We propose the physical implementation of a Grover-like search problem by means of Frenkel exciton trapping at a shallow isotopic impurity against a background of competing mechanisms. The search, culminating at the impurity molecule, designated the “winner” site, is marked by its enhanced interaction with acoustic phonons at low temperatures. The quantum search proceeds with the assistance of an oracle-like exciton-phonon interaction that addresses only the impurity site via the Dyson propagator within the Green’s function formalism. The optimum parameters of a graph lattice with long-range intersite interactions required to trap the exciton in the fastest time are determined, and estimates of error rates for the naphthalene-doped organic system are evaluated. We extend the analysis of the quantum search to a fluctuating long-range interacting cycle (LRIC) graph-lattice system.

The problem of two Coulombically interacting electrons confined in a cylindrical potential box with impenetrable walls is solved by the method of full configuration interaction (CI) for states with zero axial component of the total orbital angular momentum, including the ground state ${}^1{\Sigma }_g^{+}$. The number of variables in the Schrödinger equation is reduced from six to five by separating the ${\mathrm{L̂}}_z$ operator from the complete Hamiltonian. Two-electron basis sets of symmetry-adapted configuration state functions are constructed from real eigenfunctions of the one-electron Hamiltonian for a cylindrical potential box. Kinetic energy integrals are evaluated analytically. Electron repulsion integrals are computed by developing a separable expression for the $1/r_{12}$ operator in the form of a double Fourier-Bessel-Fourier-cosine series and using enough terms to achieve a relative accuracy of ${10}^{-10}$ for the eigenvalues. Energy-level ordering and electron densities of selected electronic states are reported for cylindrical cavities of varying size with a fixed 1:1 length-to-diameter ratio. A remarkable feature of this system is that convergence of the full CI energy with respect to the one-electron basis set varies dramatically with the extent of confinement: For small cavities, the rate of approaching the basis-set limit is similar to that for two-electron atoms but is significantly faster for large cavities. Formation of singlet and triplet Wigner molecules is observed in cylindrical cavities with radii greater than about 5 bohrs.

We study the properties of the Hooke’s law correlation energy ($E_{\mathrm{c}}$), defined as the correlation energy when two electrons interact via a harmonic potential in a $D$-dimensional space. More precisely, we investigate the ${}^{1}S$ ground-state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made that highlights the main differences of the $E_{\mathrm{c}}$ in both the weakly and strongly correlated limits. Moreover, we show that the Schrödinger equation of the spherium model is exactly solvable for two values of the dimension ($D=1 \text{and} 3$) and that the exact wave function is based on Mathieu functions.

We report results from close-coupling calculations for homonuclear ultracold collisions between potassium atoms, using the most up-to-date Born-Oppenheimer potential curves. The present study includes both of the bosonic isotopes ${}^{39}\mathrm{K}$ and ${}^{41}\mathrm{K}$. The $s$-wave scattering lengths as functions of the magnetic field strength for collisions between atoms in identical and different hyperfine states are obtained. Several Feshbach resonances are located and characterized for both isotopes. Comparison with experiments, where such data are available, show excellent agreement. We also study weakly bound molecular states of the ${\mathrm{K}}_2$ molecule in close relation to the calculated Feshbach resonances. Another objective of the present work is to study inelastic collisions in which the hyperfine states of the colliding atoms are changed. From this type of calculation we obtain transition rates as functions of the magnetic field strength. Finally, we discuss how such transition rates might be of importance for experimental work.

We consider a spinor condensate of ${}^{87}\mathrm{Rb}$ atoms in the $F=1$ hyperfine state confined in an optical dipole trap. Putting initially all atoms in the $m_F=0$ component, we find that the system evolves toward a state of thermal equilibrium with kinetic energy equally distributed among all magnetic components. We show that this process is dominated by the dipolar interaction of magnetic spins rather than spin-mixing contact potential. Our results show that because of a dynamical separation of magnetic components, the spin-mixing dynamics in the ${}^{87}\mathrm{Rb}$ condensate is governed by the dipolar interaction which plays no role in a single-component rubidium system in a magnetic trap.

We derive an effective Hamiltonian for the nonlinear process of parametric downconversion in the presence of absorption. Based upon the Green-function method for quantizing the electromagnetic field, we first set up Heisenberg’s equations of motion for a single atom driven by an external electric field and in the presence of an absorbing dielectric material. The equations of motion are then solved to second order in perturbation theory which, in rotating-wave approximation, yields the standard effective interaction Hamiltonian known from free-space nonlinear optics. In a second step, we derive the local-field-corrected Hamiltonian for an atom embedded in a dielectric host medium, that is, a nonlinear crystal. Here we show that the resulting effective Hamiltonian is found to be trilinear in the electric and noise-polarization fields and is thus capable of describing nonlinear noise processes. Furthermore, it reduces to the phenomenological nonlinear Hamiltonian for the cases where absorption, and hence the noise-polarization field, vanishes.

The stability of an optical vortex in a cell with a circular cross section containing a nematic liquid crystal is studied. A modulation theory based on an averaged Lagrangian formulation is developed to study this stability. It is found that the vortex is stable unless the radius of the cell is very small, nearly the width of the vortex itself. Based on the analysis of a stationary vortex, the stability of a low-amplitude vortex in a large cell under the influence of its orbital angular momentum and the repelling effect of the cell boundary are studied. The predictions of this modulation theory are found to be in excellent agreement with numerical simulations.

We study, both experimentally and theoretically, the underlying physics of third-harmonic generation in air by a filamented infrared femtosecond laser pulse propagating through a thin plasma channel. It is shown that the recently observed more than two-order-of-magnitude increase of the efficiency of third-harmonic generation occurs due to the plasma-enhanced third-order susceptibility. An estimate of the effective value of this susceptibility is given.

The past decade has seen frequent use of a modified nonlinear Schrödinger equation to describe ultrashort pulse propagation in materials where free-carrier plasmas are present. The optical contribution from the resulting free-current densities in this equation is often described using a classical Drude model. However, the ultimate form of this contribution in the modified nonlinear Schrödinger equation is somewhat inconsistent in the literature. We clarify this ambiguity by deriving the modified nonlinear Schrödinger equation from the classical wave equation containing a free-current density contribution. The Drude model is then used to obtain an expression for the complex free-carrier current density envelope with temporal dispersion corrections for ultrashort laser pulses. These temporal dispersion corrections to the current-density term differ, to our knowledge, from all other models in the literature in that they depend more sensitively on the value of the Drude free-carrier collision time. These corrections reduce to the current models in the literature for limiting cases. Theoretical analysis and computer simulations show that these differences can significantly affect the dynamic interactions of plasma absorption and plasma defocusing for materials with free-carrier collision times on the order of one optical cycle (or less) of the applied field.

This paper analyzes dynamical properties of small Fermi-Hubbard and Bose-Hubbard systems, focusing on the structure of the underlying Hilbert space. We evaluate time-dependent quantities such as the return probability to the initial state and the spin imbalance of spin-$1/2$ fermions. For the symmetric two-site Fermi-Hubbard model we find that the spin imbalance and the return probability are controlled by two and three frequencies, respectively. The spin imbalance and the return probability are identical for the asymmetric Falicov-Kimball limit and controlled by only one frequency. In general, the transition probabilities between the initial state and the energy eigenstates depend strongly on the particle-particle interaction. This is discussed for “self-trapping” of spinless bosons in a double-well potential. We observe that the available Hilbert space is reduced significantly by strong interaction.

We explore an alternative way of finding the link between a $\mathcal{P}\mathcal{T}$ non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of the scattering problem for an imaginary potential and its time-reversal process, it is shown that any real-energy eigenstate of a $\mathcal{P}\mathcal{T}$ tight-binding lattice with on-site imaginary potentials shares the same wave function with a resonant transmission state of the corresponding Hermitian lattice embedded in a chain. It indicates that the $\mathcal{P}\mathcal{T}$ eigenstate of a $\mathcal{P}\mathcal{T}$ non-Hermitian Hamiltonian has a connection to the resonance transmission state of the extended Hermitian Hamiltonian.

We show that the diagonal matrix elements $\langle {\mathit{Or}}^p\rangle$, where $O$$=\{1,\beta ,i\mathbf{\alpha }\mathbf{n}\beta \}$ are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, may be considered as difference analogs of the radial wave functions. Such structure provides an independent way of obtaining closed forms of these matrix elements by elementary methods of the theory of difference equations without explicit evaluation of the integrals. Three-term recurrence relations for each of these expectation values are derived as a by-product. Transformation formulas for the corresponding generalized hypergeometric series are discussed.

In this work, the steady-state or quasiequilibrium resulting from periodically modulating the Liouvillian of an open quantum system, $\widehat{\mathrm{ℒ̂}}(t)$, is investigated. It is shown that differences between the quasiequilibrium and the instantaneous equilibrium occur due to nonadiabatic contributions from the gauge field connecting the instantaneous eigenstates of $\widehat{\mathrm{ℒ̂}}(t)$ to a fixed basis. These nonadiabatic contributions are shown to result in an additional rotation and/or depolarization for a single spin-1/2 in a time-dependent magnetic field and to affect the thermal mixing of two coupled spins interacting with a time-dependent magnetic field.

Entanglement purification is a very important element for long-distance quantum communication. Different from all the existing entanglement purification protocols (EPPs) in which two parties can only obtain some quantum systems in a mixed entangled state with a higher fidelity probabilistically by consuming quantum resources exponentially, here we present a deterministic EPP with hyperentanglement. Using this protocol, the two parties can, in principle, obtain deterministically maximally entangled pure states in polarization without destroying any less-entangled photon pair, which will improve the efficiency of long-distance quantum communication exponentially. Meanwhile, it will be shown that this EPP can be used to complete nonlocal Bell-state analysis perfectly. We also discuss this EPP in a practical transmission.

A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also give a necessary condition for the adiabatic approximation that depends on local properties of the path, which is appropriate when the gap varies.

We study the structure of quantum states for a binary mixture of spin-1 atomic Bose-Einstein condensates. In contrast to collision between identical bosons, the $s$-wave scattering channel between interspecies does not conform to a fixed symmetry. The spin-dependent Hamiltonian thus contains noncommuting terms, making the exact eigenstates more challenging to obtain because they now depend more generally on both the intra- and interspecies interactions. We discuss two limiting cases, where the spin-dependent Hamiltonian reduces respectively to sums of commuting operators. All eigenstates can then be directly constructed, and they are independent of the detailed interaction parameters.

We derive the weak value deflection given in an article by Dixon et al. [P. B. Dixon et al. Phys. Rev. Lett. 102 173601 (2009)] both quantum mechanically and classically, including diffraction effects. This article is meant to cover some of the mathematical details omitted in that article owing to space constraints.

Arrays of nanoparticles whose interactions can be characterized by hopping Hamiltonians can serve as excitation transmission lines. Here we show, that in addition suitable arrangements of nanoparticles can form beam splitters, phase shifters, and crossover splitters. With these elements, any discrete unitary transformation can be implemented on input modes via a network of nanoparticles in which all the components lie in the same plane. These nanoparticle networks can produce optical functionalities at a length scale much smaller than 1 $\mu \mathrm{m}$.

We show theoretically that it is possible to optically control collective spin-exchange processes in spinor Bose condensates through virtual photoassociation. The interplay between optically induced spin exchange and spin-dependent collisions provides a flexible tool for the control of atomic spin dynamics, including enhanced or inhibited quantum spin oscillations, the optically induced ferromagnetic-to-antiferromagnetic transition, and coherent matter-wave spin conversion.

We investigate the influence of thermally activated internal molecular dynamics on the phase shifts of matter waves inside a molecule interferometer. While de Broglie physics generally describes only the center-of-mass motion of a quantum object, our experiment demonstrates that the translational quantum phase is sensitive to dynamic conformational state changes inside the diffracted molecules. The structural flexibility of tailor-made organic particles is sufficient to admit a mixture of strongly fluctuating dipole moments. These modify the electric susceptibility and through this the quantum interference pattern in the presence of an external electric field. Detailed molecular dynamics simulations combined with density-functional theory allow us to quantify the time-dependent structural reconfigurations and to predict the ensemble-averaged square of the dipole moment which is found to be in good agreement with the interferometric result. The experiment thus opens a different perspective on matter wave interferometry, as we demonstrate here that it is possible to collect structural information about molecules even if they are delocalized over more than 100 times their own diameter.