A strong nonlinear coupling between harmonic oscillators is highly desirable for quantum information processing and quantum simulation, but is difficult to achieve in many physical systems. Here, we exploit the Coulomb interaction between two trapped ions to achieve strong nonlinear coupling between normal modes of motion at the single-phonon level. We experimentally demonstrate phonon up- and down-conversion and apply this coupling to directly measure the parity and Wigner functions of the ions' motional states. Our results represent the fully quantum operation of a degenerate parametric oscillator and hold promise for quantum computation schemes that involve continuous variables.
A strong nonlinear coupling between harmonic oscillators is highly desirable for quantum information processing and quantum simulation, but is difficult to achieve in many physical systems. Here, we exploit the Coulomb interaction between two trapped ions to achieve strong nonlinear coupling between normal modes of motion at the single-phonon level. We experimentally demonstrate phonon up- and down-conversion and apply this coupling to directly measure the parity and Wigner functions of the ions' motional states. Our results represent the fully quantum operation of a degenerate parametric oscillator and hold promise for quantum computation schemes that involve continuous variables.
State measurement of a quantum harmonic oscillator is essential in quantum optics and quantum information processing. In a system of trapped ions, we experimentally demonstrate the projective measurement of the state of the ions' motional mode via an effective cross-Kerr coupling to another motional mode. This coupling is induced by the intrinsic nonlinearity of the Coulomb interaction between the ions. We spectroscopically resolve the frequency shift of the motional sideband of the first mode due to the presence of single phonons in the second mode and use it to reconstruct the phonon number distribution of the second mode.
Abstract Coupled harmonic oscillators are ubiquitous in physics and play a prominent role in quantum science. They are a cornerstone of quantum mechanics and quantum field theory, where second quantization relies on harmonic oscillator operators to create and annihilate particles. Descriptions of quantum tunneling, beamsplitters, coupled potential wells, ``hopping terms'', decoherence, and many other phenomena rely on coupled harmonic oscillators. However, the ability to couple separate quantum harmonic oscillators directly, with coupling rates that substantially exceed decoherence rates, has remained elusive. Here, we realize high-fidelity coherent exchange of single motional quanta between harmonic oscillators -- in this case, spectrally separated harmonic modes of motion of a trapped ion crystal -- where the timing, strength, and phase of the coupling are controlled through an oscillating electric potential with suitable spatial variation. The coupling rate can be made much larger than the decoherence rates, enabling demonstrations of high-fidelity quantum state transfer, entanglement of motional modes, and Hong-Ou-Mandel-type interference. We also project a harmonic oscillator into its ground state by measurement and preserve that state during repetitions of the projective measurement, an important prerequisite for non-destructive syndrome measurement in continuous-variable quantum error correction. Controllable coupling between harmonic oscillators has potential applications in quantum information processing with continuous variables, quantum simulation, and precision measurements. It can also enable cooling and quantum logic spectroscopy involving motional modes of trapped ions that are not directly accessible.
Interaction among harmonic oscillators described by a trilinear Hamiltonian $\ensuremath{\hbar}\ensuremath{\xi}({a}^{\ifmmode\dagger\else\textdagger\fi{}}bc+a{b}^{\ifmmode\dagger\else\textdagger\fi{}}{c}^{\ifmmode\dagger\else\textdagger\fi{}})$ is one of the most fundamental models in quantum optics. By employing the anharmonicity of the Coulomb potential in a linear trapped three-ion crystal, we experimentally implement it among three normal modes of motion in the strong-coupling regime, where the coupling strength is much larger than the decoherence rate of the ion motion. We use it to simulate the interaction of an atom and light as described by the Tavis-Cummings model and the process of nondegenerate parametric down-conversion in the regime of a depleted pump.
The full characterization of a continuous-variable quantum system is a challenging problem. For the trapped-ion system, a number of methods of measuring the quantum states have been developed, including the measurement of the Q quasiprobability function and the density-matrix elements in the Fock basis, but these approaches are often slow and difficult to scale to multimode states. Here, we demonstrate a novel and powerful scheme for measuring a continuous-variable quantum state that uses the direct single-shot measurement of the joint parity of the phonon states of a trapped ion. We drive a spin-dependent bichromatic beam-splitter interaction that coherently exchanges phonons between different harmonic oscillator modes of the ion. This interaction encodes the joint-parity information into the relative phase between the two spin states, enabling measurement of the combined phonon-number parity across multiple modes in a single shot. Leveraging this capability, we directly measure multimode Wigner quasiprobability distributions to perform quantum state tomography of an entangled coherent state, and calculate various quantum informational quantities with a model-based estimation of the density matrix. We further show that the single-shot joint-parity measurement can be used to detect parity-flip errors in real time. By postselecting the parity-measurement outcomes, we experimentally demonstrate the partial recovery of coherence, effectively implementing an error-mitigation technique. Lastly, we identify the various sources of error affecting the fidelity of the spin-dependent beam-splitter operation and study the feasibility of high-fidelity operations. The interaction studied in this work can be extended to more than two modes, and is highly relevant to continuous-variable quantum computing and quantum metrology.
Trapped ions are a powerful and flexible platform used for scientific applications ranging from studies of fundamental physics and precision spectroscopy to quantum simulation and quantum information processing. These applications are enabled by the excellent control that can be exerted over the long-lived atomic states of trapped ions using electromagnetic fields. In crystals containing multiple ions, the shared motion is used as a mediator to couple the atomic states of distant ions. This thesis focuses on improving the control over ion motion, and in particular for crystals that contain multiple ion species. The first set of experiments discussed in this thesis is the application of a ground-state cooling method known as electromagnetically-induced-transparency (EIT) cooling to 25Mg+. This ion has a complex hyperfine structure which enables the implementation of a long-lived qubit. By demonstrating EIT cooling on 25Mg+, it is shown that such qubit ions are also compatible with this resource-efficient sub-Doppler cooling method. The normal modes of trapped-ion motion are well-described by quantum harmonic oscillators, and the second set of experiments described in this thesis covers the direct, rapid, and parametric coupling between two harmonic oscillators in the motion modes of multi-ion crystals. This coupling can be used to indirectly cool ion modes that are otherwise difficult or inefficient to cool. It can also be used to perform mid-circuit non-destructive readout of trapped-ion motion, which is useful for the study of states encoded in the ion motion, such as those used for bosonic error correction.
Atomic ions, confined in radio-frequency Paul ion traps, are a promising candidate to host a future quantum information processor. In this article, we demonstrate a method to couple two motional modes of a single trapped ion, where the coupling mechanism is based on applying electric fields rather than coupling the ion's motion to a light field. This reduces the design constraints on the experimental apparatus considerably. As an application of this mechanism, we cool a motional mode close to its ground state without accessing it optically. As a next step, we apply this technique to measure the mode's heating rate, a crucial parameter determining the trap quality.