Spin-Dependent Forces on Trapped Ions for Phase-Stable Quantum Gates and Entangled States of Spin and Motion

Favored schemes for trapped-ion quantum logic gates use bichromatic laser fields to couple internal qubit states with external motion through a “spin-dependent force.” We introduce a new degree of freedom in this coupling that reduces its sensitivity to phase decoherence. We demonstrate bichromatic spin-dependent forces on a single trapped ${}^{111}\mathrm{Cd}^{+}$ ion, and show that phase coherence of the resulting entangled states of spin and motion depends critically upon the spectral arrangement of the optical fields. This applies directly to the operation of entangling gates on multiple ions.

Figure 1

Two possible Raman setups (top) to realize a bichromatic force on a hyperfine ion-qubit and corresponding optical spectrum (bottom) showing red (rsb) and blue (bsb) sideband beat notes. Two laser beams (solid lines, top) with optical carrier frequency $\nu$ generate (a) copropagating or (b) counterpropagating Raman running waves (dotted lines) at the ion with wave vectors $\Delta {\mathbf{k}}_{r,b}$. Symbols include optical phase fluctuation $\delta \varphi$, hyperfine splitting ${\omega }_{\mathrm{hf}}$, ion-trap frequency ${\omega }_z$, and force detuning $\delta$.

Figure 2

Single-ion evolution due to a spin-dependent bichromatic force. (a) Probability $P_{\downarrow }^c$ of measuring state $|\downarrow ⟩$ plotted vs force duration $\tau$. Ion is initially Doppler cooled, and data run-time averaged (100 shots/point). Solid line is a fit to Eq. (2) modified to include a linear change in peak and contrast (from spontaneous emission) and a detuning drift across the data. The fit gives $\delta /2\pi =-5.5\mathrm{kHz}$ and constrains $\overline{n}{\Omega }_{sb}^2$ to $\overline{n}=5–10$ for ${\Omega }_{sb}/2\pi =2.2–1.6\mathrm{kHz}$. (b),(c) Phase space sketches of the ion motion at points indicated in (a).

Figure 3

Probability $P_{\downarrow }^c$ plotted vs detuning $\delta$ of the bichromatic force for (a) a ground-state cooled and (b) a Doppler cooled ion, initially prepared in $|\uparrow ⟩$. The force is applied for $500\mu \mathrm{s}$. Data are run-time averaged with 100 shots/point. Solid lines show fits to Eq. (2) modified to include overall peak and contrast factors (for spontaneous emission) and a detuning drift across the data. An initial fit to (a) assuming $\overline{n}=0.05$ gives ${\Omega }_{sb}/2\pi =1.6\mathrm{kHz}$ and $\dot{\overline{n}}=0.4{\mathrm{ms}}^{-1}$. A subsequent fit to (b) assuming ${\Omega }_{sb}/2\pi =1.6\mathrm{kHz}$ gives $\overline{n}=6$ and $\dot{\overline{n}}=0.5{\mathrm{ms}}^{-1}$. The values of $\dot{\overline{n}}$ are reasonably consistent with the directly measured heating rate $0.2{\mathrm{ms}}^{-1}$ [18]. Phase space sketches (c)–(e) indicate ion evolution at detunings referenced in (a).

Figure 4

(a) Interferometric photon-echo sequence to test optical phase sensitivity of the MS force. Phase ${\varphi }_o$ and duration of spin-rotation pulses (unshaded) indicated. The MS pulse (with spin phase ${\varphi }_s$) and pulse for ac Stark shift compensation are shown shaded. (b),(c) Probability $P_{\downarrow }$ plotted vs applied shift in ${\varphi }_o$ where (b) and (c) correspond to setups in Figs. 1a, 1b respectively. Data are run-time averaged with 100 shots/point requiring $\sim 200\mathrm{ms}/\mathrm{\text{point}}$. Parameters include $\pi /2$ pulse time of $13\mu \mathrm{s}$, ${\Omega }_{sb}/2\pi \approx 2\mathrm{kHz}$, $\delta /2\pi \approx 5\mathrm{kHz}$, $\tau =90\mu \mathrm{s}$, and $\overline{n}\approx 7$. Solid lines are a sinusoidal fit (b) and a linear fit (c).