We present a comprehensive experimental and theoretical study on superfluorescence in the extreme ultraviolet wavelength regime. Focusing a high-intensity free-electron laser pulse in a cell filled with Xe or Kr gas, the medium is quasi instantaneously population-inverted by inner-shell ionization on the giant resonance followed by Auger decay. On the timescale of 100 ps a macroscopic polarization builds up in the medium, resulting in superfluorescent emission of several Xe and Kr lines in the forward direction. As the number of emitters in the system is increased by either raising the pressure or the pump-pulse energy, the emission shows an exponential growth of over 4 orders of magnitude and reaches saturation. With increasing yield, we observe line broadening, a manifestation of superfluorescence in the spectral domain. Our novel theoretical approach, based on a full quantum treatment of the atomic system and the irradiated field, shows quantitative agreement with the experiment and supports our interpretation.Superfluorescence [1] is the spontaneous, collective decay of an extended ensemble of atoms that have been prepared in a population-inverted state, resulting in collimated, highintensity radiation pulses. The pulses are emitted at a certain delay following excitation and have a duration that can be several orders of magnitude smaller than the typical upper-state lifetimes. Long before the advent of short-wavelength free-electron lasers (FELs), strong superfluorescence in optically thick media was proposed as a source of highly intense and pulsed extreme-ultraviolet (XUV) or X-ray radiation [2]. Strong X-ray K-α superfluorescence following ionization of the 1s shell with a focused X-ray FEL (XFEL) beam was demonstrated in neon gas [3,4], solid copper [5] and manganese salts in aqueous solution [6]. Extremely high gains were observed in these experiments [3,6], with exponential amplification factors surpassing 20 compared to spontaneous emission. In the vacuum ultraviolet and XUV regions, superfluorescence following inner-shell ionization has so far not been demonstrated.The difficulty to obtain transient gain in this wavelength regime is a consequence of the very different time scales of two competing processes: on the one hand, short (fs) Auger lifetimes of inner-valence vacancies, and, on the other hand, comparatively long (ns) radiative transition times -a highly unfavorable combination to sustain a sizable population inversion and gain. Here, we present combined experimental and theoretical work, giving strong evidence for XUV superfluorescence of Xe and Kr gases. Population inversion is

Irradiation of a medium by short intense pulses from x-ray / XUV free electron lasers can result in saturated photoionization of inner electronic shells. As a result an inversion of populations between core levels appears. The resulting fluorescent radiation can be amplified during its propagation through the inverted medium and results in intense, quasi transform-limited radiation bursts.While the optical counterpart of this phenomena, known as superfluorescence, was intensively investigated, a generalized treatment is needed in the x-ray / XUV domain, where the dynamics of pumping and evolution due to fast decay processes play a crucial role. To provide a general theoretical approach, we start from the fundamental, quantized minimal coupling Hamiltonian of light-matter interaction and after a series of approximations arrive at a closed system of equations for the two-point correlation function of atomic coherences and the two-time correlation function of the emitted field. The obtained formalism enables us to investigate collective spontaneous emission in various regimes. It is extended consistently to include incoherent processes that are relevant in the x-ray / XUV domain. These processes are introduced into the formalism by corresponding Lindblad superoperators. The connection to other approaches is discussed and numerical examples related to recent experiments are presented.

Oscillators are at the heart of optical lasers, providing stable, transform-limited pulses. Until now, laser oscillators have been available only in the infrared to visible and near-ultraviolet (UV) spectral region. In this paper, we present a study of an oscillator operating in the 5- to 12-keV photon-energy range. We show that, using theKα1line of transition metal compounds as the gain medium, an X-ray free-electron laser as a periodic pump, and a Bragg crystal optical cavity, it is possible to build X-ray oscillators producing intense, fully coherent, transform-limited pulses. As an example, we consider the case of a copper nitrate gain medium generating ∼ 5 ×1010photons per pulse with 37-fs pulse length and 48-meV spectral resolution at 8-keV photon energy. Our theoretical study and simulation of this system show that, because of the very large gain per pass, the oscillator saturates and reaches full coherence in four to six optical-cavity transits. We discuss the feasibility and design of the X-ray optical cavity and other parts of the oscillator needed for its realization, opening the way to extend X-ray–based research beyond current capabilities.

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