Simultaneous cooling of axial vibrational modes in a linear ion trap

In order to use a collection of trapped ions for experiments where a well-defined preparation of vibrational states is necessary, all vibrational modes have to be cooled to ensure precise and repeatable manipulation of the ions quantum states. A method for simultaneous sideband cooling of all axial vibrational modes is proposed. By application of a magnetic field gradient the absorption spectrum of each ion is modified such that sideband resonances of different vibrational modes coincide. The ion string is then irradiated with monochromatic electromagnetic radiation, in the optical or microwave regime, for sideband excitation. This cooling scheme is investigated in detailed numerical studies. Its application for initializing ion strings for quantum information processing is extensively discussed.

Figure 1

Illustration of the axial motional spectrum of a chain of ten ions in the presence of a spatially inhomogeneous magnetic field. The vertical lines indicate the axial position of each ion in units of ${\zeta }_0$. The corresponding horizontal lines indicate the frequencies of the spectral lines as measured at that particular ion. The thick horizontal lines indicate the ions resonance frequencies ${\omega }_j-{\omega }_1$ (in units of the secular axial frequency ${\nu }_1$) relative to the resonance frequency ${\omega }_1$ of the ion at $z_1=-2.87{\zeta }_0$. The remaining horizontal lines show the frequencies of red sideband resonances for each ion at frequencies ${\omega }_j-{\nu }_{\alpha }(j,\alpha =1,\dots ,10)$. The magnetic field is designed such that ${\omega }_1-{\nu }_1={\omega }_2-{\nu }_2=\cdots ={\omega }_N-{\nu }_N$. These resonances are highlighted by medium thick lines.

Figure 2

Schematic of the ions internal energy levels on which cooling is implemented. Indicated are the relevant Rabi frequencies (symbols $\Omega$), spontaneous decay rates $\left(\Gamma \right)$, and detunings $\left(\Delta \right)$. The corresponding equations for the dynamics are discussed in detail in the Appendix

Figure 3

(Color online) Raman sideband cooling of a chain of ten ions without a magnetic field gradient. The parameters are ${\Omega }_{12}=100\times 2\pi \mathrm{kHz}$, ${\Omega }_{01}=30\times 2\pi \mathrm{kHz}$, and ${\Delta }_{12}=-10\times 2\pi \mathrm{MHz}$. (a) Steady-state mean vibrational number $⟨n^{\left(\alpha \right)}⟩$ as a function of ${\Delta }_{01}$ in units of ${\nu }_1$. (b) Cooling rate ${\Gamma }_{\mathrm{cool}}^{\left(\alpha \right)}$ of mode $\alpha$ for ${\Delta }_{01}=-{\nu }_{\alpha }$. (c) Steady-state populations for ${\Delta }_{01}=-{\nu }_1$, corresponding to sideband cooling of mode 1. The bars indicating the vibrational excitations of modes ${\nu }_3$ and ${\nu }_4$ have been truncated.

Figure 4

(Color online) Raman sideband cooling a chain of ten ions in the presence of a spatially inhomogeneous magnetic field such that the condition ${\omega }_1-{\nu }_1=\cdots ={\omega }_N-{\nu }_N$ is fulfilled (see Fig. 1). The parameters are ${\Omega }_{12}=100\times 2\pi \mathrm{kHz}$, ${\Omega }_{01}=5\times 2\pi \mathrm{kHz}$, and ${\Delta }_{12}=-10\times 2\pi \mathrm{MHz}$. The steady-state mean vibrational excitation $⟨n^{\left(\alpha \right)}⟩$ is displayed. (a) $⟨n^{\left(1\right)}⟩$ (COM mode) as a function of ${\Delta }_{01}$ in units of ${\nu }_1$. (b) $⟨n^{\left(\alpha \right)}⟩$ with $\alpha =1,2,3,4$ as a function of ${\Delta }_{01}$. (c) $⟨n^{\left(\alpha \right)}⟩$ for $\alpha =1,\dots ,10$ as a function of ${\Delta }_{01}$. (d) $⟨n^{\left(\alpha \right)}⟩$ for $\alpha =1,\dots ,10$ when the chain is simultaneously cooled at the detuning ${\Delta }_{01}=-{\nu }_1$.

Figure 5

Cooling rates for Raman sideband cooling a chain of ten ions in the presence of a spatially inhomogeneous magnetic field such that the condition ${\omega }_1-{\nu }_1=\cdots ={\omega }_N-{\nu }_N$ is fulfilled. The same parameters have been used here as for generating Fig. 3 (${\Omega }_{12}=100\times 2\pi \mathrm{kHz}$, ${\Omega }_{01}=30\times 2\pi \mathrm{kHz}$, and ${\Delta }_{12}=-10\times 2\pi \mathrm{MHz}$). The black bars indicate the rate for each mode when simultaneously sideband cooling all modes. The grey bars give the cooling rates that are achieved if (still in the presence of the magnetic field) the Raman detuning is set to that value where the maximum cooling rate for each individual mode is obtained.

Figure 6

(Color online) Microwave sideband cooling a chain of ten ions in the presence of a spatially inhomogeneous magnetic field such that the condition ${\omega }_1-{\nu }_1=\dots ={\omega }_N-{\nu }_N$ is fulfilled—that is, simultaneous sideband cooling using microwave radiation for sideband excitation. Same parameters as for Fig. 4 (${\Omega }_{12}=100\times 2\pi \mathrm{kHz}$, ${\Omega }_{01}=5\times 2\pi \mathrm{kHz}$, and ${\Delta }_{12}=-10\times 2\pi \mathrm{MHz}$). (a) Mean vibrational numbers ${⟨n^{\left(\alpha \right)}⟩}_f$ at steady state as a function of the detuning ${\Delta }_{01}$. (b) Same as (a) for the values around ${\Delta }_{01}=-{\nu }_1$. (c) ${⟨n^{\left(\alpha \right)}⟩}_f$ at ${\Delta }_{01}=-{\nu }_1$—i.e., when all ten vibrational modes are simultaneously cooled. (d) Cooling rates for ${\Delta }_{01}=-{\nu }_1$.

Figure 7

Required magnetic field gradient to superpose the motional red sidebands of ten ${}^{171}\mathrm{Yb}^{+}$ ions (markers) in a trap characterized by a COM frequency ${\nu }_1=1\times 2\pi \mathrm{MHz}$ and calculated field gradient (solid line) produced by three single wire windings (see text).

Figure 8

The vertical bars indicate the frequencies ${\omega }_j-{\nu }_j$ of first-order sideband resonances corresponding to ten axial vibrational modes of ten ${}^{171}\mathrm{Yb}^{+}$ ions in an ion trap with the field gradient shown in Fig. 7. The frequencies are given relative to the transition frequency ${\omega }_1$ of the first ion, and the bars are labeled with $j=1,\dots ,10$. The height of the bars indicates the transition probability associated with a given sideband relative to the first sideband (COM mode) of ion 1 for a Raman transition. The relative transition probability is proportional to $\mid {\eta }_{j\alpha }^{\mathrm{eff}}\mid ∕\mid {\eta }_{11}^{\mathrm{eff}}\mid \approx S_j^{\alpha }{\nu }_{\alpha }^{-(1∕2)}∕S_1^1{\nu }_1^{-(1∕2)}$ with $j=\alpha$

Figure 9

(Color online) Raman sideband cooling a chain of ten ions in the presence of a spatially inhomogeneous magnetic field. The condition ${\omega }_1-{\nu }_1=\cdots ={\omega }_N-{\nu }_N$ is approximately fulfilled (see Fig. 8). Same parameters as in Fig. 4 (i.e., ${\Omega }_{12}=100\times 2\pi \mathrm{kHz}$, ${\Omega }_{01}=5\times 2\pi \mathrm{kHz}$, and ${\Delta }_{12}=-10\times 2\pi \mathrm{MHz}$). (a) Steady-state vibrational excitation ${⟨n^{\left(\alpha \right)}⟩}_f$ as a function of the detuning ${\Delta }_{01}$ around ${\Delta }_{01}=-{\nu }_1$. (b) ${⟨n^{\left(\alpha \right)}⟩}_f$ for each mode at that detuning where the sum of the mean vibrational quantum numbers of all ten modes is minimal.

Figure 10

(Color online) Raman sideband cooling a chain of ten ions in the presence of a spatially inhomogeneous magnetic field. The condition ${\omega }_1-{\nu }_1=\cdots ={\omega }_N-{\nu }_N$ is approximately fulfilled (see Fig. 8). (a) ${⟨n^{\left(\alpha \right)}⟩}_f$ as a function of ${\Delta }_{01}$ displaying all first-order red sidebands with all parameters unchanged compared to Fig. 9 except ${\Omega }_{12}=1\times 2\pi \mathrm{MHz}$. (b) ${⟨n^{\left(\alpha \right)}⟩}_f$ as a function of ${\Delta }_{01}$ in the vicinity of the common sideband resonance around ${\Delta }_{01}=-{\nu }_1$. The resonances are power broadened and ac Stark shifted compared to the ones in Fig. 9a. (c) Steady-state population ${⟨n^{\left(\alpha \right)}⟩}_f$ for each mode at that detuning where the sum of the mean vibrational quantum numbers of all ten modes is minimal [the analog to Fig. 9b except that ${\Omega }_{12}=1\times 2\pi \mathrm{MHz}$]. (d) ${⟨n^{\left(\alpha \right)}⟩}_f$ for each mode at that detuning where one of the modes (here mode 10) reaches the absolute minimum. Optimal cooling of all vibrational modes is achieved by first cooling mode 10 at ${\Delta }_{01}=-3.91{\nu }_1$ and subsequently modes 1 through 9 at ${\Delta }_{01}=-1.022{\nu }_1$.