Designer Spin Pseudomolecule Implemented with Trapped Ions in a Magnetic Gradient

We report on the experimental investigation of an individual pseudomolecule using trapped ions with adjustable magnetically induced $J$-type coupling between spin states. Resonances of individual spins are well separated and are addressed with high fidelity. Quantum gates are carried out using microwave radiation in the presence of thermal excitation of the pseudomolecule’s vibrations. Demonstrating controlled-NOT gates between non-nearest neighbors serves as a proof-of-principle of a quantum bus employing a spin chain. Combining advantageous features of nuclear magnetic resonance experiments and trapped ions, respectively, opens up a new avenue toward scalable quantum information processing.

Figure 1

Individual addressing of spins. (a) Spatially resolved resonance fluorescence (near 369 nm) of three ${}^{171}\mathrm{Yb}^{+}$ ions recorded with an intensified CCD camera is shown. Neighboring ions are separated by $11.9\mu \mathrm{m}$. (b) Microwave-optical double resonance spectrum of the above ions. The spectrum was recorded by applying a microwave frequency pulse of $8\mu \mathrm{s}$ to the ions initially prepared in the state $|\downarrow ⟩$. The probability to find an ion in $|\uparrow ⟩$ was determined from counting resonance fluorescence photons while probing with laser light near 369 nm. In a magnetic gradient of $\approx 18.2\mathrm{T}/\mathrm{m}$, the qubit transitions $|\downarrow ⟩\leftrightarrow |\uparrow ⟩$ of different ions are nondegenerate. The solid line is a fit to the data. Each data point accounts for 50 repetitions. Two points with error bars are displayed representing the typical statistical standard deviations.

Figure 2

$J$-type coupling of a three-spin pseudomolecule. In (a), the table lists the measured $J$-type coupling constants (below the diagonal) for a three-spin pseudomolecule together with the resonance frequencies of the microwave transitions (on the diagonal). For the nonuniform magnetic gradient present in our setup ($b_1=16.8\mathrm{T}/\mathrm{m}$, $b_2=18.7\mathrm{T}/\mathrm{m}$, $b_3=18.9\mathrm{T}/\mathrm{m}$) and the axial trap frequency (${\nu }_1=2\pi \times 123.5\mathrm{kHz}$), the calculated values are $J_{12}=2\pi \times 32.9\mathrm{Hz}$, $J_{23}=2\pi \times 37.0\mathrm{Hz}$, $J_{13}=2\pi \times 23.9\mathrm{Hz}$. (b) Dependence of the coupling strength on the trapping potential. For a pseudomolecule consisting of two ions, the coupling strength $J$ has been measured for varying c.m. frequency ${\nu }_1$. A calculated curve for a uniform magnetic gradient of $19\mathrm{T}/\mathrm{m}$ is represented by a solid line. These measurements demonstrate how $J$-type coupling can be varied by adjusting the trapping potential [16, 17].

Figure 3

Conditional quantum dynamics and CNOT gate between non-neighboring ions. The probability to find the target spin (ion 3) in state $|\uparrow ⟩$ at the end of a Ramsey-type sequence is shown as a function of the phase $\varphi$ of the second $\pi /2$ pulse applied to the target. The inset shows the input state prepared before the Ramsey experiment. In (a) the target is initially prepared in $|\uparrow ⟩$ while in (b) it is prepared in $|\downarrow ⟩$. $J$-type coupling between the control qubit (ion 1) and the target qubit (ion 3) produces a phase shift on the target population as a function of the control qubit’s state (different shades of gray). A free evolution time of $\tau =11\mathrm{ms}$ yields a phase shift of approximately $\pi$ (3.1(2) radians) between the pairwise displayed curves. For $\varphi =3\pi /2$, a CNOT gate results. The middle ion, prepared in state $|\downarrow ⟩$, does not interact with other ions. Each data point represents 50 repetitions, the error bars correspond to mean standard deviations, and solid lines are fits to the data.

Figure 4

Parity signal $\Pi (\varphi )$ showing the quantum nature of the observed correlations of ion 1 and 3. The Bell state $|{\psi }_B⟩=\frac{1}{\sqrt{2}}(|{\downarrow }_C{\downarrow }_2{\downarrow }_T⟩+e^{i\alpha }|{\uparrow }_C{\downarrow }_2{\uparrow }_T⟩)$ is the result of a CNOT operation applied to the superposition input state $|{\psi }_i⟩=\frac{1}{\sqrt{2}}(|{\downarrow }_C{\downarrow }_2{\downarrow }_T⟩+|{\uparrow }_C{\downarrow }_2{\downarrow }_T⟩)$. In order to measure the correlations along different bases [10], microwave $\pi /2$ pulses with phase $\varphi$ are applied to both ions followed by state-selective detection resulting in $\Pi (\varphi )$ oscillating as $\cos (2\varphi )$. The fidelity of creating the bipartite entangled state $|{\psi }_B⟩$ is evaluated as $F=0.57(4)$ (see text). Each data point represents 50 repetitions and error bars indicate 1 standard deviation.