Liang Hu(胡亮), Xiang-Ming Hu(胡響明), and Qing-Ping Hu(胡慶平)

College of Physical Science and Technology,Central China Normal University,Wuhan 430079,China

Keywords: resonance fluorescence,narrow spectral lines,microwave cavity

1. Introduction

Recently, in the manipulation of fluorescence radiation,attention has been paid to the narrowing of spectral lines.[1–25]This will have wide application in improving the accuracy and efficiency of high-precision measurements. The resonant fluorescence of an atom is well described in terms of dressed states, which are degeneracy-lifted composite states of the atom and the dressing fields due to their interaction. As a basic model, for a two-level atom, fluorescence is emitted from a doublet to an adjacent lower-lying doublet.[26]A particularly interesting mechanism for achieving spectral narrowing is to tune a resonant cavity with an inverted dressed-state subtransition and to create a dressed-state laser.[27–37]The essential mechanism is cavity feedback into the fluorescence emission dynamics.

In this paper, we reveal a unique feature that occurs when the cavity-feedback mechanism is applied to a threelevel atom, in which two allowed electric dipole transitions,located respectively within the microwave and optical frequency regimes, are linked to a common level. This configuration of interaction is absent for the two-level atom, because a single transition is only within either microwave or the optical regime,but not both. This is unique,because a single microwave cavity can be used to narrow all optical spectral lines. Fluorescence is emitted from a triplet to an adjacent lower-lying triplet.[38–42]Degenerate cascade fluorescence appears when the three dressed states are equally spaced from each other. A single microwave cavity that resonates with the cascade-dressed transitions is sufficient to make each dressed state enter the cavity-feedback dynamic, which gives rise to the narrowing of spectral lines from optical transitions.The novelty of the generalization from the two-to three-level atomic system lies in two aspects,as follows.

First,fluorescence emission occurs from a dressed triplet to the lower adjacent triplet, and generally exhibits a sevenpeaked structure. With equal spacing between adjacent sublevels,the fluorescence spectrum takes on a five-peaked structure and cascade degenerate fluorescence happens. The essential difference, compared with the case of the two-level atom,is the equal spacing of the adjacent dressed states,which determines the existence of the degenerate cascade fluorescence. We now focus on the structure of cascade degenerate fluorescence and consider the effect of the intrinsic cascade lasing feedback on the representative fluorescence spectra when a single cavity is tuned to be resonant with the inverted dressed transition. The lasing oscillation happens in a predictable regime only when the light amplification by the inverted dressed transition dominates,compared to the light absorbed by the uninverted dressed transition. Due to the simultaneous interactions, one- and two-photon coherences are simultaneously established between the triplet of dressed states.It is such coherences of the dressed atomic states that cause the five spectral lines from one triplet to the adjacent lower triplet to be extremely narrow and high,while,as described in previous works,narrowing only occurs at the central peak or peaks either side.

Second, since the degenerate cascade laser is sufficient for the narrowing of all spectral lines,whether from a cavitycoupled trasition or a cavity-free optical transition, a microwave cavity (even a bad cavity) can be used to narrow the optical spectral lines from the cavity-free transition. This has been a specific long-term aim for atom–photon interactions.[43–46]Unlike the two-level atom case, where a single transition is either in the microwave regime or in the optical regime, the three-level atom has two cascade transitions that belong to different regimes,e.g.,the former is in the microwave regime and the latter is in the optical regime. Because of the existence of a triplet of dressed states for a driven three-level atom,more often than not,degenerate dressed-state transitions have no global population inversion. That is, one dressed-state transition with population inversion is connected to another transition without inversion. Lasing without inversion provides an efficient mechanism for cavity-feedback dynamics.

Our scheme should be distinguished from two other kinds of three-level atomic system. One type of scheme is based on atom–cavity dynamics below the lasing threshold.[22,24]In this case, there is no lasing oscillation, even if population inversion is present. One cavity can only make two dressed states enter the atomic dynamic. More dressed states need more cavities, and thus the experimental difficulty is greatly increased.The other type of scheme is based on parallel dipole moments.[10–13]The requirements of this kind of scheme are too stringent,due to the limitations imposed by selection rules.Even an equivalent realization in dressed-state representation is not enough to avoid the strict requirements of the parallel dipole moments,[47,48]because inherent spontaneous emission of the involved control transitions itself becomes an additional obstacle that is difficult to overcome,and negates the desirable coherent effects.

The remaining part of this paper is organized as follows.In Section 2, we first present the representative fluorescence spectrum of a three-level atom without a cavity and give the locations of the spectral lines in terms of dressed atomic states.Then, in Section 3, we describe dressed-state lasing without global population inversion when the three-level atom interacts with a cavity.We include the Hamiltonian for the dressedstate laser,the possible regimes for laser gain,and the steadystate laser intensity. The main results and the underlying physics are described in Section 4. Finally, we give a summary in Section 5.

2. Representative structure of a fluorescence spectrum in free space

Our purpose is to study the fluorescence spectrum of a three-level atom that interacts with two strong dressing fields plus a weak cavity field. To study cavity effects, we select a structure representative of a fluorescence spectrum in free space, because the spectrum generally has a fairly complicated dependence on the dressing fields.[38–41]While the strong dressing fields define the entire spectral structure, the weak cavity field affects the widths and heights of the spectral lines. For a three-level atom,as shown in Fig.1,we focus on a equally spaced five-peaked structure (Fig. 3), which appears when the atom is dressed by two dressing fields with equal Rabi frequencies and opposite detunings. We use|1,2〉to denote the two ground states and|3〉stands for the excited state. We use 2?1,2to denote the real Rabi frequencies of the interactions of the two driving fields of circular frequenciesω1,2with the atom on the|1,2〉?|3〉transitions,respectively.The master equation for the density operatorρof the atomfield system is written in the dipole approximation and in an appropriate rotating frame as

where the Hamiltonian

describes the interaction of the atom with the dressing fields.is the Planck constant,σkl=|k〉〈l| (k,l= 1,2,3) are the atomic flip spin (k/=l) and projection (k=l) operators, and?l=ω3l ?ωl(l=1,2)are the detunings of the applied driving fields from the atomic resonance frequencies. The damping term in the master equation is written[2–7]

where?σl3ρdescribes the atomic decay at a rate ofγl3, and takes the standard form?oρ=2oρo??o?oρ ?ρo?o,o=σl3,l=1,2. In this article,we focus on the spectral structure under the conditions of equal Rabi frequencies and the opposite detunings

and we describe the Rabi frequencies,the circular frequencies,and the damping rates in units ofγl3=γ. After diagonalizing the HamiltonianH0, we obtain the dressed states, which are expressed in terms of the bare atomic states as[7]

Fig.1. Schematic diagram for a three-level atom that has two spontaneous transitions|3〉|1〉and|3〉|2〉(with rates of γ13 and γ23,respectively)in a Λ configuration that interacts with two dressing fields with Rabi frequencies ?1,2 and detunings ?1,2,respectively.

Fig.2. (a)A pictorial representation of the dressed transitions that are contained in one spontaneous transition,e.g,|3〉|1〉. The dressed transitions occur from a triplet to its lower adjacent triplet. The vertical lines through a common disc correspond to the same sidebands. S31+1 and S31?1 are two pairs of degenerate cascade fluorescence. (b)A cavity is set up to resonate with the transitions denoted by the red and blue lines in(a).The interaction of the dressed atom with a cavity field is established through degenerate cascade dressed-state transitions|+,n1,n2〉?|0,n1 ?1,n2〉?|?,n1 ?2,n2〉.

We obtain the steady-state populations0,1,2(where the bar“?”overzdenotes the steady state)of|0〉,|+〉,|?〉as

Once?=0,i.e.,sinθ=0,cosθ=1,we have0=1,1,2=0.This indicates that the atom is trapped in the dark state.[42]Here, we focus on the case of?/=0, where the atom is excited when|?|??. By arranging the expectation values into a column vectorX(t)= (〈σ+?〉,〈σ+0〉,〈σ?0〉,〈σ++〉,〈σ??〉,〈σ0?〉,〈σ0+〉,〈σ?+〉)T,we write the equation of motion in the compact form

where the matrixQis easily obtained from equation (6), andMis the inhomogeneous termM=(0, 0,0, γ3, γ3,0,0,0)T.The two-time correlation-function matrixS(τ) =〈δX(t+τ)δXT(t)〉can be calculated using the quantum regression theorem.[1–7]The two-time correlationS(τ)satisfies the same equation of motion as the one-time averageX(t) with a vanishing inhomogeneous term

Reversing Eq. (5), we express the fluorescence spectrum in terms of the dressed states as

where the elementsSij(ω)of the correlation matrixSin equation(10)are used,the frequency arguments

Figure 3 is a plot of the total spectrum for the spontaneous emission of a single transition|3〉|1〉for?/?=1.5. The spectrum displays asymmetry for a dressing-field frequency ofω=ω1. This is because,for transitions at symmetric frequencies, the dressed states|±〉have the same populations [see Eq.(7)],but their involved dressed-state transitions have completely different coupling strengths [see Eqs. (18) and (19)].However, because of the symmetrically opposite detunings?1=??2=?/=0, a symmetry exists between the fluorescence spectra from the two wingsS31(ω) andS32(ω). If the atomic ground states|1,2〉are degenerate and the fluorescence takes the formS31(ω)+S32(ω), it turns out to be symmetric with the central frequency.

Fig. 3. Representative five-peaked spectrum S31(ω) of spontaneous emission from the |3〉|1〉 transition of a three-level Λ atom in free space.Equal Rabi frequencies and opposite detunings are chosen, ? =5γ/2 and?=5γ/ (?/? = ＞1), such that we have d ==5γ.The five peaks are located at(ω ?ω2)/γ =0,±5,±10.

Our focus is on the nondegenerate case and to show the cavity-feedback mechanism and conditions for the narrowing of all spectral lines in Fig. 3. The essential difference,compared to the case of the two-level atom,is the equal spacing of the adjacent dressed states, which determines the existence of a pair of degenerate cascade transitions, e.g.,|+,n1,n2〉 →|0,n1?1,n2〉and|0,n1,n2〉 →|?,n1?1,n2〉.Actually, this corresponds to the degenerate two-photon process|+,n1,n2〉→|0,n1?1,n2〉→|?,n1?2,n2〉. The cavity feedback is simply based on a single-mode dressed laser generated by the system itself.This is different from cases that use external fields such as the squeezed or thermal vacuum fields,as in Refs.[49,50].

Fig. 4. Diagrammatic sketch for various frequencies and detunings along the frequency axis. The left and right parts, which center at the dressing field frequencies ω1,2 respectively, are related to the dipole-allowed transitions |1〉?|3〉 and |2〉?|3〉 respectively. The dressing field frequencies ω1,2 are very close to ω31 and ω32 by the detunings ?1,2,respectively. The five peaks locate at ωl, ωl±d, ωl±2d (l =1,2), respectively. The cavity field is tuned to be resonant with the upper inner sideband of the|1〉?|3〉transition,ωc=ω1+d,i.e.,?c=d.

3. Dressed-state lasing without global population inversion

After having chosen the representative spectral structure of the three-level atom in free space, we turn to a consideration of cavity-feedback effects. For this purpose, in addition to the interaction of the atom with the dressing fields,we need to include the interaction of the atom with the cavity field in the system dynamics. While the interaction with the dressing fields is merged into the dressed atomic states,the interaction with the cavity field is analyzed in terms of the dressed states.

3.1. Interaction of CPT atom with a cavity field

We assume that a cavity field of frequencyωcis coupled to a wing of the Λ configuration, e.g., the|1〉?|3〉transition.The Hamiltonian for the interaction of the cavity field with the atom is written in the rotating frame as[2–7]

The cavity mode, which is described by the annihilation and creation operatorsaanda?, is detuned from the driving field by?c=ωc?ω1. The strength of the atom–cavity field coupling isg. We define the operatorsσkl=|k〉〈l|andl,k=1?3 for the bare atomic states andl,k=0 and±for the dressed atomic states are the spin-flip (k/=l) and projection (k=l)operators. The master equation for the density operatorρof the atom–field system takes the conventional form[3]

as shown in Fig. 2(a) by the two vertical lines (red and blue)surrounded by the yellow circle. The case?c=?dis treated in the same way. To clearly compare the frequencies involved and the detunings, we list them along a frequency axis in Fig. 4. In our calculation, we have taken equal Rabi frequencies and the opposite detunings, as shown in Eq. (4).We assume the cavity field to have a small Rabi frequency?α=g〈a〉(real)and a small damping rate

and make a further unitary transformation with the free parts for the dressed atom and the cavity fieldH0+?ca?a. After performing the transformation, we obtain the interaction Hamiltonian

It can clearly be seen from the Hamiltonian(18)that the cavity fieldais in simultaneous resonance with two individual transitions|+〉?|0〉and|0〉?|?〉, as shown in Fig. 2(b).Not only are there two one-photon processes for the individual transitions, but also two-photon process happens for the following transition|+〉?|0〉?|?〉. Once the cavity fieldais amplified above the threshold, one- and two-photon coherences are established between the dressed atomic states.The coherences have significant effects on the fluorescence spectrum, depending on the coupled dynamics of the cavity field and the dressed atom. In what follows, we will show that the fluorescence spectrum is indeed remarkably modified under appropriate conditions. In particular, the coherences induce the narrowing of the spectral lines,both from the cavity-coupled bare-state transition|3〉|1〉, and from the non-cavity-coupled bare-state transition|3〉|2〉. In the following three subsections, we describe the dressed-state laser gain,calculate the steady-state laser intensity,and present the coherence between the dressed states.

3.2. Linear gain without global population inversion

Dressed-state lasers are a special kind of system, which are driven by strong external coherent fields.[27–37]The degenerate composite states of the atom plus the dressing fields are lifted into the dressed states, which are shifted by the generalized Rabi frequencydfrom the dressing-field frequencies.Population inversion is established for one of shifted transitions between adjacent dressed-state triplets, depending on the dressing-field Rabi frequencies?1,2=?and detunings?1,2=??. Dressed-state inversion is used to create the laser gain. The essential difference from the previous dressed-state lasers is that the laser oscillation is created from a degenerate cascade of two dressed-state transitions,|+〉 ?|0〉and|0〉 ?|?〉, and one two-photon process,|+〉 ?|0〉 ?|?〉.The onset of the dressed-state laser is described by the linear gain, which only depends on the one-photon processes. The two-photon transition is involved only when the saturation behavior is considered.

In this subsection, we first focus on the onset of the dressed-state laser. The linear gain for the laser amplitude〈a〉can be obtained from Eq.(24)or(26),and is given below,following Refs.[1–7]:

By analysis,we can identify the four regimes in which both the dressed populations and the effective coupling strengths have different dependences on the normalized detuning. For clarity,we present the regimes in Table 1.We can see in which regime dressed-state lasing is possible.

(iv) For?/? ＞1, the population relation is reversed once again1,2＞0and the coupling strengths comply with|g1|2＞|g2|2. The amplification from the|+〉?|0〉transition becomes dominant compared to the absorption from the|0〉?|?〉transition. Once again,a net gain exists and lasing is achievable.

The possible parameter regimes for dressed-state lasing are changed to different regimes,as given in Table 2.

Table 1. Possible regimes of parameters for dressed-state laser oscillation in the higher sideband ?c=d.

Table 2. Possible regimes of parameters for dressed-state laser oscillation at lower sideband ?c=?d.

3.3. Steady-state laser intensity

We are now in a position to include the nonlinear effects of the dressed-state laser. To do so, we have to calculate the intensity of the dressed-state laser. This is a necessary step for calculating the cavity-modified fluorescence spectrum. The strong dressing fields, the amplitudes of which are given initially, are assumed not to change during the interaction and are merged into the atomic dressed states. However, the cavity field,which is created via the degenerate cascade quantum transitions, has an unknown amplitude. Only when the cavity field amplitude is obtained can we have the steady-state solutions of the populations and polarizations of the dressed atom, and we can then derive the cavity- mediated fluorescence spectrum.Since we focus on the case close to the threshold, the cavity field will be much weaker than the dressing fields,?α ??. In this case, it will not be convenient to use the cavity field to redress the atom. Instead, it is most convenient for us to follow the standard methods,[2–5]where the dressed states are determined only by the strong dressing fields. For the present system, while the five-peaked structure of the fluorescence spectrum is determined by the dressing fields, the narrowing of all five peaks is based on the interaction of the generated cavity fields with the dressed atoms. Although the dressing fields and the cavity fields are coexistent,the present system is confined to the weak cavity fields, i.e.,g2I=?2α ?|?|2,as shown below in Fig.5. This guarantees that the present dressed-state approach holds good.

To obtain the cavity field intensity and further to probe its effects on the fluorescence spectrum,we have to solve the master equation (15) for the reduced density operator. To do so,we have to arrange the atomic and field operators involved into a definite sequence and collect the additional terms that originate from the operator commutation relations. This technique is well established in books.[2–5]By treating the quantummechanical operators as the correspondingc-numbers,we can transform the master equation into a set of derivative equations ofc-numbers. By doing so, we preserve the classical and quantum statistical properties, including the classical(mean) behavior and the quantum-noise properties. To the usual second order, we have Langevin equations (equivalent to a Fokker–Planck equation).

To transform to thec-number representation, we choose the normal ordering (σ+?,σ+0,σ?0,σ++,σ??,σ0?,σ0+,σ?+,a?,a), and define the correspondingc-number correspondences (υ?3,υ?1,υ2,z1,z2,υ?2,υ1,υ3,α?,α). The set of Heisenberg–Langevin equations is derived as

together with the complex conjugates and the closure relationz0+z1+z2=1.F’s are white noises,which have the vanishing means and nonvanishing second-order correlations listed in Appendix B.

We see from Eq. (24) that the cavity field amplitudeα,the atomic polarizationsυ1,2,3and the atomic populationsz1,2are strongly coupled with each other. The first equation in which two atomic polarizationsυ1,2(one-photon coherences)simultaneously contribute to the the cavity field is that forα.This originates from the degenerate cascade interactions of the dressed atom with the cavity field. The second and third equations show thatυ1,2are created, respectively, due to atomic population differences(z1,2?z0)and coupled to the third polarizationυ3(two-photon coherence) via the cavity fieldα.The fourth equation shows that the atomic polarizationυ3is supported by two atomic polarizationsυ1,2.The last two equations reflect the nonlinear couplings of the populations to the cavity fieldαvia the atomic polarizationsυ1,2. The absorptive,dispersive and fluorescent responses of the dressed atom to the intracavity field can be obtained from the set of nonlinearly coupled equations.

We can derive an equation of motion for the cavity fieldαby temporarily neglecting the noise and setting the atomic derivatives to zero. First,we expressυ1?3in terms of the populationsz0?2from a closed set of equations forυ1?3. We then useυ1?3to solve forz0?2, and insertz0?2back into the expressions ofυ1?3. Finally, substitutingυ1,2into the equation forαyields

with a nonlinear gain of

where we have defined the photon numberI=〈α?α〉, and have listed theA’s andK’s parameters in Appendix A. The nonlinear gainG(I)is simply reduced to the linear gainG(0)in Eq. (20) whenI=0. The explicit expression for the laser gain clearly displays deep nonlinearities based on the degenerate two-photon process. Except for the complicated dependence of parameters,theA0andK0,1terms originate from the two degenerate one-photon transitions|+〉?|0〉and|0〉?|?〉. TheA1,2andK2,3terms are due to the two-photon process|+〉?|0〉?|?〉,as shown in Fig.3(b),which is absent in the two-level system.[15,16,23,27–30]When the linear laser gain satisfies the different conditions

the cavity field has a different behavior. The cavity field has a zero mean below the threshold,=0 and a nonvanishing mean/=0 above the threshold.The latter is the case in which a dressed-state lasing oscillation is created. In the steady state,we can solve for the photon numberby setting ˙α=0. These steady-state solutions are given in Appendix A. Stability is guaranteed by the term. This only holds for the adiabatic case(κ ?γ1?3,Γ1?3). However, we are not confined to the adiabatic case. Beyond the adiabatic conditions,stability is guaranteed by requiring positive eigenvalues of the drift matrixBthat will appear in Eq. (28) below. This corresponds to the requisite condition that any variable has to keep its fluctuation evolution below the threshold,even when it is amplified. The curves in Fig.5 from bottom to top correspond to the laser intensity(dotted),64(dashed),and 68(solid)andκ=5γ.This figure verifies the light amplification in regime IV,as listed in Table 1. It can be seen that the cavity field Rabi frequencywhich indicates that the cavity field does not yield further splitting of the dressed states. It is for this reason that we use the dressed states only by means of the strong dressing fields. At this stage,a dressed atom behaves in the same way as a normal bare atom and interacts with the cavity field.

Once the mean field amplitude is obtained,we can easily derive from Eq.(24)the steady-state solutions of the average per atom1?3andc0?2,which are listed in Appendix A,and in which the superscript “c” denotes the saturated values in the presence of the nonvanishing cavity field as distinguished from those of the vanishing cavity field. The nonlinearity of the cavity field is fed back to the dressed atom itself and has remarkable effects on the fluorescence spectrum.

Fig.5.The intensity g2/γ2 of the dressed-state laser above threshold versus the normalized detuning ?/? for κ =5γ and C=60(dotted),64(dashed),and 68(solid).

4. Extreme narrowing for an entire fvie-peaked structure

So far, we have calculated the intensities of the dressed laser and presented a numerical verification of the establishment of weak coherence in the dressed state. In this section,we derive the cavity-mediated fluorescence spectrum and physically analyse the effects of weak coherence on the spectral lines.

4.1. Cavity feedback mediated spectrum

Having obtained the cavity field amplitude and the atomic populations and polarizations,we can proceed to the inclusion of quantum noises,F’s, and the derivation of the correlation spectrum, which is common to the absorptive and dispersive response of the dressed atom to the intracavity field,and also common to the transmitted and fluorescent light.[51–53]The incoherent fluorescent light emitted perpendicular to the direction of the cavity field propagation is shown in Eq.(11).To obtain the total spectrum,we need to calculate those spectra that come from the dressed transitions. The cavity-feedback-based spectra are obtained from the set of Heisenberg–Langevin equations(24)by following the standard techniques.[2–6,51–53]Performing lineariztion on Eq. (24) and arranging the corresponding quantities in exactly the same order asX(t)in section two,plus the laser fieldsδO(t)=δ(υ?3,υ?1,υ2,z1,z2,υ?2,υ1,υ3,α?,α,)TandF(t)=(Fυ?3,Fυ?1,Fυ2,Fz1,Fz2,Fυ?2,Fυ1,Fυ3,Fα?,Fα)T,we derive a set of linearized Heisenberg–Langevin equations in a compact form

where the drift matrixBis easily obtained from Eq.(24). Stability is guaranteed by keeping all eigenvalues of the drift matrixBpositive. The noise correlation is derived from the Einstein relation[2–6],〈F(t)FT(t＇)〉=Dδ(t ?t＇), where the nonvanishing elements of diffusion matrixDare listed in

whereS(ω)is in a 10×10 matrix form

The elementsSkl(ω)(k,l=1?8)give the atomic correlations with the cavity feedback included. Substituting the elements ofSkl(ω) into Eq. (12), we obtain the fluorescence spectra from the transitions|3〉|1〉and|3〉|2〉of the three-level atom in a cavity.

4.2. Complete narrowing for S31(ω)from cavity-mediated transition

After obtaining the fluorescence spectra from the cavitycoupled and non-cavity-coupled transitions of the three-level atom,we present the main results and the underlying physics.In Fig.6,the spectrumS31(ω)is plotted for the cavity-coupled transition|3〉|1〉when the cavity field is amplified above threshold. The parameters chosen areC=60,d=5γ,κ=5γ(Fig.6).

Fig.6. Fluorescence spectra for the|3〉|1〉transition of a dressed threelevel atom in a cavity above threshold(G(0)＞κ). The parameters used are C=60,κ=5γ,and d=5γ.We fix d=5γ to avoid an overlap of the spectral peaks. The insert uses a linear coordinate for the central peak. Narrowing happens for all five peaks.

In Fig. 7 the spectrumS32(ω) is plotted for the noncavity-coupled transition|3〉|2〉for the same parameters as in Fig.6. In all these figures,we fixd=5γto avoid an overlap of the spectral peaks. The spectraS31(ω)andS32(ω)have essentially the same structures and features under the same conditions, although the cavity is only coupled to the|3〉|1〉transition but not to the|3〉|2〉transition. The spectral lines from the|3〉|l〉transition are located at the centerω=ωl,at the inner sidebandsω=ωl±d,and at the outermost sidebandsω=ωl±2d(l=1,2).

Fig.7. Fluorescence spectra for the|3〉|2〉transition of a dressed threelevel atom in a cavity above threshold(G(0)＞κ). The parameters are the same as for Fig.6.

Third,the excitation of the atom and the level splitting are determined by the strong dressing fields, and instead the role of the weak cavity field is that it establishes the weak coherence of the well-separated dressed states. The conditions are summarized as

In the dressed-state picture, the spontaneous transition|3〉|1〉happens from a triplet to its adjacent triplet|k,n1,n2〉|l,n1?1,n2〉,k,l=0,±,respectively. As shown in Fig.2(b),the two successive degenerate dressed-state transitions|+〉 ?|0〉and|0〉 ?|?〉are in simultaneous resonance with the common cavity field. The cavity field, if it is amplified below threshold, has a vanishing amplitude, and does not induce coherence between the dressed states. The dressed-state laser,once it is created through the cascade transitions|+〉?|0〉and|0〉?|?〉, establishes coherences between the dressed states. The degenerate one-photon transitions|+〉?|0〉and|0〉?|?〉and the two-photon transition|+〉 ?|0〉 ?|?〉are coupled to each other. In subsection 3.3, we already obtained the cavity field intensityg2, whose dependence on the normalized detuning is shown in Fig. 5.With the intensity, we also obtained the coherences between the dressed states1?3of the average per atom, which are shown in Fig.8(a)for theC=60,κ=5γ. Because of the oneand two-photon resonances,the polarizations1,2have purely imaginary parts,while the polarization3is purely real. There are two characteristic features for the cavity-feedback dynamics.

As a first feature, all three dressed states are involved in the cavity feedback dynamics. This is based on the same processes as used for the laser creation. As shown in Fig. 2,the cavity field simultaneously interacts with the two degenerate cascade dressed-state transitions|+〉?|0〉and|0〉?|?〉.Such degeneracy is absent in the most widely studied two-level systems, where the dressed transitions happen from a wellseparated doublet to its adjacent lower doublet. For the threelevel atom, in sharp contrast, the dressed transitions appear from an equally spaced triplet to its adjacent lower triplet, as shown in Fig.2(a). Because of the equal spacing, degenerate transitions|+〉?|0〉and|0〉?|?〉appear in pairs. For three adjacent triplets in the two-dimensional networks of dressed states|k,n1,n2〉(k=0,±),the degenerate transitions combine to support a two-photon process|+〉?|0〉?|?〉, as shown in Fig. 2(b). It is the two one-photon processes and the twophoton process that determine the dressed atomic polarizationsυ1,2,3,which are coupled to each other and to the cavity field,as shown in Eq.(24).

Fig.8. (a)Dressed-state atomic polarizations for C=60,κ =5γ. Note thathave purely imaginary components Im, while only has the real part. (b)A comparison between the saturated population differences 1 ?and ?and the unsaturated population differences ? and ? for the same parameters as used in(a).

As a general principle, the fluctuation spectrum is determined by the behavior of the parts of the coupled system fluctuating around the steady state and below the threshold. The amplification of the fluorescent light field below the threshold should not be confused with the amplification of the cavity field above the threshold. The amplification of the cavity field acting as a dressed-state laser is based on the dominance of the inverted dressed-state transition. The dressed-state laser,after its creation,enters dressed atomic dynamics and mediates the amplification of the fluorescent light below the threshold. In other words, the fluorescence spectrum is determined by the coupled dynamics between the dressed atom and the cavity field. It is easy to deduce that the one-and two-photon coherences of the dressed atomic states will play a crucial role in determining the spectral linewidths and heights.

4.3. Complete narrowing for S32(ω)from cavity-free transition

What is most worthwhile to stress is that the cavity field coupled to the|3〉|1〉transition has remarkable effects on the fluorescence spectral lines of the non-cavity-coupled|3〉|2〉transition. Figure 7 shows the plotted spectrumS32(ω)of the atomic transition|3〉|2〉where the cavity field is amplified above the threshold. It can clearly be seen that the fluorescence spectrum from the non-cavity-coupled|3〉|2〉transition displays essentially the same structure as that of the cavity-coupled|3|1〉transition. The minor difference lies in the peak heights. For example, there are different heights of the central peaks, as shown in the inserts of Figs. 6 and 7. The spectral narrowing not only happens for the cavitycoupled transition,but also for the non-cavity-coupled transition. When the two wings of the Λ configuration are located in the microwave and the optical regime,respectively,the coupling of the cavity to the microwave transition makes the optical spectral lines extremely narrow. This establishes the possibility of coherent control of optical fields using a microwave cavity. Such a cascade of a microwave transition with an optical transition is most commonly found in various atomic and molecular systems.[43–46,54,55]

4.4. Experimental consideration

On the other hand,the present scheme can be generalized to various multilevel systems,typically including those threelevel atomic systems inVand ladder configurations. A great number of atomic structures can be used as candidates for the present scheme. For example, the rubidium 85 D1transition hyperfine structure (795 nm) is suitable for optical control.The|1〉=|52S1/2,F=1〉?|3〉=|52P1/2,F=2〉transition and the|2〉=|52S1/2,F=2〉?|3〉=|52P1/2,F=2〉transition are in the Λ configuration with two optical transitions.The ground states are separated from each other by 3 GHz and the other excited state hyperfine level|52P1/2,F=1〉, which is 362 MHz below|3〉, has a negligible influence. The hydrogen transition hyperfine structure (243 nm) is well suited to microwave control. The|1〉=|22S1/2,F= 2〉 ?|3〉=|22P3/2,F=2〉transition and|2〉=|12S1/2,F=1〉?|3〉=|22P3/2,F=2〉transition are in the Λ configuration. The separation between the levels 22P3/2and 22S1/2is 9.95 GHZ,corresponding to a microwave transition. Meanwhile, the|2〉=|12S1/2,F= 1〉 ?|3〉=|22P3/2,F= 2〉transition is an optical transition. Thus, we have a cascade of the microwave and optical transitions. So far, there has been great progress in exploring atomic coherence effects induced by microwave fields. Examples include the four-wave mixing of optical and microwave fields in warm rubidium vapor,[43]electromagnetically induced transparency controlled by a microwave field in a dense rubidium gas,[44]and an atomic interface between microwave and optical photons in superconducting resonators,ensembles of ultracold atoms.[45]Progress has also been made with various systems such as ruby,[46]artificial atoms,[54]and molecules (methanol, formaldehyde,ammonia).[55]Therefore, the present scheme is within the reach of current technology.

5. Discussion and conclusions

We should note that the electromagnetically induced transparency (EIT) spectrum is also often chosen to enhance frequency-resolved detection.[56,57]Recently, a proofof-principle experiment[58]was performed for a superheterodyne receiver based on a microwave-dressed EIT spectrum. The sensitivity was remarkably higher than in previous schemes, such as those based on Autler–Townes splitting or standard atomic electrometers. Although EIT and coherent population trapping can possibly be used to narrow spectral lines, this is usually achieved at the cost of decreasing the power.[59,60]To enhance the accuracy and efficiency of spectra-based high-precision measurements, one expects that the linewidth narrowing of fluorescence spectra should be compatible with spectral heightening. It is clear that the present scheme has the advantage of meeting the above two compatible requirements.

In conclusion,for a three-level atom we have shown that a dressed-state laser from one allowed dipole transition is sufficient to narrow all fluorescence spectral lines. The essential difference from the case of the two-level atom is that the triply degenerate atomic field composite state is lifted as a triplet of equally spaced dressed states. This corresponds to the existence of degenerate dressed-state two-photon transitions,which lead to lasing without global dressed-state population inversion and coupling of all three dressed states to the single-mode cavity field. The required conditions are easily satisfied for a wide range of parameters: symmetric Rabi frequencies and symmetrically opposite detunings. The obtainability of both a single-mode dressed-state laser and its effects show that it is possible to use a microwave cavity to manipulate the fluorescence spectral lines of a cavity-free optical transition in a multilevel atomic system.

Appendix A:Steady-state solutions

We first list theAandKparameters that appear in Eq.(26)

The steady-state laser intensityis then derived as

Appendix B:The diffusion coefficients

Here,we show the elements of the nonzero diffusion coefficientsDoo＇in terms of normalized variables. We note thatDoo＇=Do＇oandDo?o＇?=D?o＇o. The nonzero diffusion coefficients are derived as follows:

Appendix C:The parameters of the fluorescence spectrum

The parametersR1?10for the fluorescence spectra in Eqs.(12)are shown as follows: