Provided a quantum system has electric or magnetic multipole moments depending on that system's coordinates, then one can thrive to control its dynamics through suitably tailored, time-dependent external electric and/or magnetic fields. For the problem of how to cast quantum dynamics onto the below forms of bilinear control systems [1], the reader is referred to our following Wiki pages:
Because typical optimal control schemes require many propagations forth and back in time, the computational effort can become quite large, especially for open quantum systems. Hence, reduction of the dimensionality may become a key issue, see also our Wiki page on approaches to dimension reduction.
In linear time-invariant (LTI) system theory, the input equation of a control system describes the evolution of a high-dimensional system in terms of its state vector x(t)∈C^{n}. The field-free evolution is described by linear ODEs with real symmetric matrix A∈R^{n×n} (or complex Hermitian matrix A∈C^{n×n}) with simple eigenvalue 0. The (linear!) interaction with a low-dimensional control u(t)∈R^{m}, m<<n, is given by input matrix B∈R^{n×m}.
In case of quantum systems driven by external control field(s) u_{k}(t) ≡ F_{k}(t), however, we are dealing with the following bilinear version of the input equation [1]
with matrices N_{k}∈R^{n×n} where the control term is bilinear, i. e., linear both in the field components u_{k}(t) and the state vector x(t). Upon shifting the latter one, x(t) → x(t) - x_{e} by the equilibrium density x_{e} (where A x_{e} = 0, i. e., eigenvector of A corresponding to the eigenvalue 0), we end up with
where we have set b_{k} ≡ N_{k} x_{e}. On the one hand, this shift transforms a homogeneous equation to an inhomogeneous one and seems to complicate things. On the other hand, the shift establishes the basis for the balancing method by setting x(0) = 0 for an equilibrium initial condition. Note, however, that while most of control theory considers systems with linear input equations, there is only little theory for this bilinear setting.
In standard linear control theory, the output equation defines the low-dimensional observables y(t)∈R^{p}, p<<n, in terms of output matrix C ∈R^{p×n}.
However, in the case of the Schrödinger equation for closed quantum systems, we have to consider a quadratic output equation
in which case we have p matrices D_{j}∈R^{n×n}. Note again that most of control theory considers systems with linear output equations.
For a review of the theory of linear control systems, see e.g. the excellent lecture notes of Umea university and/or the books by Kemin Zhou.
In the stability theory for dynamical systems, the notion stability implies that a system is exponentially approaching a fixed point (an equilibrium) in the long time limit (rather than remaining oscillatory), and nearby points converge to it at an exponential rate. In the equations above (for u(t)=0), this requires that the spectrum of the system matrix A should be in the left half of the complex number plane (negative real part). Such matrices are also referred to as Hurwitz stable matrices.
However, matrix A has a (simple) zero eigenvalue for open quantum system dynamics described by LvNE with Lindblad corresponding to the thermal equilibrium x_{e}. In such cases, the A matrix can be stabilized by one of the following two techniques:
EVS: The diagonal values of matrix A are shifted by a small negative amount, A →A-α1, where α>0 is a (real-valued) shift parameter. Since in the solutions of the ODEs there is always a term of the form exp(At), the shift is simply equivalent to a damping proportional to exp(-αt). In the linear case, this means that the controllability functional c=min ∫_{0}^{∞}|u(t)|²dt is replaced by c=min ∫_{0}^{∞}exp(-2αt)|u(t)|²dt. (Is there a similar interpretation in the bilinear case?) In optimal control theory this is referred to as "discounting a functional", i.e., the further future is not taken quite as important as the closer future. (In financial mathematics discounting represents the interest rate or inflation rate). Note that in the case of closed quantum system dynamics described by TDSE, the complete spectrum of A lies on the imaginary axis. To be investigated: Will the shift procedure work here?
SSU: The unstable part of A is separated from the stable part by transforming matrices A and N, as well as vectors B, C and x(t), into the eigenbasis of A where the ordering of eigenvalues/eigenvectors of A is such that the eigenvalues with smallest real part (absolute!) go first. This allows for a straightforward separation of the unstable part x_{1}∈C^{M} from the stable part x_{2}∈C^{n×n-M}. The corresponding dynamics is governed by two coupled equations
Then the dynamics of the stable part is given by
where the coupling term (A_{21}+iu(t)N_{21})x_{1}(t) has been neglected. Alternatively, these couplings can be expressed in terms of additional control fields
coupling to the dynamics of the stable part x_{2}(t) through enhanced B-vector
Note that in the case of open quantum system dynamics described by LvNE with Lindblad it is sufficient to choose M=1, i.e. there is only one unstable component (eigenvalue zero) to be removed.
Wiki: Numerics.Control.Closed
Wiki: Numerics.Control.Lindblad
Wiki: Numerics.DimRed.BalTru
Wiki: Numerics.DimRed.H2Norm
Wiki: Numerics.DimRed
Wiki: Numerics.Main
Wiki: Numerics.Schroedinger