By Luc Bouten

ISBN-10: 9090187901

ISBN-13: 9789090187907

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The diagram can also be read in the Schr¨ odinger picture if we reverse the arrows: start with a state ρ of the system B in the upper right hand corner, then this state undergoes the following sequence of maps: ρ → ρ ⊗ φ⊗k → (ρ ⊗ φ⊗k ) ◦ Tˆt = Tˆt∗ (ρ ⊗ φ⊗k ) → TrF ⊗k Tˆt∗ (ρ ⊗ φ⊗k ) . This means that at t = 0, the atom in state ρ is coupled to the k channels in the vacuum state, and after t seconds of unitary evolution we take the partial trace taken over the k channels. We would now like to introduce the measurement process.

Define the integral-sum kernel of Ut (name will become apparent in a minute) to be the map ut that maps four disjoint finite subsets of R, σf , σs , τf , τs (where f and s stand for ”forward” and ”side”) to the following 2 × 2-matrix, where we write σf ∪ σs ∪ τf ∪ τs also as {t1 , t2 , . . , tk } such that t1 < t2 < . . < tk and k ∈ N: t − tk ∗ V V )Vk × 2 tk − tk−1 ∗ t1 exp(− V V )Vk−1 . . V1 exp(− V ∗ V ), 2 2 ut (σf , σs , τf , τs ) :=π(χ[0,t] )(σf ∪ σs ∪ τf ∪ τs ) exp(− 26 CHAPTER 2. THE DAVIES PROCESS OF RESONANCE FLUORESCENCE where  Vf    −V ∗ f Vj =  V s   −Vs∗ if if if if tj tj tj tj ∈ σf ∈ τf .

E. F := C ⊕ ∞ n=1 L (R) given by creation and annihilation operators on F, generating the algebra of all bounded operators. We need two copies of this algebra, which we denote by Wf , which will be the forward channel, and Ws , which will be the side channel in the field. 24 CHAPTER 2. e. the second quantization of the operator on L2 (R) which maps f (·) into f (· + t). We denote the second quantization of this operator by St . This means that in the Heisenberg picture we have an evolution on Wf ⊗ Ws mapping X into (St∗ ⊗ St∗ )X(St ⊗ St ) = (S−t ⊗ S−t )X(St ⊗ St ) , also denoted by Ad[St ⊗ St ](X).

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Filtering and control in quantum optics by Luc Bouten

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