1. Introduction

Superluminal lasers (SLLs) have attracted much attention in the last two decades due to their potential to revolutionize the field of precision measurements and metrology. Such lasers operate under the condition in which the group velocity exceeds the vacuum speed of light [1–7]. It has been shown in previous works that the sensitivity of the frequency of such a laser with respect to a change in its optical roundtrip length, due to an external perturbation, is expected to be substantially larger than that of a conventional laser. The relative increase in the sensitivity (compared to a conventional laser sensor) is designated the enhancement factor, which is inversely proportional to the group index [8–10]. This property makes SLLs highly attractive for various sensing applications, primarily for challenging tasks such as gravitational wave detection [11,12], search for dark matter [13], and navigation-grade rotation sensing [1,2].

The realization of an SLL requires the presence of a narrow dip at the center of a broad gain profile [3]. The depth and width of the dip must be tailored carefully to produce the required dispersion profile yielding enhanced sensitivity. Over the years, various approaches have been studied to generate the desired dispersion relation, such as the double Raman gain [14], coupled ring resonators with Brillouin gain [15,16], Raman gain and depletion using two isotopes of Rb [17], optically pumped Raman gain and self-pumped Raman depletion [4], and four-wave mixing employing the N-Scheme [18–20]. However, these approaches are highly complicated to implement or require the use of many lasers. For example, in the Raman gain/depletion approach, employing two isotopes of Rb, four lasers are needed to generate a unidirectional SLL. Reducing the number of lasers is essential, particularly when minimizing the scheme to a compact device as required for many applications.

In a recent publication, we presented an approach employing EIT in Raman gain [21,22], in which we have shown experimentally that the optical pump used for generating the population inversion in the low-lying states of a Λ system, via coupling to a fourth auxiliary state, can produce a dip in the gain profile in addition to generating the overall gain spectrum. This occurs when the spectral splitting produced by the so-called Autler-Townes (AT) effect [23] is averaged over the thermal velocity profile. The realization of a unidirectional SLL using this approach requires only two uncorrelated lasers: the Raman pump and the optical pump.

In this paper, we describe a further simplified approach for realizing a superluminal ring laser using a single isotope of atomic Rb vapor by producing EIT in the self-pumped Raman gain scheme. This approach constitutes an important simplification step towards the realization of compact SLL sensors as it requires a single pump source. The self-pumped Raman gain is a widely known effect [24,25]. Here, we show that it is possible to utilize the AT splitting to generate the dip in the self-pumped Raman gain profile, facilitating the realization of practical SLL-based sensors.

The proposed scheme employs a pump laser with its frequency tuned to resonate with a single transition in a Λ system. However, the laser frequency is highly detuned from the second transition due to the energy level structure of ⁸⁵Rb. Under certain conditions, specifically when the pump beam is sufficiently intense, the highly detuned coupling beam can produce an efficient Raman transition for a probe field acting along the other transition. Here, a single field fills the task of both the Raman pump and the optical pump, while simultaneously producing the AT splitting. Specifically, the resonant coupling of the pump on one transition produces population imbalance between the low-lying states and, additionally, splits each of the coupled states via the AT effect. In parallel, the detuned coupling of the pump to the other transition produces a Raman gain for the probe beam in the vicinity of the two-photon resonance frequency. As these coupling processes coincide, a dip in the self-pumped Raman gain emerges. Thus, the two processes driven by two different diode lasers, as discussed in the EIT in Raman gain approach [21,22], are now coupled and driven by a single laser source.

The rest of the paper is organized as follows. Section 2 presents a simplified theoretical model for implementing the scheme and shows simulation results of the gain and dispersion profiles for several operating parameters. In Section 3, we present the experimental configuration used to validate the approach as well as fitting to the measurements in the symmetric and asymmetric gain configurations. In Section 4, we address potential constraints inherent to this scheme, and describe how this approach may be used to realize a ring laser gyroscope with bi-directional superluminal lasers employing the same pump. We conclude with a summary and discussion in Section 5.

2. Theoretical modeling

Consider a $\Lambda $ system consisting of states $|1 \rangle $, $|2 \rangle $, and $|3 \rangle $, as depicted in Fig. 1(a). In ⁸⁵Rb, states $|1 \rangle $ and $|2 \rangle $ correspond to $F = 2$ and $F = 3$ in the $5{S_{1/2}}$ manifold, respectively, while state $|3 \rangle $ represents the $5{P_{1/2}}$ manifold. In this scheme, the Raman pump couples two transitions. For the $|2 \rangle \leftrightarrow |3 \rangle $ transition, the Raman pump is detuned by ${\delta _{p23}}$, which is defined as ${\delta _{p23}} \equiv {\omega _p} - ({{\omega_3} - {\omega_2}} )$, where ${\omega _p}$ is the frequency of the pump field and $\hbar {\omega _i}$ is the energy of state $|i \rangle $ with $i = 1,2,3$. For the $|1 \rangle \leftrightarrow |3 \rangle $ transition, the Raman pump is detuned by ${\delta _{p13}}$, which is defined as ${\delta _{p13}} \equiv {\omega _p} - ({{\omega_3} - {\omega_1}} )$. The two detunings of the Raman pump are related as ${\delta _{p13}} \equiv {\delta _{p23}} - \Delta $ where $\Delta = 3.03579\textrm{ GHz}$ is the hyperfine ground states separation. For simplification of notations, in the rest of this paper, we use ${\delta _p}$ to denote ${\delta _{p23}}$ and express ${\delta _{p13}}$ in terms of ${\delta _p}$. A probe beam is applied along the $|2 \rangle \leftrightarrow |3 \rangle $ transition with detuning ${\delta _s}$, which is defined as ${\delta _s} \equiv {\omega _s} - ({{\omega_3} - {\omega_2}} )$, where ${\omega _s}$ is the frequency of the probe field. It is important to note that the probe also couples to the $|1 \rangle \leftrightarrow |3 \rangle $ transition, but with a much higher detuning. Since the probe is very weak compared to this large detuning, this coupling can be ignored. The Rabi frequencies of the pump and the probe are denoted as ${\Omega _p}$ and ${\Omega _s}$, respectively. Here, we consider the Rabi frequency of the pump to be the same for both transitions. In practice, these would be somewhat different, as shown later in the more detailed theoretical model. State $|3 \rangle $ decays at the rate $\Gamma = 6\;\textrm{MHz}$, and we assume that the branching ratio of the decay rates is unity. In addition, each ground state decays collisionally to the other ground state at a rate of ${\Gamma _g}$ which is assumed to be ∼1 MHz based on earlier studies [22].

 

Fig. 1. (a) Energy level diagram for ⁸⁵Rb and the applied fields. Here, ${\Omega _s}$ and ${\Omega _p}$ are the Rabi frequencies of the probe and the pump respectively, ${\delta _s}$ is the probe detuning, ${\delta _{p23}}$ is the pump detuning on the right leg, and ${\delta _{p13}}$ is the pump detuning on the left leg. (b) The qualitative effective system for the case where ${\delta _{p23}} = 0$, showing the Autler-Townes splitting of the right leg caused by a strong pump. The blue and red arrows represent the pump and probe frequencies respectively. See text for details.

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In the limiting case where ${\delta _p} \equiv {\delta _{p23}} = 0$, and for a pump Rabi frequency that is stronger than the decay rates, the $|2 \rangle \leftrightarrow |3 \rangle $ transition undergoes the Autler-Townes splitting process, as illustrated schematically in Fig. 1(b). The action of the pump on the right leg of the $\Lambda $ system has two effects. First, it creates a population inversion in the ground states, which produces a Raman gain for the probe. Second, it causes the splitting of the energy levels, which produces a dip in the Raman gain profile, as needed for an SLL. Of course, this effect can occur also for ${\delta _p} \equiv {\delta _{p23}} \ne 0$, for certain ranges of parameters. In these cases, the splitting would be asymmetric, thereby introducing asymmetry in the gain profile. It should be noted that since both the probe and the pump fields excite the $|2 \rangle \leftrightarrow |3 \rangle $ transition, the diagram shown in Fig. 1(b) is only qualitative. For the same reason, the rotating wave transformation cannot be used to render the effective Hamiltonian time-independent [26]. Therefore, it is also not possible to carry out rigorously the partial diagonalization of the Hamiltonian needed to show the splitting of the energy levels along the $|2 \rangle \leftrightarrow |3 \rangle $ transition.

To address this complexity, one can use the pump-probe model, based on a perturbative approach up to the first order [14,27]. However, this model assumes that the pump and the probe beams have the same polarization and are correlated. These assumptions are not valid in our experiment, in which the gain is maximized when the pump and probe are cross-polarized. Additionally, from a practical aspect, the pump and probe are generated by two different diodes and may differ in phase randomly. As such, an accurate model requires taking into account the interaction of the fields among 24 Zeeman sublevels in the ⁸⁵Rb D1 transition, combined with the pump-probe model, leading to 1725 coupled linear equations. Of course, it would be impractical to derive the corresponding equations analytically; as such, we have developed an algorithm for solving the pump-probe model for an arbitrary number of energy levels automatically. However, the calculation time for such a complex system becomes extremely long, even for a supercomputer, when averaging the results over the velocity distribution of the atoms. This work is currently in progress and will be reported in a successive paper.

To demonstrate the basic concept under the limitations mentioned above, we developed a simplified model. Consider first the case when the excited state is only the $5{P_{1/2}}\textrm{, }F^{\prime} = 2\textrm{,}$ hyperfine level. We consider only a single Zeeman sublevel in each ground state, corresponding to $5{S_{1/2}}\textrm{, }F = 2\textrm{, }{m_F} = 0$ and $5{S_{1/2}}\textrm{, }F = 3\textrm{, }{m_F} = 0$, denoted as states $|1 \rangle $ and $|2 \rangle $, respectively. In the excited state, we consider two Zeeman sublevels: $5{P_{1/2}}\textrm{, }F^{\prime} = 2\textrm{, }{m_F} = 1$ and $5{P_{1/2}}\textrm{, }F^{\prime} = 2\textrm{, }{m_F} ={-} 1$, denoted as states $|3 \rangle $ and $|4 \rangle $, respectively. These four states are shown in Fig. 2(a). We take into account the fact that the pump and the probe are cross-linearly polarized. As such, each of these fields can be decomposed into left (${\sigma ^ - }$) and right (${\sigma ^ + }$) circular components. Specifically, the pump is an in-phase superposition of these components (${\sigma ^ + } + {\sigma ^ - }$), while the probe is an out-of-phase superposition (${\sigma ^ + } - {\sigma ^ - }$) of these components. Furthermore, we take into account the signs and the amplitudes of the dipole matrix elements coupling the two ground states to the two excited states.

 

Fig. 2. (a)Energy level scheme including the coupling fields from ground states to $F^{\prime} = 2$. (b) The modified states and the coupling fields after the transformation of two Zeeman sublevels within $F^{\prime} = 2$. (c) Energy level diagram and the coupling fields from ground states to $F^{\prime} = 3$. (d) The modified states and the coupling fields after the transformation of two Zeeman sublevels within $F^{\prime} = 3$. See text for details.

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At this point, we still have the constraint that two different frequencies are coupling the same transition (for example, the $|2 \rangle \leftrightarrow |3 \rangle $ transition). However, this constraint can be eliminated by carrying out a transformation of the basis states. Specifically, we form two linear superpositions of the two excited states, defined as follows: $|{3^{\prime}} \rangle = {{({|3 \rangle - |4 \rangle } )} / {\sqrt 2 }}$ and $|{4^{\prime}} \rangle = {{({|3 \rangle + |4 \rangle } )} / {\sqrt 2 }}$. These states and the corresponding coupling to the ground states are shown in Fig. 2(b). As can be seen, we no longer have a situation where one transition is excited by multiple frequencies. As such, it is straightforward to produce a time independent Hamiltonian by carrying out rotating wave transformations for all the transitions shown in Fig. 2(b).

Next, we consider the case where the excited state only contains the $5{P_{1/2}}\textrm{, }F^{\prime} = 3$ hyperfine level. Again, we consider only two Zeeman sublevels: $5{P_{1/2}}\textrm{, }F^{\prime} = 3\textrm{, }{m_F} = 1$ and $5{P_{1/2}}\textrm{, }F^{\prime} = 3\textrm{, }{m_F} ={-} 1$, denoted as states $|5 \rangle $ and $|6 \rangle $, respectively, as shown in Fig. 2(c). We then follow the same procedure as above, and define a pair of rotated states, $|{5^{\prime}} \rangle = {{({|5 \rangle - |6 \rangle } )} / {\sqrt 2 }}$ and $|{6^{\prime}} \rangle = {{({|5 \rangle + |6 \rangle } )} / {\sqrt 2 }}$, as shown in Fig. 2(d). As in the previous case, there is no longer a situation where a single transition is excited by multiple frequencies, and a time-independent Hamiltonian can be obtained by carrying out rotating wave transformations for all the transitions shown in Fig. 2(d). Clearly, it is also possible to take into account both of these hyperfine states: $5{P_{1/2}},F^{\prime} = 2$ and $5{P_{1/2}},F^{\prime} = 3$, simultaneously, thus producing a six-level system.

The model shown here is somewhat approximate because the transitions between all the Zeeman sublevels contribute to the interaction. However, when all the Zeeman sublevels are included, it is not possible to carry out transformations that eliminate the presence of dual frequency excitation of all transitions. As mentioned above, we have developed an algorithm which is, in principle, capable of modeling this system while taking into account all the Zeeman sublevels, and results from this model will be presented in the near future. In what follows, we show the results of the approximate model and compare them to the experimental results. As shown below, this model yields reasonably good agreement with experimental results.

The complex Hamiltonian in the 6-level system described above, under the rotating wave approximation (RWA) and on a rotative wave basis, can be written as:

The Liouville equation can thus be expressed as:

A numerical algorithm [26] is used for solving the Liouville equation in steady state for each probe detuning. The nonlinear susceptibility associated with the probe is obtained from the relevant density matrix components by the following expression:

Figure 3 shows simulation results for the imaginary part (blue) and the real part (red) of the susceptibility experienced by the probe, associated with its gain and the dispersion, respectively, as functions of the probe detuning, for various Raman pump powers. As the Raman pump power is increased, the gain profile changes from a single and relatively narrow peak in Fig. 3(a) to (a) shape containing gain plus dip, as shown in Fig. 3(b). The dip becomes larger and broader with increasing pump power, while the overall gain and its width increase as well (see Fig. 3(c) and Fig. 3(d)). Correspondingly, the slope of the dispersion curve in the vicinity of the two-photon resonance condition also changes sign from positive to negative. Tuning the Raman pump power allows for controlling the dispersion to maximize the sensitivity enhancement.

 

Fig. 3. The gain profile (blue traces) and the corresponding dispersion (red traces) as functions of the probe detuning for the following parameters: ${\Omega _s} = 0.1\Gamma $, ${\delta _p} = 51\Gamma $, ${\Gamma _g} = 1.3\textrm{ MHz}$, $\Gamma = 6\textrm{ MHz}$, ${T_{cell}} = 80^\circ \textrm{C}$, (a) ${\Omega _p} = 5\Gamma $,(b) ${\Omega _p} = 10\Gamma $,(c) ${\Omega _p} = 20\Gamma $,(d) ${\Omega _p} = 35\Gamma $, where ${T_{cell}}$ is the temperature of the Rb cell.

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Figure 4 presents simulation results for the imaginary part (blue) and the real part (red) of the probe susceptibility as a function of the probe detuning for several Raman pump detunings. As the frequency of the Raman pump is tuned, a continuous transition of the gain spectrum between a gain peak shifted to a lower frequency (Fig. 4(a)) to a different gain peak shifted to a higher frequency (Fig. 4(d)) is observed. When the frequency of the Raman pump is tuned between these two extrema, a broad gain with a narrow depletion is created (see Fig. 4(b) and Fig. 4(c)). The symmetric shape is the desired operational point of the scheme. These results stem from a combination of the light shift and the AT splitting effect [23,28].

 

Fig. 4. The gain profile (blue traces) and the corresponding dispersion (red traces) as functions of the probe detuning for the following parameters: ${\Omega _s} = 0.1\Gamma $, ${\Omega _p} = 30\Gamma $, ${\Gamma _g} = 1.3\textrm{ MHz}$, $\Gamma = 6\textrm{ MHz}$, ${T_{cell}} = 80^\circ \textrm{C}$, (a) ${\delta _p} = 20\Gamma $,(b) ${\delta _p} = 40\Gamma $,(c) ${\delta _p} = 51\Gamma $,(d) ${\delta _p} = 75\Gamma $.

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3. Experimental results

A schematic of the experimental configuration used for investigating the scheme described above is illustrated in Fig. 5. A 7.5 cm long vapor cell of naturally occurring Rubidium is heated up to 120°C, according to measurements taken at the outer surface of the cell. Two independent 795 nm laser diodes provide the probe and the Raman pump laser beams. Two saturated absorption spectroscopy setups are used to determine the probe and the Raman pump frequencies. Half-wave plates (HWPs) and polarizing beam splitters (PBSs) are used to vary the powers of each beam. The frequency of the probe beam is scanned over a range, with a center frequency tuned to 3.03579 GHz below the $5{S_{1/2}}\textrm{, }F = 3 \leftrightarrow 5{P_{1/2}}\textrm{, }F^{\prime} = 2$ transition. The probe power entering the cell is 1.3 mW, while the power of the Raman pump is 55 mW.

 

Fig. 5. The schematic description of the experimental configuration. See text for details.

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The probe and Raman pump are, respectively, polarized horizontally and vertically. The PBS at the output is configured to ensure that only the probe beam enters the photodetector. However, due to imperfections in the PBS, some of the pump power might leak into the detector. This additional power is recorded when the probe beam is blocked and subtracted from the measurement of the probe transmission signal. Another effect that may influence the result is a polarization rotation of the pump or the probe when propagating simultaneously along the Rb cell [27,29]. Polarization rotation may cause more (less) pump power to reach the photodetector, which might be misinterpreted as a higher (lower) gain level experienced by the probe. Since the frequencies of the Raman pump and the probe differ only by ∼3 GHz, a high-finesse Fabry-Perot cavity can be used to filter out the pump power to determine the probe transmission more accurately. This improvement will be implemented in the future.

It is important to mention that the pump and probe beams are not phase-locked to one another during the experiment. Typically, the beat note between such a pair of diode lasers is found to be ∼250 kHz [30]. As such, the gain spectrum observed experimentally exhibits additional broadening by this amount. However, this broadening is substantially smaller than the broadening of the gain due to the high power of the pump.

To fit the experimental results with the approximate 6-level model, we use the values of the coupling strengths as fitting parameters. Specifically, we use parameters denoted as α and β for the magnitudes of the transition matrix elements from the ground states to $5{P_{1/2}}\textrm{, }F^{\prime} = 2$ and $5{P_{1/2}}\textrm{, }F^{\prime} = 3$, respectively, while ensuring the signs and the ratios needed for the 6-level model to work. This is illustrated in Fig. 6. The parameters α and β are real numbers between 0 to1. Such a modification in the model also changes the decay rate of the excited states to ${\alpha ^2}\Gamma $(${\beta ^2}\Gamma $) for $5{P_{1/2}}\textrm{, }F^{\prime} = 2$($5{P_{1/2}}\textrm{, }F^{\prime} = 3$). This degree of freedom can be justified due to the following reasons: First, the model does not contain the interaction of the fields with all the Zeeman sublevels. The use of the effective transition rates is expected to compensate for this simplification to some extent. Second, the exact dimensions of the pump and the probe beams and their variations along the interaction regions are not known precisely. The lack of this information implies uncertainty in the precise values of the corresponding Rabi frequencies. The use of the fitting parameters for the transitions rates can compensate for this uncertainty as well.

 

Fig. 6. The schematic of the relevant energy levels and the electric fields in the 6-level model with the fitting parameters α and β for (a) the $5{S_{1/2}}$ to $5{P_{1/2}}\textrm{, }F^{\prime} = 2$ transition and (b) the $5{S_{1/2}}$ to $5{P_{1/2}}\textrm{, }F^{\prime} = 3$ transition.

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The modified Hamiltonian, in terms of the parameters α and β, can be written as:

The source term is modified accordingly as:

The nonlinear probe susceptibility under the new definition is:

The gain experienced by the probe field can be then expressed as:

Figure 7 shows a fit of the model to experimental results for the gain spectrum experienced by the probe field when the pump detuning was set to $- 3\Gamma $. The experimental method for determining the pump frequency accurately is described in the Supplement 1. This corresponds to conditions that produce a symmetric gain profile. The agreement between the model and the experimental results is reasonably good (both the shape and the measured gain), for $\alpha = 0.6$ and $\beta = 0.32$. Based on the experiment, the gain at the center of the dip is 2.3% in the symmetric case. It might be challenging to demonstrate lasing with this degree of gain, given the typical losses in the conventional optical elements. To produce a higher gain at the center of the dip where the SLL operates, a longer cell or higher atomic density can be used.

 

Fig. 7. Blue curve: The gain measured in the experiment as a function of the probe frequency. Red curve: fitting results obtained using the following parameters: ${\Omega _s} = 0.1\Gamma $, ${\Omega _p} = 42.5\Gamma $, ${\Gamma _g} = 0.32\textrm{ MHz}$, $\Gamma = 6\textrm{ MHz}$, ${T_{cell}} = 92^\circ \textrm{C}$, ${\delta _p} ={-} 3\Gamma $, $\alpha = 0.6$, $\beta = 0.32$.

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Figure 8(a) and Fig. 8(b) show the fitting of the model for the asymmetric gain profile cases obtained when the pump was detuned by $- 26.9\Gamma $ and $22.5\Gamma $, respectively. In both cases, the gain levels and profiles match reasonably well to those obtained by the model. However, it should be noted that the values of the fitting parameters used for the plots in Fig. 7, Fig. 8(a) and Fig. 8(b) are all slightly different. Given that our model is an approximate one, it is perhaps not surprising. It should also be noted that since these three cases correspond to different detunings, it is possible that there are different degrees of polarization rotations occurring in these cases, thus changing the effective strengths of the pump fields. Further investigation is necessary to determine the degree of polarization rotations and effects thereof on these gain profiles, and will be carried out in successive work.

 

Fig. 8. Blue curves: The gain measured in the experiment as a function of the probe frequency. Red curves: fitting results obtained using the following parameters: (a) ${\Omega _s} = 0.1\Gamma $, ${\Omega _p} = 50\Gamma $, ${\Gamma _g} = 0.52\textrm{ MHz}$, $\Gamma = 6\textrm{ MHz}$, ${T_{cell}} = 86^\circ \textrm{C}$, ${\delta _p} ={-} 26.9\Gamma $, $\alpha = 0.58$, $\beta = 0.6$ and (b) ${\Omega _p} = 30.0007\Gamma $, ${\Gamma _g} = 0.39996\textrm{ MHz}$, $\Gamma = 6\textrm{ MHz}$, ${T_{cell}} = 88.95^\circ \textrm{C}$, ${\delta _p} = 22.5\Gamma $, $\alpha = 0.8$, $\beta = 0.35$.

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We have also studied the case of the D2 line theoretically as well as experimentally. We used a similar approach for modeling the system. However, in the D2 line, the fields couple four hyperfine states corresponding to $5{P_{3/2}},F^{\prime\prime} = 1,2,3,4$. Therefore, we allow four fitting parameters denoted as $\varepsilon ,\alpha ,\beta ,\gamma$, representing the coupling strength of the transition from the ground state to the excited states $5{P_{3/2}},F^{\prime\prime} = 1,2,3,4$ respectively. In this model the pump detuning ${\delta _p}$ is defined with respect to the transition $5{S_{1/2}}\textrm{, }F = 3 \to 5{P_{3/2}},F^{\prime\prime} = 2$. A detailed description of the theoretical model for the D2 line is included in the supplemental document.

The experiment was carried out in a manner similar to what we described earlier for the D1 line in Fig. 5. Two independent 780 nm laser diodes were used to provide the pump and the probe beams. Similarly, we have studied the cases corresponding to both symmetric and asymmetric gain spectra. Here, for brevity, we only show the result of the symmetric case, since this is the one that is of practical interest.

Figure 9 shows the fitting of the D2 line model to the gain spectrum experienced by the probe field while the pump detuning was set to $26\Gamma $, which corresponds to condition that produces symmetric gain. The model fits reasonably well with the shape and the measured gain, for $\varepsilon = 0.3015$, $\alpha = 0.33$, $\beta = 0.3801$, and $\gamma = 0.3015$. Based on the experiment, the gain in the center of the dip is 2.6%, which is close to the gain obtained for the D1 line measurement under the same Raman pump power. The larger discrepancy in the right edge of the spectrum arises from the fact that the gain lies on the edge of the linear absorption from ⁸⁷Rb which is also contained in our cell.

 

Fig. 9. Blue curve: The gain measured in the experiment for the D2 line as a function of the probe frequency. Red curve: fitting results obtained using the following parameters: ${\Omega _s} = 0.1\Gamma $, ${\Omega _p} = 53\Gamma $, ${\Gamma _g} = 0.63\textrm{ MHz}$, $\Gamma = 6\textrm{ MHz}$, ${T_{cell}} = 115^\circ \textrm{C}$, ${\delta _p} = 26\Gamma $, $\varepsilon = 0.3015$, $\alpha = 0.33$, $\beta = 0.3801$, $\gamma = 0.3015$.

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Since the measured gain at the center of the dip is quite similar for the D1 and the D2 manifolds (2.3% for D1 and 2.6% for D2) under the same pump power and experimental conditions, there is no clear preference for one over the other in terms of gain. From the point of view of simulations, it may be easier to employ the D1 transition, since it contains only two hyperfine states, compared to four hyperfine states for the D2 line, which leads to a 24-level system for the D1 versus a 36-level system for the D2 when considering all the Zeeman sublevels. Since the simulation for the entire model is highly time-consuming, even for a supercomputer, using a smaller number of energy levels has a clear advantage when using simulation results to guide the optimal choice for experimental parameters.

4. Experimental considerations for realizing a superluminal laser

While the use of only a single laser makes the scheme described in this paper very simple, it also imposes constraints. Consider first the issue of the degree of gain that may be achievable at the bottom of the gain profile. For simplicity, we discuss the case employing an ideal three-level system shown in Fig. 1(a), although the comments would also apply to the real systems employing additional energy levels. The degree of Raman gain is dictated by two parameters. The first parameter is the population inversion between states $|2 \rangle $ and $|1 \rangle $. This is produced by the resonant interaction of the Raman pump along the $|2 \rangle \leftrightarrow |3 \rangle $ transition, since the off-resonant interaction along the $|1 \rangle \leftrightarrow |3 \rangle $ transition is highly detuned. Increasing the pump power would therefore increase the degree of population inversion, and thereby the gain. However, if the pump power is large enough to produce well-separated Autler-Townes splitting, the Raman gain will have a flat plateau with vanishingly small gain. Since we need significant gain in the middle of the gain profile, the pump power has to be kept relatively small, thus constraining the degree of population inversion and the gain at the center of the dip.

The second parameter is the effective Rabi frequency of the Raman transition, which is given approximately by ${\Omega _{eff}} = \varepsilon \,{\Omega _S}$, where $\varepsilon \approx {{{\Omega _P}} / {2\Delta }}$, when the Raman pump is resonant with the $|2 \rangle \leftrightarrow |3 \rangle $ transition, and the probe acting on the $|2 \rangle \leftrightarrow |3 \rangle $ transition is two-photon resonant with the Raman pump acting on the $|1 \rangle \leftrightarrow |3 \rangle $ transition. The Raman gain is due to the dipole moment generated at the probe frequency along the $|2 \rangle \leftrightarrow |3 \rangle $ transition. This dipole moment is proportional to $\varepsilon $. Given that the value of $\Delta $ is constrained to be the energy separation between states $|2 \rangle $ and $|1 \rangle $, it is necessary to increase the pump Rabi frequency in order to increase the Raman gain. However, again, for the same reason as discussed above, the value of the pump Rabi frequency has to be kept small in order to ensure that the Autler-Townes splitting is not too large.

In a real system (for both the D1 and D2 manifolds), there are additional hyperfine levels involved in the excited state. Consequently, the symmetric gain profile occurs for conditions when the pump is somewhat off resonant with respect to the $|2 \rangle \leftrightarrow |3 \rangle $ transition, as shown in the simulations and experimental results. Nonetheless, the general essence of the constraints discussed above apply for a real system as well.

Despite these constraints, it is indeed possible to obtain a value of Raman gain at the center of the dip that is greater than unity, albeit by a small fraction, as shown in the simulated and experimental results. For a cavity with a sufficiently high finesse, it is possible to realize a laser for any gain greater than unity. Furthermore, it should be noted that one could make use of a cell with a higher temperature to increase the atomic density, use a longer cell, or construct a cavity wherein the lasing mode passes through a single cell multiple times, and possibly a combination of these techniques, to increase the net gain.

Consider next the issue of the dispersion necessary to achieve the superluminal lasing condition, with a group index less than unity. It should be noted that what matters in this context is not the group index experienced by the probe beam, but rather the effective index experienced by the laser in steady state [3]. In order to determine this group index, it is necessary to simulate the Raman laser, using an iterative algorithm [31]. As discussed above, the model we have employed here is approximate; a more complete model that takes into account all the Zeeman sublevels employing the pump-probe approach is under development, and results from this model will be presented in the future. However, since the gain can be increased significantly via the application of the techniques mentioned above, the effective behavior of this system is expected to be similar to previous techniques demonstrated by us [4,17,21,22,32], for which it was quite easy to reach a value of the group index approaching the null value.

In Fig. 10, we illustrate schematically a possible experimental configuration for realizing a superluminal ring laser gyroscope using this scheme. As shown in Fig. 10(a), the output from the pump laser, with a frequency ${f_1}$, is split into two parts using a polarizing beam splitter (PBS). The ratio of powers of the two parts can be controlled by a half-wave plate (HWP). A small fraction is sent through an acousto-optic modulator (AOM) followed by a saturated absorption spectroscopy (SAS) system for frequency stabilization. The rest of the pump power first passes through another HWP in order ensure that it has the linear polarization that gets reflected by a PBS. After passing through this HWP, the beam is split equally by a 50/50 non-polarizing beam splitter (BS) and inserted in the cavity using two PBSs for producing superluminal lasing in each direction (clockwise and counterclockwise). Another pair of PBSs is used to eject the pump beams from the cavity. The lasing beam, with a frequency of ${f_{SL}}$, is polarized orthogonally to the pump, due to the nature of the Raman gain and the presence of the PBSs in the cavity. For each direction, by using a Rb vapor cell with large transverse area and a pair of parallel mirrors, a multiple pass scheme is employed to increase the gain experienced by the laser fields. As we have shown experimentally before [32], the Raman gain is extremely unidirectional, thus eliminating potential cross talk between the two lasers in such a geometry.

 

Fig. 10. Schematic of (a) a possible experimental configuration for realizing a superluminal ring laser gyroscope using the proposed scheme and (b) the relevant energy levels and the optical fields for the D2 manifold in ⁸⁵Rb. See text for details.

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As an explicit example, we consider the case where the D2 manifold is used for producing the Raman gain. The relevant energy levels and the optical fields in the D2 manifold are illustrated in Fig. 10(b). The part of the pump beam that enters the SAS system after passing through the AOM is downshifted to frequency ${f_2}$. A servo is then used to lock ${f_2}$ to the $5{S_{1/2}},F = 3$ to $5{P_{3/2}},F^{\prime} = 2$ transition. The frequency of the pump laser applied to the cavity will thus be detuned above this transition. Since the pump detuning for symmetric gain is $26\Gamma $ which equals ∼156 MHz, the operating frequency of the AOM would be chosen to match this value.

5. Discussion and conclusions

We have presented a highly simplified approach for realizing a superluminal ring laser based on EIT in self-pumped Raman gain. In this scheme, a single pump laser is used to produce the necessary conditions for generating a broad gain with a narrow dip. By tuning the pump power and detuning, it is possible to control the dispersion spectrum and optimize the enhancement in sensitivity. We experimentally demonstrated the realization of the gain profile, including the dip, in a Rb vapor cell, for the D1 and the D2 lines, in reasonable agreement with an approximate theoretical model. The realization of lasing in a cavity under those conditions will be carried out in successive studies. Among the configurations for SLL gain based on interaction with Rb vapor, this is the simplest approach reported so far, and has great potential to lead to realistic and practical realization of SLL-based sensor for precision measurements.