Detection-Enhanced Steady State Entanglement with Ions

Driven dissipative steady state entanglement schemes take advantage of coupling to the environment to robustly prepare highly entangled states. We present a scheme for two trapped ions to generate a maximally entangled steady state with fidelity above 0.99, appropriate for use in quantum protocols. Furthermore, we extend the scheme by introducing detection of our dissipation process, significantly enhancing the fidelity. Our scheme is robust to anomalous heating and requires no sympathetic cooling.

Figure 1

Level scheme for the model with metastable levels $|g⟩$ and $|e⟩$, and temporary (short-lived) level $|t⟩$. The state $|ab⟩$ represents the tensor product $|a⟩\otimes |b⟩$ of the individual ion states. The first motional sideband is marked by (1). Note the spontaneous emission from level $|t⟩$ marked by dotted arrows. The superpositions involving $|te⟩$ both decay to an equal classical mixture of the symmetric and antisymmetric Bell states. Motional heating and the much slower spontaneous emission from $|e⟩$ are omitted for clarity.

Figure 2

Asymptotic error ($E=1-\text{fidelity}$) for different anomalous heating rates $h_r$. Error is plotted over time for ${\mathrm{\Omega }}_r=20\mathrm{krad}/\mathrm{s}$, $\mathrm{\Omega }=26\mathrm{krad}/\mathrm{s}$, ${\gamma }_s=1{\mathrm{s}}^{-1}$ for different heating rates.

Figure 3

Asymmetric Bell state error over time for a particular trajectory of Eq. (4) for $h_r=100\mathrm{phonons}/\mathrm{s}$, ${\mathrm{\Omega }}_r=20\mathrm{krad}/\mathrm{s}$, $\mathrm{\Omega }=26\mathrm{krad}/\mathrm{s}$. The detection efficiency $\xi =10\%$.

Figure 4

For ${\mathrm{\Omega }}_r=20\mathrm{krad}/\mathrm{s}$, $\mathrm{\Omega }=26\mathrm{krad}/\mathrm{s}$, $\xi =10\%$, (a) ${\log }_{10}$ of the asymmetric Bell state infidelity (error) after a detection event, conditioned on no subsequent detections for different heating rates. Error bars mark the standard error, smaller than the data points. The dashed lines represent the steady state infidelity with no photodetector. (b) The probability of observing a single detection event and no further detections as a function of time for $h_r=1000\mathrm{phonons}/\mathrm{s}$.

Figure 5

The scheme is robust to noise: the steady state error as a function of anomalous heating rates $h_r$ and metastable excited state lifetime ${\gamma }_s$. Here ${\mathrm{\Omega }}_r=20\mathrm{krad}/\mathrm{s}$, $\mathrm{\Omega }=26\mathrm{krad}/\mathrm{s}$. The fit is a simple model based on steady state population transfer rates.

Figure 6

${\mathrm{Log}}_{10}$ of the steady state error for different energy level coupling values, $\mathrm{\Omega }$ and ${\mathrm{\Omega }}_r$. The scheme performs well, with $E<10^{-3}$ for a range of values. Low values of $\mathrm{\Omega }/{\mathrm{\Omega }}_r$ (below around 0.2) do not reach the error asymptote before our imposed time cap of 0.1 s, giving the sharp error rise in this region. High $\mathrm{\Omega }/{\mathrm{\Omega }}_r$ leads to occupation of high motional levels. We truncate at 20 motional levels and consider only parameters with population below $10^{-6}$ of the maximal level. Here ${\gamma }_s=1{\mathrm{s}}^{-1}$, and $h_r=0\mathrm{phonons}/\mathrm{s}$.