We critically examine the internal consistency of a set of minimal assumptions entering the theory of fault-tolerant quantum error correction for Markovian noise. These assumptions are fast gates, a constant supply of fresh and cold ancillas, and a Markovian bath. We point out that these assumptions may not be mutually consistent in light of rigorous formulations of the Markovian approximation. Namely, Markovian dynamics requires either the singular coupling limit (high temperature), or the weak coupling limit (weak system-bath interaction). The former is incompatible with the assumption of a constant and fresh supply of cold ancillas, while the latter is inconsistent with fast gates. We discuss ways to resolve these inconsistencies. As part of our discussion we derive, in the weak coupling limit, a new master equation for a system subject to periodic driving.

We investigate the response of an $np$ Rydberg wave packet to orthogonally polarized weak half-cycle pulses (HCPs) over the course of its field-free evolution. The population redistribution from $p$ to $s$ states is highly sensitive to the polarization of the HCP and changes with the precession of the electron orbits due to the spin-orbit coupling. We obtain the selection rules and give an intuitive classical interpretation of angular momentum redistribution by a HCP.

Electron-impact ionization cross sections for ${\mathrm{H}}_2$ are calculated using a nonperturbative time-dependent close-coupling method. In a standard frozen-core approximation, the six-dimensional wave function for the valence target electron and the incident projectile electron is expanded in products of rotational functions. The time-dependent Schrödinger equation for the two-electron system is then reduced to a set of close-coupled partial differential equations for the four-dimensional expansion functions in $(r_1,{\theta }_1,r_2,{\theta }_2)$ center-of-mass spherical polar coordinates. The nonperturbative close-coupling results are found to be over a factor of 2 lower than perturbative distorted-wave results, but in excellent agreement with experimental measurements, at incident electron energies near the peak of the total integrated cross section.

Low-energy electron collisions with ${\mathrm{O}}_2$ molecules are studied using the fixed-bond $R$-matrix method. In addition to the ${\mathrm{O}}_2$ $X{}^3\Sigma _g^{-}$ ground state, integrated cross sections are calculated for electron collisions with the $a{}^1\Delta _g$ and $b{}^1\Sigma _g^{+}$ excited states of ${\mathrm{O}}_2$ molecules. Thirteen target electronic states of ${\mathrm{O}}_2$ are included in the model within a valence configuration interaction representations of the target states. Elastic cross sections for the $a{}^1\Delta _g$ and $b{}^1\Sigma _g^{+}$ excited states are similar to the cross sections for the $X{}^3\Sigma _g^{-}$ ground state. As in case of excitation from the $X{}^3\Sigma _g^{-}$ state, the ${\mathrm{O}}_2^{-}$ ${\Pi }_u$ resonance makes the dominant contribution to excitation cross sections from the $a{}^1\Delta _g$ and $b{}^1\Sigma _g^{+}$ states. The magnitude of excitation cross sections from the $a{}^1\Delta _g$ state to the $b{}^1\Sigma _g^{+}$ state is about ten times larger than the corresponding cross sections from the $X{}^3\Sigma _g^{-}$ to the $b{}^1\Sigma _g^{+}$ state. For this $a{}^1\Delta _g\to b{}^1\Sigma _g^{+}$ transition, our cross section at $4.5\mathrm{eV}$ agrees well with the available experimental value. These results should be important for models of plasma discharge chemistry which often requires cross sections between the excited electronic states of ${\mathrm{O}}_2$.

We investigate how a two-level system inside of a planar cavity behaves as a cavity quantum electrodynamics (QED) system. Starting from a three-dimensional model, the method for determining the cavity-QED parameters $(g,\kappa ,\gamma )$ is presented, and the parameters are evaluated as functions of the input beam profile. It is shown that suppression of the radiative loss of cavity photons is possible by engineering the lateral profile of the input beam.

We report experimental observation of two dynamical regimes in operation of a $\mathrm{Tm}:{\mathrm{YAlO}}_3$ microlaser: a stable continuous wave and robust self-pulsing. The experiment makes possible the assumption of exited-state absorption at the lasing wavelength. Such excited-state absorption can be a good candidate to explain the well-known instabilities observed in some thulium-doped lasers. Our rate equation model takes account of the phonon-assisted excited-state absorption ${}^3\mathrm{H}_5\to {}^3\mathrm{H}_4$, and the cross-relaxation and up-conversion processes. The results of numerical calculations are in good agreement with experiment.

Beyond the quantum Markov approximation and the weak-coupling limit, we present a general theory to calculate the geometric phase for open systems with and without conserved energy. As an example, the geometric phase for a two-level system coupling both dephasingly and dissipatively to its environment is calculated. Comparison with the results from quantum trajectory analysis is presented and discussed.

The question how to quantize a classical system where an angle $\phi$ is one of the basic canonical variables has been controversial since the early days of quantum mechanics. The problem is that the angle is a multivalued or discontinuous variable on the corresponding phase space. The remedy is to replace $\phi$ by the smooth periodic functions $\cos \phi$ and $\sin \phi$. In the case of the canonical pair $(\phi ,p_{\phi })$, where $p_{\phi }$ is the orbital angular momentum (OAM), the phase space ${\mathcal{S}}_{\phi ,p_{\phi }}=\{\phi ∊\mathbb{R}\mathrm{mod}2\pi ,p_{\phi }∊\mathbb{R}\}$ has the global topological structure $S^1\times \mathbb{R}$ of a cylinder on which the Poisson brackets of the three functions $\cos \phi ,\sin \phi$, and $p_{\phi }$ obey the Lie algebra of the Euclidean group $\mathrm{E}\left(2\right)$ in the plane. This property provides the basis for the quantization of the system in terms of irreducible unitary representations of the group $\mathrm{E}\left(2\right)$ or of its covering groups. A crucial point is that, due to the fact that the subgroup $\mathrm{SO}\left(2\right)\cong S^1$ is multiply connected, these representations allow for fractional OAM $l=\hslash (n+\delta ),n∊\mathbb{Z},\delta ∊[0,1)$. Such $\delta \ne 0$ have already been observed in cases like the Aharonov-Bohm and fractional quantum Hall effects, and they correspond to the quasimomenta of Bloch waves in ideal crystals. The proposal of the present paper is to look for fractional OAM in connection with the quantum optics of Laguerre-Gaussian laser modes in external magnetic fields. The quantum theory of the phase space ${\mathcal{S}}_{\phi ,p_{\phi }}$ in terms of unitary representations of $\mathrm{E}\left(2\right)$ allows for two types of “coherent” states, the properties of which are discussed in detail: nonholomorphic minimal-uncertainty states and holomorphic ones associated with Bargmann-Segal Hilbert spaces.

When a three-level system is coherently driven by a strong modified standing wave, photonic band gaps are viable to realize. Theoretically detailed calculations, for ${\mathrm{Pr}}^{3^{+}}$-doped yttrium orthosilicate crystal with inhomogeneously broadening, further show that the gap structure can be dynamically controlled and optimized by modulating the Rabi frequency, the spatial periodicity of the standing wave, the reflectivity of mirror, and the angle of incidence. This adjustable photonic band gap determined by the external pump control parameters in a doped solid crystal should hold more potential for applications than that in typical photonic crystals which are determined once for all by the way the material periodic structure is grown.

We demonstrate numerically that the wavelength conversion of light is possible by the simple dynamic refractive index tuning of an optical cavity in a photonic crystal. We also clarify the mechanism and conservation rule for this conversion process. In addition, we discuss the observability of this phenomenon in realistic cavities. Our results indicate that this linear adiabatic wavelength conversion process can be observed for various high-$Q$ microcavities.

A system of diagrams is introduced that allows the representation of various elements of a quantum circuit, including measurements, in a form which makes no reference to time (hence “atemporal”). It can be used to relate quantum dynamical properties to those of entangled states (map-state duality), and suggests useful analogies, such as the inverse of an entangled ket. Diagrams clarify the role of channel kets, transition operators, dynamical operators (matrices), and Kraus rank for noisy quantum channels. Positive (semidefinite) operators are represented by diagrams with a symmetry that aids in understanding their connection with completely positive maps. The diagrams are used to analyze standard teleportation and dense coding, and for a careful study of unambiguous (conclusive) teleportation. A simple diagrammatic argument shows that a Kraus rank of 3 is impossible for a one-qubit channel modeled using a one-qubit environment in a mixed state.

A description of the excited lepton states of the hydrogen-antihydrogen quasimolecule is presented. Potential energy curves and the leptonic part of the wave functions corresponding to a variety of such states are calculated within the Born-Oppenheimer approximation employing the Ritz variational principle. Nonadiabatic corrections to the leptonic potentials are also obtained. Basis functions are constructed as products of explicitly correlated Gaussians and spherical harmonics which describe correctly the motion of leptons with arbitrary orbital angular momentum projection onto the molecular (internuclear) axis. The hadronic part of the wave function for each leptonic level of the hydrogen-antihydrogen system is calculated by solving the Schrödinger equation with the obtained leptonic potentials. Corresponding solutions are generated utilizing precise B-spline representations. Employing leptonic and hadronic parts of the wave function the electron-positron and proton-antiproton annihilation rates are computed for a number of quasimolecular states. The decay rates of the hydrogen-antihydrogen system into separate positronium and protonium atoms are also estimated for the quasimolecular levels under consideration.

An effective Hamiltonian for the Bose subsystem for the mixture of ultracold atomic clouds of bosons and fermions is obtained by integrating out the fermion degrees of freedom. Using the effective Hamiltonian, a collapse of the trapped boson-fermion mixture due to the boson-fermion attractive interaction without and in the presence of the quantized vortices is studied in the framework of variational Bose wave function and Thomas-Fermi approximation. The properties of the ${}^{87}\mathrm{Rb}$ and ${}^{40}\mathrm{K}$ mixture are analyzed quantitatively. The critical number of bosons for the collapse transition is estimated without and in the presence of the vortices as a function of the fermion number. It is shown that the critical number of bosons increases in the presence of the vortex. The vortex critical velocities are calculated as functions of the numbers of bosons and fermions.

We investigate the resonant quantum motion of spin-1 particles in a magnetic guide. A symmetry analysis is undertaken in order to reveal the complex symmetry properties of the system which lead to a twofold degeneracy in the energy-level spectrum. Lifetimes and energies of the resonance states are computed by employing the complex scaling method. By analyzing several hundreds of resonances we are able to make conclusions about the global properties of the resonance spectrum. The effect of an Ioffe field which is applied parallel to the guide is also discussed. We observe a global increase of the lifetimes with increasing Ioffe-field strength. For certain parameter regimes we find the ground-state resonance to exhibit a longer lifetime than the energetically neighbored excited states. The latter could have interesting implications on the time evolution of trapped ultracold atomic ensembles. Applications of our results to calculate the resonance energies andlifetimes of the trapped alkali-metal atoms ${}^7\mathrm{Li}$ and ${}^{87}\mathrm{Rb}$ are outlined.

A theoretical investigation is carried out into the effect of spontaneously generated coherence on the Kerr nonlinearity of general three-level systems of $\Lambda$, ladder, and V-shape types. It is found, with spontaneously generated coherence present, that the Kerr nonlinearity can be clearly enhanced. In the $\Lambda$- and ladder-type systems, the maximal Kerr nonlinearity increases and at the same time enters the electromagnetically induced transparency window as the spontaneously generated coherence intensifies. As for the V-type system, the absorption property is significantly modified and therefore enhanced Kerr nonlinearity without absorption occurs for certain probe detunings. We attribute the enhancement of Kerr nonlinearity mainly to the presence of an extra atomic coherence induced by the spontaneously generated coherence.

We measured and modeled third-harmonic generation (THG) from an $\mathrm{AlN}$ thin film on sapphire using a time-domain approach appropriate for ultrafast lasers. Second-harmonic measurements indicated that polycrystalline $\mathrm{AlN}$ contains long-range crystal texture. An interface model for third-harmonic generation enabled an analytical representation of scanning THG ($z$-scan) experiments. Using it and accounting for Fresnel reflections, we measured the $\mathrm{AlN}$–sapphire susceptibility ratio and estimated the susceptibility for aluminum nitride, ${\chi }_{xxxx}^{\left(3\right)}(3\omega ;\omega ,\omega ,\omega )=1.52\pm 0.25\times {10}^{-13}\mathrm{esu}$. The third-harmonic (TH) spectrum strongly depended on the laser focus position and sample thickness. The amplitude and phase of the frequency-domain interference were fit to the Fourier transform of the calculated time-domain field to improve the accuracy of several experimental parameters. We verified that the model works well for explaining TH signal amplitudes and spectral phase. Some anomalous features in the TH spectrum were observed, which we attributed to nonparaxial effects.

An information-theoretic temporal Bell inequality is formulated to contrast classical and quantum computations. Any classical algorithm satisfies the inequality, while quantum ones can violate it. Therefore, the violation of the inequality is an immediate consequence of the quantumness in the computation. Furthermore, this approach suggests a notion of temporal nonlocality in quantum computation.

The effect of the distribution of mass in triatomic reaction dynamics is analyzed using the geometry of the associated internal space. Atomic masses are appropriately incorporated into internal coordinates as well as the associated non-Euclidean internal space metric tensor after a separation of the rotational degrees of freedom. Because of the non-Euclidean nature of the metric in the internal space, terms such as connection coefficients arise in the internal equations of motion, which act as velocity-dependent forces in a coordinate chart. By statistically averaging these terms, an effective force field is deduced, which accounts for the statistical tendency of geodesics in the internal space. This force field is shown to play a crucial role in determining mass-related branching ratios of isomerization and dissociation dynamics of a triatomic molecule. The methodology presented can be useful for qualitatively predicting branching ratios in general triatomic reactions, and may be applied to the study of isotope effects.

Relativistic positronium (Ps) is of potential use to address fundamental questions in QED—e.g., through direct lifetime measurements of the “para” state of Ps, severe magnetic quenching of the ortho-Ps lifetime, so-called “superpenetration,” and possibly to measure Ps-atom cross sections. Existing schemes for the production of relativistic positronium have relied on pion production through, e.g., nuclear interactions. This yields a low-intensity beam with relatively poor characteristics in both longitudinal and transverse emittance. By use of positrons impinging on a thin carbon foil inside a high-frequency rf cavity, it is proposed to generate relativistic positronium ions $\left({\mathrm{Ps}}^{-}\right)$ by rapid acceleration. Relativistic Ps can be derived from the positronium negative ion by subsequent Lorentz stripping or photodetachment. Intensities of the order 100 per second and Lorentz factors $\simeq 20$ are feasible with present technology.

We consider critical space-time correlations in trapped utracold atoms and propose a method for their detection by the interference with a reference Bose-Einstein condensate. Important information about universal properties can be obtained despite nonuniformity and small sizes of atomic clouds. The advantage of this method is that the correlation properties of the interference pattern are robust with respect to expansion and are not, practically, affected by the microscopic structure of the underlying lattice including structural disorder. The nondestructive scheme allows measuring the space-time correlators of arbitrary order.