Numerics.Control.Closed

Mapping closed quantum system dynamics onto bilinear control systems

The evolution of closed quantum systems is given by the time-dependent Schrödinger equation (TDSE) in atomic units (where ħ=1)

where H₀=T+V is the unperturbed Hamiltonian describing field-free evolution. The system interacts with the electric field component(s) F_k(t) through its corresponding dipole component matrices μ_k (within the semi-classical approximation) where k normally labels spatial dimensions x, y, z (but, in principle, the number of components is not limited in WavePacket)

For a unified treatment within a bilinear control equation, we want to have the state vector describe the state relative to some "equilibrium vector". This is done by choosing the ground state ψ₀ as equilibrium vector and shifting, ψ(t) → ψ(t) - ψ₀ (with H₀ψ₀ =E₀ψ₀), yielding a modified equation

Finally, we want to have the equilibrium state being an eigenstate of H₀ with eigenvalue 0. This is done by shifting the spectrum of H₀ via H₀ → H₀ - E₀. This also makes the exponential in the last term vanish.

With all these considerations, the TDSE can be finally mapped onto the bilinear input equation by the following substitutions

For the output equations, we note that observables are obtained as expectation values

which can be re-written for shifted the state vectors, ψ(t) → ψ(t) - ψ₀ as

Note that, for a given basis expansion, every state becomes a vector, every operator a matrix, and the expectation values become vector-matrix-vector products. This output equation is then straight-forwardly mapped to the quadratic form of the output equation.