Quantum interference classical fields

S. Mukamel I would like to make a comment regarding interference effects in quantum and classical nonlinear response functions [1, 2]. Nonlinear optical measurements may be interpreted by expanding the polarization P in powers of the incoming electric field E. To nth order we have... 

Even in typical disordered metals, the classical model for MR breaks down due to quantum corrections to conductivity, especially at low temperatures [13]. In the presence of weak disorder, carriers get localized by repeated back-scattering due to constructive quantum interference, and this is called weak localization (WL). A weak magnetic field can destroy this interference process and delocalize the carrier. As a result, a negative MR (resistivity decreases with field, usually less than 3%) can be observed at temperatures around 4 K. Another quantum correction to low temperature conductivity is due to e-e interaction contributions. This is mainly due to the fact that carriers interact more often when they diffuse slowly in random disorder potentials. The resistivity increases (usually less than 3%) with field due to e-e interaction contributions. Hence, the total low-field magnetoconductance (MC, Act) due to additive contributions from WL and e-e interactions is given by... 

Recent years have also witnessed exciting developments in the active control of unimolecular reactions [30,31]. Reactants can be prepared and their evolution interfered with on very short time scales, and coherent hght sources can be used to imprint information on molecular systems so as to produce more or less of specified products. Because a well-controlled unimolecular reaction is highly nonstatistical and presents an excellent example in which any statistical theory of the reaction dynamics would terribly fail, it is instmctive to comment on how to view the vast control possibihties, on the one hand, and various statistical theories of reaction rate, on the other hand. Note first that a controlled unimolecular reaction, most often subject to one or more external fields and manipulated within a very short time scale, undergoes nonequilibrium processes and is therefore not expected to be describable by any unimolecular reaction rate theory that assumes the existence of an equilibrium distribution of the internal energy of the molecule. Second, strong deviations Ifom statistical behavior in an uncontrolled unimolecular reaction can imply the existence of order in chaos and thus more possibilities for inexpensive active control of product formation. Third, most control scenarios rely on quantum interference effects that are neglected in classical reaction rate theory. Clearly, then, studies of controlled reaction dynamics and studies of statistical reaction rate theory complement each other. 

As a consequence of the collective motion of the neutral system across the homogeneous magnetic field, a motional Stark term with a constant electric field arises. This Stark term inherently couples the center of mass and internal degrees of freedom and hence any change of the internal dynamics leaves its fingerprints on the dynamics of the center of mass. In particular the transition from regularity to chaos in the classical dynamics of the internal motion is accompanied in the center of mass motion by a transition from bounded oscillations to an unbounded diffusional motion. Since these observations are based on classical dynamics, it is a priori not clear whether the observed classical diffusion will survive quantization. From both the theoretical as well as experimental point of view a challenging question is therefore whether quantum interference effects will lead to a suppression of the diffusional motion, i.e. to quantum localization, or not. 

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]. 

In this respect the superconducting quantum interference device (SQUID) is one of the most attractive developments. Many different designs have been fabricated and studied, and modem SQUIDs on the basis of YBa2Cu307 have reached field sensitivity and performance levels not far different from those known for devices produced with classical low temperature superconductors [13.3, 13.4]. 

The dressed state picture provides a simple explanation for the phase dependent quantum interference of the four-level system coupled by the multiple laser fields. The two resonant eoupling fields create a manifold of three dressed states, the semi-classical representation of which is given by... 

The advent of lasers in spectroscopy has made possible highly precise measurements of spectroscopic as well as of fundamental interest, Particular emphasis has been put onto the elimination of the Doppler effect, which was one of the main obstacles in classical spectroscopy. This can be achieved using well collimated atomic beams or non-linear field/atom interactions, which, combined with quantum interference methods, are capable of yielding a resolution beyond the natural linewidth. In historical perspective, these methods were developed because of the problems associated with the Doppler effect, the possibilities offered by the high intensity and narrow spectral band width of lasers and, most important, an ever persistent wish to obtain very high optical resolution. 

The stability matrix carries the necessary information related to the vicinity of the trajectory and provides an efficient numerical procedure for computing the response function. It plays an important role in the field of classical chaos the sign of its eigenvalues (related to the Lyapunov exponents) controls the chaotic nature of the system. Interference effects in classical response functions have a different origin than their quantum counterparts. For each initial phase-space point we need to launch two trajectories with very close initial conditions. [For 5(n) we need n trajectories.] The nonlinear response is obtained by adding the contributions of these trajectories and letting them interfere. 

In the classical case, when the field intensity exceeds a threshold value, the electron s motion becomes chaotic and strong excitation and ionization takes place. In the quantum case instead, interference effects may lead to the suppression of the classical chaotic diffusion, the so-called quantum dynamical localization, and in order to ionize the atom, a larger field intensity is required (see Fig. 1). 

If the quantum corrections to conductivity are actual the magnetoresistance related to the influence of the magnetic field on these corrections takes place [57-59]. The interference of electrons passing the closed part of trajectory in clockwise and counter-clockwise directions causes the so-called corrections to the conductivity. The phases of the electron wave functions in this case are equal and so this interference is constructive. Therefore, the probability for electrons to come back to the initial point doubles. This leads to the interference corrections which increase the classical resistance. The external magnetic field breaks the left-right symmetry, and the phases collected by the electron wave function while it passes trajectory in clockwise and... 

The phenomenon of optical interference is commonly describable in completely classical terms, in which optical fields are represented by classical waves. Classical and quantum theories of optical interference readily explain the presence of an interference pattern, but there are interference effects that distinguish the quantum (photon) nature of light from the wave nature. In this section, we present elementary concepts and definitions of both the classical and quantum theories of optical interference and illustrate the role of optical coherence. 

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. 

---