Manuscript Hawk04-B@2 · Hawk04-B.tex · section · ba3b0713

An exact frozen realization

The commuting four-Majorana control › An exact frozen realization · lines 1832–1876

Fix

N = 4, J_{1234} = J > 0, A = \frac{S _{1} + S _{2}}{2}, g \neq = 0,

with one scalar reservoir species. For

c_{j} = \frac{χ _{2 j - 1} + i χ _{2 j}}{2},

one has

S_{j} = 2 c_{j}^{†} c_{j} - I, H_{S} = J (I - S_{1} S_{2}) .

Prepare $∣ 11 ⟩_{S}$ , which is unchanged by Euclidean filtering. Swap the two-mode diary into the entire source. The retained carrier becomes the fixed state $∣ 11 ⟩_{D}$ . The logical source states are

s_{0} = ∣ 00 ⟩_{S}, s_{1} = ∣ 11 ⟩_{S},

both with source energy zero, and with coupling eigenvalues $a_{0} = - 1$ , $a_{1} = 1$ .

For $f (ω) = 1/ 2 ω$ , define

η_{i} (t) = g a_{i} \frac{e ^{- i ω t} - 1}{ω} f (ω) .

For the collected packet space $r$ , put

η_{i, R} = P_{r} η_{i}, η_{i, U} = P_{r^{⊥}} η_{i},

and

c_{R} (t) = exp [- \frac{1}{2} ∥ η_{0, R} - η_{1, R} ∥^{2}], z_{R} (t) = 1 - c_{R} (t)^{2} .

The collected coherent states have real positive overlap $c_{R}$ .

Displayed formulas in this passage

unlabeled · line 1835
$N = 4, J_{1234} = J > 0, A = \frac{S _{1} + S _{2}}{2}, g \neq = 0,$
unlabeled · line 1841
$c_{j} = \frac{χ _{2 j - 1} + i χ _{2 j}}{2},$
unlabeled · line 1845
$S_{j} = 2 c_{j}^{†} c_{j} - I, H_{S} = J (I - S_{1} S_{2}) .$
unlabeled · line 1853
$s_{0} = ∣ 00 ⟩_{S}, s_{1} = ∣ 11 ⟩_{S},$
unlabeled · line 1860
$η_{i} (t) = g a_{i} \frac{e ^{- i ω t} - 1}{ω} f (ω) .$
unlabeled · line 1864
$η_{i, R} = P_{r} η_{i}, η_{i, U} = P_{r^{⊥}} η_{i},$
unlabeled · line 1869
$c_{R} (t) = exp [- \frac{1}{2} ∥ η_{0, R} - η_{1, R} ∥^{2}], z_{R} (t) = 1 - c_{R} (t)^{2} .$

An exact frozen realization

Displayed formulas in this passage

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