The stable operation of quantum computers will rely on error-correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator [1,2]. A powerful encoding is formed from a periodically spaced superposition of position eigenstates [3][4][5]. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach [6][7][8][9][10]. Here, we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped 40 Ca + ion, controlling and measuring the oscillator through coupling to an ancilliary internal-state qubit [11]. We prepare and reconstruct logical states with an average square fidelity of 87.3 ± 0.7%, and demonstrate a universal logical single qubit gate set which we analyze using process tomography. For Pauli gates we reach process fidelities of ≈ 97%, while for continuous rotations we use gate teleportation achieving fidelities of ≈ 89%. The control demonstrated opens a route for exploring continuous variable error-correction as well as hybrid quantum information schemes using both discrete and continuous variables [12]. The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum [13,14].

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We directly observe the quantum interference between two well-separated trapped-ion mechanical oscillator wave packets. The superposed state is created from a spin-motion entangled state using a heralded measurement. Wave packet interference is observed through the energy eigenstate populations. We reconstruct the Wigner function of these states by introducing probe Hamiltonians which measure Fock state populations in displaced and squeezed bases. Squeezed-basis measurements with 8 dB squeezing allow the measurement of interference for Δα=15.6, corresponding to a distance of 240 nm between the two superposed wave packets.

The noncommutativity of position and momentum observables is a hallmark feature of quantum physics. However, this incompatibility does not extend to observables that are periodic in these base variables. Such modular-variable observables have been suggested as tools for fault-tolerant quantum computing and enhanced quantum sensing. Here, we implement sequential measurements of modular variables in the oscillatory motion of a single trapped ion, using state-dependent displacements and a heralded nondestructive readout. We investigate the commutative nature of modular variable observables by demonstrating no-signaling in time between successive measurements, using a variety of input states. Employing a different periodicity, we observe signaling in time. This also requires wave-packet overlap, resulting in quantum interference that we enhance using squeezed input states. The sequential measurements allow us to extract two-time correlators for modular variables, which we use to violate a Leggett-Garg inequality. Signaling in time and Leggett-Garg inequalities serve as efficient quantum witnesses, which we probe here with a mechanical oscillator, a system that has a natural crossover from the quantum to the classical regime.

We demonstrate control of a trapped-ion quantum harmonic oscillator in a squeezed Fock state basis, using engineered Hamiltonians analogous to the Jaynes-Cummings and anti-Jaynes-Cummings forms. We demonstrate that for squeezed Fock states with low n the engineered Hamiltonians reproduce the √ n scaling of the matrix elements which is typical of Jaynes-Cummings physics, and also examine deviations due to the finite wavelength of our control fields. Starting from a squeezed vacuum state, we apply sequences of alternating transfer pulses which allow us to climb the squeezed Fock state ladder, creating states up to excitations of n = 6 with up to 8.7 dB of squeezing, as well as demonstrating superpositions of these states. These techniques offer access to new sets of states of the harmonic oscillator which may be applicable for precision metrology or quantum information science.The control of quantum harmonic oscillators has played a prominent role in the development of quantum state control [1,2]. A single quantum oscillator provides access to a Hilbert space with a dimension which increases rapidly as the oscillator energy increases. It is also an example of a system which has a natural transition from the quantum to the classical regimes. One of the primary methods for performing control and measurement of quantum harmonic oscillator states is by coupling the oscillator to a single spin using a JaynesCummings Hamiltonianwhere Ω and φ are real constants,â † andâ are the creation and annihiliation operators of energy quanta for the oscillator, andσ − = |↓ ↑| withσ + =σ † − . |↑ , |↓ are energy eigenstates of the spin. The Jaynes-Cummings Hamiltonian arises naturally for Cavity-QED systems [3] and can be implemented straightforwardly with trapped ions by using a laser to resonantly drive a motional sideband of an internal state transition [2,4,5]. For an ion starting in one of the basis elements |↓ |n where |n are the energy eigenstates of the oscillator, evolution as a function of the duration t of the Hamiltonian H JC results in Rabi oscillations between the two states |↓ |n ↔ |↑ |n − 1 which can be viewed as a rotation R(θ, φ) = cos(θ/2)Î + i sin(θ/2)(cos(φ)ŝ x − sin(φ)ŝ y ) where θ = Ω √ nt andÎ,ŝ x ,ŝ y are Pauli operators which act in the basis {|↓ |n , |↑ |n − 1 }. These Rabi oscillations can be observed by making projective measurements on the spin states of the ion as a function of the duration of the applied Hamiltonian. An important feature is that the Rabi frequency of the oscillations scales with the matrix element n − 1|â |n = √ n. This has played an important role in the diagnosis of energy distributions of various well-known oscillator states which are not energy eigenstates [1]. The use of coupling to a two-state system isolates pairs of states of the oscillator, which simplifies the dynamical evolution and allows simple prescriptions for creating arbitrary superpositions of states [6][7][8].Although the Jaynes-Cummings Hamiltonian arises naturally in the light-matter interaction, similar physics c...