Fault-tolerant Hahn-Ramsey interferometry with pulse sequences of alternating detuning

A scheme for efficient correction of driving-field frequency drifts in Ramsey interferometry is proposed. The two off-resonant $\pi /2$ pulses of duration $T$ used in the traditional Ramsey setup are supplemented with an additional pulse of duration $2T$ (approximate $\pi$ pulse), which is applied midway between the Ramsey pulses and has a detuning of opposite sign to theirs. This scheme, which resembles a Hahn's spin-echo pulse embedded into the Ramsey setup, corrects small-to-moderate random errors in the detuning of the driving field. This allows the observation of Ramsey fringes of high contrast even with a noisy driving field or in inhomogeneously broadened atomic ensembles. The contrast is further improved by replacing the refocusing $2T$ pulse by a composite $\pi$ pulse. We demonstrate the validity of the concept by comparing experimental results from usual Ramsey measurements with Hahn-Ramsey measurements. These experimental results are obtained from microwave-optical double-resonance spectroscopy on ${}^{171}\mathrm{Yb}{}^{+}$ ions in a segmented linear Paul trap. In the same way, we verify qualitatively the predicted advantage from using a composite $\pi$ pulse for refocusing.

Figure 1

Pulses in the traditional Ramsey scheme (top, two $\pi /2$ pulses of duration $T)$ and the fault-tolerant Hahn-Ramsey scheme proposed here (bottom, three pulses of durations $T,2T$, and $T)$. The detuning $\Delta$ is kept constant in the Ramsey scheme, while in the fault-tolerant Hahn-Ramsey scheme, its sign is flipped with the application of the intermediate $2T$ pulse, and then restored to its original value in the last $\pi /2$ pulse.

Figure 2

Ramsey signal produced by a standard two-pulse Ramsey setup. Full curve, no error, $\epsilon =0$; dotted curves, signals for detunings $\Delta$ and $-\Delta$ and error $\epsilon =0.05\Delta$; dashed curve, average of the dotted curves. The Rabi frequency is $\Omega =5\Delta$.

Figure 3

Fault-tolerant Hahn-Ramsey fringes for $\Omega =3\Delta$. Top frame: Excited-state population. Solid curve, no detuning error, $\epsilon =0$; dashed curve, $\epsilon =0.1\Delta$; dotted curve, $\epsilon =0.2\Delta$. Bottom frame: Deviation from the errorless signal. Solid curves, the second term in the approximation (12); dashed and dotted curves, exact values.

Figure 4

As Fig. 3 but for Rabi frequency $\Omega =10\Delta$. The error does not exceed 0.01.

Figure 5

Numerical comparison of the classical Ramsey scheme, the Ramsey scheme with two superposed signals for opposite detunings $\Delta$ and $-\Delta$, and the fault-tolerant Hahn-Ramsey scheme proposed here in the presence of random detuning errors. Each point is simulated by taking a random detuning error from the interval $(-{\epsilon }_0,{\epsilon }_0)$. The Ramsey fringes (top curve) and the superposed Ramsey fringes (second from top) decohere very fast even though the maximum error is rather small, ${\epsilon }_0=0.05\Delta$. The three lower curves are calculated for much larger maximum detuning errors ${\epsilon }_0=0.3\Delta$, and for three different Rabi frequencies (denoted on each curve). The solid sinusoids refer to the errorless case $\left({\epsilon }_0=0\right)$.

Figure 6

Numerical comparison of the fault-tolerant Hahn-Ramsey scheme proposed here, in the presence of random detuning errors. Each point is simulated by taking a random detuning error from the interval $(-{\epsilon }_0,{\epsilon }_0)$, with ${\epsilon }_0=0.3\Delta$. The Rabi frequency is $\Omega =3\Delta$. Top curve, three pulses of the same Rabi frequency but different durations, $T\text{−}2T\text{−}T$. Middle curve, three pulses of the same duration $T$ but different Rabi frequencies, $\Omega \text{−}2\Omega \text{−}\Omega$. Bottom curve, as the top curve, but with the middle pulse split into two pulses, each of duration $T$, the second of which is phase shifted by a phase $\varphi =2\chi$. The solid sinusoids refer to the errorless case $\left({\epsilon }_0=0\right)$.

Figure 7

Relevant levels and transitions for laser cooling and qubit preparation, manipulation, and readout of a ${}^{171}\mathrm{Yb}{}^{+}$ ion. Doppler cooling is applied on the $|S_{1/2},F=1\rangle \leftrightarrow |P_{1/2},F=0\rangle$ transition. An additional microwave field near 12.6 GHz prevents optical pumping to the $|S_{1/2},F=0\rangle$ state via off-resonant excitation of the $|P_{1/2},F=1\rangle$ state. For preparation, the cooling laser is shifted by 2.1 GHz with an AOM to resonantly pump the ion to the $|S_{1/2},F=0\rangle$ state and the microwave is switched off. Coherent manipulation can now be carried out with the microwave field near 12.6 GHz, which can be tuned by choice to a magnetic-field-sensitive or insensitive transition. Finally, the state is read out by detecting resonance fluorescence from the ion excited by the cooling laser with the microwave field switched off. Additional lasers counteracting optical pumping to metastable states are omitted for clarity.

Figure 8

Histogram of the count distribution for detection after Hahn-Ramsey spectroscopy. The gray shaded area between the two thresholds (red vertical lines) indicates the discarded events.

Figure 9

(a) Comparison of standard Ramsey spectroscopy (upper graph) with Hahn-Ramsey spectroscopy (lower graph). All data was taken with the parameters $\Omega /2\pi =53.46\left(10\right)$ kHz, $\Delta =4$ kHz, and 100 repetitions. Projection errors differ from point to point, and two example error bars are indicated for points with large and small variance, respectively. (b) Scaled deviations $[y_i-f(\vec{p},t_i)]/{\sigma }_i$ from a sinusoidal least-squares fit $f(\vec{p},t_i)$ to the data with the set of fitting parameters $\vec{p}$, the standard errors ${\sigma }_i$ calculated from projection errors, and the time $t$. The residuals for the standard Ramsey sequence grow beyond $\pm 30\sigma$ and the measured data points become uncorrelated with a sinusoidal curve for large precession times, whereas the residuals for the Hahn-Ramsey sequence in contrast remain roughly within a $\pm 3\sigma$ interval and the data remain nicely sinusoidal for the measured interval.

Figure 10

(a) Comparison of a nonrobust (upper graph) and a robust (lower graph) refocusing pulse in Hahn-Ramsey spectroscopy with precession times ranging from 2.75 to 3.55 ms. The effect of robust pulses can be observed best for longer precession times where decoherence already sets in and reduces the visible fringe contrast. As the difference even for those cases is not very pronounced, we aimed for a high signal-to-noise ratio to pinpoint the difference and used 1000 repetitions. The modulation depth for the composite $\pi$ pulse is 0.808(13) and substantially higher than 0.709(15) obtained for the simple Hahn $\pi$ pulse. (b) Scaled deviations $[y_i-f(\vec{p},t_i)]/{\sigma }_i$ from a sinusoidal least-squares fit $f(\vec{p},t_i)$ to the data with the set of fitting parameters $\vec{p}$, the standard errors ${\sigma }_i$ calculated from projection and detection errors, and the time $t$. The mean quadratic scaled deviation for the simple Hahn $\pi$ pulse is $5.5{\sigma }^2$ and about 30% larger than the value for the composite Hahn $\pi$ pulse which is $4.3{\sigma }^2$.

Figure 11

Comparison of (a) the four-pulse and (b) the three-pulse Ramsey-Bordé sequences with (c) the present Hahn-Ramsey sequence. The arrows indicate the spatial direction of each pulse and the expressions beneath them show the total frequency offset from resonance due to the driving field detuning $\Delta$, the driving field frequency drift $\epsilon$, and the Doppler shift $\delta$. The application of the pulses in opposite directions in the four-pulse Ramsey-Bordé scheme eliminates the first-order Doppler shift $\delta$ in atomic ensembles, but the frequency drift $\epsilon$ cannot be canceled because it adds to the detuning $\Delta$ in the same manner and hence it cannot be separated from $\Delta$. In the three-pulse Ramsey-Bordé scheme (b) the drift $\epsilon$ is eliminated altogether with the Doppler shift $\delta$ and the detuning $\Delta$, and hence no dependence on $\Delta$ is left in the signal. In the present Hahn-Ramsey scheme (c) both the drift $\epsilon$ and the Doppler shift $\delta$ can be canceled if all the pulses are copropagating because they add up differently to the detuning $\Delta$.