Effect of defects on phonons and the effective spin-spin interactions of an ultracold Penning-trap quantum simulator

We generalize the analysis of the normal modes for a rotating ionic Coulomb crystal in a Penning trap to allow for inhomogeneities in the system. Our formal developments are completely general, but we choose to examine a crystal of Be${}^{+}$ ions with BeH${}^{+}$ defects to compare with current experimental efforts. We examine the classical phonon modes (both transverse and planar) and we determine the effective spin-spin interactions when the system is driven by an axial spin-dependent optical dipole force. We examine situations with up to approximately 15% defects. We find that most properties are not strongly influenced by the defects, indicating that the presence of a small number of defects will not significantly affect experiments.

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 ${\varphi }_W\left(t\right)=V_W{r}^2\cos 2(\theta +\omega t)$ ($\omega >0$) 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 [7, 13, 14]).

Figure 2

Equilibrium structures for $N=217$ ions and a weak wall (${\omega }_W=0.04{\omega }_z$) with different rotating frequencies. The top panels are the faster rotating circular case and the bottom panels the slower rotating oval case. The panels show $N_d=0$, 12, 24, and 36. Black circles denote Be${}^{+}$ ions and red triangles denote BeH${}^{+}$ molecules. Note the difference in the spatial scales for the upper and lower panels.

Figure 3

Eigenfrequencies of the axial modes with defects for the fast and slow rotating cases. In general, there are about $N_d$ modes that are changed due to the defects, and these tend to be at the higher frequencies for the faster rotating case with a more circular profile and closer to the lower frequencies for the slower rotating oval case. The modes are plotted vs mode number, sorted according to increasing phonon frequencies.

Figure 4

False color image of the first four eigenvectors for the pure (top) and $N_d=36$ defect (bottom) case for the (a)–(h) fast rotating and (i)–(p) slow rotating cases. Note how the eigenvectors are quite similar for the cases without [upper, (a)–(d); (i)–(l)] and with [lower, (e)–(h); (m)–(p)] defects. Be${}^{+}$ ions are plotted with circles; BeH${}^{+}$ defects are plotted with triangles. A careful inspection shows that the center-of-mass mode (far left) is no longer uniform for the defect case. Note that the defects can also change the ordering of the eigenvectors when sorted according to their eigenvalues, as seen by comparing panels (j)–(l) with (n)–(p). The color scale for the false color image is on the right.

Figure 5

False color image showing how one of the defect modes is related to simple linear combinations of different pure modes with similar eigenvalues. The top panel is for the fast rotating circular case where a sum of the two pure case modes is essentially equal to the defect mode and the bottom panel is for the slower rotating oval case where the sum of the two modes is approximately equal to the defect mode.

Figure 6

False color image of the dot product of the pure-case eigenmodes with the corresponding defect-case eigenmodes ($\overline{b}$) with 36 defects (left) and norm of the pure-case eigenmode restricted to the defect sites (right). Two cases are shown: (i) fast rotation (circular) case on the top and (ii) slow rotation (oval) case on the bottom.

Figure 7

Axial center-of-mass frequency as a function of the number of defects for the fast (black curve) and slow (red curve) cases. One can use these curves along with the measurement of the center-of-mass frequency to determine the number of defects in the system at any given moment of time (assuming the defects rapidly move to the boundary due to centrifugal effects).

Figure 8

Visualization of some example cyclotron modes (a) and (b) and of the mode spectrum (c) for $N_d=36$ and the fast rotating case; arrows indicate the instantaneous motion of the respective ionic site. One can see that the cyclotron modes separate into motion (a) of the defects only or (b) of the Be${}^{+}$ only. The mode spectrum is shown in (c), where one can clearly see the separation of the motion according to the two different cyclotron frequencies for the defect case.

Figure 9

Spin-spin couplings vs distance for different detunings (different colored lines) and different number of defects [(a) 0, (b) 1, (c) 18, and (d) 36 defects] on a log-log plot. The straight-line fits show power-law behavior $J_{ij}\propto r_{ij}^{-\gamma }$. The detunings are $\mu ={\omega }_{\mathrm{cm}}+\delta {\omega }_z$, with $\delta ={10}^{-5}$, ${10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$, ${10}^{-1}$ and 1. This case is for the faster rotating circular crystal.

Figure 10

Spin-spin couplings vs distance for different detunings (different colored lines) and different number of defects [(a) 0, (b) 1, (c) 18, and (d) 36 defects] on a log-log plot. The straight-line fits show power-law behavior $J_{ij}\propto r_{ij}^{-\gamma }$. The detunings are $\mu ={\omega }_{\mathrm{cm}}+\delta {\omega }_z$, with $\delta ={10}^{-5}$, ${10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$, ${10}^{-1}$ and 1. This case is for the slower rotating oval-shaped crystal.