By C.W. Gardiner
This publication deals a scientific and finished exposition of the quantum stochastic tools which were constructed within the box of quantum optics. It contains new remedies of photodetection, quantum amplifier concept, non-Markovian quantum stochastic procedures, quantum input--output concept, and optimistic P-representations. it's the first publication during which quantum noise is defined via a mathematically entire idea in a sort that also is fitted to useful purposes. unique consciousness is paid to non-classical results, corresponding to squeezing and antibunching. This moment version has been enlarged in order to take account of quick growth within the box, and now contains extra chapters at the stochastic Schrödinger equation, and on cascaded quantum platforms.
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Additional info for Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics
Sample text
Commutes with the operator d of exterior differentiation. We only need to note that the form w itself is closed as a form of maximal degree. To reconstruct the differential form w = I dx l /\ dx 2 from its Radon transform 'Rw = adp + bdcp, or. equivalently, to reconstruct the function I. p). We see that the Radon transform of 2-forms has trivial kernel. in contrast to the case of I-forms. We also note that the problem of reconstructing a differential 2-form w from its Radon transform 'R w is overdetermined because one must know only one coefficient of the form 'R w to find w completely.
7) w = h (X)dx2 /\ dx3 + h(x)dx3 /\ dx l + Ia(x)dx l /\ dx 2. X = (Xl. X2. X3), with coefficients in the Schwartz space S(R3). Integrating the form w over all p0ssible orientable planes in R3, we obtain a function on the manifold of planes. We refer to this function as the Radon translorm of wand denote it by'Rw. We present the expression for 'R. w in the coordinates on the manifold of planes in R3. 8) On the manifold of planes we take the coefficients al. 02. {3 of these equations for the local coordinates.
P) satisfy the conditions of the theorem. ) = -2 11" f+oc "'({, p)e -00 ip dp. I. RADON TRANSFORM 16 It follows from the homogeneity condition 1) that this relation is equivalent to the formula F(~~) 1 = -2 j+x 'P(~. p)ei>'p dp. -oc 1r Conditions 2) and 3) imply that the function F is infinitely differentiable at any point ~ ::F 0 and infinitely differentiable in any direction at ~ = 0: hence. it is infinitely differentiable at the point ~ = 0 by Cavalieri's conditions. Further, by conditions 2) and 3).