Generation of spin-motion entanglement in a trapped ion using long-wavelength radiation

Applying a magnetic-field gradient to a trapped ion allows long-wavelength radiation to produce a mechanical force on the ion's motion when internal transitions are driven. We demonstrate such a coupling using a single trapped ${}^{171}\mathrm{Yb}^{+}$ ion and use it to produce entanglement between the spin and motional state, an essential step toward using such a field gradient to implement multiqubit operations.

Figure 1

(a) Energy-level diagram of the ground state of ${}^{171}\mathrm{Yb}^{+}$ consisting of the $F=0$ state $|0\rangle$ and three $F=1$ states $|-1\rangle ,|0^{\prime }\rangle$, and $|+1\rangle$. Due to the presence of a magnetic-field gradient a single microwave field (indicated in green) can drive motional state changing transitions from $|0\rangle$ to either $|-1\rangle$ or $|+1\rangle$. A pair of microwave fields with equal but opposite detunings $\delta$ from the two motional sidebands (shown in orange) produces a spin-dependent Mølmer-Sørensen force on the ion. (b) The Mølmer-Sørensen force displaces the two orthogonal spin states (both equal superpositions of $|0\rangle$ and in this case $|+1\rangle$ with phases related to the phases of the driving fields) in opposite directions in motional phase space. The displacements describe circular arcs in phase space, with one complete circle obtained for a driving time $\tau =2\pi /\delta$.

Figure 2

Schematic diagram showing the integration of the magnets into the trap structure. The four magnets are colored blue, and the trap electrodes are colored golden. The smaller pair of magnets have holes machined in them to allow the trap's compensation electrodes to pass through them. See [21] for a full description of the trap structure.

Figure 3

The probability to drive the $|0\rangle \leftrightarrow |-1\rangle$ transition for two ions trapped in a magnetic-field gradient. The gradient causes the transition frequencies to be different, allowing for individual addressing in frequency space. The red line is a theory curve describing the sum of two transition probabilities, with the two resonant transition frequencies and the Rabi frequency as free parameters. The axial secular frequency ${\nu }_z$ was measured to be $2\pi \times 268$ kHz, implying an axial magnetic-field gradient of $23.3\left(6\right) {\mathrm{Tm}}^{-1}$. Each point is an average of 200 measurements.

Figure 4

Probability of exciting an ion to $F=1$ as a function of frequency for a 40- $\mu \mathrm{s}$ microwave pulse, clearly showing the resolved sideband structure of the transition. The sideband peaks are separated in frequency from the carrier by the secular frequency, ${\nu }_z/2\pi =268$ kHz. The carrier Rabi frequency is $2\pi \times 46$ kHz. Each point is an average of 200 measurements. The red line is a theory curve classically summing up the responses on the carrier and two sidebands with a carrier Rabi frequency of $2\pi \times 46$ kHz, with a secular frequency of ${\nu }_z/2\pi =268$ kHz, and with $\overline{n}$ as a free parameter. The thermal distribution of phonon states determined from this curve is $\overline{n}=290\left(50\right)$.

Figure 5

Probability of an ion initially in the $|0\rangle$ state being in $F=1$ after the application of a Mølmer-Sørensen force to the $|0\rangle \leftrightarrow |+1\rangle$ transition for a fixed time $\tau =180\mu \mathrm{s}$. The carrier Rabi frequency $\Omega$ was $2\pi \times 35$ kHz. Each point is an average of 200 measurements. A theory line is also shown for an initial thermal excitation $\overline{n}=110$. The data points deviate from the theory line for a detuning of $-11$ kHz due to slow drifts in the applied magnetic field that were not compensated for when taking the data. The insets show the displacement of the orthogonal spin states in motional phase space for different detunings.