By George Boole

ISBN-10: 1148211837

ISBN-13: 9781148211831

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We will discuss later in Chapter 6 some experimental schemes for the nondirect measurement of the average value of the ﬁeld amplitude and its ﬂuctuations. The measurement schemes are diﬀerent form that discussed above, referred as homodyne or heterodyne detection techniques. In closing this section, we brieﬂy comment about the need to measure higher order correlation functions. Although the foregoing detection schemes all measure the correlation functions only to the second order, the complete knowledge of a detected ﬁeld requires measurements of the correlation functions to all orders of the ﬁeld amplitude.

B) Find the average Iˆ 3 of a quantum ﬁeld in terms of the correlation functions G(3) , G(2) and G(1) . (c) What would be the values of Iˆ 2 and Iˆ 3 in terms of the correlation functions if the ﬁeld was classical? 5 Write the quantum ﬁrst- and second-order correlation functions in terms of the creation and annihilation operators. 6 Consider the joint probability of two simultaneous detections of a stationary or slowly varying ﬁeld of intensity I (t): P2 (t) t1 t2 = η2 I (t)I (t) t1 t2 . (a) Show that the joint probability can be written as P2 (t) t1 t2 = η2 I (t) 2 [1 + λ(t)] t1 t2 , where λ(t) is the normalized intensity-ﬂuctuations correlation function ( I (t))2 .

However, for strong couplings (gk to the couplings inside optical cavities, they can have important dynamical consequences. For example, the Jaynes–Cummings model can exhibit chaotic dynamics, called “quantum chaos”, that is, the question of how classical chaos might carry over into the corresponding quantum dynamics [5]. Some other interesting eﬀects predicted when the RWA is not made include bifurcations in the phase space [6], a ﬁne structure in the optical Stern–Gerlach eﬀect [7], and entanglement between two atomic ensembles even if there is no initial excitation present in the system [8].

### A Treatise on Differential Equations by George Boole

by George

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