Gravitational decoherence effects on spontaneous emission of atomic ensembles in timed Dicke state V. Stefanov, 1, 2 I. Siutsou, 2 and D. Mogilevtsev 1 1 Institute of Physics, Belarus National Academy of Sciences, Nezavisimosti Ave. 68, Minsk 220072 Belarus 2 ICRANet-Minsk, Institute of Physics, Belarus National Academy of Sciences, Nezavisimosti Ave. 68, Minsk 220072 Belarus Here we discuss an effect of gravitational decoherence due to time dilation on the collective radiation dynamics of atomic system in timed single-photon Dicke states. We show that a photon absorbed by the stationary system of randomly placed stationary atoms is no more spontaneously emitted in the direction of the impinging photon. Time-dilation effect leads to broadening of the angular distribution of the emitted photon. Even for the spherically symmetric gravitational field, the broadening has specific non-symmetrical character. PACS numbers: 03.65.Yz, 04.62.+ I. INTRODUCTION Nowadays, interplay of quantumness and gravity at- tracts a lot of attention [1]. Recently it was recognized that even weak gravitation can have rather noticeable effects on quantum interference and on quantum correla- tions, especially on those of large systems [2–22]. These effects and their description are already well beyond the region of purely theoretical speculations. For example, gravimeters based on gravitationally induced atom inter- ference are very promising in stability and accuracy and are under active development now [23–25]. It is already established that an influence of gravita- tional field on quantum interference might be quite de- structive. Even in the linear approximation allowing to quantize in a standard way the gravitational field, quan- tum fluctuations of the gravity unavoidably lead to ap- pearance of decoherence [3–6]. Another kind of deco- herence arising due to interaction of the gravitational field with a quantum particle can be captured even when considering classical gravitational field. This is so-called ”time-dilation decoherence” [7–12]. The essence of this effect can be described with the simple interferometric example [11, 20]. If two interfering particles are moving through different arms of the interferometer, and these arms are subjected to different gravitational field inter- acting with inner degrees of freedom of the particles, the visibility of the interference measured at the output of the interferometer would be lessened. This observable effect does not depend of the frame, and can be seen even with photons. For an entangled state of a large number of particles, this decoherence can be noticeable even near the Earth surface [9, 10]. Here we discuss another effects stemming from interac- tion of a set of entangled quantum systems with classical gravity. We consider how the gravity affects creation and spontaneous decay of single timed Dicke state of an ex- tended atomic system. Emission of a photon previously absorbed by a set of randomly placed two-level atoms demonstrates quite curious and counterintuitive effect: strictly directional emission. If a photon is absorbed by sufficiently large system of identical non-interacting two-level atoms, a collective entangled state (so called ”timed Dicke state”) is formed. Then, this state leads to the spontaneous emission of the photons precisely in the direction of the photon that was absorbed [26, 27]. No- ticeably, the effect does not depend on positions of each particular atom. The prerequisite is to have the sum of phase factors stemming from different atomic positions tending to the delta-function (which is provided by ran- dom placement of sufficiently large number of atoms). In this work we show how this effect of directional emis- sion is broken by gravity. Curiously, it appears that basic features of the effect are retained in the presence of grav- ity: a pure entangled ”gravity-affected timed Dicke state” is formed after photon absorption, emitted field does not depend on the size of the atomic system. However, differ- ent time-dilation in different parts of the system leads to the broadening of the angular distribution of the emitted photon. This broadening is asymmetric with respect to the direction of the gravity gradient. The outline of this paper is as follows. In Section II we describe the process of timed Dicke states formation and spontaneous directional emission in flat time-space. Then, in Section III we consider gravitational field pro- duced by spherically symmetric mass distribution imple- menting Schwarzschild metrics and introduce corrections to electromagnetic field eigenmodes produced by grav- ity. In Section IV we derive the effective Hamiltonian describing atom-field interaction leading to the sponta- neous emission and consider the emission in the Marko- vian approximation. Finally, in Section V we analyse gravitational corrections to the directional emission ef- fect and derive the angular distribution of the emitted single-photon field. II. TIMED DICKE STATE IN FLAT TIME-SPACE We start considering the system of N stationary iden- tical two-level atoms (TLA) interacting in the dipole- dipole and rotating-wave approximations with the modes of the electromagnetic field in homogeneous vacuum [26]. arXiv:1905.12301v4 [quant-ph] 28 Oct 2019