Efficient Raman sideband cooling of trapped ions to their motional ground state

Efficient cooling of trapped ions is a prerequisite for various applications of the ions in precision spectroscopy, quantum information, and coherence control. Raman sideband cooling is an effective method to cool the ions to their motional ground state. We investigate both numerically and experimentally the optimization of Raman sideband cooling strategies and propose an efficient one, which can simplify the experimental setup as well as reduce the number of cooling pulses. Several cooling schemes are tested and compared through numerical simulations. The simulation result shows that the fixed-width pulses and varied-width pulses have almost the same efficiency for both the first-order and the second-order Raman sideband cooling. The optimized strategy is verified experimentally. A single ${{}^{25}\mathrm{Mg}}^{+}$ ion is trapped in a linear Paul trap and Raman sideband cooled, and the achieved average vibrational quantum numbers under different cooling strategies are evaluated. A good agreement between the experimental result and the simulation result is obtained.

Figure 1

The relevant energy levels of ${{}^{25}\mathrm{Mg}}^{+}$ and laser beams used. ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$, Raman beams; DC, Doppler cooling beam; and Repump, repumping beam. $|\downarrow \rangle ={}^2{S}_{1/2}|3,3\rangle , |\uparrow \rangle ={}^2{S}_{1/2}|2,2\rangle$.

Figure 2

The relative Rabi frequencies of carrier transitions (solid green line), first-order RSB transitions (dashed black line), and second-order RSB transitions (dotted red line) as a function of Fock state number $n$. The inset shows the theoretical population of a thermal distribution for $\overline{n}=17$.

Figure 3

Simulated average $n$ as a function of time with the first-order RSB cooling for $5\mu \mathrm{s}$ (short dashed blue line), $8\mu \mathrm{s}$ (long dashed red line), $10\mu \mathrm{s}$ (solid black line), and $12\mu \mathrm{s}$ (solid green [light gray] line). The inset shows the simulated population distribution after 40 times of the $8-\mu \mathrm{s}$ first-order RSB cooling. The top shows a scheme of the pulse sequence in the figure. RP, repumping pulses.

Figure 4

Simulated average $n$ as a function of time with the second-order RSB cooling for $5\mu \mathrm{s}$ (short dashed blue line), $8\mu \mathrm{s}$ (long dashed red line), $10\mu \mathrm{s}$ (solid black line), and $12\mu \mathrm{s}$ (solid green [light gray] line). The inset shows the simulated population distribution after 40 times of the $8-\mu \mathrm{s}$ second-order RSB cooling. The top shows a scheme of the pulse sequence in the figure. RP, repumping pulses.

Figure 5

Simulated average population as a function of the pulse time for the 1st-order RSB cooling and the 2nd-order RSB cooling. The numbers of cooling pulse for them are chosen to be 15 and 40.

Figure 6

Simulated average $n$ as a function of time with the second-order RSB cooling (lower lines) and the first-order RSB cooling (upper lines) with fixed-width (dashed red lines) and varied-width (solid black lines) pulses. The top shows a scheme of the pulse sequence in the figure.

Figure 7

Simulated average $n$ as a function of time with a combination of the 2nd-order RSB cooling and the 1st-order RSB cooling. Inset shows the detail of the curve after $1400\mu \mathrm{s}$.

Figure 8

Schematic of the experimental setup. ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ are the two Raman laser beams as shown in Fig. 1. DC, Doppler cooling laser; Repump, the repumping laser; $\lambda$/2, half wave plate; $\lambda$/4, quarter wave plate. The magnetic field is aligned with the ${\mathrm{R}}_1$ laser beam. The DC laser beam counterpropagates with the ${\mathrm{R}}_1$ laser beam.

Figure 9

Raman sideband spectrum of a single ${{}^{25}\mathrm{Mg}}^{+}$ ion after Doppler cooling. The average $n$ can be obtained through the BSB-RSB ratio. The calculated value is $\overline{n}=17\left(2\right)$, which is the initial average motional level for the Raman sideband cooling.

Figure 10

The thermal distribution and simulated nonthermal distribution for a same average $n$. The nonthermal distribution is Raman sideband cooled with a cooling scheme as shown in Fig. 7. From the comparison we can see that the ground-state populations are almost the same.

Figure 11

Measured average $n$ with only the first-order RSB cooling.

Figure 12

Measured average $n$ after Raman sideband cooling with optimal cooling strategy. Inset shows the red sideband and blue sideband of Raman spectrum after the Raman sideband cooling process is finished.

Figure 13

The average motional quantum number is plotted as a function of the waiting time after motional ground-state cooling. A linear fit yields a heating rate of 22(2) phonons per second.