Encoding a qubit in the motion of a trapped ion using superpositions of displaced squeezed states

Abstract

Trapped-ions form a promising platform to realize a future large scale quantum computing device. Qubits are typically stored in internal electronic states, which are coupled using their joint motion in the trap potential. In this thesis this control paradigm is reversed. The harmonic motion of a trapped calcium ion forms the main subject of studies, which is controlled via the internal electronic states. A number of new techniques are introduced and examined, primarily based on the implementation of modular variable measurements. These are realized combining an internal state dependent optical dipole force with readout of the internal states. Modular measurements are used to investigate large "Schrödinger cat'' states of the ion's motion, to violate Leggett-Garg tests of macroscopic realism, and finally to realize a logical qubit encoded in an error-correcting code based on the trapped-ion oscillator. The latter offers an alternative to the standard qubit based quantum information processing approach, which when embedded in systems of coupled oscillators could lead to a large-scale quantum computer. Measurements of a particle's modular position and momentum have been the focus of various discussions of foundational quantum mechanics. Such modular measurements of the trapped-ion's motion are studied in depth in this thesis, in particular their ability to commute, which forms a key element for the latter work on error-correcting codes. Here we make use of the ability to investigate sequences of measurements on a single harmonic oscillator, and study correlations between their results, as well as quantum measurement disturbances between the measurements. In order to achieve the major results of the thesis, it was necessary to characterize and control multiple wave packets in phase space. On the characterization side, the need to cope with states with high energy occupations led to the development of multiple new methods for quantum state tomography, including the use of a squeezed eigenstate basis, and the direct extraction of the characteristic function of the oscillator using state-dependent forces. These were used to analyze some of the largest oscillator "Schrödinger cat'' states which have been produced to date. The main result of this thesis is encoding and full control of a logical qubit in the motional oscillator space using a code proposed 18 years ago by Gottesman, Kitaev and Preskill. Logical code states are realized and manipulated using sequences of up to five modular measurements applied to an ion initially prepared in a squeezed motional state. Such sequences realize superpositions of multiple squeezed wave packets, which form the code words. The usage of the oscillator enables to encode and in principle correct a logical qubit within a single trapped ion, which when compared to typical qubit-array based approaches simplifies control and hardware. While the discussion above focuses on the new physics in this thesis, in addition the work required technical upgrades to the system, improving control of both qubit and oscillator. These form important components which have impact on all experiments in our setup, beyond the bounds of the current thesis.