Probing the interaction energy of two ${}^{85}\mathrm{Rb}$ atoms in an optical tweezer via spin-motion coupling

The inherent polarization gradients in tight optical tweezers can be used to couple the atomic spins to the two-body motion under the action of a microwave spin-flip transition, so that such a spin-motion coupling offers an important control knob on the motional states of optically trapped two colliding atoms. Here, after preparing two elastically scattering ${}^{85}\mathrm{Rb}$ atoms in the three-dimensional ground-state in the optical tweezer, we employed this control in order to probe the colliding energies of elastic and inelastic channels. The combination of microwave spectra and corresponding s-wave pseudopotential model allows us to infer the effect of the state-dependent trapping potentials on the elastic colliding energies, as well as to reveal how the presence of inelastic interactions affects elastic part of the relative potential. Our work shows that the spin-motion coupling in a tight optical tweezer expand the experimental toolbox for fundamental studies of ultracold collisions in the two body systems with reactive collisions, and potentially for that of more complex interactions, such as optically trapped atom-molecule and molecule-molecule interactions.

Figure 1

Schematic diagram of experimental setup, energy levels, and time sequence. (a) Schematic of apparatus. The movable (M) and static (S) optical traps are both from 852 nm laser and are linearly polarized along $x$ direction. These two beams are combined by a beam splitter (BS), then expanded eight times by a beam-expander group (lens with $f=-50\mathrm{mm}$ and $f=400\mathrm{mm}$), then strongly focused by a 0.6 numerical-aperture (NA) microscopic objective. The pointing of M trap is controlled by a mirror actuated by piezoelectric transducers (PZTs) to merge and split two atoms. The liquid crystal retarders are employed to dynamically adjust the polarizations of optical tweezers. The microwave (MW) horn is used to emit the MW pulse for MW spectra. (b) Level scheme for ${}^{85}\mathrm{Rb}$ Raman sideband cooling (RSC) and MW spectra. The Raman beam R1 is paired with {R2,R3} beams, respectively, so as to address the orthogonal radial directions. And the axial axis is addressed by a pair of R1 and R4. The optical pumping (OP) and repumping (RP) beams are used to efficiently pump the single atoms back to the initial spin state for RSC. (c) Schematic illustration of time sequence. The 3D magnetic-field vector $\{B_x,B_y,B_z\}$ are temporally ramped up to chosen values for the implementation of specific experimental phases, including RSC, merging two atoms, and spin-motion coupling (SMC), splitting two atoms and fluorescence detection. See the text for details.

Figure 2

Effect of Raman transition carrier frequency shift on axial cooling and the relationship between the carrier frequency shift and control voltage of LCR. (a) The relationship between RSC carrier frequency shift and the average quantum number and the ground-state preparation probability of axial direction. When the carrier frequency shift is greater than 4.4 kHz, the average axial quantum number increases to above 0.7(3) and the ground-state preparation probability decreases to below 0.6(2). (b) The relationship between the carrier frequency shift and control voltage of LCR.

Figure 3

Schematic diagram of the transition and the MW spectra via SMC. (a) Schematic diagram of the vibrational transition of ${}^{85}\mathrm{Rb}$ atom. (b) MW spectra for spin flipping of $|3,-3\rangle |2,-2\rangle \to |3,-3\rangle |3,-1\rangle$ (red filled circles) and $|2,-2\rangle \to |3,-1\rangle$ (black filled squares). The spectra set the single atom transition carrier frequency [3028.0050(8) MHz] as the reference and the solid curves are Gaussian fits of the data. The detuning frequency of the single-atom carrier peak is $c{f}_0=0.0\left(8\right)\mathrm{kHz}$, while the detuning frequency of the diatomic carrier peak is $c{f}_1=-10.0\left(9\right)\mathrm{kHz}$. Furthermore, the detuning frequency of the single-atom sideband peak is measured to be $s{f}_0=163.2\left(5\right)\mathrm{kHz}$. For a two-atom system, around the $s{f}_0$ two distinct peaks show up: $s{f}_1=153.0\left(5\right)\mathrm{kHz}$ and $s{f}_2=182.8\left(3\right)\mathrm{kHz}$. See the text for details.

Figure 4

Relationship between the interaction energy of the initial state ($|2,-2\rangle |3,-3\rangle |{\psi }_s\rangle |{\phi }_0\rangle$) and the axial trapping frequency of the optical tweezer. The black squares represent the measured values. The accompanying error bars are statistic standard deviation for the average. And the solid black line and the dashed blue line represent the theoretical calculation of Eq. (1) values with and without the correction for the perturbation of $dx$, respectively. The inset shows the measured ratios as a function of the square root of the trap frequencies ${\omega }_x$.

Figure 5

Relationship between the interaction energy of the inelastic channel {$3,-3;3,-1$} and the axial trapping frequencies. The red squares represent the measured values. The accompanying error bars are statistic standard deviation for the average. The dashed blue line represents the predictions of the energy by applying the contact pseudopotential with complex scattering length.