A ug 1 99 6 Exact dynamical response of an N - electron quantum dot subject to a time - dependent potential

We calculate analytically the exact dynamical response of a droplet of N interacting electrons in a quantum dot with an arbitrarily time-dependent parabolic confinement potential ω(t) and a perpendicular magnetic field. For a single step-like perturbation introduced at time t = 0, the droplet exhibits size oscillations for all t > 0. Over certain frequency ranges, a sinusoidal perturbation acts like an attractive effective interaction between electrons. In the absence of a time-averaged confinement potential, the N electrons can bind together to form a stable, free-standing droplet. Typeset using REVTEX 1 Quantum dots have attracted much interest recently, both from a pure and applied viewpoint. According to the fabrication of the quantum dot, the confinement length scales in the three spatial directions can, in principle, be quite different yielding quasi one-, twoor three-dimensional dots [1]. In addition the number of electrons N in the dot can be reduced down to the few-electron limit. Most theoretical quantum dot research has been concerned with the linear response of the resulting N -electron, low-dimensional system. External, time-dependent electric fields are typically treated as small perturbations which merely give rise to transitions between the eigenstates of the unperturbed dot. Given the technological possibilities for preparing stronger fields, it is interesting to consider the effects of larger, time-dependent perturbations on such dots. For example, a confinement potential with sinusoidal time-dependence could be created by applying an a.c. bias to the electrodes defining the dot in a heterostructure sample. Unfortunately, the quantum-mechanical problem of N interacting electrons in a dot subject to an arbitrarily strong, time-varying perturbation is too complicated to solve in general, even numerically. In this paper we provide an analytically solvable model for the dynamical response of N electrons in a quantum dot with an arbitrarily strong, time-varying confinement potential, in the presence of an external magnetic field of arbitrary strength. The analytic tractability of the model is made possible through a combination of a parabolic form for the dot confinement potential and an inverse-square electron-electron interaction potential (1/r with n = 2). The parabolic confinement is a reasonable approximation for many semiconductor quantum dot samples [1]. It is also known that the true repulsive interaction between electrons in the dot is likely to be better fit by n ∼ 1 at small r and n ∼ 3 at large r due to image charge effects in neighboring electrodes [2]; however the general features of our results with n = 2 for all r should still be qualitatively correct. Here we focus on the usual quasi-two-dimensional dot; however, most of our formal results can be easily generalized to both quasi-oneand quasi-three-dimensional dot systems in the absence of a magnetic field. The time-dependent Schrodinger equation describing our N -electron dot with a timedependent confinement potential in the xy-plane, subject to a constant magnetic field ap2 plied along the z-direction, is given within the effective-mass approximation by H(t)Ψ(t) = ih̄ ∂ ∂t Ψ(t) with