The control of quantum electronic states in physical systems has a host of applications such as quantum computers, control of photochemical processes, and semiconductor lasers. In a quantum computer, the energy states of a quantum system may be interpreted as logic states.
The aim of this work is to compute a time-dependent control which appears as a variable coefficient in the Schrödinger equation, such that a particle in a given initial state will have maximal probability of being in a target state at a specified time.
The approach is to formulate a PDE-constrained optimization problem where the probability of finding the particle in a target state at time T is to be maximized.
In this setting, ψ is the wavefunction which describes the time-evolution of a particle, P is an operator which projects the wavefunction onto the target state, u is the control function modeling, for example, the amplitude of a laser field, and γ is a regularization parameter. The optimization problem is subject to the equality constraint
which gives the control problem a bilinear structure. The control which minimizes the reduced cost functional
(u) = J(ψ(u),u) is computed via a globalized Newton method.
![[∇2 ˜J(uk)]p = - ∇ ˜J(uk), uk+1 = uk + αp](gvw_research3x.png)
In practice, the search direction p is computed approximately with a Krylov solver and a step length α is obtained via a robust line search based on the strong Wolfe conditions and the underlying physics.
Optimal control function |
Time evolution of |ψ|2 |
Given an open quantum system with finite potential well of width 2 and depth 50, the first two energy states are λ1 = 5.9426 and λ2 = 22.9780. Here the optimal control and corresponding wavefunction have been computed which maximizes the probability of finding the particle in the second state at time T = 3. The open boundaries are approximated using perfectly matched layers with the method of complex exterior scaling.