Three-dimensional (3D) crystals offer a route to scaling up trapped-ion systems for quantum sensing and quantum simulation applications; however, engineering coherent spin-motion couplings and effective spin-spin interactions in large crystals poses technical challenges associated with decoherence and prolonged timescales to generate appreciable entanglement. Here, we explore the possibility of speeding up these interactions in 3D crystals via parametric amplification. For this purpose, we derive a general Hamiltonian for the parametric amplification of spin-motion coupling that is broadly applicable to normal modes with motion transverse to or the spatial extent of the crystal. Unlike in lower-dimensional crystals, we find that the ability to faithfully (uniformly) amplify the spin-spin interactions in 3D crystals depends on the physical implementation of the spin-motion coupling. We consider the light-shift gate, and the so-called phase-insensitive and phase-sensitive Mølmer-Sørensen (MS) gates, and we find that only the phase-sensitive MS gate can be faithfully amplified in general 3D crystals. We discuss a situation where nonuniform amplification can be advantageous. We also reconsider the effect of counter-rotating terms on parametric amplification and find that they are not as detrimental as previous studies suggest.
Three-dimensional (3D) crystals offer a route to scaling up trapped-ion systems for quantum sensing and quantum simulation applications; however, engineering coherent spin-motion couplings and effective spin-spin interactions in large crystals poses technical challenges associated with decoherence and prolonged timescales to generate appreciable entanglement. Here, we explore the possibility of speeding up these interactions in 3D crystals via parametric amplification. For this purpose, we derive a general Hamiltonian for the parametric amplification of spin-motion coupling that is broadly applicable to normal modes with motion transverse to or the spatial extent of the crystal. Unlike in lower-dimensional crystals, we find that the ability to faithfully (uniformly) amplify the spin-spin interactions in 3D crystals depends on the physical implementation of the spin-motion coupling. We consider the light-shift gate, and the so-called phase-insensitive and phase-sensitive Mølmer-Sørensen (MS) gates, and we find that only the phase-sensitive MS gate can be faithfully amplified in general 3D crystals. We discuss a situation where nonuniform amplification can be advantageous. We also reconsider the effect of counter-rotating terms on parametric amplification and find that they are not as detrimental as previous studies suggest.
In trapped-ion quantum information processing, interactions between spins (qubits) are mediated by collective modes of motion of an ion crystal. While there are many different experimental strategies to design such interactions, they all face both technical and fundamental limitations to the achievable coherent interaction strength. In general, obtaining strong interactions and fast gates is an ongoing challenge. Here, we extend previous work [W. Ge, B. C. Sawyer, J. W. Britton, K. Jacobs, J. J. Bollinger, and M. Foss-Feig, Phys. Rev. Lett. 122, 030501 (2019)] and present a general strategy for enhancing the interaction strengths in trapped-ion systems via parametric amplification of the ions' motion. Specifically, we propose a stroboscopic protocol using alternating applications of parametric amplification and spin-motion coupling. In comparison with the previous work, we show that the current protocol can lead to larger enhancements in the coherent interaction that increase exponentially with the gate time.
Trapped atomic ion qubits or effective spins are a powerful quantum platform for quantum computation and simulation, featuring densely connected and efficiently programmable interactions between the spins.While native interactions between trapped-ion spins are typically pairwise, many quantum algorithms and quantum spin models naturally feature couplings between triplets, quartets, or higher orders of spins.Here, we formulate and analyze a mechanism that extends the standard Mølmer-Sørensen pairwise entangling gate and generates a controllable and programmable coupling between N spins of trapped ions.We show that spin-dependent optical parametric drives applied at twice the motional frequency generate a coordinate transformation of the collective ion motion in phase space, rendering displacement forces that are nonlinear in the spin operators.We formulate a simple framework that enables a systematic and faithful construction of high-order spin Hamiltonians and gates, including the effect of multiple modes of motion, and characterize the performance of such operations under realistic conditions.
Improving coherence is a fundamental challenge in quantum simulation and sensing experiments with trapped ions. Here we discuss, experimentally demonstrate, and estimate the potential impacts of two different protocols that enhance, through motional parametric excitation, the coherent spin-motion coupling of ions obtained with a spin-dependent force. The experiments are performed on two-dimensional crystal arrays of approximately 100 9Be+ ions confined in a Penning trap. By modulating the trapping potential at close to twice the center-of-mass mode frequency, we squeeze the motional mode and enhance the spin-motion coupling while maintaining spin coherence. With a stroboscopic protocol, we measure $5.4\ifmmode\pm\else\textpm\fi{}0.9\phantom{\rule{0.28em}{0ex}}\mathrm{dB}$ of motional squeezing below the ground-state motion, from which theory predicts a 10-dB enhancement in the sensitivity for measuring small displacements using a recently demonstrated protocol [K. A. Gilmore et al., Science 373, 673 (2021)]. With a continuous squeezing protocol, we measure and accurately calibrate the parametric coupling strength. Theory suggests this protocol can be used to improve quantum spin squeezing, limited in our system by off-resonant light scatter. We illustrate numerically the trade-offs between strong parametric amplification and motional dephasing in the form of center-of-mass frequency fluctuations for improving quantum spin squeezing in our setup.
Abstract Trapped-ion quantum simulators, in analog and digital modes, are considered a primary candidate to achieve quantum advantage in quantum simulation and quantum computation. The underlying controlled ion–laser interactions induce all-to-all two-spin interactions via the collective modes of motion through Cirac–Zoller or Mølmer–Sørensen schemes, leading to effective two-spin Hamiltonians, as well as two-qubit entangling gates. In this work, the Mølmer–Sørensen scheme is extended to induce three-spin interactions via tailored first- and second-order spin–motion couplings. The scheme enables engineering single-, two-, and three-spin interactions, and can be tuned via an enhanced protocol to simulate purely three-spin dynamics. Analytical results for the effective evolution are presented, along with detailed numerical simulations of the full dynamics to support the accuracy and feasibility of the proposed scheme for near-term applications. With a focus on quantum simulation, the advantage of a direct analog implementation of three-spin dynamics is demonstrated via the example of matter-gauge interactions in the U(1) lattice gauge theory within the quantum link model. The mapping of degrees of freedom and strategies for scaling the three-spin scheme to larger systems, are detailed, along with a discussion of the expected outcome of the simulation of the quantum link model given realistic fidelities in the upcoming experiments. The applications of the three-spin scheme go beyond the lattice gauge theory example studied here and include studies of static and dynamical phase diagrams of strongly interacting condensed-matter systems modeled by two- and three-spin Hamiltonians.
Trapped atomic ion qubits or effective spins are a powerful quantum platform for quantum computation and simulation, featuring densely connected and efficiently programmable interactions between the spins. While native interactions between trapped ion spins are typically pairwise, many quantum algorithms and quantum spin models naturally feature couplings between triplets, quartets or higher orders of spins. Here we formulate and analyze a mechanism that extends the standard Mølmer-Sørensen pairwise entangling gate and generates a controllable and programmable coupling between N spins of trapped ions. We show that spin-dependent optical forces applied at twice the motional frequency generate a coordinate-transformation of the collective ion motion in phase-space, rendering displacement forces that are nonlinear in the spin operators. We formulate a simple framework that enables a systematic and faithful construction of high-order spin Hamiltonians and gates, including the effect of multiple modes of motion, and characterize the performance of such operations under realistic conditions.
The construction of entangling gates with individual addressing capability represents a crucial approach for implementing quantum computation in trapped-ion crystals. Conventional entangling gate schemes typically rely on laser beam wave vectors to couple the ions' spin and motional degrees of freedom. Here, we experimentally demonstrate an alternative method that employs a polarization gradient field generated by a tightly focused laser beam-an approach theoretically proposed as the Magnus effect for quantum logic gate design [Mazzanti et al., Phys. Rev. Res. 5, 033036 (2023)PPRHAI2643-156410.1103/PhysRevResearch.5.033036]. Using this technique, we perform Raman operations on hyperfine qubits encoded in ^{171}Yb^{+} ions, generating spin-dependent forces along axial motional modes in a linear trap. By utilizing an acousto-optic deflector to create arbitrary spot pairs for individual ion addressing in two-ion (four-ion) chains, we achieve Mølmer-Sørensen gates with Bell-state fidelities exceeding 98.7(1)% [and 97.2(4)%]. Further improvements in numerical aperture (NA) and laser power could reduce gate durations while enhancing fidelity by orders of magnitude. This method is compatible with-and can significantly simplify-optical tweezer gate proposals, where motional mode engineering enables scalable trapped-ion quantum computation. The technique can be readily extended to two-dimensional ion crystals, representing a key advancement toward large-scale trapped-ion quantum processors.