Life and death of a cold ${\mathrm{BaRb}}^{+}$ molecule inside an ultracold cloud of Rb atoms

We study the evolution of a cold single ${\mathrm{BaRb}}^{+}$ molecule while it continuously collides with ultracold Rb atoms. The initially weakly bound molecule can undergo a sequence of elastic, inelastic, reactive, and radiative processes. We investigate these processes by developing methods for discriminating between different ion species, electronic states, and kinetic ion energy ranges. By analyzing the experimental data while taking into account theoretical insights, we obtain a consistent description of the typical trajectory through the manifold of available atomic and molecular states. Monte Carlo simulations describe the measured dynamics well. As a further result, we determine rates for collisional and radiative relaxation as well as photodissociation, spin-flip collisions, and chemical reactions.

Figure 1

(a) Illustration of various inelastic and reactive processes. (i) Formation of a ${\mathrm{BaRb}}^{+}$ molecule via three-body recombination, (ii) collisional relaxation of a ${\mathrm{BaRb}}^{+}$ molecule, (iii) substitution reaction, (iv) collisional dissociation, (v) collisional spin exchange, (vi) photodissociation, and (vii) radiative relaxation. (b) PECs for ${\mathrm{BaRb}}^{+}$, taken from Ref. [32]. The entrance channel $\text{Rb}\left(5s{}^{2}S\right)+{\text{Ba}}^{+}\left(6s{}^{2}S\right)$ marks zero energy. Solid black, blue, and red arrows show possible photodissociation transitions for 1064-, 493-, and 650-nm light, respectively. The dashed green arrow indicates radiative relaxation to the electronic ground state.

Figure 2

(a) Probability $P_{{\text{Ba}}^{+}}$ for detecting a ${\mathrm{Ba}}^{+}$ ion as a function of time $\tau$ after immersion into a Rb atomic cloud. Circles are measured data. Each data point is the mean value of 50 repetitions of the experiment and the error bars represent the $1\sigma$ statistical uncertainty. Curves are the results of MC simulations (see Appendix pp6). (b) Time evolution of the atomic density $n$ at the ion trap center after the ODT beams have been switched off at $\tau =-250\mu \mathrm{s}$.

Figure 3

Observation of radiative relaxation. The data points give the measured probability $P_{{\text{Ba}}^{+}}$ of detecting a ${\mathrm{Ba}}^{+}$ ion after an extensive laser cooling stage as a function of the time ${\tau }^{\prime }$ after which a 300-ms-long 1064-nm light pulse is applied. For formation of a ${\mathrm{BaRb}}^{+}$ molecule, a cold ${\mathrm{Ba}}^{+}$ ion is immersed into the atom cloud at ${\tau }^{\prime }=0$. The solid black line represents the time evolution of the atomic density $n$ at the ion trap center (see Appendix pp1). All other lines are results of MC calculations for the probabilities of finding the species as denoted in the legend.

Figure 4

Photodissociation of ground-state ${\mathrm{BaRb}}^{+}$ molecules by 493-nm (a) and 650-nm (b) laser light as a function of exposure time $\mathrm{\Delta }t$. Filled circles show the probability of detecting a ${\mathrm{BaRb}}^{+}$ molecule. Hollow diamonds give the probability of detecting a ${\mathrm{Ba}}^{+}$ ion. The solid lines are fits of an exponential decay plus offset.

Figure 5

Vibrational binding energies of a ${\mathrm{BaRb}}^{+}$ molecule for the ${\left(1\right)}^3{\mathrm{\Sigma }}^{+}$ and ${\left(2\right)}^1{\mathrm{\Sigma }}^{+}$ electronic states. The rotational quantum number is $j=0$. The brown data points result from the Lennard-Jones potential which is used for the QCT calculations. The purple and blue data points are the results from our calculated ${\left(1\right)}^3{\mathrm{\Sigma }}^{+}$ and ${\left(2\right)}^1{\mathrm{\Sigma }}^{+}$ PECs, as described in Appendix pp4-s3a.

Figure 6

Vibrational relaxation cross section as a function of the vibrational quantum number $v$, for various collision energies $E_c$. Shown are results of QCT calculations. The error bars represent $1\sigma$ standard deviation obtained by evaluating the numbers of trajectories leading to the same outcomes. The dashed lines are the Langevin cross sections.

Figure 7

PEDMs (circles) and TEDMs (solid lines) as functions of the internuclear distance of the ${\mathrm{BaRb}}^{+}$ molecule. Initial and final states are given in the plot. For $\mathrm{\Sigma }\text{−}\mathrm{\Sigma }$ transitions the dipole moment along the internuclear axis is shown, whereas for $\mathrm{\Sigma }\text{−}\mathrm{\Pi }$ transitions the dipole moment in transverse direction is shown.

Figure 8

Photodissociation cross sections as functions of the binding energy for an excited state ${\mathrm{BaRb}}^{+}$ ion exposed to 1064-nm light. The results for the most relevant transitions in our experiments are shown. Data points are calculations. Solid lines represent fits $\propto E_b^{0.75}$ to the data points (see legend).

Figure 9

Photodissociation cross sections as functions of the final energy $E^{\prime }=h\nu -E_b$. Upper panel: ${\left(2\right)}^1{\mathrm{\Sigma }}^{+}\to {\left(4\right)}^1{\mathrm{\Sigma }}^{+}$. Middle panel: ${\left(1\right)}^3{\mathrm{\Sigma }}^{+}\to {\left(3\right)}^3\mathrm{\Pi }$. Lower panel: ${\left(1\right)}^3{\mathrm{\Sigma }}^{+}\to {\left(3\right)}^3{\mathrm{\Sigma }}^{+}$. The color coding of the lines corresponds to the binding energy of the initial state as indicated on the right. The right (left) vertical dashed line marks the energy $E^{\prime }$ when a 1064-nm photon excites a molecule with $E_b=0.1\text{K}\times k_B$ ($E_b=31.5\text{K}\times k_B$).

Figure 10

(a) Photodissociation cross section as a function of final energy $E^{\prime }=h\nu -E_b$ for the transition $\left(2\right){}^1{\mathrm{\Sigma }}^{+}\to \left(4\right){}^1{\mathrm{\Sigma }}^{+}$. (b) The $\left(4\right){}^1{\mathrm{\Sigma }}^{+}$ PEC (red curve) and two energy-normalized continuum wave functions (black curves). (c) The $\left(2\right){}^1{\mathrm{\Sigma }}^{+}$ PEC (red curve) and the rovibrational wave function (black curve) for the binding energy of $E_b=10.5{\text{cm}}^{-1}$. Blue and orange dashed lines help to illustrate that a good wave function overlap at the inner turning point of the excited PEC leads to a large cross section. (d) The respective TEDM as a function of the internuclear distance $R$.

Figure 11

Radiative lifetime (blue dots) for the highest rovibrational levels $\left(v,j=0\right)$ of the ${\left(2\right)}^1{\mathrm{\Sigma }}^{+}$ electronic state, as a function of the binding energy $E_b$. The red dashed line is a fit $\propto E_b^{-0.75}$ to the data.

Figure 12

Calculated population distribution for vibrational levels $v$ of the electronic ground state ${\left(X\right)}^1{\mathrm{\Sigma }}^{+}$ of the ${\mathrm{BaRb}}^{+}$ molecule after radiative relaxation from a weakly bound level in the ${\left(2\right)}^1{\mathrm{\Sigma }}^{+}$ state with $j=1$. Here the same approach is used as described in Ref. [7].

Figure 13

Evolution of the vibrational quantum number of a ${\mathrm{BaRb}}^{+}$ molecule in collisions with Rb atoms. We only consider the “relevant” collisions which have impact parameters $b<b_{\max }$, as discussed in Appendix pp4-s1. The $y$ axis on the left shows the vibrational quantum number $v$ and the $y$ axis on the right shows the collision energy $E_c$. The data correspond to a single MC trajectory.

Figure 14

Average number of elastic collisions of Rb atoms with a ${\mathrm{BaRb}}^{+}$ ion in the vibrational state $v$ before a relaxation (excitation) to another vibrational state occurs. Data points are from MC simulations. The error bars represent $1\sigma$ standard deviations.

Figure 15

Average collision energy of the ${\mathrm{BaRb}}^{+}$ ion right after (blue filled squares) and directly before (red open squares) vibrational relaxation as a function of the vibrational quantum number. Data points are results from the MC simulation. The error bars represent $1\sigma$ standard deviations. Here, the difference between the curves describes the effect of cooling due to elastic collisions.

Figure 16

Average vibrational quantum number for a ${\mathrm{BaRb}}^{+}$ ion as a function of the number of vibrational relaxation collisions with Rb atoms. The data points represent the results of MC calculations.

Figure 17

Calculations (lines) for ion signals together with measured data (filled and open circles) of Fig. 2. Shown are the probabilities for finding different ion states or species as given in the legend. In (a) the case with ODT is considered while in (b) the case without ODT is considered. “hot ${\mathrm{Ba}}^{+}$(PD)” and “${\mathrm{Rb}}^{+}$(PD)” are populations due to photodissociation with 1064-nm light. The population “${\mathrm{Rb}}_2^{+}\mathrm{or}{\mathrm{Rb}}^{+}$” is due to secondary collisional reactions of ${\mathrm{BaRb}}^{+}{\left(X\right)}^1{\mathrm{\Sigma }}^{+}$ molecules with Rb atoms.