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We have directly determined the spectral shape of the complex conductivities of Bloch oscillating electrons by using the time-domain terahertz (THz) electro-optic sampling technique, and presented experimental evidence for a dispersive Bloch gain in superlattices. This unique dispersive gain without population inversion arises from a nonclassical nature of Bloch oscillations; that is, the phase of the Bloch oscillation is shifted by pi/2 from that of the semiclassical charged harmonic oscillation when driven by the same ac field. By increasing the bias electric field, the gain bandwidth reached approximately 3 THz in our particular sample.

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Esaki and Tsu's superlattice 1 , made by alternating two different semiconductor materials, was the first one-dimensional artificial crystal that demonstrated the ability to tailor semiconductor properties. One motivation of this work was the realization of the Bloch oscillator 2,3 and the use of its particular dispersive optical gain 4,5 to achieve a tuneable source of electromagnetic radiation. However, these superlattices were electrically unstable in the steady state 6 . Fortunately, because it is based on scatteringassisted transitions, this particular gain does not arise only in superlattices, but also more generally in semiconductor heterostructures 7,8 such as quantum cascade lasers 9 (QCLs), where the electrical stability can be controlled 10 . Here, we show the unambiguous spectral signature of Bloch gain in a special QCL designed to enhance the latter by exhibiting laser action in the condition of weak to vanishing population inversion.In solids, electrons have a fixed relation between momentum and energy: they move along energy bands, as known from condensed-matter theory. When an electric field is applied they are accelerated but the lattice forces a periodic motion at a definite Bloch frequency. This phenomenon is known as Bloch oscillations, and the idea was successfully used by Zener to explain the dielectric breakdown 3 . However, in usual solids the strong scattering due to impurities and carrier-carrier interaction prevents the observation of such oscillations, as the lattice constant is too short to allow the electrons to complete even one oscillation cycle. In superlattices, the lattice constant can be chosen and a subtle engineering may allow electrons to achieve a few oscillations before scattering. As this phenomenon is fascinating from a condensed-matter point of view, it also opens new perspectives for optics because charge oscillations naturally couple to radiation and offer a way to emit coherent radiation.Therefore, the important question is whether these oscillations can be self-sustained and provide optical gain. First, Ktitorov 4 and then Ignatov and Romanov 5 addressed the problem theoretically with Boltzmann equations and succeeded in providing a definitive signature for Bloch oscillations in superlattices in terms of a particular spectral response: the Bloch oscillations are found to amplify the electromagnetic field (optical gain) on the low-energy side of the oscillation frequency, whereas they absorb photons on the high-energy side. This particular shaped gain-Bloch gain-is the main feature of the Bloch oscillator. A series of experiments 11-13 using pulsed ultrafast techniques have successfully shown the existence of Bloch oscillations as electrons are pumped in a higher energy band and collectively oscillate over their dephasing time. However, the Bloch gain extends to zero frequencies and the structure becomes unstable in the steady state, so far preventing the observation of net gain in superlattices, although some evidence in photocurrent 14 and more recently in absorpt...