Numerics.Control.Lindblad

Mapping open quantum system dynamics (Lindblad form) onto bilinear control systems

The dynamics of open quantum systems is often described in terms of reduced density matrices ρ(t), evolving subject to a Liouville von Neumann equation (LvNE) describing a system coupled to a bath. This coupling gives gives rise to dissipation and decoherence in the system's dynamics which can be modeled within the framework of the Lindblad formalism.

Here we shall map the equations of motion onto the standard form of bilinear control systems, following the scheme ("diagonals first") outlined in Appendix A of [1]. For the example of a two-state problem, we first map the density operator on a vector by

The matrix A, which represents the system dynamics in the absence of control fields, can then be written as

where Γ and γ, γ^* are the rate constants and dephasing rates, respectively (see the page on the Lindblad formalism), and where the Bohr frequencies are given by ω_n,m≡E_n-E_m.

We note in passing that the matrix representation of A is block-diagonal. The upper left block deals with populations and has off-diagonal elements, while the lower right block concerns the coherences and is diagonal. If we increase the number of states N, the number of coherences increases much faster (∝ N²) than the number of populations (∝ N). Hence, for an increasing number of states, this matrix becomes very sparse and thereby efficient for numerics.

The elements of the control (input) matrix N represent the coupling between the external field u(t) ≡ F(t) and the vectorized density x(t) through the dipole moments μ

This matrix is obviously less sparse than the A-matrix, although this can improve substantially if certain dipole moments vanish.

Similarly, the vector b≡Nx_e can be calculated for the two-state problem as

For the output equations, we start from the calculation of expectation values ⟨O⟩ = Tr(Oρ). Casting this into a vector product form, we obtain the output matrix C. For the example of two control targets O⁽¹⁾, O⁽²⁾, the output matrix C is expressed as

C thus consists here of two row vectors c₁^T and c₂^T. Note that the ordering of the terms of the operator matrices O⁽ⁱ⁾ differs from the ordering for ρ given above, thus ensuring that expectation values ⟨O^(j)⟩ ≡ Tr (O^(j)ρ) is then straight-forwardly mapped to the linear form of the output equation.

References

1. B. Schäfer-Bung, C. Hartmann, B. Schmidt, and Ch. Schütte: J. Chem. Phys. 135, 014112 (2011)