The phase transition to mirrorless oscillation in resonantly enhanced
four-wave mixing in double-$\Lambda$ systems are studied analytically for the
ideal case of infinite lifetimes of ground-state coherences. The stationary
susceptibilities are obtained in all orders of the generated fields and
analytic solutions of the coupled nonlinear differential equations for the
field amplitudes are derived and discussed.Comment: proceedings ICLPQO'99 (Shanghai 99), 10 pages, 4 figure

Fleischhauer, M.

English

arXiv

Abstract: The phase transition to mirrorless oscillation in resonantly enhanced four-wave mixing in double-A systems are studied analytically for the ideal case of infinite lifetimes of ground-state coherences. The stationary susceptibilities are obtained in all orders of the generated fields and analytic solutions of the coupled nonlinear differential equations for the field amplitudes are derived and discussed

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arXiv:quant-ph/9910112   27 Oct 1999Mirrorless oscillation based on resonantlyenhanced 4-wave mixing: All-order analyticsolutionsM. Fleischhauer e-mail: meisch@theorie.physik.uni-muenchen.deSektion Physik, Universitat Munchen, D-80333 Munchen, GermanyAbstract. The phase transition to mirrorless oscillation in resonantly enhancedfour-wave mixing in double- systems are studied analytically for the ideal caseof innite lifetimes of ground-state coherences. The stationary susceptibilities areobtained in all orders of the generated elds and analytic solutions of the couplednonlinear dierential equations for the eld amplitudes are derived and discussed.1 IntroductionThe possibility to cancel the linear absorption in resonant atomic systems bymeans of electromagnetically induced transparency (EIT) [1] lead in recentyears to fascinating new developments in nonlinear optics [2,3]. For examplecoherently driven, resonant atomic vapors under conditions of EIT allow forcomplete frequency conversion in distances short enough, such that phasematching requirements become irrelevant [4]. Furthermore the large nonlin-earities of these systems may lead to a new regime of nonlinear quantum op-tics on the few-photon level [5,6] with potential applications to single-photonquantum control [7,8] and quantum information processing.One particularly interesting nonlinear process based on EIT is the reso-nantly enhanced 4-wavemixing in a double- system with counter-propagatingpump modes [9]. It has been shown experimentally [10] and theoretically[11,12] that this system can show a phase transition to mirrorless oscillationsfor rather low pump powers. Close to the threshold of oscillation an almostperfect suppression of quantum uctuations of one quadrature amplitude ofa combination mode of the generated elds occurs [13,14]. Also sucientlyabove threshold light elds with beat-frequencies tightly locked to the atomicRaman-transition and extremely low relative bandwidth are generated [15].All previous studies of resonantly enhanced 4-wave mixing were done inthe perturbative regime of small amplitudes of the generated elds. In thepresent paper I want to discuss the case of arbitrary amplitudes. Using asimplied open-system model I will derive stationary propagation equationsfor the eld amplitudes and present analytic solutions of these equations.It will be shown that in an ideal case complete conversion can be achievedwithin a relatively small interaction length.2 M. Fleischhauer e-mail: meisch@theorie.physik.uni-muenchen.de2 Model and Atomic PolarizationsI here consider the propagation of four electromagnetic waves in a mediumconsisting of double- atoms (see Fig.1). These waves include two counter-propagating driving elds with equal frequencies d and Rabi-frequencies 1and 2, and two probe elds (anti-Stokes and Stokes) described by the com-plex Rabi-frequencies E1 and E2, with carrier frequencies 1 = d + !0 and2 = d   !0, where !0 = !b1   !b2 is the ground-state frequency splitting.The elds interact via the long-lived coherence on the dipole-forbidden tran-sition between the metastable ground states b1 and b2. We assume that thedriving eld 1 is in resonance with the b2 ! a1 transition, whereas the sec-ond driving eld 2 has a detuning  j2j from the b1 ! a2 transition. Inthis case linear losses of the elds due to single-photon absorption processesare minimized.aEEΩ212a12b1∆2b1ΩFig. 1. Atoms in double  conguration inter-acting with two driving elds (1;2) and twogenerated elds (E1;2)Due to coherent Raman-scattering the pump elds generate counter-propagating anti-Stokes and Stokes elds. For a suciently large density-length product of the medium and for a certain pump eld intensity, thesystem shows a phase-transition to self-oscillations [10]. The feedback mech-anism required for an oscillation is provided here by the gain medium: AStokes photon spontaneously generated on the a2 ! b2 transition propagatesin the  z direction and stimulates the generation of an anti-Stokes photon.This anti-Stokes photon has a dierent frequency but a xed relative phaseand propagates in the +z direction. It stimulates the generation of anotherStokes photon upstream. The second Stokes photon will be in phase with therst one, provided that the system is approximatly phase matched and thatthere has been no decay of the Raman coherence. The phase-locked emissionof the second Stokes photon then closes the feedback loop. We have shown in[15] that phase-matching enforces a strong pulling of the beat-note of gener-ated and pump elds to the atomic Raman transition. I will therefore assumehere that both  systems are in perfect two-photon resonance.In order to calculate the medium response to the elds, one would have tosolve the atomic density matrix equations to all orders in all elds taking intoaccount all relaxation rates. Although this is in principle possible it leads toResonant 4WM: analytic solutions 3extremely involved expressions. Instead I here use a simplied open-systemmodel which allows to derive rather compact expressions for the atomic sus-ceptibilities.Since the eects of spontaneous emission are negligible in the presentsystem, we may model all relaxations out of the excited states a1 and a2by rates  out of the system. In thermal equilibrium, i.e. in the absenceof all elds, both lower states b1 and b2 are equally populated. I thereforeassume { within the open-system approach { that the atoms are pumpedinto states b1 or b2 with 50% probability respectively. The correspondingrate is denoted as r and will later be determined by the requirement thatthe total probability to nd an atom in any of the states is unity. The nitelifetime of the lower-level coherence will here be described by a decay outof all states with rate 0. Thus the open-system model corresponds to theexperimentally relevant situation of an atomic beam or a nite-temperaturevapor with time-of-ight broadening. In this case the system can be describedby generalized Schrodinger-equations for eld amplitudes instead of density-matrix equations.The interaction Hamiltonian of an atom at position z with the elds canbe written in the formHint =  hh1(z) e idtja1ihb2j+2(z) e idtja2ihb1j++E1(z) e i1tja1ihb1j+E2(z) e i2tja2ihb2j+ adj:i: (1)If we denote the state vector of the atom asj	 i = a1e ia1tja1i+ a2e i(a2 )tja2i+ b1e ib1tjb1i + b2e ib2tjb2i; (2)where h are the energies of the corresponding states, we nd the followingequations of motion of the slowly-varying state amplitudes for an atom atposition z_a1 =   (0 + ) a1 + i1b2 + iE1b1; (3)_a2 =   (0 +  + i) a2 + i2b1 + iE2b2; (4)_b1 = r1   0b1 + i2a2 + iE1a1; (5)_b2 = r2   0b2 + i1a1 + iE2a2: (6)Here I have introduced the rates r1 and r2 to distinguish the cases of pumpinginto b1 (r1 = r; r2 = 0) and into b2 (r1 = 0; r2 = r). Note that simultaneouslysetting r1 = r2 = r corresponds to a coherent preparation of the atomsin a 50{50 superposition of b1 and b2. In order to describe an incoherentpreparation in these levels one has to consider the two cases separately andadd the density matrix elements following from both cases.Solving (3{6) in steady state for the case of injection into b1, i.e. for r1 = rand r2 = 0 one ndsa(1)1 =  ir12E2   E1jE2j2j12  E1E2j2; (7)4 M. Fleischhauer e-mail: meisch@theorie.physik.uni-muenchen.dea(1)2 = irj1j22   1E1E2j12  E1E2j2; (8)b(1)1 = irj1j2j12   E1E2j2; (9)b(1)2 =  ir1E1j12  E1E2j2; (10)where I have used that     0 and have kept only the leading terms.Similarly one nds for injection into b2, i.e. for r1 = 0 and r2 = r:a(2)1 = ir1j2j2   E1E22j12  E1E2j2; (11)a(2)2 =  ir12E1   jE1j2E2j12  E1E2j2; (12)b(2)1 =  ir1E1j12  E1E2j2; (13)b(2)2 = irjE1j2j12   E1E2j2: (14)Taking into account only the leading order contribution in the above expres-sions is essentially equivalent to assuming an innitely long lived ground-statecoherence between b1 and b2. In vapor cells with coated walls or by usingbuer gases, lifetimes of Hyperne coherences in alkali vapors in the millisec-ond regime are possible. Hence neglecting contributions from nite valuesof 0 seems justied. However, in this case also linear absorption losses areneglected. As a consequence the threshold condition becomes independent onthe pump intensity and an arbitrarily small ux of pump photons is sucientto maintain oscillations [15]. If on the other hand a small but nite ground-state dephasing rate is taken into account, the threshold condition does de-pend on the pump intensity leading to a lower limit of the pump-photon ux.In the present paper I am interested only in the analytic behavior of the eldsin the ideal limit and therefore the small but nite linear losses associatedwith the ground-state dephasing will be ignored.The pump rate r can be determined from the normalization conditionP%(1) + %(2) = 1. One nds r = j12   E1E2j2= j1j2 + jE1j2:With this one obtains for the non-diagonal density matrix elements %ab =a(1) b(1) + a(2) b(2) :%a1b1 =  j1j212E2   E21E212 (j1j2 + jE1j2)2 j1j2 j2j2   jE2j2 (j1j2 + jE1j2)2E1; (15)%a1b2 =212E1E2   jE1j2E1E22 (j1j2 + jE1j2)2+jE1j2(j2j2 + jE2j2) (j1j2 + jE1j2)21; (16)%a2b1 =   j1j2 + jE1j2E1E21 (j1j2 + jE1j2)2 +j1j2 j1j2 + jE1j2 (j1j2 + jE1j2)2 2; (17)Resonant 4WM: analytic solutions 5%a2b2 =   j1j2 + jE1j212E1 (j1j2 + jE1j2)2 +jE1j2 j1j2 + jE1j2 (j1j2 + jE1j2)2 E2: (18)The rst terms in these expressions describe the nonlinear coupling betweenthe modes and the second ones ac-Stark shift induced changes in the refractiveindices. It should be noted that there are no imaginary linear susceptibilities,i.e. there is no linear dissipation despite the fact, that 1 and E1 are insingle-photon resonance.3 Stationary eld equations and analytic solutionsIn slowly-varying amplitude and phase approximation, the eld amplitudesobey the following equation of motionddzE1 = ik1E1 + i}2k12h"0N %a1b1 ; (19)ddzE2 = ik2E2 + i}2k22h"0N %a2b2; (20)ddz1 = ikd1 + i}2kd2h"0N %a1b2 ; (21)ddz2 = ikd2 + i}2kd2h"0N %a2b1 ; (22)where k1, k2 and kd are the free-space wavenumbers of the generated andpump elds, N is the atomic number density and } are the dipole momentsof the corresponding transitions, which have been assumed to be equal forsimplicity. Since the wavenumbers of the elds dier only slightly, one may ap-proximate the coupling parameter in all equations by   }2kdN=2h"0. Intro-ducing eld amplitudes which are slowly varying in space, E1 = eE1 eik1z; E2 =eE2 e ik2z; 1 = e1 eikdz and 2 = e2 e ikdz one eventually arrives atddzE1 =  ij1j212E2   E21E212 (j1j2 + jE1j2)2 i"k+ j1j2 j2j2   jE2j2 (j1j2 + jE1j2)2#E1; (23)ddzE2 =  i j1j2 + jE1j212E1 (j1j2 + jE1j2)2+ijE1j2 j1j2 + jE1j2 (j1j2 + jE1j2)2 E2 ; (24)ddz1 = i212E1E2   jE1j2E1E22 (j1j2 + jE1j2)2+ijE1j2(j2j2 + jE2j2) (j1j2 + jE1j2)2 1 (25)6 M. Fleischhauer e-mail: meisch@theorie.physik.uni-muenchen.deddz2 =  i j1j2 + jE1j2E1E21 (j1j2 + jE1j2)2+ij1j2 j1j2 + jE1j2 (j1j2 + jE1j2)2 2 ; (26)where I have dropped the tildes again for notational simplicity, and k =k2   k1 is the free-space phase mismatch. Expanding these expressions intothird order of the generated elds E1 and E2 reproduces the equations of[15]. Equations (23{26) together with the boundary-conditionsE1(0) = 0; E2(L) = 0; 1(0) = 10; and 2(L) = 20; (27)where L is the length of the interaction region and 10 and 20 are thegiven input amplitudes, form a nonlinear boundary-value problem. One easilyveries that the set of dierential equations has always the trivial solutionE1  E2  0, and 1(z)  10 and 2(z)  20.As has been discussed in detail in [15], the phase mismatch is easily com-pensated in an optically dense vapor by a small detuning from the two-photonresonance. Oscillation occurs at frequencies such that the phase-matchingcondition is automatically fullled. I therefore set this term equal to zero inthe following.Constants of Motion: The eld equations have the following constants ofmotion. From the energy-momentum conservation follow the Manley-Rowerelationsddz j1j2 + jE1j2= 0; (28)ddz j2j2 + jE2j2= 0; (29)which state that each photon taken out of the pump elds 1 or 2 is putinto the anti-Stokes and Stokes elds E1 and E2 respectively. Furthermoreone nds that the total intensity of the pump eld is constant in spaceddz j1j2 + j2j2= 0: (30)The same is true for the generated elds, which however follow already fromthe above constants of motion.ddz jE1j2 + jE2j2= 0: (31)Without the phase terms in (23{26), which represent contributions due to ac-Stark shifts, also the quartic expression Re [12E1E2 ] would be a constantof motion. In fact the boundary conditions for the generated elds imply thatRe [12E1E2 ]  0. It will be shown later on that Re [12E1E2 ] is in anycase to a very good approximation a constant of motion.Resonant 4WM: analytic solutions 7Amplitude-Phase Equations: It is convenient to rewrite the eld equa-tions in terms of amplitudes and phases. Introducing En = en e in andn = an e i n (n = 1; 2) one obtainsddze1 =a1a2e2a21 + e21sin ; (32) ddze2 =a1a2e1a21 + e21sin ; (33) ddza1 =a2e1e2a21 + e21sin ; (34)ddza2 =a1e1e2a21 + e21sin ; (35)where  = 1+2  1  2 is the relative phase between the elds. It obeysthe equationddz =a1a2e2(a21   e21)e1(a21 + e21)2 a1a2e1e2(a21 + e21)+a2e1e2(a21   e21)a1(a21 + e21)2 a1e1e2a2(a21 + e21)cos +e41   a41 + 2e21a22(a21 + e21)2: (36)Solution for Equal Input Intensities: Let me now consider the case ofequal input intensities of both pump elds, i.e. a1(0) = a10 = a20 = a2(L).Making use of the constants of motion one can writee1(z) = e sin#(z); a1(z) =qa210   e2 sin2 #(z); (37)e2(z) = e cos#(z); a2(z) =qa210   e2 cos2 #(z); (38)with the output amplitude of the generated elds e and the mixing angle #(z)as the only remaining variables. The boundary conditions are now #(0) = 0and #(L) = =2, if e 6= 0, i.e. for the non-trivial solutions.Substituting the above expressions into (32) yields the nonlinear equationddz#(z) =1  "2 +"44sin22#(z)1=2sin (z); (39)where "  e=a10. In order to solve (39) one can in principle introduce anonlinear stretch of the spatial coordinate according to(z) =Z z0dz0 sin (z0); and d = sin (z) dz (40)which removes the term sin (z) on the r.h.s. of (39). I will show later on,however, that to a very good approximation sin (z)  1. Thus  = z andsin (z) = 1 is used in the following.8 M. Fleischhauer e-mail: meisch@theorie.physik.uni-muenchen.deIntegrating (39) from z = 0 to z = L leads to an equation for the normal-ized output amplitude " = e=a10:K"44("2   1)=Lp1  "2; (41)where K is the complete elliptic integral of the rst kind [16]. One easily ver-ies that (41) has only a real-valued solution ", if L=  =2, which is thethreshold condition for mirrorless oscillations [10,12]. For smaller values ofL= the equations of motion have only the trivial solution. Figure 2 shows2Ω10E0 1 2 3 4 500.20.40.60.810 1 2 3 4 5010.20.40.60.8κ   /∆  (π/2)    L2Fig. 2. Output intensity of generated elds E  E1(L) = E2(0) normalized toinput intensity of pump elds as function of eective interaction length, 1(0) =2(L)  10the output intensity of the generated elds normalized to the input inten-sity of the pump elds as a function of the eective density length productL. One clearly recognizes that for a suciently large product L completeconversion can be achieved.The spatial behavior of the eld strength inside the vapor cell can beobtained from incomplete elliptical integrals following from (39). Figure 3shows the eld amplitudes inside the medium for " = 0:2, i.e. just abovethreshold and for " = 0:98 i.e. for almost complete conversion.Not to far above threshold, the square root in (39) can be expanded andone recovers the third-order solution obtained in [15]:#(z) z1 12"2(42)with" =p21 2L1=2forL2: (43)In order to verify the approximation sin (z)  1, I have numerically inte-grated the dierential equation (36) with the above solutions. Figure 4 showsResonant 4WM: analytic solutions 90 0.2 0.4 0.6 0.8 100.20.40.60.810 0.2 0.4 0.6 0.8 100.20.40.60.81E EΩ 12 1Ω2 E 2Ω2E1Ω10 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 100.20.40.60.81z / L z / L Fig. 3. Field amplitudes inside interaction region for small conversion " = E=10 =0:2 (left) and large conversion " = E=10 = 0:98 (right)the comparison between the nonlinear coordinate (z) and z for the case" = 0:98. One recognizes that  deviates from z by at most 1%. For smallerconversions an even smaller dierence shows up. Therefore the approxima-tion sin = 1 is very well justied. This also implies that Re [12E1E2 ] isto a very good approximation a constant of motion.0 0.2 0.4 0.6 0.8 100.20.40.60.810 0.2 0.4 0.6 0.8 1z/L00.20.40.0.81/LξFig. 4. Eective interaction distance  versus physical interaction distance z forlarge conversion (" = E=10 = 0:98). Dotted line corresponds to  = z4 SummaryIn the present paper all-order atomic susceptibilities for resonantly enhanced4-wave mixing are presented and eld equations derived. The coupled nonlin-ear dierential equations are solved analytically for the case of innitely long-lived ground-state coherences and under the assumption of negligible phasechanges due to ac-Stark shifts. Below a certain critical value of the density-length product only the trivial solution exists, where the generated Stokes10 M. Fleischhauer e-mail: meisch@theorie.physik.uni-muenchen.deand anti-Stokes components have vanishing amplitude. Above the thresh-old to mirrorless oscillations the photon conversion eciency increases veryrapidly and at a density-length product of about 3 times the threshold value,95% conversion is achieved.AcknowledgementThe author would like to thank the organizers of the International Conferenceon Laser Physics and Quantum Optics, ICLPQO'99, in particular Prof. Shi-Yao Zhu for the invitation and the hospitality in China. The nancial supportof the German Science Foundation is gratefully acknowledged.References1. for a review on EIT see: S. E. Harris, Physics Today 50, 36 (1997)2. S. E. Harris, J. E. Field, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990)3. K. Hakuta, L. Marmet, and B. P. Stoiche, Phys. Rev. Lett. 66, 596 (1991)4. M. Jain, H. Xia, G. Yin, A. J. Merriam und S. E. Harris, Phys. Rev. Lett. 77,4326 (1996)5. H. Schmidt and A. Imamoglu, Opt. Lett. 21, 1936 (1996), A. Imamoglu, H.Schmidt, G. Woods, and M. Deutsch, Phys.Rev.Lett. 79, 1467 (1997)6. S. E. Harris and L. V. Hau, Phys. Rev. Lett. 82, 4611 (1999)7. S. Harris and Y. Yamamoto, Phys.Rev.Lett. 81, 3611 (1998)8. M. Werner, and A. Imamoglu, preprint quant-ph/99020059. P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriarand P. Kumar, Opt. Lett. 20, 982 (1995)10. A. S. Zibrov, M. D. Lukin, and M. O. Scully, Phys.Rev.Lett. 83 (1999), in press11. M. D. Lukin, P. Hemmer, M. Loeer, and M. O. Scully, Phys. Rev. Lett. 81,2675 (1998)12. M. D. Lukin, P. R. Hemmer, M. O. Scully, in Adv. At. Mol. and Opt. Physics,42B, 347 (Academic Press, Boston, 1999)13. H. P. Yuen and J. H. Shapiro, Opt. Lett. 4, 334 (1979)14. M. D. Lukin, A. B. Matsko, M. Fleischhauer, M. O. Scully Phys. Rev. Lett.82, 1847 (1999)15. M. Fleischhauer, M. D. Lukin, A. B. Matsko, and M. O. Scully, preprint quant-ph/990703216. M. Abramowitz and I. A. Stegun, \Handbook of Mathematical Functions",(Thun, Frankfurt/Main, 1984)

Mirrorless oscillation based on resonantly enhanced 4-wave mixing: All-order analytic solutions

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