Sabine Wölk

Factorization of numbers with truncated Gauss sums at rational arguments

Factorization of numbers with the help of Gauss sums relies on an intimate relationship between the maxima of these functions and the factors. Indeed, when we restrict ourselves to integer arguments of the Gauss sum we profit from a one-to-one relationship. As a result, the identification of factors by the maxima is unique. However, for non-integer arguments, such as rational numbers, this powerful instrument to find factors breaks down. We develop new strategies for factoring numbers using Gauss sums at rational arguments. This approach may find application in a recent suggestion to factor numbers using a light interferometer discussed in this issue (V. Tamma et al., J. Mod. Opt. in press).

State selective detection of hyperfine qubits

In order to faithfully detect the state of an individual two-state quantum system (qubit) realized using, for example, a trapped ion or atom, state selective scattering of resonance fluorescence is well established. The simplest way to read out this measurement and assign a state is the threshold method. The detection error can be decreased by using more advanced detection methods like the time-resolved method (Myerson et al 2008 Phys. Rev. Lett. 100 200502) or the π-pulse detection method (Hemmerling et al 2012 New. J. Phys. 14 023043). These methods were introduced to qubits with a single possible state change during the measurement process. However, there exist many qubits like the hyperfine qubit of where several state change are possible. To decrease the detection error for such qubits, we develop generalizations of the time-resolved method and the π-pulse detection method for such qubits. We show the advantages of these generalized detection methods in numerical simulations and experiments using the hyperfine qubit of . The generalized detection methods developed here can be implemented in an efficient way such that experimental real time state discrimination with improved fidelity is possible.

Versatile microwave-driven trapped ion spin system for quantum information processing

A quantum register with flexible coupling topology is used to efficiently implement a quantum Fourier transform. Using trapped atomic ions, we demonstrate a tailored and versatile effective spin system suitable for quantum simulations and universal quantum computation. By simply applying microwave pulses, selected spins can be decoupled from the remaining system and, thus, can serve as a quantum memory, while simultaneously, other coupled spins perform conditional quantum dynamics. Also, microwave pulses can change the sign of spin-spin couplings, as well as their effective strength, even during the course of a quantum algorithm. Taking advantage of the simultaneous long-range coupling between three spins, a coherent quantum Fourier transform—an essential building block for many quantum algorithms—is efficiently realized. This approach, which is based on microwave-driven trapped ions and is complementary to laser-based methods, opens a new route to overcoming technical and physical challenges in the quest for a quantum simulator and a quantum computer.

Maximal coherence and the resource theory of purity

The resource theory of quantum coherence studies the off-diagonal elements of a density matrix in a distinguished basis, whereas the resource theory of purity studies all deviations from the maximally mixed state. We establish a direct connection between the two resource theories, by identifying purity as the maximal coherence which is achievable by unitary operations. The states that saturate this maximum identify a universal family of maximally coherent mixed states. These states are optimal resources under maximally incoherent operations, and thus independent of the way coherence is quantified. For all distance-based coherence quantifiers the maximal coherence can be evaluated exactly, and is shown to coincide with the corresponding distance-based purity quantifier. We further show that purity bounds the maximal amount of entanglement and discord that can be generated by unitary operations, thus demonstrating that purity is the most elementary resource for quantum information processing.

Quantum dynamics of trapped ions in a dynamic field gradient using dressed states

Novel ion traps that provide either a static or a dynamic magnetic gradient field allow for the use of radio frequency (rf) radiation for coupling internal and motional states of ions, which is essential for conditional quantum logic. We show that the coupling mechanism in the presence of a dynamic gradient is the same, in a dressed state basis, as in the case of a static gradient. Then, it is shown how demanding experimental requirements arising when using a dynamic gradient could be overcome. Thus, using dressed states in a dynamic gradient field could decisively reduce experimental complexity on the route towards a scalable device for quantum information science based on rf-driven trapped ions.

Optimized parameter estimation in the presence of collective phase noise

We investigate phase and frequency estimation with different measurement strategies under the effect of collective phase noise. First, we consider the standard linear estimation scheme and present an experimentally realizable optimization of the initial probe states by collective rotations. We identify the optimal rotation angle for different measurement times. Second, we show that subshot noise sensitivity\char22{}up to the Heisenberg limit\char22{}can be reached in presence of collective phase noise by using differential interferometry, where one part of the system is used to monitor the noise. For this, not only Greenberger-Horne-Zeilinger states but also symmetric Dicke states are suitable. We investigate the optimal splitting for a general symmetric Dicke state at both inputs and discuss possible experimental realizations of differential interferometry.

Unified approach to entanglement criteria using the Cauchy-Schwarz and Hölder inequalities

We present unified approach to different recent entanglement criteria. Although they were developed in different ways, we show that they are all applications of a more general principle given by the Cauchy-Schwarz inequality. We explain this general principle and show how to derive with it not only already known but also new entanglement criteria. We systematically investigate its potential and limits to detect bipartite and multipartite entanglement.

Radio frequency sideband cooling and sympathetic cooling of trapped ions in a static magnetic field gradient

We report a detailed investigation on near-ground state cooling of one and two trapped atomic ions. We introduce a simple sideband cooling method for confined atoms and ions, using RF radiation applied to bare ionic states in a static magnetic field gradient, and demonstrate its application to ions confined at secular trap frequencies, kHz. For a single Yb ion, the sideband cooling cycle reduces the average phonon number, from the Doppler limit to 0.30(12). This is in agreement with the theoretically estimated lowest achievable phonon number in this experiment. We extend this method of RF sideband cooling to a system of two Yb ions, resulting in a phonon number of 1.1(7) in the centre-of-mass mode. Furthermore, we demonstrate the first realization of sympathetic RF sideband cooling of an ion crystal consisting of two individually addressable identical isotopes of the same species.

Coherent Quantum Fourier Transform Using 3-Qubit Conditional Gates and Ultrasensitive Magnetometry with RF-Driven Trapped Ions

Using long-range magnetic gradient induced coupling between three effective spins, a coherent QFT is efficiently realized with trapped Yb⁺ ions. With a single Yb⁺ ion, RF magnetic fields are measured close to the quantum limit.

Generalized effective operator formalism for decaying systems

Systems of neutral kaons can be used to observe entanglement and the violation of Bell inequalities. The decay of these particles poses some problems, however, and recently an effective formalism for treating such systems has been derived. We generalize this formalism and make it applicable to other quantum systems that can be made to behave in a similar manner. As examples, we discuss two possible implementations of the generalized formalism using trapped ions such as