1. Introduction

Multilevel atoms have been extensively used to host nonlinear wave mixing processes. The atomic coherence established between dipole-forbidden energy levels by two coherent optical fields gives rise to important phenomena such as coherent population trapping and electromagnetically induced transparency (EIT) [1–3]. Depending upon the relative position of the intermediate energy level, such coherence creates a polarization in the medium at the sum or difference frequency of the coherence-inducing waves. The driven oscillation efficiently mixes with additional fields, giving rise to higher-order such as four- and six-wave mixing processes [4–8]. The efficient matter-field coupling and the suppressed absorption by virtue of the EIT make it possible to perform nonlinear optical processes using low-power continuous-wave (CW) beams. This feature is important in obtaining high-resolution spectra, and has been an important alternative to using high-powered pulses that, while enhancing the higher-order nonlinearities, can have large frequency bandwidths and can significantly alter the atomic energy levels via effects such as power-broadening and Autler-Townes (AT) splitting.

Wave-mixing processes facilitated by atomic coherence have been explored and characterized in many energy-level configurations such as lambda, double-lambda, ladder, Y and inverted-Y type systems [4–8]. Many important and useful properties have been identified, including multi-mode quantum correlations and entanglement, photonic memory and phase-controllable interference between different higher-order nonlinear processes [9–11]. In these processes, the signal is always generated at the transition between the first excited state and the ground state.

In the current experimental work, we parametrically amplify the vacuum mode between two excited states in a three-level ladder-type configuration (Fig. 1(a) ) and study it in the frequency domain. Atomic coherence and EIT mechanism are implemented, allowing the amplification to occur using low-power CW beams. We have investigated the spectral response of the generated waveform to the various contributing parameters, such as the spectroscopic properties (and multi-level structures) of the atomic energy levels, the vicinity of the laser frequencies to the various atomic resonances, and the powers of the driving beams. We consider their contributions to the properties of the generated radiations (such as the efficiency, line shape and linewidth) and discuss the optimum conditions suitable for this process. The generated radiation, containing high-resolution narrow-linewidth spectroscopic information, is background-free. These features can make this process desirable over other spectroscopic methods relying on two-photon fluorescence where the signals are typically very weak since the photons are scattered in all spatial directions and in any given detection direction, the background noise can be of comparable intensity with the signal.

 

Fig. 1 (a) Simplified three-level ladder-type configuration in ⁸⁵Rb that is coherently driven in the FWM process; |g>, |i> and |e> stand for the ground, intermediate and excited states, respectively. E_1, E₁′ and E₂ are external driving beams from laser sources, while E_f is the atom-radiated FWM signal that is parametrically amplified from the vacuum mode. (b) Schematic of the experimental configuration showing the directions and polarizations of the four fields. (c)-(f) Realistic energy level diagram showing the hyperfine (hf) levels of each driven state, as well as the incoherent decay channels (wavy arrows) between various combinations of driven hf levels: [f, f′, f′′] = (c) [3,2,2], (d) [3,3,4], (e) [3,4,3], and (f) [3,4,4]. The single-resonance decay channels are drawn first, followed by the double-resonance decay channels. The number of single-resonance and double-resonance optical pumping channels is different in the various cases. The spacings between the energy levels are not drawn to scale.

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Recently, it was shown that the FWM radiation generated in this atomic configuration can be effectively phase-modulated in the frequency domain by implementing interferometric control in the driving beams [12]. The system was used to demonstrate various features such as line shape symmetrization, linewidth narrowing and bandwidth switching. The versatile potential of this FWM configuration for Fourier-domain applications is one of our main motivations for experimentally characterizing it, and we believe that the results will be of general interest in the study of nonlinear wave mixing phenomena in multi-level atoms. An energy level configuration similar to the current work was studied in the time domain in Ref [13]. Analytical solutions to the system were derived in [14] with approximations such as very weak driving beams and small hyperfine coupling, both of which are different from our current experimental conditions.

In Fig. 1(a), we have only shown the closed three-level atomic system consisting of the fine structures being driven by the laser beams. In reality, each of these structures consists of a myriad of hyperfine (hf) levels due to coupling with the nuclear magnetic moment, most of which can radiatively decay to energy levels not being driven in the FWM process, making this a mixed system (i.e. consisting of closed as well as open sub-systems.) How open a driven sub-system is depends on the selection rules for the associated hf levels. The hf levels f, f′ and f′′ of the ground, intermediate and excited states, respectively, as well as the various decay channels associated with four different three-level subsystems, are shown in Figs. 1(c)-1(f). In each diagram, the decay channels of the single-photon transition are drawn first, followed by the decay channels due to the two-photon process. From these examples, it can be seen that some FWM subsystems have more decay channels than the others, and that various channels exist via which the atomic population gets optically pumped into the undriven ground state f = 2. The single-photon transition f = 3 → f′ = 4 is closed, whereas f = 3 → f′ = 2, 3 can also radiatively decay to f = 2. A peculiar feature of the ladder-type configuration consisting of multiple sublevels is the so-called double resonance optical pumping (DROP) effect [15]. Beacause of DROP, even when the single-photon transition is closed, the two-photon process opens various optical pumping channels as shown in Figs. 1(e)-1(f). The 5D state also has decay channels via the 6P state, which are not shown in Fig. 1.

Furthermore, due to the close proximity of the hf energies in the 5D and 5P states, many subsystems are simultaneously resonant for a given pair of driving beam frequencies. The hf levels further consist of varying numbers of Zeeman sublevels with different transition strengths. As will be shown below, the multi-level nature as well as the fact that most of the driven subsystems can radiatively decay and lose population to the environment have important consequences to the Fourier-domain waveforms of the atom-radiated coherent FWM signal including the line shape and efficiency at various spectral positions, giving rise to a rich array of spectra. Accurately reproducing the line shapes analytically or numerically thus involves the bookkeeping of all the subsystem parameters, and is beyond the scope of this current experimental work. The intention of this current article is to illuminate the experimentally observed properties of this interesting FWM system and to qualitatively assess the underlying causes of the most important features.

2. Experimental methods

The energy level configuration and the experimental geometry are shown in Fig. 1. The probe beam E₁ (frequency ω₁, wavelength λ₁, wave vector k₁) is generated by a CW diode laser DL. The wavelength λ₁ is scanned around 780 nm in order to probe the Doppler-broadened spectral bandwidth of the ⁸⁵Rb isotope’s D₂ transition. The vapor cell is 5 cm long, and is magnetically shielded and heated to 60° C. The transmitted probe beam intensity is monitored by a photodiode PD. A strong beam E₂ (frequency ω₂, wavelength λ₂, wave vector k₂) from a CW Ti-Sapphire laser is aligned to counterpropagate with E₁. The wavelength of E₂ is fixed but can be tuned around 776.158 nm, the wavelength of the upper transition in the cascade scheme. When both one-photon resonance and two-photon resonance (TPR) are satisfied, the coupling beam renders the atomic medium transparent for the probe beam by virtue of EIT. Even though the D₂ absorption line is Doppler broadened in the hot atomic medium, the two-beam counterpropagating geometry allows the two-photon EIT process to be basically Doppler-free for the cascade configuration [16]. When the probe beam’s frequency detuning is larger than the Doppler-broadened linewidth of the D2 transition, the single photon absorption is vanishingly small. Here, when the strong coupling beam is present satisfying TPR, we no longer have EIT. Instead, a direct two-photon transition is driven between |g> and |e> resulting in a two-photon absorption (TPA) peak. This Doppler-free TPA resonance has a much narrower linewidth than the Doppler-broadened D2 line’s absorption linewidth, and the TPA depth can be tuned via the coupling beam’s intensity. For intermediate frequency detunings lying between the EIT and TPA regimes, both stepwise (via |i>) and direct transitions from |g> to |e> are driven. At these frequency detunings, we observe a convolution of EIT and TPA in the transmission of the probe beam. The analytical solution showing the evolution of the TPR from EIT to TPA as the intermediate frequency detuning is increased can be found in [16].

The output of the diode laser DL is split to create a third beam E₁′ (frequency ω₁, wavelength λ₁, wave vector k₁′) that intersects with E₁ and E₂ inside the vapor cell at a small angle of θ (typically 0.4°) with k₁. The polarizations and powers of the beams can be altered independently. The third-order nonlinearity of the atomic medium, made efficient by the induced resonant coherences, leads to the generation of a new FWM radiation E_f ∝ χ⁽³⁾E₁′E₂E₁ which counterpropagates with E₁′ (due to conservation of linear momentum satisfying k_f = k₂ + k₁ - k₁′) but has the frequency of E₂ (due to conservation of energy satisfying ω_f = ω₂ + ω₁- ω₁′). In the EIT regime, the FWM process is enhanced because the transitions can be driven near the atomic resonances, while dissipation from |i> as well as |e> are vanishingly small. This allows the χ⁽³⁾ nonlinear optical process to be driven and measured at low intensities. In the TPA regime, the dissipation from |i> is small due to the large intermediate frequency detuning. Here, when only E₁ and E₂ are present but E₁′ is absent, the TPR coherence between |g> and |e> radiatively decays causing incoherent fluorescence scattering. The presence of E₁′ stimulates coherent FWM radiation in the phase-matched direction. As will be shown below, the spectral linewidth of the FWM radiation is similar to the TPA linewidth, governed basically by the linewidth of the |e> state.

By appropriate choices of the polarizations of the driving beams, the polarization of the FWM signal E_f is made to be orthogonal to that of E₁′. This allows for an effective isolation of the weak signal E_f using a polarization beam splitter, and is monitored with an avalanche photodiode APD. The voltage measurements of PD and APD are monitored simultaneously using a multi-channel oscilloscope, along with a reference Fabry-Perot cavity signal used for frequency calibration of the scanned DL output.

3. Experimental observations and discussions

Figure 2 shows the line-shapes of the generated FWM signal at various frequency detunings of the intermediate resonance, Δ₁, where Δ₁ = ω₁ – ω_ig, with ω_ig being the transition frequency between the ground state and the first excited state. The Doppler-broadened absorption profile of the probe beam is also shown in the figure for reference, and gives information about the spectral position of the intermediate resonance used in the two-photon and FWM processes. Here, the absorption and FWM signals correspond to the f = 3 ground state of the ⁸⁵Rb isotope. We note that while the intermediate-state detuning is different in each signal, all of them are two-photon-resonant (TPR); that is, the signal occurs only when Δ₁ + Δ₂ = 0, where Δ₂ = ω₂ – ω_ei and ω_ei is the transition frequency between the two excited states in the cascade configuration. As a result, even though the absorption profile is Doppler-broadened for the heated atomic ensemble, the generated FWM radiation has a line-shape that is Doppler-free. For convenience, the cases Δ₁ < 0, Δ₁ = 0 and Δ₁ > 0 will be referred to as “red-detuned”, “zero-detuned” and “blue-detuned”, respectively.

 

Fig. 2 FWM signal line shape, linewidth and efficiency at four different intermediate frequency detunings, placed at intervals of 500 MHz. All other experimental parameters are constant in the four cases. The Doppler-broadened absorption linewidth of the corresponding ground state (hf = 3 of ⁸⁵Rb, 5S_1/2) is also shown for reference.

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In obtaining the four signals shown in Fig. 2, the only experimental parameter being varied is the value of ω₂, which causes TPR to occur at different values of Δ₁ as ω₁ is being scanned. Except for their spectral positions, occurring at 500 MHz intervals, all the other experimental conditions, such as the vapor cell temperature (60 °C), beam powers (P₁ = 7.4 mW, P₁′ = 11.6 mW and P₂ = 40 mW) and beam geometry (θ = 0.4°), are identical. The importance of the intermediate frequency detuning is evident in the line shape, linewidth and efficiency of the FWM process. Far from intermediate state resonance, the signal has a narrow linewidth. As the condition Δ₁ = 0 is approached, the signal’s linewidth becomes broader and the line shape becomes significantly convoluted. In the region slightly red-detuned from center, the FWM signal intensity also sharply decreases, experiencing a local minima.

FWM signals occurring at 100 MHz intervals are presented in Fig. 3(a) , showing the frequency-detuning dependent trends in more detail. All other experimental parameters are the same as those used in Fig. 2. At each spectral position, the signal is a convolution of a sharp “right” peak and a broad “left” peak. Each pair of dots connected by a line corresponds to the maximum intensities of the sharp and broad peaks occurring within a signal at a given detuning. Below, we will discuss the dependence of the FWM signal’s (1) linewidth, (2) line shape and (3) efficiency upon the driving beam parameters as well as upon the internal structure of the atoms, and also (4) consider the dual role of the driving beam E₁′.

 

Fig. 3 (Color online) TPR FWM signals corresponding to different values of Δ₁, separated by 100 MHz each. The energy levels driven are |g> = 5S_1/2, hf = 3, |i> = 5P_3/2 and |e> = 5D_3/2. At each value of the frequency detuning, the FWM signal is a convolution of a sharp, strong peak and a broad, weak peak, the maximum intensities of which are denoted by a blue square and a red dot, respectively. Each of the two signal peak trends are connected by lines to aid the eye. The two peak values corresponding to a given convolution at a given frequency detuning are connected by a solid black line. The horizontal (vertical) dotted lines identify the intensities (intermediate frequency detunings) of the maximas and minimas of the signal convolution’s two peaks. (a) P₁ = 7.4 mW, P₁′ = 11.6 mW, P₂ = 38 mW (b) P₁ = 3 mW, P₁′ = 4 mW, P₂ = 55 mW. Note the change of scale in the intensity axes.

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3.1. Linewidth variations

Towards the center of the Doppler-broadened linewidth, as the condition Δ₁ = 0 is approached, power-related effects, such as power broadening and AT splitting of the atomic energy levels, become dominant [17]. These effects are also revealed in the broadening and splitting of the FWM signal in the zero-detuned region. For large powers of E₁ and E₁′, when Δ₁ = 0 is satisfied, the signal occurs at the AT-satellites [18] of the energy levels. There is a decrease in signal intensity in the spectral region occurring between the power-broadened AT satellites; that is, the signal maximum is displaced around Δ₁ + Δ₂ = 0. When |Δ₁| >> 0 as in the edges of the Doppler-broadened absorption linewidth and outside it, the EIT evolves into a two-photon-absorption (TPA) [16] having a narrow linewidth since power broadening is substantially reduced. Here the signal occurs within the linewidth of the TPA resonance and the signal maxima occurs at Δ₁ + Δ₂ = 0. In particular, the power- broadening or splitting of the intermediate level is minimal in this two-photon resonant condition, and the linewidth of the FWM signal is mainly limited by the linewidth of the upper-excited state. At intermediate detunings |Δ₁| > 0, the probe beam experiences a convolution of EIT and TPA effects, and the generated FWM signal also displays the contributions due to these two mechanisms. Here, both direct two-photon transition from |g> to |e>, as well as stepwise transitions via |i> exist; the direct two-photon transition’s linewidth is narrower as it depends on the relatively long-lived 5D state (natural linewidth 0.97 MHz), whereas the stepwise transition is broader because it also depends on the 5P state (natural linewidth 6 MHz).

3.2. Line shape asymmetries

In order to understand the asymmetries in the spectral line-shape of the generated FWM radiation, one needs to consider the multi-level structure of the atoms as shown in Figs. 1(c)-1(f). For different values of ω₂, different hf levels of the intermediate state are closest to satisfy TPR and contribute to the FWM process most effectively. The hf levels of the upper excited state are sufficiently close and lie within the power-broadened linewidth of the intermediate level, and all contribute to the TPR, whereas the intermediate state hf levels are further apart and dispersed within the Doppler-broadened absorption window. The hf levels have varying multiplicities and disparate transition strengths; it is these spectroscopic characteristics of the atomic energy levels that contribute to the sharp asymmetries in the signal line-shape.

The TPR effects involving f′ = 4 has the biggest contribution to the convoluted FWM line-shape, due to its large multiplicity and larger Clebsch-Gordon coefficients. More importantly, as shown in Fig. 1, the FWM pathways involving f′ = 4 have the fewest decay channels. This is why the signal generation (Fig. 2 and Fig. 3) is strongest in the blue-detuned region of the Doppler width, as this is where the f′ = 4 level lies. This is also why the sharp peak lies towards the right edge of each signal convolution for the chosen spin levels in this configuration. The effects of the branching ratios of the energy levels in the ladder-type system have been analyzed by Noh and Moon [19], showing signal convolutions due to the presence of closed and open subsystems.

To make these facts more evident, we have also generated signals by using other spin-levels having different constraints. First, in Figs. 4(a) -4(b), we change |g> to the other ground state hf level 2, while using the same |i> and |e> fine structures as used for Fig. 3. Here, we observe that the position of the sharp peak within the signal convolution occurs at the red-detuned side. Also, the position within the Doppler width where signal intensity is at a maximum, is in the red-detuned region. These changes occur because here, it is the transition f = 2 → f′ = 1 that is closed. Next, we use the same |g> and |i> as that used in Fig. 3, but change the upper excited state |e> to the other fine structure of 5D, i.e. 5D_5/2 (Figs. 4(c)-4(d)). Here, similar to Fig. 3, the maximum signal intensity occurs in the blue-detuned region of the Doppler-width since the transition f = 3 → f′ = 4 is closed. However, in this case, the maximum peak within the signal convolution occurs in the red-detuned side. This change occurs because the energies of the hf levels in 5D_5/2 are inverted; that is, in this fine structure, the hf levels with higher values have lower energies [20]. This causes the contribution to the FWM signal from the higher f′′ levels, which have allowed transitions from f′ = 4 of |i> as well as larger Zeeman multiplicities, to shift to the lower-energy side of the signal convolution. This is in contrast to Fig. 3, where |e> corresponded to 5D_3/2 in which the hf levels are not inverted. The stronger FWM signal intensity observed when |e> is 5D_5/2 may be attributed to it having fewer decay channels; while 5D_3/2 can decay to both J = 3/2 and 1/2 of the 5P and 6P levels, the 5D_5/2 fine structure can decay only to J = 3/2, due to selection rules. From Figs. 3 and 4, it is clear that the FWM efficiency is largest at frequency detunings where the single-resonance and double-resonance optical pumping effects are the weakest.

 

Fig. 4 (The meanings of the dots, squares and lines in (b) and (d) are the same as in Fig. 3.) The beam powers are P₁ = 7.4 mW, P₁′ = 11.6 mW, P₂ = 38 mW. (a) and (b) The ground state used is hf = 2 of ⁸⁵Rb, 5S_1/2, with |i> = 5P_3/2 and |e> = 5D_3/2. (c) and (d) The ground state hf = 3 of ⁸⁵Rb, 5S_1/2 is used with |i> = 5P_3/2, but with |e> = 5D_5/2 and ω₂ = 775.978 nm, where the hf levels are inverted. Note the change of scale in the intensity axes. In (a) and (c), the FWM transitions with the least number of decay channels are shown.

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3.3. Variations in the FWM signal efficiency

At high beam powers for the lower transitions, saturation effects begin to occur, reducing the FWM efficiency. The associated power-broadening effects also contribute to the reduction of the maximum signal intensity. The minima in the signal intensity towards the red-detuned region deserves some attention. The decrease is the largest when the powers of E₁ and E₁′ are large. For instance, as shown in Fig. 3(a) for the conditions of P₁ = 7.4 mW, P₁′ = 11.6 mW, P₂ = 38 mW, the percent decrease is 1030% (480%) for the strong (weak) peak of the FWMsignal convolution. When the beam powers are changed to P₁ = 3 mW and P₁′ = 4 mW, the percent decrease is only 230% (180%) for the strong (weak) peak (Fig. 3(b)). These values remained constant as P₂ was changed from 20 mW to 60 mW. We note that the hf levels 2 and 3 of the intermediate state lie in the red detuned region of the Doppler-broadened linewidth, as can be observed from saturation absorbtion spectroscopy. At high powers of E₁ and E₁′, these hf levels are power-broadened and there is a significant overlap between them. When Δ₁ lies in this overlapped region, the contributions of these hf levels to the total two-photon transition amplitude becomes comparable in strength but with opposite signs. The sign of the individual phases has contributions from the signs of the dispersions due to the opposite detunings. Such destructive interference due to multiple intermediate states [21–23] causes the total transition amplitude to decrease, suppressing the FWM efficiency. Moreover, in the high intensity regime of the ground-state coupling beams driving to f’ = 2 or 3, the atomic population gets optically pumped out of the system to the f = 2 ground state as shown in Figs. 1(c)-1(d), leading to reduced signal generation.

3.4. Dual role of the driving beam E₁′

Finally, we note the dual role of the coupling beam E₁′ in this configuration. As a stimulant to the FWM process, it gains a photon whenever a photon is generated in the FWM signal, as shown in Fig. 1(a). However, because it has access to the ground state population, E₁′ also contributes to the depletion of the ground state population into incoherent channels, which becomes especially important at large beam powers. Also, because both E₁ and E₁′ haveaccess to the ground state population, the coherences induced by these two beams between |g> and |i> are both significant. A mismatch between the strengths of these two coherences is detrimental to the FWM efficiency [24, 25]. As a result, increasing P₁′ indefinitely does not help the FWM efficiency. In fact, increasing the strength of the coherence due to E₁′ beyond the coherence due to E₁ begins to extinguish the FWM intensity. The rates of initial growth and subsequent extinction of the FWM radiation with increasing P₁′ is different at different values of Δ₁, and is shown in Fig. 5 . This behavior distinguishes this FWM process from the traditional cascade FWM configuration in which the signal is generated in the lower transition with two coupling beams in the upper transition and one probe beam in the lower transition. There, as the coupling beam intensity is increased, the FWM intensity grows until it reaches a maximum value where it remains constant, and signal extinction does not occur.

 

Fig. 5 Dependence of the FWM signal strength on the power P₁′ at three different values of Δ₁ within the Doppler-broadened D₂ absorption linewidth. The powers of the other two beams are held fixed at P₁ = 3 mW and P₂ = 22 mW. The three chosen values of Δ₁, also shown in the inset, correspond to where (i) signal maxima occurs at the blue detuned region (blue dots), (ii) signal maxima occurs at the red detuned region (red triangles), and (iii) signal minima occurs towards the center-red detuned region (black squares). The EIT peak visible in the inset corresponds to case (iii). The three signal trends are connected by lines to aid the eye.

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4. Summary

The vacuum mode between the upper excited states in a ladder-type configuration is parametrically amplified using atomic coherence mechanisms to enhance the third-order nonlinear response, and studied in the frequency domain. The generated radiation is background free, and its Doppler-free spectral waveform contains high-resolution information about the spectroscopic properties of the atomic energy levels. The line-shape, linewidth and intensity of the generated FWM radiation are determined by various factors such as the beam powers and associated power-broadening effects, the multilevel nature of the atoms and selection rules, the frequency detunings, the destructive interference effects due to contributions by multiple intermediate states, and the dual role of one of the coupling beams. The new radiation could find use in FWM-based applications, and the method can be used to improve procedures using two-photon fluorescence that typically have weak signals and large background noises.