Brijender Dahiya*, Kriti Batra and Vinod Prasad *Department of Physics, Swami Shraddhanand College (University of Delhi) Delhi, INDIA

INTRODUCTION The Basic Problem of Quantum Mechanics: Electron in a quantum well (finite) has well defined and quantized energy levels. Also the solution of Hydrogen Atom is defined as electron in a quantum well (Electron trapped in the potential well)

Quantum Well Bound States Quasi-bound/ Continuum states Atom Electron in valance band Electron in conduction band/ionization

Quantum Heterostructures Quantum wells(1D): (a) Finite/Infinite wells (b) Square/rectangular/triangular/parabolic etc. Quantum wells(2D): a) Quantum Rings b) Quantum wires c) Quantum nanotubes

Quantum wells(3D): a)Cubical Quantum Dots[1] b)Spherical Quantum Dots[2] c)Cylindrical Quantum Dots[3] All these types of quantum heterostructures are possible to fabricate using the available recent techniques. 1.D. Lagarde, A. Balocchi, H. Carrere, P. Renucci, T. Amand, X. Marie, S. Founta and H. Mariette,, Phys. Rev. B, 77, (2008) 2.İbrahim Karabulut and Sotirios Baskoutas, J. Appl. Phys. 103, (2008) 3.Philippe Matagne and Jean-Pierre Leburton, Phys. Rev. B 65, (2002)

Recent developments in LASER Technology  Wide Range of frequency/wavelength (nanometer- micrometer)  Wide Range of Intensity  Different techniques of laser shaping * Chirped Lasers * Optimized lasers

Quantum Dots=Artificial Atoms # As mentioned by U. Banin [4], quantum dots due to their small size and more the transition between molecular and solid state regimes are described as “artificial atoms”. # Quantum dots termed as artificial atoms, due to the similarities of their energy level structure, transition matrix elements can be used in many predominant effects such as quantum hall effect, quantum chaos[5] [4] U. Banin, Y.W. Cao, D. Katz and O. Millo, Nature, vol. 400, pp , August 1999 [5] A.D. Stone and H. Bruus, Physica B, vol. 189, pp , June 1993

ARTIFICIAL ATOMS* The charge and energy of a sufficiently small particle of metal or semiconductor are quantized just like those of an atom. The current through such a quantum dot or one-electron transistor reveals atom-like features in a spectacular way. *Marc A. Kasfner, Phys. Today, vol. 46(1), pp , January 1993

Importance of the work  Interband transitions  Intersubband transitions  Optical Properties: Linear and Non- linear Refractive index

The efficient and well tested non- perturbative Floquet theory that was originally developed to describe the atomic behavior in the presence of the intense laser fields proposed by Shirley* *J. H. Shirley, Phys. Rev., vol. 138, pp. B979-B987, May 1965 Theory

with H 0 = The basis Hamiltonian, E s = The static electric field strength U(r, t)= The interaction potential energy of the electron with the laser field Where

The wave function, in terms of Floquet states Where is the Floquet characteristic exponent or the quasi energy is the Floquet wavefunction

Expanding Φ(t) and H(t) in terms of Floquet state basis

and Corresponding Fourier amplitudes are Substituting Φ(t) and H(t) in Hamiltonian equation, we get the recursion relations for

With is the time independent Floquet Hamiltonian We get the Floquet Hamiltonian Equation given by

The quasi energies are the Eigen values of Floquet equation By diagonalising H f we can obtain corresponding Eigen vectors

The absorption spectra probability are given by the equation Where U(t,t 0 ) is the evolution operator The absorption spectra probability averaged over the elapsed time are given by

Computation The Eigen energies and the wavefunctions of the conduction band of Artificial Atom in static field are obtained using finite difference method. The Schrödinger equation can be written as follows: With

In the limit the Schrödinger equation gives the Eigen values and wavefunctions for the unperturbed system The Hamiltonian (H 0 +erE s ) is reduced to tridiagonal matrix and is diagonalised using standard Fortran subroutines to obtain the Eigen values and the wavefunctions of a AA.

The Floquet theory is then implemented to obtain the Floquet Hamiltonian in the form of infinite matrix H f. The Floquet Hamiltonian so obtained is

where With is the energy Eigen value denotes DME in presence of static electric fields.

The transition between various energy levels of the artificial atom defined as e i is the energy of i th level and μ ij is the dipole matrix element given by

RESULTS AND DISCUSSION Considering various levels of the CQD as different levels of an artificial atom we can give nomenclature to the various levels depending upon the value of “nlm” is given in Table 1. The value of “n” gives the principal quantum number. From next two indices, the greater value defines the state (i.e. s, p or d and the difference between them define the sublevel index i.e. 0,±1, ±2*) *V. Prasad and P. Silotia, Phys. Lett. A, vol. 375, 3910 (2011)

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CONCLUSION Due to the removal of the degeneracy new absorption peaks are observed in the absorption spectra of the artificial atom. This may give rise to various other features such as linear and non-linear absorption coefficients and refractive index changes of the material of which the atom is made of. (Work in this regard is in progress and will be published soon).

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