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ARTICLE OPEN Proposal to test quantum - superposition on massive mechanical resonators

Wei Qin1,2, Adam Miranowicz 1,3, Guilu Long4,5,6,J.Q.You2,7 and Franco Nori 1,8

We present and analyze a proposal for a macroscopic quantum delayed-choice experiment with massive mechanical resonators. In our approach, the electronic of a single nitrogen-vacancy impurity is employed to control the coherent coupling between the mechanical modes of two carbon nanotubes. We demonstrate that a mechanical can be in a coherent superposition of wave and particle, thus exhibiting both behaviors at the same . We also discuss the mechanical noise tolerable in our proposal and predict a critical below which the morphing between wave and particle states can be effectively observed in the presence of environment-induced fluctuations. Furthermore, we describe how to amplify single-phonon excitations of the mechanical-resonator superposition states to a macroscopic level, via squeezing the mechanical modes. This approach corresponds to the -covariant cloning. Therefore, our proposal can serve as a test of macroscopic quantum superpositions of massive objects even with large excitations. This work, which describes a fundamental test of the limits of quantum at the macroscopic scale, would have implications for quantum and processing. npj Quantum Information (2019) 5:58 ; https://doi.org/10.1038/s41534-019-0172-9

INTRODUCTION superconducting circuits.24,25 However, all these experiments Wave-particle duality lies at the heart of quantum . were performed essentially at the microscopic scale. According to Bohr’s complementarity principle,1 a quantum Here, as a step in the macroscopic test for a coherent wave- system may behave either as a wave or as a particle depending particle superposition on massive objects, we propose and analyze on the measurement apparatus, and both behaviors are never an approach for a mechanical quantum delayed-choice experi- observed simultaneously. This can be well demonstrated via a ment. Mechanical systems are not only being explored now for 26,27 single Mach–Zehnder interferometer, as depicted in Fig. 1a. potential quantum technologies, but they also have been considered as a promising candidate to test fundamental An incident photon is split, at an input beam-splitter BS1, into an 28 equal superposition of being in the upper and lower paths. This is principles in quantum theory. In this manuscript, we demon- followed by a phase shift ϕ in the upper path. At the output beam- strate that, similar to a single photon, the mechanical phonon can be prepared in a of both a wave and a splitter BS2, the paths are recombined and the detection particle. The basic idea is to use a single nitrogen-vacancy (NV) probability in the detector D1 or D2 depends on the phase ϕ, center in diamond to control the coherent coupling between two heralding the wave nature of a single photon. If, however, BS2 is 29,30 absent, the photon is detected with probability 1/2 in each separated carbon nanotubes (CNTs). We focus on the = detector, and thus, shows its particle nature. In Wheeler’s delayed- electronic of the NV center, which is a spin S 1 fi ≃ π choice experiment,2,3 the decision of whether or not to insert BS triplet with a zero- eld splitting D 2 × 2.87 GHz between spin 2 〉 〉 〉 is randomly made after a photon is already inside the states |0 and |±1 [see Fig. 1b]. If the spin is in |0 , the mechanical interferometer. The arrangement rules out a hidden-variable modes are decoupled, and otherwise are coupled. Moreover, the theory, which suggests that the photon may determine, in mechanical noise tolerated by our proposal is evaluated and we advance, which behavior, wave or particle, to exhibit through a show a critical temperature, below which the coherent signal is hidden variable.4–11 Recently, a quantum delayed-choice experi- resolved. ment, where BS2 is engineered to be in a quantum superposition of being present and absent, has been proposed.12 Such a version allows a single system to be in a quantum superposition of a wave RESULTS and a particle, so that both behaviors can be observed in a single Physical model measurement apparatus at the same time.13,14 This extends the We consider a hybrid system31,32 consisting of two (labelled as conventional boundary of Bohr’s complementarity principle. The k = 1, 2) parallel CNTs and an NV electronic spin, as illustrated in quantum delayed-choice experiment has already been implemen- Fig. 1c. The CNTs, both suspended along the ^x-direction, carry dc 15–17 18–23 ted in nuclear magnetic resonance, optics, and currents I1 and I2, respectively, while the spin is placed between

1Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan; 2Quantum Physics and Quantum Information Division, Beijing Computational Science Research Center, 100193 Beijing, China; 3Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland; 4State Key Laboratory of Low- Dimensional Quantum Physics and Department of Physics, Tsinghua University, 100084 Beijing, China; 5Beijing Academy of Quantum Information Science, 100193 Beijing, China; 6Beijing National Research Center for Information Science and Technology, 100084 Beijing, China; 7Department of Physics and State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, 310027 Hangzhou, China and 8Department of Physics, The University of Michigan, Ann Arbor, MI 48109-1040, USA Correspondence: J. Q. You ([email protected]) or Franco Nori ([email protected]) Received: 20 December 2018 Accepted: 17 June 2019

Published in partnership with The University of New South Wales W. Qin et al. 2 field

BxðÞ¼t B0 cosðÞω0t ; (3) ^ with B0 and ω0, along the x-direction, to drive the |0〉 → |±1〉 transitions with Rabi frequency μ g B Ω ¼ Bpsffiffiffi 0 : (4) 2 2h We apply a static magnetic field X k ; Bz ¼ ðÞÀ1 dkGk (5) k¼1;2 along the ^z-direction to eliminate the Zeeman splitting between the spin states |±1〉.36 This causes the same Zeeman shift, 3Ω2 Δ ¼ ΔÀ þ ; (6) Δþ

where Δ± = D ± ω0, to be imprinted on |±1〉, and a coherent 2 coupling, of strength Ω /Δ+, between them, as shown in Fig. 1b. We can, thus, introduce a dark state pffiffiffi jDi¼ðÞ jþ1iÀjÀ1i = 2; (7) and a bright state Fig. 1 a Demonstration of the wave-particle duality using a pffiffiffi Mach–Zehnder interferometer. A single photon is first split at the jBi¼ðÞ jþ1iþjÀ1i = 2; (8) input beam-splitter BS , then undergoes a phase shift ϕ and finally is 1 ≃ Ω2 Δ 〉 observed at detectors D1 and D2. The photon behaves as a wave if with an splitting 2 / +. In this case, the spin state |0 is the output beam-splitter BS2 is inserted, or as a particle if BS2 is decoupled from the dark state, and is dressed by the bright state. 1234567890():,; removed. In quantum delayed-choice experiments, BS2 is set in a Under the assumption of Ω=Δ  1, the dressing will only increase quantum superposition of being present and absent, and conse- the energy splitting between the dark and bright states to quently, the photon can simultaneously exhibit its wave and particle  nature. b Level structure of the driven NV spin in the electronic 2 1 1 ωq ’ 2Ω þ : (9) ground state. Here we have assumed that the Zeeman splitting Δ Δþ between the spin states |±1〉 is eliminated by applying an external field. c Schematic representation of a mechanical quantum delayed- This yields a spin with |D〉 as the ground state and |B〉 as the choice experiment with an NV electronic spin and two CNTs. The exited state. The spin-CNT coupling Hamiltonian is accordingly mechanical of the CNTs are completely decoupled or transformed to coherently coupled, depending, respectively, on whether or not the X 〉 Hint ’ hg kσxqk; intermediate spin is in the spin state |0 , with the dc current Ik (10) k 1;2 through the kth CNT, and the distance dk between the spin and the ¼ kth CNT σ = σ + σ σ = 〉〈 σ σy where x + −, with − |D B| and þ ¼ À. When we further restrict our discussion to a dispersive regime ωq ± ωm jgkj, the spin qubit becomes a quantum data bus, them, at a distance d1 from the first CNT and at a distance d2 from ^ allowing for mechanical excitations to be exchanged between the the CNT. When vibrating along the y-direction, the CNTs CNTs. By using a time-averaging treatment,38,39 the unitary can parametrically modulate the Zeeman splitting of the dynamics of the system is then described by an effective intermediate spin through the magnetic field, yielding a magnetic 33–37 Hamiltonian (see Supplementary Section 1 for a detailed coupling to the spin. For simplicity, below we assume that the derivation), H = H ⊗ σ , where CNTs are identical such that they have the same vibrational eff "#cnt z X  frequency ωm and the same vibrational mass m. The mechanical 2hω H ¼ q g2by b þ g g b by þ H:c: ; (11) vibrations are modelled by quantized harmonic oscillators with a cnt ω2 À ω2 k k k 1 2 1 2 Hamiltonian q m k¼1;2 X and σ = |B〉〈B| − |D〉〈D|. The Hamiltonian H includes a coherent H ¼ hω by b ; z cnt mv m k k (1) spin-mediated CNT–CNT coupling in the beam-splitter form, ; k¼1 2 which is conditioned on the spin state. Here, we neglect the y – where bk (bk) denotes the phonon annihilation (creation) . direct CNT CNT coupling much smaller than the spin-mediated The Hamiltonian characterizing the coupling of the mechanical coupling, as is described in Supplementary Section 1. Further- modes to the spin is more, we find that the decoupling of one CNT from the spin gives X rise to a spin-induced shift of the vibrational resonance of the Hint ¼ hg kSzqk; (2) other CNT. Hence, the dynamics described by Heff can be used to k¼1;2 implement controlled Hadamard and phase gates.

where Sz = |+1〉〈+1| − |−1〉〈−1| is the z-component of the spin, y qk ¼ bk þ bk represents the canonical phonon position operator, Quantum delayed-choice experiment with mechanical resonators = μ ħ and gk BgsyzpGk/ refers to the Zeeman shift corresponding to Let us first discuss the Hadamard gate. Having Ik = I and dk = d = ħ ω 1/2 μ the zero-point yzp [ /(2m m)] . Here, BÀÁis the Bohr gives a symmetric coupling gk = g, and a mechanical beam-splitter ≃ μ = π 2 magneton, gs 2 is the Landé factor, and Gk ¼ 0Ik 2 dk is the coupling of strength fi μ magnetic- eld gradient, where 0 is the permeability. In 2g2ω order to mediate the coherent coupling of the CNT mechanical J ¼ q : (12) ω2 ω2 modes through the spin, we apply a time-dependent magnetic q À m

npj Quantum Information (2019) 58 Published in partnership with The University of New South Wales W. Qin et al. 3

Unitary evolution for a time τ0 = π/(4J) then leads to Next, we consider how to initialize and measure the mechanical pffiffiffi 〉 τ = ; system. Initially, the NV spin needs to be in the state |D (i.e., the b1ðÞ¼0 ðÞb1 À ib2 2 (13) ground state of the spin qubit), one CNT, e.g., the first CNT, needs pffiffiffi to be in its single-phonon state, and the other CNT, e.g., the τ = : b2ðÞ¼0 ðÞb2 À ib1 2 (14) second CNT, needs to be in its vacuum state. To prepare such an ρ = ρ ⊗ ρ ⊗ For the phase gate, we can turn off the current, for example, of the initial state, we can begin with an arbitrary state ini 1 2 ρspin, where ρk (k = 1, 2) and ρspin are the density matrices of the second CNT, so that g1 = g and g2 = 0. In this case, a dispersive shift of ≃J is imprinted into the vibrational resonance of the first kth CNT resonator and the spin, respectively. One can apply a 〉 CNT, which in turn introduces a relative phase ϕ ≃ Jτ after a time 532 nm pulse to initialize the spin qubit in the state |0 , and 1 then apply a microwave π/2-pulse to it, to obtain the super- τ1 under unitary evolution. Note that, here, both Hadamard and position state p1ffiffi ðÞj0iþjÀ1i , which is followed by a microwave phase gates are controlled operations conditional on the spin 2 state, as mentioned before. The two gates and their timing errors π-pulse to obtain the spin-qubit |B〉. By using the are analyzed in detail in the Supplementary Section 2. sideband-cooling technique,43–47 the CNT resonators can be We now turn to the quantum delayed-choice experiment with cooled down to their quantum ground state, i.e., the acoustic the macroscopic CNTs. We assume that the hybrid system is vacuum |vac〉. For example, one can couple an auxiliary qubit with initially prepared in the state a large spontaneous-emission rate to the CNT resonators.48 Once  the mechanical ground state is achieved, one can tune the spin- jΨi ¼ by I jvaci jDi; (15) i 1 2 qubit transition frequency ωq to be close to the CNT resonance frequency ω , such that the spin-CNT coupling is then approxi- where |vac〉 refers to the phonon vacuum and is the identity m I k mately given by a Jaynes–Cummings-type Hamiltonian operator for the kth CNT. After the initialization, the currents are  = τ tuned to be Ik I, to drive the system for a time 0, and the H ’ hg σ b þ σ by : (20) resulting Hadamard operation splits the single phonon into an int þ 1 À 1 equal superposition across both CNTs. Then, we turn off I2 for a When acting for a time equal to π/(2g), such a Hamiltonian can, τ time 1 to accumulate a relative phase between the CNTs. While with the spin qubit in the excited state |B〉, transfer a mechanical achieving the desired phase ϕ, we turn on I2 following a spin 49 – excitation to the left CNT. Meanwhile, the spin qubit goes to its single-qubit rotation |D〉 → cos(φ)|0〉 + sin(φ)|D〉40 42 with φ a ground state |D〉. The desired initial state jΨi ¼ by I jvaci rotation angle, and hold for another τ0 for a Hadamard operation. i 1 2 Therefore, this Hadamard gate is in a quantum superposition of jDi is then obtained. For the phonon number measurement, we being both present and absent. The three steps correspond, still need ωq ≃ ωm as in the initialization, but the spin qubit is respectively, to the input beam-splitter, the phase shifter and the required to be in the ground state |D〉. In this situation, the Rabi quantum output beam-splitter acting in sequence on a single frequency between the spin and the mechanical resonator photon in the Mach–Zehnder interferometer, as shown in Fig. 1a. depends on the number of in the resonator.49–53 Thus The final state of the system therefore becomes by directly measuring the occupation probability of |B〉, the jΨi ¼ cosðÞjφ particleij0iþsinðÞjφ waveijDi; (16) phonon number in each CNT can be obtained. The measurement f of the spin state is enabled by the different fluorescence of the where states |0〉 and |±1〉.54 To measure the state of the spin qubit, one hi 1ffiffi 1 can first apply a microwave π pulse to map jDi!p ðÞj0iÀjÀ1i jparticlei¼pffiffiffi expðÞiϕ by þ iby jvaci; (17) 2 1 2 and jBi!p1ffiffi ðÞj0iþjÀ1i , and then apply a microwave π/2 pulse 2 2 noto map p1ffiffi ðÞ!jj0iÀjÀ1i 0i and p1ffiffi ðÞ!jÀj0iþjÀ1i 1i.By 1 2 2 ϕ y ϕ y ; 〉 − 〉 jwavei¼ ½ŠexpðÞÀi 1 b1 þ i½ŠexpðÞþi 1 b2 jvaci (18) measuring the Rabi oscillations between the states |0 and | 1 2 according to spin-state-dependent fluorescence,55 one can read- describe the particle and wave behaviors, respectively. The out the spin-qubit state. If one employs the repetitive-readout coherent evolution of the system is given in more detail in technique with auxiliary nuclear spins, the readout fidelity can be Supplementary Section 2. We find from Eq. (16) that the further improved.56 mechanical phonon is in a quantum superposition of both a wave and a particle, and thus can exhibit both characteristics Mechanical noise simultaneously. By applying microwave pulse sequences to tune the rotation angle φ, an arbitrary wave-particle superposition state Before discussing the mechanical noise, we need to analyze the τ = τ + τ can be prepared on demand. In the case of φ = 0, the single total operation time, T 2 0 1, required for our quantum τ phonon behaves completely as a particle, but as a wave for φ = π/ delayed-choice experiment. Note that during T, we have 2. The morphing between them can also be observed by tuning neglected the spin single-qubit operation time due to the driving 57,58 ≤ τ ≤ π the rotation angle φ. The probability, P ,offinding a phonon in the pulse length ~ns. Since 0 1 2 /J, we focus on the k τ τmax π= π kth CNT is given by maximum T: T ¼ 5 ðÞ2J . A modest spin-CNT coupling g/2 = 100 kHz, which can be obtained by tuning the current I and the 1 k1 2 Pk ¼ þÀðÞ1 sin ðÞφ cosðÞϕ ; (19) distance d (see Supplementary Section 1), is able to mediate an 2 2 – π ≃ τmax : effective CNT CNT coupling J/2 12 kHz, thus giving T ’ 0 1 which includes two physical contributions, one from the particle ms. The relaxation time T1 of a single NV spin at low nature and the other from the wave nature. Note that the spin in a can reach up to a few minutes. Moreover, with spin echo mixed state cos2ðÞjφ 0ih0jþsin2ðÞjφ DihDj is capable of reprodu- techniques, a single spin in an ultra-pure diamond example 11 cing the same measured statistics as in Eq. (19). Thus, in order to typically has a dephasing time T2 ≃ 2 ms even at room tempera- 59 exclude the classical interpretation and prove the existence of the ture, corresponding to a dephasing rate γs/2π ≃ 80 Hz. When coherent wave-particle superposition, the quantum dynamical decoupling pulse sequences are employed, the between the states |0〉 and |D〉 should be verified.19,20,24,25 dephasing time can be made even close to one second at low Experimentally, such a verification can be implemented by temperatures.60 These justify neglecting the spin decoherence. In performing tomography to show all elements of this case, the mechanical noise dominates the dissipative the of the spin.42 processes. The dynamics of the system is therefore governed by

Published in partnership with The University of New South Wales npj Quantum Information (2019) 58 W. Qin et al. 4 the following master equation,  γ P ρ_ i ρ ; m y ρ ðÞ¼t h ½ŠÀðÞt HtðÞ 2 nth L bk ðÞt k¼1;2 γ P (21) m ρ ; À 2 ðÞnth þ 1 LðÞbk ðÞt k¼1;2

where ρ(t) is the density operator of the system, γm is the −1 mechanical decay rate, nth = [exp(ħωm/kBT) − 1] is the equili- brium phonon occupation at temperature T, and LðÞo ρðÞ¼t oyoρðÞÀt 2oρðÞt oy þ ρðÞt oyo is the Lindblad superoperator. Here, H(t) is a binary Hamiltonian of the form,  H0; 0

at time t = τT. For a realistic CNT, we can set the mechanical 61 linewidth to be γm/2π = 0.4 Hz, leading to a single-phonon lifetime of τm = 1/γm ≃ 400 ms. In this situation, τm is much longer τ γ τ than the total operation time T, m T  1 and, thus, we obtain γ τ : nk ¼ Pk þ nth m T (25)

This shows that, in addition to the coherent signal Pk, the final occupation has a thermal contribution nthγmτT. In Fig. 2,we demonstrate the morphing behavior between particle and wave at T ≃ 10 mK, according to Eq. (25). To confirm this, we also plot numerical simulations, which are in exact agreement with our analytical expression. The thermal occupation, nthγmτT, increases as the phase ϕ, because such a phase arises from the dynamical Fig. 2 Morphing between particle and wave characteristics of a CNT accumulation as discussed above. However, an extremely long mechanical phonon. Phonon occupation a n1 and b n2 as a function phonon lifetime causes it to become negligible even at finite of the relative phase ϕ and the rotation angle φ. The analytical results (colored surfaces) are in excellent agreement with the temperatures, as shown in Fig. 2. γ π = We now consider the fluctuation noise. In the limit γ τ 1, numerical simulations (black symbols). Here, in addition to s/2 m T  200γ /2π = 80 Hz, we assume that g/2π = 100 kHz, ω /2π = 2 MHz, the fluctuation noise δnnoise in the phonon occupation n is m m k k Ω = 10ω , and Δ− = 142ω , resulting in ω ≃ 1.5ω and then J/2π expressed, according to the analysis in the Supplementary Section m m q m ≃ × 12 kHz, and that nth = 100, corresponding to an environmental 4, as temperature of ≃10 mK ÀÁ δ noise 2 γ τ γ τ ; nk ¼ PkðÞ2Pk À 1 m m þ ðÞ2Pk þ 1 nth m T (26) where the first term is the vacuum fluctuation, which can be neglected, and the second term is the thermal fluctuation, which and R2 greater than 1. In this case, at least one CNT signal is increases with temperature. To quantitatively describe the ability resolved for each measurement. The conservation of the coherent to resolve the coherent signal from the fluctuation noise, we phonon number equal to 1 ensures that the unresolved signal can typically employ the signal-to-noise ratio defined as be inferred from the resolved one, which allows the morphing between wave and particle to be effectively observed from the P R ¼ k : fluctuation noise. To quantify this, we define a signal visibility as, k δ noise (27) nk pffiffiffi The signal-resolved regime often requires Rk>1 for any Pk. 2 However, the probability P in the range zero to unity indicates R¼ ; (30) k 2B that there always exist some Pk such that Rk<1, in particular, at finite temperatures. Nevertheless, we find that the total fluctuation noise in analogy to the signal-to-noise ratio Rk. The ratio R describes ÀÁ2 ÀÁ2 the visibility of the total signal rather than the single CNT signals. S2 ¼ δnnoise þ δnnoise (28) 1 2 At zero temperature (nth = 0), the noise originates only from the is kept below an upper bound vacuum fluctuation, and this yields R1. However, at finite temperatures, n increases as T, causing a decrease in R,as 2 γ τmax γ τmax; th B ¼ m T þ 4nth m T (29) shown in Fig. 3. Therefore, the requirement of R>1 sets an upper 2 and further that assuming B <1=2 can make either or both of R1 bound on the temperature, and as a result, leads to a critical

npj Quantum Information (2019) 58 Published in partnership with The University of New South Wales W. Qin et al. 5 states, as, e.g., those generated and measured in ref. 66. In this sense, the single-phonon wave-particle superposition, given in Eq. (16), can be referred to as a Schrödinger kitten state, since the excitations of the macroscopic mechanical systems are small, i.e., at the single-phonon level. Indeed, the of single- phonon excitations are not large enough to satisfy criterion (i). However, such superpositions of single phonons are large enough that the constituent states of the superposition, given in Eqs. (17) and (18), are experimentally distinguishable, thus satisfying criterion (ii). Therefore, such a test of a quantum principle at the low-excitation level of massive mechanical objects can also be viewed as a test at the macroscopic scale, as claimed, e.g., in refs. 67–69 and references therein. We note that a collective degree of freedom of many does not necessarily imply that the system is in a macroscopic quantum state. However, we showed that the studied system of macroscopic resonators can be in a maximally entangled two- Fig. 3 Signal visibility R as a function of the temperature T. The mode state. This state is described by a non-positive yellow shaded area represents the signal-resolved regime, where Glauber–Sudarshan P function. This implies that the system itself the morphing between wave and particle can be effectively fl is quantum. Below we describe the method to amplify the small- observed in the uctuation noise. The vertical line corresponds to excitation kitten states, given in Eqs. (17) and (18), to a cat state the critical temperature Tc. The inset shows a linear increase in Tc with increasing the ratio of the spin-mediated CNT–CNT coupling with large excitation. strength J to the mechanical-mode decay rate γm. Here, all parameters are set to be the same as in Fig. 2 Amplification of the Schrödinger kitten states Here we apply the idea and method of ref. 70 to show how to temperature, amplify the phonon numbers of the single-phonon superposition hω states |particle〉 and |wave〉, given in Eqs. (17) and (18), by T ¼ m : (31) c k ln½ŠðÞ 1 þ 15πγ =J =ðÞ1 À 5πγ =J squeezing the mechanical modes b1 and b2. Thus, these states can B m m become Schrödinger’s cat-like states. For simplicity, but without γ The critical temperature linearly increases with J/ m, as plotted in loss of generality, here we consider a squeezing operator the inset of Fig. 3. To increase J, we can increase the current I hi r y2 2 ; through the CNTs, decrease the distance d between the CNTs, or Uk ¼ exp 2 bk À bk (32) decrease the spin-qubit transition frequency ωq. Furthermore, the acting on the mode bk (k = 1, 2), with r being a squeezing increase in the CNT resonance frequency ωm or the decrease in γ parameter. This squeezing leads to the CNT loss rate m can also lead to an increase in the critical  temperature. For modest parameters of J/2π = 12 kHz and y ; jS10i¼ U1b1 U2 jvaci¼jS1i1jS0i2 (33) γm/2π = 0.4 Hz, a critical temperature Tc of ≃47 mK, which is routinely accessible in current experiments, can be achieved.  y ; jS01i¼ U1 U2b2 jvaci¼jS0i1jS1i2 (34) Test of macroscopicity where we have defined the phonon squeezed Fock states |S0〉k = We have described the implementation of a quantum paradox 〉 y 〉 Uk|0 k and jS1ik ¼ Ukbkj0ik, with |0 k being the vacuum state of with massive mechanical objects with experimentally distinguish- the mechanical-mode bk. As a result, the states |particle〉 and able single-phonon excitations. The question arises whether this |wave〉 become 62,63 proposal can be considered as a test of macroscopicity. 1 Typical proposals of such tests (as cited below) have been based jPri¼pffiffiffi ½ŠexpðÞjiϕ S10iþijS01i ; (35) on implementing superpositions of macroscopically distinguish- 2 able states of classical-like systems, which are often referred to as 64 1 Schrödinger’s cat states (see, e.g., ref. ). Sometimes, the meaning jWri¼ fg½ŠjexpðÞÀiϕ 1 S10iþi½ŠjexpðÞþiϕ 1 S01i ; (36) of Schrödinger’s cat states is limited to “superposition states of 2 fi Ψ〉 macroscopic systems, where the amplitude of their excitations is respectively. The nal state | f becomes ” 65 “ ” large . Note, however, that the term large amplitude can be jΨi ¼ cosðφ ÞjPrij0iþsinðφ ÞjWrijDi: (37) understood in various ways. These include the cases (criteria) f = when (i) the amplitudes of the constituent states of a given The modes bk for k 1, 2 are transformed, via squeezing, to the superposition are large as in classical systems, or (ii) when these Bogoliubov modes described by amplitudes are large enough concerning their experimental y y U bkUk ¼ coshðÞr bk þ sinhðÞr b : (38) distinguishability (i.e., compared to the resolution of detectors). k k Strictly speaking, a state satisfying one of these conditions, does By using this unitary transformation, one obtains the average 〉 〉 not necessarily satisfy the other. For example, a superposition of phonon numbers of |S0 k and |S1 k equal to ψ α β coherent states, j i¼Nðj iþj iÞ with N being a normalization hS jby b jS i ¼ sinh2ðÞr ; (39) constant, is a cat state according to criterion (i) if jαj; jβj1, but k 0 k k 0 k cannot be considered as a cat state according to criterion, (ii) if hS jby b jS i ¼ 3 sinh2ðÞþr 1: (40) ϵ jα À βj1 is beyond the resolution of detectors. Conversely, k 1 k k 1 k |ψ〉 is a cat state according to criterion (ii) if ϵ can be resolved We note that by applying this unconditional amplification experimentally even if |α|, |β| ≈ 1, i.e., when criterion (i) is not method, one can exponentially increase the distinguishability of satisfied. In the latter case, when the amplitude of such excitations the states |S10〉 and |S01〉. Although, a single-shot distinguishability is not large in classical terms, but still macroscopically distinguish- of the mechanical-mode states jPri and jWri is not increased, a able, the states are sometimes referred to as Schrödinger’s kitten tomographic distinguishability of these states in the phase

Published in partnership with The University of New South Wales npj Quantum Information (2019) 58 W. Qin et al. 6 corresponds to approximate quantum cloning, i.e., phase- covariant cloning by .

DISCUSSION We have presented a proposal for a quantum delayed-choice experiment with nanomechanical resonators, which enables a macroscopic test of an arbitrary quantum wave-particle super- position. The ability to tolerate the mechanical noise has also been given here, demonstrating that our proposal can be implemented with current experimental techniques. While we have chosen to focus on a spin-nanomechanical setup, the present method could be directly extended to other hybrid systems, for example, mechanical devices coupled to a superconducting .32,49,72 Recently, an experimental work reported that can be entangled in their wave-particle degree of freedom.22 This indicates that the wave-particle nature of photons may be used to encode flying for long-distance quantum communica- tion. Photons are ideal quantum information carriers, but they are difficult to store. In contrast to photons, long-lived phonons could be used for optical information storage.73 Our study shows that phonons can also be prepared in a wave-particle superposition state, and that the wave-particle nature of phonons is not more special than their other degrees of freedom. Thus, the wave- particle degree of freedom of phonons may be exploited for Fig. 4 Wigner function for the states |S 〉 = U |0〉 ( curves) and storing quantum information encoded in the wave-particle degree 0 k k k of freedom of photons. In addition, optomechanical interactions |S1〉k = Uk|1〉k (dashed curves) for r = 0ina and 1 in b. Here, k = 1, 2. TheR Wigner functionÂà is deÀÁfined as WðÞα = can couple a mechanical mode to optical modes at different 74 πÀ2 d2β expðÞÀiβαà À iβÃα Tr expðÞiβÃa exp iβay ϱ . For both plots, . Thus, the -particle degree of we have assumed Re(α) = 0. It is seen that the region marked in freedom may be employed to map quantum information encoded yellow, corresponding to the negative Wigner function for |S1〉k, in the wave-particle degree of freedom from photons at a given increases with increasing squeezing parameter r frequency to photons at any desired frequency. The mechanical wave-particle nature, as a new degree of freedom, may find various applications in quantum information. is increased with the amplified amplitudes of the mechanical- We believe that the macroscopicity of our single-phonon wave- particle superposition is highly counter-intuitive, as based on a mode excitations. Indeed, the distinguishability of jPir and jWri, 2 fi as measured by the infidelity, IF ¼ 1 À jhWrjPrij ¼ re ned version of the quantum paradox, even if the mechanical 1 Àjhwavejparticleij2, is independent of the squeezing parameter resonators are in the single-phonon-excitation regime. Indeed, we r for a given ϕ. For any ϕ ≠ ±π/2, the states are distinguishable, analyzed a “nested” kitten state, as given in Eq. (16), where the and the highest distinguishability is for ϕ = 0,π, for which the particle and wave states, given in Eqs. (17) and (18), are purely infidelity is IF = 1/2. Thus, even for such optimal values of ϕ,itis mechanical kitten states for ϕ ≠ ±π/2. Moreover, we have impossible to deterministically distinguish the states jP i and described a method, based on mechanical-mode squeezing, r fi jWri from each other in a single-shot experiment. We refer to this which enables the ampli cation of small-excitation Schrödinger property as a single-shot distinguishability. Anyway, these kitten states, given in Eqs. (17) and (18), to large-excitation mechanical states can be macroscopically distinguished by Schrödinger cat states of the massive mechanical resonators. For performing, e.g., Wigner-function tomography on a number of these reasons, an experimental realization of our proposal can be their copies. Such tomographic distinguishability in phase space a fundamental test of a coherent wave-particle superposition of indeed increases with the squeezing parameter r, as shown in massive objects with phonon excitations, which can be increased Fig. 4. exponentially by squeezing. Hence, this proposed quantum Finally, we note that the famous optical prototypes of the delayed-choice experiment of massive mechanical resonators Schrödinger’s cat states, which are given by the odd and even not only leads to a better understanding of quantum theory at the macroscopic scale, but also indicates that, like the vertical and coherent states, |ψ±〉 = N (|α〉 ±|−α〉), cannot be distinguished deterministically in a single-shot experiment either. This is horizontal polarizations of photons, the mechanical wave-particle because the coherent states |α〉 and |−α〉 are not orthogonal for nature, as an additional degree of freedom of phonons, may be finite values of α. Their overlap decreases exponentially with widely exploited for quantum information applications. increasing α,so|α〉 and |−α〉 become orthogonal in the limit of large |α|. However, this amplification of α cannot be done DATA AVAILABILITY deterministically, because this process is prohibited by the no- cloning and no-signalling theorems. Indeed, non-orthogonal The data that supports the findings of this study are available in the Supplementary fi states cannot be deterministically transformed to orthogonal Information le. Additional data are also available from the corresponding authors upon reasonable request. (thus, completely distinguishable) states. Note that popular methods of amplifying small-amplitude states are based on either (i) probabilistic but accurate amplification or (ii) deterministic but ACKNOWLEDGEMENTS inaccurate cloning. For example, the method described, e.g., in 66,71 W.Q. thanks Peng-Bo Li for valuable discussions. W.Q. and J.Q.Y. were supported in refs. is probabilistic, because it is based on conditional part by the National Key Research and Development Program of China (Grant No. measurements performed on two copies of |ψ±〉. In contrast to 2016YFA0301200), the China Postdoctoral Science Foundation (Grant No. 70 this, the amplification method in ref. , as applied here, 2017M610752), and the NSFC (Grant No. 11774022). G.L.L. is supported in part by

npj Quantum Information (2019) 58 Published in partnership with The University of New South Wales W. Qin et al. 7 National Key Research and Development Program of China (2017YFA0303700); 23. Long, G.-L., Qin, W., Yang, Z. & Li, J.-L. Realistic interpretation of quantum Beijing Advanced Innovation Center for Future Chip (ICFC). A.M. and F.N. mechanics and encounter-delayed-choice experiment. Sci. China Phys. Mech. acknowledge the support of a grant from the John Templeton Foundation. F.N. Astron. 61, 030311 (2018). is supported in part by the: MURI Center for Dynamic Magneto-Optics via the Air 24. Zheng, S.-B. et al. Quantum delayed-choice experiment with a beam splitter in a Office of Scientific Research (AFOSR) (FA9550-14-1-0040), Army Research quantum superposition. Phys. Rev. Lett. 115, 260403 (2015). Office (ARO) (Grant No. Grant No. W911NF-18-1-0358), Asian Office of Aerospace 25. Liu, K. et al. A twofold quantum delayed-choice experiment in a superconducting Research and Development (AOARD) (Grant No. FA2386-18-1-4045), Japan circuit. Sci. Adv. 3, e1603159 (2017). Science and Technology Agency (JST) (Q-LEAP program, ImPACT program, and 26. Blencowe, M. Quantum electromechanical systems. Phys. Rep. 395, 159–222 CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (2004). (JSPS-RFBR Grant No. 17-52-50023, and JSPS-FWO Grant No. VS.059.18N), and 27. Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. . Rev. RIKEN-AIST Challenge Research Fund. Mod. Phys. 86, 1391 (2014). 28. Poot, M. & van der Zant, H. S. J. Mechanical systems in the quantum regime. Phys. Rep. 511, 273–335 (2012). AUTHOR CONTRIBUTIONS 29. Iijima, S. Helical microtubules of graphitic carbon. Nature 354, 56 (1991). W.Q. and A.M. developed the theory and performed the calculations. G.L.L., J.Q.Y., and 30. Liu, D. E. Sensing Kondo correlations in a suspended F. N. supervised the project. 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