Ion-assisted ground-state cooling of a trapped polar molecule

We propose and analyze a scheme for sympathetic cooling of the translational motion of polar molecules confined in the wells of a deep optical lattice and interacting one by one with laser-cooled ions in a radio-frequency trap. The energy gap between the excitation spectra of the particles in their respective trapping potentials is bridged by means of a parametric resonance, provided by the additional modulation of the rf field. We analyze two scenarios: simultaneous laser cooling and energy exchange between the ion and the molecule, and a scheme where these two processes take place separately. We calculate the lowest final energy of the molecule and the cooling rate depending on the amplitude of the parametric modulation. For small parametric modulation, the dynamics can be solved analytically within the rotating wave approximation.

Figure 1

Specific considered setup consisting of a trapped polar molecule and an ion interacting via long-range polarization forces.

Figure 2

Energy spectrum of the relative motion for a molecule and an ion confined in spherically symmetric traps versus the distance $d$ between traps, calculated for some example parameters: $\phi =-\pi /8$, and $R^{*}=3.48l$, with $l=\sqrt{\hslash /\mu \omega }$. The exact numerical results (black lines) are compared with the prediction of the coupled-oscillator Hamiltonian (17) (thick red dashed lines). The vertical blue dashed line shows the minimal distance at which the Hamiltonian can be expanded quadratically around the equilibrium points, whereas the green dash-dotted line shows the distance at which the barrier between particles becomes equal to the relative energy of the particles.

Figure 3

(a) Modulation frequency ${\omega }_{\mathrm{f}}$ that leads to the resonant coupling between the molecule and the ion (solid line with dots), and width of the resonance (dashed lines), calculated numerically from the continued fraction solution, compared with the analytical results (22) (solid line). (b) Effective coupling frequency ${\Omega }_c$ calculated numerically, compared with the analytical result of Eq. (15) (solid line). For both panels the calculations are performed with ${\omega }_m=0.1{\omega }_i$ and ${\omega }_c=0.01{\omega }_i$.

Figure 4

Typical dynamics of molecule and ion energies for the cooling scheme where the ion is cooled separately from the molecule. The Roman number corresponds to the different cooling phases described in the text.

Figure 5

(a) Length of the rectangular $\pi$ pulse (solid line) and lower bound for the pulse length (dashed line) due to the physical coupling ${\omega }_c$. (b) Mean phonon number for the molecule $\langle n_m\rangle$ at the end of the cooling process as a function of the amplitude $A$ and of the initial mean phonon number $\langle n_m(0)\rangle$. Calculations are performed with ${\omega }_m=0.1{\omega }_i$, ${\omega }_c=0.01{\omega }_i$, and $\langle n_i(0)\rangle =0$.

Figure 6

Cooling rate of the molecule ${\Gamma }_m$ as a function of the cooling rate of the ion $\Gamma$ for different values of the parameter $\omega$ (see text for details). Cooling rates are scaled by the effective coupling ${\Omega }_c$.

Figure 7

Cooling rate of the molecule ${\Gamma }_m$ versus the amplitude of the parametric modulation $A$, for different values of the Rabi frequency $\Omega$. For comparison we include a line showing the values of $2{\Omega }_c$ (solid line without symbols), and the cooling rate of the ion $\Gamma$ (horizontal lines). The results are obtained for ${\omega }_m=0.2{\omega }_i$ and ${\omega }_c=0.01{\omega }_i$; $m_m=m_i$; and for cooling parameters corresponding to the sideband cooling regime $\gamma =0.1{\omega }_i$, $\Delta =-{\omega }_i$, and $\eta =0.1$.

Figure 8

Dynamics of the mean phonon number of the molecule $n_{\mathrm{mol}}(t)$ and the ion $n_{\mathrm{ion}}(t)$. The calculations are performed for the parameters of Fig. 7 and $\Omega =0.1{\omega }_i$. The dotted horizontal line shows the final mean phonon number for the molecule and the ion.

Figure 9

The final mean phonon number of the molecule and the ion as a function of the amplitude $A$. The calculations are performed for the same parameters as in Fig. 8.

Figure 10

The final mean phonon number of the ion as a function of the detuning $\Delta$ and for different values of the amplitude $A$. The results are calculated for the parameters of Fig. 7.

Figure 11

The final mean phonon number of the ion as a function of the Rabi frequency $\Omega$ and for different values of the amplitude $A$. The results are calculated for the parameters of Fig. 7.

Figure 12

The final mean phonon number of ion as a function of the spontaneous emission rate $\gamma$ and for different values of the amplitude $A$. The results are calculated for the parameters of Fig. 7.

Figure 13

Stability diagram of Eq. (C1) for ${\omega }_{\mathrm{f}}/{\omega }_{\mathrm{rf}}=0.05$ and $A=0$, $0.1$, $0.2$, and $0.3$. The amplitude $W$ of the component varying with the frequency ${\omega }_{\mathrm{f}}$ is determined from the parameters $a$, $q$, and $A$ (see text for details).