Quantum theory of superfluorescence based on two-point correlation functions

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.

Figure 1

Sketch of the two levels of interest subject to pumping with rates $r_e,r_g$, nonradiative decay with a rate ${\gamma }_n$, depletion with rates ${\gamma }_g,{\gamma }_e$, decoherence with a rate $q$, and radiative transitions, which are treated self-consistently within our approach—the typical value determining the evolution rate being ${\mathrm{\Gamma }}_{\text{sp}}$.

Figure 2

(a) Simulation of emitted photon number versus system length (red line, correlation function calculation; blue line, Maxwell-Bloch approach); (b) simulation of temporal intensity profiles for systems of different length $z$, calculation according to the correlation function Eqs. (24), (25), and (27), dashed white line shows the position of the intensity maximum; (c) the same figure obtained from the Maxwell-Bloch Eqs. (48, 49, 50, 51) with noise terms Eqs. (52) and (F5), averaged over 100 realizations; (d) section of the same data taken at $z=0$, the correlation function calculation (red curve) coincides with $e^{-{\mathrm{\Gamma }}_{\text{sp}}\tau }$, while the Maxwell-Bloch result averaged over 1000 realizations (blue curve) approaches ${\mathrm{\Gamma }}_{\text{sp}}\tau e^{-{\mathrm{\Gamma }}_{\text{sp}}\tau }e$ (dashed blue line).

Figure 3

Evolution of collective spontaneous emission induced by an XFEL pump in Ne gas. (a) Profile of emitted superfluorescence intensity normalized to 1 for each $z$ value; (b) population inversion; (c) profile of the XFEL pump pulse normalized to maximal value; (d) diagonal part of correlation function of atomic coherences; (e), (f) values of correlation function of atomic coherences taken at time moments 55 and 70 fs.

Figure 4

Evolution of collective spontaneous emission induced by an XFEL pump in Xe gas. (a), (b) Evolution of the excited and ground state populations; (c) temporal evolution of the correlation function of the atomic coherences taken at close spatial points $z_1=z_2=z$; (d) profile of emitted superfluorescence intensity normalized to 1 for each $z$ value; (e) spatial evolution of emitted radiation spectrum, normalized to 1 for each $z$ value; (f) correlation function of the atomic coherences at the time moment τ = 0.5 ns.