Superluminal and slow light in L-type three-level atoms via squeezed vacuum and spontaneously generated coherence F. Carreño,1 Oscar G. Calderón,1 M. A. Antón,1 and Isabel Gonzalo2 1Escuela Universitaria de Óptica, Universidad Complutense de Madrid, C/ Arcos de Jalón s/n, 28037 Madrid, Spain 2Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Ciudad Universitaria s/n, 28040, Spain sReceived 26 November 2004; revised manuscript received 24 February 2005; published 15 June 2005d We study the dispersion and absorption spectra of a weak probe in aL-type three-level atomic system with closely ground sublevels driven by a strong field and damped by a broadband squeezed vacuum. We analyze the interplay between the spontaneous generated coherence and the squeezed field on the susceptibility of the atomic system. We find that by varying the intensity of the squeezed field the group velocity of a weak pulse can change from subluminal to superluminal. In addition we exploit the fact that the properties of the atomic medium can be dramatically modified by controlling the relative phase between the driving field and the squeezed field, allowing us to manipulate the group velocity at which light propagates. The physical origin of this phenomenon corresponds to a transfer of the atomic coherence from electromagnetically induced trans- parency to electromagnetically induced absorption. Besides, this phenomenon is achieved under nearly trans- parency conditions and with negligible distortion of the propagation pulse. DOI: 10.1103/PhysRevA.71.063805 PACS numberssd: 42.50.Gy, 32.80.2t, 42.50.Dv I. INTRODUCTION In recent years, there has been a renewed interest in the study of subluminal and superluminal light propagation in atomic media having very special controlled optical proper- ties. Slow group velocity in coherent media has been shown to provide new regimes of nonlinear interaction with highly increased efficiency even for very weak light fields. It has been demonstrated that electromagnetically induced trans- parencysEITd is accompanied by large frequency dispersion f1–4g, thus the group velocity can be slowed down up to 10–102 m/s f5,6g. Harris and co-workers have found that large positive dispersion of refractive index in the EIT win- dow can be used to reduce dramatically the group velocity of light pulsesf5g, which leads to nonlinear optics at low light level f7g, and quantum memoryf8g. Kocharosvkayaet al. f9g showed how the spatial dispersion can even stop light in a hot gas. Using EIT and adiabatic following ofdark state polaritons, the group velocity of light pulses can be dramati- cally decelerated and their quantum state can be mapped into a metastable collective state of atomic ensemblesf10g, which is of application in quantum information. On the other side, superluminal propagation occurs in optical media with anomalous dispersionf11g. Wang et al. f12g demonstrated superluminal light propagation using the region of lossless anomalous dispersion between two closely spaced gain lines in a double-peaked Raman gain medium. The gain doublet is created by applying two intense detuned cw pumps with slightly different frequencies to one transition of aL-type three-level system in atomic cesium. Another important con- sequence of quantum coherence, the so-called electromag- netically induced absorptionsEIAd, can lead to a highly anomalous dispersion where enhanced absorption occurs f13–15g. Akulshin et al. f16g and Kim et al. f17g reported experiments where negative group velocities ofvg =−c/14 400 where achieved in Cs vapor cells. It is worth noting that anomalous dispersion obtained in these systems occurs, in general, with some absorption or gain, a nondesir- able feature for propagation of pulses with negligible distor- tion. Recently, light propagation from subluminal to superlu- minal in L-type and V-type three-level atoms has been shownf18–20g. The control of the dispersion from normal to anomalous has been obtained bysid applying a field connect- ing the two closed sublevels, andsii d considering spontane- ously generated coherencesSGCd. Bigelow et al. f21g have observed experimentally superluminal and ultraslow light propagation in an alexandrite crystal measuring group ve- locities as slow as 91 ms−1 to as fast as −800 ms−1. In this experiment the phenomenon was attributed to coherent popu- lation oscillations. In recent works, subluminal and superlu- minal propagation have been theoretically predicted. Agar- wal and Dasguptaf22g have shown superluminal propagation via coherent manipulation of Raman gain processes in N-type atoms. Sahraiet al. f23g have shown tunable phase control from subluminal to superluminal light propagation in an L-type atom with an extra energy level. Experimental evidences of fast and slow light propagation in four-level atoms have been obtainedf24,25g. In view of the many potential applications of subluminal and superluminal light propagation, a natural question is how one can have a controlling parameter for switching from one to the other regime of light propagation. In this paper we show theoretically the possibility of light propagation from subluminal to superluminal inL-type atoms with closely spaced lower levels, when the atom is driven by a strong coherent field and damped by a broadband squeezed vacuum sSVd. The combined effect of the SV field and the SGC produces a small gain or absorption over a large bandwidth which makes this configuration very attractive for propaga- tion of pulses without strong distortion in both the sublumi- nal and superluminal regimes. We find that this phenomenon arises from the transition from EIT to EIA when the intensity of the squeezed vacuum increases. This transition can also be induced by varying the relative phase between the driving field to the squeezed field. PHYSICAL REVIEW A 71, 063805s2005d 1050-2947/2005/71s6d/063805s11d/$23.00 ©2005 The American Physical Society063805-1 The existence of SGC effects inL-type atoms, which is not obvious due to the fact that there are two final possible lower states, was predicted by Javananienf26g. SGC has been proved to significantly modify the optical properties of the atomic systemf26–31g. The role played by the squeezed vacuumsSVd fluctuations in the atomic dynamics has been a subject of intense activity since the seminal work due to Gardinerf32g. Ficeket al. f33g have shown that a two-level atom damped by a broadband squeezed vacuum can exhibit a strong emission peaksgaind at the central frequency of the atomic levels, which is not attributed to population inversion, and results from the so-calledcoherent population oscilla- tions f34g. Akram et al. f35g have studied the index of re- fraction of a two-level atom replacing the usually applied coherent field by a squeezed vacuum field. They showed that the system can produce a large, although not maximum, in- dex of refraction accompanied by vanishing absorption when the carrier frequency of squeezed vacuum is detuned from the atomic resonance. So far, however, the driven multilevel systems interacting with squeezed vacuum have been consid- ered only towards exploiting the fluorescent properties f36,37g. The paper is organized as follows: Section II establishes the model, i.e., the Hamiltonian of the system and the evo- lution equation of the atomic operators assuming the rotating wave approximation. Section III is devoted to present the effects of SGC and the SV on the absorptive and dispersive properties of the atomic medium, and in the group velocity of a weak pulse. A discussion of the numerical results is present in Sec. IV. Finally, Sec. V summarizes the main con- clusions. II. ATOMIC MODEL AND DENSITY-MATRIX EQUATIONS We consider a closed,L-type three-level system with two near-degenerate levelsu1l andu2l, and an excited levelu3l as shown in Fig. 1sad. Transition u3l↔ u2l is driven by an ex- ternal coherent fieldEW given by EW = 1 2 EW 0stde−isvLt+fd + c.c., s2.1d EW 0std being the slowly varying field envelope, whereasvL, and f are the angular frequency and the phase of the field, respectively. In order to take into account the induced-coherence ef- fects by spontaneous emission, the upper levelu3l is coupled to lower levelsu2l and u1l by the same vacuum modes. The resonant frequencies between the upper levelu3l and the ground levelsu2l and u1l arev32 andv31, respectively. Note thatv32−v31=v21, v21 being the frequency separation of the lower levels. The Hamiltonian of the system in the rotating wave ap- proximation is given byf38,39g H = "o m=1 3 vmumlkmu + "o kl vklakl † akl − "o m=1 2 o kl gmkaklu3lkmu − H.c. −"Ve−isvLt+fdu3lk2u − H.c., s2.2d where"vm are the energies of the atomic levels, andaklsakl † d is the annihilationscreationd operator of thekth mode of the vacuum field with polarizationeWkl sl=1,2d and angular fre- quencyvkl. The parametergmk is the coupling constant of the atomic transitionuml↔ u3l with the vacuum electromag- netic mode gmk=Î vkl 2"e0V smW 3m ·eWkld, s2.3d wheremW 3m is the dipolar moment of the transitionuml↔ u3l. V=mW 32·EW 0/ s2"d is the Rabi frequency of the transition u2l↔ u3l. We consider the polarization arrangement shown in Fig. 1sbd, i.e., mW 31·EW 0=0. We now assume that the quantized radiation field is in a broadband squeezed vacuum state with carrier frequencyvv, which is tuned close to the frequency of the atomic transi- tions u3l↔ u1l and u3l↔ u2l, that is, 2vv.v31+v32. The bandwidth of the squeezed field is assumed to be broad enough so that the squeezed vacuum appears asd-correlated squeezed white noise to the atom. The correlation function for the field operatorsasvkld anda†svkld can be written as f32,36g kasvklda†svkl8 dl = fNsvkld + 1gdsvkl − vkl8 d, ka†svkldasvkl8 dl = Nsvklddsvkl − vkl8 d, kasvkldasvkl8 dl = Msvkldds2vv − vkl − vkl8 d, s2.4d Nsvkld and Msvkld being slowly varying functions of the frequency that characterize the squeezing. Note thatM is a complex magnitude so thatMsvkld= uMsvkldueifv, wherefv is the phase of the squeezed vacuum. ForMsvkld=0, Eq. s2.4d describes a thermal field at a finite temperatureT, whereNsvkld is the mean occupation number of the modekl with frequencyvkl. FIG. 1. sad L-type atom driven by a coherent fieldE0 of fre- quencyvL and phasef. g1 and g2 are the decay rates from the excited levelu3l to levels u1l, and u2l, respectively.sbd The polar- ization arrangement. CARREÑOet al. PHYSICAL REVIEW A 71, 063805s2005d 063805-2 The system is studied using the density-matrix formalism. By following the traditional approach of Weisskopf and Wigner f38,40g, we have re-derived the master equation, in an appropriate interaction picture, for the reduced density matrix of the atomic system,rs sId, in the Born and Markov approximation, which read as ]rs I ]t = − i " fHex I ,rs Ig − N + 1 2 sg1 + g2dss33rs I + rs Is33d − N 2 fg1ss11rs I + rs Is11d + g2ss22rs I + rs Is22d + g12ss12rs Ie−iv21t + rs Is21e iv21td + g12ss21rs Ieiv21t + rs Is12e −iv21tdg + sN + 1dfg1s13rs Is31 + g2s23rs Is32 + g12s13rs Is32e −isv31−v32dt + g12s23rs Is31e isv31−v32dtg + Nfg1s31rs Is13 + g2s32rs Is23 + g12s31rs Is23e isv31−v32dt + g12s32rs Is13e −isv31−v32dtg + Mfg1s31rs Is31e −is2vv−2v31dt + g2s32rs Is32e −is2vv−2v32dt + g12s31rs Is32e −is2vv−v31−v32dt + g12s32rs Is31e −is2vv−v32−v31dtg + M*fg1s13rs Is13e is2vv−2v31dt + g2s23rs Is23e is2vv−2v32dt + g12s13rs Is23e is2vv−v31−v32dt + g12s23rs Is13e is2vv−v31−v32dtg, s2.5d where Hex I = − "D1s33 − "sD1 − D2ds22 − "sVs32 + H.c.d s2.6d represents the interaction between the atom and the external driving field in the interaction picture.D1=vL−v31 and D2 =vL−v32 are the optical detunings. The notationsmn = umlknu for the atomic operators has been introduced, and g1,g2 are the decay rates for theu3l↔ u1l and u3l↔ u2l tran- sitions, respectively. The damping terms proportional tog12 in Eq. s2.5d are particularly important whenv21.g1,g2, and they arise due to the coupling of the two transitionsu3l↔ u1l and u3l↔ u2l with the same vacuum mode. They are respon- sible for the quantum interference between the two decay channelsf41g. It can be seen that these terms oscillate at the frequency differencev31−v32, thus when this difference is large enough, they may be dropped. This is the case treated in Ref. f36g. The present discussion is based on the situation wherev31.v32, so such nonsecular terms must be retained. In addition, it can be shownf38g that g12 = Îg1g2p s2.7d and p=mW 31·mW 32/ umW 31uumW 32u=cosu denotes the alignment of the dipole momentsmW 31 andmW 32. Through an appropriate unitary transformation we elimi- nate the explicit temporal dependence in the master equation, and obtain the evolution equations of the density-matrix elements, ]r33 ]t = − fsN + 1dsg1 + g2d + Ng1gr33 + Ng12sr12 + r21d + iVsr23 − r32d + Nsg2 − g1dr22 + Ng2, ]r22 ]t = sN + 1dg2r33 − N g12 2 sr12 + r21d − Ng2r22 − iVsr23 − r32d, ]r31 ]t = − F31r31 + iVr21 − N g12 2 r32 + uMug1e −iFr13 + uMug12e −iFr23, ]r21 ]t = − F21r21 + s3N + 2d g12 2 r33 + iVr31 − N g12 2 , ]r32 ]t = − F32r32 + uMug12e −iFr13 − N g12 2 r31 − iVsr33 − r22d + uMug2e −iFr23, s2.8d SUPERLUMINAL AND SLOW LIGHT IN L-TYPE … PHYSICAL REVIEW A 71, 063805s2005d 063805-3 where F31 = F sN + 1d 2 sg1 + g2d + N 2 g1 − iD1G , F21 = FN 2 sg1 + g2d − isD1 − D2dG , F32 = F sN + 1d 2 sg1 + g2d + N 2 g2 − iD2G , s2.9d and the relative phase F = fv − 2f, s2.10d which represents the phase difference between the coherent field and the squeezing field. The terms related top =g12/Îg1g2=cosu in Eq. s2.8d represent the effect of the quantum interference arising from the cross coupling be- tween spontaneous emissionsu3l↔ u1l andu3l↔ u2l. The pa- rameterp depends on the angleu between the two transition dipole moments. Due to the polarization arrangement consid- ered in Fig. 1sbd p,1 and p=0 represents the case of no quantum interference. In the context of inversionless gain studies, the main difference between our equations and those analyzed by other authorsf26,29g are the terms proportional to N andM, which account for the presence of the squeezed vacuum field. This fact modifies the equations in an impor- tant way: in the absence of squeezed vacuum, the interfer- ence term only appears in the equation of the coherencer21, and it is proportional to the population of the excited level r33. However, the presence of squeezed vacuum makes the interference parameterp appear in the equation of motion of all coherences and populationsfsee Eq.s2.8dg. We can also see that coherences depend on the correlations between pairs of modes in the reservoir which leads to a phase sensitivity in the optical response. III. EFFECT OF SGC AND SV ON THE ABSORPTIVE AND DISPERSIVE PROPERTIES OF ATOMIC MEDIUM We are interested in analyzing the behavior of a probe signal which drives transitionu3l→ u1l. To do that, we sup- pose that after the system has reached the steady state, the atom is perturbed by a weak probe field of frequencyvp. The probe intensity is assumed to be sufficiently weak that does not produce a noticeable perturbation of the atom-driving- field system. This will allow us to determine the dispersive and absorptive properties of the medium and to analyze how a pulse propagates inside the medium in the presence of both the driving field and the SV field. Thus we will need to determine the linear susceptibility which is defined as Asvpd =E 0 ` lim t→` kfD−st8 + td,D+stdgle−isvp−vLdt8dt8, s3.1d where D−std=d13s13std is the slowly varying part of the atomic polarization operator in the direction of the probe field polarization vectoreWp, with d13=eWp·mW 31. The commuta- tor is calculated in the absence of the probe field, but the squeezed vacuum and the driving field are always present. We can obtain the linear susceptibility with the help of the quantum regression theorem and the Bloch Eq.s2.8d. The details of the calculation are provided in the Appendix. We proceed to analyze how SGC and SV modifies the absorptive-dispersive properties of the probe field by analyz- ing Asvpd given by Eq.s3.1d. The linear probe absorption and the index of refraction are related with the real and imaginary parts ofAsvpd, respectively. In this way, a nega- tive value of RefAsvpdg means that the system exhibits gain. In the calculations we setg1=g2;g. For the sake of sim- plicity, we consider a degenerateL-type atom, i.e.,v21=0, and the driving field at resonance with transitionu2l→ u3l sD2=0d. We also assume a perfect squeezing condition, i.e., uMu=ÎNsN+1d. The absorptive-dispersive properties of the probe field are of interest when considering the propagation of a weak probe pulse. The group velocity can be modified by changing the slope of the dispersion curve. It is well known that when the probe absorption spectrum is characterized by a strong deep at resonance the phenomenon is called EIT. However, when the absorption spectrum presents a sharp peak at resonance, the phenomenon is called electromagnetically induced ab- sorptionsEIAd f16g. In order to carry out all kinds of propa- gation velocities we have to control atomic coherence of EIT and EIA. Generally, EIA cannot be generated in conventional three-level atomic systems. However, we will show in this work that SGC and SV can lead the atomic system from EIT to EIA. Thus a probe pulse will propagate in the subluminal ssuperluminald regime if nearly EITsEIAd conditions hold. Figure 2 displays the real and imaginary parts ofAsvpd as a function of the dimensionless frequency differencesvp −vLd /g for different values of the mean photon numberN, where we assumep=0.99, V=0.5g, and F=0. Note that when the system is damped by the standard vacuum, i.e., FIG. 2. Realsdashed lined and imaginaryssolid lined parts of the absorption spectrumAsvpd of the weak probe field versus the di- mensionless frequency differencesvp−vLd /g. The parameters are g1=g2;g, V=0.5g, D2=0, v21=0, p=0.99, andF=0. The mean photon number issad N=0, sbd N=0.1, scd N=0.7, andsdd N=1. A perfect squeezing conditionfuMu=ÎNsN+1dg is assumed. CARREÑOet al. PHYSICAL REVIEW A 71, 063805s2005d 063805-4 N=M =0, the probe line shape shows absorption at the Rabi sidebands, and transparency at the center, where the slope of the dispersion becomes very steep and positivefsee Fig. 2sadg. This is a well-known result, in agreement with Ref. f1g, and leads to subluminal group velocity. The behavior of the system for small values ofN resembles that obtained in the case withN=0, although it is obvious that SV partially destroys EIT and negligible absorption atvp=vL takes place. For a weak SV,N=0.1, the absorption peaks are lowered and the slope of the dispersion curve atvp=vL has decreased in comparison with the caseN=0, but it remains positivefsee Fig. 2sbdg. By increasing the photon numberN up to N =0.7, the optical response of the system is completely modi- fied. A sharp absorption peaksEIAd at vp=vL appearsfsee Fig. 2scdg. Furthermore, the slope of the index of refraction at vp=vL changes its sign from positive to negative. Thus de- pending on the value ofN we can change from the sublumi- nal to the superluminal regime of pulse propagation. This striking physical situation arises from the fact that the line- width of the absorption spectrum in the EIA regime is much narrower than the natural linewidth, allowing a large nega- tive slope of the refraction index atvp=vL. We remind here that according to the Kramers-Kroning relations, any change in the absorption of the medium will be accompanied by the corresponding change in the dispersion. In Fig. 2sdd we show that a further increase ofN up to N=1 leads to a moderate absorption atvp=vL whereas the slope of the dispersion becomes steeper than in Fig. 2scd. In summary, Fig. 2 clearly reveals that the squeezing field can convert an EIT atomic medium into an EIA atomic medium by increasing the pho- ton number. In other words, the squeezed vacuum can act as a knob for changing pulse propagation from subluminal to superluminal. The group indexng of the probe field is given by the following expression: ng ; c vg = 1 + x8svpd 2 + vp 2 ] ]vp x8svpd, s3.2d wherevg is the group velocity, andx8svpd is the real part of the susceptibility, which is related toAsvpd according to x8svpd = Na 2e0"g ImfAsvpdg, s3.3d Na being the atomic density. The group indexng can be larger than unityssubluminal lightd when the dispersion is positive in the region ofx8svpd.0, and can be less than unity ssuperluminal lightd when the dispersion is negative in this region. Let us analyze the time delayDT in the propa- gation of a pulse through the medium with regard to free space, which is defined as DT = L c sng − 1d = L c Navp 4e0"g ImF ]Asvpd ]vp G , s3.4d L being the length of the sample. In order to calculate the time delay of the transmitted pulse in a realistic situation, we consider the data of the experiment of Hauet al. f6g carried out in a medium consist- ing of cold three-level atoms. The parameters are Nam31 2 / s4e0g"d=0.013, vL /g=1.023108, and g /2p =5 MHz. The sample length wasL=229310−6 m. In Fig. 3sad, we plot DT versus the mean photon number of the squeezed vacuumsNd for different values of the Rabi fre- quency of the driving fieldV. We considervp=vL and the rest of parameters are those used in Fig. 2. We see that with the increase ofN, the pulse delay changes from positive to negative. Thus the group velocity can be controlled by ad- justing the photon number of the squeezed field and the propagation can change from subluminal to superluminal. The time interval at which delay can be varied increases as the Rabi frequency of the driving field decreasesfsee Fig. 3sadg. It is to be noted that the change from the subluminal to the superluminal regime of pulse propagation cannot be pro- duced when the atom is damped by a thermal field at finite temperaturesM =0d, i.e., the two-photon correlation is essen- tial to produce the change in the sign of the dispersion. The change in the slope of the dispersion is always ac- companied by a modification in the value of the absorption at FIG. 3. sad Time delay defined in Eq.s3.4d versus the mean photon number of the squeezed vacuumN for different values of the Rabi frequency of the driving fieldV: V=0.5g sdashed lined, V=1g sdashed-dotted lined, andV=1.5g ssolid lined. The horizon- tal thick solid line is a reference line that allows to determine the crossing of the curves from the subluminal to the superluminal re- gime. We consideruMu=ÎNsN+1d, and F=0. sbd Peak reduction versus the mean photon number for the Rabi frequencies considered in sad. SUPERLUMINAL AND SLOW LIGHT IN L-TYPE … PHYSICAL REVIEW A 71, 063805s2005d 063805-5 vp=vL. Thus a compromise between the absorption and the value of the slope of the dispersion must be reached in order to get observable effects: the largest the values in the slope of the dispersionfsee dashed line in Fig. 3sadg, the largest the values in the absorptionsnot shownd, thus producing a reduc- tion in the peak height of the transmitted pulse. In order to analyze the propagation of the pulse, we consider again the data of the experiment of Hauet al. f6g. We present in Fig. 3sbd the peak reduction of a Gaussian pulseEistd=expf−st −L /cd2/ s2t2dg with t=1 ms, at the output of the medium as a function of the mean photon number of the squeezed vacuumN, for the different values ofV considered in Fig. 3sad. We have used the analytical approach given by Garrett and McCumberf42g, thus we have considered in the compu- tation the dispersion in both the real and imaginary parts of the refractive index. Note that the case withV=0.5g sdashed lined allows us to obtain the largest negative value in the time delayDT. The counterpart is that in the superluminal regime, the peak height of the transmitted pulse is reduced up to 1/2500 with regard to that of the incident pulse. By consid- ering V=1g sdashed-dotted lined, the interval of variation in the time delay is smaller in width than in the previous case, whereas the peak reduction of the output pulse is not larger than 1/50 with regard to that of the incident pulse. A further increase of the Rabi frequency toV=1.5g ssolid lined pro- duces less attenuation throughout all the values ofN, al- though the range of time delay achievable is reduced in com- parison to the previous cases. We have obtained a rough estimation of the absorption at vp=vL as a function of the mean photon number by using a symbolic mathematical package. The absorption can be ap- proximated to RefAsvp = vLdg < 135 2 N3 fsV/gd2 + 18N2g . s3.5d From Eq.s3.5d we can see that the linear absorption presents the maximum value of 0.32/sV /gd for a mean photon num- ber given byNm