Manuscript Em03-B@2 · Em03-B-v2.tex · section · c1a0977d

A declared control set and an exact pulse sequence

B4 extraction: explicit swaps and control certificates › A declared control set and an exact pulse sequence · lines 1249–1272

Em03-B@2#eq:swapPulse

Define the orthogonal rank-one projections

R_{n} = \frac{1}{2} (A_{n} - B_{n}) (A_{n}^{†} - B_{n}^{†}) .

The available extraction controls in our effective model are

H_{E} (t) = n = 1 \sum^{K} u_{n} (t) R_{n} .

These are the specified number-selective matrix elements; their availability is part of the model.

For a common duration $T_{E} > 0$ , choose

u_{n} (t) = \frac{2 π}{T _{E}} sin^{2} (\frac{π t}{T _{E}}), 0 \leq t \leq T_{E} .

The pulses are zero outside this interval. They have area $π$ , maximum $2 π / T_{E}$ , and derivative bounded by $2 π^{2} / T_{E}^{2}$ .

Displayed formulas in this passage

Em03-B@2#eq:swapPulse · line 1263
$u_{n} (t) = \frac{2 π}{T _{E}} sin^{2} (\frac{π t}{T _{E}}), 0 \leq t \leq T_{E} .$
unlabeled · line 1252
$R_{n} = \frac{1}{2} (A_{n} - B_{n}) (A_{n}^{†} - B_{n}^{†}) .$
unlabeled · line 1256
$H_{E} (t) = n = 1 \sum^{K} u_{n} (t) R_{n} .$