Research Express@NCKU - Articles Digest
Research Express@NCKU Volume 12 Issue 1 - December 25, 2009 [ http://research.ncku.edu.tw/re/articles/e/20091225/4.html ]
Spontaneous emission of quantum dot excitons into
surface plasmons in a nanowire
Yueh-Nan Chen
Assistant Professor of Department of Physics, College of Sciences, National Cheng Kung University [email protected]
Optics Letters 33, 2212, Oct. 1 (2008)
T
he issue of spontaneous emission (SP) can be traced back to the early age of constructing Quantum Mechanics. Albert Einstein is the first one who proposed the idea of spontaneous emission. With the perturbation theory and the Markovian assumption, on can easily calculate the SP rate (or lifetime) and frequency shift of a two-level atom. In general, the value of this frequency shift is divergent under perturbation theory and has to be renormalized with a proper way. During the progress of physics in the 20th century, the idea of'renormalization' has great influence on the development of Quantum Electro- Dynamics (QED). One of the most well known characters in this field is Richard Feynman, who got the Nobel Prize in 1965 together with Schwinger
and Tomanaga. The famous book," Surely you're joking, Mr. Feynman", is almost read by every physics student.
Due to the rapid developments of science and technology, people are now able to fabricate the so called 'artificial atoms (quantum dots)' in solid state systems. If a laser pulse is applied to a quantum dot, an exciton (pair of electron and hole) can be created. Similar to that for a real atom, an exciton can recombine and spontaneously emit a photon into free space. In addition to the quantum dots fabricated on a solid-state substrate, experimentalists recently has developed a way to create the so called “colloidal quantum dots" in a chemical solvent. This kind of quantum dots are usually made of the II and VI elements in the periodic table. Due to its characteristic of solvable, the most well known application is that putting the dots inside human bodies and tracing the positions of them by the emitted lights. If the dots are attached to cancer cells, for example, one can know the locations of the cancer cells exactly.
Fig 1: Schematic view of a colloidal quantum dot sitting on top of a silver nanowire. [From A. V.
Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S.
Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, Nature 450, 402 (2007)]
Of course, the important issue for the physicists is the examination of QED with quantum dots. In a recent paper [Nature 450, 402 (2007)], the group in Harvard University has demonstrated the following experiment: putting a colloidal quantum dot on top of a silver nanowire (Fig. 1).
The exciton in the dot can either emit into free space or spontaneously emit into a surface plasmon. If the dot is very close to the wire, the probability of becoming a surface plasmon is greatly enhanced. In fact, the surface plasmons can be viewed as the photons propagating on the metal surface. Increasing attention has been focused on the field of 'plasmonics' due to the following reason: Since the devices can nowadays be shrunk to nano-size, the size of propagating surface plasmons is also reduced actually. In another word, this reduction in principle breaks the diffraction limit in traditional optics.
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Research Express@NCKU - Articles Digest
Inspired by the experiment mentioned above, we started to consider the SP rate of quantum dot excitons into nanowire surface plasmons. To solve this problem, one has to write down the solutions of electric and magnetic fields satisfying Maxwell equations in cylindrical symmetry. With appropriate boundary conditions, one can obtain the dispersion relations of the surface plasmons by solving the following equation:
where a is the wire radius and c is the velocity of surface plasmon. Once we have the dispersion relations, Fermi's Golden rule can be applied to calculate the SP rate:
From Eq. (2), we can know that the decay rate is inversely proportional to the slope of the dispersion relations. We thus plot in Fig. 2 the dispersion relations of surface plasmons in a nanowire. As cane be seen, the dispersion relation for n=0 mode is a monotonic increasing function. For higher modes, however, the dispersion relations show highly non-linear behavior. For example, the dispersion relation of n=1 mode has a local minimum at some point of kz. If one uses Eq. (2) to calculate the decay rate into this mode, one can obtain that the rate goes to infinity at some of ω0 as shown in Fig. 3. The reason that is that, at those points, the slope is zero, such that the vanishing group velocity of the surface plasmons can induce the trapping phenomenon.
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Fig 2: Dispersion relations of nanowire surface plasmons. Fig 3: Spontaneous emission rates of quantum dot excitons into nanowire surface plasmons.
In addition to the trapping state, one can also put another dot close to the wire. Due to the 'one dimensional' transport property of the nanowire surface plasmons, one can easily obtain the occupation probabilities of the two dots:
As can be seen from above equation, there is a 50% chance for the system to evolve into the maximum entangled states, which is a crucial requirement in quantum information science (QIS). In another word, the nanowire surface plasmons not only benefits to quantum optics, but also can have some advantages in QIS. Further investigations in this field certainly will draw great attention in the near future.
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