1. Introduction

Spontaneous emission is a basic phenomenon resulting from atom-light interactions and has been one of the most attractive topics in the field of quantum optics for the last decades. By focusing on spontaneous emission, we can draw a lot of information of quantum systems. For example, Das and Agarwal showed that the photon-photon correlations of the radiation can be viewed as a probe of vacuum-induced coherence effects in a four-level system [1]. In Ref [2], Temnov and Woggon found giant photon bunching in the cooperative spontaneous emission. In Ref [3], Norris et al observed ground-state quantum beats in spontaneous emission from a continuously driven atomic ensemble. Depending on difference between the emission spectrum peak position and the eigen frequency of the two-level system, one can find both suppression and acceleration of the decay of the two-level system [4]. Via controlled spontaneous emission, the authors studied the two-dimensional atom localization in the subwavelength domain in a five-level M-type atomic system [5]. In particular, one can alter the emission of atoms by placing them in different environments such as cavities [6], free space [7], waveguide [8], photonic crystals [9], and nanostructures [10].

Resonance fluorescence, which refers to the detection of an atom in open space by means of resonant absorption and reemission of electromagnetic waves, is a good example of spontaneous emission. So the observation of resonance fluorescence spectrum can be an effective way to study the property of spontaneous emission. The resonance fluorescence spectrum of a two-level atom (TLA’s) driven by a strong near-resonant monochromatic field was predicted [11] by Mollow and later observed by many groups [12–14]. Resonance fluorescence of TLA’s Ba atoms driven by bichromatic fields in different situations [15–19] was also studied theoretically and experimentally. And there are quite a few further researches on resonance fluorescence of a three-level system [20–26]. Especially, on the basis of density matrix equation and quantum regression theorem [25, 26], the authors investigate the steady-state spontaneous emission spectrum of V, Λ and cascade models of three-level atom driven by two coherent fields.

Apart from the interest of fundamentals, modification and control of resonance fluorescence can also find applications in many fields. In Ref [27], the authors theoretically investigate the fluorescence from a J = 1/2 to J = 1/2 transition that is driven by a monochromatic laser field. They showed that the spectrum and intensity of the resonance fluorescence enforced by the principle of time-energy complementarity, which can be used for providing evidence of the quantum interference. In Ref [28], the resonance fluorescence from regular atomic systems is shown to represent a continuous radiation source emitting two multimode light beams in different spatial directions, which show bipartite entanglement. Based on the narrowing of fluorescence spectrum, the determination of atomic multipole moments by means of the detection of the fluorescence spectrum is anticipated to increase in accuracy by several orders of magnitude [29].

We can also study the resonance fluorescence in different environments [30–33]. In Ref [30], the authors showed that resonance fluorescence can be realized with a semiconductor quantum dot in a Cavity. The observation of the Mollow fluorescence triplet from a single solid-state emitter was demonstrated and can be used for studying nonlinear effects with only a few photons [31]. In Ref [32], the authors reported on the observation of resonance fluorescence from a single artificial atom. The strong atom-field interaction as revealed in a high degree of extinction of propagating waves will allow applications of controllable artificial atoms in quantum optics and photonics. The resonance fluorescence of a single molecular system interacting with a plasmonic nanostructure exhibited the Mollow triplet, which can pave the way for applications in nanoscale quantum devices and quantum information processing [33].

In the presence of spontaneously generated coherence (SGC), the spectrum of resonance fluorescence can also be altered and be demonstrated to exhibit narrowing [34], quenching [35], squeezing [36] and superfluorescence [37]. We can also use quantum coherence generated by coherent fields to effectively modify the properties of resonance fluorescence. Ficek and Swain simulated SGC with the coupling of a DC field under the condition of two perpendicular dipole-allowed transitions and showed that its resonance fluorescence spectrum can display ultranarrow lines [38]. In Ref [39, 40], via a microwave field and a laser field, the authors simulated SGC and showed the narrowing of the resonance fluorescence spectrum. Nevertheless, most works of this type are theoretic and need experimental verification. Applying the method of simulating SGC with a laser field, we reported an experimental investigation of the absorption features in a four-level system with SGC [41, 42].

Coupling different energy levels of atoms with laser field has been shown to be an efficient way of controlling fluorescence spectrum [20–26, 43]. It may produce enriched phenomena and provide flexibility for the control of fluorescence spectrum. However, most of the present researches on fluorescence spectrum are theoretical and there is no experimental work deal with the resonance fluorescence of the excited two-level atom coupled by a nonresonant field to a third state. On the other hand, no experimental work has demonstrated the effect of SGC can modify the spontaneous emission. In this paper we experimentally study the resonance fluorescence from an excited two-level atom when the atomic upper level is coupled by a nonresonant field to a higher-lying state in a rubidium atomic beam. We demonstrate phenomena such as narrow lines of the central peak in the fluorescence spectrum, and show the possibility of controlling the spectrum by the detuning of the auxiliary field. And with the auxiliary field our system emulates to a large degree a V-type atom with SGC effect [34] and the corresponding results convincingly demonstrate that the effect of SGC can modify the resonance fluorescence spectrum.

The paper is organized as follows. In Sec. 2, we introduce the model and describe the observed results. In Sec. 3, we explain the corresponding features with the transition properties of the dressed states generated by the laser fields. Sec. 4 contains a summary of the results.

2. Experiments and results

The experiments are conducted in the hyperfine levels of ${}^{85}\text{R}\text{b}$ and the level structure is depicted in Fig. 1 . The coupling field ${\omega }_1$ couples the $5{\text{S}}_{1/2}\text{,F=}3\to 5{\text{P}}_{3/2}\text{,F=4}$ transition, and the Rabi frequency and the detuning of ${\omega }_1$ are ${\Omega }_1$ and ${\Delta }_1$, respectively. The controlling field ${\omega }_2$ couples the $5{\text{P}}_{3/2}\text{,F=4}\to 5{\text{D}}_{5/2}\text{,F=5}$ transition with Rabi frequency ${\Omega }_2$ and detuning ${\Delta }_2$, respectively. The spontaneous decay rate ${\gamma }_1$ from level $|2〉$ to level $|1〉$ is about 6MHz, the spontaneous decay rate ${\gamma }_2$ from level $|3〉$ to level $|2〉$ is about 0.43MHz, and owing to the selection rule there is no spontaneous decay from level $|3〉$ to level $|1〉$.

 

Fig. 1 Energy-level scheme for${}^{85}\text{R}\text{b}$.

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The schematic of the experimental apparatus are shown in Fig. 2 . The field ${\omega }_1$ is provided by an extended cavity diode laser which has maximum powers of 30mw and linewidth of 1MHz. The controlling field ${\omega }_2$ is provided by the coherent −899 Ti: sapphire laser. It has maximum powers of 800mw with the linewidth of 0.5 MHz. Both the lasers are frequency stability through the laser frequency stabilization system. The two laser beams counter-propagate nearly linearly. To minimize the effect of Doppler broadening we carry out the experiment in a rubidium atomic beam. In the interaction region, the diameter of the atomic beam, laser beam ${\omega }_1$, and laser beam ${\omega }_2$ are 1mm, 2mm, and 3mm, respectively.

 

Fig. 2 Schematic diagram of the experimental setup. ECDL: external cavity diode laser; Ti:Sapphire: 
coherent-899 Ti:sapphire laser ; A1 and A2: apertures; L: lens; FP: Fabry-Pérot interferometer.

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We use spatial and angular filtering of Fabry-Pérot interferometer to obtain the maximal spectral resolution. First we use two apertures to select a portion of fluorescence that enters the F-P interferometer. The selected portion is emitted nearly orthogonal to the atomic beam and originated from the interaction area of the lasers and the atomic beam. The piezoelectrically scannable F-P interferometer has a resolution better than 2MHz and a free spectral range of 680 MHz. The laser beams, the atomic beam, and the interferometer axis are mutually orthogonal. Spectra are measured by monitoring the light power emitted out the end of the F-P interferometer as a function of its length. Only a single-interferometer mode, whose frequency we denote by$\omega$, fell within the atomic emission profile. The spectral sweep rate of the interferometer mode is 100MHz/min. The fluorescence transmitted through the F-P interferometer is detected by a cooled photomultiplier, followed by a photon counter. Experimental spectral resolution is deduced from the observed width of weak signal elastic scattering and is found to be less than 10MHz.

We observe the fluorescence spectrum of the three-level cascade system from level $|2〉$ to level $|1〉$ and obtain the results shown in Fig. 3 . The experimental data are plotted in solid blue lines, while the calculated results are presented in dashed red lines. When the Rabi frequency ${\Omega }_2=0$, which means only ${\omega }_1$ is applied, a classic Mollow-type resonance fluorescence spectrum of TLA’s shows up [see Fig. 3(a)]. Then we apply the laser field ${\omega }_2$ and study the resonance fluorescence influenced by the detuning ${\Delta }_2$. In the case of ${\Delta }_2=0$, the fluorescence from level $|2〉$ to level $|1〉$is nearly suppressed. For this is not a desirable feature for the purpose of observation, we do not show this result in the paper.

 

Fig. 3 (a) Observed fluorescence spectrum of two-level system for ${\Omega }_1=10\text{MHz}$, ${\Omega }_2=0$; (b)-(e) Observed fluorescence spectrum of three-level cascade system for ${\Omega }_1=10\text{MHz}$, ${\Omega }_2=6\text{MHz}$. (b) ${\Delta }_2=4\text{MHz}$, (c) ${\Delta }_2=8\text{MHz}$, (d) ${\Delta }_2=20\text{MHz}$, (e) ${\Delta }_2=30\text{MHz}$. Other parameters are ${\Delta }_1\approx 0$, ${\gamma }_2=6\text{MHz}$, ${\gamma }_3=0.43\text{MHz}$. Solid blue curves are the experimental results; dotted red curves are the theoretical simulation.

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When we set the detuning ${\Delta }_2\text{=}4\text{MHz}$, the fluorescence spectrum acquires additional peaks, as shown in Fig. 3(b). The spectrum is actually seven peaked in Fig. 3(b), but the spacing of the two pair of inner sidebands is too small to be barely separated in the figure (especially in the lower frequency side of the inner peaks). When we increase the value of the detuning to ${\Delta }_2\text{=}8\text{MHz}$, it is clear that the fluorescence spectrum has seven peaks and the distance between two pairs of inner sidebands is longer. Compared with Fig. 3(b), the total intensity of the fluorescence is increased. Also the outer sidebands are removed from the center peak a little farther. In particular, in both case, the central peak acquires linewidth narrowing, which is about 2MHz less than the one in Fig. 3(a).

If the increase of ${\Delta }_2$ continues, we can observe less peaks as shown in Fig. 3(d) and Fig. 3(e). In Fig. 3(d) there remains only one pair of inner sidebands. And the intensity of this pair of inner sidebands grows larger, while the intensity of the outer sidebands grows smaller. With an increase of ${\Delta }_2$, the total intensity of the fluorescence is increased compared with Fig. 3(b) and 3(c). And with much larger value of ${\Delta }_2$, the fluorescence spectrum reverts to a three-peaked one as shown in Fig. 3(e). We can also notice that the total intensity of the fluorescence is almost unchanged relative to Fig. 3(d). In Fig. 3(d) and 3(e), the narrowing of the central peak cannot be observed clearly.

In our experiment, Doppler broadening caused by angular spread of the atomic beam undoubtedly changes the fluorescence line shape. The linewidths of the lasers also modifies the spectrum. We consider the residual Doppler broadening of the atomic beam in the one-dimensional case by neglecting the divergences of the two laser beams and calculate the spectrum of the fluorescence spectrum with the method described by in Ref [24]. including the above two effects. The calculated results are presented in dashed red lines in Fig. 3. And the experimental results are in accordance with the theoretical simulations.

3. Theoretical analysis

It is well known that positions, heights, and widths of fluorescence peaks are determined by the energies, steady-state populations, and electronic dipole moments of dressed states. In order to interpret the numerical results, we investigate the properties of the dressed levels. First we give the form of Hamiltonian of energy-level scheme in frame rotating, which is written as

The eigenstates of the interaction Hamiltonians is the set of three linear combinations of the energy eigenstates $|1〉$, $|2〉$and $|3〉$, and they are given by the formulas

For simplicity, we write Eq. (2) in the following form:

We can see that both the state $|2〉$ and ground state $|1〉$ [see Fig. 4(a) ] are split into three dressed levels [see Fig. 4(b)], which are $|a〉$, $|b〉$ and $|c〉$ for the bare-state level $|2〉$, while $|a^{\prime }〉$, $|b^{\prime }〉$ and $|c^{\prime }〉$ for the bare-state level $|1〉$. Therefore the fluorescence from the state $|2〉$ to the ground state $|1〉$ has 9 dipole transitions in the dressed-state representation. This result is similar to that in Ref [44], where a two-level system has 4 dipole transitions in the dressed-state representation.

 

Fig. 4 The energy scheme under consideration. (a) In the bare-state basis. (b) In the dresse-dstate basis of the two laser fields. (c) In the dressed-state basis of the laser field ${\omega }_2$.

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Note that though $|i〉$ and $|i^{\prime }〉$ are different in constant energy by the energy difference between level $|2〉$ and $|1〉$, the dressed levels $|i〉$ has the same expressions and the eigenvalues as the dressed levels $|i^{\prime }〉$ with $i=a,b,c$. In the general case, the spectrum consists of seven spectral components and their positions are determined by the eigen energies of the dressed levels. The central fluorescence comes from transitions between identical dressed levels, which are $|a〉\to |a^{\prime }〉$, $|b〉\to |b^{\prime }〉$ and $|c〉\to |c^{\prime }〉$; one pair of inner sidebands placed at frequencies $\pm \left({\lambda }_a-{\lambda }_b\right)$, which is the result of the transitions between the dressed states $|a〉\to |b^{\prime }〉$ and $|b〉\to |a^{\prime }〉$; the other pair of inner sidebands placed at frequencies $\pm ({\lambda }_b-{\lambda }_c)$, which is the result of the transitions between the dressed states $|b〉\to |c^{\prime }〉$ and $|c〉\to |b^{\prime }〉$; and the outer sidebands located at frequencies $\pm ({\lambda }_a-{\lambda }_c)$, which is the result of the transitions between the dressed states $|a〉\to |c^{\prime }〉$ and $|c〉\to |a^{\prime }〉$.

First we plot the eigen energies $({\lambda }_i,i=a,b,c)$ as functions of the detuning ${\Delta }_2$ [see Fig. 5(a) ] to see the position of the fluorescence peaks. We can see that as the detuning ${\Delta }_2$ is increased, the value of $\left|{\lambda }_a-{\lambda }_c\right|$ is increased, thus the position of the outer peaks is becoming far from the central peak. And the dependence of $\left|{\lambda }_a-{\lambda }_b\right|$ and $\left|{\lambda }_b-{\lambda }_c\right|$ on ${\Delta }_2$ are responsible for the variation of the splitting between the two pairs of inner peaks. Under the small detuning ${\Delta }_2$, the difference between $\left|{\lambda }_a-{\lambda }_b\right|$ and $\left|{\lambda }_b-{\lambda }_c\right|$ is smaller than the linewidths of the inner peaks, so the two pairs of the inner sidebands can hardly be separated [see Fig. 3(b)]. And as the detuning ${\Delta }_2$ is increased, the difference between $\left|{\lambda }_a-{\lambda }_b\right|$ and $\left|{\lambda }_b-{\lambda }_c\right|$ is becoming large, as a result, we can see two separate pairs of the inner sidebands clearly from the fluorescence spectrum [see Fig. 3(c)].

 

Fig. 5 Properties as a function of the detuning${\Delta }_2$, (a) The eigen energies, (b) steady-state population of the bare-state, (c) decay rates ${\Gamma }_{ij}$, (d) steady-state population of the dressed-state, (e) decay rates ${\Gamma }_{i{i}^{\prime }}$.

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Second we talk about the physical mechanism of the fluorescence intensity. As we know, the intensity of the total fluorescence is relative to the population of the state $|2〉$. So we plot the steady-state population of the bare states as a function of ${\Delta }_2$ in Fig. 5(b). From the figure we can see that when ${\Delta }_2$ is not large, the population of the bare state $|2〉$ $({\rho }_{22})$ is sensitive to ${\Delta }_2$. Thus the intensity of the total fluorescence is greatly enhanced with the increase of ${\Delta }_2$ [see Fig. 3(b)–3(d)]. While when ${\Delta }_2$ is larger, we can see that the total fluorescence intensity of Fig. 3(d) and 3(e) are almost the same owing to the fact that ${\rho }_{22}$ depends weakly on ${\Delta }_2$.

The decay rate of the transition between the dressed levels $|i〉$ to $|j^{\prime }〉$ is proportional to the squared dipole moments$R_{ij}=\left|〈j|P|i〉\right|$, where $P={\mu }_{12}|1〉〈2|$ is the transition dipole moment operator between $|1〉$ and $|2〉$ in the bare state basis. $R_{ij}$ can be calculated with the expression

We plot the decay rate of the transition between the dressed levels $|i〉$ to $|j^{\prime }〉$ as a function of ${\Delta }_2$ in Fig. 5(c). When ${\Delta }_2$ is small, all the decay rate of ${\Gamma }_{a{c}^{\prime }}({\Gamma }_{c{a}^{\prime }})$, ${\Gamma }_{b{c}^{\prime }}({\Gamma }_{c{b}^{\prime }})$ and ${\Gamma }_{a{b}^{\prime }}({\Gamma }_{b{a}^{\prime }})$ are fast, thus all of the fluorescence peaks can be obtained [Fig. 3(b) and 3(c)]. While under the larger detuning ${\Delta }_2$, the decay rate ${\Gamma }_{a{c}^{\prime }}({\Gamma }_{c{a}^{\prime }})$ and ${\Gamma }_{a{b}^{\prime }}({\Gamma }_{b{a}^{\prime }})$are becoming very slow, and result in the vanishing of the outer peaks and one pair of inner peaks [Fig. 3(d) and 3(e)].

Last we talk about the linewidths of the central peak. The central peak comes from transitions of $|a〉\to |a^{\prime }〉$, $|b〉\to |b^{\prime }〉$ and $|c〉\to |c^{\prime }〉$. And its linewidths is related to the decay rates of $|a〉\to |a^{\prime }〉$, $|b〉\to |b^{\prime }〉$ and $|c〉\to |c^{\prime }〉$, which are proportional to the squared dipole moment $R_{ij}$. In order to explain the linewidths of the central peak, we plot the steady-state population of the dressed states ${\rho }_{ii}$ as a function of ${\Delta }_2$ in Fig. 5(d) and the decay rate of the dressed states ${\Gamma }_{i{i}^{\prime }}$ as a function of ${\Delta }_2$ in Fig. 5(e). From Fig. 5(d) we can see that in the case of small value of ${\Delta }_2$, the population of the dressed state $|b〉$ is much larger than that of $|a〉$ and $|c〉$. Therefore the linewidth of the central peak is mainly relative to the decay rate ${\Gamma }_{b{b}^{\prime }}$. In this case, from Fig. 5(e) we can see that the decay rate ${\Gamma }_{b{b}^{\prime }}$ is very slow, which leads to the linewidth narrowing of the central peak [Fig. 3(b) and 3(c)]. And when ${\Delta }_2$ becomes large, both the state $|b〉$ and $|c〉$ are well populated. Thus the width of the central peak is relative to the decay rate ${\Gamma }_{b{b}^{\prime }}$ and ${\Gamma }_{c{c}^{\prime }}$. In this case, the decay rate ${\Gamma }_{b{b}^{\prime }}$ and ${\Gamma }_{c{c}^{\prime }}$ are much faster from Fig. 5(e). As a result, there is no significant spectral narrowing occurs [Fig. 3(d) and 3(e)].

The observed results can also be viewed as the combined effects of the coupling field ${\omega }_1$ and SGC between dressed levels. In the dressed-state representation of the laser field ${\omega }_2$, the system turns to be a V-type scheme with two close-lying excited levels [see Fig. 4(c)], where $|+〉=\cos \theta |2〉-\sin \theta |3〉$ and $|-〉=\sin \theta |2〉+\cos \theta |3〉$ with $\tan \theta =({\Delta }_2+\sqrt{{\Delta }_2{}^2+4{\Omega }_2{}^2})/2{\Omega }_2$. As is well known, it is very difficult to realize SGC in real atoms owing to the rigorous requirements: there are at least two near degenerate levels subject to the condition that the dipole moments from them to another level are not orthogonal. In Ref [45], the authors studied the influence of quantum interference on the spontaneous emission from an excited two-level atom when the atomic upper level is coupled by a coherent field to a higher-lying state and showed that destructive quantum interference between two competing decay amplitudes can produce a dark line in the emission spectrum. Thus the system presented here emulates to a large degree a V-type atom with SGC [34] and no stringent condition is required.

When the detuning of the field ${\omega }_2$ is small, the SGC effect could be very strong, resulting in the interesting fluorescence spectrum such as narrow central peaks. As detuning of the field ${\omega }_2$ gets larger the SGC effect would become weaker and we obtain a Mollow-type resonance fluorescence spectrum with no width narrowing or sidebands. So we can realize the control of the intensity of quantum interference and modification of fluorescence spectrum by tuning the detuning of the field ${\omega }_2$.

4. Conclusion

In conclusion, we have experimentally studied the resonance fluorescence from an excited two-level atom when the atomic upper level is coupled by a nonresonant field to a higher-lying state in a rubidium atomic beam. We have obtained a few interesting features such as the narrowing of the central peak in the fluorescence spectrum. We can control the positions, heights, and linewidths of the fluorescence peaks by tuning the detuning of the auxiliary field. We have analyzed the fluorescence spectrum in the dressed state basis of the two laser fields. We can also attribute the observed phenomena to the effects of SGC between the close-lying levels in the dressed state picture of the auxiliary field. The experimental measurements agree quantitatively with theoretical calculations.