Journal of Superconductivity: Incorporating Novel Magnetism, Vol. 16, No. 2, April 2003 ( C  2003) Measure of Entanglement States of Two Interacting Electrons in Vertically Coupled Quantum Dots induced by a Time-Dependent Electric Field W. B. Chouikha, 1 S. Jaziri, 1,2 and R. Bennaceur 1 Received September 30, 2002 We study the dynamics of two electrons located in two vertically tunnel-coupled quantum dots in the presence of an oscillatory electric field. By solving the time-dependent Schr ¨ odinger equation, we predict the dynamical generation of entangled electron states, such as the EPR (Einstein, Podolsky, and Rosen) pairs or Bell states. The Schmidt rank and the von Neumann entropy are evaluated to characterize the degree of entanglement of the two electron states. KEY WORDS: spintronic; entanglement; double quantum dot. 1. INTRODUCTION Entangled states are of paramount importance in quantum information. This phenomenon has im- portant applications in quantum communication and quantum computation. The Einstein–Podolsky– Rosen (EPR) state [1,2] is an interesting example of two-particle entanglement, because of its potential use in secure quantum communication protocol [3,4], quantum information processing [5], and fundamen- tal tests of quantum mechanics [6]. Recently, there has been a growing interest in solid-state implemen- tations of quantum computation using the electron spin as the qubit [7], as well as quantum communica- tion schemes based on spin-entangled electron states. Coupled semiconductor quantum dots are candidates for controlling quantum superposition and entangle- ment of electron states. In this paper, we concentrate on a theoretical proposal to use coupled semiconduc- tor quantum dots, in which the presence of one elec- tron confined in the double dot represents 1 qubit. We consider the case of two electrons located in two vertically tunnel-coupled quantum dots in a time- dependent electric field, and study entangled pure 1 Laboratoire de Physique de la Mati ` ere Condens ´ ee, Facult´ e des Sciences de Tunis, Tunisie. 2 Department de Physique, Facult´ e des Sciences de Bizerte, Jarzouna 7021, Bizerte, Tunisie. states of 2 qubits. This allows us to predict the dy- namical generation of entangled EPR pairs, using two interacting electrons in a system of coupled quantum dots in the presence of an oscillatory electric field, and give a measure of the resulting entanglement by the Schmidt rank and the partial von Neumann entropies in a short-time scale. 2. MODEL AND RESULTS We consider two electrons in a vertically coupled InAs double quantum dot structure [9]. The complete Hamiltonian in the presence of time-dependant elec- tric field F (t ) = F 0 cos(ω 0 t ) is given by H(t ) = H 0 + eF (t )(z 1 + z 2 ) (1) H 0 =  i =1,2 h i + V c (2) h i = P 2 i 2m + mω 2 8d 2 (z 2 i − d 2 ) + m 2 ω 2 u (x 2 i + y 2 i )θ (z ) + m 2 ω 2 1 (x 2 i + y 2 i ) θ (−z ), i = 1, 2 (3) where V c = e 2 /ε| r 1 − r 2 | is the Coulomb potential, m = 0.08 m e is the electron effective mass in InAs, m e is the bare electron mass, hω = 20 meV is the vertical confinement, hω u(1) = 6 meV is the lateral 313 0896-1107/03/0400-0313/0 C  2003 Plenum Publishing Corporation