Weak-To-Strong Transition of Quantum Measurement in a Trapped-Ion System

Articles https://doi.org/10.1038/s41567-020-0973-y

Weak-to-strong transition of quantum measurement in a trapped-ion system

Yiming Pan 1,5 ✉ , Jie Zhang 2,3,5, Eliahu Cohen 4, Chun-wang Wu 2,3 ✉ , Ping-Xing Chen2,3 and Nir Davidson 1

Quantum measurement remains a puzzle through its stormy history from the birth of quantum mechanics to state-of-the-art quantum technologies. Two complementary measurement schemes have been widely investigated in a variety of quantum systems: von Neumann’s projective ‘strong’ measurement and Aharonov’s weak measurement. Here, we report the observation of a weak-to-strong measurement transition in a single trapped 40Ca+ ion system. The transition is realized by tuning the interaction strength between the ion’s internal electronic state and its vibrational motion, which play the roles of the measured system and the measuring pointer, respectively. By pre- and post-selecting the internal state, a pointer state composed of two of the ion’s motional wavepackets is obtained, and its central-position shift, which corresponds to the measurement outcome, demonstrates the transition from the weak-value asymptotes to the expectation-value asymptotes. Quantitatively, the Γ 2=2 weak-to-strong measurement transition is characterized by a universal transition factor eÀ , where Γ is a dimensionless parameter related to the system–apparatus coupling. This transition, which continuously connectsI weak measurements and strong measurements, may open new experimental possibilities to test quantum foundations and prompt us to re-examine and improve the measurement schemes of related quantum technologies.

o date, quantum mechanics has succeeded in describing a The weak value is defined as A f Ai^ = f i , where A^ is the h iw ¼ hjji h j i variety of physical, chemical and even biological phenom- measured observable. Note thatI the weak value is, in general, Ia com- Tena with unprecedented precision. However, fundamental plex number and can sometimes be ‘superweak’3 or ‘anomalous’4–8 challenges remain, for example, the quantum measurement prob- lying well outside the spectrum of the measured operator. This has lem is still considered to be an unsolved puzzle, persisting from led to successful demonstrations of weak-value amplification tech- the birth of quantum mechanics. The mathematical formalism of niques9–11. However, anomalous weak values demand the price of a quantum measurement was set forth by von Neumann in 19321, small success rate of post-selection p = ∣〈f∣i〉∣2 due to the approximate by treating both the measured system and measuring apparatus orthogonality of the pre- and post-selected states. Alongside prac- as being quantum and coupling them through a simple interac- tical applications, weak values and weak measurements have also tion Hamiltonian. This process, also called ‘pre-measurement’, is been linked to many fundamental topics and quantum paradoxes, unitary and is followed by a non-unitary macroscopic amplifica- in theory and experiment12–18. When the measured system and the tion that selects only a single outcome an, being an eigenvalue of measuring apparatus interact strongly, the post-selection will offer the measured operator A^ with eigenstate n . By repeating this a measurement outcome characterized by the expectation-value ji measurement procedure Iover a large ensembleI of similarly pre- asymptotics. Note that the weak value is quantitatively and concep- pared systems with state i , the expectation value of A^ can be tually different from the expectation value19,20. Consequently, the ji 2 represented as A Ii Ai^ = i i i n A , I where observation of these two values can provide a direct test of whether hii  hj ji hj i¼ njhj ij hin A n An^ = n nI . Then, in 1988, von Neumann’s measure- the performed measurement is weak or strong, especially in the hin ¼ hjjihij mentI scheme, the so-called projective (‘strong’)P measurement, was context of experimental realization19–24. extended by Aharonov, Albert and Vaidman to the weak-coupling A natural question to ask is how does the measurement outcome regime, corresponding to the situation where the measuring appa- transition from the weak value to the expectation value when the ratus interacts weakly with the measured system2. A weak mea- system–apparatus interaction is tuned from weak to strong? In this surement can only obtain a small amount of information about the work, we experimentally demonstrate the weak-to-strong mea- measured system. However, by increasing the number of measure- surement transition using a single trapped 40Ca+ ion to unify the ment trials, we can still approach 〈A〉i with any desired precision. weak-value and expectation-value predictions. The trapped-ion In ref. 2, Aharonov, Albert and Vaidman also showed that, system has been widely developed as one of the most promising by introducing a post-selection procedure, a weak measure- candidates for achieving large-scale quantum simulation and com- ment enables one to record information regarding a pre- and putation25–27. In this context, it offers us a well-designed measure- post-selected ensemble using the concept of a ‘weak value’. This pro- ment set-up with variable measurement strength. The ion’s internal cedure requires both pre-selection (that is, i , the prepared initial states and vibrational motion are identified as the measured system ji state) and post-selection (that is, f , the finalI state being strongly and measuring apparatus, respectively. Following von Neumann’s ji projected or filtered) of the measuredI system as shown in Fig. 1a. measurement scheme1,2, we introduce a generic system–apparatus

1Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel. 2Department of Physics, College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, Hunan, China. 3Interdisciplinary Center for Quantum Information, National University of Defense Technology, Changsha, Hunan, China. 4Faculty of Engineering and the Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat Gan, Israel. 5These authors contributed equally: Yiming Pan, Jie Zhang. ✉e-mail: [email protected]; [email protected]

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a b c t von Neumann’s coupling Set-up and measurement Post-selection |ƒ〉 (final state) ωz – + P ↑ 1/2 ωz Read-out Ion pointer D5/2 m

e 729 nm t AOM y s

Read-in 729 nm s t Pre-selection 397 nm i

|i〉 RSB + BSB (initial state) Qu b ↑

S1/2 System Apparatus

d e 0 Expectation-value limit 0.20

t WM 0 / γ 0.15 Overlap region 2 z 〉 θ –1 0.10

Eigenstate projection 0.05 –2 Γ↓ 0 0.20 SM Separated wavepackets –3 0.15 0.10 Cat state probability, ∣ ϕ –4 Weak-value limit 0.05 Cat state displacement, 〈 δ WVA 0 0 π π 3π π –5 0 5 8 4 8 2 Coordinate (Δz) Post-selection (θ)

Fig. 1 | The weak-to-strong quantum measurement set-up in a trapped-ion system and its weak-value to expectation-value transition. a, The measurement procedure consists of pre-selection, system–apparatus coupling and post-selection steps. b,c, The single trapped 40Ca+ ion can offer a well-designed measurement set-up with variable measurement strength with its internal electronic states and axial vibrational motion playing the roles of the measured system and measuring apparatus, respectively. The acousto-optical modulators (AOM) are used as frequency shifters and switches of the laser pulses with both the red- and blue-sideband (RSB and BSB) transitions. d, The relative spatial displacement of the ion’s motional state 〈δz〉θ/γ0t as a function of the post-selection angle θ with different Γ = γ0t/Δz. The weak-value amplification (WVA) emerges in the nearly orthogonal regime θ → 0, and the weak-coupling limit Γ → 0. e, Two spatially superposed wavepackets in the motional cat state overlap, with an interference contribution, in the weak-measurement (WM) regime (top), but are well separated in the strong-measurement (SM) regime (bottom). Gray shaded regions represent the potentials for quantizing the ion’s motional state into harmonic oscillators (that is, phonon modes).

coupling Hamiltonian H^ I γ A^^p using a bichromatic light field, the creation and annihilation operators for the vibrational motion. ¼ 0 where the measured observableI A^ and the momentum operator ^p The von Neumann coupling between the measured system and the belong to the system and the apparatus,I respectively. By tuning the measuring apparatus is realized by a bichromatic laser beam simul- system–apparatus coupling time t, we can control the measurement taneously resonant with both the red- and blue-sideband transi- 26,28 strength γ0t in our experiment. After the pre-selection step, the tions , as depicted in Fig. 1b,c (see Methods and Supplementary system–apparatus coupling, and the post-selection step, we finally Fig. A1 and Supplementary Section 2). The corresponding interac- generate a quantum superposition of two wavepackets of the appa- tion Hamiltonian in the Lamb–Dicke approximation reads ratus (Fig. 1e). Its central-position shift relative to the initial state H^ γ σ^ p^: 1 (the vibrational ground state) gives us the measurement outcome I ¼ 0 x ð Þ transitioning from the weak value to the expectation value when the measurement strength is tuned. Here σ^x is the Pauli-x operator of the qubit and ^p is the momentum I operator of the oscillator. The coupling parameter is γ0 = ηΩΔz with Set-up and modelling the Lamb–Dicke parameter η = 0.08 and the Rabi frequency Ω = 2π In our experimental demonstration (see Supplementary Section × 19 kHz. A third rapidly decaying level P1/2 (lifetime about 7.1 ns) is 1 for further details), the Zeeman sublevels S1/2(mJ = −1/2) and used for laser cooling and qubit state readout with a laser field at 397 40 + D5/2(mJ = −1/2) of a single trapped Ca ion in a magnetic field nm as shown in Fig. 1b. Also, the wavepacket and the average spatial of 5.3 G are chosen as system states and , which compose displacement of the axial motion after post-selection are measured #j i "j i a qubit with energy spacing ℏω0. The resonantI Itransition between indirectly by means of photon fluorescence detection (see Methods and , that is, the so-called carrier transition, is realized using and Supplementary Section 4). #j i "j i aI narrow-linewidthI laser at 729 nm. The ion’s axial vibrational Using Doppler cooling, resolved sideband cooling and opti- motion (along the z direction), which is treated as a quantum har- cal pumping27,28, the internal electronic state of the ion is initial- monic oscillator with a frequency of ωz = 2π × 1.41 MHz, is cho- ized in i and the axial vibrational motion is prepared in j i ¼ #j i 1 2 sen as the measuring apparatus. Its ground-state wavepacket has I 1 4 z the ground state ϕ z Δ2 exp �Δ2 . After applying the sys- _ ð Þji¼ð2π z Þ ð4 z Þ the size Δz 9:47 nm, where m is the ion’s mass. Laser ¼ 2mωz ¼ tem–apparatus interactionI in equation (1) for a controllable time fields withI frequenciesqffiffiffiffiffiffiffiffi ω0 − ωz, ω0 + ωz can induce the so-called duration t, we post-selected the qubit system in the final state red-sideband and blue-sideband transitions, which are related to f cos θ sin θ by a projective measurement (see Methods j i¼ "j i � #j i a^ h:c: and a^y h:c: respectively, where a^y and a^ are andI Supplementary Section 3). On the Bloch sphere of the qubit "jI i #h j þ "jI i #h j þ I Nature Physics | www.nature.com/naturephysics NATure PHysiCs Articles

After normalization, the final pointer state of the ion’s vibrational 1.0 1.0

z motion has a form of the cat state as Δ

/ 0.8 〉

z 0.6 π π δ θ γ θ γ 〈 sin z t cos z t 0.8 0.4 4 ϕ 0 4 ϕ 0 catθ � ð þ Þ ð þ Þj i þ ð þ Þ ð � Þj i : 5 0.2 j i¼ 1 cos 2θ ϕ z γ t ϕ z γ t ð Þ 0 � ð Þh ð þ 0 Þj ð � 0 Þi sin(2 θ ) 0.6 0 100 200 300 t 0

cos(2 θ ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi θ t (µs) 〉 By defining a dimensionless factor Γ = γ0t/Δz, that is, the ratio + γ z θ δ 〉 0.4 〈

z between the interaction strength and the measuring pointer’s width,

δ Post-selection 〈 the overlap between the two spatially separated Gaussian wavepack- = 0.05 0.2 θ = 0.1 ets of the cat state, which contributes to the quantum interference θ effect in this measurement procedure, can be quantified as θ = 0.2 0 Γ2=2 ϕ z γ t ϕ z γ t e� : 6 h ð þ 0 Þj ð � 0 Þi ¼ ð Þ 0 0.5 1.0 1.5 2.0 2.5 3.0

Γ (γ0t/Δz) Therefore, Γ can be simply regarded as the ‘interference factor’ in our measurement setting, which plays an important role in the tran- Fig. 2 | The weak-to-strong measurement transition is exactly sition from weak to strong (projective) measurement. characterized by the overlapping extent of the motional cat state’s The central-position shift of the motional cat state relative to the Γ2=2 superposed wavepackets ϕ z γ0t ϕ z γ0t e� . Based on initial ground state, corresponding to the measurement outcome, hI ð þ Þj ð � Þi ¼ can be obtained as the experimentally measurable pointer’s spatial displacement 〈δz〉θ, this transition factor can be observed indirectly for different θ using Γ2=2 catθ ^z catθ γ0t sin 2θ e� δz θ γ0t sin 2θ = δz θ cos 2θ . The purple curve is the δz : ¼ ðh i þ ð ÞÞ ðh i Þ θ hjji Γ2= 7 I h i ¼ catθ catθ ¼�1 cos 2θ e 2 ð Þ theoretical prediction. The points are experimental data. Inset: the ion’s h j i � ð Þ � axial displacement 〈δz〉θ/δz). The agreement between prediction and data indicates that the weak-to-strong measurement transition has a universal In the weak-coupling regime Γ ≪ 1, we found transition factor regardless of the post-selection procedure. The error bars δz θ Γ 0 γ0t cot θ γ0t σx W, corresponding to the h i j ! ¼� ¼ h i represent the fitting errors using the weighted fitting method. weakI value of the measured Pauli operator (equation (2)). Whereas in the strong-coupling regime Γ ≫ 1, we obtain δz θ Γ γ0t sin 2θ γ0t σx S, in accordance with the h i j !1 ¼� ð Þ¼ h i system, the vector angle between the pre- and post-selected states is expectationI value (equation (3)). The correspondence in both limits π − 2θ. Considering the pre- and post-selection performed on the is not a coincidence, it reveals that the process of quantum measure- system, we can anticipate the measurement outcome of the Pauli ment corresponds to creating entanglement between the measured operator σ^x in both the weak- and strong-coupling configurations. system and the measuring apparatus via von Neumann coupling. In the contextI of the weak-measurement regime, the weak value of The interference factor Γ affects the outcomes of the measurement, the Pauli-x observable is obtained as and as a result, determines the weak value and expectation value in a unified way (equation (7)). At the selection angle θ = π/4, we notice that 〈δz〉π/4 = −γ0t and σx W σx S, since the final apparatus state f σ^x i h i ¼h i σ cot θ: 2 reduces to a single shiftedI wavepacket ϕ z γ0t , and correspond- x W hj ji ð þ Þj i h i ¼ f i ¼� ð Þ ingly the final qubit system state, thatI is =p2, is h j i �j i ¼ ð "j i � #j i Þ one eigenstate of the Pauli-x operator. I ffi ffiffi On the other hand, in the strong-measurement regime, the expectation value is given by Weak value versus expectation value Figure 1d shows the relative spatial displacement of the ion’s f σ^x f σ hj ji sin 2θ: 3 motional state 〈δz〉θ/γ0t as a function of the post-selection angle h xiS ¼ f f ¼� ð Þ h j i θ with different Γ. As indicated by the correspondence between 〈δz〉θ/γ0t of the pointer and 〈σx〉 of the system, all the possible out- Usually, the weak value in equation (2) is not equal to the expec- comes of the measured observable, lying in between the weak-value tation value in equation (3), except for the eigenstate projection asymptotics (red dashed line) and the expectation-value asymptot- (for example, at θ = π/4). If we represent the qubit state as a vector ics (black dot-dashed line), are presented in the shaded region of on the Bloch sphere, the post-selected final state is orthogonal at Fig. 1d. When the interference factor is decreased Γ → 0, the relative angle θ = 0 and parallel at θ = π/2 to the initial state. Again, we spatial displacement shows ‘blow-up’ shift characteristics that indi- claim based on our experimental demonstration that the essential cate an anomalous weak-value amplification at the nearly orthogo- difference between the weak- and strong-measurement schemes nal post-selection regime θ → 0. This anomalous phenomenon is completely captured by the observable’s measurement out- exhibited by the weak measurement has been applied to the detec- come, being a weak value in equation (2) or an expectation value tion of extremely weak signals, such as the spin Hall effect of light9, in equation (3). and could provide practical advantages in the presence of detector To verify the above expectation and claim, the measurement pro- saturation and technical noise29,30. cess as depicted in Fig. 1a–c can be further described in an exact Figure 1e explains the measurement results in both the limits of mathematical form weak measurement and strong measurement based on the proper- ties of the cat state. Note that the overlap of the two superposed wavepackets is essential for the survival of the weak-value amplifi- i t f exp H^ I t0 dt0 i ϕ z cation. Under the condition γ0t > Δz, the overlap effect diminishes, hj ð� _ 0 ð Þ Þ ji ð Þj i and the two wavepackets are well separated, eventually resulting post selection Z pre selection 4 � von Neumann coupling � ð Þ in the expectation value. Both the interaction strength γ0t and the |{z} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} size (standard deviation) of the measuring pointer Δz are relevant sin θ cos|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}θ ϕ z γ t cos θ sin θ ϕ z γ t : ¼� þ2 ð þ 0 Þj i þ �2 ð � 0 Þj i to determine the weak-to-strong measurement transition. The ratio

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(i) 0 1.0 (iv) 0 1.0 (vi) –0.5 0 1.0

3 3 3 WF θ = 1.5, Γ = 0.1 θ = 1.5, Γ = 1 θ = 1.5, Γ = 2.9 0 0 0 –3 –3 –3 Momentum |ϕ|2 0.2 0.2 0.2

0 0 0 –5 0 5 –5 0 5 –7 0 7 Coordinate Coordinate Coordinate (ii) 0 1.0 (vii) 0 1.0

WVA (i) 3 3 θ = π / 4, Γ = 0.1 (ii) (iv) θ = π / 4, Γ = 2.9 0 (vi) 0 –3 –3 Momentum 0.2 (vii) 0.3 (iii)

0 (v) Coupling strength 0 –5 0 5 (viii) –7 0 7 Post-selection angle Coordinate Coordinate

(iii) –0.5 0 1.0 (v) –1.0 0 0.5 (viii) –1.0 0 0.5

3 θ = 0.02, Γ = 0.04 3 θ = 0.02, Γ = 1 3 θ = 0.02, Γ = 2.9 0 0 0 –3 –3 –3 Momentum 0.3 0.2 0.2

0 0 0 –5 0 5 –7 0 7 –7 0 7 Coordinate Coordinate Coordinate

Fig. 3 | The measurement regimes of 〈δz〉θ/(γ0t) in the full parameter space (Γ, θ). The probability density distributions of the typical cat states are reconstructed in the weak-measurement regime with coupling strength Γ = 0.1, 0.04 ((i)–(iii)), the intermediate-measurement regime with Γ = 1 ((iv),(v)) and the strong-measurement regime with Γ = 2.9 ((vi)–(viii)). The post-selection angles θ = 0.02, π/4, 1.5 correspond to nearly orthogonal, eigenstate projection and nearly parallel between the pre- and post-selected states, respectively. The Wigner functions (WF) are plotted based on the theoretical results (see Methods) using Δz and ℏ/2Δz as the units of coordinate and momentum. For the probability density distributions, the difference between the theoretical predictions (red curves) and the experimental data (yellow histograms) originates from the spin-state detection error and the instability of experimental parameters during the long timescale of data acquisition. between these two parameters can characterize the measurement experimentally in three ways, each corresponding to tuning one of regime adequately. the three parameters γ0, t and Δz. The parameters γ0 and t determine the coupling strength and duration between the qubit and The weak-to-strong measurement transition the ion’s axial vibrational motion, respectively. More interestingly, The weak-to-strong transition of measurement is exactly character- since Γ is inversely proportional to the wavepacket’s width, Δz deter- Γ2=2 ized by the transition factor eÀ , which quantifies the overlap of mines the sensitivity of the measuring apparatus. The decrease in I the two wavepackets and shows a general law for weak-to-strong Δz, implying the high sensitivity of the measuring apparatus and transition in different coupling regimes that were preliminarily the rapid loss of the wavepackets’ overlap, leads to easy access to investigated in previous works31–33. In experiment, this transition the strong-measurement regime. In a single-trapped-ion system, property is difficult to observe directly. However, based on the tuning the magnitude of Δz can be realized by applying the squeez- 25,38 pointer’s experimentally measurable spatial displacement 〈δz〉θ ing technique to the ion’s axial motion, as discussed in refs. . under different post-selection angles θ and ratios Γ, we can infer Nevertheless, the most straightforward approach to implement the information on the transition factor by rewriting equation (7) as weak-to-strong transition is to tune the coupling duration t as we Γ2=2 e� δz γ t sin 2θ = δz cos 2θ . Figure 2 compares did in this work. ¼ ðh iθ þ 0 ð ÞÞ ðh iθ Þ theI theoretical prediction (purple curve) and the experimental data (points) as a function of Γ ranging from the weak-coupling State reconstruction of measurement regimes regime (Γ ~ 0.02) to the strong-coupling regime (Γ ~ 3.0). The In Fig. 3, we demonstrate the measurement regimes in the full nearly perfect agreement between the theoretical prediction and parameter space (Γ, θ). The typical superposition states in the weak the experimental data indicates that the weak-to-strong transition (also called kitten state), intermediate and strong (cat state) mea- is universal for the Gaussian apparatus, regardless of the specific surement regimes are reconstructed. pre- and post-selections of the measured qubit system19. It should be Figure 3(i)–(iii) present the probability density distributions and emphasized that for other types of measuring pointer, for example, Wigner functions of the pointer states in the weak-measurement Lorentzian pointer and exponential pointer34–37, the transition fac- regime with Γ = 0.1, 0.04 and post-selection angles θ = 0.02, π/4, 1.5. tors have different mathematical forms. The three angles correspond to nearly orthogonal, eigenstate Because of the critical role of Γ = γ0t/Δz in determining the mea- projection and nearly parallel pre- and post-selected states, surement regime, the measurement transition can be implemented respectively. Note that the remarkable weak-value amplification

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Methods distribution for ^z can be extracted by using the Fourier transform of the function Interaction between the measured system and the measuring pointer. In our g k cos k^z i sin k^z , which is Ið Þ¼h ð Þi þ h ð Þi experiment, the two energy states, that is, S1/2(mJ = −1/2) and D5/2(mJ = −1/2), and 1 ikz 2 the axial motional states of a single 40Ca+ ion are chosen as the system and pointer G z g k e� dk 2πδ ^z z 2π φ z ; 13 ð Þ¼ ð Þ ¼h ð � Þi ¼ j ð Þj ð Þ degrees of freedom, respectively. Tree basic quantum operations are used in our Z �1 measurement procedure: where ∣φ(z)∣2 is the probability distribution of the measured wavepacket. _Ω iϕ iϕ H^ ^σþe car ^σ�e� car ; However, there are some problems with using this method. Instead, we employed car ¼ 2 ð þ Þ i_ηΩ iϕ iϕ the constrained least-squares optimization method to obtain the probability H^ a^^σþe red a^y^σ�e� red ; 8 red ¼ 2 ð � Þ ð Þ distribution of the wavepacket (Supplementary Fig. A2 and Supplementary Section i_ηΩ iϕ iϕ H^ blue a^y^σþe blue a^^σ�e� blue ; 4). ¼ 2 ð � Þ d ^ From equation (12) we have dk O k t 0 ^z^σy . By setting the internal state h ð Þij ¼ ¼h i ^ ^ ^ of the ion to the eigenstate of ^σy andI probing O^ k for different k, the expectation where Hcar, Hred and Hblue are the Hamiltonians of the carrier, the red sideband h ð Þi I I I value of ^z can be extracted viaI the slope of theI data, see Supplementary Fig. A3 as and the blue sideband transitions, ϕcar, ϕred and ϕblue are the corresponding laser hI i phases, η is the Lamb–Dicke parameter, Ω is the Rabi frequency, and a^y and a^ an example. are the creation and annihilation operators for the motional degree of Ifreedom respectively. A bichromatic laser field resonant with the blue and red sidebands Wigner function distribution of the cat state. In Fig. 3, we show the typical of the ion gives a combined operation H^ bic H^ red H^ blue. The interaction cat states of the vibrational motion in a phase-space representation with their ¼ þ Hamiltonian can be recast into a new formI Wigner functions. For a wavepacket φ z , its Wigner function is defined as 1 ð Þj i ipz =_ W z; p φ z z0=2 φ z z0I=2 e� 0 dz0. By substituting equation (5), ηΩ ð Þ¼2π_ ð þ Þ ð � Þ ^ _ oneI can obtain the Wigner function of the motional cat state Hbic ^σx sin ϕ ^σy cos ϕ a^y a^ cos ϕ i a^y a^ sin ϕ ; 9 R ¼ 2 ½ þ þ þ  ½�ð þ Þ � þ ð � Þ � ð Þ Γ2=2 1 2 0 W z; p 1 cos 2θ e� � sin θ π=4 Wð Þ z γ t; p ð Þ¼ð � ð Þ Þ ð þ Þ ð þ 0 Þ 14 where ϕ+ = (ϕred + ϕblue)/2 and ϕ− = (ϕred − ϕblue)/2 indicate the sum and difference 2 0 0 cos θ π=4 Wð Þ z γ t; p cos 2θ cos 2γ pt Wð Þ z; p ; ð Þ of the red-sideband phase and the blue-sideband phase. It can be seen that the þ ð þ Þ ð � 0 Þ�È ð Þ ð 0 Þ ð Þ system operator is determined by ϕ+ and the pointer operator is selected by ϕ−. 0 1 z2 =2Δ2 2Δ2p2 = 2 É ; z z _ where the Wigner function Wð Þ z p π_ e� � describes the By setting ϕ+ = π/2, ϕ− = π/2, the interaction Hamiltonian is simplified to ð Þ¼ i_ ^ay a^ initial ground state of the ion’sI axial motion in (z, p) phase space. And the spatial H^ bic ηΩΔz ^σxp^, where p^ ð � Þ, which is the von Neumann coupling we have ¼ ¼ 2Δz usedI throughout this work (equation (1)). When we choose /2, , probability density distribution of the cat state can be obtained as I ϕ+ = π ϕ− = π 2 equation (9) reduces to ∣φ(z)∣ = ∫W(z, p)dp.

_ηΩ _ηΩ=Δz H^ ^σ a^ a^y σ^ ^z: 10 Data availability bic x x ð Þ ¼ 2 ð þ Þ¼ 2 Data that support the plots within this paper and other findings of this study are The corresponding unitary evolution operator is available from the corresponding authors upon reasonable request. Source data are provided with this paper. U^ exp ik^z^σ =2 ; 11 z ¼ ð� x Þ ð Þ Acknowledgements where k = ηΩt/Δz. The evolution operator (equation (11)) was used in the reconstruction of the motional wavepacket and the measurement of ^z . This work was supported in part by DIP (German–Israeli Project Cooperation) and hI i by the I-CORE Israel Center of Research Excellence programme of the ISF and by Pre- and post-selection. In the experiment, the pre-selected state ϕ z was the Crown Photonics Center, and was also supported by the National Basic Research #j i  ð Þj i created by using the optical pumping and resolved sideband coolingI techniques. Program of China under grant no. 2016YFA0301903 and the National Natural Science The post-selected internal state can only be the dark state with the motional Foundation of China under grant nos. 61632021 and 11574398. E.C. acknowledges "j i state not destroyed, where no fluorescence is detected whenI irradiating with the support from the Israeli Innovation Authority under project no. 70002 and from the 397 nm laser field. However, an arbitrary internal state f cos θ sin θ Quantum Science and Technology Program of the Israeli Council of Higher Education. j i¼ "j i � #j i can effectively be post-selected by using a unitary rotationI Ry(2θ) before post-selecting , where Ry(2θ) indicates the unitary rotation of a single qubit with "j i Author contributions angle 2θ aroundI the y axis of the Bloch sphere. Ry(2θ) can be implemented with the Y.P. and E.C. proposed the concept and the modelling. Y.P., J.Z., C.W. and P.C. carrier transition. contributed to the theoretical analysis, design and setting up of the experiments. J.Z. and C.W. performed the atom experiment, and analysed the data together with Y.P. and E.C. Reconstruction of the motional wavepackets and measurement of z^ . To obtain h i All authors contributed to the writing and revision of the manuscript. the information about the ion’s final pointer state, the motional wavepacket’sI probability distribution should be measured. However, the only observable that Competing interests can be measured directly for the trapped ion is ^σz. With this in mind, we applied a The authors declare no competing interests. unitary operation U^ z exp ik^zσ^x=2 (equationI (11)) before measuring ^σz, the ¼ ð� Þ effective measured Iobservable will be I Additional information O^ k U^ y^σ U^ cos k^z ^σ sin k^z ^σ ; 12 ð Þ¼ z z z ¼ ð Þ z þ ð Þ y ð Þ Supplementary information is available for this paper at https://doi.org/10.1038/ s41567-020-0973-y. where k t/ . It can be seen from this formula that cos k^z and sin k^z = ηΩ Δz h ð Þi h ð Þi can be obtained by first preparing the internal state of theI ion in the eigenstateI Correspondence and requests for materials should be addressed to Y.P. or C.-w.W. of ^σz and ^σy, respectively, and then measuring O^ k . The probability Reprints and permissions information is available at www.nature.com/reprints. I I hI ð Þi

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