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Dynamics of slow-light formation
1 July 2002
Optics Communications 208 (2002) 125–130 www.elsevier.com/locate/optcom
Dynamics of slow-light formation E. Cerboneschi a,b, F. Renzoni c,*, E. Arimondo b a Istituto di Tecniche Spettroscopiche del CNR, Via La Farina 237, I-98123 Messina, Italy INFM and Dipartimento di Fisica, Universit a di Pisa, Via F. Buonarroti 2, I-56127 Pisa, Italy Laboratoire Kastler Brossel, D epartement de Physique de l’Ecole Normale Sup erieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France b
c
Received 17 February 2002; accepted 23 May 2002
Abstract We investigate theoretically the phenomenon of slow light in a cold sample of open three-level atoms interacting with the two light fields in the K configuration. We consider a cold atomic sample geometry such that the photon reabsorption is greatly reduced, and therefore the medium properties are determined by the single atom response. The dynamics of the slow propagating light pulse is examined, and the typical length scales for the propagation derived. We demonstrate that STIRAP is the mechanism behind the slow-light phenomenon. Furthermore, in the considered geometry a significant occupation of the excited state in the early phase of the propagation does not necessarily imply a decay of the ground-state coherence, and therefore does not inhibit the slow-light formation. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50.Gy; 32.80.-t; 42.50.Vk
1. Introduction The linear absorption of an atomic medium, as measured by a weak probe laser beam, corresponds to the quantum structure of the probed atoms, with absorption lines at the atomic transition frequencies. Low-frequency coherences created by applying one additional laser field (the coupling field) may modify completely the absorptive properties of the medium, which can be made transparent whenever the probe and the coupling fields couple two ground states to a *
Corresponding author. Tel.: +3301-4432-3568; fax: +33014432-3434. E-mail address: [email protected] (F. Renzoni).
common excited state with the laser frequency difference matching the ground-state splitting. This is the so-called electromagnetically induced transparency (EIT) [1,2]. The stationary state of the medium in the regime of EIT corresponds to atoms prepared in a linear superposition of the ground states which is decoupled from the light fields (non-coupled or dark state [3]), as a result of the destructive quantum interference between transition amplitudes. The probe light group velocity is greatly reduced by the very steep dispersion associated with the EIT resonance. After the pioneering work on adiabatic pulse propagation by Grobe et al. [4], remarkable results for the reduction of the group velocity in three-level systems were recently reported in cold samples [5–7] as well
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as in samples with thermal velocity distributions [8–10]. Many applications based on the enhancement of the optical non-linearities in a EIT medium, as a result of the reduced probe light group velocity and cancellation of the absorption, have been proposed, as reviewed in [11]. A key question for the efficient slow-light propagation is the initial atomic preparation in the dark state. The central role of STIRAP (Stimulated Raman Adiabatic Passage [12]) in establishing EIT for laser pulses, whence their slow-light propagation, was pointed out by Fleischhauer [13] and recently reexamined [14]. Indeed, in his work it was shown that STIRAP is the mechanism underlying the slow-light atomic preparation, and that the conditions for efficient slow-light propagation coincide with those of STIRAP. Those conclusions were derived from the minimization of the reabsorption of spontaneously emitted photons, which is the dominating damping mechanism of the ground-state coherence for a dense threedimensional atomic sample. That minimization required, obviously, a very small occupation of the excited state and led naturally to the counterintuitive coupling/probe pulse sequence of STIRAP. In the present work we examine a different situation: we assume a geometry of the cold atomic sample such that the photon reabsorption is greatly reduced and the properties of the atomic medium are determined by the single atom response. We will show that also for the considered geometry the STIRAP conditions of adiabatic following of the dark state, imposed all along the propagation distance, define the mechanism behind the slowlight phenomenon. This generalizes the results of [13] to geometries of cold atomic samples characterized by a greatly reduced photon reabsorption. In this case a significant occupation of the excited state in the early phase of the propagation does not necessarily imply a decay of the ground-state coherence, and, as we will show, does not inhibit the slow-light formation.
configuration. This open system corresponds to the conditions of the cold atoms experiment by Hau et al. [5]. We neglect atom–atom interactions [16]. We assume a cigar shape geometry of the cold atomic sample [17], so that the photon reabsorption, which destroys the ground-state coherence, is greatly reduced. The interaction scheme is shown in Fig. 1. The ground (or metastable) states jci and jpi, with energies Ec and Ep , respectively, are excited to a common state jei, with energy Ee , by two laser fields of frequencies xa , electric field amplitudes Ea , and wavevectors ka ¼ xa =c ða ¼ c; pÞ, c being the light speed in vacuum. The laser detunings are denoted by da ¼ xa xea and the Raman detuning from the two-photon resonance is dR ¼ dc dp . The Rabi frequencies are given by Xa ¼
Dea Ea ; h
ð1Þ
with Dea the atomic dipole moment, supposed to be real for the sake of simplicity. The evolution of the atomic density matrix q is described by the optical Bloch equations (OBE) for an open system (see for example [18]). We assume the excited-state relaxation, denoted by C, due to spontaneous emission, as for a dilute atomic sample. The rate C is composed by the rates Cc and Cp for the decays into the ground states jci and jpi, and by the decay Cout into a sink state jouti, not excited by the lasers C ¼ Cc þ Cp þ Cout :
ð2Þ
The decay rate of the optical coherences, qep and qec , is equal to C=2.
2. Level scheme and Maxwell–Bloch equations We consider a cold sample of open three-level atoms interacting with two laser fields in the K
Fig. 1. Interaction scheme of an open three-level atomic system coupled with two laser fields in the K configuration.
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To describe the propagation of laser pulses through a cold sample of open three-level atoms, the electric field amplitudes, whence the Rabi frequencies, are assumed dependent on the time t and the coordinate z along the propagation direction. For slowly varying electric field amplitudes, the Maxwell equations for the coupling/probe envelope Rabi frequencies expressed in terms of the pulse-localized coordinates z and s ¼ t z=c, Xp ðz; sÞ and Xc ðz; sÞ reduce to [19] o Xa ðz; sÞ ¼ ija q~ea ðz; sÞ ð3Þ oz with a ¼ ðc; pÞ and the slowly varying density matrix element given by q~ep ¼ qep expðiðxe xp ÞtÞ:
ð4Þ
The parameter ja is given by ja ¼
xa nD2ea ; c0 h
ð5Þ
where n is the atomic density and 0 the vacuum susceptibility. The Beer’s absorption length for the probe field fp in the absence of coupling laser is fp ¼ C=jp . In the numerical calculations we considered an initial Gaussian pulse on the probe transition 2 t Xp ðz ¼ 0; tÞ ¼ Xop exp : ð6Þ 2T 2 The coupling laser, with constant Rabi frequency Xc , was assumed switched on at earlier times t 6 0, in order to realize the STIRAP counterintuitive pulse sequence. The atoms were initially prepared in the ground jpi state. This corresponds to an initial preparation of the atoms in the non-coupled state. Both probe and coupling fields are taken to be resonant with the corresponding atomic transitions: dc ¼ dp ¼ dR ¼ 0. The Maxwell–Bloch equations were solved for different strength of the coupling field and different probe pulse amplitude and length. To study the mechanism behind the slow-light propagation, it is useful to examine to which extent the initial probe/coupling configuration and the timeevolved one satisfy the condition of adiabaticity. We introduce here the basic concepts and the relevant quantities.
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In STIRAP [12,13,15] the condition for adiabatic following of the dark state is: X_ p Xc X_ c Xp X; ð7Þ X2 where the dot denotes the time derivative and X is defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ¼ jXp j2 þ jXc j2 : ð8Þ
X
This corresponds to a small non-adiabatic coupling between non-coupled and coupled states. In [15] it was shown that condition (7) corresponds also to a formstable probe pulse propagation. In this work the dynamics of slow-light formation is studied by considering initial probe/coupling configurations for which condition (7) is not strictly satisfied.
3. Numerical results Numerical results for the propagation of the probe pulse within a sample of open three level atoms are shown in Fig. 2. We used the parameters of the experiment in [5], i.e., Na D2 line parameters (C ¼ 2p 9:9 MHz, k ¼ 589:0 nm), cold atom
Fig. 2. Probe Rabi frequency Xp ðz; sÞ=C as a function of the pulse-localized reduced time s, normalized to 1=C, at different penetration distances for parameters Cc ¼ C=3, Cp ¼ C=2, and Cout ¼ C=6. The dashed line represents a reference probe pulse with no atoms in the propagation medium, scaled down by a factor two. Initial Gaussian probe pulse width T ¼ 80=C, and initial coupling laser Rabi frequency Xc =C ¼ 0:18.
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density of 3:3 1012 atoms=cm3 , propagation length through the medium up to a distance z equal to 63 times the Beer’s length fp , initial Rabi frequencies Xop and Xc around a tenth of C. The delay time in the propagation through the medium was measured in units of 1=C. The numerical results show a pulse peak height decreasing with the penetration distance, a broadening of the probe pulse, and a propagation with a reduced group velocity. Results for the probe group velocity are in the range of the values measured in [5]. The data of Fig. 2 indicate that for the simulation parameters an attenuation of the propagating probe pulse occurs. An example of the probe pulse transmission fraction Ft ðzÞ ¼ Xp ðzÞ= Xp ð0Þ as a function of the propagation distance z is plotted in Fig. 3, both Xp calculated at the pulse peak. The probe pulse amplitude decreases with
Fig. 3. In the top, ratio X =X, continuous line, and transmission fraction Ft of the probe pulse, dashed line, as a function of the propagation distance z. Parameters as those of Fig. 2, except T ¼ 10=C. The z decrease is described through a doubleexponential with absorption lengths reported in the inset. In the bottom, dependence of fS , close dots, and of fL , open squares, on the square of the coupling Rabi frequency Xc .
the propagation distance, and that decrease may be described through a double exponential dependence on the propagation distance, with absorption lengths fS and fL . The fS propagation length, the short one, is required in order to modify the probe pulse into that required for the EIT propagation. The second propagation length, the long one, describes the weak attenuation of the probe pulse in the EIT regime. The fL dependence on X2c shown in the bottom part of Fig. 3 has been predicted in [20]. We found that fS presents a similar dependence. The overall behavior can be understood by noting that for the parameters of the initial coupling/probe configuration chosen for Fig. 3 the adiabaticity conditions are not fully satisfied, as it appears clearly from Fig. 4(a). Correspondingly we studied to what extent the evolved coupling/ probe configuration satisfies the adiabaticity conditions, with numerical results for X =X as a function of the propagation distance in Fig. 3. Because X =X depends on the time t, we considered its maximum value with respect of t. We found that X =X decays mainly with the absorption length fS .
Fig. 4. (a) Non-adiabaticity test comparison between X and X for parameters of Fig. 3 at z ¼ 0. (b) Dependence of the transmission fraction Ft for the probe Rabi frequency through the sample for different pulse lengths T at z ¼ 63fp in the conditions of the open system of Fig. 3.
E. Cerboneschi et al. / Optics Communications 208 (2002) 125–130
Analytic results for the propagation of laser pulses through a medium of three-level atoms under conditions of quasi-adiabaticity have been derived by Fleischhauer and Manka [15]. In that work, analytic expressions for the spatial evolution of the laser fields were obtained through a perturbation expansion in X =X, after Fourier transforming the equations of motions in the frequency domain. Such results are appropriate for the understanding of the absorptive properties of the K system in the formation of slow light, as detailed by our simulations. Equation (78) in [15] shows that the z-dependence of the Fourier components of the ratio X =X is an exponential, that includes a dispersive term accounting for the quasi-formstable propagation with reduced group velocity and an absorptive term explaining the attenuation of the high-frequency components of the propagating pulse. In their analysis, Fleischhauer and Manka focused on the formstable dispersive propagation arising when all relevant Fourier frequencies x are sufficiently small, such that x X2 =C. In this work we have examined the opposite case. The light pulses at the entry of the medium have a broad Fourier spectrum and the absorption cannot be neglected. According to the analytic results in [15], the high-frequency components, for x X2 =C are absorbed linearly, within a Beer’s one-photon absorption length. A longer penetration distance is required to attenuate Fourier components at lower frequency. The absorption lengths fS for the initial fast decay derived from our numerical analysis are of the order of the Beer’s absorption length and, therefore, are quantitatively consistent with the analytical prediction of [15]. We must point out that the fast decay in our simulations involves the high frequency Fourier components of the fields, as demonstrated by the broadening of the time profile of the propagating pulses. In order to ascertain that the short decay length fS that we identified in the zdependence of the maximum of the temporal profile of X =X corresponds to the absorption of the highest frequency components, we performed a fast Fourier transform analysis of the ratio X =X at the entry of the medium and after some decay lengths. In fact, this analysis, carried out for the same parameters as in Fig. 3, evidences that the
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maximum relevant frequency in the power spectrum of the incoming fields exceeds X2 =C and only the highest frequencies are suppressed after propagating by a few fS ’s through the medium. Therefore, in agreement with the analytic theory developed in [15], the fast initial decay of the amplitude of the light pulses corresponds to the absorption of the high frequency Fourier components. This shows that the probe pulse is modified while propagating so as to satisfy the adiabaticity conditions, and demonstrates that STIRAP is the mechanism determining the slowlight propagation. In our cigar shape geometry, the reabsorption of the spontaneously emitted photons is strongly reduced. Thus a significant occupation of the excited state does not imply a decay of the ground-state coherence. In this way, even if the initial probe/coupling configuration does not satisfy the adiabaticity condition, a ground-state coherence can be formed and via stimulated Raman scattering the probe pulse is modified so as to enter the EIT regime [21]. The role of the adiabatic following for the dark state in the slow-light phenomenon was also derived from the probe pulse transmission fraction as a function of the initial pulse width T. The coupling X strongly depends on the value of T, with the condition of adiabaticity Eq. (7) better satisfied for longer pulses. Our numerical calculations (Fig. 4(b)) show that the transmission fraction Ft increases with T. These results confirm the previous analysis: for a larger value of T, whence for initial probe/coupling configurations better satisfying the adiabaticity conditions, smaller modifications of the probe pulse are required to enter the EIT regime.
4. Conclusions In conclusion we have examined theoretically the propagation of a probe pulse through a cold sample of open three-level K atoms in the presence of a coupling laser acting on the adjacent transition of the K system. We have considered a cold atomic sample geometry where the photon reabsorption is greatly reduced. Correspondingly, we did not introduce in the Maxwell–Bloch equations
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any non-linear relaxation term describing the photon reabsorption [13]. We have found that the pulse amplitude decays with two very different characteristic length (fS and fL , with typically fL fS ). In a first phase of the pulse propagation, corresponding to fS , the pulse is significantly modified to better satisfy the adiabaticity conditions. Thus the second part of the propagation, characterized by fL , corresponds to better conditions for the probe pulse propagation. This demonstrates that STIRAP is the mechanism behind the slow-light phenomenon, and generalizes the results of [13] to geometries in which a significant occupation of the excited state does not necessarily imply a decay of the ground-state coherence, and the properties of the atomic sample are determined by the single atom response. Our investigation clarifies also why it is often possible to describe slow-light experiments on alkali atoms with a closed K system despite the presence of sink states. In the initial phase of propagation through the medium the probe pulse is modified via Raman scattering so as to satisfy the STIRAP conditions. Therefore the occupation of the excited state is negligible, and the presence of sink states becomes irrelevant. Obviously this behavior applies only for the case of limited modifications of the probe/coupling. Otherwise the excited-state occupation following the non-adiabatic coupling produces a pumping of the population in the sink state.
Acknowledgements E.A. acknowledges stimulating discussions with M. Fleischhauer on the relations between STIRAP and EIT. F.R. is grateful to I. Carusotto for critical reading the manuscript.
References [1] For a review S.E. Harris, Phys. Today 50 (1997) 36. [2] See also M.O. Scully, M.S. Zubairy, Quantum Opt., University Press, Cambridge, 1997. [3] For a review see E. Arimondo, in: E. Wolf (Ed.), Progress in Optics, vol. 35, Elsevier, Amsterdam, 1996, p. 257. [4] R. Grobe, F.T. Hioe, J.H. Eberly, Phys. Rev. Lett. 73 (1994) 3183. [5] L.V. Hau, S.E. Harris, Z. Dutton, C.H. Behroozi, Nature (London) 397 (1999) 594. [6] S. Inouye, R.F. Low, S. Gupta, T. Pfau, A. G€ orlitz, T.L. Gustavson, D.E. Pritchard, W. Ketterle, Phys. Rev. Lett. 85 (2000) 4225. [7] C. Liu, Z. Dutton, C.H. Behroozi, L.V. Hau, Nature (London) 409 (2001) 490. [8] M.M. Kash, V.A. Sautenkov, A.S. Zibrov, L. Hollberg, G.R. Welch, M.D. Lukin, Y. Rostovtsev, E.S. Fry, M.O. Scully, Phys. Rev. Lett. 82 (1999) 5229. [9] D. Budker, D.F. Kimball, S.M. Rochester, V.V. Yashchuk, Phys. Rev. Lett. 83 (1999) 1767. [10] D.F. Phillips, A. Fleischhauer, A. Mair, R.L. Walsworth, M.D. Lukin, Phys. Rev. Lett. 86 (2001) 783. [11] M.D. Lukin, A. Imamoglu, Nature 413 (2001) 273. [12] K. Bergmann, H. Theuer, B.W. Shore, Rev. Mod. Phys. 70 (1998) 1003. [13] M. Fleischhauer, Opt. Expr. 4 (1999) 107. [14] R.N. Shakhmuratov, J. Odeurs, R. Coussement, P. Megret, G. Kozyreff, P. Mandel, Phys. Rev. Lett. 87 (2001) 153601. [15] M. Fleischhauer, A.S. Manka, Phys. Rev. A 54 (1996) 794. [16] For the role of the atom–atom interactions in slow light see G. Morigi, G. Agarwal, Phys. Rev. A 62 (2000) 013801. [17] See investigations on Bose–Einstein condensates reviewed by W. Ketterle, D.S. Durfee, D.M. Stamper-Kurn, in: M. Inguscio, S. Stringari, C. Wieman (Eds.), Bose–Einstein condensation in atomic gases, IOS Press, Amsterdam, 1999, pp. 67–176. [18] F. Renzoni, E. Arimondo, Opt. Commun. 178 (2000) 345. [19] M. Sargent III, M.O. Scully, W.E. Lamb Jr., Laser Physics, Addison-Wesley, Reading, 1974. [20] S.E. Harris, J.E. Field, A. Kasapi, Phys. Rev. A 46 (1992) R29. [21] See also S.E. Harris, Z.-F. Luo, Phys. Rev. A 52 (1995) R928, for the role of stimulated Raman scattering in the matched pulse propagation.
Optics Communications 208 (2002) 125–130 www.elsevier.com/locate/optcom
Dynamics of slow-light formation E. Cerboneschi a,b, F. Renzoni c,*, E. Arimondo b a Istituto di Tecniche Spettroscopiche del CNR, Via La Farina 237, I-98123 Messina, Italy INFM and Dipartimento di Fisica, Universit a di Pisa, Via F. Buonarroti 2, I-56127 Pisa, Italy Laboratoire Kastler Brossel, D epartement de Physique de l’Ecole Normale Sup erieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France b
c
Received 17 February 2002; accepted 23 May 2002
Abstract We investigate theoretically the phenomenon of slow light in a cold sample of open three-level atoms interacting with the two light fields in the K configuration. We consider a cold atomic sample geometry such that the photon reabsorption is greatly reduced, and therefore the medium properties are determined by the single atom response. The dynamics of the slow propagating light pulse is examined, and the typical length scales for the propagation derived. We demonstrate that STIRAP is the mechanism behind the slow-light phenomenon. Furthermore, in the considered geometry a significant occupation of the excited state in the early phase of the propagation does not necessarily imply a decay of the ground-state coherence, and therefore does not inhibit the slow-light formation. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50.Gy; 32.80.-t; 42.50.Vk
1. Introduction The linear absorption of an atomic medium, as measured by a weak probe laser beam, corresponds to the quantum structure of the probed atoms, with absorption lines at the atomic transition frequencies. Low-frequency coherences created by applying one additional laser field (the coupling field) may modify completely the absorptive properties of the medium, which can be made transparent whenever the probe and the coupling fields couple two ground states to a *
Corresponding author. Tel.: +3301-4432-3568; fax: +33014432-3434. E-mail address: [email protected] (F. Renzoni).
common excited state with the laser frequency difference matching the ground-state splitting. This is the so-called electromagnetically induced transparency (EIT) [1,2]. The stationary state of the medium in the regime of EIT corresponds to atoms prepared in a linear superposition of the ground states which is decoupled from the light fields (non-coupled or dark state [3]), as a result of the destructive quantum interference between transition amplitudes. The probe light group velocity is greatly reduced by the very steep dispersion associated with the EIT resonance. After the pioneering work on adiabatic pulse propagation by Grobe et al. [4], remarkable results for the reduction of the group velocity in three-level systems were recently reported in cold samples [5–7] as well
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 5 8 2 - 1
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as in samples with thermal velocity distributions [8–10]. Many applications based on the enhancement of the optical non-linearities in a EIT medium, as a result of the reduced probe light group velocity and cancellation of the absorption, have been proposed, as reviewed in [11]. A key question for the efficient slow-light propagation is the initial atomic preparation in the dark state. The central role of STIRAP (Stimulated Raman Adiabatic Passage [12]) in establishing EIT for laser pulses, whence their slow-light propagation, was pointed out by Fleischhauer [13] and recently reexamined [14]. Indeed, in his work it was shown that STIRAP is the mechanism underlying the slow-light atomic preparation, and that the conditions for efficient slow-light propagation coincide with those of STIRAP. Those conclusions were derived from the minimization of the reabsorption of spontaneously emitted photons, which is the dominating damping mechanism of the ground-state coherence for a dense threedimensional atomic sample. That minimization required, obviously, a very small occupation of the excited state and led naturally to the counterintuitive coupling/probe pulse sequence of STIRAP. In the present work we examine a different situation: we assume a geometry of the cold atomic sample such that the photon reabsorption is greatly reduced and the properties of the atomic medium are determined by the single atom response. We will show that also for the considered geometry the STIRAP conditions of adiabatic following of the dark state, imposed all along the propagation distance, define the mechanism behind the slowlight phenomenon. This generalizes the results of [13] to geometries of cold atomic samples characterized by a greatly reduced photon reabsorption. In this case a significant occupation of the excited state in the early phase of the propagation does not necessarily imply a decay of the ground-state coherence, and, as we will show, does not inhibit the slow-light formation.
configuration. This open system corresponds to the conditions of the cold atoms experiment by Hau et al. [5]. We neglect atom–atom interactions [16]. We assume a cigar shape geometry of the cold atomic sample [17], so that the photon reabsorption, which destroys the ground-state coherence, is greatly reduced. The interaction scheme is shown in Fig. 1. The ground (or metastable) states jci and jpi, with energies Ec and Ep , respectively, are excited to a common state jei, with energy Ee , by two laser fields of frequencies xa , electric field amplitudes Ea , and wavevectors ka ¼ xa =c ða ¼ c; pÞ, c being the light speed in vacuum. The laser detunings are denoted by da ¼ xa xea and the Raman detuning from the two-photon resonance is dR ¼ dc dp . The Rabi frequencies are given by Xa ¼
Dea Ea ; h
ð1Þ
with Dea the atomic dipole moment, supposed to be real for the sake of simplicity. The evolution of the atomic density matrix q is described by the optical Bloch equations (OBE) for an open system (see for example [18]). We assume the excited-state relaxation, denoted by C, due to spontaneous emission, as for a dilute atomic sample. The rate C is composed by the rates Cc and Cp for the decays into the ground states jci and jpi, and by the decay Cout into a sink state jouti, not excited by the lasers C ¼ Cc þ Cp þ Cout :
ð2Þ
The decay rate of the optical coherences, qep and qec , is equal to C=2.
2. Level scheme and Maxwell–Bloch equations We consider a cold sample of open three-level atoms interacting with two laser fields in the K
Fig. 1. Interaction scheme of an open three-level atomic system coupled with two laser fields in the K configuration.
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To describe the propagation of laser pulses through a cold sample of open three-level atoms, the electric field amplitudes, whence the Rabi frequencies, are assumed dependent on the time t and the coordinate z along the propagation direction. For slowly varying electric field amplitudes, the Maxwell equations for the coupling/probe envelope Rabi frequencies expressed in terms of the pulse-localized coordinates z and s ¼ t z=c, Xp ðz; sÞ and Xc ðz; sÞ reduce to [19] o Xa ðz; sÞ ¼ ija q~ea ðz; sÞ ð3Þ oz with a ¼ ðc; pÞ and the slowly varying density matrix element given by q~ep ¼ qep expðiðxe xp ÞtÞ:
ð4Þ
The parameter ja is given by ja ¼
xa nD2ea ; c0 h
ð5Þ
where n is the atomic density and 0 the vacuum susceptibility. The Beer’s absorption length for the probe field fp in the absence of coupling laser is fp ¼ C=jp . In the numerical calculations we considered an initial Gaussian pulse on the probe transition 2 t Xp ðz ¼ 0; tÞ ¼ Xop exp : ð6Þ 2T 2 The coupling laser, with constant Rabi frequency Xc , was assumed switched on at earlier times t 6 0, in order to realize the STIRAP counterintuitive pulse sequence. The atoms were initially prepared in the ground jpi state. This corresponds to an initial preparation of the atoms in the non-coupled state. Both probe and coupling fields are taken to be resonant with the corresponding atomic transitions: dc ¼ dp ¼ dR ¼ 0. The Maxwell–Bloch equations were solved for different strength of the coupling field and different probe pulse amplitude and length. To study the mechanism behind the slow-light propagation, it is useful to examine to which extent the initial probe/coupling configuration and the timeevolved one satisfy the condition of adiabaticity. We introduce here the basic concepts and the relevant quantities.
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In STIRAP [12,13,15] the condition for adiabatic following of the dark state is: X_ p Xc X_ c Xp X; ð7Þ X2 where the dot denotes the time derivative and X is defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ¼ jXp j2 þ jXc j2 : ð8Þ
X
This corresponds to a small non-adiabatic coupling between non-coupled and coupled states. In [15] it was shown that condition (7) corresponds also to a formstable probe pulse propagation. In this work the dynamics of slow-light formation is studied by considering initial probe/coupling configurations for which condition (7) is not strictly satisfied.
3. Numerical results Numerical results for the propagation of the probe pulse within a sample of open three level atoms are shown in Fig. 2. We used the parameters of the experiment in [5], i.e., Na D2 line parameters (C ¼ 2p 9:9 MHz, k ¼ 589:0 nm), cold atom
Fig. 2. Probe Rabi frequency Xp ðz; sÞ=C as a function of the pulse-localized reduced time s, normalized to 1=C, at different penetration distances for parameters Cc ¼ C=3, Cp ¼ C=2, and Cout ¼ C=6. The dashed line represents a reference probe pulse with no atoms in the propagation medium, scaled down by a factor two. Initial Gaussian probe pulse width T ¼ 80=C, and initial coupling laser Rabi frequency Xc =C ¼ 0:18.
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density of 3:3 1012 atoms=cm3 , propagation length through the medium up to a distance z equal to 63 times the Beer’s length fp , initial Rabi frequencies Xop and Xc around a tenth of C. The delay time in the propagation through the medium was measured in units of 1=C. The numerical results show a pulse peak height decreasing with the penetration distance, a broadening of the probe pulse, and a propagation with a reduced group velocity. Results for the probe group velocity are in the range of the values measured in [5]. The data of Fig. 2 indicate that for the simulation parameters an attenuation of the propagating probe pulse occurs. An example of the probe pulse transmission fraction Ft ðzÞ ¼ Xp ðzÞ= Xp ð0Þ as a function of the propagation distance z is plotted in Fig. 3, both Xp calculated at the pulse peak. The probe pulse amplitude decreases with
Fig. 3. In the top, ratio X =X, continuous line, and transmission fraction Ft of the probe pulse, dashed line, as a function of the propagation distance z. Parameters as those of Fig. 2, except T ¼ 10=C. The z decrease is described through a doubleexponential with absorption lengths reported in the inset. In the bottom, dependence of fS , close dots, and of fL , open squares, on the square of the coupling Rabi frequency Xc .
the propagation distance, and that decrease may be described through a double exponential dependence on the propagation distance, with absorption lengths fS and fL . The fS propagation length, the short one, is required in order to modify the probe pulse into that required for the EIT propagation. The second propagation length, the long one, describes the weak attenuation of the probe pulse in the EIT regime. The fL dependence on X2c shown in the bottom part of Fig. 3 has been predicted in [20]. We found that fS presents a similar dependence. The overall behavior can be understood by noting that for the parameters of the initial coupling/probe configuration chosen for Fig. 3 the adiabaticity conditions are not fully satisfied, as it appears clearly from Fig. 4(a). Correspondingly we studied to what extent the evolved coupling/ probe configuration satisfies the adiabaticity conditions, with numerical results for X =X as a function of the propagation distance in Fig. 3. Because X =X depends on the time t, we considered its maximum value with respect of t. We found that X =X decays mainly with the absorption length fS .
Fig. 4. (a) Non-adiabaticity test comparison between X and X for parameters of Fig. 3 at z ¼ 0. (b) Dependence of the transmission fraction Ft for the probe Rabi frequency through the sample for different pulse lengths T at z ¼ 63fp in the conditions of the open system of Fig. 3.
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Analytic results for the propagation of laser pulses through a medium of three-level atoms under conditions of quasi-adiabaticity have been derived by Fleischhauer and Manka [15]. In that work, analytic expressions for the spatial evolution of the laser fields were obtained through a perturbation expansion in X =X, after Fourier transforming the equations of motions in the frequency domain. Such results are appropriate for the understanding of the absorptive properties of the K system in the formation of slow light, as detailed by our simulations. Equation (78) in [15] shows that the z-dependence of the Fourier components of the ratio X =X is an exponential, that includes a dispersive term accounting for the quasi-formstable propagation with reduced group velocity and an absorptive term explaining the attenuation of the high-frequency components of the propagating pulse. In their analysis, Fleischhauer and Manka focused on the formstable dispersive propagation arising when all relevant Fourier frequencies x are sufficiently small, such that x X2 =C. In this work we have examined the opposite case. The light pulses at the entry of the medium have a broad Fourier spectrum and the absorption cannot be neglected. According to the analytic results in [15], the high-frequency components, for x X2 =C are absorbed linearly, within a Beer’s one-photon absorption length. A longer penetration distance is required to attenuate Fourier components at lower frequency. The absorption lengths fS for the initial fast decay derived from our numerical analysis are of the order of the Beer’s absorption length and, therefore, are quantitatively consistent with the analytical prediction of [15]. We must point out that the fast decay in our simulations involves the high frequency Fourier components of the fields, as demonstrated by the broadening of the time profile of the propagating pulses. In order to ascertain that the short decay length fS that we identified in the zdependence of the maximum of the temporal profile of X =X corresponds to the absorption of the highest frequency components, we performed a fast Fourier transform analysis of the ratio X =X at the entry of the medium and after some decay lengths. In fact, this analysis, carried out for the same parameters as in Fig. 3, evidences that the
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maximum relevant frequency in the power spectrum of the incoming fields exceeds X2 =C and only the highest frequencies are suppressed after propagating by a few fS ’s through the medium. Therefore, in agreement with the analytic theory developed in [15], the fast initial decay of the amplitude of the light pulses corresponds to the absorption of the high frequency Fourier components. This shows that the probe pulse is modified while propagating so as to satisfy the adiabaticity conditions, and demonstrates that STIRAP is the mechanism determining the slowlight propagation. In our cigar shape geometry, the reabsorption of the spontaneously emitted photons is strongly reduced. Thus a significant occupation of the excited state does not imply a decay of the ground-state coherence. In this way, even if the initial probe/coupling configuration does not satisfy the adiabaticity condition, a ground-state coherence can be formed and via stimulated Raman scattering the probe pulse is modified so as to enter the EIT regime [21]. The role of the adiabatic following for the dark state in the slow-light phenomenon was also derived from the probe pulse transmission fraction as a function of the initial pulse width T. The coupling X strongly depends on the value of T, with the condition of adiabaticity Eq. (7) better satisfied for longer pulses. Our numerical calculations (Fig. 4(b)) show that the transmission fraction Ft increases with T. These results confirm the previous analysis: for a larger value of T, whence for initial probe/coupling configurations better satisfying the adiabaticity conditions, smaller modifications of the probe pulse are required to enter the EIT regime.
4. Conclusions In conclusion we have examined theoretically the propagation of a probe pulse through a cold sample of open three-level K atoms in the presence of a coupling laser acting on the adjacent transition of the K system. We have considered a cold atomic sample geometry where the photon reabsorption is greatly reduced. Correspondingly, we did not introduce in the Maxwell–Bloch equations
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any non-linear relaxation term describing the photon reabsorption [13]. We have found that the pulse amplitude decays with two very different characteristic length (fS and fL , with typically fL fS ). In a first phase of the pulse propagation, corresponding to fS , the pulse is significantly modified to better satisfy the adiabaticity conditions. Thus the second part of the propagation, characterized by fL , corresponds to better conditions for the probe pulse propagation. This demonstrates that STIRAP is the mechanism behind the slow-light phenomenon, and generalizes the results of [13] to geometries in which a significant occupation of the excited state does not necessarily imply a decay of the ground-state coherence, and the properties of the atomic sample are determined by the single atom response. Our investigation clarifies also why it is often possible to describe slow-light experiments on alkali atoms with a closed K system despite the presence of sink states. In the initial phase of propagation through the medium the probe pulse is modified via Raman scattering so as to satisfy the STIRAP conditions. Therefore the occupation of the excited state is negligible, and the presence of sink states becomes irrelevant. Obviously this behavior applies only for the case of limited modifications of the probe/coupling. Otherwise the excited-state occupation following the non-adiabatic coupling produces a pumping of the population in the sink state.
Acknowledgements E.A. acknowledges stimulating discussions with M. Fleischhauer on the relations between STIRAP and EIT. F.R. is grateful to I. Carusotto for critical reading the manuscript.
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