We present analytic descriptions of atomic interaction at ultracold temperatures using both single-channel and multichannel quantum-defect theories. In the case of a single channel, addressed in this paper, the expansion of Gao [Phys. Rev. A 58, 4222 (1998)] is generalized to higher orders for angular momentum $l\ge 2$ to give a more complete description of ultracold scattering, including an analytic description of ultracold shape resonances of arbitrary $l$. We also introduce a generalized scattering length that is well defined and useful for all partial waves to replace the traditional definition that fails for $l\ge 2$ due to the long-range van der Waals interaction. The results are used in a companion paper to derive analytic descriptions of atomic interaction around a magnetic Feshbach resonance of arbitrary angular momentum $l$.

The coupled-wave equations of three-wave interactions are generalized into three dimensions. The reciprocal lattice vectors (RLVs) of three-dimensional (3D) nonlinear photonic crystals (NPCs) which show some unique properties are analyzed in detail as compared to those in one- and two-dimensional cases. It is proposed as an example that the cross-distributing conical harmonic beams can be obtained from the 3D NPC by intentionally using the proper RLVs. Some other possible applications are suggested at the end of discussion.

The paper continues the analysis of vacuum Rabi oscillations we started in part I [Phys. Rev. A 79, 033836 (2009)]. Here we concentrate on experimental consequences for cavity QED of two different classes of representations of harmonic-oscillator Lie algebras. The zero-temperature master equation, derived in part I for irreducible representations of the algebra, is reformulated in a reducible representation that models electromagnetic fields by a gas of harmonic-oscillator wave packets. The representation is known to introduce automatic regularizations that in irreducible representations would have to be justified by ad hoc arguments. Predictions based on this representation are characterized in thermodynamic limit by a single parameter $\varsigma$, responsible for collapses and revivals of Rabi oscillations in exact vacuum. Collapses and revivals disappear in the limit $\varsigma \to \infty$. Observation of a finite $\varsigma$ would mean that cavity quantum fields are described by a non-Wightmanian theory, where vacuum states are zero-temperature Bose-Einstein condensates of a $N$-particle bosonic oscillator gas and, thus, are nonunique. The data collected in the experiment of Brune et al. [Phys. Rev. Lett. 76, 1800 (1996)] are consistent with any $\varsigma >400$.

The binding energy and fine-structure splittings of the arsenic negative ion $\left({\text{As}}^{-}\right)$ have been measured using infrared photodetachment threshold spectroscopy. The relative cross section for neutral atom production was measured with a crossed ion-beam-laser-beam apparatus over selected photon energy ranges between 630–810 meV. An $s$-wave threshold was observed due to the opening of the ${\text{As}}^{-}\left(4p^4{{}^{3}P}_2\right)$ to $\text{As}\left(4p^3{{}^{4}S}_{3/2}\right)$ ground-state to ground-state transition, which yields the electron affinity of As to be 804.8(2) meV. $s$-wave thresholds were also observed for detachment from the $J=1$ and $J=0$ excited levels of ${\text{As}}^{-}$, permitting accurate determination of the fine-structure splittings of 127.6(2) meV for ${{}^{3}P}_1-{{}^{3}P}_2$ and 164.3(10) meV for ${{}^{3}P}_0-{{}^{3}P}_2$. The present values are consistent with previous measurements and substantially reduce the uncertainties.

We develop a general approach for monitoring and controlling evolution of open quantum systems. In contrast to the master equations describing time evolution of density operators, here, we formulate a dynamical equation for the evolution of the process matrix acting on a system. This equation is applicable to non-Markovian and/or strong-coupling regimes. We propose two distinct applications for this dynamical equation. We first demonstrate identification of quantum Hamiltonians generating dynamics of closed or open systems via performing process tomography. In particular, we argue how one can efficiently estimate certain classes of sparse Hamiltonians by performing partial tomography schemes. In addition, we introduce an optimal control theoretic setting for manipulating quantum dynamics of Hamiltonian systems, specifically for the task of decoherence suppression.

We present the analytical solution in closed form for the semiclassical limit of the quantum-mechanical Coulomb Green’s function in position space in $n$ dimensions. We utilize a projection method which has its roots in Lambert’s theorem and which allows us to treat the system as an essentially one-dimensional problem. The semiclassical result assumes a simple analytical form and is well suited for a numerical evaluation. The method can also be extended to classically forbidden space regions. Already for moderately large principal quantum numbers $\nu \ge 5$, the semiclassical Green’s function is found to be an excellent approximation to the quantum-mechanical Green’s function.

The set of entanglement measures proposed by Hein, Eisert, and Briegel for $n$-qubit graph states [Phys. Rev. A 69, 062311 (2004)] fails to distinguish between inequivalent classes under local Clifford operations if $n\ge 7$. On the other hand, the set of invariants proposed by van den Nest, Dehaene, and De Moor (VDD) [Phys. Rev. A 72, 014307 (2005)] distinguishes between inequivalent classes, but contains too many invariants (more than $2\times {10}^{36}$ for $n=7$) to be practical. Here we solve the problem of deciding which entanglement class a graph state of $n\le 8$ qubits belongs to by calculating some of the state’s intrinsic properties. We show that four invariants related to those proposed by VDD are enough for distinguishing between all inequivalent classes with $n\le 8$ qubits.

We have further extended the two-step approach developed by Chung and Chew [N. N. Chung and L. Y. Chew, Phys. Rev. A 76, 032113 (2007)] to the solution of the quantum dynamics of general systems of $N$-coupled anharmonic oscillators. The idea is to employ an optimized basis set to represent the dynamical quantum states of these oscillator systems. The set is generated via the action of the optimized Bogoliubov transformed bosonic operators on the optimal squeezed vacuum product state. The procedure requires (i) applying the two-step approach to the eigendecomposition of the time evolution operator and (ii) transforming the representation of the initial state from the original to the optimal bases. We have applied the formalism to examine the dynamics of squeezing and entanglement of several anharmonic oscillator systems.

The situation where the quantum Cramér-Rao inequality for a general measurement becomes equality is analyzed in some detail in the case of a family of pure states. In particular, it turns out that under some natural assumptions, the measurement in question is simple, and the states must have a special form. This fact in turn allows us to obtain in the two-dimensional case a characterization of the pure states for which the quantum Cramér-Rao equality holds.

We investigate the feasibility of carrying out quantum walks with cold atoms in a double optical lattice. Monte Carlo simulations of time-of-flight (TOF) detection and absorption imaging were carried out, focusing on a specific experimental implementation. These indicate that absorption imaging would be best suited for detection of quantum walks. With typical experimental parameters a few hundred quantum walk steps will be needed for an unambiguous detection of the quantum walk signature. We show that in special cases, few-step quantum walks can also be detected in our system if one measures the relative population of the atoms in their internal states rather than their displacement in space, that is, measurements are made in the space of the coin operator rather than in that of the displacement operator.

We consider a two-step process that may occur in relativistic ion-atom collisions. In this process, within a single ion-atom collision, an atomic electron is captured by the ion via the radiative and/or nonradiative capture channels, while the colliding nuclei of the atom and ion simultaneously produce an electron-positron pair with the electron created in a bound state of the ion. As a result, the ion picks up two electrons and a positron is emitted. We calculate the total cross section for this process in collisions of incident $1–4\text{GeV}/\text{u}$ ${\text{U}}^{92+}$ projectiles with various atomic targets and discuss its dependence on the collision energy and target atomic number. We also analyze whether this process can compete with electron recombination proceeding via the excitation of the negative-continuum states, in which an atomic electron incident on a bare nucleus is captured into a bound state of the ion by transferring the energy excess to a negative-continuum electron with the latter undergoing a transition into a bound state. According to our estimates, the electron recombination dominates the two-step process in collisions with light and intermediately heavy atoms $\left(Z_a\lesssim 50\right)$, while in collisions with heavy atoms the latter leads to much larger cross sections.

The quantitative rescattering theory (QRS) for high-order harmonic generation (HHG) by intense laser pulses is presented. According to the QRS, HHG spectra can be expressed as a product of a returning electron wave packet and the photorecombination differential cross section of the laser-free continuum electron back to the initial bound state. We show that the shape of the returning electron wave packet is determined mostly by the laser. The returning electron wave packets can be obtained from the strong-field approximation or from the solution of the time-dependent Schrödinger equation (TDSE) for a reference atom. The validity of the QRS is carefully examined by checking against accurate results for both harmonic magnitude and phase from the solution of the TDSE for atomic targets within the single active electron approximation. Combining with accurate transition dipoles obtained from state-of-the-art molecular photoionization calculations, we further show that available experimental measurements for HHG from partially aligned molecules can be explained by the QRS. Our results show that quantitative description of the HHG from aligned molecules has become possible. Since infrared lasers of pulse durations of a few femtoseconds are easily available in the laboratory, they may be used for dynamic imaging of a transient molecule with femtosecond temporal resolutions.

Full three-dimensional (3D) calculations of small two-component Fermi gases under highly elongated confinement, in which unlike fermions interact through short-range potentials with variable atom-atom $s$-wave scattering length, are performed using the correlated Gaussian approach. In addition, microscopic one-dimensional (1D) calculations are performed for effective “atomic” and “molecular” 1D model Hamiltonian. Comparisons of the 3D and 1D energies and excitation frequencies establish the validity regimes of the effective 1D Hamiltonian. Our numerical results for three- and four-particle systems suggest that the effective 1D atom-dimer and dimer-dimer interactions are to a good approximation determined by simple analytical expressions. Implications for the description of quasi-1D Fermi gases within strict 1D frameworks are discussed.

The $2{{}^{3}P}_1–2{{}^{3}P}_2$ interval in helium is measured using microwave separated oscillatory fields. Our measured result is $2291177.53\pm 0.35\text{kHz}$ . A disagreement with theory of over 15 kHz indicates a major problem with two-electron QED calculations. If this disagreement is resolved, measurements of both $2{}^{3}P$ fine-structure intervals at this accuracy would lead to a six-parts-per-billion determination of the fine-structure constant.

A quantum-shift-register circuit acts on a set of input qubits and memory qubits, outputs a set of output qubits and updated memory qubits, and feeds the memory back into the device for the next cycle (similar to the operation of a classical shift register). Such a device finds application as an encoding and decoding circuit for a particular type of quantum error-correcting code called a quantum convolutional code. Building on the Ollivier-Tillich and Grassl-Rötteler encoding algorithms for quantum convolutional codes, I present a method to determine a quantum-shift-register encoding circuit for a quantum convolutional code. I also determine a formula for the amount of memory that a Calderbank-Shor-Steane (CSS) quantum convolutional code requires. I then detail primitive quantum-shift-register circuits that realize all of the finite- and infinite-depth transformations in the shift-invariant Clifford group (the class of transformations important for encoding and decoding quantum convolutional codes). The memory formula for a CSS quantum convolutional code then immediately leads to a formula for the memory required by a CSS entanglement-assisted quantum convolutional code.

A simple and completely general representation of the exact exchange-correlation functional of density-functional theory is derived from the universal Lieb-Oxford bound, which holds for any Coulomb-interacting system. This representation leads to an alternative point of view on popular hybrid functionals, providing a rationale for why they work and how they can be constructed. A similar representation of the exact correlation functional allows to construct fully nonempirical hyper-generalized-gradient approximations (HGGAs), radically departing from established paradigms of functional construction. Numerical tests of these HGGAs for atomic and molecular correlation energies and molecular atomization energies show that even simple HGGAs match or outperform state-of-the-art correlation functionals currently used in solid-state physics and quantum chemistry.

In order to explain discrepancies between theoretical predictions and experimental data for the helium fine structure, we check and recalculate all theoretical contributions up to orders $m{\alpha }^7$ and $m^2/M{\alpha }^6$. The previous result for the $m{\alpha }^7$ correction is improved by a more accurate calculation of relativistic corrections to the Bethe logarithm. The theoretical values of the $2{{}^{3}P}_0-2{{}^{3}P}_1$ and $2{{}^{3}P}_1-2{{}^{3}P}_2$ fine-structure intervals in helium are, correspondingly, ${\nu }_{01}=29616946.2\left(1.6\right)\text{kHz}$ and ${\nu }_{12}=2291177.3\left(1.6\right)\text{kHz}$, with the uncertainties being due to higher-order effects. For the small interval ${\nu }_{12}$, the theoretical value agrees with the experimental data, whereas for the large interval ${\nu }_{01}$, a discrepancy of about three standard deviations is present.

We have performed a comprehensive study of the singlet ground state of two electrons on the surface of a sphere of radius $R$. We have used electronic structure models ranging from restricted and unrestricted Hartree-Fock theories to explicitly correlated treatments, the last of which leads to near-exact wave functions and energies for any value of $R$. Møller-Plesset energy corrections (up to fifth-order) are also considered, as well as the asymptotic solution in the large-$R$ regime.

We have developed a method within the Dirac $R$-matrix theory to calculate x-ray line polarization resulting from electron-impact excitation to excited-state magnetic sublevels of complex ions. We apply this method to the excitation of two well-known x-ray lines of Fe XVII, $3C$ $\left(\lambda =15.015\text{Å}\right)$ and $3D$ $\left(\lambda =15.262\text{Å}\right)$. We find strong resonance effects which contribute to complicated energy dependence in the polarization. Our results show large differences, up to 30% or more, from earlier theoretical work, and thus provide a stimulus for direct experimental measurement of the $3C$ and $3D$ line polarizations and further theoretical studies.

We have simultaneously measured the kinetic energy release and the angular distributions for the multiply charged fragmentation of diatomic molecular ions, ${{\text{N}}_2}^{+}$, ${\text{CO}}^{+}$, ${\text{NO}}^{+}$, and ${{\text{O}}_2}^{+}$, in intense ultrashort laser pulses. Using a coincidence three-dimensional momentum imaging method we unambiguously separate all fragmentation channels, including dissociation, since we measure both neutral and ionic products. Based on the experimental results, we conclude that the fragmentation mechanism that explains the multielectron dissociative ionization of these molecular ions entails a stairstep process. This mechanism, which involves stretching the molecules prior to each sequential ionization step, is similar to that proposed in some earlier studies that use neutral molecules as targets. We find that structure in the breakup of the molecules to lower charge states is molecule specific, while, for breakup to higher charge states, all species tend to follow a universal trend.