Numerics.Control

Controlling quantum dynamics by external fields

The dynamics of quantum systems with coordinate-dependent electric or magnetic multipole moments can be controlled 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:

• Closed quantum system dynamics decribed by TDSE
• Open quantum system dynamics described by LvNE with Lindblad

Because typical optimal control schemes require many propagations back and forth in time, the computational effort can become large, especially for open quantum systems, without the option of parallelisation. Hence, reduction of the dimensionality may become a key issue, see also our Wiki page on approaches to dimension reduction.

General concepts: input/control

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) ∈ Cⁿ. 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 0 as a simple eigenvalue. The interaction with a low-dimensional control u(t) ∈ R^m, m<<n, is given by the input matrix B ∈ R^n×m.

Note that the interaction is linear, that is the derivative of the state vector depends linearly on the control!

However, for quantum systems driven by external control field(s) u_k(t) ≡ F_k(t), we are dealing with a bilinear input equation [1]

with x as the coefficient vector of the wave function or density operator, and N_k ∈ R^n×n some e.g. dipole matrices. Bilinear means that now the control term depends (linearly) on both the field components u_k(t) and the state vector x(t). The latter can be shifted, x(t) → x(t) - x_e, by the equilibrium state x_e defined as A x_e = 0, yielding a new equation

where we have set b_k ≡ N_k x_e. The shifted equation is now inhomogeneous and therefore more complicated. However, for an equilibrium initial condition, we have x(0) = 0, which allows us to use the balancing method.

Note: While most of control theory considers systems with linear input equations, there is only little theory for this bilinear setting.

General concepts: output/observation

In linear time-invariant (LTI) system theory, the output equation defines the low-dimensional observables y(t) ∈ R^p, p<<n, in terms of an output matrix C ∈ R^p×n.

For use with open quantum systems (Liouville-von Neumann equation, we rewrite this in terms of components (see here)

where every single observable is represented by a vector c_j ∈ Rⁿ. We have also shifted the state vectors again, x(t) → x(t) - x_e similar to the input equation.

For closed quantum systems (time-dependent Schrödinger equation), we have to consider quadratic output equations (see here)

where every single observable is represented by a Hermitian matrix D_j ∈ R^n×n. Note that the last term on the r. h. s. can be rewritten as 2ℜ(e_j^Tx(t)) with a coefficient vector e_j≡D_j^Tx_e which allows to treat it in close analogy to the linear case given above.

Note: While there is a substantial body of literature for linear systems, there is hardly anything in the literature on control theory for the above bilinear output equation.

Classifications

For the above control systems, there is the following classification:

• If m=1, p>1 the system is called SIMO (single input)
• If m>1,p=1 the system is called MISO (single output)
• If m=1 and p=1, the system is called a SISO system
• If both input and output are bounded, the system is called BIBO

Stabilization

In stability theory, a stable system approaches a fixed point (an equilibrium) in the long time limit, 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. Solutions of ODEs always contain a term of the form exp(At), hence this shift introduces a damping of the form exp(-αt). The damping forces the system towards x=0, i.e., to the equilibrium state, even for the case of closed quantum system dynamics described by TDSE, where this procedure may violate norm conservation.

• SSU: The unstable part of A is separated by transforming the matrices A and N and the vectors B, C and x(t), into the eigenbasis of A. If we order the eigenvalues by the absolute value of the real part, we can directly separate the unstable part x₁∈C^M from the stable part x₂∈C^n×n-M. The corresponding dynamics is governed by two coupled equations

  

  Then the dynamics of the stable part is given by

  

  if we neglect the coupling term (A₂₁+iu(t)N₂₁)x₁(t). Alternatively, these couplings to the unstable part can be expressed in terms of additional control fields

  

  coupling to the dynamics of the stable part x₂(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.

References

1. B. Schäfer-Bung, C. Hartmann, B. Schmidt, and Ch. Schütte: J. Chem. Phys. 135, 014112 (2011)
2. 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.