Spontaneous emission

Spontaneous emission is the process in which a quantum mechanical system (such as an atom, molecule or subatomic particle) transitions from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantised amount of energy in the form of a photon. Spontaneous emission is ultimately responsible for most of the light we see all around us; it is so ubiquitous that there are many names given to what is essentially the same process. If atoms (or molecules) are excited by some means other than heating, the spontaneous emission is called luminescence. For example, fireflies are luminescent. And there are different forms of luminescence depending on how excited atoms are produced (electroluminescence, chemiluminescence etc.). If the excitation is affected by the absorption of radiation the spontaneous emission is called fluorescence. Sometimes molecules have a metastable level and continue to fluoresce long after the exciting radiation is turned off; this is called phosphorescence. Figurines that glow in the dark are phosphorescent. Lasers start via spontaneous emission, then during continuous operation work by stimulated emission. Spontaneous emission cannot be explained by classical electromagnetic theory and is fundamentally a quantum process. The first person to derive the rate of spontaneous emission accurately from first principles was Dirac in his quantum theory of radiation, the precursor to the theory which he later coined quantum electrodynamics. Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field. In 1963 the Jaynes-Cummings model was developed describing the system of a two-level atom interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave the nonintuitive prediction that the rate of spontaneous emission could be controlled depending on the boundary conditions of the surrounding vacuum field. These experiments gave rise to cavity quantum electrodynamics (CQED), the study of effects of mirrors and cavities on radiative corrections. If a light source ('the atom') is in an excited state with energy E 2 {displaystyle E_{2}} , it may spontaneously decay to a lower lying level (e.g., the ground state) with energy E 1 {displaystyle E_{1}} , releasing the difference in energy between the two states as a photon. The photon will have angular frequency ω {displaystyle omega } and an energy ℏ ω {displaystyle hbar omega } : where ℏ {displaystyle hbar } is the reduced Planck constant. Note: ℏ ω = h ν {displaystyle hbar omega =h u } , where h {displaystyle h} is the Planck constant and ν {displaystyle u } is the linear frequency. The phase of the photon in spontaneous emission is random as is the direction in which the photon propagates. This is not true for stimulated emission. An energy level diagram illustrating the process of spontaneous emission is shown below: If the number of light sources in the excited state at time t {displaystyle t} is given by N ( t ) {displaystyle N(t)} , the rate at which N {displaystyle N} decays is: where A 21 {displaystyle A_{21}} is the rate of spontaneous emission. In the rate-equation A 21 {displaystyle A_{21}} is a proportionality constant for this particular transition in this particular light source. The constant is referred to as the Einstein A coefficient, and has units s − 1 {displaystyle s^{-1}} . The above equation can be solved to give: where N ( 0 ) {displaystyle N(0)} is the initial number of light sources in the excited state, t {displaystyle t} is the time and Γ rad {displaystyle Gamma _{ ext{rad}}} is the radiative decay rate of the transition. The number of excited states N {displaystyle N} thus decays exponentially with time, similar to radioactive decay. After one lifetime, the number of excited states decays to 36.8% of its original value ( 1 e {displaystyle {frac {1}{e}}} -time). The radiative decay rate Γ rad {displaystyle Gamma _{ ext{rad}}} is inversely proportional to the lifetime τ 21 {displaystyle au _{21}} :