Encoding a qubit in a trapped-ion mechanical oscillator.

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 oscillator1-3. In such a system, a powerful encoding has been devised based on periodically spaced superpositions of position eigenstates4-6. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach7-11. Here we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped 40Ca+ ion, controlling and measuring the mechanical oscillator through coupling to an ancillary internal-state qubit12. We prepare and reconstruct logical states with an average squared fidelity of 87.3 ± 0.7 per cent. Also, we demonstrate a universal logical single-qubit gate set, which we analyse using process tomography. For Pauli gates we reach process fidelities of about 97 per cent, whereas for continuous rotations we use gate teleportation and achieve fidelities of approximately 89 per cent. This control method opens a route for exploring continuous variable error correction as well as hybrid quantum information schemes using both discrete and continuous variables13. The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum14,15.

Repeated multi-qubit readout and feedback with a mixed-species trapped-ion register.

Quantum error correction is essential for realizing the full potential of large-scale quantum information processing devices1,2. Fundamental to its experimental realization is the repetitive detection of errors via projective measurements of quantum correlations among qubits, as well as corrections using conditional feedback3. Repetitive application of such tasks requires that they neither induce unwanted crosstalk nor impede further control operations, which is challenging owing to the need to dissipatively couple qubits to the classical world for detection and reinitialization. For trapped ions, state readout involves scattering large numbers of resonant photons, which increases the probability of stray light causing errors on nearby qubits and leads to undesirable recoil heating of the ion motion. Here we demonstrate up to 50 sequential measurements of correlations between two beryllium ion microwave qubits using an ancillary optical qubit in a calcium ion, and implement feedback that allows us to stabilize two-qubit subspaces as well as Bell states, a class of maximally entangled states. Multi-qubit mixed-species gates are used to transfer information within the register from the qubit to the ancilla, enabling readout with negligible crosstalk to the data qubits. Heating of the ion motion during detection is mitigated by recooling all three ions using light that interacts with only the calcium ion, known as sympathetic cooling. A key element of our experimental setup is a powerful classical control system that features flexible in-sequence processing for feedback control. The methods employed here provide essential tools for scaling trapped-ion quantum computing, quantum-state control and entanglement-enhanced quantum metrology4.

Trapping and Ground-State Cooling of a Single H_{2}

We demonstrate co-trapping and sideband cooling of a H_{2}^{+}-^{9}Be^{+} ion pair in a cryogenic Paul trap. We study the chemical lifetime of H_{2}^{+} and its dependence on the apparatus temperature, achieving lifetimes of up to 11_{-3}^{+6} h at 10 K. We demonstrate cooling of two of the modes of translational motion to an average phonon number of 0.07(1) and 0.05(1), corresponding to a temperature of 22(1) and 55(3) µK, respectively. Our results provide a basis for quantum logic spectroscopy experiments of H_{2}^{+}, as well as other light ions such as HD^{+}, H_{3}^{+}, and He^{+}.

Phonon Laser in the Quantum Regime.

We demonstrate a trapped-ion system with two competing dissipation channels, implemented independently on two ion species cotrapped in a Paul trap. By controlling coherent spin-oscillator couplings and optical pumping rates we explore the phase diagram of this system, which exhibits a regime analogous to that of a (phonon) laser but operates close to the quantum ground state with an average phonon number of n[over ¯]<10. We demonstrate phase locking of the oscillator to an additional resonant drive, and also observe the phase diffusion of the resulting state under dissipation by reconstructing the quantum state from a measurement of the characteristic function.

Generation of a Maximally Entangled State Using Collective Optical Pumping.

We propose and implement a novel scheme for dissipatively pumping two qubits into a singlet Bell state. The method relies on a process of collective optical pumping to an excited level, to which all states apart from the singlet are coupled. We apply the method to deterministically entangle two trapped ^{40}Ca^{+} ions. Within 16 pumping cycles, an initially separable state is transformed into one with 83(1)% singlet fidelity, and states with initial fidelity of ⪆70% converge onto a fidelity of 93(1)%. We theoretically analyze the performance and error susceptibility of the scheme and find it to be insensitive to a large class of experimentally relevant noise sources.

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

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.

Direct Characteristic-Function Tomography of Quantum States of the Trapped-Ion Motional Oscillator.

We implement direct readout of the symmetric characteristic function of quantum states of the motional oscillation of a trapped calcium ion. Suitably chosen internal state rotations combined with internal state-dependent displacements, based on bichromatic laser fields, map the expectation value of the real or imaginary part of the displacement operator to the internal states, which are subsequently read out. Combining these results provides full information about the symmetric characteristic function. We characterize the technique by applying it to a range of archetypal quantum oscillator states, including displaced and squeezed Gaussian states as well as two and three component superpositions of displaced squeezed states. For each, we discuss relevant features of the characteristic function and Wigner phase-space quasiprobability distribution. The direct reconstruction of these highly nonclassical oscillator states using a reduced number of measurements is an essential tool for understanding and optimizing the control of oscillator systems for quantum sensing and quantum information applications.

Sustained State-Independent Quantum Contextual Correlations from a Single Ion.

We use a single trapped-ion qutrit to demonstrate the quantum-state-independent violation of noncontextuality inequalities using a sequence of randomly chosen quantum nondemolition projective measurements. We concatenate 53×10^{6} sequential measurements of 13 observables, and unambiguously violate an optimal noncontextual bound. We use the same data set to characterize imperfections including signaling and repeatability of the measurements. The experimental sequence was generated in real time with a quantum random number generator integrated into our control system to select the subsequent observable with a latency below 50 µs, which can be used to constrain contextual hidden-variable models that might describe our results. The state-recycling experimental procedure is resilient to noise and independent of the qutrit state, substantiating the fact that the contextual nature of quantum physics is connected to measurements and not necessarily to designated states. The use of extended sequences of quantum nondemolition measurements finds applications in the fields of sensing and quantum information.

Quantum Harmonic Oscillator State Control in a Squeezed Fock Basis.

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 sqrt[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.

Observation of Quantum Interference between Separated Mechanical Oscillator Wave Packets.

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.

Entangled mechanical oscillators.

Hallmarks of quantum mechanics include superposition and entanglement. In the context of large complex systems, these features should lead to situations as envisaged in the 'Schrödinger's cat' thought experiment (where the cat exists in a superposition of alive and dead states entangled with a radioactive nucleus). Such situations are not observed in nature. This may be simply due to our inability to sufficiently isolate the system of interest from the surrounding environment-a technical limitation. Another possibility is some as-yet-undiscovered mechanism that prevents the formation of macroscopic entangled states. Such a limitation might depend on the number of elementary constituents in the system or on the types of degrees of freedom that are entangled. Tests of the latter possibility have been made with photons, atoms and condensed matter devices. One system ubiquitous to nature where entanglement has not been previously demonstrated consists of distinct mechanical oscillators. Here we demonstrate deterministic entanglement of separated mechanical oscillators, consisting of the vibrational states of two pairs of atomic ions held in different locations. We also demonstrate entanglement of the internal states of an atomic ion with a distant mechanical oscillator. These results show quantum entanglement in a degree of freedom that pervades the classical world. Such experiments may lead to the generation of entangled states of larger-scale mechanical oscillators, and offer possibilities for testing non-locality with mesoscopic systems. In addition, the control developed here is an important ingredient for scaling-up quantum information processing with trapped atomic ions.

Coherent diabatic ion transport and separation in a multizone trap array.

We investigate the dynamics of single and multiple ions during transport between and separation into spatially distinct locations in a multizone linear Paul trap. A single 9Be+ ion in a ~2 MHz harmonic well was transported 370 µm in 8 µs, corresponding to 16 periods of oscillation, with a gain of 0.1 motional quanta. Similar results were achieved for the transport of two ions. We also separated chains of up to 9 ions from one potential well to two distinct potential wells. With two ions this was accomplished in 55 µs, with excitations of approximately two quanta for each ion. Fast transport and separation can significantly reduce the time overhead in certain architectures for scalable quantum information processing with trapped ions.

Randomized benchmarking of multiqubit gates.

We describe an extension of single-qubit gate randomized benchmarking that measures the error of multiqubit gates in a quantum information processor. This platform-independent protocol evaluates the performance of Clifford unitaries, which form a basis of fault-tolerant quantum computing. We implemented the benchmarking protocol with trapped ions and found an error per random two-qubit Clifford unitary of 0.162±0.008, thus setting the first benchmark for such unitaries. By implementing a second set of sequences with an extra two-qubit phase gate inserted after each step, we extracted an error per phase gate of 0.069±0.017. We conducted these experiments with transported, sympathetically cooled ions in a multizone Paul trap-a system that can in principle be scaled to larger numbers of ions.

Estimation of a general time-dependent Hamiltonian for a single qubit.

The Hamiltonian of a closed quantum system governs its complete time evolution. While Hamiltonians with time-variation in a single basis can be recovered using a variety of methods, for more general Hamiltonians the presence of non-commuting terms complicates the reconstruction. Here using a single trapped ion, we propose and experimentally demonstrate a method for estimating a time-dependent Hamiltonian of a single qubit. We measure the time evolution of the qubit in a fixed basis as a function of a time-independent offset term added to the Hamiltonian. The initially unknown Hamiltonian arises from transporting an ion through a static laser beam. Hamiltonian estimation allows us to estimate the spatial beam intensity profile and the ion velocity as a function of time. The estimation technique is general enough that it can be applied to other quantum systems, aiding the pursuit of high-operational fidelities in quantum control.

Generation of large coherent states by bang-bang control of a trapped-ion oscillator.

Fast control of quantum systems is essential to make use of quantum properties before they degrade by decoherence. This is important for quantum-enhanced information processing, as well as for pushing quantum systems towards the boundary between quantum and classical physics. 'Bang-bang' control attains the ultimate speed limit by making large changes to control fields much faster than the system can respond, but is often challenging to implement experimentally. Here we demonstrate bang-bang control of a trapped-ion oscillator using nanosecond switching of the trapping potentials. We perform controlled displacements with which we realize coherent states with up to 10,000 quanta of energy. We use these displaced states to verify the form of the ion-light interaction at high excitations far outside the usual regime of operation. These methods provide new possibilities for quantum-state manipulation and generation, alongside the potential for a significant increase in operational clock speed for trapped-ion quantum information processing.

An ion trap built with photonic crystal fibre technology.

We demonstrate a surface-electrode ion trap fabricated using techniques transferred from the manufacture of photonic-crystal fibres. This provides a relatively straightforward route for realizing traps with an electrode structure on the 100 micron scale with high optical access. We demonstrate the basic functionality of the trap by cooling a single ion to the quantum ground state, allowing us to measure a heating rate from the ground state of 787 ± 24 quanta/s. Variation of the fabrication procedure used here may provide access to traps in this geometry with trap scales between 100 µm and 10 µm.

Quantum optics. Quantum harmonic oscillator state synthesis by reservoir engineering.

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.

Long-lived mesoscopic entanglement outside the Lamb-Dicke regime.

We create entangled states of the spin and motion of a single 40Ca+ ion in a linear ion trap. We theoretically study and experimentally observe the behavior outside the Lamb-Dicke regime, where the trajectory in phase space is modified and the motional coherent states become squeezed. We directly observe the modification of the return time of the trajectory, and infer the squeezing. The mesoscopic entanglement is observed up to Deltaalpha=5.1 with coherence time 170 micros and mean phonon excitation n = 16.

High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping.

We propose and demonstrate experimentally the discrimination between two spin states of an atom purely on the basis of their angular momentum. The discrimination relies on angular momentum selection rules and does not require magnetic effects such as a magnetic dipole moment of the atom or an applied magnetic field. The central ingredient is to prevent by coherent population trapping an optical pumping process which would otherwise relax the spin state before a detectable signal could be obtained. We detected the presence or absence of a single quantum (h) of angular momentum in a trapped calcium ion in a single observation with success probability 0.86. As a practical technique, the method can be applied to read out some types of quantum computer.

Oxford ion-trap quantum computing project.

We describe recent progress in the development of an ion-trap quantum information processor. We discuss the choice of ion species and describe recent experiments on read-out for a ground-state qubit and photoionization trap loading.