Manuscript En04-A@1 · En04-A.tex · section · bab77211

The independently specified model

The four-Majorana control is not a magic certificate › The independently specified model · lines 1431–1498

Take four Majoranas with

{χ_{i}, χ_{j}} = 2 δ_{ij} I, S_{j} = i χ_{2 j - 1} χ_{2 j},

and freeze

H_{S} = J (I - S_{1} S_{2}), A = \frac{S _{1} + S _{2}}{2}, J > 0.

For $0 < μ < Λ < \infty$ , let

h = L^{2} ([μ, Λ], d ω), (h_{B} f) (ω) = ω f (ω),

H_{B} = d Γ (h_{B}), f (ω) = \frac{1}{2 ω}, H = H_{S} + H_{B} + g A \otimes Φ (f) .

The Hamiltonian is specified independently of Enter.

The field estimates

∥ a (f) ψ ∥ \leq h_{B}^{- 1/2} f H_{B}^{1/2} ψ,

a^{†} (f) ψ^{2} = ∥ a (f) ψ ∥^{2} + ∥ f ∥^{2} ∥ ψ ∥^{2}

give arbitrarily small relative bound of the interaction with respect to $H_{B}$ . Since $H_{S}$ and $A$ are bounded, Kato–Rellich gives self-adjointness on $D (H_{B})$ Kato. The finite cutoffs make the displayed norms finite.

Prepare the source in $∣ 11 ⟩_{S}$ and the bath in vacuum. Swap an even-parity two-mode diary into the source:

∣ 0 ⟩_{C} \mapsto ∣ 00 ⟩_{S}, ∣ 1 ⟩_{C} \mapsto ∣ 11 ⟩_{S} .

The displaced carrier is the fixed state $∣ 11 ⟩_{D}$ . The logical source states $s_{0}, s_{1}$ have zero source energy and coupling eigenvalues $a_{0} = - 1, a_{1} = 1$ .

For a declared packet projection $P_{r}$ , set

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

η_{i, R} = P_{r} η_{i}, η_{i, U} = (I - P_{r}) η_{i},

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

Radiation is the collected bath; the source, carrier, and uncollected bath form the complete residual.

Displayed formulas in this passage

unlabeled · line 1434
${χ_{i}, χ_{j}} = 2 δ_{ij} I, S_{j} = i χ_{2 j - 1} χ_{2 j},$
unlabeled · line 1440
$H_{S} = J (I - S_{1} S_{2}), A = \frac{S _{1} + S _{2}}{2}, J > 0.$
unlabeled · line 1445
$h = L^{2} ([μ, Λ], d ω), (h_{B} f) (ω) = ω f (ω),$
unlabeled · line 1449
$H_{B} = d Γ (h_{B}), f (ω) = \frac{1}{2 ω}, H = H_{S} + H_{B} + g A \otimes Φ (f) .$
unlabeled · line 1458
$∥ a (f) ψ ∥ \leq h_{B}^{- 1/2} f H_{B}^{1/2} ψ,$
unlabeled · line 1462
$a^{†} (f) ψ^{2} = ∥ a (f) ψ ∥^{2} + ∥ f ∥^{2} ∥ ψ ∥^{2}$
unlabeled · line 1474
$∣ 0 ⟩_{C} \mapsto ∣ 00 ⟩_{S}, ∣ 1 ⟩_{C} \mapsto ∣ 11 ⟩_{S} .$
unlabeled · line 1483
$η_{i} (t) = g a_{i} \frac{e ^{- i ω t} - 1}{ω} f (ω),$
unlabeled · line 1486
$η_{i, R} = P_{r} η_{i}, η_{i, U} = (I - P_{r}) η_{i},$
unlabeled · line 1490
$c_{R} = exp [- \frac{1}{2} ∥ η_{0, R} - η_{1, R} ∥^{2}], z_{R} = 1 - c_{R}^{2} .$

The independently specified model

Displayed formulas in this passage

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