Abstract

Cooling a sample of atoms into their center—of—mass ground state leads to systems which can be described with the help of so called macroscopic wave-functions. These samples then provide a clean and highly controllable environment to perform tests on fundamental questions arising from the basis of quantum mechanics. We consider a cold thermal cloud of harmonically trapped, two-level, fermionic atoms, in the internal ground state, whose number and motional states are described by a grand canonical ensemble. We then add one internally excited atom, whose motional state is also thermal and at the same temperature. Since fermions cannot perform s-wave scattering, we neglect inter-atomic interactions. Also we will scale all energies and lengths in the usual trap units ћ ω 0 and 2Mω0/h−−−−−−√ ;we allow an anisotropic trap by defining the vector frequency = ω 0 . The initial probability for a motional state |n⃗ ⟩ to be occupied by a ground state atom is given by the Fermi-Dirac distribution Fn⃗ =+1)−1 and the chemical potential μ is fixed by demanding that the mean number of particles be N=/6. The probability that the excited atom is initially in the motional state |m⃗ ⟩ is Pm⃗ =P0e−βv⃗ .m⃗ The initial rate of decay of the excited atom by spontaneous emission of a photon with wave vector k⃗ is given by Fermi’s Golden Rule. In the dipole approximation for the atom-photon interaction, this yields 1Γ=Γ∑m⃗ ,n⃗ =0∞pm⃗ N|⟨n⃗ |e−ik⃗ .r⃗ |m⃗ ⟩2where Γ0=λω30d2/ is the decay rate for an isolated atom, and d is the dipole moment for the e ↔ g transition. N is the dipole pattern of the transition . For an isotropic trap the emission rate becomes independent of the photon direction and Eqn. reduces to 2Mf=∑n,m=0∞Pm∑t=0min[n,m]|⟨n−l|eikx|−l⟩|2