The validity of the few-level approximation is investigated in a system of two dipole-dipole interacting four-level atoms. Each atom is modelled by two complete sets of angular momentum multiplets. We provide two independent arguments demonstrating that the few-level approximation in general leads to incorrect predictions if it is applied to the Zeeman sublevels of the atomic level scheme. First, we show that the artificial omission of sublevels generally leads to incorrect eigenenergies of the system. The second counterexample involves an external laser field and illustrates that the relevant states in each atom are not only determined by the laser field polarization, but also by the orientation of the atomic separation vector. As the physical origin of this outcome, we identify the dipole-dipole interaction between orthogonal dipole transitions of different atoms. Our interpretation enables us to identify conditions on the atomic level structure as well as special geometries in which (partial) few-level approximations are valid.
The validity of the few-level approximation is investigated in a system of two dipole-dipole interacting four-level atoms. Each atom is modelled by two complete sets of angular momentum multiplets. We provide two independent arguments demonstrating that the few-level approximation in general leads to incorrect predictions if it is applied to the Zeeman sublevels of the atomic level scheme. First, we show that the artificial omission of sublevels generally leads to incorrect eigenenergies of the system. The second counterexample involves an external laser field and illustrates that the relevant states in each atom are not only determined by the laser field polarization, but also by the orientation of the atomic separation vector. As the physical origin of this outcome, we identify the dipole-dipole interaction between orthogonal dipole transitions of different atoms. Our interpretation enables us to identify conditions on the atomic level structure as well as special geometries in which (partial) few-level approximations are valid.
The validity of the few-level approximation in dipole-dipole interacting collective systems is discussed. As an example system, we study the archetype case of two dipole-dipole interacting atoms, each modeled by two complete sets of angular momentum multiplets. We establish the breakdown of the few-level approximation by first proving the intuitive result that the dipole-dipole induced energy shifts between collective two-atom states depend on the length of the vector connecting the atoms, but not on its orientation, if complete and degenerate multiplets are considered. A careful analysis of our findings reveals that the simplification of the atomic level scheme by artificially omitting Zeeman sublevels in a few-level approximation generally leads to incorrect predictions. We find that this breakdown can be traced back to the dipole-dipole coupling of transitions with orthogonal dipole moments. Our interpretation enables us to identify special geometries in which partial few-level approximations to two- or three-level systems are valid.
The interaction of an atomic system with an externally applied electromagnetic field can be treated in the electric dipole approximation by means of either the minimal coupling (p · A) or direct coupling (d · E) Hamiltonian. It is shown that both methods lead to identical and unambiguous predictions for observable quantities as long as the atomic wavefunctions are transformed when used in the minimal-coupling formulation. The physical meaning of kinetic momentum is used to show that the atomic states must be described by wavefunctions calculated in the absence of an electromagnetic field when using the d · E (but not the p · A) form of the interaction Hamiltonian. When, however, observables are calculated using the common approximations of resonance atomic physics – the two-level approximation and the rotatingwave approximation – the two formulations can lead to measurably different results. This point is illustrated by calculating the induced polarization (and hence the refractive index) of an atomic system for the two exactly soluble cases of the harmonic oscillator and the hydrogen atom.
The Wigner representation of angular-momentum orientation in the classical limit on $J\ensuremath{\gg}1$ has been used to derive an equation of angular-momentum motion in the case of a closed two-level system interacting with an arbitrary polarized radiation. This equation enables us to carry out a theoretical analysis of angular-momentum motion and consider final states of the quantum system for all types of dipole transitions. We have found that in the case of a zero magnetic field in the final state of the system with $\stackrel{\ensuremath{\rightarrow}}{J}J$ or $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ optical transition, the angular momentum is directed along or opposite to the wave vector of the radiation depending on the sign of the polarization ellipsity, while for a linear polarization the angular momentum is isotropically distributed on the plane orthogonal to the polarization vector. For the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ transition two directions of angular momentum are possible in the final state. These directions are determined by the ellipsity of polarization. The spatial size of the final distribution of the angular momentum has been defined by the quantum uncertainty, 1/J, of the angular-momentum orientation. In the case of the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ or $\stackrel{\ensuremath{\rightarrow}}{J}J,$ transition particles come into ``dark'' states while for the $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ transition they occupy the ``brightest'' state. In the presence of a nonzero magnetic field, particles with $\stackrel{\ensuremath{\rightarrow}}{J}J$ transition have in the final state only one angular-momentum orientation, i.e., along or opposite to the direction of the magnetic field depending on the polarization ellipsity sign. For the $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ transition both directions are possible, i.e., along and opposite to the magnetic field. In the case of the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ transition the directions of the angular momentum form a cone around the magnetic field.
The limits of the well-established single-atom model for describing photon-pair generation via four-wave mixing in a diamond configuration in atomic ensembles are experimentally tested. Using a cold-atom source, biphotons are generated and detected through polarization analyzers that resolve the emitted light into horizontal and vertical components in the laboratory frame. The pump lasers driving the first and second excitation transitions are horizontally and vertically polarized, respectively. To predict single-photon counts and coincidence rates in all polarization channels, the first- and second-order correlation functions are calculated directly using a comprehensive model that accounts for all Zeeman sublevels of the relevant hyperfine states. Furthermore, the population dynamics with and without the re-pump laser of the magneto-optical trap are compared, revealing substantial differences in the populations of the Zeeman sublevels. This motivates the inclusion of two additional hyperfine levels and their corresponding Zeeman sublevels in the final model. The resulting density matrix is used to calculate expectation values of the far-field electric-field operators and compare them with experimental measurements over a range of pump-laser powers. Excellent agreement with the measured photon counts is obtained over most of this range. For the coincidence measurements, good agreement is found when the polarization of the photon generated by the first (second) decay is parallel to that of the first (second) pump beam. In contrast, the model consistently underestimates the experimental coincidence rates for the opposite polarization configuration. These findings indicate that effects beyond the internal level dynamics of individual atoms, most notably collective phenomena, are required to fully account for photon coincidences generated by four-wave mixing in atomic ensembles.
In this thesis we investigate theoretically two distinct systems of dipole-coupled atoms. In Part I we investigate light scattering from a system of N ultracold atoms, with laser induced electric dipole interactions. We derive the equations of motion in the Heisenberg picture for the atomic operators, for an arbitrary dipole transition. Then we take the specific case of a jl = 0 → ju = 1 transition and derive the second and third-order inter-atomic correlations. We solve the resulting set of linear equations for two and three atoms exactly, with particular focus on the exact two-atom system. We find a steady-state analytic solution for the two atoms in a specific geometric configuration, and use this to derive explicit expressions for the spatial and coherence properties of the far-field scattered light intensity, valid for the full range of system parameters. A comprehensive survey of the scattering behaviour is given, with key features precisely characterised, including subradiant scattering. We examine in detail a decorrelation approximation that has potential applications for larger systems of atoms that are intractable in a full quantum treatment. We introduce the concept of an effective driving field and show that it can provide a direct and intuitive physical interpretation of key aspects of the system behaviour, including subradiant scattering. Finally, we show that this effective field interpretation, and the decorrelation approximation, is readily extended to a three atom system. In Part II of this thesis we present our investigation of the quantum many-body dynamics of a large ensemble of bosonic S = 3 chromium atoms in a three-dimensional lattice. We implement an extended Bose-Hubbard model which includes the tunnelling, onsite spinor in- teractions and dipole-dipole interactions (giving interactions between atoms on neighbouring sites). We study the dynamics of the population of the different Zeeman levels as a function of lattice depth following a sudden spin rotation, and provide comparison to experimental res- ults. We are able to identify two distinct regimes: At low lattice depths, where atoms are in the superfluid regime, we observe the spin dynamics are strongly determined by the competi- tion between particle transport, onsite interactions and external magnetic field gradients. On the contrary, at high lattice depths, transport is largely frozen out and the spin populations are mainly driven by the long range dipolar interactions and quantum fluctuations.
We study the two-body bound states of a model Hamiltonian that describes the interaction between two field-oriented dipole moments. This model has been used extensively in the many-body physics of ultracold polar molecules and magnetic atoms, but its few-body physics has been explored less fully. With a hard-wall short-range boundary condition, the dipole-dipole bound states are universal and exhibit a complicated pattern of avoided crossings between states of different character. For more realistic Lennard--Jones short-range interactions, we consider parameters representative of magnetic atoms and polar molecules. For magnetic atoms, the bound states are dominated by the Lennard--Jones potential, and the perturbative dipole-dipole interaction is suppressed by the special structure of van der Waals bound states. For polar molecules, we find a dense manifold of dipole-dipole bound states with many avoided crossings as a function of induced dipole or applied field, similar to those for hard-wall boundary conditions. This universal pattern of states may be observable spectroscopically for pairs of ultracold polar molecules.