Spectrum Estimation of Density Operators with Alkaline-Earth Atoms

We show that Ramsey spectroscopy of fermionic alkaline-earth atoms in a square-well trap provides an efficient and accurate estimate for the eigenspectrum of a density matrix whose $n$ copies are stored in the nuclear spins of $n$ such atoms. This spectrum estimation is enabled by the high symmetry of the interaction Hamiltonian, dictated, in turn, by the decoupling of the nuclear spin from the electrons and by the shape of the square-well trap. Practical performance of this procedure and its potential applications to quantum computing and time keeping with alkaline-earth atoms are discussed.

Figure 1

Spectrum estimation with alkaline-earth atoms. (a) $n$ copies of a $d$-dimensional density matrix $\widehat{\rho }$ are stored in the nuclear spin of $n$ fermionic alkaline-earth atoms trapped in a single square-well trap and prepared in their ground electronic state $|g⟩$. (b) A Ramsey sequence is applied consisting of two pulses of area $\beta$ and $-\beta$, respectively, coupling $|g⟩$ to the first excited electronic state $|e⟩$. (c) The number $⟨{\widehat{n}}_e⟩$ of $e$ atoms is measured for different dark times $\tau$ (red circles) between the pulses, allowing one to extract the spectrum of $\widehat{\rho }$.

Figure 2

(a) For spectrum $\overrightarrow{p}=(0.7,0.2,0.1)$ and $n=30$, we compare the true expectation value $⟨{\widehat{n}}_e(\tau ,\overrightarrow{p})⟩/n$ (solid line) with that estimated using mean-field theory (dashed line). The blue region indicates outcomes that are within 1 standard deviation of $⟨{\widehat{n}}_e(\tau ,\overrightarrow{p})⟩/n$, where the standard deviation is estimated using the mean field result Eq. (5). (b) The normalized probability distribution $\mathrm{Pr}(S|n,p)$ for measurement outcome $S$ (and the estimate $S/n+1/2$ for $p$) for $n=30$ and $n=300$ copies of $\widehat{\rho }$ with spectrum $(p,1-p)$ with $p=0.8$. (c) For $n=30$, the probability distribution is shown for different outcomes $({\lambda }_1,{\lambda }_2,n-{\lambda }_1-{\lambda }_2)$ given spectrum (0.7,0.2,0.1).

Figure 3

The Young diagrams $\overrightarrow{\lambda }=({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_d)$ for $n=6$, $d=3$. With all atoms in $|g⟩$, the interaction Hamiltonian $|G|{\widehat{H}}_D|G⟩=U\sum _{j<k}(1-{\widehat{s}}_{jk})$ has $S_n\times SU(d)$ symmetry and is therefore diagonal in $\overrightarrow{\lambda }$ subspaces. The energy in $⟨G|{\widehat{H}}_D|G⟩$ is displayed above each Young diagram. Notice two of the Young diagrams correspond to the same energy.