Collective Eigenstates Research Articles

Microscopic dynamics and an effective Landau-Zener transition in the quasiadiabatic preparation of spatially ordered states of Rydberg excitations

We examine the adiabatic preparation of spatially-ordered Rydberg excitations of atoms in finite one-dimensional lattices by frequency-chirped laser pulses, as realized in a number of recent experiments simulating quantum Ising model. Our aims are to unravel the microscopic mechanism of the phase transition from the unexcited state of atoms to the antiferromagnetic-like state of Rydberg excitations by traversing an extended gapless phase, and to estimate the preparation fidelity of the target state in a moderately sized system amenable to detailed numerical analysis. We find that, in the basis of the bare atomic states, the system climbs the ladder of Rydberg excitations predominantly along the strongest-amplitude paths towards the final ordered state. We show that, despite its complexity, the interacting many-body system can be described as an effective two-level system involving a pair of lowest-energy instantaneous collective eigenstates of the time-dependent Hamiltonian. The final preparation fidelity of the target state can then be well approximated by the Landau-Zener formula, while the nonadiabatic population leakage during the passage can be estimated using a perturbative approach applied to the instantaneous collective eigenstates.

Probing few-excitation eigenstates of interacting atoms on a lattice by observing their collective light emission in the far field

The collective emission from a one-dimensional chain of interacting two-level atoms coupled to a common electromagnetic reservoir is investigated. We derive the system's dissipative few-excitation eigenstates, and analyze their static properties, including the collective dipole moments and branching ratios between different eigenstates. Next, we study the dynamics, and characterize the light emitted or scattered by such a system via different far-field observables. Throughout the analysis, we consider spontaneous emission from an excited state as well as two different pump field setups, and contrast the two extreme cases of non-interacting and strongly interacting atoms. For the latter case, the two-excitation submanifold contains a two-body bound state, and we find that the two cases lead to different dynamics and far-field signatures. Finally we exploit these signatures to characterize the wavefunctions of the collective eigenstates. For this, we identify a direct relation between the collective branching ratio and the momentum distribution of the collective eigenstates' wavefunction. This provides a method to proof the existence of certain collective eigenstates and to access their wave function without the need to individually address and/or manipulate single atoms.

Collective eigenstates of emission in an N-entity heterostructure and the evaluation of its Green tensors and self-energy components

Optical emission from emitters strongly interacting among themselves and also with other polarizable matter in close proximity has been approximated by emission from independent emitters. This is primarily due to our inability to evaluate the self-energy matrices and radiative properties of the collective eigenstates of emitters in heterogeneous ensembles. A method to evaluate self-energy matrices that is not limited by the geometry and material composition is presented to understand and exploit such collective excitations. Numerical evaluations using this method are used to highlight the significant differences between independent and the collective modes of emission in nanoscale heterostructures. A set of N Lorentz emitters and other polarizable entities is used to represent the coupled system of a generalized geometry in a volume integral approach. Closed form relations between the Green tensors of entity pairs in free space and their correspondents in a heterostructure are derived concisely. This is made possible for general geometries because the global matrices consisting of all free-space Green dyads are subject to conservation laws. The self-energy matrix can then be assembled using the evaluated Green tensors of the heterostructure, but a decomposition of its components into their radiative and nonradiative decay contributions is nontrivial. The relations to compute the observables of the eigenstates (such as quantum efficiency, power/energy of emission, radiative and nonradiative decay rates) are presented. A note on extension of this method to collective excitations, which also includes strong interactions with a surface in the near-field, is added.

Collective interactions in an array of atoms coupled to a nanophotonic waveguide

A lattice of trapped atoms strongly coupled to a one-dimensional nanophotonic waveguide is investigated in exploiting polaritons as natural collective eigenstates. We derive polariton-polariton kinematic interactions by applying a bosonization procedure to transform excitation spin-$\frac{1}{2}$ operators into interacting bosons. In solving the scattering problem we extract the effective potential, which is shown to be modulated by using the excitation-photon detuning as a control parameter. We examine the regime in which polaritons behave as a dilute degenerate boson gas and in the limit where polaritons can be treated as weakly interacting photons we propose the system for realizing superfluidity of photons. We implement the kinematic interaction as a mechanism for nonlinear optical processes that provide an observation tool for the system properties, e.g., the interaction strength produces a blue shift in pump-probe experiments.

Spontaneous emission from cylindrical atomic cloud: Collective eigenstates and non-local effects

We consider collective emission of a single photon stored in a cloud of N two-level atoms (with energy ћω) confined inside an infinite cylinder and discuss eigenstates of this system, their decay rates and collective frequency shifts. We found that states with wave number k z ≥ ω/ c do not decay and analogous to guiding modes in dielectric waveguides. Evolution of such states is qualitatively different in local (Markovian) and non-local regimes. We found that in the Markovian regime there is no photon emission. In contrast, non-local (memory) effects result in emission and reabsorption of the photon so that probability to find atoms excited oscillates with a collective Rabi frequency. Cross-over between local and non-local behavior can be observed by increasing radius of the cylinder or wave number k z of the excited atomic state. Similar behavior can also be observed in slab geometry and tested in synchrotron experiments on collective excitation of solid-state samples by increasing thickness of the nuclear layer.

Physical interpretation and quantization of periodic time-dependent Hartree-Fock solutions.

We outline several sets of physical conditions which lead to periodic or multiply periodic solutions of the time-dependent Hartree-Fock equations. In every instance the Fourier components can be identified with transition matrix elements of the density fluctuation operator between collective eigenstates of the system. A simple variational derivation of the Bohr-Sommerfeld quantization condition for such solutions is described.

Variational principles for collective motion: Relation between invariance principle of the Schrödinger equation and the trace variational principle

The invariance principle of the Schrödinger equation provides a basis for theories of collective motion with the help of the time-dependent variational principle. It is formulated here with maximum generality, requiring only the motion of intrinsic state in the collective space. Special cases arise when the trial vector is generalized coherent state and when it is a uniform superposition of collective eigenstates. The latter example yields variational principles uncovered previously only within the framework of the equations of motion method.

An ensemble of random particle-hole matrices with collective eigenstates

An ensemble of random particle-hole matrices is studied as a function of the signs of the matrix elements. When all matrix elements have the same sign, a theorem due to Perron ensures the existence of a coherent state: the collective particle-hole state. The example of Gamow-Teller transitions between208Bi(1+) and208Pb(0+) is considered and the strength of the Gamow-Teller giant resonance as well as the fluctuations in the ensemble are studied. When a fractionp of matrix elements are of opposite sign, we find that a giant resonance may still exist providedp is small enough. Its magnitude and position in the spectrum are studied. The method proposed is valid for any particle hole excitation. The two body matrix elements of several interactions are analysed and are shown to fit into a unique valuep=0.2 for the 1 + states.

Collective excitations of nuclear matter

A study of the collective motions of nuclear matter has been made. We first give a purely classical macroscopic description of hydrodynamic waves in nuclear matter, and suggest some experimental consequences of their excitation. Next a quantum mechanical study of the collective eigenstates of nuclear matter is taken up. The starting point of this discussion is the theory of the nuclear ground state as given by Brueckner and his collaborators. The excited states are described by means of the method developed by Sawada to apply to an electron gas. We generalize this method so as to include the internal degrees of freedom associated with spin and i-spin and to handle the momentum dependence of the level-shift operator K used by Brueckner. The connection between the quantum-mechanical eigenstates and the classical hydrodynamic motion is established. As a consequence of the internal degrees of freedom, there exist not only the usual compressive waves, but spin, i-spin, and coupled spin- i-spin waves. The i-spin waves can be associated with the Goldhaber-Teller oscillations. We have investigated the corrections to the Sawada theory. This gives rise to the damping of the stable Sawada collective eigenmodes, analogous to the viscous damping of a plasma oscillation. The spin and i-spin waves are only slightly damped, whereas the compressional mode is unstable in an exponentially growing sense.