49 research outputs found

### Quantum State Tomography of a Single Qubit: Comparison of Methods

The tomographic reconstruction of the state of a quantum-mechanical system is
an essential component in the development of quantum technologies. We present
an overview of different tomographic methods for determining the
quantum-mechanical density matrix of a single qubit: (scaled) direct inversion,
maximum likelihood estimation (MLE), minimum Fisher information distance, and
Bayesian mean estimation (BME). We discuss the different prior densities in the
space of density matrices, on which both MLE and BME depend, as well as ways of
including experimental errors and of estimating tomography errors. As a measure
of the accuracy of these methods we average the trace distance between a given
density matrix and the tomographic density matrices it can give rise to through
experimental measurements. We find that the BME provides the most accurate
estimate of the density matrix, and suggest using either the pure-state prior,
if the system is known to be in a rather pure state, or the Bures prior if any
state is possible. The MLE is found to be slightly less accurate. We comment on
the extrapolation of these results to larger systems.Comment: 15 pages, 4 figures, 2 tables; replaced previous figure 5 by new
table I. in Journal of Modern Optics, 201

### Tighter quantum uncertainty relations follow from a general probabilistic bound

Uncertainty relations (URs) like the Heisenberg-Robertson or the time-energy
UR are often considered to be hallmarks of quantum theory. Here, a simple
derivation of these URs is presented based on a single classical inequality
from estimation theory, a Cram\'er-Rao-like bound. The Heisenberg-Robertson UR
is then obtained by using the Born rule and the Schr\"odinger equation. This
allows a clear separtion of the probabilistic nature of quantum mechanics from
the Hilbert space structure and the dynamical law. It also simplifies the
interpretation of the bound. In addition, the Heisenberg-Robertson UR is
tightened for mixed states by replacing one variance by the so-called quantum
Fisher information. Thermal states of Hamiltonians with evenly-gapped energy
levels are shown to saturate the tighter bound for natural choices of the
operators. This example is further extended to Gaussian states of a harmonic
oscillator. For many-qubit systems, we illustrate the interplay between
entanglement and the structure of the operators that saturate the UR with
spin-squeezed states and Dicke states.Comment: 8 pages, 1 figure. v2: improved presentation, references added,
results on the connection between saturated inequality and entanglement
structure for multi-qubit states adde

### Optimised surface-electrode ion-trap junctions for experiments with cold molecular ions

We discuss the design and optimisation of two types of junctions between
surface-electrode radiofrequency ion-trap arrays that enable the integration of
experiments with sympathetically cooled molecular ions on a monolithic chip
device. A detailed description of a multi-objective optimisation procedure
applicable to an arbitrary planar junction is presented, and the results for a
cross junction between four quadrupoles as well as a quadrupole-to-octupole
junction are discussed. Based on these optimised functional elements, we
propose a multi-functional ion-trap chip for experiments with translationally
cold molecular ions at temperatures in the millikelvin range. This study opens
the door to extending complex chip-based trapping techniques to
Coulomb-crystallised molecular ions with potential applications in mass
spectrometry, spectroscopy, controlled chemistry and quantum technology.Comment: 19 pages, 10 figure

### Using Mathematica for Quantum Mechanics: A Student's Manual

This book is an attempt to help students transform all of the concepts of
quantum mechanics into concrete computer representations, which can be
constructed, evaluated, analyzed, and hopefully understood at a deeper level
than what is possible with more abstract representations. It was written for a
Master's and PhD lecture given yearly at the University of Basel, Switzerland.
The goal is to give a language to the student in which to speak about quantum
physics in more detail, and to start the student on a path of fluency in this
language. On our journey we approach questions such as: -- You already know how
to calculate the energy eigenstates of a single particle in a simple
one-dimensional potential. How can such calculations be generalized to
non-trivial potentials, higher dimensions, and interacting particles? -- You
have heard that quantum mechanics describes our everyday world just as well as
classical mechanics does, but have you ever seen an example where such behavior
is calculated in detail and where the transition from classical to quantum
physics is evident? -- How can we describe the internal spin structure of
particles? How does this internal structure couple to the particles' motion? --
What are qubits and quantum circuits, and how can they be assembled to simulate
a future quantum computer?Comment: 164 pages, 5 chapters; third edition with major additions: included
executable Mathematica notebooks; included solutions to exercises; included
Quantum Circuits, Rashba model, Jaynes-Cummings mode

### Conformal carpet and grating cloaks

We introduce a class of conformal versions of the previously introduced
quasi-conformal carpet cloak, and show how to construct such conformal cloaks
for different cloak shapes. Our method provides exact refractive-index profiles
in closed mathematical form for the usual carpet cloak as well as for other
shapes. By analyzing their asymptotic behavior, we find that the performance of
finite-size cloaks becomes much better for metal shapes with zero average
value, e.g., for gratings.Comment: added Ref. 12; added 2 figures; reformatte

### Quantum metrology with nonclassical states of atomic ensembles

Quantum technologies exploit entanglement to revolutionize computing,
measurements, and communications. This has stimulated the research in different
areas of physics to engineer and manipulate fragile many-particle entangled
states. Progress has been particularly rapid for atoms. Thanks to the large and
tunable nonlinearities and the well developed techniques for trapping,
controlling and counting, many groundbreaking experiments have demonstrated the
generation of entangled states of trapped ions, cold and ultracold gases of
neutral atoms. Moreover, atoms can couple strongly to external forces and light
fields, which makes them ideal for ultra-precise sensing and time keeping. All
these factors call for generating non-classical atomic states designed for
phase estimation in atomic clocks and atom interferometers, exploiting
many-body entanglement to increase the sensitivity of precision measurements.
The goal of this article is to review and illustrate the theory and the
experiments with atomic ensembles that have demonstrated many-particle
entanglement and quantum-enhanced metrology.Comment: 76 pages, 40 figures, 1 table, 603 references. Some figures bitmapped
at 300 dpi to reduce file siz

### Bell correlations in a many-body system with finite statistics

A recent experiment reported the first violation of a Bell correlation
witness in a many-body system [Science 352, 441 (2016)]. Following discussions
in this paper, we address here the question of the statistics required to
witness Bell correlated states, i.e. states violating a Bell inequality, in
such experiments. We start by deriving multipartite Bell inequalities involving
an arbitrary number of measurement settings, two outcomes per party and one-
and two-body correlators only. Based on these inequalities, we then build up
improved witnesses able to detect Bell-correlated states in many-body systems
using two collective measurements only. These witnesses can potentially detect
Bell correlations in states with an arbitrarily low amount of spin squeezing.
We then establish an upper bound on the statistics needed to convincingly
conclude that a measured state is Bell-correlated.Comment: 5+12 pages, 3+4 figure

### Quantum simulation of the hexagonal Kitaev model with trapped ions

We present a detailed study of quantum simulations of coupled spin systems in
surface-electrode ion-trap arrays, and illustrate our findings with a proposed
implementation of the hexagonal Kitaev model [A. Kitaev, Annals of Physics
321,2 (2006)]. The effective (pseudo)spin interactions making up such quantum
simulators are found to be proportional to the dipole-dipole interaction
between the trapped ions, and are mediated by motion which can be driven by
state-dependent forces. The precise forms of the trapping potentials and the
interactions are derived in the presence of a surface electrode and a cover
electrode. These results are the starting point to derive an optimized
surface-electrode geometry for trapping ions in the desired honeycomb lattice
of Kitaev's model, where we design the dipole-dipole interactions in a way that
allows for coupling all three bond types of the model simultaneously, without
the need for time discretization. Finally we propose a simple wire structure
that can be incorporated in a microfabricated chip to generate localized
state-dependent forces which drive the couplings prescribed by this particular
model; such a wire structure should be adaptable to many other situations.Comment: 24 pages, 7 figures. v2: simplified the derivation of (28) without
changing conclusions; minor edits. v3: minor edit