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Physics Leiters A 165 (1992) 148-152North-Holland

PHYSICS LETTERS A

Nonlinear polariton excitations in quantum dot arrays

Pawel Hawrylaklns titu te for Microstructural Sciences . National Research Council of Canada. Ot tawa. Canada KiA OR6

Marek Grabowski IFrank 1. Seiler Research Laboratory, US Air Force Academy. Colorado Springs . CO 80840, USA

and

Jacek A. TuszynskiDepartment of Physics, University of Alberta, Edmonton. Alberta. Canada T6G 2JI

Received 3 February 1992; accepted for publication 3 March 1992

Communicated by A.A. Maradudin

We investigate the tunneling of photons in a quasi-one-dimensional array of polarizable quantum dots using Maxwell-Blochequations. This allows a first principle study of nonlinear polaritons in saturable, dispersive and absorptive, periodic media. Thenonlinear field equation for the steady state generates linearized excitations which form photonic bands and nonlinear excitations:gap solitons. The possibility of manipulating the photonic gaps and gap solitons byseparately contacting the dots isexamined.

The interaction of photons with electronic excitations in an artificially structured dielectric medium leadsto many interesting and important phenomena; photonic gaps and gap solitons [1,2] providing a good ex-ample. The modulation of the dielectric medium is on the order of the wavelength of light, typically muchgreater than the length scale associated with both individual atoms and their separation. Hence a classical de-scription of the medium in terms of its polarizability is sufficient. If one considers the interaction of individualatoms with the field, the quantum mechanical description of this interaction leads to many interesting phe-nomena such as Rabi oscillations, photon echos, and self-induced transparency [3]. The dielectric mediumand individual atom phenomena can be combined by creating arrays of rnesoscopic atoms [4] (quantum dots)frozen in a semiconductor matrix. A typical separation a between dots can be comparable to the wavelengthcorresponding to the lowest radiative transition of the dot at frequency Woo Hence one must describe self-con-sistently spatial and temporal collective behaviour of the field and quantum dots. This behaviour can be mod-ified by selectively contacting individual dots and changing their equilibrium state by, for example, electricallyinjecting electron-hole pairs. To describe the essential features of such a complex system we shall adopt a modelone-dimensional array of two-level dots. Each quantum dot is characterized by its complex polarization P(x,t) and population inversion ~n(x, r ). The time t is measured in units of the characteristic transition frequencyW oI and the distance x in units of the separation of dots a. The collective state of quantum dots and the linearlypolarized complex electromagnetic field Et x, t) are described by the Maxwell-Bloch equations [3,5]:

I p(x)ExAx, 1)- -:;_E,,(x,t)= -~2-Prt(X,t),

v vP = -i{1-iy)P+!i(E+ E*)on ,

I Permanent address: Department of Physics, University of Colorado, Colorado Springs,CO 80933, USA.

148 Elsevier Science Publishers B. V.

8/3/2019 Pawel Hawrylak et al- Nonlinear polariton excitations in quantum dot arrays

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