We develop the theory of continuous-variable quantum secret sharing and propose its interferometric realization using passive and active optical elements. In the ideal case of infinite squeezing, a fidelity $\mathcal{F}$ of unity can be achieved with respect to reconstructing the quantum secret. We quantify the reduction in fidelity for the (2,3) threshold scheme due to finite squeezing and establish the condition for verifying that genuine quantum secret sharing has occurred.

In this paper, we study decoherence in Grover’s quantum searching algorithm using a perturbative method. We assume that each two-state system (qubit) suffers ${\sigma }_z$ error with probability $p(0\mathrm{}<~p<~1)$ independently at every step in the algorithm. Considering an n-qubit density operator to which Grover’s operation is applied M times, we expand it in powers of $2Mnp$ and derive its matrix element order by order under the $\overrightarrow{n}\infty$ limit. (In this large-n limit, we assume p is small enough, so that $2Mnp(>~0)$ can take any real positive value or $0.)$ This approach gives us an interpretation about creation of new modes caused by ${\sigma }_z$ error and an asymptotic form of an arbitrary-order correction. Calculating the matrix element up to the fifth-order term numerically, we investigate a region of $2Mnp$ (perturbative parameter), where the algorithm finds the correct item with a threshold of probability $P_{\mathrm{th}}$ or more. It satisfies $2Mnp<\mathrm{}(8/5)\left(1\mathrm{}-P_{\mathrm{th}}\right)$ around $2Mnp\simeq 0$ and $P_{\mathrm{th}}\simeq 1,$ and this linear relation is applied to a wide range of $P_{\mathrm{th}}$ approximately. This observation is similar to a result obtained by Bernstein and Vazirani concerning accuracy of quantum gates for general algorithms. We cannot investigate a quantum-to-classical phase transition of the algorithm, because it is outside the reliable domain of our perturbation theory.

The general principle for a quantum-signature scheme is proposed and investigated based on ideas from classical signature schemes and quantum cryptography. The suggested algorithm is implemented by a symmetrical quantum key cryptosystem and Greenberger-Horne-Zeilinger (GHZ) triplet states and relies on the availability of an arbitrator. We can guarantee the unconditional security of the algorithm, mostly due to the correlation of the GHZ triplet states and the use of quantum one-time pads.

We explore the sensitivity of an interferometer based on a quantum circuit for coherent states. We show that its sensitivity is at the Heisenberg limit. Moreover, we show that this arrangement can measure very small length intervals.

We present a class of observables which are suitable for determining the fidelity of a state to the multipartite Greenberger-Horne-Zeilinger (GHZ) state. Given an expectation value of an observable belonging to the class, we give a simple formula that gives a lower bound and an upper bound for the fidelity. Applying the formula to the GHZ-state preparation experiment by Pan et al. [Nature (London) 403, 515 (2000)], we show that the observed state lies outside of the class of biseparable mixed three-qubit states. We also show that for this class of operators, adopting the principle of minimum variance [Phys. Rev. A 60, 4338 (1999)] in the state estimation always results in the state with the minimum fidelity.

Using nuclear magnetic resonance (NMR) techniques with a three-qubit sample, we have experimentally implemented the highly structured algorithm for the satisfiability problem with one variable in each clause proposed by Hogg. A simplified temporal averaging procedure was employed to prepare the three-qubit pseudopure state. The algorithm was completed with only a single evaluation of the structure of the problem and the solutions were found theoretically with probability $100\%,$ results that outperform both unstructured quantum and the best classical search algorithms. However, about $90\%$ of the corresponding experimental fidelities can be attributed to the imperfections of manipulations.

The ground-state properties of ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$ trimers and tetramers in infinite and restricted two-dimensional and quasi-two-dimensional (2D) space are studied. Using Monte Carlo calculation in two stages successively, simple variational Monte Carlo and diffusion Monte Carlo, we find that all trimers (except one of ${}^3\mathrm{He}$ with spin 3/2) and tetramers are bound. In infinite 2D space the binding energy of ${}^4\mathrm{He}$ trimer is $-183\left(1\right)\mathrm{}\hspace{0.5em}\mathrm{mK}$ and of tetramer $-435\left(1\right)\mathrm{}\hspace{0.5em}\mathrm{mK}.$ In the same environment for ${}^3\mathrm{He},$ using Jastrow-Feenberg-Bruch wave functions trimer (spin 1/2) is bound with $-0.013\left(1\right)\mathrm{}\hspace{0.5em}\mathrm{mK}$ and tetramer with $-0.021\left(1\right)\mathrm{}\hspace{0.5em}\mathrm{mK}.$ Employing in the same procedure “dimerlike” wave functions for ${}^3\mathrm{He}$ molecules in infinite 2D space it is discovered that trimer prefers the structure of one dimer and one separate particle while a composition of two separate dimers is preferable for tetramer. In this case binding energies are $-0.020\left(1\right)\mathrm{}\hspace{0.5em}\mathrm{mK}$ for trimer and $-0.040\left(1\right)\mathrm{}\hspace{0.5em}\mathrm{mK}$ for tetramer. The binding energies of mixed molecules in infinite 2D space are $-74.3\left(4\right)\mathrm{}$ mK ${(}^3\mathrm{He}{-}^4{\mathrm{He}}_2);$ $-14\left(1\right)\mathrm{}$ mK ${(}^3{\mathrm{He}}_2{-}^4\mathrm{He});$ $-254\left(4\right)\mathrm{}$ mK ${(}^3\mathrm{He}{-}^4{\mathrm{He}}_3);$ $-118\left(2\right)\mathrm{}$ mK ${(}^3{\mathrm{He}}_2{-}^4{\mathrm{He}}_2);$ $-11\left(1\right)\mathrm{}$ mK ${(}^3{\mathrm{He}}_3{-}^4\mathrm{He}).\mathrm{}$ In holding potentials with well-adjusted parameters there is a drastic increase of binding for trimer and tetramer in both sorts of helium atoms.

Upper bounds to the nonrelativistic ground-state energies of beryllium isoelectronic ions with $3<~Z<~10$ were computed from 1600-term exponentially correlated Gaussian functions. The energies obtained substantially improve over previous variational results known from literature. The mass polarization corrections to the energy are also presented.

When energetic protons and hydrogen molecular ions interact with several different gases (i.e., ${\mathrm{N}}_2,$ ${\mathrm{N}}_2\mathrm{O},$ and ${\mathrm{O}}_2),$ excited hydrogen atoms are produced that release Balmer-series radiation. Balmer-β and -γ radiations thus produced with beam energies ranging from 3 to 10 keV and scanned wavelengths ranging from 3000 to 6000 Å are detected and analyzed. The velocity of projectiles is considered as the most important factor that influences the emission cross section of Balmer radiations. Under different bombarding conditions, the data indicate the following three characteristics: (a) Given the same gas target and the same beam energy, the magnitude of the light emission varies with different projectiles according to the following order: ${\mathrm{H}}_3^{+}>{\mathrm{H}}_2^{+}>{\mathrm{H}}^{+};$ (b) independent of the incident ion, the largest emission cross section for the Balmer radiation is induced by oxygen gas and the smallest by nitrogen gas; (c) the emission cross sections of all Balmer radiations are inversely proportional to the beam energy. A striking feature is observed: the emission cross section for the Balmer radiation depends only on the velocity of the incident ion. Namely, the cross sections for, say, incident ions ${\mathrm{H}}^{+},$ ${\mathrm{H}}_2^{+},$ and ${\mathrm{H}}_3^{+}$ with incident energies $E_1,$ ${2E}_1,$ and ${3E}_1,$ respectively, are the same. A velocity model is used to describe this phenomenon: Upon approaching a gas molecule X, the incident proton or the proton fragment of the incident molecule ion will form a temporary compound $({\mathrm{H}}^{+}+X)$ with the target. The proton will capture an electron from the target and leave the compound in an excited state of the hydrogen atom. An electron-promotion model is suggested for the interaction.

Accurate values of the electron-impact ionization cross sections for the rare gases are needed in a variety of contexts. However, despite numerous investigations over many decades, uncertainty as to the correct values has persisted. The pioneering total-cross-section measurements of Rapp and Englander-Golden are generally regarded as the most reliable but no comprehensive study has independently verified their correctness. In this paper, measurements of electron-impact ionization cross sections of helium, neon, argon, krypton, and xenon are reported for energies ranging from the first ionization threshold to 1000 eV. These data confirm the essential correctness of Rapp and Englander-Golden’s total measurements and at the same time provide a complete set of consistent absolute partial cross sections.

We present a time-dependent formulation of the exterior complex scaling method that has previously been used to treat electron-impact ionization of the hydrogen atom accurately at low energies. The time-dependent approach solves a driven Schrödinger equation, and scales more favorably with the number of electrons than the original formulation. The method is demonstrated in calculations for breakup processes in two dimensions (2D) and three dimensions for systems involving short-range potentials and in 2D for electron-impact ionization in the Temkin-Poet model for electron-hydrogen atom collisions.

Using the eigenchannel R-matrix method we calculate partial photoionization cross sections from the ground state of the helium atom for incident photon energies up to the $N=9\mathrm{}$ manifold. The wide energy range covered by our calculations permits a thorough investigation of general patterns in the cross sections which were first discussed by Menzel and coworkers [Phys. Rev. A 54, 2080 (1996)]. The existence of these patterns can easily be understood in terms of propensity rules for autoionization. As the photon energy is increased the regular patterns are locally interrupted by perturber states until they fade out indicating the progressive breakdown of the propensity rules and the underlying approximate quantum numbers. We demonstrate that the destructive influence of isolated perturbers can be compensated with an energy-dependent quantum defect.

A detailed investigation of positron annihilation in ethylene gas at room temperature and at densities $\rho$ below 1 amagat $(\equiv 2.69\times {10}^{25}\hspace{0.5em}{\mathrm{molecules}\mathrm{}\mathrm{m}}^{-3})$ has revealed that the mean free positron annihilation rate $〈{\lambda }_{\mathrm{f}}〉$ displays a density dependence of the form $〈{\lambda }_{\mathrm{f}}〉=a{\rho }^2+b\rho ,$ with a and b constants. This implies that positrons can annihilate in the gas as a result of three-body interactions and is an unambigous observation of this effect. The physical interpretation of this result is discussed, particularly in the light of related work on positron annihilation on single molecules. It is shown that earlier work on positron annihilation on molecular gases has displayed effects similar to those described here, but their significance has not previously been appreciated.

The Levinson theorem for the Schrödinger equation with a spherically symmetric potential in D dimensions is uniformly established by the Sturm-Liouville theorem. It is shown that the Levinson theorem for the cases without a half bound state does not depend on the spatial dimension D, namely, the phase-shift ${\delta }_l\mathrm{}\left(0\right)\mathrm{}$ of the scattering state with angular momentum l at zero momentum is equal to the total number $n_l$ of bound states multiplied by $\pi .$ When a half bound state occurs the Levinson theorem may be modified.

We present a study of the interaction of a relativistically planar channeled electron with an intense electromagnetic field. Using a S-matrix approach in the strong-field approximation, it is shown that the crystal periodicity affects drastically the excitation process, suppressing the possibility of multiphoton absorption except for some particular cases. This selective excitation opens the possibility to control the dynamics of the channeling process by means of an external field. Explicit expressions for the S-matrix N-photon excitation rates together with the corresponding conservation laws are obtained from the relativistic quantum-mechanical Dirac equation.

This work explores temporal and spatial aspects of coherent population transfer in a three-level atom through a synergic combination of experimental measurements and theoretical calculations. Experimental measurements exploit the broad bandwidth of a femtosecond laser pulse to initiate simultaneous excitation of the $5p{}^2{P}_{1/2}\leftarrow 5s{}^2{S}_{1/2}$ and $5p{}^2{P}_{3/2}\leftarrow 5s{}^2{S}_{1/2}$ components of the doublet line of atomic rubidium. By adjusting positive or negative frequency sweeps the pump pulse favors either one of the two transitions and eventually even decouples the two excited states. The population of the excited spin-orbit levels is monitored in real time by stimulated emission probing under conditions of different intensity, chirp, and pulse width of the driving field, giving detailed information on the coupling between the three levels and their interactions with the driving and probe fields at different points in spacetime. Both pump and probe pulses are carefully characterized after the interaction region by frequency-resolved optical gating. In order to interpret and understand the experimental results it is essential to consider the close relationship between pulse propagation and time evolution of the atomic system via the coupled Maxwell-Bloch equations. This analysis highlights the importance of spatial propagation of the light fields, as well as their temporal dependence, in understanding the dynamical population evolution, and quantitatively reproduces all aspects of the experimental measurements.

We consider a Bose-Einstein condensate that is illuminated by a short resonant light pulse that coherently couples two internal states of the atoms. We show that the subsequent time evolution prepares the atoms in an interesting entangled state called a spin-squeezed state. This evolution is analyzed in detail by developing a Bogoliubov theory that describes the entanglement of the atoms. Our calculation is a consistent expansion in $1/\sqrt{N},$ where N is the number of particles in the condensate, and our theory predicts that it is possible to produce spin-squeezing by at least a factor of $1/\sqrt{N}.$ Within the Bogoliubov approximation this result is independent of temperature.

Generation, interaction, and detection of dark solitons in Bose-Einstein condensates are studied. In particular, we focus on the dynamics resulting from phase imprinting and density engineering. We show that solitons slow down significantly when the trap is opened and that soliton phase shifts after binary interactions cannot be observed with present experiments. Finally, motivated by the recent experimental results of Cornish et al. [Phys. Rev Lett. 85, 1795 (2000)], we analyze the stability of dark solitons under changes of the scattering length and thereby demonstrate a new way to detect them. Our theoretical and numerical results compare well with the existing experimental ones and provide guidance for future experiments.

The connection between quantized vortices and dark solitons in a waveguidelike trap geometry is explored in the framework of the nonlinear Schrödinger equation. Variation of the transverse confinement leads from the quasi-one-dimensional (1D) regime, where solitons are stable, to 2D (or 3D) confinement, where soliton stripes are subject to a transverse modulational instability known as the “snake instability.” We present numerical evidence of a regime of intermediate confinement where solitons decay into single, deformed vortices with solitonic properties rather than vortex pairs as associated with the “snake” metaphor. Further relaxing the transverse confinement leads to the production of two and then three vortices, which correlates perfectly with a Bogoliubov stability analysis. The decay of a stationary dark soliton (or, planar node) into a single solitonic vortex is predicted to be experimentally observable in a 3D harmonically confined dilute-gas Bose-Einstein condensate.

In this paper we calculate the energy levels of two trapped atoms interacting through a spherical two-body potential $V\left(r\right)\mathrm{}$ under external confinement quantum mechanically. We then replace $V\left(r\right)\mathrm{}$ by either a regularized energy-independent or an energy-dependent δ-function potential. A comparison shows that the use of an energy-dependent pseudopotential improves significantly upon the use of an energy-independent pseudopotential. The accuracy of the pseudopotential approximation is approximately determined by the ratio ${\beta }_6{/a}_{\mathrm{ho}},$ where ${\beta }_6$ and $a_{\mathrm{ho}}$ denote the characteristic length scale of the two-body interaction potential $V\left(r\right)\mathrm{}$ and of the trapping potential, respectively.