Spontaneous emission

Introduction

If a light source ('the atom') is in an excited state with energy E_2, it may spontaneously decay to a lower lying level (e.g., the ground state) with energy E_1, releasing the difference in energy between the two states as a photon. The photon will have angular frequency \omega and an energy \hbar \omega:

:E_2 - E_1 = \hbar \omega,

where \hbar is the reduced Planck constant. Note: \hbar \omega = h\nu, where h is the Planck constant and \nu 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 is given by N(t), the rate at which N decays is:

:\frac{\partial N(t)}{\partial t} = -A_{21} N(t),

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.R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University Press Inc., New York, 2001). The above equation can be solved to give:

:N(t) =N(0) e^{ - A_{21}t }= N(0) e^{ - \Gamma_{!\text{rad}}t },

where N(0) is the initial number of light sources in the excited state, t is the time and \Gamma_{!\text{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_{\text{rad}} is inversely proportional to the lifetime \tau_{21}:

:A_{21}=\Gamma_{21}=\frac{1}{\tau_{21}}.

Theory

Spontaneous transitions were not explainable within the framework of the Schrödinger equation, in which the electronic energy levels were quantized, but the electromagnetic field was not. Given that the eigenstates of an atom are properly diagonalized, the overlap of the wavefunctions between the excited state and the ground state of the atom is zero. Thus, in the absence of a quantized electromagnetic field, the excited state atom cannot decay to the ground state. In order to explain spontaneous transitions, quantum mechanics must be extended to a quantum field theory, wherein the electromagnetic field is quantized at every point in space. 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 QED vacuum, which can mix with the excited stationary states of the atom. 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. |name-list-style=amp |title=Spontaneous emission and laser oscillation in microcavities |name-list-style=amp |title=Quantum optics

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 different 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. This infinite degree of freedom for the emission of the photon results in the apparent irreversible decay, i.e., spontaneous emission.

In the presence of electromagnetic vacuum modes, the combined atom-vacuum system is explained by the superposition of the wavefunctions of the excited state atom with no photon and the ground state atom with a single emitted photon:

: |\psi(t)\rangle = a(t)e^{-i\omega_0 t}|e;0\rangle + \sum_{k,s} b_{ks}(t)e^{-i\omega_k t}|g;1_{ks}\rangle

where |e;0\rangle and a(t) are the atomic excited state-electromagnetic vacuum wavefunction and its probability amplitude, |g;1_{ks}\rangle and b_{ks}(t) are the ground state atom with a single photon (of mode ks ) wavefunction and its probability amplitude, \omega_0 is the atomic transition frequency, and \omega_k = c|k| is the frequency of the photon. The sum is over k and s , which are the wavenumber and polarization of the emitted photon, respectively. As mentioned above, the emitted photon has a chance to be emitted with different wavenumbers and polarizations, and the resulting wavefunction is a superposition of these possibilities. To calculate the probability of the atom at the ground state ( |b(t)|^2), one needs to solve the time evolution of the wavefunction with an appropriate Hamiltonian. To solve for the transition amplitude, one needs to average over (integrate over) all the vacuum modes, since one must consider the probabilities that the emitted photon occupies various parts of phase space equally. The "spontaneously" emitted photon has infinite different modes to propagate into, thus the probability of the atom re-absorbing the photon and returning to the original state is negligible, making the atomic decay practically irreversible. Such irreversible time evolution of the atom-vacuum system is responsible for the apparent spontaneous decay of an excited atom. If one were to keep track of all the vacuum modes, the combined atom-vacuum system would undergo unitary time evolution, making 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 Victor Weisskopf and Eugene Wigner in 1930 in a landmark paper. The Weisskopf-Wigner calculation remains the standard approach to spontaneous radiation emission in atomic and molecular physics. Dirac had also developed the same calculation a couple of years prior to the paper by Wigner and Weisskopf.

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:

: \Gamma_{\text{rad}}(\omega)= \frac{\omega^3n|\mu_{12}|^2} {3\pi\varepsilon_{0}\hbar c^3} = \frac{4 \alpha \omega^3n| \langle 1|\mathbf{r}|2\rangle |^2} {3 c^2}

: \frac{|\mu_{12}|^2} {\pi\varepsilon_{0}\hbar c} = 4 \alpha | \langle 1|\mathbf{r}|2\rangle |^2

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 the reduced Planck constant, c is the vacuum speed of light, and \alpha is the fine-structure constant. The expression |\langle 1|\mathbf{r}|2\rangle| stands for the definition of the transition dipole moment |\mu_{12}|=|\langle 1|\mathbf{d}|2\rangle| for dipole moment operator \mathbf{d}=q\mathbf{r}, where q is the elementary charge and \mathbf{r} 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 \omega^3.

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 \omega^3-frequency dependence of the spontaneous emission rate as described by Fermi's golden rule.A. F. van Driel, G. Allan, C. Delerue, P. Lodahl, W. L. Vos and D. Vanmaekelbergh, Frequency-dependent spontaneous emission rate from CdSe and CdTe nanocrystals: Influence of dark states, Physical Review Letters, 95, 236804 (2005).Phys. Rev. Lett. 95, 236804 (2005) - Frequency-Dependent Spontaneous Emission Rate from CdSe and CdTe Nanocrystals: Influence of Dark States (aps.org)

Radiative and nonradiative decay: the quantum efficiency

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_{\text{tot}}, radiative and nonradiative rates should be summed:

:\Gamma_{\text{tot}}=\Gamma_{\text{rad}} + \Gamma_{\text{nrad}}

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

: \text{QE}=\frac{\Gamma_{\text{rad}}}{\Gamma_{\text{nrad}} + \Gamma_{\text{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 cannot 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 deliberately piled up in this state without too much loss and then stimulated emission can be used to boost an optical signal.

Radiative cascade

If emission leaves a system in an excited state, additional transitions can occur, leading to atomic radiative cascade. For example, if calcium atoms in a low-pressure atomic beam are excited by ultraviolet light from the 41S0 ground state to the 61P1 state, they can decay in three steps, first to 61S0 then to 41P1 and finally to the ground state. The photons from the second and third transitions have correlated polarizations demonstrating quantum entanglement. These correlations were used by John Clauser{{rp|880|q=A proposed extension of the experiment of Kocher and Commins, on the polarization correlation of a pair of optical photons, will provide a decisive test between quantum mechanics and local hidden-variable theories.}} and Alain Aspect{{cite journal | doi-access = free }} in work that contributed to their 2022 Nobel prize in physics.

References

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