Control of inhomogeneous atomic ensembles of hyperfine qudits

We study the ability to control $d$-dimensional quantum systems (qudits) encoded in the hyperfine spin of alkali-metal atoms through the application of radio- and microwave-frequency magnetic fields in the presence of inhomogeneities in amplitude and detuning. Such a capability is essential to the design of robust pulses that mitigate the effects of experimental uncertainty and also for application to tomographic addressing of particular members of an extended ensemble. We study the problem of preparing an arbitrary state in the Hilbert space from an initial fiducial state. We prove that inhomogeneous control of qudit ensembles is possible based on a semianalytic protocol that synthesizes the target through a sequence of alternating rf and microwave-driven SU(2) rotations in overlapping irreducible subspaces. Several examples of robust control are studied, and the semianalytic protocol is compared to a brute-force, full numerical search. For small inhomogeneities, $<$$1\%$, both approaches achieve average fidelities greater than 0.99, but the brute-force approach performs superiorly, reaching high fidelities in shorter times and capable of handling inhomogeneities well beyond experimental uncertainty. The full numerical search is also applied to tomographic addressing whereby two different nonclassical states of the spin are produced in two halves of the ensemble.

Figure 1

Radio-frequency (rf) magnetic fields (in red) and microwave magnetic fields (in blue) control the ground-electronic hyperfine manifold of magnetic sublevels in ${}^{133}$Cs. SU(2) rotations in the irreducible subspaces, $F=3,4$, are driven by rf-magnetic fields. Because of the sign difference of the $g$ factors for the upper and lower manifold, the two rotations are in opposite directions. The microwave field is shown to be resonant only with the transition between the states $|F=4,m_F=4\rangle$ and $|F=3,m_F=3\rangle$ due to the application of a bias magnetic field that breaks the degeneracy between the magnetic sublevels. Microwave-driven SU(2) rotations on this qubit, together with the rf control, can generate an arbitrary unitary transformation on the full 16-dimensional manifold.

Figure 2

Average fidelity vs inhomogeneity for (a) semianalytic state preparation and (b) fully numerical state preparation. The $x$ axis represents the fractional variation in each parameter for the three cases: power inhomogeneity in rf and micorwave, ${\epsilon }_{\mathrm{rf},\mathrm{mw}}$, and detuning as a fraction of ${\Omega }_{\mathrm{mw}}$. We show the performance as a function of one inhomogeneity and averaged over the the other two. Each curve is an average of 20 random states in the 8D Hilbert space. In the semianalytic case, each wave form consists of a series of alternating rf and microwave SU(2) pulses for a total of 3.98 ms. The optimized pulses are found through a numerical search of the time varying amplitude and phase of the relevant fields. In the fully numerical case, rf and microwave fields are applied simultaneously for a total of 1 ms. The microwave and rf powers are set to their maximum values, and a we perform a numerical optimization of the phases. For both the semianalytic and the fully numerical case, the wave form consisted of a series of piecewise constant 125 $\mu$s pulses.

Figure 3

(a) Fidelity of fully numerical state preparation vs 5$\%$ errors in ${\epsilon }_{\mathrm{rf}}$, ${\epsilon }_{\mathrm{mw}}$, and $\Delta /{\Omega }_{\mathrm{rf}}$. The $x$ axis represents the fractional variation in the each parameter for the three cases: power inhomogeneity in rf and microwave, ${\epsilon }_{\mathrm{rf},\mathrm{mw}}$, and detuning as a fraction of ${\Omega }_{\mathrm{mw}}$. We show the performance as a function of one inhomogeneity and averaged over the the other two. Each curve is an average of 20 random states in the 8D Hilbert space. (b) Fidelity of fully numerical state preparation vs 10$\%$ errors in $\Delta /{\Omega }_{\mathrm{rf}}$.

Figure 4

Spatially selective control. (a) Plot of the spatial variation in the detuning, $\Delta$, imposed by a light-shift gradient. The gradient separates the gas into two spatial regions in order to generate two distinct target states: For $x<0.5$ mm, the target state is $|{\psi }_1\rangle =\frac{1}{\sqrt{2}}(|F=3,m_F=-3\rangle +|F=3,m_F=3\rangle )$ and for $x>0.5$ mm the target state is $|{\psi }_2\rangle =|F=3,m_F=0\rangle$. Because a perfectly sharp transition between the two detunings is not possible in the laboratory, there is a transition region around $x=0.5$ mm. The microwave and rf fields are applied simultaneously with maximum power and numerically optimized phases. (b) Results of the synthesized states. The left axis shows $\langle F_z^2\rangle$ vs position and the right axis shows $\langle F_z\rangle$ vs position. If the target states are synthesized perfectly, then $\langle {\psi }_1|F_z|{\psi }_1\rangle =\langle {\psi }_2|F_z|{\psi }_2\rangle =0$ and $\langle {\psi }_1|F_z^2|{\psi }_1\rangle =9$ while $\langle {\psi }_2|F_z^2|{\psi }_2\rangle =0$. With the exception of the transition region at $x=0.5$ mm, these goals are achieved.