Fine-Structure Qubit Encoded in Metastable Strontium Trapped in an Optical Lattice

We demonstrate coherent control of the fine-structure qubit in neutral strontium atoms. This qubit is encoded in the metastable ${}^3{P}_2$ and ${}^3{P}_0$ states, coupled by a Raman transition. Using a magnetic quadrupole transition, we demonstrate coherent state initialization of this THz qubit. We show Rabi oscillations with more than 60 coherent cycles and single-qubit rotations on the $\mathrm{\mu }\mathrm{s}$ scale. With spin echo, we demonstrate coherence times of tens of ms. Our results pave the way for fast quantum information processors and highly tunable quantum simulators with two-electron atoms.

Figure 1

Level scheme, schematic of experimental setup, and coherent state initialization. (a) Relevant ${}^{88}\mathrm{Sr}$ energy levels. The fine-structure qubit is encoded in the metastable states $|{{}^{3}P}_2,m_J=0⟩$ and $|{{}^{3}P}_0,m_J=0⟩$, which are coupled via a two-photon Raman transition with a one-photon detuning $\mathrm{\Delta }$ from $|{{}^{3}S}_1,m_J=0⟩$. The 689 and 461-nm light is used for cooling and imaging of the atoms, respectively. (b) Schematic of the experimental setup. Atoms are trapped in a horizontal (vertical) lattice formed at 914 nm (1064 nm). The magnetic bias field and the 671-nm state-preparation beam point along the $z$ axis. The Raman beams and the imaging beam propagate along the $x$ axis. (c) Coherent transfer of atoms from $|g⟩$ to $|\uparrow ⟩$ using a Landau-Zener sweep with an efficiency of up to 97.5(6)%. The transfer-laser frequency is swept over 4 kHz with the indicated ramp speeds.

Figure 2

Characterization of the $\mathrm{\Lambda }$ system. (a) Autler-Townes splitting probed on the $|\uparrow ⟩–|s⟩$ transition in the presence of a strong resonant down-laser field, as illustrated by the level scheme. The color scale indicates the number of atoms that decayed into $|g⟩$ through ${}^3{P}_1$ after excitation to $|s⟩$. The blue data points show an example of a spectrum for fixed down-laser power $P_{\downarrow }=3.6\mathrm{mW}$, which is fitted with an electromagnetically induced transparency model [31] (blue line) to extract the level splitting of $|s⟩$. The red line represents a fit of the level splittings used to calibrate the one-photon Rabi frequency of the down transition. (b) The analog calibration of the one-photon Rabi frequency of the up transition. (c) Excitation spectrum at low coupling strength with resonant up-laser field. A Lorentzian fit (solid line) to the narrow dip in the data with a full width at half maximum of $2\pi \times 0.71(19)\mathrm{kHz}$ demonstrates a large degree of coherence in our system.

Figure 3

Fine-structure qubit Rabi oscillations. (a) Measurement of the spatial distribution of the number of atoms in $|\uparrow ⟩$. (b) Spatial distribution of two-photon Rabi frequency $\mathrm{\Omega }$. The data for the Rabi oscillations are taken at the red cross. (c) Rabi oscillations with Raman lasers about $2\pi \times 6\mathrm{GHz}$ red-detuned from $|s⟩$. Blue circles (orange squares) show the number of atoms $N_{\uparrow }(t)$ and $N_{\downarrow }(t)$ in $|\uparrow ⟩$ and $|\downarrow ⟩$, respectively, both normalized with interleaved measurements of $N_{\uparrow }(0)$ and averaged over 10 experimental runs. The lines are fits with a sinusoidal oscillation. (d) A similar measurement as shown in (c), with data from individual runs of the experiment. The lines represent a damped sinusoidal-oscillation model illustrating that $\mathrm{\Omega }=2\pi \times 100.92(4)\mathrm{kHz}$ and the exponential decay time $\tau =0.68(1)\mathrm{ms}$, both found with our analysis method [22], provide a good description of the data. The error bars correspond to the standard deviation of the data in a $3\times 3\mathrm{pixel}$ region. (e) Rabi oscillations at various one-photon detunings $|\mathrm{\Delta }|/2\pi$. Upper panel: the Rabi frequency (orange squares) shows a $1/\mathrm{\Delta }$ dependence (orange line). Middle panel: The $1/e$ decay time $\tau$ of the envelope of the oscillations (blue dots) increases with $\mathrm{\Delta }$. The green stars and the solid line show the limit of the decay time given by the one-photon scattering rate. Lower panel: the number of cycles $\mathrm{\Omega }\tau /(2\pi )$ (dots) increases with $\mathrm{\Delta }$ and saturates at about 69 cycles (see text). The blue line shows the scattering-limited number of cycles inferred from the fits in the upper and middle panels.

Figure 4

Coherence measurements of the Sr fine-structure qubit. (a) Ramsey measurement. Two $\pi /2$ pulses at $|\mathrm{\Delta }|=2\pi \times 6\mathrm{GHz}$ are applied with a dark time $T$ in between. The phase of the second $\pi /2$ pulse is scanned. The resulting oscillations of the $|\uparrow ⟩$ population (dots) are fitted with a sinusoidal function (line). (b) Spin-echo measurement. A $\pi$ pulse at $T/2$ lets the spins rephase at $T$. (c) Ramsey (blue dots) and spin-echo (orange squares) measurements for various $T$. We extract the contrast from the fits in (a) and (b) and fit a Gaussian decay with $1/e$ decay times of $T_2^{*}=2.03(7)\mathrm{ms}$ (blue line) and $T_2^{\prime }=38(3)\mathrm{ms}$ (orange line) limited by the about $3\mathrm{\mu }\mathrm{K}$ deep nonmagic vertical lattice. (d) Ramsey fringes as a function of $T$ to extract the differential lattice light shift for vertical potential depths of 21(6) $\mathrm{\mu }\mathrm{K}$ (blue dots) and $52(15)\mathrm{\mu }\mathrm{K}$ (orange squares), as experienced by $|\uparrow ⟩$. The Raman lasers are tuned to a two-photon detuning of about $2\pi \times 10\mathrm{kHz}$ with respect to the free-space resonance. The frequency $f_{\text{Ramsey}}$ of the $|\uparrow ⟩$ population oscillations over the dark time $T$ is determined with a damped sinusoidal fit model (lines). (e) $f_{\text{Ramsey}}$ for different vertical potential depths of $|\uparrow ⟩$. A linear fit yields a differential light shift of $192(82)\mathrm{Hz}/\mathrm{\mu }\mathrm{K}$ between $|\uparrow ⟩$ and $|\downarrow ⟩$ at the lattice-light wavelength of 1064 nm.