Quantum control of the hyperfine-coupled electron and nuclear spins in alkali-metal atoms

We study quantum control of the full hyperfine manifold in the ground-electronic state of alkali-metal atoms based on applied radio frequency and microwave fields. Such interactions should allow essentially decoherence-free dynamics and the application of techniques for robust control developed for NMR spectroscopy. We establish the conditions under which the system is controllable in the sense that one can generate an arbitrary unitary map on the system. We apply this to the case of ${}^{133}\mathrm{Cs}$ with its $d=16$ dimensional Hilbert space of magnetic sublevels in the $6S_{1∕2}$ state, and design control wave forms that generate an arbitrary target state from an initial fiducial state. We develop a generalized Wigner function representation for this space consisting of the direct sum of two irreducible representations of SU(2), allowing us to visualize these states. The performance of different control scenarios is evaluated based on the ability to generate a high-fidelity operation in an allotted time with the available resources. We find good operating points commensurate with modest laboratory requirements.

Figure 1

(Color online) The ground state hyperfine manifold of ${}^{133}\mathrm{Cs}$. Radio-frequency magnetic fields (rf, in red) lead to independent rotations in the two manifolds. Microwaves ($\mu \mathrm{w}$, in blue) are the generators of rotation in a two-dimensional subspace between states in the two manifolds, here the stretched state transition $\mid 4,4⟩\to \mid 3,3⟩$.

Figure 2

(Color online) Table exploring controllability of the system for a variety of configurations: one microwave field driven on different two-level transitions, $\mid F=3,M⟩\to \mid F=4,M^{\prime }⟩$, amplitude and/or phase control, one or two sets of orthogonal rf coils (rf polarizations), and resonant vs detuned fields. The different configurations yield one of four different outcomes: (green circle) all microwave transitions provide full controllability, (yellow square) all transitions but the clock transition $\mid 3,0⟩\to \mid 4,0⟩$ provide full controllability, (orange pentagon) only the transitions of the form $\mid 3,\pm 3⟩\to \mid 4,\pm 4⟩$ and $\mid 3,\pm 3⟩\to \mid 4,\pm 2⟩$ provide full controllability, and (red octagon) no transitions yield controllable Hamiltonian dynamics. In this calculation we consider all valid microwave transitions that can be selected with polarization and/or frequency.

Figure 3

(Color online) Sample evolutions that result from our state preparation algorithm. Starting in the spin coherent state ∣4, 4⟩, we simulate preparation of the state $\frac{1}{\sqrt{2}}(\mid 4,4⟩+\mid 3,-3⟩)$ and obtain a fidelity 0.993. We control the amplitudes and phases of rf coils in both the $x$ and $y$ directions, as well as the amplitude and phase of a resonant microwave that couples the states $\mid 4,-4⟩$ and $\mid 3,-3⟩$. We show the Cartesian components of the three control fields ($\Omega \cos \varphi$ and $\Omega \sin \varphi$) over the entire state preparation time of $150\mu \mathrm{s}$. We show snapshots of the evolved state at five different times, identified as times (0)–(4). Two different representations of the state are shown: bar charts of the absolute values of the density matrix elements, and a generalized spherical Wigner function. The spheres on the diagonal represent the Wigner functions in the irreducible subspaces $F^{\pm }$ and the off-diagonal spheres represent the coherences between the manifolds. For details see Appendix .

Figure 4

(Color online) Same as Fig. 3, preparing the state $\frac{1}{\sqrt{2}}\mid 4,4⟩+\frac{1}{2}(\mid 3,3⟩+\mid 3,-3⟩)$ with a fidelity of 0.995.

Figure 5

(Color online) Plots of the average fidelity of state preparation for different control configurations and total preparation times. Each point represents the fidelity averaged over a set of ten states randomly chosen from the Harr measure. For each state and configuration, the gradient search was performed with 20 random seeds and we chose the protocol that generated the highest fidelity.

Figure 6

(Color online) Representations of states with bar charts of the absolute values of the density matrix elements. (a.i) is a spin coherent state ${\mid \psi ⟩}_{ai}=\mid 4,4⟩$ and (a.ii) is a superposition of two oppositely oriented spin coherent states, one for each of the two manifolds, ${\mid \psi ⟩}_{aii}=\frac{1}{\sqrt{2}}(\mid 4,4⟩+\mid 3,-3⟩\phantom{)}$. In (b.i), (b.ii) we show the effects of rotations on a superposition of spin squeezed states, each determined as the ground state of $F_z^2-F_y$ in the respective irreducible manifold. Finally, in (c.i) we have a coherent superposition of the state ∣4,0⟩ and a cat state $\frac{1}{\sqrt{2}}(\mid 3,3⟩+\mid 3,-3⟩)$ and in (c.ii) we have an incoherent mixture of those two states.

Figure 7

(Color online) Representations of the six states shown in Fig. 6 by the generalized spherical Wigner functions. Each state is represented by four spheres. The spheres on the diagonal are the standard SU(2) Wigner functions in the $F=4$ (upper diagonal) and $F=3$ (lower diagonal) irreducible subspaces. The radius of these spheres, ranging from 0 to 1, determines the total population in that subspace. The off-diagonal spheres represent the coherences between the two subspaces (see text).