Optical dipole-force cooling of anions in a Penning trap

We discuss the possibility of using optical dipole forces for Sisyphus cooling of ions stored in a Penning trap by addressing the specific case of the molecular cooling candidate ${{\mathrm{C}}_2}^{-}$. Using a GPU accelerated code for Penning trap simulations, which we extended to include the molecule-light interaction, we show that this scheme can decrease the time required for cooling by an order of magnitude with respect to Doppler cooling. In our simulation we found that a reduction of the axial anion temperature from $10\mathrm{K}$ to $50\mathrm{mK}$ in around $10\mathrm{s}$ is possible. The temperature of the radial degrees of freedom was seen to thermalize to $150\mathrm{mK}$. Based on the laser-cooled ${{\mathrm{C}}_2}^{-}$, a study on the sympathetic cooling of anions with masses 1–50 nucleon was performed, covering relevant candidates for investigations of chemical anion reactions at ultracold temperatures as well as for antimatter studies.

Figure 1

(a) Overview of the electronic and vibrational states. The relevant transitions from the $A{}^2\mathrm{\Pi }$ state for our cooling scheme are indicated. The electron affinity is equal to $3.27\mathrm{eV}$ [9]. (b) Sketch of the Sisyphus cooling cycle to scale with simulation parameters. The geometry of the ion cloud (gray) is shown together with laser beam profiles (red, green, and yellow) and an overlay of the involved electronic levels (see also Fig. 2). A $6-\mathrm{W}$ laser (red), detuned from the resonance of the $A{}^2\mathrm{\Pi }(v^{\prime }=0,N^{\prime }=1,J^{\prime }=0.5,M_J^{\prime }=1/2)\leftrightarrow X{}^2\mathrm{\Sigma }(v^{\prime \prime }=0,N^{\prime \prime }=0,J^{\prime \prime }=0.5,M_J^{\prime \prime }=1/2)$ transition by $\delta =1\mathrm{GHz}$ and with a waist of $w_{\mathrm{dip}}=185\mu \mathrm{m}$, shifts the levels locally around $z=0$, thereby creating a potential $U_{\mathrm{dip}}=12.4\mathrm{mK}$. A second laser (green) on resonance with the maximally shifted levels causes transitions to the upper state in particles in the maximum of the dipole potential, by absorbing a photon with energy $h{\nu }^{\prime }$. Due to the long lifetime of $50\mu \mathrm{s}$ of the $A{}^2\mathrm{\Pi }(v=0)$ state in relation to their thermal energy, the particles are likely to leave the region of the shift and then decay to the initial state by emitting a photon with energy $h\nu =h{\nu }^{\prime }+U_{\mathrm{dip}}$. Repump lasers (orange) on all other levels prevent losses from the cooling cycle by particles decaying to dark states. The whole cycle effectively removes the energy $\mathrm{\Delta }E=h(\nu -{\nu }^{\prime })$ from the particles.

Figure 2

Detail of the $A{}^2\mathrm{\Pi }(v^{\prime }=0)\leftrightarrow X{}^2\mathrm{\Sigma }(v^{\prime \prime }=0)$ transition and the intermediate $X{}^2\mathrm{\Sigma }(v=1)$ states in a $5-\mathrm{T}$ magnetic field. Several arrows indicate the transitions that are addressed by the different lasers described in the text.

Figure 3

Plasma temperature plot of 1024 ${{\mathrm{C}}_2}^{-}$ molecules subjected to dipole force cooling, once without (blue lower symbols) and once including a SWP (yellow upper symbols). The temperature is derived by fitting a Boltzmann distribution to the velocity histogram of the axial and radial degrees of freedom in cylindrical coordinates. The two solid lines show linear fits to the initial temperature evolution in order to illustrate the deviation from linear behavior as the molecules get colder. The insets show projected snapshots of the plasma taken from the simulation without a SWP at two points in time.

Figure 4

Results of the sympathetic cooling of 200 negatively charged species with a mass $m^{\prime }$ immersed in an ion cloud of 824 ${{\mathrm{C}}_2}^{-}$. For reference, the temperature of the ${{\mathrm{C}}_2}^{-}$ is given from the simulation involving the largest mass with $m^{\prime }=50$ nucleon. The cooling of the ${{\mathrm{C}}_2}^{-}$ was implemented using an artificial drag force that resembles a dipole cooling scheme.