Modeling near ground-state cooling of two-dimensional ion crystals in a Penning trap using electromagnetically induced transparency

Penning traps, with their ability to control planar crystals of tens to hundreds of ions, are versatile quantum simulators. Thermal occupations of the motional drumhead modes, transverse to the plane of the ion crystal, degrade the quality of quantum simulations. Laser cooling using electromagnetically induced transparency (EIT cooling) is attractive as an efficient way to quickly initialize the drumhead modes to near ground-state occupations. We numerically investigate the efficiency of EIT cooling of planar ion crystals in a Penning trap, accounting for complications arising from the nature of the trap and from the simultaneous cooling of multiple ions. We show that, in spite of challenges, the large bandwidth of drumhead modes (hundreds of kilohertz) can be rapidly cooled to near ground-state occupations within a few hundred microseconds. Our predictions for the center-of-mass mode include a cooling time constant of tens of microseconds and an enhancement of the cooling rate with increasing number of ions. Successful experimental demonstrations of EIT cooling in the NIST Penning trap [Jordan et al., [Phys. Rev. Lett. 122, 053603 (2019)] validate our predictions.

Figure 1

EIT absorption spectrum with laser parameters relevant to the NIST EIT cooling experiment. Two strong dressing lasers, with equal Rabi frequencies and equal detunings from their respective transitions, couple two long-lived states to an excited state in a closed three-level system. The absorption from a weak probe is plotted as the probe detuning ${\mathrm{\Delta }}_{\text{P}}$ is scanned across the dressing detuning ${\mathrm{\Delta }}_{\text{D}}$. The motion-removing (blue impulses) and motion-adding (red impulses) sidebands can be interpreted as weak probes that sample this spectrum. Inset: Close-up near $({\mathrm{\Delta }}_{\text{P}}-{\mathrm{\Delta }}_{\text{D}})/2\pi =0$, with a magnified $y$ axis, showing the zero at the transparency point as well as the asymmetry in the spectrum on either side of this point. The blue dashed line is a mirror image of the spectrum on the blue-detuned side, drawn on the red-detuned side to highlight the asymmetric growth of the absorption away from the transparency point. The dressing lasers have equal detuning ${\mathrm{\Delta }}_{\text{D}}/2\pi \equiv {\mathrm{\Delta }}^0/2\pi =360\mathrm{MHz}$, and equal Rabi frequency ${\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)/2\pi \approx 33.9\mathrm{MHz}$. The Rabi frequency of the weak probe is ${\mathrm{\Omega }}_{\text{P}}=0.05{\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)$. The decay rates from the excited state to the two long-lived states are ${\mathrm{\Gamma }}_1/2\pi =6\mathrm{MHz}$, ${\mathrm{\Gamma }}_2/2\pi =12\mathrm{MHz}$. [See the discussion in Sec. 2 and Eq. (14) for a detailed explanation of the parameters.]

Figure 2

Experimental setup to cool ions in a Penning trap using electromagnetically induced transparency (EIT). Two EIT lasers address the ion crystal at angles $\pm \theta$ with respect to the $x$ axis, with one driving the $|g_1\rangle \leftrightarrow |e\rangle$ transition and the other driving the $|g_2\rangle \leftrightarrow |e\rangle$ transition in a blue-detuned regime ($\mathrm{\Delta }>0$). The curved arrow indicates the rotation direction in the $x\text{−}y$ plane.

Figure 3

Cooling of the transverse motion over time, for a single ion revolving around the trap center at different radii. The cooling is slower at larger radii because of time-varying Doppler shifts modulating the detunings of the EIT lasers as seen by the ion. Inset: The steady-state occupation also increases with distance from the trap center. Here, ${\mathrm{\Delta }}^0/2\pi =180\mathrm{MHz}$, ${\mathrm{\Omega }}_1/2\pi ={\mathrm{\Omega }}_2/2\pi ={\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)/2\pi \approx 24\mathrm{MHz}$.

Figure 4

Cooling curve for an ion at the trap center that experiences no Doppler shifts. The separation of the time scales for the electronic and motional degrees of freedom is not large enough to validate the adiabatic elimination of the electronic degrees of freedom. The cooling is therefore much slower than the result predicted from such an elimination procedure [12]. Here, ${\mathrm{\Delta }}^0/2\pi =180\mathrm{MHz}$, ${\mathrm{\Omega }}_1/2\pi ={\mathrm{\Omega }}_2/2\pi ={\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)/2\pi \approx 24\mathrm{MHz}$.

Figure 5

Cooling curves for the center-of-mass (COM) mode for crystals with different ion numbers, calculated using the Gaussian model (GM), showing rapid near ground-state cooling within 100 $\mu \mathrm{s}$. However, the cooling rates are almost identical for all of these crystals. Here, ${\mathrm{\Delta }}^0/2\pi =360\mathrm{MHz}$, ${\mathrm{\Omega }}_1/2\pi ={\mathrm{\Omega }}_2/2\pi ={\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)/2\pi \approx 33.9\mathrm{MHz}$.

Figure 6

Cooling curve for a single revolving ion, located at $r=0,20,40$, and $60\mu \mathrm{m}$ from the trap center, computed using three numerical approaches: (i) Time evolution of the full density matrix (full DM), (ii) the Gaussian model (GM), and (iii) the sampling model (SM) using 2096 trajectories. The cooling curves from the GM do not agree with the full DM curves. However, sampling the initial noise systematically (SM) accounts for beyond-Gaussian properties of the phase-space distribution of the system degrees of freedom, and reproduces the full DM curves very well. Here, ${\mathrm{\Delta }}^0/2\pi =360\mathrm{MHz}$, ${\mathrm{\Omega }}_1/2\pi ={\mathrm{\Omega }}_2/2\pi ={\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)/2\pi \approx 33.9\mathrm{MHz}$.

Figure 7

Cooling curves for the center-of-mass (COM) mode for crystals with different ion numbers, calculated using the sampling model (SM) with 2048 trajectories. The SM predicts that the cooling rate increases with ion number $N$. The cooling curves from the GM (Fig. 5) are also shown for comparison, where the $N$ dependency of the cooling rate does not manifest. Inset: Cooling rate of an $N$-ion crystal $R_N$ relative to the single-ion rate $R_1$ extracted from the SM (markers). A power-law fit (solid line) shows that the cooling rate scales as $\sim N^{0.3}$ for the parameters used. Here, ${\mathrm{\Delta }}^0/2\pi =360\mathrm{MHz}$, ${\mathrm{\Omega }}_1/2\pi ={\mathrm{\Omega }}_2/2\pi ={\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)/2\pi \approx 33.9\mathrm{MHz}$.

Figure 8

(a) Cooling curves for all the drumhead modes of a 37-ion crystal with bandwidth $B/2\pi \approx 185$ kHz, computed using the SM, showing efficient cooling within 100 $\mu \mathrm{s}$. Here, ${\mathrm{\Delta }}^0/2\pi =360\mathrm{MHz}$, ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2={\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)\approx 33.9\mathrm{MHz}$. (b) GM cooling curves for all the drumhead modes of a 120-ion crystal with bandwidth $B/2\pi \approx 376$ kHz, showing near ground-state, steady-state occupations of all the modes after few hundred $\mu \mathrm{s}$ of EIT cooling. Note that the $y$ axis is plotted in log scale. Here, ${\mathrm{\Delta }}^0/2\pi =400\mathrm{MHz}$, ${\mathrm{\Omega }}_1/2\pi ={\mathrm{\Omega }}_2/2\pi ={\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)/2\pi \approx 35.7\mathrm{MHz}$. The same rotating wall frequency, ${\omega }_r/2\pi =180$ kHz, was used in both cases.

Figure 9

Steady-state occupation as a function of the misalignment angle $\delta \theta$ for a single ion revolving around the trap center at different radii. The final occupation is not very sensitive to small misalignments ($\delta \theta \le 1^{\circ }$) of the EIT wave vectors. Here, ${\mathrm{\Delta }}^0/2\pi =400\mathrm{MHz}$, ${\mathrm{\Omega }}_1/2\pi ={\mathrm{\Omega }}_2/2\pi ={\mathrm{\Omega }}_{\text{opt}}\left({\mathrm{\Delta }}^0\right)/2\pi \approx 35.7\mathrm{MHz}$.