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Dt i If the Hamiltonian is constant, this can be solved straightforwardly via the operator exponential, i U (t, ti ) = exp [−iH (t − ti ) / ] . Limit theorem for QSDEs; cavity QED relay model 19 We note that any time-dependent operator moment can thus be written O t = ψ (t)| O | ψ (t) = ψ (ti )| U † (t, ti ) OU (t, ti ) | ψ (ti ) , and we note that we can reproduce all such observable quantities in the Heisenberg picture, where the state vector | ψ (ti ) is considered to be constant (held to its initial condition) while the operators evolve according to O (t) = U † (t, ti ) OU (t, ti ) .
Note that this QSDE corresponds to Fig. 5 in this chapter, but Bouter et al. ) We will first take the limit k1 → ∞ and then k2 → ∞. For the first elimination, K = k 2 Y + kA + B, Li = kFi + Gi , Ni j = Wi j , ∗ Y = −κa a∗ a − ga a∗ σge − aσge , A = 0, ∗ ∗ B = −k22 κb b∗ b − k22 κc c∗ c − k2 gb b∗ σgr − bσgr − k2 gb (b∗ σhe − bσhe ) ∗ ∗ −k2 gc (c σhr − cσhr ) , ⎛ √ κa a∗ √ ⎜ κa a∗ F =⎜ ⎝ 0 0 ⎞ ⎟ ⎟, ⎠ ⎛ ⎞ 0 ⎜ 0 ⎟ ⎟ √ G=⎜ ⎝ k2 2κb b∗ ⎠ , √ k2 2κc c∗ Wi j = δi j . We choose H0 = span {| g 0a nb nc , | h 0a nb nc , | r 0a nb nc }, which clearly lies within the kernel of Y .
5 (a) Level diagram for the “atomic” system coupled to cavity modes in the relay construction. (b) Arrangement of resonator modes and correspondence of inputs/outputs to logical signals. (c) Component diagram for the relay, showing logical signals only. Limit theorem for QSDEs; cavity QED relay model 27 This ensures that Y Y˜ P0 = Y˜ Y P0 = 0. We next note that, for na ≥ 1, √ Y | g na nb nc = −κa na | g na nb nc + ga na | e (na − 1)a nb nc , √ Y | e (n − 1)a nb nc = −κa (na − 1) | e (na − 1)a nb nc − ga na | g na nb nc , Y | h na nb nc = −κa na | h na nb nc , Y | r na nb nc = −κa na | r na nb nc .