the decay process reversible. Cavity quantum electrodynamics is one such system where the vacuum modes are modified resulting in the reversible decay process, see also Quantum revival. The theory of the spontaneous emission under the QED framework was first calculated by Weisskopf and Wigner. Rate of spontaneous emission The rate of spontaneous emission (i.e., the radiative rate) can be described by Fermi's golden rule. The rate of emission depends on two factors: an 'atomic part', which describes the internal structure of the light source and a 'field part', which describes the density of electromagnetic modes of the environment. The atomic part describes the strength of a transition between two states in terms of transition moments. In a homogeneous medium, such as free space, the rate of spontaneous emission in the dipole approximation is given by: where is the emission frequency, is the index of refraction, is the transition dipole moment, is the vacuum permittivity, is the reduced Planck constant, is the vacuum speed of light, and is the fine-structure constant. The expression stands for the definition of the transition dipole moment for dipole moment operator , where is the elementary charge and stands for position operator. (This approximation breaks down in the case of inner shell electrons in high-Z atoms.) The above equation clearly shows that the rate of spontaneous emission in free space increases proportionally to . In contrast with atoms, which have a discrete emission spectrum, quantum dots can be tuned continuously by changing their size. This property has been used to check the -frequency dependence of the spontaneous emission rate as described by Fermi's golden rule. Radiative and nonradiative decay: the quantum efficiency In the rate-equation above, it is assumed that decay of the number of excited states only occurs under emission of light. In this case one speaks of full radiative decay and this means that the quantum efficiency is 100%. Besides radiative decay, which occurs under the emission of light, there is a second decay mechanism; nonradiative decay. To determine the total decay rate , radiative and nonradiative rates should be summed: where is the total decay rate, is the radiative decay rate and the nonradiative decay rate. The quantum efficiency (QE) is defined as the fraction of emission processes in which emission of light is involved: In nonradiative relaxation, the energy is released as phonons, more commonly known as heat. Nonradiative relaxation occurs when