Manuscript Em04-A@2 · Em04-A-v2.tex · definition · f5b28abc

Definition: Specified zero-temperature chain

Scope, evidence categories, and model specification › Notation · lines 183–227

Em04-A@2#def:model, Em04-A@2#eq:H, Em04-A@2#eq:collapse, Em04-A@2#eq:master

Let $N \geq 2$ , $J > 0$ , $γ_{r}, γ_{d} \geq 0$ , and $ϵ_{k} \in R$ , $0 \leq k < N$ . On $H = (C^{2})^{\otimes N}$ , use

X = (0110), Y = (0 i - i 0), Z = (10 0 - 1),

and

a = ∣0 ⟩ ⟨ 1∣, a^{†} = ∣1 ⟩ ⟨ 0∣, n = a^{†} a = \frac{I - Z}{2} .

Subscripts denote tensor-local operators. Set

H L_{k}^{r} = \frac{J}{2} k = 0 \sum^{N - 2} (X_{k} X_{k + 1} + Y_{k} Y_{k + 1}) + \frac{1}{2} k = 0 \sum^{N - 1} ϵ_{k} Z_{k}, = γ_{r} a_{k}, L_{k}^{d} = γ_{d} /2 Z_{k} .

For any operator $L$ , define

D [L] (M) = L M L^{†} - \frac{1}{2} {L^{†} L, M} .

The master equation is

ρ ˙ = L (ρ) = - i [H, ρ] + k = 0 \sum^{N - 1} (D [L_{k}^{r}] (ρ) + D [L_{k}^{d}] (ρ)) .

There are no bath-raising operators. The default initial state is

ρ (0) = ∣ e_{0} ⟩ ⟨ e_{0} ∣, ∣Ω ⟩ = ∣0 ⟩^{\otimes N},

where $∣ e_{k} ⟩$ has its unique excitation at site $k$ .

Displayed formulas in this passage

Em04-A@2#eq:H · line 199
$H L_{k}^{r} = \frac{J}{2} k = 0 \sum^{N - 2} (X_{k} X_{k + 1} + Y_{k} Y_{k + 1}) + \frac{1}{2} k = 0 \sum^{N - 1} ϵ_{k} Z_{k}, = γ_{r} a_{k}, L_{k}^{d} = γ_{d} /2 Z_{k} .$
Em04-A@2#eq:collapse · line 199
$H L_{k}^{r} = \frac{J}{2} k = 0 \sum^{N - 2} (X_{k} X_{k + 1} + Y_{k} Y_{k + 1}) + \frac{1}{2} k = 0 \sum^{N - 1} ϵ_{k} Z_{k}, = γ_{r} a_{k}, L_{k}^{d} = γ_{d} /2 Z_{k} .$
Em04-A@2#eq:master · line 213
$ρ ˙ = L (ρ) = - i [H, ρ] + k = 0 \sum^{N - 1} (D [L_{k}^{r}] (ρ) + D [L_{k}^{d}] (ρ)) .$
unlabeled · line 188
$X = (0110), Y = (0 i - i 0), Z = (10 0 - 1),$
unlabeled · line 194
$a = ∣0 ⟩ ⟨ 1∣, a^{†} = ∣1 ⟩ ⟨ 0∣, n = a^{†} a = \frac{I - Z}{2} .$
unlabeled · line 209
$D [L] (M) = L M L^{†} - \frac{1}{2} {L^{†} L, M} .$
unlabeled · line 222
$ρ (0) = ∣ e_{0} ⟩ ⟨ e_{0} ∣, ∣Ω ⟩ = ∣0 ⟩^{\otimes N},$