Phonon-mediated quantum spin simulator employing a planar ionic crystal in a Penning trap

We derive the normal modes for a rotating Coulomb ion crystal in a Penning trap, quantize the motional degrees of freedom, and illustrate how they can be driven by a spin-dependent optical dipole force to create a quantum spin simulator on a triangular lattice with hundreds of spins. The analysis for the axial modes (oscillations perpendicular to the two-dimensional crystal plane) follow a standard normal-mode analysis, while the remaining planar modes are more complicated to analyze because they have velocity-dependent forces in the rotating frame. After quantizing the normal modes into phonons, we illustrate some of the different spin-spin interactions that can be generated by entangling the motional degrees of freedom with the spin degrees of freedom via a spin-dependent optical dipole force. In addition to the well-known power-law dependence of the spin-spin interactions when driving the axial modes blue of the phonon band, we notice certain parameter regimes in which the level of frustration between the spins can be engineered by driving the axial or planar phonon modes at different frequencies. These systems may allow for the analog simulation of quantum spin glasses with large numbers of spins.

Figure 1

Electrostatic potentials provided by electrodes in a Penning trap have contributions from both the end-cap electrodes [with the trapping potential ${\varphi }_E\left(\mathbf{r}\right)=V_0{z}^2$ that pushes the atoms towards $z=0$] and the cylindrical electrodes [with the radial quadratic potential ${\varphi }_C\left(\mathbf{r}\right)=-\frac{V_0}{2}(x^2+y^2)$, which tends to push the ions out radially]. In addition, the repulsive Coulomb potential between the ions tends to destabilize the system in the trap. A static magnetic field $\mathbf{B}=B_z\widehat{z}$($B_z>0$) is thus applied along the axial direction to provide the radial confinement of the ions. To lock the rotational angular frequency of the ions at a specific rotational speed, a time-dependent clockwise quadrupole potential ${\varphi }_W\left(t\right)=V_W{\rho }^2\cos 2(\theta +\Omega t)$ ($\Omega >0$) is applied to the ions through ring electrodes so that the steady state of the ions (with a rigid-body rotational speed $\Omega$) can be phase locked. Note that the rotating quadrupole potential is well implemented by ring electrodes with just six sectors (as employed in the NIST Penning traps [8, 18, 19]).

Figure 2

One-to-two plane transition frequency for different numbers of ions $N$. The vertical dashed line shows the deconfinement frequency for $V_W=0$. For larger values of $V_W$, the vertical line moves to the right, reducing the frequency range for the rotating wall, where the single plane is stable, while the one-to-two plane transition line hardly changes.

Figure 3

Spatial configuration for a seed lattice with $N=217$ ions. The spatial dimension is scaled by the characteristic length in the axial direction $\widehat{z}$ as $l_0={\left(\frac{k_{e}e}{V_0}\right)}^{\frac{1}{3}}={\left(\frac{2k_e{e}^2}{m{\omega }_z^2}\right)}^{\frac{1}{3}}$. The red cross denotes the point where the external potential due to the electrodes vanishes. The minimal ion configuration often, but not always, has an ion located at this energy minimum.

Figure 4

Equilibrium structures for different rotating frequencies. (a) Low effective trapping frequency ${\omega }_l^W=0.06{\omega }_z$. (b) Medium effective trapping frequency ${\omega }_m=0.16{\omega }_z$. (c) High effective trapping frequency ${\omega }_h=0.21{\omega }_z$. The wall potential is ${\omega }_W=0.04{\omega }_z$ for all cases.

Figure 5

Dependence of the crystal distortion ratio (or aspect ratio) on the rotational frequency ${\omega }_{\mathrm{eff}}$ and the rotating wall potential ${\omega }_W$. The black solid curve represents a strong rotating wall with ${\omega }_W=0.07{\omega }_z$. The green dashed curve is a weak rotating wall potential with ${\omega }_W=0.04{\omega }_z$. The red dotted curve is a very weak rotating wall potential with ${\omega }_W=0.01{\omega }_z$. The colored dots are the principle cases we examine with more detailed calculations for much of the remaining numerical work.

Figure 6

Ion spacing dependence as a function of the distance $\rho$ from the origin. The black, green, and the red symbols denote, respectively, the data with a strong rotating wall potential ${\omega }_W=0.07{\omega }_z$, a weak rotating wall potential ${\omega }_W=0.04{\omega }_z$, and a very weak rotating wall potential ${\omega }_W=0.01{\omega }_z$. The values of the corresponding effective trapping frequencies ${\omega }_{\mathrm{eff}}$ are shown in the legend. The same parameter sets as in Fig. 5 are used.

Figure 7

Eigenfrequencies of the upper (cyclotronlike) branch of the planar modes. The same parameter sets and notation are used as in Fig. 5.

Figure 8

Eigenfrequencies of the axial modes. The same parameter sets and notation are used as in Fig. 5.

Figure 9

Eigenfrequencies of the lower (magnetronlike) branch of the planar modes. The same parameter sets and notation are used as in Fig. 5.

Figure 10

Comparison of the phonon bandwidths for the axial (red) and lower (magnetronlike) planar (blue) branches for a fixed weak rotating wall potential. As the effective trap frequency increases, the lowest frequency of the axial mode decreases as the largest (magnetronlike) lower-branch planar frequency increases until they overlap very near the one-to-two plane transition. The uppermost axial mode is fixed, while the lowest planar mode does not change significantly with ${\omega }_{\mathrm{eff}}$.

Figure 11

Four highest-frequency axial eigenvectors for three different effective trapping frequencies with a weak rotating wall potential ${\omega }_W=0.04{\omega }_z$. Subplots (a)–(d) present the case with a low effective trapping frequency ${\omega }_l^W=0.06{\omega }_z$, (e)–(h) present the case with a mean effective trapping frequency ${\omega }_m=0.16{\omega }_z$, and (i)–(l) represent the case with a high effective trapping frequency ${\omega }_h=0.21{\omega }_z$. The color in the color bar represents the scaled value of the normalized eigenvector $b_j^{\nu z}$ with respect to its maximal positive component for the mode $\nu$.

Figure 12

A snapshot of the spatial eigenvectors for the lowest two modes in each planar branch at the fixed rotating wall potential ${\omega }_W=0.04{\omega }_z$. Subplots (a) and (b), (e) and (f), and (i) and (j) show the lowest two planar modes in the upper branch with the effective trapping frequencies ${\omega }_l^W=0.05{\omega }_z$, ${\omega }_m=0.06{\omega }_z$, and ${\omega }_h=0.09{\omega }_z$, respectively. Subplots (c) and (d), (g) and (h), and (k) and (l) on the right panel are the corresponding data for the lower branch of the planar phonon modes.

Figure 13

Detuning above the axial modes (blue of the c.m. mode). As in Fig. 2b of Ref. 8, a detuning $\mu =\delta +{\omega }_z$ above ${\omega }_z$ yields a power-law-like decay of $J_{ij}\propto r_{ij}^{-\alpha }$ as a function of the ion distance, $r_{ij}$. The legend presents the values for the effective trapping frequencies in units of ${\omega }_z$. Here we plot the exponent of the fitted power law as a function of the detuning.

Figure 14

Detuning inside the axial mode branch with an effective trapping ${\omega }_m=0.16{\omega }_z$ and the weak rotating wall potential ${\omega }_W={\omega }_l^W=0.06{\omega }_z$ for $N=217$. (a) Detuning $\delta =0.00567{\omega }_z$ below ${\omega }_z$ exactly between the two highest modes with $99.93\%$ of the $J_{ij}$ in the plot. (b) Detuning $\delta =0.1174{\omega }_z$ below ${\omega }_z$ exactly between the $\lfloor 2N/3\rfloor$ and $\lfloor 2N/3\rfloor +1$ modes with $99.28\%$ of the $J_{ij}$ in the plot. (c) Detuning $\delta =0.1921{\omega }_z$ below ${\omega }_z$ exactly between the $\lfloor N/3\rfloor$ and $\lfloor N/3\rfloor +1$ modes with $59.73\%$ of the $J_{ij}$ in the plot. (d) Detuning $\delta =0.2926{\omega }_z$ below ${\omega }_z$ exactly between the two lowest modes with $63.65\%$ of the $J_{ij}$ in the plot. The inset green curves are generated from Fig. 8, with the arrow indicating where the detuning lies.

Figure 15

Dependence of the spin-spin interaction as a function of the distance $r$ between the ions for a detuning below the axial modes (red of the lowest axial mode). The effective trapping ${\omega }_m=0.16{\omega }_z$ and the weak rotating wall potential ${\omega }_W={\omega }_l^W=0.06{\omega }_z$ are used.

Figure 16

Dependence of planar spin-spin interaction as a function of the distance $r$ between ions for various detunings: (a) $\mu =1\times {10}^{-5}{\omega }_z$; (b) $\mu ={\omega }_z$; (c) $\mu =10{\omega }_z$. The darker colors illustrate positive $J_{ij}$ and the lighter colors illustrate negative $J_{ij}$. The effective trapping ${\omega }_m=0.16{\omega }_z$ and the weak rotating potential ${\omega }_W={\omega }_l^W=0.06{\omega }_z$ are used.

Figure 17

Angular correlation of $J_{ij}$ with $\mu =1\times {10}^{-5}{\omega }_z$. (a),(c) The subshell definition for the ion at the center and for an outer ion. (b),(d) The spin-spin interaction $J_{ij}$ of these subshells as a function of radius $r$. The effective trapping ${\omega }_m=0.16{\omega }_z$ and the weak rotating wall potential ${\omega }_W={\omega }_l^W=0.06{\omega }_z$ are used.