a pair of distinct laser lines or of a single frequency component). The realization of lasing without inversion is the consequence of the appearance of atomic coherence between the two lower levels which reduces the rate of absorption for laser light. In the open system configuration, the atomic coherence can be achieved through a suitable initial preparation before the atoms enter the cavity. In the closed system, we assumed that the atoms developed coherence between the two lower levels in the absence of a laser field. This coherence can be achieved through a Raman process or by the interactions with a microwave field. In this paper we focus on the Raman process-created coherence, and in a subsequent paper we will study the microwaveinduced coherence. It is known that a resonant interaction between a three-level A system and one (or two) strong electromagDepartment of Physics, Drexel University, Philadelphia, PA 19104, USA * Permanent address: netic field(s) (Raman processes) can lead to atomic population trapping in a linear combination of the lower levels [-2-6] and consequently leads to the reduction or even exclusion of transitions from the lower levels to a higher lying level. The trapping is due to the electromagnetic field-induced atomic coherence between the two lower states. With zero absorption by the lower levels, the possibility of laser emission from the upper level down to the two lower levels is enhanced. In this paper we show how to use the Raman processes to create the atomic coherence, and to induce lasing without inversion in a four-level closed system. We also establish the necessary conditions for lasing without inversion and the relation between the laser intensities and the intensities of the Raman fields. In a previous analysis limited to the linear regime, [7] we have shown that lasing without inversion (resulting from complete or partial cancellation of absorption from the lower-levels) is possible also with the help of the A quantum-beat laser scheme, even in the collisiondominated regime. More recent work has shown that a double A-system may also display gain without inversion at optical transitions [8]. The nonlinear analysis of [8] is conceptually identical to the Raman case considered in this work. Here, however, we wish to account for the effects of the off-diagonal atomic density matrix element Pan which couples the upper level ]a) to the auxiliary Raman level ]d). For this reason we do not neglect this contribution to the atomic polarization. We consider lasing without inversion in the four-level closed systems shown schematically in Fig. 1. The transitions between the upper level Ia) and the two lower levels [c) and Ib) produce two fields in the nondegenerate case or one field in the degenerate case. The two lower levels are coupled by Raman processes through an auxiliary level Id). The unperturbed Hamiltonian for the atomic system is N t¢o= d Y h o l S>< il, j=l ~=a (1) 484 la> la) --ld) V~ V3 ---7 ~ with wave vectors kl, kz, k3 and k4, respectively. We assume that the phase matching condition is fulfilled (kl - k 2 - k 3 + k 4 ) L ~ 1, where L is the cavity length. The equations of motion for the density matrix elements of the total atomic system are ld) 1 - , ivt i,,t p,ta = --~Pdd-l-i[o~ e 3 pac_o~ e- , ivt ~ Pcd+fl e ' Pdb - fie -iv4~pba], (~1 (3 a) Ic> t lc> 1 Jr"a-- (b) Fig. la, b. Four level closed system. ]a), tb) and lc) are lasing levels, while Id) is an auxiliary levelfor Raman processes, a Nondegenerate case, b Degenerate case T1 Paa+2(Pbb+Pcc)+i[g* E*eiV"P"c --gl El e-iVltpca + s"*z E*2 e'~2tPab--g2 E2 e-~2tpba], 1 where hco~ is the energy of the c~-th level, and led) the corresponding stationary state for the j-th atom; N is the total number of atoms in the ring cavity. The interaction Hamiltonians between atoms and fields are different in the nondegenerate and degenerate configurations. First we discuss the nondegenerate case in Sect. II, then the degenerate case in Sect. III. Section IV contains a general discussion and an overview of the results. Finally, the Appendix outlines some of the details of the calculations. 1 (3b) 1 + i[g 2 E 2 e-iv2tpb a - g * E~ e i ~ Pab+fle-i~"tpbd--fl*ei"4tpj, • 1 1 (3c) I + i [ga E1 e-i"ltPca --g* E* ei"'tp~c+O~e-i"3~pca-o:*ei~3tpac], 1 p~,+ie),cpc,-i[g* P~'-- - - -T2 (3d) E'~(p,,-p~c) e ~ t --g2 E2 e'~2tP~b+c~*e'"~tPej1, Pba = - - ~ II. The nondegenerate configuration Pba + i 69ab Pba -- i [g~ E* (Paa -- Pbb) eiv2t -- g* E* eiVlt Pbc+ fl* ei~4tp aa] , A. Equations of motion The interaction Hamiltonian is comprised of two parts, v=v 1 (3e) (30 1 Pea = -- ~ P~a + i mac Pea-- i [(Pea - Pcc) c~*e 1"3~ • _fl*e~,4tpc b + g l* Et• ei~'~Pad], + where V1 denotes the interaction between the laser fields and the atoms, and V2 the interaction between the input Raman fields and the atoms. The two Raman fields couple level ld) to levels lc) and Ib), as shown in Fig. l a and induce a Raman coherence between them. Each Raman field (or laser field) couples with only one atomic transition due to some selection roles. The explicit forms of these interaction energies are given by N (3g) 1 Pbd = - - T 2 Pbd + i ('Odb Pbd -- i [(Dad-- Pbb) fi* ci"4t --o~*e~Vatpb c +g2* E2* ei~2'p,d], • Pcb= 1 Z--2 Pcb+iOgbcPcb--i[g~ E* e i ~ t p a b --g2 E2e-iVZtpca+~*eiV3tpab--fle-iV4tpcd] Pda-- (3h) , (3i) T1 + 7~ i[o~e_i~3tpc" 27"1 T~ Pa"+im~dPe"-+ fle-i"" Pb.--g * ~t~*""i'~t"~'ac .52r~*~21g*eiVztpab] ' - - (3j) j=l + H.A., N V2= ~ h[c~e-i"'tld~)(c~l+fle-i~JldJ)(bJI]+g.A., (2a) (2b) j=t where e and fl are the Rabi frequencies of the Raman fields whose carrier frequencies are v3 and v4, respectively. E~ and E2 are the lasing field strengths with carrier frequencies v~ a n d v2, respectively, g~ and g2 are the coupling constants between the laser fields and the corresponding transitions. The two laser fields and the Raman fields, in the ring cavity, propagate in the same directions where we have included incoherent pump contributions from levels [c> and Ib> to level la>, with the same rate 2, T1 (T~) and zl are population lifetimes from the upper level la> (the auxiliary level Id>) to the lower levels Ic> and [b), and between the two lower levels, respectively, and the corresponding polarization liftetimes are T2(T~) and z2. Here c%p = (o~-- cot~where ~, fl = a, b, c, d. We introduce the slowly varying functions of time P,B defined by P c a = ~ c a e iv~t, (4a) Pba = Pba eiv2t, (4 b)

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