# Purcell effect

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The rate of spontaneous emission depends partly on the environment of a light source. This means that by placing the light source in a special environment, the rate of spontaneous emission can be modified. In the 1940s Edward Mills Purcell discovered the enhancement of spontaneous emission rates of atoms when they are matched in a resonant cavity (the Purcell Effect).[1] The enhancement is given by the Purcell factor[2]

$F_{P}=\frac{3}{4\pi^2}\left(\frac{\lambda_{c}}{n}\right)^3\left(\frac{Q}{V}\right)\,,$

where $(\lambda_{c}/n)$ is the wavelength within the material, and $Q$ and $V$ 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 free space density of states[4]

$\rho_f = \frac{8\pi n^3 \nu^2}{c^3}$

to that of the cavity

$\rho_c = \frac{1}{\Delta \nu V}$

Using

$Q=\nu / \Delta \nu$

we get that

$\rho_c / \rho_f = \frac{c^3}{8\pi n^3 \nu^2}\frac{Q}{\nu V} = \frac{1}{8 \pi} \left ( \frac{\lambda}{n} \right )^3 \left ( \frac{Q}{V} \right )$

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. ^ http://qwiki.stanford.edu/index.php?title=Purcell_Factor
3. ^ S. Haroche, D. Kleppner (1989). "Cavity Quantum Dynamics". Physics Today: 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. ^ V. P. Bykov, Spontaneous emission from a medium with a band spectrum, Soviet Journal of Quantum Electronics 4, 861 (1975).
6. ^ E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Physical Review Letters 58, 2059(1987).
7. ^ A. Kress, F. Hofbauer, N. Reinelt, M. Kaniber, H. J. Krenner, R. Meyer, G. Bohm, and J. J. Finley, Manipulation of the spontaneous emission dynamics of quantum dots in two-dimensional photonic crystals, Physical Review B 71, 241304 (2005).
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. ^ D. Bimberg, E. Stock; A. Lochmann; A. Schliwa, J. A. Tofflinger; W. Unrau. M. C. Münnix, S. Rodt, V. A. Haisler, A. I. Toropov, A. Bakarov, A. K. Kalagin (2009). "Quantum Dots for Single- and Entangled-Photon Emitters". IEEE Photonics Journal 1 (1): 58–68. doi:10.1109/JPHOT.2009.2025329.