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Physical Review A

Superradiance from a cylindrically shaped cloud of N two-level atoms

Jamal T. Manassah

Published Nov 15, 2010• 7 citations
Publisher

Abstract

Using the eigenmodes of the scalar-photon kernel of interacting $N$ two-level atoms in a cylindrical configuration, I obtain the time dependence of the superradiant emission for this geometry in two cases of interest. In particular, I consider first the case when one or two eigenmodes dominate the system's dynamics for an inverted system which has been initially prepared in an isotropic state and show that for certain values of the radius of the cylinder ($r=m\ensuremath{\lambda},m \mathrm{integer}$) one obtains temporal modulation in the state function. In the second case of interest, I show that the angular distribution of the state function of a system initially prepared in a phased state appears to be chaotic; however, I show that this behavior is due to the high multiplicity of modes having close values for their amplification coefficients.

DOI
10.1103/physreva.82.053816

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Using the eigenmodes of the scalar-photon kernel of interacting NNN two-level atoms in a cylindrical configuration, I obtain the time dependence of the superradiant emission for this geometry in two cases of interest. In particular, I consider first the case when one or two eigenmodes dominate the system's dynamics for an inverted system which has been initially prepared in an isotropic state and show that for certain values of the radius of the cylinder (r=m\ensuremathλ,mintegerr=m\ensuremath{\lambda},m \mathrm{integer}r=m\ensuremathλ,minteger) one obtains temporal modulation in the state function. In the second case of interest, I show that the angular distribution of the state function of a system initially prepared in a phased state appears to be chaotic; however, I show that this behavior is due to the high multiplicity of modes having close values for their amplification coefficients.
Previous numerical work on the spatiotemporal distributions of the atomic polarization and of the difference in levels population of a one-dimensional system of two-level atoms initially inverted strongly suggested that the dynamics of the slab atomic ensemble in the linear regime of superradiance may best be treated by an eigenmode analysis. This method proved effective in obtaining both the time dependence of the atomic polarization and the spectral distribution of the emitted radiation from the slab in a number of physically interesting problems in the linear regime. In the present paper, I use the eigenfunction expansion technique to reduce the Maxwell-Bloch system of partial differential equations into a system of coupled first-order ordinary differential equations whose solutions clearly detail the contributions of the different modes to the system dynamics in the nonlinear regime of superradiant emission. I compute also the delay in time of the peak of the superradiant burst and its magnitude as functions of the ratio of the longitudinal decay rate to the transverse decay rate of the system.
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%.
Interaction with a multimode quantized radiation field of an assembly of NNN two-level atoms contained in a small volume is considered for the special cases of (i) highly excited, (ii) weakly excited, and (iii) superradiant atomic assemblies, and solutions of equations of motion are obtained in closed form. For (i) and (ii) we show that, if the atomic assembly is in the Dicke state ∣r,m>|r,m>∣r,m>, the spontaneously emitted radiation is Lorentzian, and its spectral width and the numerical value of the Lamb shift are r+∣m∣r+|m|r+∣m∣ times that for a single excited atom. For (i) the sign of Lamb shift is opposite to that for (ii) and for a single excited atom. Change in the frequency shift gives chirping in the pulse radiation emitted spontaneously by an assembly of atoms. We obtain the reduced density operator for a single mode and for the complete spontaneous emitted radiation, and discuss its characteristic function, weight function in diagonal representation, coherence functions, and normalized variance. We find that the emitted radiation is chaotic if all atoms are excited or if the assembly is weakly excited and is in thermal equilibrium. For a superradiant assembly, the radiation is coherent; it is only amplitude coherent if the initial atomic density operator is nondiagonal. We also study stimulated emission and scattering for the cases (i) and (ii). For case (iii) these phenomena do not take place.
The collective nature of light emission by atomic ensembles yields fascinating effects such as superradiance and radiation trapping even at the single-photon level.Light emission is influenced by virtual transitions and the collective Lamb shift which yields peculiar features in temporal evolution of the atomic system.We study how two-dimensional atomic structures collectively emit a single photon.Namely, we consider spherical, cylindrical, and spheroidal shells with two-level atoms continuously distributed on the shell surface and find exact analytical solutions for eigenstates of such systems and their collective decay rates and frequency shifts.We identify states which undergo superradiant decay and states which are trapped and investigate how size and shape of the shell affects collective light emission.Our findings could be useful for quantum information storage and the design of optical switches.
The collective nature of light emission by atomic ensembles yields fascinating effects such as superradiance and radiation trapping even at the single-photon level. Light emission is influenced by virtual transitions and the collective Lamb shift which yields peculiar features in temporal evolution of the atomic system. We study how two-dimensional atomic structures collectively emit a single photon. Namely, we consider spherical, cylindrical, and spheroidal shells with two-level atoms continuously distributed on the shell surface and find exact analytical solutions for eigenstates of such systems and their collective decay rates and frequency shifts. We identify states which undergo superradiant decay and states which are trapped and investigate how size and shape of the shell affects collective light emission. Our findings could be useful for quantum information storage and the design of optical switches.
The eigenmodes of a nanosphere configuration where the core is an ensemble of two-level atoms with resonance frequency ${\ensuremath{\omega}}_{C}$, and the shell is an ensemble of two-level atoms but with shifted resonant frequency ${\ensuremath{\omega}}_{B}$ are obtained for different values of the ratio of the core radius to the shell outer radius as function of $({\ensuremath{\omega}}_{C}\ensuremath{-}{\ensuremath{\omega}}_{B})$. I show that the eigenmodes belong to one of two branches. The dominant decay mode for superradiant emission may belong to either branch, depending on the system's parameters. In instances of configurations where the cooperative decay rate values for the two branches become equal, the cooperative exponential amplification (decay) is accompanied by temporal oscillations.