Modulating carrier and sideband coupling strengths in a standing-wave gate beam

We control the relative coupling strength of carrier and first-order motional sideband interactions of a trapped ion by placing it in a resonant optical standing wave. Our configuration uses the surface of a microfabricated chip trap as a mirror, avoiding technical challenges of in-vacuum optical cavities. Displacing the ion along the standing wave, we show a periodic suppression of the carrier and sideband transitions with the cycles for the two cases ${180}^{\circ }$ out of phase with each other. This technique allows for the suppression of off-resonant carrier excitations when addressing the motional sidebands, and has applications in quantum simulation and quantum control. Using the standing-wave fringes, we measure the relative ion height as a function of applied electric field, allowing for a precise measurement of ion displacement and, combined with measured micromotion amplitudes, a validation of trap numerical models.

Figure 1

(a) Energy diagram of an ion in a harmonic potential of frequency $\nu$, with arbitrary electronic states $|g\rangle$ and $|e\rangle$ and motional states $|n\rangle$. Carrier ($\mathrm{\Delta }n=0$) and first-order ($\mathrm{\Delta }n=\pm 1$) sideband transitions are shown. In a running-wave configuration, the coupling strengths of the red and blue sidebands relative to the carrier are (to first order) proportional to $\eta \sqrt{n}$ and $\eta \sqrt{n+1}$, respectively, where $\eta$ is the Lamb-Dicke parameter. (b) Schematic of our experimental configuration. ${}^{40}\mathrm{Ca}^{+}$ ions are confined $\sim 60\mu \mathrm{m}$ above the trap surface by rf and dc electric fields. 729 nm light is used to excite the $\left|S_{1/2},m_F=-\frac{1}{2}\right.〉\to \left|D_{5/2},m_F=-\frac{5}{2}\right.〉$ quadrupole transition. The incident 729 nm laser (with incidence angle $\alpha$) is retroreflected from the aluminum trap surface to produce a standing wave. The ion is displaced through the standing-wave fringes by changing the trap fields. In this configuration, the resulting carrier and sideband coupling strengths are represented by Eqs. (1) and (2).

Figure 2

$D_{5/2}$ populations vs applied $E_y$ field measured after (a) carrier and (b) red sideband transitions. The sideband's gate beam power is 9.5 dB greater than the carrier's, and the interaction time is fixed for both cases (13 $\mu \mathrm{s}$). The overall envelope is generated by the micromotion modulation $J_0\left(\kappa \right)$ (see text). (c) The carrier and sideband populations oscillate $180°$ out of phase as the ion is transported through the standing-wave fringes. The solid lines represent a fit to the data using the model described in the text. The deviation between the data and fit at $E_y\approx 0.05\mathrm{kV}/\mathrm{m}$ may be due to scattering of the reflected light from the trap surface.

Figure 3

$D_{5/2}$ populations vs relative gate beam power measured after (a) carrier and (b) red sideband transitions. Interaction time is fixed for both cases (13 $\mu \mathrm{s}$). The populations were measured with the ion on either a node (blue circles, $E_y=-0.08$ kV/m) or an adjacent antinode (red triangles, $E_y=-0.01$ kV/m) near the rf null $[J_0\left(\kappa \right)\approx 1]$. An effective suppression of 8.4 dB for the carrier and 11 dB for the sideband is achieved. The solid lines represent a simultaneous fit to the data using the model described in the text. The results from Figs. 2 and 2 were acquired at $-13.5$ and $-4$ dB, respectively.

Figure 4

Dependence of the pseudopotential ${\mathrm{\Phi }}_{\text{pp}}$ and ion displacement $y$ on the applied field $E_y$, along with predictions from a numerical model of the trapping potentials. Deviations from the model can be attributed to the nonuniformity of the stray fields in the trapping region.