Editing Spontaneous emission (section)

==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.<ref name=Milonni /> 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 [[quantum fluctuation|vacuum fluctuation]]s to get started.<ref name=Yokoyama,>
{{cite book 
|author1=Hiroyuki Yokoyama  |author2=Ujihara K
 |name-list-style=amp |title=Spontaneous emission and laser oscillation in microcavities
|publisher= CRC Press
|location=Boca Raton
|page=6
|year=1995
|isbn=0-8493-3786-0
|url=https://books.google.com/books?id=J_0ZAwf6AQ0C&q=%22spontaneous+emission%22}}
</ref><ref name=Scully1>
{{cite book 
|author1=Marian O Scully  |author2=M. Suhail Zubairy
 |name-list-style=amp |title=Quantum optics
|publisher= Cambridge University Press
|location=Cambridge UK
|page=§1.5.2 pp. 22–23
|year=1997
|isbn=0-521-43595-1
|url=https://books.google.com/books?id=20ISsQCKKmQC&dq=atom+transition+photon&pg=PA430}}
</ref>

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:

:<math> |\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 </math>

where <math> |e;0\rangle </math> and <math> a(t) </math> are the atomic excited state-electromagnetic vacuum wavefunction and its probability amplitude, <math> |g;1_{ks}\rangle </math> and <math> b_{ks}(t)</math> are the ground state atom with a single photon (of mode <math> ks </math>) wavefunction and its probability amplitude, <math>\omega_0</math> is the atomic transition frequency, and <math>\omega_k = c|k|</math> is the frequency of the photon.  The sum is over <math> k </math> and <math> s </math>, 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 (<math> |b(t)|^2</math>), one needs to solve the time evolution of the wavefunction with an appropriate Hamiltonian.<ref name=Dirac>{{cite journal|last1=Dirac|first1=Paul Adrien Maurice|title=The Quantum Theory of the Emission and Absorption of Radiation|journal=Proc. R. Soc.|date=1927|volume=A114|issue=767|pages=243–265|doi=10.1098/rspa.1927.0039|bibcode=1927RSPSA.114..243D|doi-access=free}}</ref> 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.<ref>{{Cite journal |last1=Weisskopf |first1=V. |last2=Wigner |first2=E. |date=1930-01-01 |title=Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie |url=https://doi.org/10.1007/BF01336768 |journal=Zeitschrift für Physik |language=de |volume=63 |issue=1 |pages=54–73 |doi=10.1007/BF01336768 |bibcode=1930ZPhy...63...54W |issn=0044-3328}}</ref><ref>{{Citation |last1=Berman |first1=Paul R. |title=Chapter 5 - Spontaneous Decay, Unitarity, and the Weisskopf–Wigner Approximation |date=2010-01-01 |volume=59 |pages=175–221 |editor-last=Arimondo |editor-first=E. |url=https://www.sciencedirect.com/science/article/pii/S1049250X10590050 |access-date=2024-06-21 |series=Advances in Atomic, Molecular, and Optical Physics |publisher=Academic Press |last2=Ford |first2=George W. |chapter=Spontaneous Decay, Unitarity, and the Weisskopf–Wigner Approximation |doi=10.1016/S1049-250X(10)59005-0 |isbn=978-0-12-381021-2 |editor2-last=Berman |editor2-first=P. R. |editor3-last=Lin |editor3-first=C. C.}}</ref><ref>{{Cite journal |last1=Sharafiev |first1=Aleksei |last2=Juan |first2=Mathieu L. |last3=Gargiulo |first3=Oscar |last4=Zanner |first4=Maximilian |last5=Wögerer |first5=Stephanie |last6=García-Ripoll |first6=Juan José |last7=Kirchmair |first7=Gerhard |date=2021-06-10 |title=Visualizing the emission of a single photon with frequency and time resolved spectroscopy |url=https://quantum-journal.org/papers/q-2021-06-10-474/ |journal=Quantum |language=en-GB |volume=5 |pages=474 |doi=10.22331/q-2021-06-10-474|arxiv=2001.09737 |bibcode=2021Quant...5..474S }}</ref> The Weisskopf-Wigner calculation remains the standard approach to spontaneous radiation emission in atomic and molecular physics.<ref>{{Cite journal |last1=Stenholm |first1=Stig Torsten |last2=Suominen |first2=Kalle-Antti |date=1998-04-27 |title=Weisskopf-Wigner decay of excited oscillator states |url=https://opg.optica.org/oe/abstract.cfm?uri=oe-2-9-378 |journal=Optics Express |language=en |volume=2 |issue=9 |pages=378–390 |doi=10.1364/OE.2.000378 |pmid=19381205 |bibcode=1998OExpr...2..378S |issn=1094-4087|doi-access=free }}</ref> Dirac had also developed the same calculation a couple of years prior to the paper by Wigner and Weisskopf.<ref>{{Cite journal |last=Gottfried |first=Kurt |date=2011-03-01 |title=P. A. M. Dirac and the discovery of quantum mechanics |url=https://pubs.aip.org/ajp/article/79/3/261/398648/P-A-M-Dirac-and-the-discovery-of-quantum-mechanics |journal=American Journal of Physics |language=en |volume=79 |issue=3 |pages=261–266 |doi=10.1119/1.3536639 |issn=0002-9505|arxiv=1006.4610 |bibcode=2011AmJPh..79..261G }}</ref>