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arXiv:2505.19979 · Physical Review A

High-entanglement regimes in the Weisskopf-Wigner theory for spontaneous decay

J. C. C. Capella, Alejandro Fonseca, Pablo L. Saldanha, D. Felinto

Published Oct 1, 2025• 0 citations
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Abstract

In this work we review the Weisskopf-Wigner formalism for spontaneous emission considering the spatial modes of light as well as external atomic degrees of freedom, which we introduce in the theory by modeling the atom as a wave packet in momentum space with a given initial uncertainty. We perform a purity calculation in order to quantify the entanglement encoded in the momentum variables of the atom-photon system. Our purity calculations reveal two high-entanglement regimes depending on the initial atomic momentum uncertainty: (1) the Recoil entanglement regime (which arises in the small momentum uncertainty region) where recoil effects dominate the mechanisms that originate entanglement, and (2) the Doppler entanglement regime (in the large momentum uncertainty region) where homogeneous Doppler shifts in the emitted photon's frequency play the fundamental part in the build up of quantum correlations in the system. Physical considerations are made to explain the nature of each entanglement regime as well as provide their respective thresholds.

DOI
10.1103/wmy8-jpvp

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In this work we review the Weisskopf-Wigner formalism for spontaneous emission considering the spatial modes of light as well as external atomic degrees of freedom, which we introduce in the theory by modeling the atom as a wave packet in momentum space with a given initial uncertainty. We perform a purity calculation in order to quantify the entanglement encoded in the momentum variables of the atom-photon system. Our purity calculations reveal two high-entanglement regimes depending on the initial atomic momentum uncertainty: (1) the Recoil entanglement regime (which arises in the small momentum uncertainty region) where recoil effects dominate the mechanisms that originate entanglement, and (2) the Doppler entanglement regime (in the large momentum uncertainty region) where homogeneous Doppler shifts in the emitted photon's frequency play the fundamental part in the build up of quantum correlations in the system. Physical considerations are made to explain the nature of each entanglement regime as well as provide their respective thresholds.
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Spontaneous emission of a photon by an atom is described theoretically in three dimensions with the initial wave function of a finite-mass atom taken in the form of a finite-size wave packet. Recoil and wave-packet spreading are taken into account. The total atom-photon wave function is found in the momentum and coordinate representations as the solution of an initial-value problem. The atom-photon entanglement arising in such a process is shown to be closely related to the structure of atom and photon wave packets which can be measured in the coincidence and single-particle schemes of measurements. Two predicted effects, arising under the conditions of high entanglement, are anomalous narrowing of the coincidence wave packets and, under different conditions, anomalous broadening of the single-particle wave packets. Fundamental symmetry relations between the photon and atom single-particle and coincidence wave-packet widths are established. The relationship with the famous scenario of Einstein-Podolsky-Rosen is discussed.
The extraction of Schmidt modes for continuous-variable systems is considered. An algorithm based on the singular-value decomposition of a matrix is proposed. It is applied to the entanglement in (i) an atom—photon system with spontaneous emission and (ii) a system of biphotons with spontaneous parametric downconversion (SPDC) of type II. For the atom—photon system, the evolution of entangled states is found to be governed by a parameter approximately equal to the fine-structure constant times the atom-to-electron mass ratio. An analysis is made of the dynamics of atom—photon entanglement on the assumption that the system’s evolution is determined by the superposition of an initial and a final state. It is shown that in the course of emission the entanglement entropy first rises on a timescale of order the excited-state lifetime and then falls, approaching asymptotically a residual level due to the initial energy spread of the atomic packet (momentum spread squared). SPDC of type II is analyzed by means of the polarization density matrix and a newly introduced coherence parameter for two spatially separated modes. The loss of intermodal coherence is addressed that results from the difference in behavior between ordinary-and extraordinary-ray photons in a nonlinear crystal. The degree of intermodal coherence is investigated as a function of the product of crystal length and pump bandwidth.
A theoretical study of the phenomenon of spontaneous emission has been carried out, using as a model the Wigner-Weisskopf atom in a one-dimensional radiation field. The calculation is performed within the framework of the Prigogine theory of nonequilibrium statistical mechanics. When the model is solved exactly to first order in the coupling parameter α and the evolution in time of the diagonal elements of the density matrix ρ is studied, it is found that the relaxation to equilibrium is characterized in part by a sequence of slowly damped oscillations. This result seems to be in agreement with the observation made by Zwanzig, namely, that exponential decay in time seems not to be universal, and may, in fact, be hidden behind some other kind of time dependence. An approximate theory is developed alongside the exact one, and corresponding terms in each treatment are compared numerically. It is found that, for small values of the coupling parameter α (α ≤ 0.1) and for sufficiently large values of τ, defined as τ = αEt where E is frequency and t is time, the approximate theory gives a satisfactory representation of the exact solution to first order in α. The importance and relevance of the model introduced by Van Hove and coworkers, in which nonexponential behavior was also observed, will be noted but not stressed in this paper, as this relationship will be developed in considerable detail in a subsequent contribution. Finally, the possible relevance of the theory to a problem of interest in magnetic resonance is mentioned.
Photonically interconnected matter qubit systems have wide-ranging applications across quantum science and technology, with entanglement between distant qubits serving as a universal resource. While state-of-the-art performance in heralded entanglement generation has thus far been achieved in trapped atomic systems modeled as stationary emitters, the improvements to fidelities and generation rates demanded by large-scale applications require consideration of their motional degrees of freedom. Here, we derive the effects of atomic motion on spontaneous emission coupled into arbitrary optical modes and study the implications for commonly used atom-atom entanglement protocols. We arrive at a coherent physical picture in the form of “kick operators” associated with each instant in the photonic wavepackets, which also suggests a method to mitigate motional errors by disentangling qubit and motion postherald. This proposed correction technique removes overheads associated with the thermal motion of atoms and may greatly increase entanglement rates in long-distance quantum network links by allowing single-photon-based protocols to be used in the high-fidelity regime.