We propose methods of fidelity estimation and entanglement verification for experimentally produced four-qubit cluster states. We show that we can obtain a high lower bound of the fidelity using only four local projective measurement settings. The lower bound is close to the exact fidelity, which is determined only by at least nine local projective measurement settings. We also present witness operators for distinguishing entanglement around a four-qubit cluster state from specific classes of genuine four-qubit entanglement, e.g., a class including GHZ and $W$ types of entanglement.

Following a key idea of unconventional geometric quantum computation developed earlier [S. L. Zhu and Z. D. Wang, Phys. Rev. Lett. 91, 187902 (2003)], here we propose a more general scheme in such an intriguing way: ${\gamma }_d={\alpha }_g+\eta {\gamma }_g$, where ${\gamma }_d$ and ${\gamma }_g$ are respectively the dynamic and geometric phases accumulated in the quantum gate operation, with $\eta$ as a constant and ${\alpha }_g$ being dependent only on the geometric feature of the operation. More interestingly, we demonstrate an experiment to implement a universal set of such kind of generalized unconventional geometric quantum gates with high fidelity in an NMR system.

The narrow optical frequency resonances of the $6s{}^{2}S_{1∕2}\to 5d{}^{2}D_{3∕2}$ and $6s{}^{2}S_{1∕2}\to 5d{}^{2}D_{5∕2}$ forbidden transitions in ${\mathrm{Ba}}^{+}$ have been proposed as suitable frequencies for a new optical frequency standard. The major sources of errors in measurements of singly ionized systems are due to the quadratic Zeeman and electric quadrupole shifts. We report here the most accurate calculations to date for the hyperfine structure constants and electric-quadrupole moments of the $5d{}^{2}D_{3∕2}$ and $5d{}^{2}D_{5∕2}$ states in ${\mathrm{Ba}}^{+}$, which determine the quadratic Zeeman and electric quadrupole shifts. Relativistic coupled-cluster theory has been employed to calculate these quantities and large electron correlation effects are observed. It is also shown that for the high accuracy calculation of the $5d{}^{2}D_{5∕2}$ state in ${\mathrm{Ba}}^{+}$, all order core polarization effects play a significant role.

Prepare and measure quantum key distribution protocols can be decomposed into two basic steps: delivery of the signals over a quantum channel and distillation of a secret key from the signal and measurement records by classical processing and public communication. Here we formalize the distillation process for a general protocol in a purely quantum-mechanical framework and demonstrate that it can be viewed as creating an “effective” quantum channel between the legitimate users Alice and Bob. The process of secret key generation can then be viewed as entanglement distribution using this channel, which enables application of entanglement-based security proofs to essentially any prepare and measure protocol. To ensure secrecy of the key, Alice and Bob must be able to estimate the channel noise from errors in the key, and we further show how symmetries of the distillation process simplify this task. Applying this method, we prove the security of several key distribution protocols based on equiangular spherical codes.

We present a universal description of quantum entanglement using group theory and noncommutative characteristic functions. It leads to reformulations of the separability problem, which allows us to generalize the latter, thus connecting the theory of entanglement and harmonic analysis. As an example, we translate and analyze the positivity of partial transpose criterion and a simple criterion for pure states into the group-theoretical language. We also show that when applied to finite groups, our formalism embeds the separability problem in a given dimension into a higher dimensional but highly symmetric one. Finally, our formalism reveals a connection between the very existence of entanglement and group noncommutativity.

Optimal control of a driven single dissipative qubit is formulated as an inverse problem. We show that direct inversion is possible which allows an analytic construction of optimal control fields. Exact inversion is shown to be possible for dissipative qubits which can be described by a Lindblad equation. It is shown that optimal solutions are not unique. For a qubit with weak coupling to phonons we choose, among the set of exact solutions for the dissipationless qubit, one which minimizes the dissipative contribution in the kinetic equations. Examples are given for state trapping and $Z$-gate operation. Using analytic expressions for optimal control fields, favorable domains for dynamic stabilization in the Bloch sphere are identified. In the case of approximate inversion, the identified approximate solution may be used as a starting point for further optimization following standard methods.

We show how to efficiently simulate a quantum many-body system with tree structure when its entanglement (Schmidt number) is small for any bipartite split along an edge of the tree. As an application, we show that any one-way quantum computation on a tree graph can be efficiently simulated with a classical computer.

As shown by Pauli, [Z. Phys. 36, 336 (1926)], the electric dipole operator $\mathbf{r}$ can be replaced by the Runge-Lenz vector $\mathbf{A}$ when operating within the $n^2$ degenerate manifold of hydrogenic states of principal quantum number $n$. We seek to develop similar rules for higher multipole operators by expressing equivalent operators in terms only of the two vector constants of motion—the orbital angular momentum $\mathbf{L}$ and the Runge-Lenz vector $\mathbf{A}$—appropriate to the degenerate hydrogenic shell. Equivalence of two operators means here that they yield identical matrix elements within a subspace of Hilbert space that corresponds to fixed $n$. Such equivalent-operator techniques permit direct algebraic calculation of perturbations of Rydberg atoms by external fields and often exact analytical results for transition probabilities. Explicit expressions for equivalent quadrupole and octupole operators are derived, examples are provided, and general aspects of the problem are discussed.

We investigate the excitation of an atom or molecule via two-photon absorption induced by an ultrashort laser pulse in the perturbative intensity regime. We show that the probability for a resonant transition, i.e., with an intermediate state within the spectral width of the pulse, can be reduced to zero by pulses having a $\theta$-phase step at a certain detuning $\delta \omega$ from the central frequency. Our theoretical predictions are confirmed by numerical calculations for the $4s\to 4d$ transition in the potassium atom.

We present a theoretical model describing recently observed collective effects in large magneto-optically trapped atomic ensembles. Based on a kinetic description, we develop an efficient test particle method, which, in addition to the single-atom light pressure, accounts for other relevant effects such as laser attenuation and forces due to multiply scattered light with position-dependent absorption cross sections. Our calculations confirm the existence of a dynamical instability and provide deeper insights into the observed system dynamics.

We propose a toroidal trap designed for ultracold atoms. It relies on a combination of a magnetic trap for rf-dressed atoms, which creates a bubble-like trap, and a standing wave of light. This trap is well-suited for investigating questions of low dimensionality in a ring potential. We study the trap characteristics for a set of experimentally accessible parameters. A loading procedure from a conventional magnetic trap is also proposed. The flexible nature of this ring trap, including an adjustable radius and adjustable transverse oscillation frequencies, will allow the study of superfluidity in variable geometries and dimensionalities.

We study the dissociation of Feshbach molecules by a magnetic field sweep across a zero-energy resonance. In the limit of an instantaneous magnetic field change, the distribution of atomic kinetic energy can have a peak indicating dominance of the molecular closed-channel spin configuration over the entrance channel. The extent of this dominance influences physical properties such as stability with respect to collisions, and so the readily measurable presence or absence of the corresponding peak provides a practical method of classifying zero-energy resonances. Currently achievable ramp speeds, e.g., those demonstrated by Dürr et al. [Phys. Rev. A 70, 031601 (2005)], are fast enough to provide magnetic field changes that may be interpreted as instantaneous. We study the transition from sudden magnetic field changes to asymptotically wide, linear ramps. In the latter limit, the predicted form of the atomic kinetic energy distribution is independent of the specific implementation of the two-body physics, provided that the near-resonant scattering properties are properly accounted for.

In many experiments involving conversion of quantum-degenerate atomic gases into molecular dimers via a Feshbach resonance, an external magnetic field is linearly swept from above resonance to below resonance. In the adiabatic limit, the fraction of atoms converted into molecules is independent of the functional form of the sweep and is predicted to be 100%. However, for nonadiabatic sweeps through resonance, Landau-Zener theory predicts that a linear sweep will result in a negligible production of molecules. Here we employ a genetic algorithm to determine the functional time dependence of the magnetic field that produces the maximum number of molecules for sweep times that are comparable to the period of resonant atom-molecule oscillations, $2\pi {\Omega }_{\mathit{\text{Rabi}}}^{-1}$. The optimal sweep through resonance indicates that more than 95% of the atoms can be converted into molecules for sweep times as short as $2\pi {\Omega }_{\mathit{\text{Rabi}}}^{-1}$ while the linear sweep results in a conversion of only a few percent. We also find that the qualitative form of the optimal sweep is independent of the strength of the two-body interactions between atoms and molecules and the width of the resonance.

We employ a generalized Dicke model to study theoretically the quantum criticality of an extended two-level atomic ensemble interacting with a single-mode quantized light field. Effective Hamiltonians are derived and diagonalized to investigate numerically their eigenfrequencies for different quantum phases in the system. Based on the analysis of the eigenfrequencies, an intriguing quantum-phase transition from a normal phase to a superradiant phase is revealed clearly, which is quite different from that observed with a standard Dicke model.

We show that nonclassical intensity correlations and quadrature entanglement can be generated by frequency doubling in a resonator with two output ports. We predict twin-beam intensity correlations $6\mathrm{dB}$ below the coherent-state limit, and that the product of the inference variances of the quadrature fluctuations gives an Einstein-Podolsky-Rosen correlation coefficient of $V_{\mathrm{EPR}}=0.64<1$, using the criterion of Reid [Phys. Rev. A 40, 913 (1989)]. Comparison with an entanglement source based on combining two frequency doublers with a beam splitter shows that the dual-ported resonator provides stronger entanglement at lower levels of individual beam squeezing. Calculations are performed using a self-consistent propagation method that does not invoke a mean-field approximation. Results are given for physically realistic parameters that account for the Gaussian shape of the intracavity beams, as well as intracavity losses.

With the creation of ultracold atoms and molecules, a new type of chemistry—“resonance” chemistry—emerges: chemical reactions can occur when the energy of colliding atoms and molecules matches a bound state of the combined molecule (Feshbach resonance). This chemistry is rather similar to reactions that take place in nuclei at low energies. In this paper we suggest some problems for future experimental and theoretical work related to the resonance chemistry of ultracold molecules. Molecular Bose-Einstein condensates are particularly interesting because in this system collisions and chemical reactions are extremely sensitive to weak fields; also, a preferred reaction channel may be enhanced due to a finite number of final states. The sensitivity to weak fields arises due to the high density of narrow compound resonances and the macroscopic number of molecules with kinetic energy $E=0$ (in the ground state of a mean-field potential). The high sensitivity to the magnetic field may be used to measure the distribution of energy intervals, widths, and magnetic moments of compound resonances and study the onset of quantum chaos. A difference in the production rate of right-handed and left-handed chiral molecules may be produced by external electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields and the finite width $\Gamma$ of the resonance (correlation $\Gamma \mathbf{E}\bullet \mathbf{B}$). The same effect may be produced by the parity-violating energy difference in chiral molecules.

We show how recent results of Lieb and Seiringer can be obtained from repeated use of the monotonicity of relative entropy under partial traces, and explain how to use their approach to obtain tighter bounds in many situations.

In the framework of quantum field theory we discuss the emergence of a phase locking among the electromagnetic modes and the matter components on an extended space-time region. We discuss the formation of extended domains exhibiting in their fundamental states nonvanishing order parameters, whose existence is not included in the Lagrangian. Our discussion is motivated by the interest in the study of the general problem of the stability of mesoscopic and macroscopic complex systems arising from fluctuating quantum components in connection with the problem of defect formation during the process of non-equilibrium symmetry breaking phase transitions characterized by an order parameter.

We derive an analytical formula for the optimal trade-off between the mean estimation fidelity and the mean fidelity of the qubit state after a partial measurement on $N$ identically prepared qubits. We also conjecture an analytical expression for the optimal fidelity trade-off in case of a partial measurement on $N$ identical copies of a $d$-level system.

A computational scheme has been implemented to evaluate magnetic hypershielding at the nuclei of a molecule in the presence of an external spatially uniform, time-independent magnetic field, accounting for cubic response contributions via Rayleigh-Schrödinger perturbation theory. Numerical estimates have been obtained for hydrogen and heavier atoms in ${\mathrm{H}}_2$, HF, ${\mathrm{H}}_2\mathrm{O}$, $\mathrm{N}{\mathrm{H}}_3$, and $\mathrm{C}{\mathrm{H}}_4$ molecules at the coupled Hartree-Fock level of accuracy within the conventional common-origin approach. Gaugeless basis sets of increasing size and flexibility have been employed in a numerical test, to estimate the degree of convergence of theoretical tensor components. A further test for convergence has been carried out by evaluating properties for two different coordinate systems.