Four- and six-photon stimulated Raman transitions for coherent qubit and qudit operations

Abstract

We experimentally demonstrate transitions between electronic angular momentum states with a difference in magnetic quantum numbers $Δ\mathrm{m_J} = $ 3, 4, and 5 via resonant four- and six-photon stimulated Raman transitions in a single trapped atom. Derivation of the corresponding Rabi frequencies, which are verified experimentally, follows the standard treatment of two-photon transitions including the adiabatic elimination of intermediate states. Finally, we discuss pathways to increase the observed multi-photon transition fidelities to $>99.99\%$, providing a tool for efficient, high-fidelity control of qu\textit{d}its and single-atom logical qubits.

Figures (19)

• Figure 1: a) Illustration of four-photon transition pathways between $\ket{0}\equiv\ket{\mathrm{m_J} = +5/2}$ and $\ket{3}\equiv\ket{\mathrm{m_J} = -1/2}$ in the $\mathrm{D_{5/2}}$ manifold of $^{40}$Ca$^+$. They are driven by two 976 nm beams with $\sigma^-$ (red) and equal components of $\sigma^+$, $\pi$ and $\sigma^-$ (blue) polarizations and a relative detuning $\omega_r$. b) Six-photon transition between $\ket{0}$ and $\ket{4}$. Solid and transparent lines represent (three of the five) interfering pathways that drive the multi-photon transitions, with solid lines highlighted as an example pathway. Energies are not to scale.
• Figure 2: a) (Top) Rabi-flopping driven by the resonant four-photon process displayed in Fig. \ref{['fig:paths_flopping']}a. (Bottom) Rabi flopping driven by the six-photon resonant transition shown in Fig. \ref{['fig:paths_flopping']}b. b) Two- (gray), four- (blue) and six-photon (red) transition Rabi frequencies as a function of $R_\perp$ beam power, with $R_\parallel$ held fixed at 195 mW. Blue and red solid lines correspond to the predictions of Equations \ref{['4photon_rabi']} and \ref{['6photon_rabi']}, respectively, plus corrections for $\mathrm{F}$-state couplings and polarization impurities (see SM \ref{['appendix:calibrations']} and \ref{['appendix:fstatecoupling']}). Shaded regions represent 68% confidence intervals from fits to beam intensities and polarizations. Dashed lines are predictions from numerical simulations. c) Fractional residuals of data and analytic predictions (colored curves) to numeric predictions of the Rabi frequencies.
• Figure 3: a). Four-photon $\pi$ pulse transfer infidelity at different four-photon $\pi$ times, corresponding to different $R_\perp$ beam powers, with expected contributions due to intermediate state population (solid red), spin decoherence (solid blue), and spontaneous Raman scatter error (solid green). Solid black line represents the sum of each expected contribution. Dashed blue line is projected spin dephasing infidelity with coherence demonstrated in coherenceImprovement. Dashed and dotted red lines are the numerically-simulated maximum leakage to intermediate states with the $R_\perp$ beam amplitude ramped with $\sin^2$ and $\sin^4$ envelopes respectively. b). Two-beam pulse envelopes with $R_\perp$ amplitude without pulse shaping (black) and shaped with a $\sin^2$ ramp (cyan). c). Numerically simulated resonant population dynamics driven by pulses shown in b). d). Power spectral density of pulses in b).
• Figure 4: a). Rabi spectroscopy of transitions within $\mathrm{D_{5/2}}$ with $\Delta \mathrm{m_J} \geq 3$ as driven by four (circles) and six (squares) photon processes. b). Diagram illustrating full direct unitary connectivity in the $\mathrm{D_{5/2}}$ manifold enabled by four- and six-photon transitions.
• Figure 5: a) Geometry of 976 nm Raman beams used to drive four- and six-photon transitions relative to quantization magnetic field direction. b) Relevant atomic structure of $^{40}$Ca$^+$ used in this work. 397 nm beam drives fluorescent $S_{1/2}\leftrightarrow P_{1/2}$ cycling transition with 866 nm beam to depump the $D_{3/2}$ manifold. 729 nm beam drives $\pi$ pulses between $S$ and $m_J$ qudit states encoded in $D_{5/2}$ for state-preparation and readout of the qudit states. An optional RF $\pi$ pulse between $S_{1/2}$ ground states is used to maximize the 729 transition Rabi frequency to each qudit state.