# Publications by Andrew Steane

## Probing qubit memory errors at the part-per-million level

Physical Review Letters American Physical Society 123 (2019) 110503

MA Sepiol, AC Hughes, JE Tarlton, DP Nadlinger, TG Ballance, CJ Ballance, TP Harty, AM Steane, D Lucas, JF Goodwin

<p>Robust qubit memory is essential for quantum computing, both for near-term devices operating without error correction, and for the long-term goal of a fault-tolerant processor. We directly measure the memory error ε<sub>m</sub> for a <sup>43</sup>Ca<sup>+</sup> trapped-ion qubit in the small-error regime and find ε<sub>m</sub>&lt;10<sup>−4</sup> for storage times <em>t</em> ≲ 50  ms. This exceeds gate or measurement times by three orders of magnitude. Using randomized benchmarking, at <em>t</em> = 1  ms we measure ε<sub>m</sub>=1.2(7)×10<sup>−6</sup>, around ten times smaller than that extrapolated from the T<sup>∗</sup><sub>2</sub> time, and limited by instability of the atomic clock reference used to benchmark the qubit.</p>

## Fast quantum logic gates with trapped-ion qubits

Nature Nature Publishing Group 555 (2018) 75-78

VM Schäfer, C Ballance, K Thirumalai, LJ Stephenson, TG Ballance, AM Steane, D Lucas

Quantum bits (qubits) based on individual trapped atomic ions are a promising technology for building a quantum computer. The elementary operations necessary to do so have been achieved with the required precision for some error-correction schemes. However, the essential two-qubit logic gate that is used to generate quantum entanglement has hitherto always been performed in an adiabatic regime (in which the gate is slow compared with the characteristic motional frequencies of the ions in the trap), resulting in logic speeds of the order of 10 kilohertz. There have been numerous proposals of methods for performing gates faster than this natural 'speed limit' of the trap. Here we implement one such method, which uses amplitude-shaped laser pulses to drive the motion of the ions along trajectories designed so that the gate operation is insensitive to the optical phase of the pulses. This enables fast (megahertz-rate) quantum logic that is robust to fluctuations in the optical phase, which would otherwise be an important source of experimental error. We demonstrate entanglement generation for gate times as short as 480 nanoseconds-less than a single oscillation period of an ion in the trap and eight orders of magnitude shorter than the memory coherence time measured in similar calcium-43 hyperfine qubits. The power of the method is most evident at intermediate timescales, at which it yields a gate error more than ten times lower than can be attained using conventional techniques; for example, we achieve a 1.6-microsecond-duration gate with a fidelity of 99.8 per cent. Faster and higher-fidelity gates are possible at the cost of greater laser intensity. The method requires only a single amplitude-shaped pulse and one pair of beams derived from a continuous-wave laser. It offers the prospect of combining the unrivalled coherence properties, operation fidelities and optical connectivity of trapped-ion qubits with the submicrosecond logic speeds that are usually associated with solid-state devices.

## The effect of atomic response time in the theory of Doppler cooling of trapped ions

Journal of Modern Optics Taylor and Francis 65 (2018) 577-584

H Janacek, AM Steane, D Lucas, DN Stacey

<p>We describe a simple approach to the problem of incorporating the response time of an atom or ion being Doppler-cooled into the theory of the cooling process. The system being cooled does not in general respond instantly to the changing laser frequencies it experiences in its rest frame, and this ‘dynamic effect’ can affect significantly the temperatures attainable. It is particularly important for trapped ions when there is a slow decay out of the cooling cycle requiring the use of a repumping beam. We treat the cases of trapped ions with two and three internal states, then apply the theory to <sup>40</sup>Ca<sup>+</sup>. For this ion experimental data exist showing the ion to be cold under conditions for which heating is predicted if the dynamic effect is neglected. The present theory accounts for the observed behaviour.</p>

## Science and humanity: A humane philosophy of science and religion

, 2018

A Steane

© Andrew Steane 2018. All rights reserved. This volume offers an in-depth presentation of the structure of science and the nature of the physical world, with a view to showing how it complements and does not replace other types of human activity, such as the arts and humanities, spirituality and religion. The aim is to better inform scientists, science educators, and the general public. Many think that science can and does establish that the natural world is a vast machine, and this is the whole truth of ourselves and our environment. This is wrong. In fact, scientific models employ a rich network of interconnecting concepts, and the overall picture suggests the full validity of further forms of truth-seeking and truth-speaking, such as art, jurisprudence, and the like. In fundamental physics, the equations that describe physical behaviour interact in a subtle symbiotic way with symmetry principles which describe overarching guidelines. The relationship between physics and biology is similar, and so is the relationship between biology and the humanities. Darwinian evolution is an exploratory mechanism which allows richer patterns and truths to come to be expressed; it does not negate or replace those truths. The area of values, of what can or should command our allegiance, requires a different kind of response, a response that is not completely captured by logical argument, but which is central to human life. Religion, when it is understood correctly and done well, is the engagement with the idea that we have a meaningful role to play, and much to learn.

## High-fidelity elementary qubit operations with trapped ions

Optics InfoBase Conference Papers Part F73-QIM 2017 (2017)

DM Lucas, TP Harty, CJ Ballance, DPL Aude Craik, MA Sepiol, VM Schäfer, K Thirumalai, JE Tarlton, L Stephenson, J Wolf, JF Goodwin, A Hughes, C Loschnauer, TG Ballance, AM Steane

© OSA 2017. I will describe recent experimental work at Oxford on implementing elementary one- and two-qubit operations with the high fidelities (99.9% or more) required for the implementation of quantum error correction, using trapped-ion “atomic clock“ qubits.

## High-fidelity spatial and polarization addressing of Ca-43 qubits using near-field microwave control

Physical Review A American Physical Society 95 (2017) 022337-

DPL Aude Craik, NM Linke, MA Sepiol, TP Harty, JF Goodwin, CJ Ballance, DN Stacey, AM Steane, DM Lucas, DTC Allcock

Individual addressing of qubits is essential for scalable quantum computation. Spatial addressing allows unlimited numbers of qubits to share the same frequency, whilst enabling arbitrary parallel operations. We demonstrate addressing of long-lived $^{43}\text{Ca}^+$ "atomic clock" qubits held in separate zones ($960\mu$m apart) of a microfabricated surface trap with integrated microwave electrodes. Such zones could form part of a "quantum CCD" architecture for a large-scale quantum information processor. By coherently cancelling the microwave field in one zone we measure a ratio of Rabi frequencies between addressed and non-addressed qubits of up to 1400, from which we calculate a spin-flip probability on the qubit transition of the non-addressed ion of $1.3\times 10^{-6}$. Off-resonant excitation then becomes the dominant error process, at around $5 \times 10^{-3}$. It can be prevented either by working at higher magnetic field, or by polarization control of the microwave field. We implement polarization control with error $2 \times 10^{-5}$, which would suffice to suppress off-resonant excitation to the $\sim 10^{-9}$ level if combined with spatial addressing. Such polarization control could also enable fast microwave operations.

## Detecting continuous spontaneous localization with charged bodies in a Paul trap

Physical Review A American Physical Society 95 (2017) 032112

Y Li, AM Steane, D Bedingham, GAD Briggs

Continuous spontaneous localisation (CSL) is a model that captures the effects of a class of extensions to quantum theory which are expected to result from quantum gravity, and is such that wavefunction collapse is a physical process. The rate of such a process could be very much lower than the upper bounds set by searches to date, and yet still modify greatly the interpretation of quantum mechanics and solve the quantum measurement problem. Consequently experiments are sought to explore this. We describe an experiment that has the potential to extend sensitivity to CSL by many orders of magnitude. The method is to detect heating of the motion of charged macroscopic objects confined in a Paul trap. We discuss the detection and the chief noise sources. We find that CSL with standard parameters could be observed using a vibration-isolated ion trap of size 1\,{\rm cm} at ultra-low pressure, with optical interferometric detection.

## Matter-wave coherence limit owing to cosmic gravitational wave background

PHYSICS LETTERS A 381 (2017) 3905-3908

AM Steane

## Dark-resonance Doppler cooling and high fluorescence in trapped Ca-43 ions at intermediate magnetic field

NEW JOURNAL OF PHYSICS 18 (2016) ARTN 023043

DTC Allcock, TP Harty, MA Sepiol, HA Janacek, CJ Ballance, AM Steane, DM Lucas, DN Stacey

## On determining absolute entropy without quantum theory or the third law of thermodynamics

New Journal of Physics IOP Publishing 18 (2016) 043022-043022

A Steane

We employ classical thermodynamics to gain information about absolute entropy, without recourse to statistical methods, quantum mechanics or the third law of thermodynamics. The Gibbs-Duhem equation yields various simple methods to determine the absolute entropy of a fluid. We also study the entropy of an ideal gas and the ionization of a plasma in thermal equilibrium. A single measurement of the degree of ionization can be used to determine an unknown constant in the entropy equation, and thus determine the absolute entropy of a gas. It follows from all these examples that the value of entropy at absolute zero temperature does not need to be assigned by postulate, but can be deduced empirically.

## Thermodynamics A Complete Undergraduate Course

Oxford University Press, 2016

AM Steane

This book aims to convey the style and power of thermodynamic reasoning, along with applications such as Joule-Kelvin expansion, the gas turbine, magnetic cooling, solids at high pressure, chemical equilibrium, radiative heat exchange and ...

## Reduced-order Abraham-Lorentz-Dirac equation and the consistency of classical electromagnetism

AMERICAN JOURNAL OF PHYSICS 83 (2015) 256-262

AM Steane

## Tracking the radiation reaction energy when charged bodies accelerate

AMERICAN JOURNAL OF PHYSICS 83 (2015) 703-710

AM Steane

## Self-force of a rigid ideal fluid, and a charged sphere in hyperbolic motion

PHYSICAL REVIEW D 91 (2015) ARTN 065008

AM Steane

## Hybrid quantum logic and a test of Bell's inequality using two different atomic isotopes.

Nature 528 (2015) 384-386

CJ Ballance, VM Schäfer, JP Home, DJ Szwer, SC Webster, DTC Allcock, NM Linke, TP Harty, DPL Aude Craik, DN Stacey, AM Steane, DM Lucas

Entanglement is one of the most fundamental properties of quantum mechanics, and is the key resource for quantum information processing (QIP). Bipartite entangled states of identical particles have been generated and studied in several experiments, and post-selected or heralded entangled states involving pairs of photons, single photons and single atoms, or different nuclei in the solid state, have also been produced. Here we use a deterministic quantum logic gate to generate a 'hybrid' entangled state of two trapped-ion qubits held in different isotopes of calcium, perform full tomography of the state produced, and make a test of Bell's inequality with non-identical atoms. We use a laser-driven two-qubit gate, whose mechanism is insensitive to the qubits' energy splittings, to produce a maximally entangled state of one (40)Ca(+) qubit and one (43)Ca(+) qubit, held 3.5 micrometres apart in the same ion trap, with 99.8 ± 0.6 per cent fidelity. We test the CHSH (Clauser-Horne-Shimony-Holt) version of Bell's inequality for this novel entangled state and find that it is violated by 15 standard deviations; in this test, we close the detection loophole but not the locality loophole. Mixed-species quantum logic is a powerful technique for the construction of a quantum computer based on trapped ions, as it allows protection of memory qubits while other qubits undergo logic operations or are used as photonic interfaces to other processing units. The entangling gate mechanism used here can also be applied to qubits stored in different atomic elements; this would allow both memory and logic gate errors caused by photon scattering to be reduced below the levels required for fault-tolerant quantum error correction, which is an essential prerequisite for general-purpose quantum computing.

## Nonexistence of the self-accelerating dipole and related questions

PHYSICAL REVIEW D 89 (2014) ARTN 125006

AM Steane

## Pulsed force sequences for fast phase-insensitive quantum gates in trapped ions

NEW JOURNAL OF PHYSICS 16 (2014) ARTN 053049

AM Steane, G Imreh, JP Home, D Leibfried

## The fields and self-force of a constantly accelerating spherical shell

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470 (2014)

AM Steane

We present a partial differential equation describing the electromagnetic potentials around a charge distribution undergoing rigid motion at constant proper acceleration, and obtain a set of solutions to this equation. These solutions are used to find the self-force exactly in a chosen case. The electromagnetic self-force for a spherical shell of charge of proper radius R undergoing rigid motion at constant proper acceleration a0 is, to high-order approximation, (2e2a0/R) ∑∞n=0 (a0R)2n((2n - 1)(2n + 1)2(2n + 3))-1, and this is conjectured to be exact. © 2013 The Author(s) Published by the Royal Society. All rights reserved.

## Faithful to Science The Role of Science in Religion

Oxford University Press, USA, 2014

A Steane

What I want to do in this chapter is show why the above statements, though partly understandable, are largely mistaken, and I will comment on how the combination of science and faith, in the case of one trying to follow Jesus of Nazareth,&nbsp;...

## Microwave control electrodes for scalable, parallel, single-qubit operations in a surface-electrode ion trap

APPLIED PHYSICS B-LASERS AND OPTICS 114 (2014) 3-10

DPLA Craik, NM Linke, TP Harty, CJ Ballance, DM Lucas, AM Steane, DTC Allcock