Long-lived qubit from three spin-$\frac{1}{2}$ atoms

A system of three spin-$\frac{1}{2}$ atoms allows the construction of a reference-frame-free (RFF) qubit in the subspace with total angular momentum $j=1/2$. The RFF qubit stays coherent perfectly as long as the spins of the three atoms are affected homogeneously. The inhomogeneous evolution of the atoms causes decoherence, but this decoherence can be suppressed efficiently by applying a bias magnetic field of modest strength perpendicular to the plane of the atoms. The resulting lifetime of the RFF qubit can be many days, making RFF qubits of this kind promising candidates for quantum information storage units. Specifically, we examine the situation of three ${}^6\text{Li}$ atoms trapped in a ${\text{CO}}_2$-laser-generated optical lattice and find that, with conservatively estimated parameters, a stored qubit maintains a fidelity of 0.9999 for two hours.

Figure 1

Three ${}^6\text{Li}$ atoms are trapped at the corners of an equilateral triangle. The probability clouds indicate the center-of-mass distributions, the spread $w$ of which is about 1/12 of $a$, the distance between the atoms.

Figure 2

Ground-state hyperfine levels of the neutral ${}^6\text{Li}$ atom. The $f=3/2$ quartet is separated from the $f=1/2$ doublet by a transition frequency of $228.2\hspace{0.5em}\mathrm{MHz}$. Three ${}^6\text{Li}$ atoms confined to their $f=1/2$ ground states serve as the spin-$\frac{1}{2}$ particles from which the RFF qubit is constructed.

Figure 3

Level scheme for the effective three-atom Hamiltonian of Eq. (32). The separations are not drawn to scale: $\hslash {\omega }_0$ is many orders of magnitude larger than $\hslash \Omega$. The two $j=1/2$ levels are degenerate doublets; this degeneracy is exploited for the encoding of the robust signal qubit.

Figure 4

Comparison of the data from a numerical simulation with the analytical results of Eqs. (56). Curve “a” displays ${\langle P_{j=1/2}\rangle }_t$, curves “b” show ${\langle {\Sigma }_1\rangle }_t$ for ${\langle {\Sigma }_1\rangle }_0=1$ (or ${\langle {\Sigma }_2\rangle }_t$ for ${\langle {\Sigma }_2\rangle }_0=1$), and curve “c” is for ${\langle {\Sigma }_3\rangle }_t$ with ${\langle {\Sigma }_3\rangle }_0=1$. The crosses are from a simulation of the dynamics, averaged over 1000 runs. The solid-line curves represent the analytical results of Eqs. (56). The dotted “b” curve shows what one would get for ${\langle {\Sigma }_1\rangle }_t$ if $\Omega$ vanished rather than being large on the scale set by $\tau$; we observe that the interatomic dipole-dipole interaction accelerates the decay of ${\langle {\Sigma }_1\rangle }_t$. For the parameter values used throughout the paper, the time range is $3\times {10}^{10}$ s (roughly 1000 years); see Eq. (45) for the value of $\tau$.

Figure 5

Fidelity of the RFF qubit. For ${\Omega }_2/{\Omega }_1={10}^{-4}$, the plots show $F(t)$ of Eq. (90) and its lower bound of Eq. (91) for $t<45\times 2\pi /{\Omega }_1$ (top plot), for $t<150\times 2\pi /{\Omega }_1$ (inset in the top plot), and for $t<2\pi /{\Omega }_2$ (bottom plot). Curve “a” is the lower bound on $F(t)$; the other three curves are for $\langle {\Sigma }_1^{(\phi )}{\rangle }_0=0.4$ and $s(0)=1$ (curve “b”), $s(0)=0.8$ (curve “c”), and $s(0)=0.6$ (curve “d”). One can clearly see the small-amplitude short-period oscillations and the large-amplitude long-period oscillation. For the parameter values used throughout the paper, the respective time ranges are 2, 7, and 450 h. We have ${\Omega }_1\approx \Omega$, with $\Omega =2\pi \times 6$ mHz given in Eq. (30).

Figure 6

RFF qubit constructed from four spin-$\frac{1}{2}$ atoms. (a) Two-dimensional square configuration. The dipole-dipole interaction is unavoidably unbalanced here because the distance for the two diagonal pairs is larger than the distance for the four edge pairs. (b) Three-dimensional pyramidal configuration. Here, if the height $h$ is chosen such that $b/a=0.661$, the effective dipole-dipole interaction has equal strength for all six pairs of atoms.

Figure 7

Lower bounds for the fidelity $F(t)$ (solid curve) and the expectation value ${\langle P_{j=0}\rangle }_t$ (dashed curve) for the RFF qubit constructed from four spin-$\frac{1}{2}$ atoms at the corners of a perfect square. The fidelity of Fig. 5 (three atoms with nonideal geometry) would be indiscernible from $F(t)=1$ here.

Figure 8

Six coplanar laser beams consist of two sets of three coherent beams; the angle between beams within each set is $2\pi /3$. The respective wave vectors have lengths $\left|{\mathit{k}}_1\right|=\left|{\mathit{k}}_2\right|=\left|{\mathit{k}}_3\right|=2\pi /\lambda$ and $\left|{\mathit{k}}_4\right|=\left|{\mathit{k}}_5\right|=\left|{\mathit{k}}_6\right|=2\pi /{\lambda }^{\prime }$. Different lattice structures can be created by alternating the phases of the laser beams. For the lattice of our design, we keep the set of beams with wavelength $\lambda$ to be in phase and the phases for the other three beams with wavelength ${\lambda }^{\prime }$ are $2\pi /3$, $0$, and $-2\pi /3$.

Figure 9

The top figure shows the contour plot of the overall optical potential, and the three global minima at each lattice site of the big triangular lattice indicate where the trio of atoms can be trapped. The bottom left figure gives a more detailed view of the site where one trio is trapped. The dashed curve in the bottom right figure is a plot of the potential along the vertical line cutting through the global minima $a$ and $b$; and the solid curve is a plot of the potential along the horizontal line cutting through the other global minima $c$. The two cuts intersect at the saddle point $s$.