# Purcell effect

The Purcell effect is the enhancement of a quantum system's spontaneous emission rate by its environment. In the 1940s Edward Mills Purcell discovered the enhancement of spontaneous emission rates of atoms when they are incorporated into a resonant cavity (the Purcell Effect).[1] The magnitude of the enhancement is given by the Purcell factor[2]

${\displaystyle F_{P}={\frac {3}{4\pi ^{2}}}\left({\frac {\lambda _{free}}{n}}\right)^{3}\left({\frac {Q}{V}}\right)\,,}$

where ${\displaystyle (\lambda _{free}/n)}$ is the wavelength within the cavity material of refractive index ${\displaystyle n}$, and ${\displaystyle Q}$ and ${\displaystyle V}$ are the quality factor and mode volume of the cavity, respectively.

One way of seeing why this enhancement arises is by using cavity quantum electrodynamics.[3] Fermi's golden rule dictates that the transition rate for the atom-vacuum (or atom-cavity) system is proportional to the density of final states. In a cavity at resonance, the density of final states is enhanced (though the number of final states may not be). The Purcell factor is then just the ratio of the cavity

${\displaystyle \rho _{c}={\frac {1}{\Delta \nu V}}}$

to that of the free space density of states[4]

${\displaystyle \rho _{f}={\frac {8\pi n^{3}\nu ^{2}}{c^{3}}}}$

Using

${\displaystyle Q=\nu /\Delta \nu }$

we get that

${\displaystyle \rho _{c}/\rho _{f}={\frac {c^{3}}{8\pi n^{3}\nu ^{2}}}{\frac {Q}{\nu V}}={\frac {1}{8\pi }}\left({\frac {\lambda _{free}}{n}}\right)^{3}\left({\frac {Q}{V}}\right)}$

which is correct up to a constant.

It has been predicted theoretically [5][6] that a 'photonic' material environment can control the rate of radiative recombination of an embedded light source. A main research goal is the achievement of a material with a complete photonic bandgap: a range of frequencies in which no electromagnetic modes exist and all propagation directions are forbidden. At the frequencies of the photonic bandgap, spontaneous emission of light is completely inhibited. Fabrication of a material with a complete photonic bandgap is a huge scientific challenge. For this reason photonic materials are being extensively studied. Many different kinds of systems in which the rate of spontaneous emission is modified by the environment are reported, including cavities, two,[7][8] and three-dimensional[9] photonic bandgap materials.

The Purcell effect can also be useful for modeling single-photon sources for quantum cryptography.[10] Controlling the rate of spontaneous emission and thus raising the photon generation efficiency is a key requirement for quantum dot based single-photon sources.[11]

## References

1. ^ E. M. Purcell, Phys. Rev. 69, 681 (1946). DOI web link
2. ^ "Archived copy". Archived from the original on 2011-07-17. Retrieved 2010-09-21.CS1 maint: Archived copy as title (link)
3. ^ S. Haroche; D. Kleppner (1989). "Cavity Quantum Dynamics". Physics Today. 42 (1): 24–30. Bibcode:1989PhT....42a..24H. doi:10.1063/1.881201.
4. ^ D. Kleppner (1981). "Inhibited Spontaneous Emission". Physical Review Letters. 47 (4): 233–236. Bibcode:1981PhRvL..47..233K. doi:10.1103/PhysRevLett.47.233.
5. ^ Bykov, Vladimir P (1975). "Spontaneous emission from a medium with a band spectrum". Soviet Journal of Quantum Electronics. 4 (7): 861–871. Bibcode:1975QuEle...4..861B. doi:10.1070/QE1975v004n07ABEH009654. ISSN 0049-1748.
6. ^ Yablonovitch, Eli (1987). "Inhibited Spontaneous Emission in Solid-State Physics and Electronics". Physical Review Letters. 58 (20): 2059–2062. Bibcode:1987PhRvL..58.2059Y. doi:10.1103/PhysRevLett.58.2059. ISSN 0031-9007.
7. ^ Kress, A.; Hofbauer, F.; Reinelt, N.; Kaniber, M.; Krenner, H. J.; Meyer, R.; Böhm, G.; Finley, J. J. (2005). "Manipulation of the spontaneous emission dynamics of quantum dots in two-dimensional photonic crystals". Physical Review B. 71 (24): 241304. arXiv:quant-ph/0501013. Bibcode:2005PhRvB..71x1304K. doi:10.1103/PhysRevB.71.241304. ISSN 1098-0121.
8. ^ D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, J. Vuckovic, Controlling the Spontaneous Emission Rate of Single Quantum Dots in a 2D Photonic Crystal, Physical Review Letters 95 013904 (2005)
9. ^ P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh andW. L. Vos, Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals, Nature, 430, 654 (2004).http://cops.tnw.utwente.nl/pdf/04/nature02772.pdf
10. ^ M. C. Münnix; A. Lochmann; D. Bimberg; V. A. Haisler (2009). "Modeling Highly Efficient RCLED-Type Quantum-Dot-Based Single Photon Emitters". IEEE Journal of Quantum Electronics. 45 (9): 1084–1088. Bibcode:2009IJQE...45.1084M. doi:10.1109/JQE.2009.2020995.
11. ^ Bimberg, D.; Stock, E.; Lochmann, A.; Schliwa, A.; Tofflinger, J.A.; Unrau, W.; Munnix, M.; Rodt, S.; Haisler, V.A.; Toropov, A.I.; Bakarov, A.; Kalagin, A.K. (2009). "Quantum Dots for Single- and Entangled-Photon Emitters". IEEE Photonics Journal. 1 (1): 58–68. doi:10.1109/JPHOT.2009.2025329. ISSN 1943-0655.