Quantum simulation with Rydberg ions in a Penning trap

Abstract

Quantum simulation of interacting many-body spin systems is routinely performed with cold trapped ions, and systems with hundreds of spins have been studied in one and two dimensions. In the most common realizations of these platforms, spin degrees of freedom are encoded in low-lying electronic levels, and interactions among the spins are mediated through crystal vibrations. Here we propose a new approach which enables the quantum simulation of two-dimensional spin systems with interaction strengths that are increased by orders of magnitude. This, together with the unprecedented longevity of trapped ions, opens an avenue for the exploration of phenomena that take place on long timescales, e.g., slow and collective relaxation in frustrated and kinetically constrained systems. Our platform makes use of the strong dipolar interactions among electronic Rydberg states and planar confinement provided by a Penning trap. We investigate how the strong electric and magnetic fields that form this trap affect the properties of the Rydberg states and show that spin-spin interaction strengths on the order of MHz are achievable under experimentally realistic conditions. As a brief illustration of the capabilities of this quantum simulator, we study the entanglement in a frustrated spin system realized by three ions.

Figures (7)

• Figure 1: Rydberg ions in a Penning trap. Length scales and coordinate system. To describe a single ion, we use $\mathbf{R}$ and $\mathbf{r}$, the center-of-mass and the relative coordinate, respectively. The distance between the ion core and the Rydberg electron scales with the principal quantum number as $\sim n^{2}$, reaching typical values $\langle r \rangle\approx100\, \text{nm}$. For typical trap frequency magnitudes, the harmonic confinement yields an oscillator length of approximately $\ell \approx 10\, \text{nm}$. The equilibrium interparticle spacing, determined by the balance between Coulomb repulsion and harmonic confinement, is on the order of $R_0 \approx 10\,\mu\text{m}$.
• Figure 2: Energy spectrum of Rydberg states and the Paschen-Back regime: (a) Spectrum of Rydberg states for $^{40}\text{Ca}^{+}$ ions as a function of the magnetic field $B$. The spectrum exhibits Zeeman splitting, quadratic energy shifts arising from the diamagnetic coupling. (b, c) Magnified view of the Paschen–Back regime for $S$ and $P$ states, where we highlight the dominance of the quantum numbers $m_{s}$ and $m_{l}$. These states are used to construct MW-dressed Rydberg states that generate non-vanishing dipole moments. The colors are used to highlight the dominant $l$-character of each state, where states with $l>2$ are altogether represented by gray lines. The technical approach to obtain the Rydberg energy spectrum here and in the remainder of the manuscript is described in the App. \ref{['app:numerics']}.
• Figure 3: Microwave dressing scheme and relevant dipole matrix element: (a) Spectrum of Rydberg states for $^{40}\text{Ca}^{+}$ ions as a function of the magnetic field strength $B$ for principal quantum number $n = 45$. Here, we highlight the states $S$ and $P$, which are combined to form the MW-dressed Rydberg states. The purple lines indicate the coupling between selected states, where each of them corresponds to a spin magnetic quantum number $m_{s} = \pm 1/2$. (b) Two-level system consisting of the ground state $\ket{\downarrow}$ and the dressed Rydberg state $\ket{\uparrow}$. (c) Dipole matrix element of Rydberg states as a function of the magnetic field strength $B$, for $m_{s} = -1/2$. Here, we highlight states $S$ and $P$, represented in blue and red, respectively.
• Figure 4: Two-body interaction coordinates for linear and planar ion crystals. (a) Two interacting Rydberg ions in a Penning trap. The schematic depicts the normalized interparticle axis vector between ions $i$ and $j$, $\mathbf{n}_{ij}$, resulting in the angles $\theta_{ij}$ for the magnetic field $\mathbf{B}$ along the $z$-direction, i.e., $\mathbf{B} = B \mathbf{e}_{z}$; the schematic also highlights that the $\pi$-polarized field along the $z$-axis results in $m_{l}$-conserving transitions. (b) Strong axial confinement. (c) Strong radial confinement. In the latter, we have defined the azimuth angle $\phi_{ij} = \arctan(\tfrac{Y_{i} - Y_{j}}{X_{i} - X_{j}})$.
• Figure 5: Dipole-dipole interaction strength in a planar three-ion crystal. The figure shows the dipole-dipole interaction strength versus magnetic field strength $B$, for different values of the principal quantum number $n$. The ratio between axial and radial frequencies is set $\omega_{z}/\omega_{\rho} = 2$ (orange) and $\omega_{z}/\omega_{\rho} = 4$ (blue), respectively. The inset shows axial (dashed lines) and radial (solid lines) frequencies as a function of the magnetic field strength $B$. The principal quantum numbers are $n = 30, 35, 40$ and $45$. Values here are obtained with the eigenfunctions for $^{40}\text{Ca}^{+}$ ions.