Second-order correlation function

By rewriting all of the (5-functions in this manner, the problem of taking the expectation of equation 7.110 has been conveniently reduced to taking expectations of simple products of ct s. Schulman and Seiden go on to show exactly how such expectations can be expressed as functions of second-order correlation functions (or cumulants), <0 02>c = — <02> [schul78] ... 

The physical meaning of is a two-photon detection amplitude, through which one can express the second-order correlation function G j = y l J vIh7 [Scully 1997]. The knowledge of the two-photon wavefunction allows one to calculate the amplitudes of state vector (24) via the two dimensional Fourier transform of 4, at t = t ... 

Being of fourth order in the particle operators, the static structure factor provides a measure of second order correlation. Defining the normalized second order correlation function as... 

The second order correlation functions can be obtained in the same manner giving... 

In this review we discuss the major effects resulting from the modification of spontaneous emission by quantum interference. We begin in Section II by presenting elementary concepts and definitions of the first- and second-order correlation functions, which are frequently used in the analysis of the inter-... 

The analysis of the interference phenomenon can be extended into higher-order correlation functions. The first experimental demonstration that such correlations exist in optical fields was given by Hanbury-Brown and Twiss [14], who measured the second-order correlation function of a thermal held. 

We can dehne the normalized second-order correlation function as... 

In the plane-wave approximation, the second-order correlation function (20) can be written as... 

The second-order correlation function has coherence properties completely different from those of the hrst-order correlation function. An interference pattern can be observed in the second-order correlation function, but in contrast to the hrst-order correlation function, the interference appears between two points located at Ri and R2. Moreover, an interference pattern can be observed even if the helds are produced by two independent sources for which the phase difference completely random [15]. In this case the second-order correlation function (22) is given by... 

Clearly, the second-order correlation function exhibits a cosine modulation with the separation Ri — R2 of the two detectors. This is an interference, although it involves a correlation function that is of the second order in the intensity. Hence an interference pattern can be observed even for two completely independent fields. Similar to the first-order correlation function, the sharpness of the fringes depends on the relative intensities of the fields. For equal intensities, I = I2 = /o, the correlation function (23) reduces to... 

In the case of the quantum description of the field, the first- and second-order correlation functions are defined in terms of the normally ordered field operators E(+) and E as... 

The correlation functions (28) described by the field operators are similar to the correlation functions (6) and (20) of the classical field. A closer look into Eqs. (6), (20), and (28) could suggest that the only difference between the classical and quantum correlation functions is that the classical amplitudes E (R, f) and E(R, f) are replaced by the field operators E (R, t) and eW(R,(). This is true as long as the first-order correlation functions are considered, where the interference effects do not distinguish between the quantum and classical theories of the electromagnetic field. However, there are significant differences between the classical and quantum descriptions of the field in the properties of the second-order correlation function [16]. 

It follows from Eq. (31) that the second-order correlation function vanishes when... 

If we assume that initially the field is in the vacuum state, then the free-field part Eq+) (R, t) does not contribute to the expectation values of the normally ordered field operators, and we obtain the following expressions for the first-and second-order correlation functions... 

It is seen from Eq. (36) that the second-order correlation function of the EM field emitted from the two systems depends on various two-time dipole correlation functions of the form S t Sf (t2)Sj (t2)SY h)). The functions are proportional to the probabilities of detecting two photons emitted from the same (i = j) or different (/ / /) bare systems. For example, the correlation function (S (t )S2 (t2)S2 (t2)S] (fi)) is proportional to the probability of detecting a photon at time t2 emitted from system 2 if a photon emitted from the system 1 was detected at time fr. 

The second-order correlation function (36) also depends on the dipole correlation functions of the form (S (fi)Sj(t2)Si(t2)S2 (t ), which result from correlations of photons emitted from a superposition of the bare systems. 

As we have shown in Sec. III. A, the second-order correlation function of the fluorescence field depends on correlation functions of the atomic dipole moments (S+(f)S+(f + x)Sy(t)Sj (t)), which correspond to different processes including photon emissions from a superposition of the excited levels. Therefore, we write the correlation functions G (R, t) and G (R, t R, t + x) in terms of the symmetric and antisymmetric superposition states as... 

It is seen that in the bases of the symmetric and antisymmetric states, there are three terms contributing to the first- and the second-order correlation functions. The first term is from the transition. v) > 2), the second is from... 

If the photons emitted from the excited states to the ground state are distinguishable, such as by having significantly different polarizations or frequencies, then the following normalized second-order correlation functions of the steady-state fluorescence intensity can be written as [57]... 

As we have seen from Figs. 14-16, the effect of quantum interference on the second-order correlation function, is very sensitive to the splitting A of the excited levels. For degenerate excited levels (A = 0), the photon emissions are similar to those of a two-level atom, independent of quantum interference. For large splittings, the correlation functions g (x) and gj (x). ij = 1,3 are smaller than unity for all times x, while (x) exhibits strong correlations (gj (x) 3> 2) for x (2 20,)-17i, which decay at a very low rate. 

Substituting the interaction Hamiltonian (7), we find that the evolution of the density operator depends on the second order correlation functions of the reservoir operators. We assume that a part of the reservoir modes is in a squeezed vacuum state for which the correlation functions are given by Eq. (93). 

One of the other useful quantities to characterize dynamical processes in SMS is the fluorescence intensity correlation function, also called the second-order correlation function, defined by [73, 74]... 

Fig. 6.66 Background-free measurement of the second-order correlation function by...

The second-order correlation function is a constant G r) = 1 for cases of both completely uncorrelated light and strictly monochromatic light. 

This shows that the autocorrelation function C(t) of the photoelectron current is directly related to the second-order correlation function G r) of the light field. 

As proved by Siegert [939] for optical fields with a Gaussian intensity distribution (7.60), the second-order correlation function G x) is related to G x) by the Siegert relation... 

We begin this section by considering a two-quantum absorption detector initially in the ground state. The detector response at the space-time point X. = t, may be written in terms of the second-order correlation function [7.19, 34, 36, 41], and is given by... 

It is not difficult to associate various second-order correlation functions G - x Xf,XcX ) = abcd] with (7.15). (When two beams are present, we must consider a space-time point for each of the beams so that the index in takes on two values [7.34].) Thus, the first term, ls may be associated with [1111], the second with [2222], the third with [1221] and [2112], the fourth with the four permutations of [1112], the fifth with the four permutations of [2221], and the sixth with the four permutations of [1212], with h + c. The coefficient of each term in (7.15) is there fore equal to the number o f permutations in the appropriate form of the correlation function for that term. The physical interpretation follows immediately the first two dc terms in (7.15) arise from the absorption of two monochromatic photons, both from the same beam. The third dc term, which exists in two permutations with b = c, arises from the two ways in which two single monochromatic photons can be absorbed, one from each beam. The fourth and fifth terms correspond to the absorption of a single monochromatic photon from one of the beams plus a single nonmonochromatic photon which must be associated with both beams. These terms therefore contribute currents at the difference frequency (tOi —analogy with the single-quantum heterodyne interference term [7.12-14]. The final term corresponds to the absorption of two nonmonochromatic photons, and therefore varies at double the difference frequency, i.e., at 2(o), —coj) clearly there is no analogous process possible in the one-quantum case. 

From the foregoing, it is clear that the double-quantum current can be calculated for a variety of configurations involving different relative time scales, angular separations, polarization properties, and statistical characteristics. Some additional examples are treated in [7.34]. Clearly, the second-order correlation functions of the field play an important role in determining the magnitude of the signal, in distinction to the one-quantum case. 

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