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Physical Review A

Field distribution in a generalized geometric radiation state

Hanan Ali Batarfi, M. Sebawe Abdalla, A.‐S. F. Obada, Saad S. M. Hassan

Published Mar 1, 1995• 11 citations
Publisher

Abstract

Both the Wigner function and the field distribution function in the generalized geometric state of the radiation field, which were recently introduced by us [A.-S. F. Obada, S. S. Hassan, R. R. Puri, and M. S. Abdalla, Phys. Rev. A 48, 3174 (1993)], are calculated exactly and examined in both the phase-averaged and the nonaveraged cases.
DOI
10.1103/physreva.51.2644

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We propose a variation of the Jaynes-Cummings Model (JCM), which consists of a Two-Level Atom (TLA) within a cross cavity configuration. This new geometry is a suitable tool to understand the origin of new QED features that the original JCM does not reveal, showing revivals in cases where they are not expected to appear. Full Text: PDF References E.T. Jaynes, F.W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser", Proc. IEEE 51, 1 (1963). CrossRef M. Wilkens, P. Meystre, "Nonlinear atomic homodyne detection: A technique to detect macroscopic superpositions in a micromaser", Phys. Rev. A 43, 7 (1991). CrossRef I.V. Jyotsna, G.S. Agarwal, "The Jaynes-Cummings model with continuous external pumping", Opt. Commun. 99, 344 (1993). CrossRef M.S. Abdalla, M. Abdel-Aty, A.-S.F. Obada, "Quantum entropy of isotropic coupled oscillators interacting with a single atom", Opt. Comm. 211, 225 (2001). CrossRef M.A. Marchiolli, R.J. Missori, "Qualitative aspects of entanglement in the Jaynes?Cummings model with an external quantum field", J. Phys. A: Math. Gen. 36, 49 (2003). CrossRef J.C. García-Melgarejo, Ph.D. Thesis. INAOE (2015). J.R. Ackerhalt, K. Rzazewski, "Heisenberg-picture operator perturbation theory", Phys. Rev. A 12, 6 (1975) CrossRef J.H. Eberly, N.B. Narozhny, J.J. Sanchez-Mondragon, "Periodic Spontaneous Collapse and Revival in a Simple Quantum Model", Phys. Rev. Lett. 44, 20 (1980). CrossRef N.B. Narozhny, J.J. Sanchez-Mondragon, J.H. Eberly, "Coherence versus incoherence: Collapse and revival in a simple quantum model", Phys. Rev. A 23, 1 (1981). CrossRef L. Allen, J.H. Eberly, Optical resonance and two-level atoms (New York, Wiley 1975). A. Joshi, R.R. Puri, "Effects of the Binomial Field Distribution on Collapse and Revival Phenomena in the Jaynes-Cummings Model", J. Modern Optics 34, 11 (1987). CrossRef
Using the Wigner-Vlasov formalism, an exact three-dimensional solution of the Schr\"odinger equation for a scalar particle in an electromagnetic field is constructed. Electric and magnetic fields are nonuniform. According to the exact expression for the wave function, the search for two types of Wigner function is conducted. The first function is the usual Wigner function with a kinetic momentum. The second Wigner function is constructed on the basis of the Weyl-Stratonovich transform [Javanainen et al., Phys. Rev. A 35, 2791 (1987); Nedjalkov et al., Phys. Rev. B 99, 014423 (2019)]. It turns out that the second function, unlike the first one, has areas of negative values for wave functions with the Gaussian distribution (Hudson's theorem). An example of an electromagnetic quantum system described by a non-Gaussian wave function has successfully been found. The second Wigner function is positive over the whole phase space for the non-Gaussian wave function. This result is analogous to the Hudson theorem for the gauge invariant Wigner function. On the one hand, knowing the Wigner functions allows one to find the distribution of the mean momentum vector field and the energy spectrum of the quantum system. On the other hand, within the framework of the Wigner-Vlasov formalism, the mean momentum distribution and the magnitude of the energy are initially known. Consequently, the mean momentum distributions and energy values obtained according to the Wigner functions can be compared with the exact momentum distribution and energy values. This paper presents this comparison and describes the differences. The Vlasov-Moyal approximation of average acceleration flow has been built in phase space for a quantum system with an electromagnetic field. The obtained approximation makes it possible to cut the Vlasov chain off at the second equation and also to analyze the Boltzmann H-function evolution. By averaging the Vlasov-Moyal approximation over momentum space we can derive an expression for the electromagnetic force of the classical system. This averaging makes the high-order quantum terms disappear from the ``motion equation.''
The Fourier series representation of the equilibrium quasiprobability density function ${W}_{S}(\ensuremath{\vartheta},\ensuremath{\varphi})$ or Wigner function of spin ``orientations'' for arbitrary spin Hamiltonians in a representation (phase) space of the polar angles $(\ensuremath{\vartheta},\ensuremath{\varphi})$ (analogous to the Wigner function for translational motion) arising from the generalized coherent state representation of the density operator is evaluated explicitly for some nonaxially symmetric problems including a uniaxial paramagnet in a transverse external field, a biaxial, and a cubic system. It is shown by generalizing transition state theory to spins [i.e., calculating the escape rate using the equilibrium density function ${W}_{S}(\ensuremath{\vartheta},\ensuremath{\varphi})$ only] that one may evaluate the reversal time of the magnetization. The quantum corrections to the transition state theory escape rate equation for classical magnetic dipoles appear both in the prefactor and in the exponential part of the escape rate and exhibit a marked dependence on the spin number. Furthermore, the phase-space representation allows us to estimate the switching field curves and/or surfaces for spin systems because quantum effects in these fields can be estimated via Thiaville's geometrical method [Phys. Rev. B 61, 12221 (2000)] for the study of the magnetization reversal of single-domain ferromagnetic particles. The calculation is accomplished (just as the determination of the equilibrium quasiprobability distributions in the phase space of the polar angles) by calculating switching field curves and/or surfaces using the Weyl symbol (ccc-number representation) of the Hamiltonian operator for given magnetocrystalline-Zeeman energy terms. Examples of such calculations for various spin systems are presented. Moreover, the reversal time of the magnetization allows us to estimate thermal effects on the switching fields for spin systems.