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Physical Review A

Two-ion superradiance theory

Richard G. Brewer

Published Oct 1, 1995• 25 citations
Publisher

Abstract

A superradiance theory is developed for two identical hydrogenic ions (four states each) in a microtrap, as in recent experiments. The ions oscillate (micromotion) due to the trap's radio frequency (rf) electric quadrupole field. One signature of superradiance is a deviation of the two-ion average upper state decay rate \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{} from the one-ion value \ensuremath{\gamma}. A master equation is derived, giving a fractional change in the upper state lifetime \ensuremath{\gamma}/\ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}-1=sinkR/2kR [${\mathit{J}}_{0}^{2}$(z)-2${\mathit{J}}_{1}^{2}$(z)+...]cos\ensuremath{\Phi}, where k=2\ensuremath{\pi}/\ensuremath{\lambda}, \ensuremath{\lambda} is the emission wavelength, R is the ion-ion distance, ${\mathit{J}}_{\mathit{n}}$(z) is a Bessel function of integer order n, and z=ka, a is the ion amplitude of motion, and \ensuremath{\Phi} is the two-ion relative phase due to the preparation. In the Lamb-Dicke regime, a, ${\mathit{J}}_{0}^{2}$(z)\ensuremath{\approxeq}1 and thus superradiance is not influenced significantly by ion motion. This damped sinusoid is diluted by the factor of 1/2 due to destructive interference. Superradiance vanishes in the absence of coherent preparation, e.g., with inversion, as indicated by the time evolution of the two-ion dipole correlation function. Fringes and a beat at the rf are predicted in forward scattering, the most elementary form of optical free induction decay.

DOI
10.1103/physreva.52.2965

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A superradiance theory is developed for two identical hydrogenic ions (four states each) in a microtrap, as in recent experiments. The ions oscillate (micromotion) due to the trap's radio frequency (rf) electric quadrupole field. One signature of superradiance is a deviation of the two-ion average upper state decay rate \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{} from the one-ion value \ensuremath{\gamma}. A master equation is derived, giving a fractional change in the upper state lifetime \ensuremath{\gamma}/\ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}-1=sinkR/2kR [J02{\mathit{J}}_{0}^{2}J02​(z)-2J12{\mathit{J}}_{1}^{2}J12​(z)+...]cos\ensuremath{\Phi}, where k=2\ensuremath{\pi}/\ensuremath{\lambda}, \ensuremath{\lambda} is the emission wavelength, R is the ion-ion distance, Jn{\mathit{J}}_{\mathit{n}}Jn​(z) is a Bessel function of integer order n, and z=ka, a is the ion amplitude of motion, and \ensuremath{\Phi} is the two-ion relative phase due to the preparation. In the Lamb-Dicke regime, a, J02{\mathit{J}}_{0}^{2}J02​(z)\ensuremath{\approxeq}1 and thus superradiance is not influenced significantly by ion motion. This damped sinusoid is diluted by the factor of 1/2 due to destructive interference. Superradiance vanishes in the absence of coherent preparation, e.g., with inversion, as indicated by the time evolution of the two-ion dipole correlation function. Fringes and a beat at the rf are predicted in forward scattering, the most elementary form of optical free induction decay.
We report the first observation of a microscopically resolved interaction between two trapped atoms. This is also the first observation of two-ion superradiance and subradiance, the most elementary example of Dicke's theory. Its signature is the variation of the spontaneous emission decay rate of a laser-cooled two-ion crystal, +1.5+1.5%+1.5 and $\ensuremath{-}1.2%$, as the ion-ion distance is changed from 1380 to 1540 nm, in agreement with a no-free-parameter theory. The experiment uses an ion microtrap (80 \ensuremath{\mu}m radius) with planar electrodes and a new few-atom technique for measuring lifetimes.
Bragg scattering and superradiance, always considered independently, are examined jointly for the case of two trapped ions. The two phenomena are seen to be different representations of the same basic two-ion interference resulting from a common coherent optical preparation. Solutions of a master equation for two four-level atoms show that fringes (spatial interference) and a variation of the spontaneous emission lifetime with ion-ion distance (temporal interference) can occur either simultaneously or separately. Unlike a classical calculation, not all scatterings result in interference, a quantum effect not considered in interpreting recent experiments.
We discuss a radio-frequency (rf) ion trap, the rotating-radio-frequency (rotating-rf) trap, in which the motion of a charged particle is described by trigonometric functions rather than the usual Mathieu functions of a normal rf trap. In the rotating-rf trap, a rotating quadrupole electric field confines charged particles, whereas in a normal rf trap, an oscillating quadrupole electric field does. Ion motion in a rotating-rf trap is a superposition of two nondegenerate circular secular motions and two corresponding circular micromotions. The cases of applying a uniform dc magnetic field or a quadrupole dc electric field in addition to the rotating-rf field are also discussed. Confinement in a rotating-rf trap can be tighter than in a normal linear rf trap with the same experimental parameter values.
We discuss a radio-frequency (rf) ion trap, the rotating-radio-frequency (rotating-rf) trap, in which the motion of a charged particle is described by trigonometric functions rather than the usual Mathieu functions of a normal rf trap. In the rotating-rf trap, a rotating quadrupole electric field confines charged particles, whereas in a normal rf trap, an oscillating quadrupole electric field does. Ion motion in a rotating-rf trap is a superposition of two nondegenerate circular secular motions and two corresponding circular micromotions. The cases of applying a uniform dc magnetic field or a quadrupole dc electric field in addition to the rotating-rf field are also discussed. Confinement in a rotating-rf trap can be tighter than in a normal linear rf trap with the same experimental parameter values.
The problem of ion motion and laser cooling in a radio frequency quadrupole trap is considered. It is shown by a simple numerical calculation that for ion motion in the absence of laser cooling the effect of the time varying potential v(R-bar)cos(Omega x t) can be approximated by a time independent effective potential whenever the frequency of oscillation of the electric field, Omega is sufficiently greater than a resonant frequency, (Omega)squared = (sq. root of (2) times eVo)/M were e is the electronic charge, Vo is the maximum potential applied to the trap electrodes, and M is the mass of the ion. Equations for the Hamiltonian matrix elements in the presence of laser cooling, in the Lambe-Dicke limit are derived and studied to determine the cooling transitions. The equations of motion for the density matrix in the Lambe-Dicke limit are derived using a simple representation of spontaneous emission. (Theses) (Author)
Advances in laser cooling and trapping have made possible a new kind of microscopic atomic physics experiment in which each atom is detected individually and in which the interatomic distances are known and controlled on a sub-micron scale. This chapter shows that the triplet and singlet states represent states in which the dipole moments of the two atoms are either correlated or anticorrelated. Since the two atoms’ wave-functions are identical, so are all their physical observables, including the dipole moments. The chapter describes the theory of superradiant and subradiant spontaneous emission in sufficient detail to understand current experiments. An experiment will in general prepare a partially coherent state described by a density matrix. The chapter discusses a simple physical picture of singlet and triplet states in both wave function and density matrix approaches. It also describes a classical model of the decay rate oscillations, and also shows how the experimental signal depends on the excitation conditions.