# Spontaneous emission

Spontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to the ground state and emits a photon. Spontaneous emission of light or luminescence is a fundamental process that plays an essential role in many phenomena in nature and forms the basis of many applications, such as fluorescent tubes, television screens, plasma display panels, lasers and light emitting diodes.

## Introduction

If a light source ('the atom') is in the excited state with energy $E_{2}$ , it may spontaneously decay to the ground state, with energy $E_{1}$ , releasing the difference in energy between the two states as a photon. The photon will have frequency $\omega$ and energy $\hbar \omega$ :

$E_{2}-E_{1}=\hbar \omega$ ,

where $\hbar$ is Dirac's constant. The phase of the photon in spontaneous emission is random as is the direction the photon propagates in. 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 is given by $N$ , the rate at which $N$ decays is:

${\frac {\partial N}{\partial t}}=-A_{21}N$ ,

where $A_{21}$ is the rate of spontaneous emission. In the rate-equation $A_{21}$ 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 $s^{-1}$ . The above equation can be solved to give:

$N(t)=N(0)e^{-A_{21}t}=N(0)e^{-\Gamma _{rad}t}$ ,

where $N(0)$ is the initial number of light sources in the excited state, $t$ is the time and $\Gamma _{rad}$ is the radiative decay rate of the transition. The number of excited states $N$ 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 (${\frac {1}{e}}$ -time). The radiative decay rate $\Gamma _{rad}$ is inversly proportional to the lifetime $\tau _{12}$ : $A_{21}=\Gamma _{12}={\frac {1}{\tau _{21}}}$ .

## Theory

Quantum mechanics explicitly prohibits spontaneous transitions. That is, using the machinery of ordinary first-quantized quantum mechanics and one computes the probability of spontaneous transitions from one stationary state to another, one finds that it is zero. In order to explain spontaneous transitions, quantum mechanics must be extended to a second-quantized theory, wherein the electromagnetic field is quantized at every point in space. Such a theory is known as a quantum field theory; the quantum field theory of electrons and electromagnetic fields is known as quantum electrodynamics.

In quantum electrodynamics (or QED), the electromagnetic field has a ground state, the vacuum state, which can mix with the excited stationary states of the atom (for more information, see Ref. ). As a result of this interaction, the "stationary state" of the atom is no longer a true eigenstate of the combined system of the atom plus electromagnetic field. In particular, the electron transition from the excited state to the electronic ground state mixes with the transition of the electromagnetic field from the ground state to an excited state, a field state with one photon in it. Spontaneous emission in free space depends upon vacuum fluctuations to get started.

Although there is only one electronic transition from the excited state to ground state, there are many ways in which the electromagnetic field may go from the ground state to a one-photon state. That is, the electromagnetic field has infinitely more degrees of freedom, corresponding to the directions in which the photon can be emitted. Equivalently, one might say that the phase space offered by the electromagnetic field is infinitely larger than that offered by the atom. Since one must consider probabilities that occupy all of phase space equally, the combined system of atom plus electromagnetic field must undergo a transition from electronic excitation to a photonic excitation; the atom must decay by spontaneous emission. The time the light source remains in the excited state thus depends on the light source itself as well as its environment.

## 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 is given by:

$\Gamma _{rad}(\omega )={\frac {\omega ^{3}n|\mu _{12}|^{2}}{3\pi \varepsilon _{0}\hbar {c_{0}}^{3}}}$ where $\omega$ is the emission frequency, $n$ is the index of refraction, $\mu _{12}$ is the transition dipole moment, $\varepsilon _{0}$ is the vacuum permittivity, $\hbar$ is Dirac's constant and $c_{0}$ is the vacuum speed of light. Clearly, the rate of spontaneous emission in free space increases with $\omega ^{3}$ . In contrast with atoms, which have a discrete emission spectrum, quantum dots form an ideal model system to probe the frequency dependence: the emission frequency of quantum dots can be tuned continuously by their size. In fact, it was confirmed that the rate of spontaneous emission of quantum dots follows the $\omega ^{3}$ -frequency dependence as described by Fermi's golden rule.

In the rate-equation above, it is assumed that decay of the number of excited states $N$ 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 $\Gamma _{tot}$ , radiative and nonradiative rates should be summed:

$\Gamma _{tot}=\Gamma _{rad}+\Gamma _{nrad}$ where $\Gamma _{tot}$ is the total decay rate, $\Gamma _{rad}$ is the radiative decay rate and $\Gamma _{nrad}$ the nonradiative decay rate. The quantum efficiency (QE) is defined as the fraction of emission processes in which emission of light is involved:

$QE={\frac {\Gamma _{rad}}{\Gamma _{nrad}+\Gamma _{rad}}}$ In nonradiative relaxation, the energy is released as phonons, more commonly known as heat. Nonradiative relaxation occurs when the energy difference between the levels is very small, and these typically occur on a much faster time scale than radiative transitions. For many materials (for instance, semiconductors), electrons move quickly from a high energy level to a meta-stable level via small nonradiative transitions and then make the final move down to the bottom level via an optical or radiative transition. This final transition is the transition over the bandgap in semiconductors. Large nonradiative transitions do not occur frequently because the crystal structure generally can not support large vibrations without destroying bonds (which generally doesn't happen for relaxation). Meta-stable states form a very important feature that is exploited in the construction of lasers. Specifically, since electrons decay slowly from them, they can be piled up in this state without too much loss and then stimulated emission can be used to boost an optical signal. 