We present a theoretical analysis of a Bose-Einstein condensate (BEC) enclosed in an optical cavity driven by an external laser beam. In the limit where the cavity-field dynamics is adiabatically eliminated, the interaction between the condensate and the cavity displays a range of nonlinear dynamical behaviors such as multistability, quasiperiodicity, and conservative chaotic oscillations. We show that chaotic oscillations are ubiquitous when repulsive atom-atom interactions are included in the model. Despite the complexity of the full coupled BEC-cavity system, the essential features of its nonlinear and chaotic behavior are low-dimensional and are well described by considering the evolution of only a few motional states of the BEC.

We report on the observation of quantum interference of a collective single spin excitation with a spin ensemble of $N_{\mathrm{a}}\approx {10}^5$ atoms. Detection of a single photon scattered from the atoms creates the single spin excitation, a Fock state embedded in the collective spin of the ensemble. The state of the atomic ensemble is then detected by a quantum nondemolition measurement of the collective spin. A macroscopic difference of the order of ${\sqrt{N}}_{\mathrm{a}}$ in the marginal distribution of the collective spin state arises from the interference between the single excited spin and $N_{\mathrm{a}}$ atoms. These hybrid discrete-continuous manipulation and measurement procedures of collective spin states in an atomic ensemble pave the road towards generation of even more exotic ensemble states for quantum information processing, precision measurements, and communication.

A scheme for quantum-state tomography is presented that can be performed for polarized light with an arbitrary photon-number distribution. The proposed method fills the gap between existing polarization-tomography schemes for single-photon states and for optical fields with very large photon numbers. It consists of an optical homodyne setup triggered by the outcome of a single-photon counting module. The configuration space of polarization is parametrized by Euler angles and by an additional interference parameter, whose assessment requires conditional homodyne measurements.

In this paper, we derive a microscopic master equation for a pair of $XY$-coupled two-level systems interacting with the same memoryless reservoir. In particular, we apply this master equation to the case of a pair of two-level atoms in free space where we can clearly contrast the predictions made with the microscopic master equation obtained here and the phenomenological approaches where the atom-atom coupling is included just a posteriori, i.e., not taking into account in the derivation of the open system equation of motion. We show, for instance, that the phenomenological approach fails completely in the assessment of the role played by the symmetric and antisymmetric decay channels. As a consequence, the predictions related to collective effects such as superradiance, for instance, are misleading in the phenomenological approach. We also obtain the fluorescence spectrum using the microscopic model developed here.

Implementing a single-qubit unitary is often hampered by imperfect control. Systematic amplitude errors $\epsilon$, caused by incorrect duration or strength of a pulse, are an especially common problem. But a sequence of imperfect pulses can provide a better implementation of a desired operation, as compared to a single primitive pulse. We find optimal pulse sequences consisting of $L$ primitive $\pi$ or $2\pi$ rotations that suppress such errors to arbitrary order $O\left({\epsilon }^n\right)$ on arbitrary initial states. Optimality is demonstrated by proving an $L=O\left(n\right)$ lower bound and saturating it with $L=2n$ solutions. Closed-form solutions for arbitrary rotation angles are given for $n=1,2,3,4$. Perturbative solutions for any $n$ are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to $n=12$. The derivation proceeds by a novel algebraic and nonrecursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.

In the graph isomorphism (GI) problem two $N$-vertex graphs $G$ and $G^{\prime }$ are given and the task is to determine whether there exists a permutation of the vertices of $G$ that preserves adjacency and transforms $G\to G^{\prime }$. If yes, then $G$ and $G^{\prime }$ are said to be isomorphic; otherwise they are nonisomorphic. The GI problem is an important problem in computer science and is thought to be of comparable difficulty to integer factorization. In this paper we present a quantum algorithm that solves arbitrary instances of GI and which also provides an approach to determining all automorphisms of a given graph. We show how the GI problem can be converted to a combinatorial optimization problem that can be solved using adiabatic quantum evolution. We numerically simulate the algorithm's quantum dynamics and show that it correctly (i) distinguishes nonisomorphic graphs; (ii) recognizes isomorphic graphs and determines the permutation(s) that connect them; and (iii) finds the automorphism group of a given graph $G$. We then discuss the GI quantum algorithm's experimental implementation, and close by showing how it can be leveraged to give a quantum algorithm that solves arbitrary instances of the NP-complete subgraph isomorphism problem. The computational complexity of an adiabatic quantum algorithm is largely determined by the minimum energy gap $\Delta \left(N\right)$ separating the ground and first-excited states in the limit of large problem size $N\gg 1$. Calculating $\Delta \left(N\right)$ in this limit is a fundamental open problem in adiabatic quantum computing, and so it is not possible to determine the computational complexity of adiabatic quantum algorithms in general, nor consequently, of the specific adiabatic quantum algorithms presented here. Adiabatic quantum computing has been shown to be equivalent to the circuit model of quantum computing, and so development of adiabatic quantum algorithms continues to be of great interest.

Calculations of the positron binding energies to all atoms in the Periodic Table are presented and atoms where the positron-atom binding actually exists are identified. The results of these calculations and accurate calculations of other authors (which existed for several atoms only) are used to evaluate recommended values of the positron binding energies to the ground states of atoms. We also present the recommended energies of the positron excited bound levels and resonances (due to the binding of positron to excited states of atoms) which cannot emit positronium and have relatively narrow widths. Such resonances in positron annihilation and scattering may be used to measure the positron binding energy.

Ab initio scattering calculations for low-energy electron collisions with an open-shell molecule H${}_2$CN have been carried out using the $R$-matrix method. The elastic integral, differential, and momentum transfer cross sections and the excitation cross sections from the ground state to the three low-lying electron excited states are presented in the energy region of 0–10 eV. The results of the static exchange, correlated 1-state and 14-state close-coupling approximations are investigated to identify the low-lying resonant states of the anions formed due to electron capture by the radical. Four shape resonances and three core-excited shape resonances have been detected in the 14-state model. The Born-closure approximation is applied for the elastic and dipole-allowed transitions to account for the $l$ > 4 partial waves excluded from the $R$-matrix calculations.

A transport theory that explicitly incorporates loss of flux due to annihilating collisions is developed and applied to low-energy positron diffusion and annihilation. The use of more complete momentum transfer and annihilation cross sections for helium has resulted in improved descriptions of the time dependence of $\langle Z_{\mathrm{eff}}\rangle$ for positrons injected into gaseous helium. Similarly, the variation of $\langle Z_{\mathrm{eff}}\rangle$ versus $E/n_0$ for experiments where the annihilation region is immersed in an electric field is in closer agreement with experimental data. Inclusion of loss of flux due to annihilation was found to have a very small effect on the derived $\langle Z_{\mathrm{eff}}\left(t\right)\rangle$ for helium.

We propose an efficient method for the generation of ultrabroadband supercontinuum spectra and isolated ultrashort attosecond laser pulses from He atoms with two-color midinfrared laser fields. High-order harmonic generation (HHG) is obtained by solving the time-dependent Schrödinger equation accurately by means of the time-dependent generalized pseudospectral method. We found that the optimizing two-color midinfrared laser pulse allows the HHG cutoff to be significantly extended, leading to the production of an ultrabroadband supercontinuum. As a result, an isolated 18-attosecond pulse can be generated directly by the superposition of the supercontinuum harmonics. To facilitate the exploration of the ultrashort attosecond generation mechanisms, we perform both a semiclassical simulation and a wavelet time-frequency transform.

The Compton scattering of x-ray photons, assisted by a short intense optical laser pulse, is discussed. The differential scattering cross section reveals the interesting feature that the main Klein-Nishina line is accompanied by a series of side lines forming a broad plateau where up to $O\left({10}^3\right)$ laser photons participate simultaneously in a single scattering event. An analytic formula for the width of the plateau is given. Due to the nonlinear mixing of x-ray and laser photons a frequency-dependent rotation of the polarization of the final-state x-ray photons relative to the scattering plane emerges. A consistent description of the scattering process with short laser pulses requires to work with x-ray pulses. An experimental investigation can be accomplished, e.g., at LCLS or the European XFEL, in the near future.

We studied the superfluid-to-Mott-insulator transition for bosonic hard spheres loaded in asymmetric three-dimensional optical lattices by means of diffusion Monte Carlo calculations. The onset of the transition was monitored through the change in the chemical potential around the density corresponding to one particle per potential well. With this method, we were able to reproduce the results given in the literature for three-dimensional symmetric lattices and for systems whose asymmetry makes them equivalent to a set of quasi-one-dimensional tubes. The location of the same transition for asymmetric systems akin to a stack of quasi-two-dimensional lattices will be also given. Our results were checked against those given by a Bose-Hubbard model for similar arrangements.

We consider a Bose-Fermi mixture in the molecular limit of the attractive interaction between fermions and bosons. For a boson density smaller than or equal to the fermion density, we show analytically how a T-matrix approach for the constituent bosons and fermions recovers the expected physical limit of a Fermi-Fermi mixture of molecules and atoms. In this limit, we derive simple expressions for the self-energies, the momentum distribution function, and the chemical potentials. By extending these equations to a trapped system, we determine how to tailor the experimental parameters of a Bose-Fermi mixture in order to enhance the indirect Pauli exclusion effect on the boson momentum distribution function. For the homogeneous system, we present finally a diffusion Monte Carlo simulation which confirms the occurrence of such a peculiar effect.

We investigate the ground-state properties of trapped fermion systems described by the Hubbard model with an external confining potential. We discuss the universal behaviors of systems in different regimes: from few particles, i.e., in dilute regimes, to the trap thermodynamic limit. The asymptotic trap-size (TS) dependence in the dilute regime (increasing the trap size $\ell$ keeping the particle number $N$ fixed) is described by a universal TS scaling controlled by the dilute fixed point associated with the metal-to-vacuum quantum transition. This scaling behavior is numerically checked by DMRG simulations of the one-dimensional (1D) Hubbard model. In particular, the particle density and its correlations show crossovers among different regimes: for strongly repulsive interactions they approach those of a spinless Fermi gas, for weak interactions those of a free Fermi gas, and for strongly attractive interactions they match those of a gas of hard-core bosonic molecules. The large-$N$ limit keeping the ratio $N/\ell$ fixed corresponds to a 1D trap thermodynamic limit. We address issues related to the accuracy of the local density approximation (LDA). We show that the particle density approaches its LDA in the large-$\ell$ limit. When the trapped system is in the metallic phase, corrections at finite $\ell$ are $O\left({\ell }^{-1}\right)$ and oscillating around the center of the trap. They become significantly larger at the boundary of the fermion cloud, where they get suppressed as $O\left({\ell }^{-1/3}\right)$ only. This anomalous behavior arises from the nontrivial scaling at the metal-to-vacuum transition occurring at the boundaries of the fermion cloud.

Many works are based on the steady-state analysis of mean-value dynamics in electro- or optomechanical systems to explore vibration cooling, squeezing, and quantum-state controlling of massive objects. These studies are always conducted in a red-detuned pumping field under a lower power to maintain a stable situation. In this paper we consider self-sustained oscillations of a cavity-field-driven oscillator combined with quadratic coupling in a blue-detuned regime above a pumping threshold. Our study finds that the oscillator will be far away from its steady-state behavior by conducting a self-sustained oscillation with a discrete amplitude locking effect producing a rich energy-balanced structure. The dynamical backaction of this self-oscillation on the field mode induces a multipeak field spectrum, which implies an efficient harmonic generation with its intensity modified not only by the displacement $x_0$ but also by the amplitude $A$ of the mechanical oscillation. The corresponding nonlinear field spectrum and its magnitude are analytically analyzed with quadratic coupling when the mechanical oscillator is dynamically locked to a self-sustained oscillation.

Measuring a weak force is an important task for micromechanical systems, both when using devices as sensitive detectors and, particularly, in experiments of quantum mechanics. The optimal strategy for resolving a weak stochastic signal force on a huge background (typically given by thermal noise) is a crucial and debated topic, and the stability of the mechanical resonance is a further, related critical issue. We introduce and analyze the parametric control of the optical spring, which allows us to stabilize the resonance and provides a phase reference for the oscillator motion, yet conserving a free evolution in one quadrature of the phase space. We also study quantitatively the characteristics of our micro-optomechanical system as detector of stochastic force for short measurement times (for quick, high-resolution monitoring) as well as for the longer-term observations that optimize the sensitivity. We compare a simple strategy based on the evaluation of the variance of the displacement which is a widely used technique) with an optimal Wiener-Kolmogorov data analysis. We show that, due to the parametric stabilization of the effective susceptibility, we can more efficiently implement Wiener filtering, and we investigate how this strategy improves the performance of our system. We finally demonstrate the possibility to resolve stochastic force variations well below 1$\%$ of the thermal noise.

We propose a scheme for the generation of a robust stationary squeezed state of a mechanical resonator in a quadratically coupled optomechanical system, driven by a pulsed laser. The intracavity photon number presents periodic intense peaks suddenly stiffening the effective harmonic potential felt by the mechanical resonator. These “optical spring kicks” tend to squeeze the resonator position, and due to the interplay with fluctuation-dissipation processes one can generate a stationary state with more than 10 dB of squeezing in a realistic scenario, even starting from moderately “precooled” initial thermal states.

We predict a paradigm shift in the nonlinear physics associated with ultrashort long-wavelength pulse propagation in gaseous and condensed media. Optical carrier shock formation, which profoundly modifies the underlying optical wave form, emerges prior to the onset of the well-known self-focusing collapse singularity. The canonical description of pulse propagation in this regime, where all nonlinear envelope descriptions fail, is identified as the full field carrier resolved modified Kadomtsev-Petviashili (MKP) equation.