Reply to: “Comment on: ‘Effects of including the counterrotating term and virtual photons on the eigenfunctions and eigenvalues of a scalar photon collective emission theory’ [Phys. Lett. A 372 (2008) 2514]” [Phys. Lett. A 372 (2008) 5732]

We compute the emission amplitude for the collective emission from a sphere of identical atoms in the scalar photon theory for both the cases of the complex kernel (i.e. including virtual photons) and real kernel. We explicitly show that the single mode theory based on the real kernel neglects the effects of the different decay rates and frequency shifts associated with the eigenfunctions belonging to the same angular index but with different radial indices. We show that these effects modify, for k₀R?1, both the time dependence of the emission amplitude and its angular distribution, in clear contradiction to the assertions made by the Comment's authors.     

The metamorphosis in the emission angular profile from an inverted two-level atoms system

A rich variety of angular distributions in the cooperative emission from a sphere of inverted N two-level atoms are shown to result from the eigenstructure of the complex kernel of scalar photon theory exp(ik₀R)/(ik₀R). This angular distribution is sensitive both to the size of the sphere and to the instant of observation of the emission.     

Time-dependent directionality of cooperative emission after short pulse excitation

Cooperative emission from a sphere of N two-level atoms excited by a short pulse is shown to have a time-dependent angular distribution, with the possibility of reversal in the dominant direction of emission. This is a result of the different values of the frequency shifts and decay rates associated with the various collective eigenmodes of the atomic system obtained in a scalar photon model.     

Experimental Realization of Perfect Discrimination for Two Unitary Operations

We experimentally demonstrate perfect discrimination between two unitary operations by using the sequential scheme proposed by Duan et al.[Phys. Rev. Lett. 98 (2007) 100503] Also, we show how to understand the scheme and to calculate the parameters for two-dimensional operations in the picture of the Bloch sphere.     

Theory of nonlinear optical response of gauge invariant currents to linear and nonlinear couplings

When a gauge field interacts with a quantum condensed matter system, at first order of the gauge field it couples to the current operator of the electrons. Higher orders of the gauge field couple to electrons through other operators such as the stress tensor, etc. On the other hand, when one performs a measurement on a quantum system, not only the current operator, but also stress tensor operator of the electrons, etc. are hidden in the measurement, as they contribute to the gauge invariant current. We formulate a general problem of nonlinear optical response of the gauge invariant currents in presence of nonlinear couplings. We show that the new couplings along with new responses arising from field current have a very simple structure which can be formulated as time ordered multi-particle correlation functions. We also obtain their Lehman representation and thereby show that one need not use non-equilibrium formulations to deal with them. These new correlation functions suggest that in nonlinear optical response many new processes are possible. The experimental detection of the new terms in the current operator, and application corresponding multi-photon processes needs further theoretical and experimental investigations.     

Control of group velocity of light via magnetic field

We have theoretically studied the effects of quantum coherence in a driven quasi-degenerate two-level atomic system. We have shown that the quantum interference, which can be destructive or constructive, can be controlled by an externally applied magnetic field allowing one to implement both electromagnetically induced transparency and electromagnetically induced absorption in the same atomic system. Determined by frequency dispersion of the index of refraction of the system, the group velocity of light pulses ranges from ultra-slow to superluminal with changing of the magnitude of the magnetic field.     

Reconciling the eigenmode analysis with the Maxwell-Bloch equations approach to superradiance in the linear regime

The superradiance from a slab of inverted two-level atoms is theoretically analyzed in the linear regime from both the perspective of the expansion in eigenfunctions of the integral equation with the Lienard-Wiechert potential as kernel, and that of linearizing the Maxwell-Bloch equations. We show the equivalence of both approaches. We show that the so-called Reduced Maxwell-Bloch equations do not yield even approximately the correct solution when applied in the obvious way, but that they can be made to give the correct solution by adding a fictitious input field.     

Coherent control of laser pulse temporal duration: An experimental proposal

We present a study of temporal compression resulting from the coherent control peculiarities of electromagnetically induced transparency propagation dynamics. We discuss the crucial conditions required to accomplish temporal compression in an experiment with a sample of hot atoms.     

Interaction of a two-level atom with squeezed light

We consider a degenerate parametric oscillator whose cavity contains a two-level atom. Applying the Heisenberg and quantum Langevin equations, we calculate in the bad-cavity limit the mean photon number, the quadrature variance, and the power spectrum for the cavity mode in general and for the signal light and fluorescent light in particular. We also obtain the normalized second-order correlation function for the fluorescent light. We find that the presence of the two-level atom leads to a decrease in the degree of squeezing of the signal light. It so turns out that the fluorescent light is in a squeezed state and the power spectrum consists of a single peak only.     

Dynamics of quantum observables in entangled states

We examine the dynamics of a radiation field propagating through a nonlinear medium. A time series analysis of the mean photon number illustrates how an open quantum system interacting with a quantum environment can exhibit remarkably diverse ergodicity properties, both nonlinearity and departure from coherence playing a crucial role.     

Eigenvalues of collective emission in multi-slice slab configurations

We compute the eigenmodes of collective emission from multi-slice slab configurations, using the transfer matrix formalism. We elucidate within this formalism the phenomena of “Invisible Gaps” in multiple-slice configuration and of “Precocious Superradiance” in periodic structures previously observed in numerical solutions of Maxwell-Bloch equations.     

Quantum interferences in coherent population trapping of a cold double Λ-type atom

We investigate quantum interferences in coherent population trapping of a cold double Λ-type four-level atomic system driven by two counterpropagating laser fields. We study both decoherence and enhanced-coherence actions resulting from the multi-transition pathways in building up the trapping state, and analyze the system operating with and without external coherences in various configurations of the atomic dipole moments.     

Anomalous Effects of Driving Field Linewidth on a One-Atom Dressed-State Laser

We examine the effects of driving field linewidth on a one-atom dressed state laser. Unexpectedly, the linewidth leads to anomalous effects on the cavity field. The mean photon number of the cavity field is raised or the normalized variance is reduced to a certain degree as the linewidth increases for an appropriate range of parameters. The responsible mechanism is attributed to the fluctuation-induced modification of the electromagnetic reservoir where the atom stays.     

Quantum theory of super-resolution for optical systems with circular apertures

We present the two-dimensional quantum theory of super-resolution, applicable for a large variety of optical systems with circular pupils. Our theory is formulated in terms of circular prolate spheroidal functions which form the eigen basis of two-dimensional imaging system with circular pupils. We provide, in particular, analytical and numerical results for the point-spread function characterizing reconstruction of optical objects with super-resolution from diffraction-limited images. We evaluate the super-resolution factor as a function of the signal-to-noise ratio in the input object for coherent light and multimode squeezed light.     

The single photon superradiance from the eigenmode analysis

Using the eigenmode analysis of the scalar photon theory, I compute the probability of the atoms remaining excited and the probability for the atoms remaining in the initial quantum state of a system of two-level atoms cloud in a sphere initially prepared to radiate in the forward direction, i.e., the single photon superradiance problem. The convergence in the results obtained for increasingly larger radii for the sphere suggests that the asymptotic limits for these quantities are obtained for a sphere with a radius equal to six times the resonant wavelength. I predict the maximal value of the probability of secondary excited states from large spheres at 17.1%.     

Evolution of population in an equispaced three-level ladder-type atomic system

Transient response of nearly equispaced three-level ladder-type atomic system with a broad-band squeezed vacuum (SV) is investigated. We focus our attention in the interplay between the quantum interference and the squeezed field on the population distribution. It is shown that an atomic population inversion can be attained on one of the optical transitions due to the SV. Additionally, we show, with the proper value of the relative phase, the SV can also lead to unexpected population inversion on the transition between two different levels.     

Duration-Dependent Asymmetry in Photoionization of H atoms in Few-Cycle Laser Pulses

The photoionization of H atoms irradiated by few-cycle laser pulses is studied numerically. The variations of the total ionization, the partial ionizations in opposite directions, and the corresponding asymmetry with the carrier-envelope phase in several pulse durations are obtained. We find that besides a stronger modulation on the partial ionizations, the change of pulse duration leads to a shift along carrier-envelope （CE） phase in the calculated signals. The phase shift arises from the nonlinear property of ionization and relates closely to the Coulomb attraction of the parent ion to the ionized electron. Our calculations show good agreement with the experimental observation under similar conditions.