Optimizing inhomogeneous spin ensembles for quantum memory

We propose a method to maximize the fidelity of quantum memory implemented by a spectrally inhomogeneous spin ensemble. The method is based on preselecting the optimal spectral portion of the ensemble by judiciously designed pulses. This leads to significant improvement of the transfer and storage of quantum information encoded in the microwave or optical field.

Figure 1

Level scheme of dopants: $|g\rangle$ and $|e\rangle$ are the ground and excited states of the spin-$\frac{1}{2}$ subspace, and $|a\rangle$ is the auxiliary metastable state. The subensemble with desired spectrum $n^{\prime }\left(\omega \right)$ is selected by the $\Omega \left(t\right)$ field, which is followed by transferring all the remaining dopants to $|a\rangle$.

Figure 2

(a) Square (blue dashed line) and optimal (black solid line) preparation pulses. (b) The corresponding selection spectra $P\left(\omega \right)$ calculated numerically. The approximate selection spectrum of Eq. (10) for the optimal pulse is also shown (red dotted line); the exact and approximate spectra differ significantly around resonance but start to coincide for $\omega T/2\pi \gtrsim 1$. (c) Second moments ${\omega }^{2}P\left(\omega \right)$ for the corresponding selection spectra.

Figure 3

Fidelity $F^{\prime }\left(\tau \right)$ for subensemble selected by square preparation pulse (blue dashed line) and by optimal pulse of Eq. (14): exact solution (black solid line) and approximate analytical solution of Eq. (15) (red dotted line). The duration of the pulses is $T=10{\tau }_{\mathrm{tr}}$. (Inset) Rabi oscillations of a single excitation between the field and the correspondingly selected subensemble.

Figure 4

(Left column) Preparation fields $\Omega \left(t\right)$ of Eq. (17), optimized for storage time ${\tau }_{\mathrm{s}}$, consisting of (a) series of instantaneous pulses $\xi \left(t\right)=\delta (t-{\tau }_{\mathrm{s}}/2)$, (b) series of short pulses of width ${\tau }_{\mathrm{s}}/5$, and (c) step function constant during each ${\tau }_{\mathrm{s}}$ interval $\xi \left(t\right)=1/{\tau }_{\mathrm{s}}$. The preparation time is $T=5{\tau }_{\mathrm{s}}$. (Right column) The resulting selection spectrum $P\left(\omega \right)$ of Eq. (18) given by the product of comb and envelope. All the spectra have the same fidelity $F^{\prime }\left({\tau }_{\mathrm{s}}\right)$ but a different number of spins $N^{\prime }\propto {\int }_0^{{\tau }_{\mathrm{s}}}{\xi }^2\left(t\right)dt$.

Figure 5

(Left column) Preparation fields $\Omega \left(t\right)$ optimized for (a) long and (b) short preparation time $T$. In (a) the dashed (red) curve is the truncated $\mathrm{sinc}\left(t\right)$ function, and the solid (black) curve is the numerically optimized pulse. In (b) the dashed (red) curve is the $\sin \left(t\right)$ function of Eq. (14) and the solid (black) curve is the numerically optimized pulse. (Right column) The resulting exact selection spectra $P\left(\omega \right)$.