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We observe 30% two-photon optical amplification of a probe laser-field propagating through a laser-pumped potassium vapor. This amplification is spectrally isolated and substantially larger than that of previously reported continuous-wave two-photon amplifiers. The combination of large amplification and spectral isolation of the two-photon gain feature will greatly facilitate precise studies of the photon statistics of this highly nonlinear quantum amplifier and the development and characterization of a two-photon laser based on this gain medium. We also observe spectrally-distinct three-photon amplification (ϳ5%͒ in the same system under different experimental conditions. We present a simple model of the interaction that gives qualitative agreement with our observations and explains the dependence of the two-photon gain on the various system parameters. This model predicts that the size of the two-photon gain is quite sensitive to an interference between two different quantum pathways.

We present a rate-equation model for two-photon lasers that, despite its simplicity, captures the essential physics of their behavior and affords an intuitive understanding of their novel threshold and stability behavior. We use the model to investigate the steady-state behavior of the laser, explore the stability of the steady-state solutions, and predict the injected pulse strength needed to initiate lasing.Modeling the behavior of two-photon lasers 1 is challenging because such lasers are based on the twophoton stimulated-emission (SE) process and hence operate in a highly nonlinear manner under all conditions. Recall that in the two-photon SE process two incident photons stimulate an inverted atom to a lower energy state, and four photons are scattered coherently by the atom. We have developed a model for two-photon lasers based on a set of self-consistent rate equations that predict many of their crucial attributes without being overly complex. In this Letter we present our rate-equation model and use it to make predictions of the behavior of two-photon optical lasers. Although more comprehensive treatments that include interactions ignored in our rateequation model, such as coherent effects, 3 4 singlephoton processes, 5 and dynamical Stark shifts, 6 may be needed to make quantitative comparisons with experimental results, we feel that our model is useful for developing an intuitive understanding of twophoton lasers.A key test for any model of two photon lasers is whether it predicts the novel threshold behavior of the laser that we briefly describe below. The threshold condition for all lasers is that the round-trip gain must equal the round-trip loss. For one-photon lasers this criterion yields the well-known result that lasing will commence when a uniquely defined minimum inversion (proportional to the gain) is attained by means of sufficient pumping. The situation is more complicated for the two-photon laser because the gain increases with increasing inversion AN and with increasing cavity photon number q (until the atoms are saturated); thus the threshold condition must be specified by two parameters.We define a threshold inversion AN th as the inversion needed to satisfy the threshold condition with cavity photon number qsat just sufficient to saturate the two-photon gain. When AN > ANth there is a corresponding cavity photon number (which is less than qsat) that must be present in the cavity before the laser can be turned on. Hence, if the laser is initially off, it cannot be turned on unless some perturbation, such as an externally injected field, brings it above the nec-Our rate-equation model of the two-photon laserfollows from the standard model of one-photon lasers with the exception that the one-photon SE rate (1) is replaced by the two-photon SE rate"where BM') [B (2)] is the one-(two-) photon rate coefficient. For simplicity we have assumed that the twophoton laser operates in the degenerate mode, that the laser oscillates in a single plane-wave mode, and that the cavity (population) decay r...

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Gain arising from the interaction of intense laser fields and atoms has attracted recent attention because it has been found, for example, that two-photon lasing1 and lasing without inversion2 occurs in these systems. Physical and intuitive models of these interactions can be developed by using the dressed-atom states, which, unfortunately, tend to be difficult to detect directly with high precision. We will report on a new modulation spectroscopy for the quantum limited detection of gain in dressed-atom systems. The spectroscopy involves weak modulation of the dressed-state splitting and lockin detection of the intensity of the probing beam. It differs from classic frequency-modulation spectroscopies in that the probe beam is not modulated before it interacts with the sample of strongly driven atoms and hence does not suffer from the effects of residual amplitude modulation.