Abstract
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Unitary Control in Quantum Ensembles: Maximizing Signal Intensity in Coherent Spectroscopy
S. J. Glaser, T. Schulte-Herbrüggen, M. Sieveking, O. Schedletzky, N. C. Nielsen, O. W. Sørensen, and C. Griesinger
[Web Site] Additional information also available at Dr. Glaser's home page

Supplementary Material


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1) Derivation of the gradient of |f(U)|2

Let Mat(m) designate the space of complex m x m matrices equipped with the scalar product (9), U(m) the space of all unitary m x m matrices U, and f resp. F functions defined by

f(U) := <UAU|C> resp. F(U) = |f(U)|2 = f(U)f*(U), UsubsetU(m)

By definition, a matrix TsubsetMat(m) is tangent at the group U(m) in the point UsubsetU(m) if there exists a curve tÆV(t) subsetU(m) which is continuously differentiable and satisfies

V(t0) = U and d/dt|t=t0V(t)= T

A useful relation is TU = -UT which follows from

d/dt|t=t0Id = 0 = d/dt|t=t0 {V(t) V (t)} = TU + UT

The gradientnabla F(U) subsetMat(m) of F is by definition a matrix which is tangent at UsubsetU(m) and for all curves tÆV(t) defined above satisfies

d/dt|t=t0F(V(t))= <nabla F(U)|T>

Using the product rule of calculus we obtain:

d/dt|t=t0F(V(t)) = d/dt|t=t0 { f(V(t)) f*(V(t))}
= 2 Re{f*(U) d/dt|t=t0f(V(t))}
= 2 Re{f*(U) tr{TAUC + UATC}}
= 2 Re{f*(U) tr{ U [UAU, C] T}}
= <{(f*(U) [ UAU,C])-f*(U) [ UAU,C] } U | T>

Hence it follows that nabla F(U) ={(f*(U) [ UAU,C])-f*(U) [ UAU,C] } U. nabla F(U) is the tangent in U of the curve tÆW(t)= exp{(t-t0) nabla F(U)U} UsubsetU(m) since W(t0) = U and d/dt|t=t0W(t) = nabla F(U).

2) An Algorithm to trace the boundary of WC(A)

The boundary of the C-numerical range WC(A) can be traced by the following strategy based on the fact that WC(A) is always star-shaped (17). (A subset of the complex plane is said to be star-shaped with respect to a center, if the respective line connecting any point of the subset with the center is entirely in the subset.) The star center is given by tr(A) tr(C)/m if the operators A and C are represented by m x m matrices (17). For the traceless auxiliary matrix At={A- tr(A)/m}, the C-numerical range WC(At) is shifted in the complex plane by -tr(A) tr(C)/m relative to WC(A) such that the star center of WC(At) is located at the origin of the complex plane. Hence the boundary of the C-numerical range of At intersects the positive real axis at one and only one point. If this point can be found, the entire boundary of WC(At) can be traced if the intersection between WC(A) and the positive real axis is determined for 0 �ϕ� 2 π with A=e-i ϕAt, because a multiplication of At by ei ϕ rotates WC(At) by ϕ about the origin of the complex plane. Finally, the boundary of WC(A) is obtained by shifting the boundary of WC(At) by tr(A) tr(C)/m.

The point where the positive real axis intersects the boundary of WC(A) can be approached by optimizing the real part Re{f (U)} subject to the constraint that the imaginary part Im{f (U)} = 0 with f(U):= <UAU| C>.

For the cases studied in this report (compare Figures 2 - 4), this was achieved by the following preliminary iterative algorithm which consists of two steps:

  1. Move one step in the direction of -nabla Re{f(U)} to find Uk+1 in analogy to Eq. 5 and reduce the step size δ whenever Re{f(Uk+1)} < Re{f(Uk)}.
  2. In analogy to the iterative scheme defined in Eq. 5 follow nabla (Im{f(U)} )2 until the real axis is reached. The unitary operator U for which Im{f(U)}=0 defines Uk+2.

Starting from an arbitrary unitary operator U0, steps (a) and (b) are repeated until convergence is reached. The gradients of Re{f(U)} and ( Im{f(U)} )2 are given by

nabla Re{f(U)}= 1/2([UAU, C]- [UAU, C])U and
nabla (Im{f(U)} )2 = i Im{f(U)} ([ UAU,C] + [ UAU,C]) U, respectively.

If WC(A) is convex, a simplified algorithm can be applied. In this case, for each angle ϕ the maximum of Re{f(U)} is also well defined and can be found using nabla Re{f(U)}.