Observation of Quantum Interference between Separated Mechanical Oscillator Wave Packets

We directly observe the quantum interference between two well-separated trapped-ion mechanical oscillator wave packets. The superposed state is created from a spin-motion entangled state using a heralded measurement. Wave packet interference is observed through the energy eigenstate populations. We reconstruct the Wigner function of these states by introducing probe Hamiltonians which measure Fock state populations in displaced and squeezed bases. Squeezed-basis measurements with 8 dB squeezing allow the measurement of interference for $\mathrm{\Delta }\alpha =15.6$, corresponding to a distance of 240 nm between the two superposed wave packets.

Figure 1

Experimental data from measurements of $P(\downarrow ,t_p)$ using ${\widehat{H}}_r$ [${\widehat{H}}_b$ for (a)] (left panels), fitted using Eq. (3) to obtain the motional energy eigenstate populations (right panels). The data are taken by preparation of state $|{\psi }_{\mathrm{ent}}⟩$, followed by an analysis which is performed after (a) repumping the spin to $|\downarrow ⟩$, (b) detection of the ion in the state $|\uparrow ⟩$, producing the state $|{\psi }_{-}⟩$, and (c) detection of the ion in the state $|\uparrow ⟩$ after applying a spin flip using the carrier transition, which produces $|{\psi }_{+}⟩$. The vertical dashed lines indicate the revival times $t_r$ and $t_{r,\text{mix}}$. The reconstructed population distributions show the effect of the postselected measurement for the latter two cases. The blue points in the population graphs correspond to the ideal cases for $\alpha =3$. Spin populations are the result of 250 repeats of the full experimental sequence, which corresponds to roughly 125 analysis detections for the postselected cases. Error bars are estimated from quantum projection noise. The population errors are given as the standard error of the mean (SEM).

Figure 2

Comparison between the energy eigenbasis and the squeezed basis for $\alpha =4$ and $r=0.54$. The Wigner function of the cat state is overlayed by lines which indicate the maximal quasiprobability of the three closest Fock states to the mean value of the cat state in the relevant basis. This is approximately given by $\beta =(n+1/2{)}^{1/2}[\cos (\theta )e^r+i\sin (\theta )e^{-r}]$ [17]. The use of the squeezed basis reduces both the mean value and the variance, simplifying the extraction of the motional populations from the spin oscillation data.

Figure 3

Experimental data for the spin evolution $P(\downarrow ,t_p)$ obtained by probing in the squeezed basis with ${\widehat{H}}_s$ (left panels). These are fitted by using Eq. (3) to obtain the squeezed Fock state populations (right panels). The parameters for the cat size and the squeezing amplitude are given, along with the parity value obtained from the extracted populations. The blue points in the Fock state population plots are the populations obtained from a fit using a weighted mixture of $|{\psi }_{+}⟩$ and $|{\psi }_{-}⟩$. Experimental data points are an average of 750 repeats of the full experimental sequence for (a) and (b) [1000 for (c)], which corresponds to roughly 375 (500) analysis detections for the postselected cases. Error bars are estimated from quantum projection noise. Errors on population and parity estimates are given as the SEM.

Figure 4

Reconstructed Wigner functions (a) for a cat with $\alpha \simeq 2.0$, extracted using 289 settings of the displaced Fock state basis, (b) on the imaginary axis in phase space for a cat with $\alpha =4.2$, using a displaced basis, (c) for $\alpha \simeq 4.3$ performed using a displaced-squeezed basis with $r=0.5$, (d) for a cat with $\alpha \simeq 6$ performed using a displaced-squeezed basis with $r=0.8$. In (b)–(d) the fits are a theoretical form which can be derived from the Wigner function of a well-separated cat state (for further details, see the Supplemental Material [10]). Errors bars are given as the SEM. The $\mathrm{Im}(\beta )$ axis is extracted from periodic calibration scans, and we observe drifts in the scaling at the 10% level over a day. Thus there is some uncertainty in the values of $\mathrm{Im}(\beta )$ due to drifts between calibrations.