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The Journal of Chemical Physics

Cooperative single-quantum excitations of a closed-ring polymer chain

Helen S. Freedhoff

Published Nov 15, 1986• 14 citations
Publisher

Abstract

In this paper, we study the collective single-quantum excitations of a ring of N identical molecular units, of arbitrary radius R relative to the radiation wavelength λ. We first calculate the single-quantum eigenstates of the system for N=2,...,6 and N large, their energies and their lifetimes due to interaction with the radiation field. For large N, we study the collective emission by the ring as a function of direction, comparing the pattern with that created by a collectively excited linear chain of molecular units. Finally, we calculate the emission spectrum of the ring for arbitrary initial excitation, and the spectrum of the radiation absorbed by the ring from an external electromagnetic field.

DOI
10.1063/1.451476

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Physical Review A•Jul 1, 1987•30 citations
In this paper, we study the collective single-quantum excitations of a ring of N identical molecular units, of arbitrary radius R relative to the radiation wavelength λ. We first calculate the single-quantum eigenstates of the system for N=2,...,6 and N large, their energies and their lifetimes due to interaction with the radiation field. For large N, we study the collective emission by the ring as a function of direction, comparing the pattern with that created by a collectively excited linear chain of molecular units. Finally, we calculate the emission spectrum of the ring for arbitrary initial excitation, and the spectrum of the radiation absorbed by the ring from an external electromagnetic field.
A quantum emitted by any of a collection of identical atoms may be absorbed and re-emitted by other atoms many times before it eventually emerges. The radiation process is thus best described as collective or cooperative in nature. The atomic excitations are shown to attenuate as linear combinations of certain characteristic decay modes that lend a complex structure to the spectrum radiated. Instead of a single line, it becomes a closely-spaced multiplet of lines, the elements of which have a variety of lifetimes, line-shifts and line-widths. We calculate these quantities, first with an abstract two-state model for the atoms and then with an isotropic four-state model that accommodates the full polarization dependence of the radiation.
We investigate the topological and collective radiative properties of a one-dimensional diatomic chain of identical quantum emitters with alternating spacing, coupled to the electromagnetic vacuum. The system realizes an extended, non-Hermitian Su-Schrieffer-Heeger-type model with vacuum-mediated long-range interactions and collective dissipation. We focus on the single-excitation manifold, where the dynamics are described by an effective Hamiltonian. For an infinite chain, the complex band structure reveals subradiant modes associated with wave vectors outside the light line, as well as parameter regimes where real-part band crossings occur. The bulk topology is characterized by a complex Berry phase, which remains quantized in the presence of inversion symmetry and a spectral gap. For finite chains, using exact diagonalization, we identify superradiant, subradiant, and edge states with distinct decay rates and spatial profiles. Edge states emerge in the topologically nontrivial regime when a bulk gap in the real part of the energy spectrum is present, while subradiant states exhibit strongly suppressed decay with system-size dependence. Finally, we analyze the far-field radiation patterns associated with different classes of eigenstates and show that the emission characteristics reflect the decay properties, spatial localization, and the parity of the unit-cell amplitudes.
The spontaneous emission of a set of NNN identical two-level atoms interacting with a quantized electromagnetic field is studied. The atoms are assumed to be close together compared to the mean wavelength of the emitted radiation, and their initial state is either a Dicke state or a superposition of the ground and excited state for each atom. Approximate expressions are obtained for transition probabilities, expectation values, and correlations as functions of time. The spontaneous emission for a very large separation of the atoms is also studied. In this case the directional properties of the emitted radiation, as well as the time within which the atoms decay to their ground state, depend on the position of each atom.
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 study the collective radiation properties of cold, trapped ensembles of atoms. We consider the high-density regime with the mean interatomic distance being comparable to, or smaller than, the wavelength of the resonant optical radiation emitted by the atoms. We find that the emission rate of a photon from an excited atomic ensemble is strongly enhanced for an elongated cloud. We analyze collective single-excitation eigenstates of the atomic ensemble and find that the absorption-emission spectrum is broadened and shifted to lower frequencies as compared to the noninteracting (low-density) or single-atom spectrum. We also analyze the spatial and temporal profile of the emitted radiation. Finally, we explore how to efficiently excite the collective superradiant states of the atomic ensemble from a long-lived storage state in order to implement matter-light interfaces for quantum computation and communication applications.
The time-resolved cooperative emission from a system of correlated neutral dissociation fragments, or molecular collision products in beams, is investigated. The investigation is focused on emission at large fragment separations (between 1 nm and a few emission wavelengths), exceeding the domain of short-range interactions within the reactive or collisional molecular complex. A master-equation approach is used to obtain a general expression for the cooperative emission rate, which consists of nonexponential decay factors multiplied by temporal ringing patterns. These features result from the time-dependent radiative coupling between the receding fragments; they depend in an essential manner on the initial electronic state of the parent molecular complex and its symmetry which determine the correlations between the fragments. In the model system of a pair of identical two-level fragments two cases are considered separately. (a) A single photon shared by the fragments, where the emission is initially superradiant or subradiant (radiation trapping), depending on the spin and inversion symmetry of the parent molecular system and of the nascent fragments. The ringing pattern depends on the electronic angular momentum state of the parent molecule and on the polarization of the emitted light. (Such a ringing has been observed recently by Grangier, Aspect, and Vigu\'e [Phys. Rev. Lett. 54, 418 (1985)] in the emission of photodissociated Ca2{\mathrm{Ca}}_{2}Ca2​.) (b) Two initially excited fragments, where the ringing pattern is of smaller amplitude, and is weakly dependent on the electronic angular momentum of the parent molecule. All the aforementioned cooperative features generally last until the fragments recede several radiation wavelengths away from each other. The application of this time-resolved analysis to various diagnostic problems is discussed, especially with regard to the identification of excited electronic states of the parent molecular complex, and the stereospecificity of the fragmental orbitals.