By Luc Bouten
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Additional info for Filtering and control in quantum optics
Example text
For all t, s ≥ 0 and E ∈ Σs ⊗ Σs , F ∈ Σt ⊗ Σt and all X ∈ M2 we have: ˜ E](X), E t [F ] ◦ E s [E](X) = E s+t [F − s∪ where F − s ∈ Σ [−t − s, −s) ⊗ Σ [−t − s, −s) is given by: ˜ is defined by: F − s = {(ff − s, fs − s); (ff , fs ) ∈ F )} and ∪ ˜ A∪B = {(ωf ∪ σf , ωs ∪ σs ); (ωf , ωs ) ∈ A, (σf , σs ) ∈ B}. Proof. The only point where there is really something to prove is point 5. Let us first introduce some short notation which we shall only use in this proof. Let π(zt , 0) denote π(zχ[0,t] ) ⊗ π(0) and denote St ⊗ St just by St .
In a similar fashion, only extracting two integrals now, we find that for i, j = 2, 3, . . : x y FXi ,Xj (x, y) = 0 0 z(x′ )z(y ′ )dx′ dy ′ . If i or j is 1 we again have to substitute zf irst for z. It is now obvious that the random variables Xi and Xj are independent. We conclude that the family of random variables {Xi }i=1,2,... is a (modified) renewal process. Chapter 3 Stochastic Schr¨ odinger equations Luc Bouten† M˘ad˘alin Gut¸˘a†† Hans Maassen† † Mathematisch Instituut, Katholieke Universiteit Nijmegen Toernooiveld 1, 6526 ED Nijmegen, The Netherlands †† EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands Abstract1 A derivation of Belavkin’s stochastic Schr¨odinger equations is given using quantum filtering theory.
Tracing out the local oscillator yields the following jump operation for the atom in the homodyne setup: Ja (ρ) = Trlo Ja⊗lo ρ ⊗ ψ(αt ) ψ(αt ) = κs V + wt wt ρ κs V ∗ + . 8) where the extra ε’s are introduced for future convenience. 7). e. we are interested in the limit ε → 0 [9], [27], [92]. Then the number of detected photons becomes very large and it makes sense to scale and center Nt , obtaining in this way the process with differential dWtε := εdNt − dt/ε and W0ε = 0. We find the following Itˆo rules for dWtε : 1 1 dWtε dWtε = εdNt − dt εdNt − dt = ε2 dNt = εdWtε + dt, ε ε ε dWt dt = 0.
