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.
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.
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 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.
The problem of the interaction of N atoms with a laser beam and vacuum modes can give rise to many interesting phenomena concerning the spontaneous emission of light and its propagation in the medium. The cooperative effects, for example, such as superadiance and subradiance, are effects related to the coherence created between the atoms when a photon is emitted spontaneously by a single excited atom. Superradiance can be defined as the enhancement of the spontaneous emission due to constructive interference of the scattered light. Its counterpart, subradiance, is the trapping of some remaining light due to destructive interference. In cold atoms, some previous theoretical works predict and characterize these two cooperative effects in a large and diluted atomic cloud, in the regime of low intensities and large detunings of the incident laser. The theoretical model is a coupled-dipole model for two-level atoms driven by a low-intensity field and in the scalar approach. The experiment consists in measuring the super- and subradiant decay rates from the temporal emitted intensity after the switch off of the incident laser in the steady state. Our experimental setup consists in a magneto-optical trap of rubidium 87 atoms at large resonant optical thicknesses. A probe beam excites the atoms close to the D2 line. The intensity emitted is detected by a single photon detector with no afterpulsing and a calibration procedure allows us to determine the resonant optical thickness of the cloud and its temperature. In this work, we report the experimental observation of super- and subradiance in a large cloud of cold atoms. For subradiance, the main result is the linear evolution of the characteristic time with the resonant optical thickness of the cloud and its independence of the detuning. For superradiance, we observe superradiance out of the forward direction. We verify the validity of our interpretations with the predictions of the coupled-dipole model. Finally, we discuss the interplay of subradiance and radiation trapping, as well as theoretical predictions for: a setup of a phased cloud, to control the subradiant amplitude emission; and temperature effects, where subradiance is shown to be robust in a large range of temperatures.
The usual form of electromagnetic forces is given by but for neutral atoms q = 0. The next order of force is the dipole term, but this also vanishes because neutral atoms have no inherent dipole moment. However, a dipole moment can be induced by a field, and this is most efficient if the field is alternating near the atomic resonance frequency. Since these frequencies are typically in the optical range, dipole moments are efficiently induced by shining nearly resonant light on the atoms. Laser cooling and trapping rely on the interaction between laser light and atoms to exert a controllable force on the atoms, and many sophisticated schemes have been developed using the special properties of the interaction. This chapter begins with the simplest schemes for exerting optical forces on atoms, namely a single-frequency light field interacting with a two-level atom. Although this is the simplest possible scheme, it is pedagogically valuable because it shows many of the features that will be encountered further on. The ultimate temperature using such a scheme is limited, but lower temperatures can be achieved with multilevel atoms. This topic is addressed later in the chapter. Two kinds of optical forces When an atom absorbs nearly resonant light, it makes a transition to the excited state, and its subsequent return to the ground state can be either by spontaneous or by stimulated emission. The nature of the optical force that arises from these two different processes is quite different, and will be described separately [1]. The spontaneous emission case is different from the familiar quantum mechanical calculations using state vectors to describe the state of the system, because spontaneous emission causes the state of the system to evolve from a pure state into a mixed state (see Exercise 6.9). Spontaneous emission is an essential ingredient for the dissipative nature of the optical forces in monochromatic light, and the density matrix is needed to describe it (see Chap. 6). When an atom absorbs light, its energy excites the atom and its angular momentum changes the electron's orbit. Both of these are internal atomic properties. However, the linear momentum of the light, can change only the external, translational motion of the atom.
Laser-cooled trapped ion crystals offer a unique opportunity to study the interactions between atoms. We have previously developed a microscopic planar Paul trap1 which is small and strong enough to bring the ion separation into the sub-micron region where quantum collective effects become important. A search is underway for the superradiant coupling of two atoms which manifests itself as a variation of the spontaneous emission rate with the ion-ion distance. We have produced an ion crystal with a separation of 1.1 microns in an 80 micron radius planar Paul trap. At this distance, equal to 2.2 wavelengths of the 493 nm Ba + transition, superradiant effects should alter the decay rate by about 10%. The two-ion crystal is imaged with sub-micron resolution by a new diffraction-limited vacuum-compatible microscope. The spontaneous emission rate is measured by a high repetition-rate transient spectrometer with an error of less than 1% for a single ion in a few minutes. Efficient extinction of the exciting laser light is provided by a new nonlinear electrooptic sideband switch.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms A1, A2, whose separation r21 can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only A1 inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators O1, O2. The atoms exhibit superradiant behavior, whereas O1, O2 tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that A2 and A2 radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
Sub- and superradiant dynamics of spontaneously decaying atoms are manifestations of collective many-body systems. We study the internal dynamics and the radiation properties of two atoms in free space. Interesting results are obtained when the atoms are separated by less than half a wavelength of the atomic transition, where the dipole-dipole interaction gives rise to new coherent effects, such as (a) coherence between two intermediate collective states, (b) oscillations in the two-photon correlation G(2), (c) emission of two photons by one atom, and (d) the loss of directional correlation. We compare the population dynamics during the two-photon emission process with the dynamics of single-photon emission in the cases of a $\ensuremath{\Lambda}$ and a V scheme. We compute the temporal correlation and angular correlation of two successively emitted photons using the G(2) for different values of atomic separation. We find antibunching when the atomic separation is a quarter wavelength $\ensuremath{\lambda}∕4$. Oscillations in the temporal correlation provide a useful feature for measuring subwavelength atomic separation. Strong directional correlation between two emitted photons is found for atomic separation larger than a wavelength. We also compare the directionality of a photon spontaneously emitted by the two atoms prepared in phased-symmetric and phased-antisymmetric entangled states ${\ensuremath{\mid}\ifmmode\pm\else\textpm\fi{}⟩}_{{\mathbf{k}}_{0}}={e}^{i{\mathbf{k}}_{0}∙{\mathbf{r}}_{1}}\ensuremath{\mid}{a}_{1},{b}_{2}⟩\ifmmode\pm\else\textpm\fi{}{e}^{i{\mathbf{k}}_{0}∙{\mathbf{r}}_{2}}\ensuremath{\mid}{b}_{1},{a}_{2}⟩$ by a laser pulse with wave vector k0. Photon emission is directionally suppressed along k0 for the phased-antisymmetric state. The directionality ceases for interatomic distances less than $\ensuremath{\lambda}∕2$.