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We demonstrate a slow-light augmented unbalanced Mach-Zehnder interferometer (MZI) which can be used to enhance very significantly the sensitivity of measuring the frequency shift in a laser. The degree of enhancement depends on the group index of the slow-light medium, the degree of imbalance between the physical lengths of the two arms of the MZI, and the spectral width of the laser. For a laser based on a high-finesse cavity, yielding a narrow quantum noise limited spectral width, the group index has to be larger than the finesse in order to achieve enhancement in measurement sensitivity. For the reported results, strong slow-light effect is produced by employing electro-magnetically induced transparency via coherent population trapping in a buffer-gas loaded vapor cell of Rb atoms, with a maximum group index of ~1759. The observed enhancement in sensitivity for a range of group indices agrees well with the theoretical model. The maximum enhancement factor observed is ~22355, and much larger values can be obtained using cold atoms for producing the slow-light effect, for example. The sensitivity of any sensor that relies on measuring the frequency shift of a laser can be enhanced substantially using this technique. These include, but are not limited to, gyroscopes and accelerometers based on a conventional ring laser or a superluminal ring laser, and detectors for virialized ultra-light field dark matter. We also show how similar enhancements can be achieved in a slow-light augmented unbalanced Michelson interferometer.

We describe a generalized algorithm for evaluating the steady-state solution of the density matrix equation of motion, for the pump-probe scheme, when two fields oscillating at different frequencies couple the same set of atomic transitions involving an arbitrary number of energy levels, to an arbitrary order of the harmonics of the pump-probe frequency difference. We developed a numerical approach and a symbolic approach for this algorithm. We have verified that both approaches yield the same result for all cases studied, but require different computation time. The results are further validated by comparing them with the analytical solution of a two-level system to first order. We have also used both models to produce results up to the third order in the pump-probe frequency difference, for two-, three- and four-level systems. In addition, we have used this model to determine accurately, for the first time, the gain profile for a self-pumped Raman laser, for a system involving 16 Zeeman sublevels in the D1 manifold of 87Rb atoms. We have also used this model to determine the behavior of a single-pumped superluminal laser. In many situations involving the applications of multiple laser fields to atoms with many energy levels, one often makes the approximation that each field couples only one transition, because of the difficulty encountered in accounting for the effect of another field coupling the same transition but with a large detuning. The use of the algorithm presented here would eliminate the need for making such approximations, thus improving the accuracy of numerical calculations for such schemes.

We show that the frequency of a Raman laser is highly correlated or anti-correlated with the frequency of the Raman pump laser, depending on whether the dispersion experienced by the Raman laser is positive or negative. For a subluminal laser, corresponding to a positive dispersion with a group index that is much larger than unity, the shift in its frequency is approximately the same as that in the Raman pump laser. In contrast, for a superluminal laser, corresponding to a negative dispersion with a group index that is close to zero, its frequency shifts in the direction opposite to that of the Raman pump lasers, and has an amplitude that is larger by a factor approximately equaling the inverse of the group index. These findings would play a critical role in determining the maximum achievable sensitivity of sensors employing such lasers, especially under conditions where the pump laser linewidth is broadened significantly beyond the Schawlow-Townes linewidth due to classical fluctuations., Comment: 28 pages, 8 figures

We have developed a conceptual design for an Active Fast Light Fiber Optic Sensor (AFLIFOS) that can perform simultaneously or separately as a gyroscope (differential mode effect) and a sensor for acceleration, strain, and other common mode effects. Two Brillouin lasers in opposite directions and separated in frequency by several free spectral ranges are used for this sensor. By coupling two auxiliary resonators to the primary fiber resonator, we produce superluminal effects for two laser modes. We develop a detailed theoretical model for optimizing the design of the AFLIFOS, and show that the enhancement factor of the sensitivity is $\sim{187}$ and $\sim{-187}$, respectively for the two Brillouin lasers under the optimized condition, when the effective change in perimeter of the primary fiber resonator is 0.1nm, corresponding to a rotation rate of 0.4 deg/sec for a ring resonator with radius 1m. It may be possible to get much higher enhancement by adjusting the parameters such as the perimeters and the coupling coefficients., Comment: 18 pages, 8 figures

Previously, we had proposed an optically-pumped five-level Gain EIT (GEIT) system, which has a transparency dip superimposed on a gain profile and exhibits a negative dispersion suitable for the white-light-cavity signal-recycling (WLC-SR) scheme of the interferometeric gravitational wave detector [Phys. Rev. D. 92, 082002 (2015)]. Using this system as the negative dispersion medium (NDM) in the WLC-SR, we get an enhancement in the quantum noise (QN) limited sensitivity-bandwidth product by a factor of ~18. Here, we show how to realize this GEIT system in a realistic platform, using non-degenerate Zeeman sublevels in alkali atoms. Specifically we choose $^{87}$Rb atoms, which produce the negative dispersion around 795nm. The current LIGO operates at 1064nm but future LIGO may operate at a wavelength that is consistent with this atomic system. We present a theoretical analysis for the susceptibilities of the system. To account for the QN from the GEIT system, it is necessary to use the master equation (ME) approach. However, due to the number of energy levels involved, applying the full ME approach to this system is very complex. We have also shown earlier, in the reference cited above, that under GEIT condition, the net enhancement in the sensitivity-bandwidth product, as predicted by the ME, is close to that predicted by applying the Caves model for a phase-insensitive linear amplifier. Therefore we here use the Caves model for the QN from the NDM and this simplified numerical model shows that the enhancement of the sensitivity-bandwidth product as high as 17 is possible., Comment: 14 pages, 6 figures

When considering the effect of quantum noise (QN) in a phase-insensitive linear amplifier or attenuator, it is customary to use the single channel Caves model (SC-CM). Although this model is valid in simple situations, such as the presence of a beam splitter, it is not necessarily valid when a system with many degrees of freedom is involved. In order to address this issue, we consider in this paper various atomic transitions corresponding to amplification or attenuation using the master-equation- (ME-) based approach to model the QN and to compare the results with the SC-CM. For a four-level system that consists of a transition producing a broad gain peak and a transition producing an absorption dip, which results in perfect transparency at the center, we observe a catastrophic breakdown of the SC-CM. We also show that for a general two-level atomic system, the SC-CM does not apply, except in the limiting case when only either amplification or attenuation exists. A special case where the two models predict the same result is a {\Lambda}-type three-level electromagnetically induced transparency (EIT) system in which the QN at zero detuning vanishes while the system is in the dark state. We also study an optically pumped five-level gain EIT system which has a perfect transparency dip superimposed on a gain profile, and yields the negative dispersion suitable for use in enhancing the sensitivity-bandwidth product of an interferometric gravitational wave detector. In this case, we find that, for some set of parameters, the QN is vanishingly small at the center of the dip, and the SC-CM agrees closely with the ME model. However, we also find that for some other set of parameters, the SC-SM model disagrees strongly with the ME model. All these cases illustrate a wide range of variations in the degree of disagreement between the predictions of the SC-CM and the ME approaches., Comment: 38 pages, 16 figures

We describe a subluminal laser which is extremely stable against perturbations. It makes use of a composite gain spectrum consisting of a broad background along with a narrow peak. The stability of the laser, defined as the change in frequency as a function of a change in the cavity length, is enhanced by a factor given by the group index, which can be as high as 105 for experimentally realizable parameters. We also show that the fundamental linewidth of such a laser is expected to be smaller by the same factor. We first present an analysis where the gain profile is modeled as a superposition of two Lorentzian functions. We then present a numerical study based on a physical scheme for realizing the composite gain profile. In this scheme, the broad gain is produced by a high pressure buffer-gas loaded cell of rubidium vapor. The narrow gain is produced by using a Raman pump in a second rubidium vapor cell, where optical pumping is used to produce a Raman population inversion. We show close agreement between the idealized model and the explicit model. A subluminal laser of this type may prove to be useful for many applications., Comment: 37 pages, 17 figures

Previously, we had proposed a gravitational wave detector that incorporates the white-light-cavity (WLC) effect using a compound cavity for signal recycling (CC-SR). Here, we first use an idealized model for the negative dispersion medium (NDM) and use the so-called Caves model for a phase-insensitive linear amplifier to account for the quantum noise (QN) contributed by the NDM, in order to determine the upper bound of the enhancement in the sensitivity-bandwidth product. We calculate the quantum noise limited sensitivity curves for the CC-SR design, and find that the broadening of sensitivity predicted by the classical analysis is also present in these curves, but is somewhat reduced. Furthermore, we find that the curves always stay above the standard quantum limit. To circumvent this limitation, we modify the dispersion to compensate the non-linear phase variation produced by the optomechanical resonance effects. We find that the upper bound of the factor by which the sensitivity-bandwidth product is increased, compared to the highest-sensitivity result predicted by Bunanno and Chen [Phys. Rev. D 64, 042006 (2001)], is ~14. We also present a simpler scheme (WLC-SR) where a dispersion medium is inserted into the SR cavity. For this scheme, we found the upper bound of the enhancement factor to be ~18. We then consider an explicit system for realizing the NDM, which makes use of five energy levels in M-configuration to produce gain, accompanied by electromagnetically induced transparency (the GEIT system). For this explicit system, we employ the rigorous approach based on Master Equation to compute the QN contributed by the NDM, thus enabling us to determine the enhancement in the sensitivity-bandwidth product definitively rather than the upper bound thereof. Specifically, we identify a set of parameters for which the sensitivity-bandwidth product is enhanced by a factor of 17.66., Comment: 35 pages, 19 figures