Collective radiation spectrum for ensembles with Zeeman splitting in single-photon superradiance

Abstract

In an ensemble of identical atoms, cooperative effects like sub-or superradiance may alter the decay rates and the energy of specific transitions may be shifted from the single-atom value by the so-called collective Lamb shift. While such effects in ensembles of two-level systems are by now well understood, realistic multi-level systems are more difficult to handle. In this work we show that in a system of atoms or nuclei under the action of an external magnetic field, the collective contribution to the level shifts can amount to seizable deviations from the single-atom Zeeman or magnetic hyperfine splitting picture. We develop a formalism to describe single-photon superradiance in multi-level systems in the small sample limit and quantify the parameter regime for which the collective Lamb shift leads to measurable deviations in the magnetic-field-induced splitting. In particular, we show that this effect should be observable in the nuclear magnetic hyperfine splitting in Mössbauer nuclei embedded in thin-film x-ray cavities. PACS numbers: 78.70.Ck, 42.50.Nn, 76.80.+y 32.30.Dx In 1947, a famous experiment by Lamb and Retherford [1] confirmed that the 2s 1/2 and 2p 1/2 levels in hydrogen are not degenerate, leading to increased efforts in the theoretical understanding of radiation quantization [2,3] and to the development of quantum electrodynamics (QED) [4]. Today, it is known that this small shift has to do with emission and reabsorption of virtual photons within the atom, mainly with selfenergy and vacuum-polarization corrections. Interestingly, it could be shown that an additional contribution arises if many identical atoms are interacting collectively with resonant photons, and virtual photons are exchanged between different atoms [5][6][7][8][9][10]. This additional contribution has been termed in analogy collective Lamb shift, although the exact underlying processes share with the single-atom Lamb shift mechanism only the virtual character of the photons being exchanged.The collective Lamb shift has been investigated theoretically for ensembles of two-level systems in the small and large sample limits [5,[11][12][13][14][15]. Collective scattering of a weakintensity laser off a cold ensemble of rubidium atoms with Zeeman splitting has been recently investigated both theoretically and experimentally [16][17][18][19], with results that partly contradict predictions of the standard cooperative Lamb shift theory. In the context of ensembles of atoms in magnetic fields, a legitimate question is to which extend can the Zeeman splitting be considered independently from the cooperative effects and in particular the collective Lamb shift. In this Letter, we show that in ensembles of atoms or nuclei with magneticfield induced level multiplets, the collective magnetic splitting can show significant deviations of the Zeeman splitting compared to the single-atom behaviour and cannot be justified by mere two-level system collective Lamb shift contributions for each transition independently, as done for instance in Re...

Cooperative spontaneous emission from indistinguishable atoms in arbitrary motional quantum states

“…They can also be observed in dilute atomic systems for which the near-field contributions of the dipole-dipole interactions are insignificant. Recent studies concern single-photon superradiance [11][12][13][14], subradiance in cold atomic gases [15,16], collective Lambshift [17,18] and localization of light [19,20]. Moreover, super-and subradiance can be explained using a quantum multipath interference approach and can be simulated from the measurement of higher-order-intensitycorrelation functions on atoms separated by a distance larger than the emission wavelength [21,22].…”

Superradiance and anomalous hyperfine splitting in inhomogeneous ensembles

“…[57] and later on discussed in more detail for the general case in Ref. [26]. We note that in particular, Ref.…”

“…We note that in particular, Ref. [26] has addressed the specific case of uniform magnetic hyperfine splitting, using the atomic cloud model of Svidzinsky et al [49] to describe the collective coupling. This method derives an interaction kernel for the effective inter-atomic interactions, which turns out to be identical to the free space Green's function.…”

“…The mechanism behind superradiance and collective emission also applies to multilevel systems, and extensions to the Dicke model for multilevel atoms have been explored [21][22][23][24][25][26][27]. Like the Dicke model, these assume completely uniform illumination of the ensemble of atoms, and that the latter are all completely identical.…”

Superradiance and subradiance in inverted atomic arrays

“…In extended samples larger than one atomic transition wavelength, dipoledipole interactions between different atoms become relevant and the aforementioned symmetry is broken. As a result, the whole Hilbert space with dimension 2 N needs to be considered and most theoretical studies of superradiance and subradiance are consequently restricted to the emission of few photons [12][13][14][15][16][17] or to systems with a small number of atoms [18][19][20] such that numerical Monte-Carlo type methods can be applied.…”