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arXiv:quant-ph/0402065 · Physical Review A

Collective excitations in circular atomic configurations and single-photon traps

Hanno Hammer

Published Aug 1, 2004• 3 citations
arXivHTMLPDF

Abstract

Correlated excitations in a plane circular configuration of identical atoms with parallel dipole moments are investigated. The collective energy eigenstates, which are formally identical to Frenkel excitons, can be computed together with their level shifts and decay rates by decomposing the atomic state space into carrier spaces for the irreducible representations of the symmetry group ${\mathbb{Z}}_{N}$ of the circle. It is shown that the index $p$ of these representations can be used as a quantum number analogously to the orbital angular momentum quantum number $l$ in hydrogenlike systems. Just as the hydrogen $s$ states are the only electronic wave functions which can occupy the central region of the Coulomb potential, the quasiparticle corresponding to a collective excitation of the atoms in the circle can occupy the central atom only for vanishing ${\mathbb{Z}}_{N}$ quantum number $p$. If a central atom is present, the $p=0$ state splits into two and shows level crossing at certain radii; in the regions between these radii, damped quantum beats between two ``extreme'' $p=0$ configurations occur. The physical mechanisms behind super- and subradiance at a given radius are discussed. It is shown that, beyond a certain critical number of atoms in the circle, the lifetime of the maximally subradiant state increases exponentially with the number of atoms in the configuration, making the system a natural candidate for a single-photon trap.

DOI
10.1103/physreva.70.023803

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Correlated excitations in a plane circular configuration of identical atoms with parallel dipole moments are investigated. The collective energy eigenstates, which are formally identical to Frenkel excitons, can be computed together with their level shifts and decay rates by decomposing the atomic state space into carrier spaces for the irreducible representations of the symmetry group ZN{\mathbb{Z}}_{N}ZN​ of the circle. It is shown that the index ppp of these representations can be used as a quantum number analogously to the orbital angular momentum quantum number lll in hydrogenlike systems. Just as the hydrogen sss states are the only electronic wave functions which can occupy the central region of the Coulomb potential, the quasiparticle corresponding to a collective excitation of the atoms in the circle can occupy the central atom only for vanishing ZN{\mathbb{Z}}_{N}ZN​ quantum number ppp. If a central atom is present, the p=0p=0p=0 state splits into two and shows level crossing at certain radii; in the regions between these radii, damped quantum beats between two ``extreme'' p=0p=0p=0 configurations occur. The physical mechanisms behind super- and subradiance at a given radius are discussed. It is shown that, beyond a certain critical number of atoms in the circle, the lifetime of the maximally subradiant state increases exponentially with the number of atoms in the configuration, making the system a natural candidate for a single-photon trap.
We study correlated states in circular and linear-chain configurations of identical two-level atoms containing the energy of a single quasi-resonant photon in the form of a collective excitation, where the collective behavior is mediated by exchange of transverse photons between the atoms. For a circular atomic configuration containing N atoms, the collective energy eigenstates can be determined by group-theoretical means making use of the fact that the configuration possesses a cyclic symmetry group Z N . For these circular configurations, the carrier spaces of the various irreducible representations of the symmetry group are at most two-dimensional, so that the effective Hamiltonian on the radiationless subspace of the system can be diagonalized analytically. As a consequence, the radiationless energy eigenstates carry a Z N quantum number p = 0, 1, …, N, which is analogous to the angular momentum quantum number l = 0, 1, … carried by particles propagating in a central potential, such as a hydrogen-like system. Just as the hydrogen s states are the only electronic wave functions that can occupy the central region of the Coulomb potential, the quasi-particle corresponding to a collective excitation of the circular atomic sample can occupy the central atom only for vanishing Z N quantum number p. When a central atom is present, the p = 0 state splits into two, showing level crossing at certain radii; in the regions between these radii, damped oscillations between two “ extreme” p = 0 states occur, where the excitation occupies either the outer atoms or the central atom only. For large numbers of atoms in a maximally subradiant state, a critical interatomic distance of λ/2 emerges both in the linear-chain and in the circular configuration of atoms. The spontaneous decay rate of the linear configuration exhibits a jumplike “critical” behavior for next-neighbor distances close to a half-wavelength. Furthermore, both the linear-chain and the circular configurations exhibit exponential photon trapping once the next-neighbor distance becomes less than a half-wavelength, with the suppression of spontaneous decay being particularly pronounced in the circular system. In this way, circular configurations containing sufficiently many atoms may be natural candidates for single-photon traps.
We present a theoretical study of the collective optical effects which can occur in groups of three and four quantum dots. We define conditions for stable subradiant (dark) states, rapidly decaying super-radiant states, and spontaneous trapping of excitation. Each quantum dot is treated like a two-level system. The quantum dots are, however, realistic, meaning that they may have different transition energies and dipole moments. The dots interact via a short-range coupling which allows excitation transfer across the dots, but conserves the total population of the system. We calculate the time evolution of single-exciton and biexciton states using the Lindblad equation. In the steady state the individual populations of each dot may have permanent oscillations with frequencies given by the energy separation between the subradiant eigenstates.
We consider a pair of identical two-level atoms interacting with a scalar field in one dimension, separated by a distance x21{x}_{21}x21​. We obtain collective decaying states, belonging to a complex spectral representation of the Hamiltonian. The imaginary parts of the eigenvalues give the decay rates, and the real parts give the average energy of the collective states. In one dimension there is strong interference between the fields emitted by the atoms, producing cooperative effects even when the atoms are far apart. The decay rates and the energy oscillate with the distance x21{x}_{21}x21​. Depending on x21{x}_{21}x21​, the decay rates will either decrease, vanish, or increase as compared with the one-atom decay rate. We have sub- and superradiance at periodic intervals. Our model may be used to study two-cavity electron waveguides. The vanishing of the collective decay rates then suggests the possibility of obtaining stable configurations, where an electron is trapped inside the two cavities.
Abstract We theoretically investigate the light scattering of super- and subradiant states of an atomic ring prepared by single excitation with a photon which carries an orbital angular momentum (OAM). For excitations with linear polarizations, the helical phase imprinted (HPI) atomic ring presents a discrete C4 rotational symmetry when number of atoms N = 4n with integers n, while for circular polarizations with arbitrary N, the continuous and C N symmetries emerge for the super- and subradiant modes, respectively. The HPI superradiant modes predominantly scatter photons in the forward-backward direction, and the forward scattering can be further enhanced as atomic rings are stacked along the excitation direction. The HPI subradiant modes then preferentially scatter photons in the transversal directions, and when rings are stacked concentrically and on a plane, crossover from sub- to superradiance is observed which leads to splitting and localization of the far-field scattering patterns in the polar angle. The HPI super- and subradiant states are thus detectable through measuring the far-field radiation patterns, which further allow quantum storage and detection of a single photon with an OAM.
Energy and lifetime of collective optical excitations in regular arrays of atoms and molecules are significantly influenced by dipole-dipole interaction. While the dynamics of closely positioned atoms can be approximated well by the Dicke superradiance model, the situation of finite regular configurations is hard to access analytically. Most treatments use an exciton based description limited to the lowest excitation manifold. We present a general approach studying the complete decay cascade of a finite regular array of atoms from the fully inverted to the ground state. We explicitly calculate all energy shifts and decay rates for two generic cases of a three-atom linear chain and an equilateral triangle. In numerical calculations we show that despite fairly weak dipole-dipole interactions, collective vacuum coupling allows for superradiant emission as well as subradiant states in larger arrays through multi-particle interference. This induces extra dephasing and modified decay as important limitations for Ramsey experiments in lattice atomic clock setups as well as for the gain and frequency stability of superradiant lasers.
The properties of coupled emitters can differ dramatically from those of their individual constituents. Canonical examples include sub- and super-radiance, wherein the decay rate of a collective excitation is reduced or enhanced due to correlated interactions with the environment. Here, we systematically study the properties of collective excitations for regularly spaced arrays of quantum emitters coupled to a one-dimensional waveguide. We find that, for low excitation numbers, the modal properties are well-characterized by spin waves with a definite wavevector. Moreover, the decay rate of the most subradiant modes obeys a universal scaling with a cubic suppression in the number of emitters. Multi-excitation subradiant eigenstates can be built from fermionic combinations of single excitation eigenstates; such 'fermionization' results in multiple excitations that spatially repel one another. We put forward a method to efficiently create and measure such subradiant states, which can be realized with superconducting qubits. These measurement protocols probe both real-space correlations (using on-site dispersive readout) and temporal correlations in the emitted field (using photon correlation techniques).