Cavity-enhanced optical readout and control of nuclear spin qubits

Abstract

Their exceptional coherence makes nuclear spins in solids a prime candidate for quantum memories in quantum networks and repeaters. Still, the direct all-optical initialization, coherent control, and readout of individual nuclear spin qubits have been an outstanding challenge. Here, this is achieved by embedding 167-Er dopants in yttrium orthosilicate in a cryogenic Fabry-Perot cavity, whose linewidth of 65 MHz is much smaller than the 0.9 GHz separation of neighboring hyperfine levels. Frequency-selective emission enhancement thus enables a single-shot readout fidelity of 91(2)%. Furthermore, a large magnetic field freezes paramagnetic impurities, leading to coherence times exceeding 0.2 s. The combination of nuclear-spin qubits with frequency-multiplexed addressing and lifetime-limited photon emission in the minimal-loss telecommunications C-band establishes 167-Er as a leading platform for long-range, fiber-based quantum networks.

Figures (4)

• Figure 1: Concept of the experiment. a, Left panel: Quantum bits are encoded in the nuclear spin (gold) of $^{167}\textrm{Er}$ dopants in a crystal. At cryogenic temperatures of $<2K$, the electronic spin of the emitters (red) can be aligned, e.g., by optical pumping, eliminating its precession. However, the electronic spins of other dopants and paramagnetic impurities (orange, green) ---potentially unknown and/or without optical transitions--- would be random. Their precession (black arrows) and flipping lead to fluctuating magnetic fields and thus to decoherence. Right panel: When a field of several Tesla is applied, all spins align, which eliminates magnetic field fluctuations even at substantial impurity concentrations. b,$^{167}\textrm{Er}$ dopants exhibit an optical transition at a wavelength of 1536.4nm (green arrow) between their optical ground $|g\rangle$ and excited $|e\rangle$ state manifolds. Each of them comprises eight nuclear spin levels $|m_I\rangle$, whose energy is shifted by the nuclear Zeeman and hyperfine interactions, such that neighboring levels are separated by up to $0.9GHz$ at the applied magnetic field. Using an optical resonator, the nuclear-spin-preserving optical transitions (green arrows) can be enhanced selectively, without altering the decays that increase (red dotted) or reduce (blue dotted) $m_I$ by one quantum. c, Selective enhancement of photon (green curly arrow) emission on the $\Delta{m_I}=0$ transitions requires a resonator with a small mode volume and a very narrow linewidth, $\ll 0.9GHz$. This is achieved by integrating a $10.9µm$ thin membrane of the YSO host crystal (yellow) containing the erbium dopants (red spin symbols) into the optical mode (green) of a frequency-tunable (grey arrow) Fabry-Perot resonator (blue cylinders). d, Via the Purcell effect, the decay rate into the cavity mode on the transition $|-7/2\rangle_e \rightarrow |-7/2\rangle_g$ (green line) is enhanced up to a factor of $P=95(10)$, while detuned optical transitions can exhibit several orders of magnitude lower relative Purcell factors $P/P_\textrm{max}$. The colored areas indicate the spectral bands containing the spin-preserving (light green) and spin-flip (blue and red) transitions.
• Figure 2: Spectroscopy of single erbium dopants. In pulsed resonant fluorescence spectroscopy, many peaks are observed that originate from individual erbium dopants (blue bars), both with and without hyperfine structure. The shaded gray area below 0.36arb. u. indicates the detector dark counts. When applying an optical pumping sequence to initialize an $^{167}\textrm{Er}$ dopant into the $|-7/2\rangle_g$ state, it exhibits a stronger fluorescence intensity on the respective spin-preserving transition $|-7/2\rangle_e \rightarrow |-7/2\rangle_g$ (orange bar). Initialization in the other spin states (not shown) allows assigning all eight spin-preserving transitions of this dopant, highlighted in green in the fluorescence trace and marked by green dots below. A steady increase in frequency is observed from $|-7/2\rangle_e \rightarrow |-7/2\rangle_g$ to $|+7/2\rangle_e \rightarrow |+7/2\rangle_g$ (left to right).
• Figure 3: Optical nuclear-spin qubit readout. a, Inset: The qubit is encoded in the two lowest-energy states $|-7/2\rangle_g$ (orange) and $|-5/2\rangle_g$ (blue) of the ground state $|g\rangle$, and the resonator is tuned to selectively enhance the optical readout transition $|-7/2\rangle_g \leftrightarrow |-7/2\rangle_e$ (green). After initialization, a readout is performed by measuring the fluorescence photons (green curly arrow) after resonant laser excitation (straight arrow). Main panel: After 110 readout pulses, the number of detector photons differs depending on the qubit initialization. The distributions are clearly separated. When assigning the $|-7/2\rangle_g$ nuclear spin state in case at least $n=5$ photons are detected, an average readout fidelity of $91(2)\,\%$ is achieved. This value is limited by photons detected when the spin is prepared in $|-5/2\rangle_g$, with a distribution (blue) that is dominated by detector dark counts that are observed even without excitation laser pulses (black open bars). b, The achieved fidelity depends on the number of readout pulses and the threshold photon number $n$ (dotted line: 4, solid red line: 5, dashed line: 6). The optimal value (black circle) is found at 110 pulses and $n=5$ (as shown in panel a).
• Figure 4: All-optical coherent control and hyperfine coherence. a, For coherent control of the nuclear-spin qubit (gray box), the resonator is temporarily detuned by $\Delta_\textrm{C}=-400MHz$ from the $|-7/2\rangle_g\leftrightarrow |-7/2\rangle_e$ frequency. Then, Raman transitions are driven using square laser pulses with Rabi frequencies $\Omega_1$ and $\Omega_2$ that are irradiated simultaneously with a frequency difference that matches the qubit transition when $\delta=0$. Scattering is avoided by choosing a detuning of $\Delta = -90MHz$ from the readout transition. b, When varying the Raman pulse duration, Rabi oscillations are observed at $\delta = 0$. A cosine function with a stretched-exponential envelope (solid line) fits the data well. c, In a Hahn-Echo experiment, consisting of a pulse sequence of $\pi/2 - \pi - \pi/2$ (see inset), a coherence time of $T_\mathrm{Hahn}=14.8(9)ms$ is obtained from a fit to a stretched-exponential function (solid line). d, The coherence time can be further extended by dynamical decoupling, in which $N$ equidistant $\pi$-pulses are applied sequentially. To reduce the influence of pulse errors, the phase of the pulses is altered between rotations around the $X$ and $Y$ axes, forming an XY$(N)$ sequence. With this, the coherence time increases with the number of pulses (purple data) according to $T_\mathrm{DD} \propto N^{0.82(2)}$ (black fit curve) up to a value of $T_\mathrm{DD}=0.28(8)\s$. The dotted gray line shows the expected increase for a slowly varying spin bath, $\propto N^{2/3}$.