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].

The robust generation of quantum states in the presence of decoherence is a primary challenge for explorations of quantum mechanics at larger scales. Using the mechanical motion of a single trapped ion, we utilize reservoir engineering to generate squeezed, coherent, and displaced-squeezed states as steady states in the presence of noise. We verify the created state by generating two-state correlated spin-motion Rabi oscillations, resulting in high-contrast measurements. For both cooling and measurement, we use spin-oscillator couplings that provide transitions between oscillator states in an engineered Fock state basis. Our approach should facilitate studies of entanglement, quantum computation, and open-system quantum simulations in a wide range of physical systems.

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Mesoscopic superpositions of distinguishable coherent states provide an analog to the Schrödinger's cat thought experiment [1,2]. For mechanical oscillators these have primarily been realised using coherent wavepackets, for which the distinguishability arises due to the spatial separation of the superposed states [3][4][5]. Here, we demonstrate superpositions composed of squeezed wavepackets, which we generate by applying an internal-state dependent force to a single trapped ion initialized in a squeezed vacuum state with 9 dB reduction in the quadrature variance. This allows us to characterise the initial squeezed wavepacket by monitoring the onset of spin-motion entanglement, and to verify the evolution of the number states of the oscillator as a function of the duration of the force. In both cases, we observe clear differences between displacements aligned with the squeezed and anti-squeezed axes. We observe coherent revivals when inverting the state-dependent force after separating the wavepackets by more than 19 times the ground state root-mean-square extent, which corresponds to 56 times the r.m.s. extent of the squeezed wavepacket along the displacement direction. Aside from their fundamental nature, these states may be useful for quantum metrology [6] or quantum information processing with continuous variables [7][8][9].The creation and study of nonclassical states of spin systems coupled to a harmonic oscillator has provided fundamental insights into the nature of decoherence and the quantum-classical transition. These states and their control form the basis of experimental developments in quantum information processing and quantum metrology [1,2,10]. Two of the most commonly considered states of the oscillator are squeezed states and superpositions of coherent states of opposite phase, which are commonly referred to as "Schrödinger's cat" (SC) states. Squeezed states involve reduction of the fluctuations in one quadrature of the oscillator below the ground state uncertainty, which has been used to increase sensitivity in interferometers [11,12]. SC states provide a complementary sensitivity to environmental influences by separating the two parts of the state by a large distance in phase space. These states have been created in microwave and optical * Electronic address:hylo@phys.ethz.ch; Electronic address: jhome@phys.ethz.ch cavities [2,13], where they are typically not entangled with another system, and also with trapped ions [1,[3][4][5], where all experiments performed have involved entanglement between the oscillator state and the internal electronic states of the ion. SC states have recently been used as sensitive detectors for photon scattering recoil events at the single photon level [14].In this Letter, we use State-Dependent Forces (SDFs) to create superpositions of distinct squeezed oscillator wavepackets which are entangled with a pseudo-spin encoded in the electronic states of a single trapped ion. We will refer to these states as Squeezed Wavepacket Entangled States (SWES) in the rest of ...

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.