Correlation functions quantum interference

The focus of this chapter is exploration of the ability of mixed quantum classical approaches to capture the effects of interference and coherence in the approximate dynamics used in these different mixed quantum classical methods. As outlined below, the expectation values of computed observables are fundamentally non-equilibrium properties that are not expressible as equilibrium time correlation functions. Thus, the chapter explores the relationship between the approximations to the quantum dynamics made in these different approaches that attempt to capture quantum coherence. 

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 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]. 

The visibility of the interference pattern of the intensity correlations provides a means of testing for quantum correlations between two light fields. Mandel et al. [18] have measured the visibility in the interference of signal and idler modes simultaneously generated in the process of degenerate parametric downconver-sion, and observed a visibility of about 75%, that is a clear violation of the upper bound of 50% allowed by classical correlations. Richter [19] has extended the analysis of the visibility into the third-order correlation function and also found significant differences in the visibility of the interference pattern of the classical and quantum fields. 

We concentrate on the role of quantum interference in the correlation of photons emitted from a coherently driven V-type atom, recently analyzed by Swain et al. [58]. We calculate the normalized second-order two-time correlation function g (R, t R, t + x) for the fluorescent field emitted from a three-level V-type atom driven by a coherent laser field coupled to both atomic transitions. The fluorescence field is observed by a single detector located at a point R = RR, where R is the unit vector in the direction of the observation. 

In Fig. 16, we plot the correlation functions (148) and (149) for nondegenerate transitions with A = 5T. Again, the solid line represents p = 0.99 and the dashed line, p 0. It is apparent from the graphs that with quantum interference (p = 0.99), there are very strong correlations of photons on the, v) —> 12) transition, whereas the photons are strongly anticorrelated on the ja) > 2) transition. The correlation function g (x) oscillates with 2 /2i l and attains the maximum value at time r = (2 /20) V. Moreover, the correlations decay at a very low rate and it takes a time in excess of 30071 before it gets close to unity. The correlation function 15 (x) oscillates with /2il and in the presence of quantum interference is smaller than unity for all times, whereas the values can be larger than unity, with the maximum value of around 2.8, for... 

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. 

In Fig. 14 the correlation function (q (t)q (O) is shown for the nonlinear potential in Eq. (3.85) at /3 = 10. This correlation function presents another nontrivial test of the various approximate methods because, classically, it can have no negative values while, quantum mechanically, it can be negative due to interference effects. Clearly, only the cumulant method can describe the latter effects. The classical result is extremely poor for this low-temperature correlation function. The CMD with semiclassical operators method also cannot give a correlation function with negative values in this case. This feature of the latter method arises because the correlation of the two operators at different times is ignored when the Gaussian averages are performed. Consequently, the semiclassical operator approximation underestimates the quantum real-time interference of the two operators and thus fails to... 

To test the methods outlined in Section III.B.3 for calculating general correlation functions in the phase-space centroid perspective, the correlation function >l(f)B(0)), where A= pq and B = qp, was studied [5], The results of this calculation are shown in Fig. 15 for the nonlinear potential in Eq. (3.85) at /8 = 10. The classical MD result is, as expected, extremely inaccurate for this low temperature. The CMD with semiclassical operators method does not reproduce the amplitude and negative values of this correlation function as well. On the other hand, the cumulant method can describe the quantum interference effects for this correlation function, and it appears to do so quite well. 

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. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

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