In order to check finite propagation speed Fermi, in 1932, had considered two atoms A and B separated by some distance R. At time t=0, A is in an excited state, B in its ground state, and no photons are present. Fermi's idea was to calculate the excitation probability of B. In a model-independent way and with minimal assumptions - Hilbert space and positive energy only - it is proved, not just for atoms but for any systems A and B, that the excitation probability of B is nonzero immediately after t=0. Possible ways out to avoid a contradiction to finite propagation speed are discussed. The notions of strong and weak Einstein causality are introduced.
A critical analysis of the problem of causality in the energy transfer between a pair of atoms is carried out. One of the pair, the source, is initially excited and the other, the detector, is in its ground state. The probabilities are calculated for the detector to be excited at a later time
t under different experimental conditions. In the first, a precise specification is made of the state of the source and that of the radiation field at the observation time
t. In the second, the final state of the field is unspecified, whereas in the third, the state of neither the field nor the source is specified at time
t. These three cases correspond to predicting the outcomes of different experimental measurements. It is shown that of the three cases, only the third exhibits exact causal behavior. The other two are noncausal. The noncausal contributions to the probability arise from the correlation of vacuum fluctuations at the two atoms at different times in the interval 0 to
t. The probabilities for these cases can be significant for times slightly less than
R/c, where
R is the interatomic separation. Explicit expressions, correct to fourth order in the transition moments, for the probabilities in the three cases are given. A proof of strict causality to all orders is presented for the third case where the final measurement is solely made on the detector. In Appendix A the related problem of the time dependence of the electric energy density associated with the source is analyzed and shown to be causal.
We propose a feasible experimental test of a 1D version of the Fermi problem using superconducting qubits. We give an explicit nonperturbative proof of strict causality in this model, showing that the probability of excitation of a two-level artificial atom with a dipolar coupling to a quantum field is completely independent of the other qubit until signals from it may arrive. We explain why this is in perfect agreement with the existence of nonlocal correlations and previous results which were used to claim apparent causality problems for Fermi's two-atom system.