| 1.E?ect of the relative phase in a closed loop on theoptical properties of the media in Er3+:YAG crystalA new method to control the gain (absorption), dispersion properties andfurther more the group velocity of a weak probe light by use of the total closed in-teraction phaseΦis proposed in erbium- doped yttrium aluminum garnet crystal.Firstly we discuss the e?ect of the total closed interaction phaseΦon the probegain and absorption and show that the phenomenon of ?attened gain (absorption)can be achieved in a closed three-level system when the total interaction phaseequals toÏ€/2 (3Ï€/2). We also can realize a ?exible control of the gain equal- ization with di?erent amplitudes and frequency regions by varying intensities ofstrong coupling fields. In particular, we point out that the Raman gain alwaysplays an important role in achieving and controlling the ?attened gain. Secondlywe demonstrate that a ?at dispersion, an normal dispersion and a anomalous dis-persion can be achieved separately by controlling phases of the applied fields. Onthis basis, the corresponding variations of the group velocity are discussed andwe demonstrate that the subluminal group velocity and the superluminal groupvelocity can be separately achieved with di?erent light phases and strengths.A three-level closed-loop system is shown in fig.1. The energy-level scheme isrelevant to the Er3+ ions in YAG crystal, where levels |1 , |2 and |3 respectivelycorrespond to the energy levels of Er3+ ions 4I15/2, 4I13/2 and 4S3/2. Transition|1â†â†’|2 is driven by a weak probe laser field Ep of frequencyωp with Rabi fre-quencyΩp. A strong coupling laser field E1 of frequencyωc1 with Rabi frequencyΩ1 is applied to the transition |1â†â†’|3 and transition |2â†â†’|3 is coupledby another strong coupling laser field E2 of frequencyωc2 with Rabi frequencyΩ2. In this way, a closed interaction contour is formed.In the interaction picture, with the rotating-wave approximation and the electric-dipole approximation, the density matrix equations can be derived fromthe Semi-classical interaction Hamiltonian of this three-level closed-loop system.It is the closed-loop that makes the optical properties of the media become sensi-tive to the relative phases of the applied fields. Considering these relative phasesand making a standard transfer for the elements of the density matrix such as,Ï12 =σ12e-iφp,Ï13 =σ13e-iφ1,Ï23 =σ23e-iφ2,Ïii =σii,Ïij =Ï*ji and the relativephaseΦ=φ2 +φp -φ1, the equation for the density matrix can be expressedas those relevant to the total closed interaction phaseΦ, from which the opticalproperties of the media such as probe gain (absorption) and dispersion can bederived. Different from the phenomenon of EIT observed in a normal three-level modelunder the action of strong coupling laser, it is found that, in a three-level modelwith closed-loop, the optical properties of the media depend on the total closedinteraction phase and, with the variation of the probe detuning, the gain (absorp-tion) curve behaves as the ?at one (Φ=Ï€/2(3Ï€/2))or similar to the dispersionone (Φ= 0,Ï€/4,Ï€,5Ï€/4).Based on the analysis of the dressed-state, it is pointed out that, the reasonthat the optical properties of the media behave differently is that the e?ect ofthe"e?ective coherent field"is diffeerent under the condition of the varied totalclosed interaction phase, for example, when the"effective coherent field"modifiesthe e?ective magnitude of the probe detuning, the media behaves similar to thedispersion curve; however, when the"e?ective coherent field"modifies the inco-herent pumping or decay, the media behaves as the flat gain or absorption curve;when both e?ects of the"effective coherent field"exist, the media behaves similarto the dispersion curve with controllable magnitude of the dispersion. Furthermore, we discuss the origin of the gain equalization in this model, and point outthat Raman gain plays a most important role whether in the amplitude or in thefrequency region of the flattened gain.Also we discuss the controllable dispersion properties and the group velocityof a probe field in this three-level system with a closed interaction contour (asshown in fig.3). When two strong coupling fields are applied, the steady statedispersion properties of the weak probe field are sensibly affected by the relativephase of all lasers. By modifying the relative phase, the flat dispersion, the nor-mal dispersion and the anomalous dispersion can be achieved separately. Thevalue and the spectrum range of the flat dispersion can be modified by the Rabifrequencies of the applied strong coupling fields. The slope gradient of the nor-mal and the anomalous dispersion can also be modified optionally not only bythe relative phase but also by the Rabi frequencies of the coupling fields. Fur-thermore, we have discussed the corresponding variations of the group velocity and demonstrated them with pulsed lasers.2.Optical gain properties in a coherently preparedfour-level cold atomic system with the technique ofStimulated Raman Adiabatic passageA remarkable optical gain properties of probe pulse in a coherently preparedfour-level 87Rb cold atomic system via STIRAP-like process are demonstrated.With the application of the STIRAP in a three-level T model, the populationis transferred firstly from the ground state and then to the highest excited stateusing pulses in the counterintuitive order, realizing the coherent population in-version with the highest inversion e?ciency. Based on this, we demonstrate theproperties of the probe amplification. The contributions of the amplitudeαp0,the delay timeÏ„p, the duration Tp of the probe pulse, and the atomic density Non the probe gain are explored numerically in detail. The physical mechanism forthe probe gain is also discussed and point out the e?ect of the Raman gain on theprobe amplification. From the numerical results and the discussions, it can beseen that, without the consideration of incoherent pumping laser but consideringthe technique of STIRAP, the probe gain in this scheme is much larger than theone in earlier scheme.A diagram of this four-level system in 87Rb atom is shown in fig.4. Thelevel |1 >, |2 >, |3 > and |4 > correspond to the 5S1/2, 5P3/2, 6S1/2 and 5P1/2state of 87Rb cold atom, respectively. The pump laserΩ1 is detuned from the5S1/2 - 5P3/2 transition byδ1, while the Stokes laserΩ2 is detuned from the5P3/2 - 6S1/2 transition byδ2. The probe laserΩp is detuned from the 5P1/2-6S1/2transition byδp. In this work, about 90% of the population reaches level |3 > dueto the spontaneous decay from level |3 > to level |2 > and |4 >. When the probepulse arrives after the pump and the Stokes pulses, a remarkable probe gain willbe presented as illustrated typically in fig.5. The coming problem should be when and how the probe pulse is applied tothe system so that the largest probe gain would be achieved. The parametersof interest are the amplitudeαp0, the delay timeÏ„p, and the duration Tp of theprobe pulse and the atomic density N.Fig.6 illustrates the maximum probe gain Max.?2p/αp20 versus the input am-plitudeαp0 and the delay timeÏ„p of probe pulse. As shown in fig.6, when aweak probe pulse is employed, for fixed input amplitudeαp0, the probe gain isdecreased with the increasing delay timeÏ„p. This result is intuitive because thepopulation di?erence (Ï33 -Ï44) reduces with the time evolution. So, the laterthe input probe pulse is applied, the smaller the probe gain would be obtained.On the other hand, the probe gain becomes small with increasing input probeamplitude when the delay time of probe pulse is fixed. It is because the popula- tion provided by the atomic system is limited. As known, the probe gain is theresult that the pumping energy is absorbed by the probe pulse. So, comparedwith the pumping energy, the larger the input amplitude is employed, the smallerthe probe pulse is amplified. Furthermore, the probe pulse will reach the gainsaturation especially when the amplitude of probe pulse is large enough. Whena strong probe pulse is employed, it is seen that there exists several regions inwhich the probe pulse is absorbed rather than amplified. This phenomenon isthe result of Rabi oscillation, which leading to the rising edge and the trailingedge of the input probe pulse are amplified and the peak of input probe pulse isabsorbed and the maximum value of output probe pulse is smaller than the peakof input probe pulse.For each of fixed pulse duration, the probe gain is seen to decrease with theincreasing input amplitude of probe pulse. On the other hand, for fixed inputamplitude, the maximum probe gain increases while the probe pulse duration is enlarged. It is known that the probe pulse duration is also the time for the probefield interacting with the atom. Therefore, the longer the probe pulse durationis, the more the pumping energy would be absorbed by probe pulse especiallywhen a weak probe pulse is employed. Then, a large probe gain is presented.For the limitation of pump energy, the probe gain decreases with the increasinginput amplitude of probe pulse and the di?erence among the probe gain becomessmall.Finally, we would like to discuss the probe gain versus the atomic densityN. It is obvious that the larger the atomic density of atom cloud is employed,the more atoms participate in the process of probe gain. As thus, a larger probegain can be expected when the atomic density employed is large. However, theprobe gain is not amplified linearly with the increase of the atomic density N,and there exist a reduced region for probe gain. Since a weak probe pulse isemployed in this case, it is obvious that the most populations in level |3 > can not be transferred to level |4 > due to the amplification of the probe pulse. Infact, the population inversion between level |3 > and level |2 > is achieved aftercoherently preparation of the atom cloud. As a result, the population transfersfrom level |3 > to level |2 > and a pulse field ?2 corresponding to the transition|2 >→|4 > is generated and amplified. In this sense, it can be concluded thatthe reduced probe gain relates to the pulse field ?2. If the pulseΩ'2 is largeenough, it can cause the Rabi oscillation between level |3 > and level |2 > andthe caseÏ33 -Ï44 < 0 occurs especially when the most populations in level |3 >is transferred to level |2 >. So, the probe pulse is absorbed and a reduced probegain is presented.3.Enhancement of the four-wave mixing conversionefficiency in a four levelâ—‡system In a four-levelâ—‡model Rb87 cold atomic system, we proposed a schemeto demonstrate the enhancement of the nonlinear four-wave mixing conversionefficiency via the method of quantum interference. Two strong coupling lasersΩ1 andΩ2 are used to prepare the atomic coherence among three different levels,e.g. the ground state and both middle excited states in thisâ—‡system. Basedon this, the nonlinear four-wave mixing signal is generated under the effect ofthe stimulated Raman scattering of the weak probe laser. Two methods areproposed to make the preparation of atomic coherence: in the first method twostrong pulse lasers are applied to three lower states inâ—‡system (formulating astandard V model), however in the second method two strong continuous lasersare considered. It is found that the conversion e?ciency is relevant to the atomiccoherence between both middle excited states and the association between bothatomic coherence of another two transitions in this V model. Only when these twoconditions are both large, high conversion e?ciency can be observed. In the firstmethod, Such efficiencies compare with those obtained by fractional stimulatedRaman adiabatic passage in rubidium vapor cells and are also twice as large asthose obtained in different atomic configurations by resonant four-wave mixingin the ultra-slow propagation regime. In the second method, it is found that thestrength of the generated four-wave mixing signal could be larger than that ofinitial input probe laser under a certain condition.Consider a medium of four-level atoms with a ground state |1 and threeexcited states |2 , |3 and |4 as shown in Fig.7. The two intermediate levelsare coupled to the common ground by two strong coupling lasersΩ1 andΩ2,resonant with the corresponding |1 >â†â†’|2 > and |1 >â†â†’|3 > transitions. Anadditional short and weak probe pulseΩ3, resonantly couples instead level 3 and4 and generates a short four-wave mixed signal pulseΩ4.What we are interested in is how to achieve high nonlinear four-wave mixingconversion efficiency in this system, which is found being relevant to the atomiccoherence between both middle excited states and the association between both atomic coherence of another two transitions in the V model. When these twovalues are both large, a high conversion e?ciency could be obtained. To obtainthis optimized atomic coherence, three kinds of methods are proposed: the first,apply two strong pulses with fully overlapped time scale in the three-level Vmodel to prepare the atomic coherence; the second, apply two strong pulses withthe time di?erence in the time scale about half pulse width in the three-level Vmodel to prepare the atomic coherence; the third, apply two strong continuouslasers in the V model to prepare the atomic coherence.For the both cases in the former to prepare the optimized atomic coherence,with two pulses fully overlapped as example, as shown in Fig.8, it is found thatthe inputting time of the probe pulse is important to the conversion efficiency,which compare with those obtained by fractional stimulated Raman adiabatic passage in rubidium vapor cells and are also twice as large as those obtained indifferent atomic configurations by resonant four-wave mixing in the ultra-slowpropagation regime. Further more, a maximum value of the generated four-wavemixing signal is always observed at the point ofÏ„=Ï„i0 +Ï„3/2 for each inputtingtimeÏ„i0 of the weak probe pulse, which shows that the properties of the generatedlaserΩ4 such as the pulse width and the group velocity are both same to those ofthe initial probe pulseΩ3. Finally, it is shown in Fig.8 that the pulse distortion isso small to ignore. In summary, whenÏ„0 is fixed, with the variation of the inputtime of the weak probe pulseÏ„i0, a good four-wave mixing pulse could alwaysbe achieved. The variation of the input time of the weak probe pulseÏ„i0 onlychanges the strength of the generated nonlinear four-wave mixing laser.For the third case using two continuous lasers to prepare optimized atomiccoherence(as shown in Fig.9 and Fig.10), although the conversion e?ciency is notrelevant to the input time of the probe pulse yet, it depends on the probe detuneand propagation distance. As shown in Fig.9, in the case of small probe detune,with the increase of the propagation distance, the strength of the generated four-wave signal increases too, accompanied with the decrease of the strength of theprobe pulse. This kind of transient process goes to steady until to the case that the propagation is large enough and the steady value is larger than the transientvalues. According to the case of large probe detune, as shown in Fig.10,threepoints should be noted: Firstly, with the increase of the propagation distance,the oscillation of the strength of both probe pulse and the generated four-wavemixing pulse is a damped one and finally achieves the tranquilization with themagnitude similar to that attained in the case of small probe detuneδ≈0.Secondly, in a certain point z0 during the first oscillation periodicity, there is amaximum magnitude of the square magnitude of the signal pulseΩ4(z0,t) whichis about 160% of the initial input probe pulseΩ3(0,t). It means that, accordingto the definition of the Rabi frequency and the dipole moment, with the contribu-tion of constructive quantum interference between two Gaussian components theintensity of the nonlinear signal pulseΩ4(z,t) is about 80% of the initial inputprobe pulse. Finally we'd like to point out that there exists no pulse distortionwhether in the case of small probe detune or in the case of large probe detune. 4.Dynamical adjustment of the coherent forbiddenband induced by standing waveIn a three-levelΛmodel, when a standing wave is applied to the transitionfrom a meta-stable state to an excited state, the optical properties of the mediavary with the traveling distance. In this case, the optical properties of the mediaseems similar to that of the photonic crystal, for an example, the laser withwavelength during a special frequency region, named as coherent forbidden band,is reflected. The frequency width of the coherent forbidden band is controllable:with a larger peak Rabi frequency of the standing wave, a wider frequency regionis observed. However, it is found that, in this model, accompanied with theincrease of the frequency region of the coherent forbidden band, the re?ection tothe laser during the frequency region of the coherent forbidder band is decreaseddue to the reason that the coherent forbidden band is observed in the transparentwindow. It means that when the frequency region increased to a certain value,the frequency region can't be named as coherent forbidden band due to the smallre?ectivity. To solve this problem, based on the variation of the refractivity,we demonstrate the scheme to adjust synchronously the frequency width andthe corresponding re?ectivity in the coherent forbidden band. Two methods arerepresented, the first is to apply a coupling laser onto the transition driven by thestanding wave; the other is apply a weak incoherent pumping onto the transitiondriven by the probe laser.As shown in Fig. 11, we solve the problem of adjusting the frequency widthof the forbidden band and corresponding reflectivity with two methods. In thefirst way, a coupling laser ?c and a standing waveΩs are together applied to thetransition |2â†â†’|3 , a weak probe laserΩp is applied to the transition |1 >â†â†’|3>to probe the optical properties of the media. If the Rabi frequency of the couplinglaserΩc is larger than that of the probe laser but small enough not to create adeep transparent window, the e?ect of the standing wave is to further modulate the refractivity of the media periodically based on the fact that the refractivity ofthe media is already modified by the coupling laserΩc, which can be seemed asthe floor of the created coherent forbidden band. In this case, a good forbiddenband with wide frequency region can be observed with a smallη, which is usefulto increase the reflectivity synchronously. In the second way, di?erent from thefirst method, we don't consider the effect of the coupling laser ?c but apply anincoherent pumpΛto the transition |1â†â†’|3 to minish the absorption of themedia. In this case, a large Rabi frequency of the standing wave at note is notnecessary yet to eliminate the absorption of the media, which is useful for create aforbidden band with wide frequency region and large reflectivity with a relativelysmall Rabi frequency of standing wave.Now let's consider the first case, e.g., the modulation of the coherent for-bidden band under the e?ect of coupling laserΩc (as shown in Fig.11(a)). Wefirst point out that under the action of the coupling laserΩc, even the periodicstructure of the media is created by the perfect standing wave, the probe laser isnot absorbed any more and the coherent forbidden band, as shown in Fig.12(a),is observed. As far as this coherent forbidden band, whether about the frequencyregion or about the corresponding re?ectivity, it is parallel to that created witha not-so-prefect standing wave. Further more, we demonstrate that a better co-herent forbidden band is observed with not-so-perfect standing wave under the action of the coupling laser ?c. The superiority of combing the e?ects of cou-pling laser and not-so-perfect standing wave behaves not only in the fact that agood coherent forbidden band with wide frequency region and large correspond-ing reflectivity is observed, as shown as in Fig.12(b), but also in the fact that thedependence of the coherent forbidden band on the value ofηis decreased, whichis useful to observe a band with wider frequency region and large correspondingreflectivity through increasing the Rabi frequency of the standing wave, as shownin Fig.12(c).Next we turn our attention to the second case, e.g., the modulation of thecoherent forbidden band under the e?ect of incoherent pumpΛ(as shown in Fig.11(a)). As shown in the solid curve of Fig.13(a), under the effect of a weakincoherent pumpΛ, the refraction ratio between two different layers, respectivelycorresponding to the note and the anti-note of the standing wave, is large andthus a good coherent forbidden band with wide frequency region, as shown inFig.13(b), is observed. Further more, it is seen from Fig.13(b) that, under thee?ect of weak incoherent pumpλ, a good forbidden band with wide frequencyregion and large reflectivity is observed with a relatively smaller peak Rabi fre-quency of standing wave compared with the first case, which is useful to observethe coherent forbidden band in solid, in which the heating effect caused by largestrength of applied lasers can destroy the quantum coherence.
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