Sequential Modular Position and Momentum Measurements of a Trapped Ion Mechanical Oscillator

The noncommutativity of position and momentum observables is a hallmark feature of quantum physics. However, this incompatibility does not extend to observables that are periodic in these base variables. Such modular-variable observables have been suggested as tools for fault-tolerant quantum computing and enhanced quantum sensing. Here, we implement sequential measurements of modular variables in the oscillatory motion of a single trapped ion, using state-dependent displacements and a heralded nondestructive readout. We investigate the commutative nature of modular variable observables by demonstrating no-signaling in time between successive measurements, using a variety of input states. Employing a different periodicity, we observe signaling in time. This also requires wave-packet overlap, resulting in quantum interference that we enhance using squeezed input states. The sequential measurements allow us to extract two-time correlators for modular variables, which we use to violate a Leggett-Garg inequality. Signaling in time and Leggett-Garg inequalities serve as efficient quantum witnesses, which we probe here with a mechanical oscillator, a system that has a natural crossover from the quantum to the classical regime.

Popular Summary

At the heart of quantum theory is the fundamental impossibility of measuring both position and momentum. However, in measurements that simultaneously sample the position (or momentum) at regular intervals, this incompatibility goes away—a scenario that has not previously been explored in experiments. Doing so, however, would allow researchers to test some fundamental aspects of quantum physics and could lead to resilient approaches to quantum computing. We experimentally study these modular position and momentum measurements using a mechanical oscillator formed by a single trapped calcium ion.

Using sequences of multiple periodic position and momentum measurements, we demonstrate that varying the period controls whether one measurement disturbs the state of the next. At specific values of the period, we confirm that such measurements can avoid disturbance. Other choices for the period produce a strong disturbance that allows us to certify the quantum nature of the oscillator, which we also confirm by examining correlations between the measurements. Both methods allow us to study the crossover from quantum to classical physics using mesoscopic mechanical oscillator states. The states created during our measurements are among the most complex quantum oscillator states produced to date. They generalize Schrödinger’s famous cat to eight distinct mesoscopic states, analogous to a cat finding itself at various stages of illness rather than being alive or dead.

These modular position and momentum measurements are central components of a number of proposals for quantum computing, precision measurement, and fundamental tests with quantum harmonic oscillators. Our results provide the fundamental ingredient—measurement—that brings these possibilities into reach.

Figure 1

(a) Circuit model of the measurement type used in this paper. In general, a measurement proceeds by coupling the system of interest to a meter via a unitary transformation $\widehat{V}$ (enclosed in the dashed box). Our measurements use a qubit as the meter system, and a common coupling is given by a controlled-unitary ($\widehat{S}$). This gives a phase kickback on the meter qubit depending on the eigenvalue of the operator $\widehat{S}$. The phase change is converted to a qubit population by applying the controlled operation between two Hadamard gates ($\widehat{H}$). The action of $\widehat{V}$ may involve additional unitary transformations $\widehat{U}$, which affect the post-measurement state but not the measurement outcome. (b) In our experiments, the system of interest is a harmonic oscillator. The core element is a spin-state-dependent force (SDF), which realizes a phase-space displacement $\widehat{\mathcal{D}}(\pm \alpha /2)$, with the displacement sign dependent on the ${\widehat{\sigma }}_x$ eigenstate of the qubit. This is denoted with the diamond control. This realizes a circuit of the type shown in diagram (a) with $\widehat{S}=-\widehat{\mathcal{D}}(\alpha )$ and $\widehat{U}=\widehat{\mathcal{D}}(-\alpha /2)$. Shown as input is the Wigner function of the oscillator ground state, which is projected into a “Schrödinger’s cat” like superposition by the measurement with the relative phase of the two components correlated with the measurement outcome.

Figure 2

Dependence of SIT of measurement $A$ to measurement $B$ on (a) interference and (b) geometrical phases. Solid lines show the expectations for an ideal experimental system, and the error bars of $S$ are propagated from the shot-noise standard errors of the mean (SEM) of the directly measured probabilities $P_B(b)$, $P_A(a)$, and $P_{B|A}(b|a)$. (a) The SIT measurement settings ${\alpha }_B=3.1i$, ${\alpha }_A=3.02\approx 3\pi /|{\alpha }_B|$ are applied to squeezed input states $\widehat{S}(r)|0⟩$, where the squeezed axis is aligned with position. (b) The geometric phase is varied by sweeping the displacement amplitude ${\alpha }_A$ of measurement $A$, and using a Schrödinger’s cat input superposition $\mathbf{(}\widehat{\mathcal{D}}(-{\alpha }_B/2)+\widehat{\mathcal{D}}({\alpha }_B/2)\mathbf{)}|0⟩$ with ${\alpha }_B=i\pi$.

Figure 3

NSIT of modular position and momentum measurements for a variety of input states. Rows (i)–(iii) show NSIT of $A$ to $B$ for the measurement settings ${\alpha }_B=i\pi$, ${\alpha }_A=4.09\approx 4\pi /|{\alpha }_B|$, and input superposition states $\mathbf{(}\widehat{\mathcal{D}}(-|{\alpha }_B|e^{i{\varphi }_I}/2)+\widehat{\mathcal{D}}(|{\alpha }_B|e^{i{\varphi }_I}/2)\mathbf{)}|\varphi ⟩$ with the phase ${\varphi }_I$ varied. Here, $|\varphi ⟩$ is chosen to be (i) the ground state $|0⟩$, (ii) a squeezed vacuum state $\widehat{S}(-0.82)|0⟩$, or (iii) the first excited state $|1⟩$. We observe qualitative agreement between column (a), showing measurement of $B$ alone, and column (b), showing $B$ measured after $A$. Solid lines show the expectations for an ideal experiment, which are identical in both columns. Errors are given as SEM and propagation of SEM. The 150 measured $S$ values are quantified in red in histogram (c) and compared to a theoretical histogram for the SIT settings ${\alpha }_A=3$ and ${\alpha }_B=i\pi$ using the same set of input states (see main text). (d) Theoretical Wigner function plots of the input state ($|\varphi ⟩=|0⟩$ and ${\varphi }_I=1.22\mathrm{rad}$) as well as its post-measurement states with result $+1$ during the experimental sequence. The red circles show the locations of the multiple displaced coherent states; their radius denotes the r.m.s. wave-packet size.

Figure 4

Two time correlators and violation of a Leggett-Garg inequality. (a) Two-time correlation measurement in the asymmetric implementation performed on an initial ground-state cooled ion as a function of the second displacement ${\alpha }_B$. The fixed experimental settings were ${\alpha }_A=2.1$, ${\varphi }_A=0$, ${\varphi }_B=\pi /2$. The full data set is shown in the false-color plot to the right. Two cuts through this data set with fixed $|{\alpha }_B|$ are indicated in this false-color plot and explicitly plotted to the left, where solid lines show the expectations for an ideal experiment. (b) Violation of a Leggett-Garg inequality for increased modular measurement displacements $|\alpha |$ and three different initial temperatures. Solid lines show the expected violation including simulated qubit and motional dephasing as well as phase calibration errors (see Appendix pp14); the dashed lines instead are the exceptions for an ideal experiment. Violations are observed over a wide range of $\alpha$ for all investigated temperatures. With a ground-state cooled ion, violations are observed up to $\alpha =3$. The points highlighted with a diamond violate the LGI when being penalized by the built-in theoretical amount of SIT (Appendix pp15). The discrepancy between the data and the simulation at $\overline{n}=0.42$ is due to additional experimental fluctuations in the preparation of this higher thermal occupation. All error bars are propagated from the SEM errors because of quantum projection noise.

Figure 5

Calibrations. Panel (a) shows a calibrated SDF pulse applied to a ground-state cooled ion and a fit to the expected behavior $P(\downarrow )=\frac{1}{2}(1+e^{-2(c{t}_{\mathrm{SDF}}{)}^2})$. From this fit, we extract the proportionality constant $c$ between displacement size and $t_{\mathrm{SDF}}$, which we use in all analytic calculations of expected measurement results. Panel (b) shows the phase matching between addressing transition 2 with single frequency or with a bichromatic pulse. The sequence consists of a single $\pi /2$ pulse followed by a SDF pulse, here with $t_{\mathrm{SDF}}=200\mu \mathrm{s}$. Whenever the $\pi /2$ pulse phase $\phi$ is matched to the SDF, no superposition will be created; instead, the full wave packet is displaced, and therefore, $P(\downarrow )\equiv 1$ for any SDF duration. Finally, we need to calibrate the phase of the $R_1(\varphi )$ pulse using an experimental calibration, as shown in panel (c). We use a sequence of two modular measurements with the same duration and opposite SDF directions in order to be able to observe a signal. We sweep both phases ${\varphi }_A$ and ${\varphi }_B$ simultaneously and read out the relevant minimum.

Figure 6

Additional SIT and NSIT data compared to observable commutation. Columns (a) and (b) show the direct-detection data obtained by first measuring $A$ and then $B$. The blue line corresponds to $a=+1$, while purple stands for $a=-1$. Column (c) shows the result of measuring $B$ alone. Here, $P_B$ is given as a single point plotted at $\arg ({\alpha }_A)=2.5$, and it is compared to $P_{B(A)}$, which is calculated from the results of columns (a) and (b). The blue line is for $b=+1$, while purple is for $b=-1$. Finally, the classical fidelity $\kappa \equiv \sum _b\sqrt{P_B(b)P_{B(A)}(b)}$ [35] is shown in column (d). The classical fidelity can be used as an alternative to $S$ in order to quantify the amount of SIT. For NSIT measurements, $\kappa =1$. The black vertical lines in column (d) indicate settings for which the underlying observables commute. (i) Asymmetric implementation, SIT as a function of $\arg ({\alpha }_A)$, with $|{\alpha }_A|=\sqrt{\pi }$, ${\alpha }_I=\sqrt{\pi }$, ${\alpha }_B=-\sqrt{\pi }$, ${\varphi }_A={\varphi }_B=0$. The data vary between SIT and NSIT, where NSIT is observed for the settings where the observables commute. (ii) The same experiment with the symmetric implementation, ${\varphi }_A={\varphi }_B=\pi$. We observe that this implementation can be SIT even in cases where the observables commute. (iii) SIT as a function of ${\alpha }_A$ in the symmetric implementation with ${\alpha }_B={\alpha }_I\approx \sqrt{\pi }$. Again, SIT is observed at points where the observables commute.

Figure 7

Theoretical Wigner function plots of two more examples of input states for the measurements presented in Fig. 3 of the main text. The displayed input states are $\mathbf{(}\widehat{\mathcal{D}}(-|{\alpha }_B|e^{i{\varphi }_I}/2)+\widehat{\mathcal{D}}(|{\alpha }_B|e^{i{\varphi }_I}/2)\mathbf{)}|\varphi ⟩$ with ${\varphi }_I=1.22\mathrm{rad}$ and (a) $|\varphi ⟩=|1⟩$ and (b) $\widehat{S}(-0.82)|0⟩$. Further, their post-measurement states with the result $+1$ during the experimental sequence are shown.

Figure 8

Two-time correlation measurement in the asymmetric implementation. The fixed experimental settings were ${\alpha }_A=2.1$, ${\varphi }_A=0$, ${\varphi }_B=\pi /2$. The blue points show $a=+1$ measurement results, while purple points shows $a=-1$. Solid lines show the expectations for an ideal experiment. Error bars of $P_A(a),P_{B|A}(+1|a)$ are given as SEM, from these the error bars of $C_{AB}$ are propagated. The data sets shown here together create the false-color plot of Fig. 4 in the main text.

Figure 9

Expected results for a Ramsey measurement coupling to $x(t)=A\cos (2\pi ft)$, a classical single-frequency noise source. The amplitude $A$ of this single-frequency noise fluctuates on time scales slower than an experimental shot with a Gaussian probability distribution $P(A)=[1/(\sigma \sqrt{2\pi })]e^{-\frac{1}{2}[(A-A_0)/\sigma {]}^2}$. The coupling constant is assumed to be 1: $\widehat{H}=|\uparrow ⟩⟨\uparrow |x(t)$. Thus, we find $⟨\widehat{Q}⟩=⟨P(+1)-P(-1)⟩=-e^{\frac{1}{2}\{[\sin (2\pi fT)\sigma ]/2\pi f\}}\cos \mathbf{(}\varphi +[A_0/(2\pi f)]\sin (2\pi fT)\mathbf{)}$, with $T$ given by the Ramsey wait time and $\varphi$ the second Ramsey pulse phase. We show three different amplitudes $A_0$: (a) $A_0=8000$, (b) $A_0=5000$, and (c) $A_0=2000$, of the noise, with the noise frequency fixed to 50 Hz, $\sigma =1000$, $\varphi =0$. The oscillations resemble characteristic traces of Schrödinger cat states, such as those found in Ref. [28].

Figure 10

A sample of individual detection data and measurement settings for the Leggett-Garg violations presented in Fig. 4 of the main text. The data are from the set $\overline{n}\approx 0$ and $\overline{n}\approx 0.23$. Data are shown as red points, while the expectations for an ideal experiment are shown as bars. Blue bars are the detection data, and the three red bars show the correlations calculated from these detections. We see qualitatively good agreement for smaller $\alpha$, which decreases for higher displacements, mainly due to dephasing noise in the experimental system. The settings ${\theta }_1$ and ${\theta }_2$ correspond to ${\theta }_1={\theta }_B-{\theta }_A$, ${\theta }_2={\theta }_C-{\theta }_B$.

Figure 11

Penalization of $L$ for built-in two-time SIT. (a) Analytic calculation of the total amount of SIT $TS=2(|{\tilde{S}}_{AB}|+|{\tilde{S}}_{BC}|+|{\tilde{S}}_{AC}|)$ due to incompatible settings used to violate the LGI. The $TS$ approaches zero for the larger displacement $\alpha$. (b) The measured data penalized by the theoretical amount of built-in SIT. Several points around $\alpha =2.25$ are still able to violate the LGI in this penalized fashion. (c) ${\tilde{S}}_{AB}$ extracted from the experimental data. The amount of SIT is higher than expected from the analytic calculation, which predicts values up to 0.02. But the values stay close to zero. Note that the total SIT is dominated by $|{\tilde{S}}_{BC}|$ and $|{\tilde{S}}_{AC}|$.

Figure 12

Theoretical comparison of a LGI experiment using a ground-state cooled or a squeezed oscillator state. Results are presented for a squeezing parameter $r\approx 0.9$. (a) Analytic calculation of the achievable $L$ values. (b) Simulated $L$ values including the same motional dephasing and linewidths as in the simulations in the main part of the paper Fig. 4. (c) Analytic calculation of the amount of $TS$ present during the measurements.