Coherence-preserving cooling of nuclear-spin qubits in a weak magnetic field

Nuclear-spin memories of divalent neutral atoms can allow spin-preserving resolved-sideband cooling in a strong magnetic field [I. Reichenbach and I. H. Deutsch, Phys. Rev. Lett. 99, 123001 (2007)]. We present a theory for cooling ${}^{87}\mathrm{Sr}$ nuclear-spin qubits in a weak magnetic field. The theory depends on laser excitation of $5s5p{}^{3}P_1$ to a nearby state which results in $m_J$-dependent AC Stark shifts large compared to the hyperfine interaction. This effectively suppresses the nuclear-spin mixing due to the hyperfine interaction. Sideband cooling via the clock state quenched by the AC Stark-shifted ${}^{3}P_1$ state leads to nuclear-spin-preserving spontaneous emission back to the ground state. More than being compatible with low magnetic fields, the theory is applicable when the nuclear-spin qubits are defined by the two lowest Zeeman substates.

Figure 1

Cooling scheme illustrated with (a) atomic levels and (b) vibrational states; the inset of (b) shows a simplified process about removing one vibrational quantum number. Altogether five energy levels are involved in the nuclear-spin-preserving sideband cooling of ${}^{87}\mathrm{Sr}$ atoms, namely, the ground state, the clock state, $5s5p{}^1{P}_1$, $5s6s{}^1{S}_0$, and $5s15d{}^1{D}_2$. The directions of the arrows do not indicate polarization of photons. Cooling starts from a narrow-line laser excitation of the clock transition from the ground state to $5s5p{}^3{P}_0$ when the vibrational quantum number reduces by one. A two-photon transition via an intermediate state (not shown here) transfers the state $5s5p{}^3{P}_0$ to $5s5p{}^1{P}_1$ which decays back to the ground state rapidly. The hyperfine interaction in $5s5p{}^1{P}_1$ mixes nuclear spins by nature; by inducing a transition between $5s5p{}^1{P}_1$ and $5s6s{}^1{S}_0$ with a strong Rabi frequency which is large compared to the hyperfine interaction, the nuclear-spin mixing is suppressed, so that polarization resolution is removed in the spontaneous emission to the ground state. A small diagonal-hyperfine-interaction-induced energy difference between the two nuclear-spin states is compensated by the AC Stark shift via off-resonantly exciting $5s5p{}^1{P}_1$ to $5s15d{}^1{D}_2$, which removes the frequency resolution in the spontaneous emission.

Figure 2

State dynamics in the cooling simulated by the master equation in Eq. (8) with $({\mathrm{\Omega }}_{\text{eff}},{\mathrm{\Omega }}_{\text{pd}},{\mathrm{\Omega }}_{\text{ps}},{\mathrm{\Delta }}_{\text{pd}},\mathrm{\Delta },{\mathrm{\Gamma }}_{\text{p}},{\mathrm{\Gamma }}_{\text{s}},{\mathrm{\Gamma }}_{\text{d}})/2\pi =(1,144.27,300,-1700,3.8826,32,3,0.47)$ MHz, $B=1$ G, and initial state $|{\psi }_0\rangle$. The hyperfine constants $(A,Q)/2\pi$ are $(-3.4,39)$ MHz for $5s5p{}^1{P}_1$ and $(-194,-75)\mathrm{MHz}$ for $5s15d{}^1{D}_2$. (a) The solid, long-dashed, dash-dotted, short-dashed, and dotted curves show the populations of ${log}_{10}$ scale in the states $|\perp \rangle =\left(\right|\left[5s^2{}^{1}S_0\right]0,\uparrow \rangle -|\left[5s^2{}^{1}S_0\right]0,\downarrow \rangle )\otimes |n\rangle /\sqrt{2}$, $|\mathcal{A}\rangle$, $5s5P{}^{1}P_1$, $5s15d{}^{1}D_2$, and $5s6s{}^{1}S_0$, respectively. $|\perp \rangle$ is the state perpendicular to the final target state $|{\psi }_{\text{f}}\rangle =\left(\right|\left[5s^2{}^{1}S_0\right]0,\uparrow \rangle +|\left[5s^2{}^{1}S_0\right]0,\downarrow \rangle )\otimes |n\rangle /\sqrt{2}$. The final populations in $|\perp \rangle$ and $|\mathcal{A}\rangle$ are $1.0\times {10}^{-4}$ and $2.9\times {10}^{-4}$, respectively. (b) Evolution of the population of the states $|{\psi }_0\rangle =\left(\right|\left[5s5p{}^{3}P_0\right]0,\uparrow \rangle +|\left[5s5p{}^{3}P_0\right]0,\downarrow \rangle )\otimes |n\rangle /\sqrt{2}$ and $|{\psi }_{\text{f}}\rangle$. The populations in $|{\psi }_0\rangle$ and $|{\psi }_{\text{f}}\rangle$ at 20 $\mu \mathrm{s}$ are $7\times {10}^{-6}$ and 0.9996, respectively, and the main population loss is in the reservoir state and the state perpendicular to $|{\psi }_{\text{f}}\rangle$ as shown in (a).