Abstract
We present and analyze a proposal for a macroscopic quantum delayedchoice experiment with massive mechanical resonators. In our approach, the electronic spin of a single nitrogenvacancy impurity is employed to control the coherent coupling between the mechanical modes of two carbon nanotubes. We demonstrate that a mechanical phonon can be in a coherent superposition of wave and particle, thus exhibiting both behaviors at the same time. We also discuss the mechanical noise tolerable in our proposal and predict a critical temperature below which the morphing between wave and particle states can be effectively observed in the presence of environmentinduced fluctuations. Furthermore, we describe how to amplify singlephonon excitations of the mechanicalresonator superposition states to a macroscopic level, via squeezing the mechanical modes. This approach corresponds to the phasecovariant 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 mechanics at the macroscopic scale, would have implications for quantum metrology and quantum information processing.
Introduction
Waveparticle duality lies at the heart of quantum physics. According to Bohr’s complementarity principle,^{1} a quantum system may behave either as a wave or as a particle depending on the measurement apparatus, and both behaviors are never observed simultaneously. This can be well demonstrated via a single photon Mach–Zehnder interferometer, as depicted in Fig. 1a. An incident photon is split, at an input beamsplitter BS_{1}, into an equal superposition of being in the upper and lower paths. This is followed by a phase shift ϕ in the upper path. At the output beamsplitter BS_{2}, the paths are recombined and the detection probability in the detector D_{1} or D_{2} depends on the phase ϕ, heralding the wave nature of a single photon. If, however, BS_{2} is absent, the photon is detected with probability 1/2 in each detector, and thus, shows its particle nature. In Wheeler’s delayedchoice experiment,^{2,3} the decision of whether or not to insert BS_{2} is randomly made after a photon is already inside the interferometer. The arrangement rules out a hiddenvariable theory, which suggests that the photon may determine, in advance, which behavior, wave or particle, to exhibit through a hidden variable.^{4,5,6,7,8,9,10,11} Recently, a quantum delayedchoice experiment, where BS_{2} 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 and a particle, so that both behaviors can be observed in a single measurement apparatus at the same time.^{13,14} This extends the conventional boundary of Bohr’s complementarity principle. The quantum delayedchoice experiment has already been implemented in nuclear magnetic resonance,^{15,16,17} optics,^{18,19,20,21,22,23} and superconducting circuits.^{24,25} However, all these experiments were performed essentially at the microscopic scale.
Here, as a step in the macroscopic test for a coherent waveparticle superposition on massive objects, we propose and analyze an approach for a mechanical quantum delayedchoice experiment. Mechanical systems are not only being explored now for potential quantum technologies,^{26,27} but they also have been considered as a promising candidate to test fundamental principles in quantum theory.^{28} In this manuscript, we demonstrate that, similar to a single photon, the mechanical phonon can be prepared in a quantum superposition of both a wave and a particle. The basic idea is to use a single nitrogenvacancy (NV) center in diamond to control the coherent coupling between two separated carbon nanotubes (CNTs).^{29,30} We focus on the electronic ground state of the NV center, which is a spin S = 1 triplet with a zerofield splitting D ≃ 2π × 2.87 GHz between spin states 0〉 and ±1〉 [see Fig. 1b]. If the spin is in 0〉, the mechanical modes are decoupled, and otherwise are coupled. Moreover, the mechanical noise tolerated by our proposal is evaluated and we show a critical temperature, below which the coherent signal is resolved.
Results
Physical model
We consider a hybrid system^{31,32} consisting of two (labelled as k = 1, 2) parallel CNTs and an NV electronic spin, as illustrated in Fig. 1c. The CNTs, both suspended along the \(\hat x\)direction, carry dc currents I_{1} and I_{2}, respectively, while the spin is placed between them, at a distance d_{1} from the first CNT and at a distance d_{2} from the second CNT. When vibrating along the \(\hat y\)direction, the CNTs can parametrically modulate the Zeeman splitting of the intermediate spin through the magnetic field, yielding a magnetic coupling to the spin.^{33,34,35,36,37} For simplicity, below we assume that the CNTs are identical such that they have the same vibrational frequency ω_{m} and the same vibrational mass m. The mechanical vibrations are modelled by quantized harmonic oscillators with a Hamiltonian
where b_{k} (\(b_k^\dagger\)) denotes the phonon annihilation (creation) operator. The Hamiltonian characterizing the coupling of the mechanical modes to the spin is
where S_{z} = +1〉〈+1 − −1〉〈−1 is the zcomponent of the spin, \(q_k = b_k + b_k^\dagger\) represents the canonical phonon position operator, and g_{k} = μ_{B}g_{s}y_{zp}G_{k}/ħ refers to the Zeeman shift corresponding to the zeropoint motion y_{zp} = [ħ/(2mω_{m})]^{1/2}. Here, μ_{B} is the Bohr magneton, g_{s} ≃ 2 is the Landé factor, and \(G_k = \mu _0I_k{\mathrm{/}}\left( {2\pi d_k^2} \right)\) is the magneticfield gradient, where μ_{0} is the vacuum permeability. In order to mediate the coherent coupling of the CNT mechanical modes through the spin, we apply a timedependent magnetic field
with amplitude B_{0} and frequency ω_{0}, along the \(\hat x\)direction, to drive the 0〉 → ±1〉 transitions with Rabi frequency
We apply a static magnetic field
along the \(\hat z\)direction to eliminate the Zeeman splitting between the spin states ±1〉.^{36} This causes the same Zeeman shift,
where Δ_{±} = D ± ω_{0}, to be imprinted on ±1〉, and a coherent coupling, of strength Ω^{2}/Δ_{+}, between them, as shown in Fig. 1b. We can, thus, introduce a dark state
and a bright state
with an energy splitting ≃2Ω^{2}/Δ_{+}. In this case, the spin state 0〉 is decoupled from the dark state, and is dressed by the bright state. Under the assumption of \(\Omega /\Delta \ll 1\), the dressing will only increase the energy splitting between the dark and bright states to
This yields a spin qubit with D〉 as the ground state and B〉 as the exited state. The spinCNT coupling Hamiltonian is accordingly transformed to
where σ_{x} = σ_{+} + σ_{−}, with σ_{−} = D〉〈B and \(\sigma _ + = \sigma _  ^\dagger\). When we further restrict our discussion to a dispersive regime \(\omega _q \pm \omega _m \gg g_k\), the spin qubit becomes a quantum data bus, allowing for mechanical excitations to be exchanged between the CNTs. By using a timeaveraging treatment,^{38,39} the unitary dynamics of the system is then described by an effective Hamiltonian (see Supplementary Section 1 for a detailed derivation), H_{eff} = H_{cnt} ⊗ σ_{z}, where
and σ_{z} = B〉〈B − D〉〈D. The Hamiltonian H_{cnt} includes a coherent spinmediated CNT–CNT coupling in the beamsplitter form, which is conditioned on the spin state. Here, we neglect the direct CNT–CNT coupling much smaller than the spinmediated coupling, as is described in Supplementary Section 1. Furthermore, we find that the decoupling of one CNT from the spin gives rise to a spininduced shift of the vibrational resonance of the other CNT. Hence, the dynamics described by H_{eff} can be used to implement controlled Hadamard and phase gates.
Quantum delayedchoice experiment with mechanical resonators
Let us first discuss the Hadamard gate. Having I_{k} = I and d_{k} = d gives a symmetric coupling g_{k} = g, and a mechanical beamsplitter coupling of strength
Unitary evolution for a time τ_{0} = π/(4J) then leads to
For the phase gate, we can turn off the current, for example, of the second CNT, so that g_{1} = g and g_{2} = 0. In this case, a dispersive shift of ≃J is imprinted into the vibrational resonance of the first CNT, which in turn introduces a relative phase ϕ ≃ Jτ_{1} after a time τ_{1} under unitary evolution. Note that, here, both Hadamard and phase gates are controlled operations conditional on the spin state, as mentioned before. The two gates and their timing errors are analyzed in detail in the Supplementary Section 2.
We now turn to the quantum delayedchoice experiment with the macroscopic CNTs. We assume that the hybrid system is initially prepared in the state
where vac〉 refers to the phonon vacuum and \({\cal{I}}_k\) is the identity operator for the kth CNT. After the initialization, the currents are tuned to be I_{k} = I, to drive the system for a time τ_{0}, and the resulting Hadamard operation splits the single phonon into an equal superposition across both CNTs. Then, we turn off I_{2} for a time τ_{1} to accumulate a relative phase between the CNTs. While achieving the desired phase ϕ, we turn on I_{2} following a spin singlequbit rotation D〉 → cos(φ)0〉 + sin(φ)D〉^{40,41,42} with φ a rotation angle, and hold for another τ_{0} for a Hadamard operation. Therefore, this Hadamard gate is in a quantum superposition of being both present and absent. The three steps correspond, respectively, to the input beamsplitter, the phase shifter and the quantum output beamsplitter acting in sequence on a single photon in the Mach–Zehnder interferometer, as shown in Fig. 1a. The final state of the system therefore becomes
where
describe the particle and wave behaviors, respectively. The coherent evolution of the system is given in more detail in Supplementary Section 2. We find from Eq. (16) that the mechanical phonon is in a quantum superposition of both a wave and a particle, and thus can exhibit both characteristics simultaneously. By applying microwave pulse sequences to tune the rotation angle φ, an arbitrary waveparticle superposition state can be prepared on demand. In the case of φ = 0, the single phonon behaves completely as a particle, but as a wave for φ = π/2. The morphing between them can also be observed by tuning the rotation angle φ. The probability, P_{k}, of finding a phonon in the kth CNT is given by
which includes two physical contributions, one from the particle nature and the other from the wave nature. Note that the spin in a mixed state \(\cos ^2\left( \varphi \right)0\rangle \langle 0 + \sin ^2\left( \varphi \right)D\rangle \langle D\) is capable of reproducing the same measured statistics as in Eq. (19).^{11} Thus, in order to exclude the classical interpretation and prove the existence of the coherent waveparticle superposition, the quantum coherence between the states 0〉 and D〉 should be verified.^{19,20,24,25} Experimentally, such a verification can be implemented by performing quantum state tomography to show all elements of the density matrix of the spin.^{42}
Next, we consider how to initialize and measure the mechanical system. Initially, the NV spin needs to be in the state D〉 (i.e., the ground state of the spin qubit), one CNT, e.g., the first CNT, needs to be in its singlephonon state, and the other CNT, e.g., the second CNT, needs to be in its vacuum state. To prepare such an initial state, we can begin with an arbitrary state ρ_{ini} = ρ_{1} ⊗ ρ_{2} ⊗ ρ_{spin}, where ρ_{k} (k = 1, 2) and ρ_{spin} are the density matrices of the kth CNT resonator and the spin, respectively. One can apply a 532 nm laser pulse to initialize the spin qubit in the state 0〉, and then apply a microwave π/2pulse to it, to obtain the superposition state \(\frac{1}{{\sqrt 2 }}\left( {0\rangle +   1\rangle } \right)\), which is followed by a microwave πpulse to obtain the spinqubit excited state B〉. By using the sidebandcooling technique,^{43,44,45,46,47} the CNT resonators can be cooled down to their quantum ground state, i.e., the acoustic vacuum vac〉. For example, one can couple an auxiliary qubit with a large spontaneousemission rate to the CNT resonators.^{48} Once the mechanical ground state is achieved, one can tune the spinqubit transition frequency ω_{q} to be close to the CNT resonance frequency ω_{m}, such that the spinCNT coupling is then approximately given by a Jaynes–Cummingstype Hamiltonian
When acting for a time equal to π/(2g), such a Hamiltonian can, with the spin qubit in the excited state B〉, transfer a mechanical excitation to the left CNT.^{49} Meanwhile, the spin qubit goes to its ground state D〉. The desired initial state \(\Psi \rangle _i = \left( {b_1^\dagger \otimes {\cal{I}}_2{\mathrm{vac}}\rangle } \right) \otimes D\rangle\) is then obtained. For the phonon number measurement, we still need ω_{q} ≃ ω_{m} as in the initialization, but the spin qubit is required to be in the ground state D〉. In this situation, the Rabi frequency between the spin and the mechanical resonator depends on the number of phonons in the resonator.^{49,50,51,52,53} Thus by directly measuring the occupation probability of B〉, the phonon number in each CNT can be obtained. The measurement of the spin state is enabled by the different fluorescence of the states 0〉 and ±1〉.^{54} To measure the state of the spin qubit, one can first apply a microwave π pulse to map \(D\rangle \to \frac{1}{{\sqrt 2 }}\left( {0\rangle    1\rangle } \right)\) and \(B\rangle \to \frac{1}{{\sqrt 2 }}\left( {0\rangle +   1\rangle } \right)\), and then apply a microwave π/2 pulse to map \(\frac{1}{{\sqrt 2 }}\left( {0\rangle    1\rangle } \right) \to 0\rangle\) and \(\frac{1}{{\sqrt 2 }}\left( {0\rangle +   1\rangle } \right) \to   1\rangle\). By measuring the Rabi oscillations between the states 0〉 and −1〉 according to spinstatedependent fluorescence,^{55} one can readout the spinqubit state. If one employs the repetitivereadout technique with auxiliary nuclear spins, the readout fidelity can be further improved.^{56}
Mechanical noise
Before discussing the mechanical noise, we need to analyze the total operation time, τ_{T} = 2τ_{0} + τ_{1}, required for our quantum delayedchoice experiment. Note that during τ_{T}, we have neglected the spin singlequbit operation time due to the driving pulse length ~ns.^{57,58} Since 0 ≤ τ_{1} ≤ 2π/J, we focus on the maximum τ_{T}: \(\tau _T^{{\mathrm{max}}} = 5\pi /\left( {2J} \right)\). A modest spinCNT coupling g/2π = 100 kHz, which can be obtained by tuning the current I and the distance d (see Supplementary Section 1), is able to mediate an effective CNT–CNT coupling J/2π ≃ 12 kHz, thus giving \(\tau _T^{{\mathrm{max}}} \simeq 0.1\) ms. The relaxation time T_{1} of a single NV spin at low temperatures can reach up to a few minutes. Moreover, with spin echo techniques, a single spin in an ultrapure diamond example typically has a dephasing time T_{2} ≃ 2 ms even at room temperature,^{59} corresponding to a dephasing rate γ_{s}/2π ≃ 80 Hz. When dynamical decoupling pulse sequences are employed, the dephasing time can be made even close to one second at low temperatures.^{60} These justify neglecting the spin decoherence. In this case, the mechanical noise dominates the dissipative processes. The dynamics of the system is therefore governed by the following master equation,
where ρ(t) is the density operator of the system, γ_{m} is the mechanical decay rate, n_{th} = [exp(ħω_{m}/k_{B}T) − 1]^{−1} is the equilibrium phonon occupation at temperature T, and \({\cal{L}}\left( o \right)\rho \left( t \right) = o^\dagger o\rho \left( t \right)  2o\rho \left( t \right)o^\dagger + \rho \left( t \right)o^\dagger o\) is the Lindblad superoperator. Here, H(t) is a binary Hamiltonian of the form,
with
and \(H_1 = Jb_1^\dagger b_1\sigma _z\). In Eq. (22), we did not include the spin singlequbit operation before the third time interval because the length of the driving pulse is very short, as mentioned above. The master equation in Eq. (21) drives the phonon occupation of the kth CNT to be
at time t = τ_{T}. For a realistic CNT, we can set the mechanical linewidth to be γ_{m}/2π = 0.4 Hz,^{61} leading to a singlephonon lifetime of τ_{m} = 1/γ_{m} ≃ 400 ms. In this situation, τ_{m} is much longer than the total operation time τ_{T}, \(\gamma _m\tau _T \ll 1\) and, thus, we obtain
This shows that, in addition to the coherent signal P_{k}, the final occupation has a thermal contribution n_{th}γ_{m}τ_{T}. 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, n_{th}γ_{m}τ_{T}, increases as the phase ϕ, because such a phase arises from the dynamical accumulation as discussed above. However, an extremely long phonon lifetime causes it to become negligible even at finite temperatures, as shown in Fig. 2.
We now consider the fluctuation noise. In the limit \(\gamma _m\tau _T \ll 1\), the fluctuation noise \(\delta n_k^{{\mathrm{noise}}}\) in the phonon occupation n_{k} is expressed, according to the analysis in the Supplementary Section 4, as
where the first term is the vacuum fluctuation, which can be neglected, and the second term is the thermal fluctuation, which increases with temperature. To quantitatively describe the ability to resolve the coherent signal from the fluctuation noise, we typically employ the signaltonoise ratio defined as
The signalresolved regime often requires \({\cal{R}}_k > 1\) for any P_{k}. However, the probability P_{k} in the range zero to unity indicates that there always exist some P_{k} such that \({\cal{R}}_k < 1\), in particular, at finite temperatures. Nevertheless, we find that the total fluctuation noise
is kept below an upper bound
and further that assuming \({\cal{B}}^2 < 1/2\) can make either or both of \({\cal{R}}_1\) and \({\cal{R}}_2\) greater than 1. In this case, at least one CNT signal is resolved for each measurement. The conservation of the coherent phonon number equal to 1 ensures that the unresolved signal can be inferred from the resolved one, which allows the morphing between wave and particle to be effectively observed from the fluctuation noise. To quantify this, we define a signal visibility as,
in analogy to the signaltonoise ratio \({\cal{R}}_k\). The ratio \({\cal{R}}\) describes the visibility of the total signal rather than the single CNT signals. At zero temperature (n_{th} = 0), the noise originates only from the vacuum fluctuation, and this yields \({\cal{R}} \gg 1\). However, at finite temperatures, n_{th} increases as T, causing a decrease in \({\cal{R}}\), as shown in Fig. 3. Therefore, the requirement of \({\cal{R}} > 1\) sets an upper bound on the temperature, and as a result, leads to a critical temperature,
The critical temperature linearly increases with J/γ_{m}, as plotted in the inset of Fig. 3. To increase J, we can increase the current I through the CNTs, decrease the distance d between the CNTs, or decrease the spinqubit transition frequency ω_{q}. Furthermore, the increase in the CNT resonance frequency ω_{m} or the decrease in the CNT loss rate γ_{m} can also lead to an increase in the critical temperature. For modest parameters of J/2π = 12 kHz and γ_{m}/2π = 0.4 Hz, a critical temperature T_{c} of ≃47 mK, which is routinely accessible in current experiments, can be achieved.
Test of macroscopicity
We have described the implementation of a quantum paradox with massive mechanical objects with experimentally distinguishable singlephonon excitations. The question arises whether this proposal can be considered as a test of macroscopicity.^{62,63} Typical proposals of such tests (as cited below) have been based on implementing superpositions of macroscopically distinguishable states of classicallike systems, which are often referred to as Schrödinger’s cat states (see, e.g., ref. ^{64}). Sometimes, the meaning of Schrödinger’s cat states is limited to “superposition states of macroscopic systems, where the amplitude of their excitations is large”.^{65} Note, however, that the term “large amplitude” can be understood in various ways. These include the cases (criteria) when (i) the amplitudes of the constituent states of a given superposition are large as in classical systems, or (ii) when these amplitudes are large enough concerning their experimental distinguishability (i.e., compared to the resolution of detectors). Strictly speaking, a state satisfying one of these conditions, does not necessarily satisfy the other. For example, a superposition of coherent states, \(\vert \psi\rangle = {\cal{N}}(\vert \alpha \rangle + \vert\beta\rangle)\) with \(\cal{N}\) being a normalization constant, is a cat state according to criterion (i) if \(\alpha ,\beta  \gg 1\), but cannot be considered as a cat state according to criterion, (ii) if \(\epsilon \equiv \alpha  \beta  \ll 1\) is beyond the resolution of detectors. Conversely, ψ〉 is a cat state according to criterion (ii) if \(\epsilon\) can be resolved experimentally even if α, β ≈ 1, i.e., when criterion (i) is not satisfied. In the latter case, when the amplitude of such excitations is not large in classical terms, but still macroscopically distinguishable, the states are sometimes referred to as Schrödinger’s kitten states, as, e.g., those generated and measured in ref. ^{66}. In this sense, the singlephonon waveparticle 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 singlephonon level. Indeed, the amplitudes of singlephonon 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 lowexcitation level of massive mechanical objects can also be viewed as a test at the macroscopic scale, as claimed, e.g., in refs. ^{67,68,69} and references therein.
We note that a collective degree of freedom of many atoms 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 twomode state. This state is described by a nonpositive Glauber–Sudarshan P function. This implies that the system itself is quantum. Below we describe the method to amplify the smallexcitation kitten states, given in Eqs. (17) and (18), to a cat state with large excitation.
Amplification of the Schrödinger kitten states
Here we apply the idea and method of ref. ^{70} to show how to amplify the phonon numbers of the singlephonon superposition states particle〉 and wave〉, given in Eqs. (17) and (18), by squeezing the mechanical modes b_{1} and b_{2}. Thus, these states can become Schrödinger’s catlike states. For simplicity, but without loss of generality, here we consider a squeezing operator
acting on the mode b_{k} (k = 1, 2), with r being a squeezing parameter. This squeezing leads to
where we have defined the phonon squeezed Fock states S_{0}〉_{k} = U_{k}0〉_{k} and \(S_1\rangle _k = U_kb_k^\dagger 0\rangle _k\), with 0〉_{k} being the vacuum state of the mechanicalmode b_{k}. As a result, the states particle〉 and wave〉 become
respectively. The final state Ψ〉_{f} becomes
The modes b_{k} for k = 1, 2 are transformed, via squeezing, to the Bogoliubov modes described by
By using this unitary transformation, one obtains the average phonon numbers of S_{0}〉_{k} and S_{1}〉_{k} equal to
We note that by applying this unconditional amplification method, one can exponentially increase the distinguishability of the states S_{10}〉 and S_{01}〉. Although, a singleshot distinguishability of the mechanicalmode states \({\cal{P}}_r\rangle\) and \({\cal{W}}_r\rangle\) is not increased, a tomographic distinguishability of these states in the phase space is increased with the amplified amplitudes of the mechanicalmode excitations. Indeed, the distinguishability of \({\cal{P}}\rangle _r\) and \({\cal{W}}_r\rangle\), as measured by the infidelity, \({\mathrm{IF}} = 1  \langle {\cal{W}}_r{\cal{P}}_r\rangle ^2 = 1  \langle {\mathrm{wave}}{\mathrm{particle}}\rangle ^2\), is independent of the squeezing parameter r for a given ϕ. For any ϕ ≠ ±π/2, the states are distinguishable, and the highest distinguishability is for ϕ = 0,π, for which the infidelity is IF = 1/2. Thus, even for such optimal values of ϕ, it is impossible to deterministically distinguish the states \({\cal{P}}_r\rangle\) and \({\cal{W}}_r\rangle\) from each other in a singleshot experiment. We refer to this property as a singleshot distinguishability. Anyway, these mechanical states can be macroscopically distinguished by performing, e.g., Wignerfunction tomography on a number of their copies. Such tomographic distinguishability in phase space indeed increases with the squeezing parameter r, as shown in Fig. 4.
Finally, we note that the famous optical prototypes of the Schrödinger’s cat states, which are given by the odd and even coherent states, ψ_{±}〉 = \(\cal{N}\)(α〉 ± −α〉), cannot be distinguished deterministically in a singleshot experiment either. This is because the coherent states α〉 and −α〉 are not orthogonal for finite values of α. Their overlap decreases exponentially with increasing α, so α〉 and −α〉 become orthogonal in the limit of large α. However, this amplification of α cannot be done deterministically, because this process is prohibited by the nocloning and nosignalling theorems. Indeed, nonorthogonal states cannot be deterministically transformed to orthogonal (thus, completely distinguishable) states. Note that popular methods of amplifying smallamplitude states are based on either (i) probabilistic but accurate amplification or (ii) deterministic but inaccurate cloning. For example, the method described, e.g., in refs. ^{66,71} is probabilistic, because it is based on conditional measurements performed on two copies of ψ_{±}〉. In contrast to this, the amplification method in ref. ^{70}, as applied here, corresponds to approximate quantum cloning, i.e., phasecovariant cloning by stimulated emission.
Discussion
We have presented a proposal for a quantum delayedchoice experiment with nanomechanical resonators, which enables a macroscopic test of an arbitrary quantum waveparticle superposition. 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 spinnanomechanical setup, the present method could be directly extended to other hybrid systems, for example, mechanical devices coupled to a superconducting atom.^{32,49,72} Recently, an experimental work reported that photons can be entangled in their waveparticle degree of freedom.^{22} This indicates that the waveparticle nature of photons may be used to encode flying qubits for longdistance quantum communication. Photons are ideal quantum information carriers, but they are difficult to store. In contrast to photons, longlived phonons could be used for optical information storage.^{73} Our study shows that phonons can also be prepared in a waveparticle superposition state, and that the waveparticle nature of phonons is not more special than their other degrees of freedom. Thus, the waveparticle degree of freedom of phonons may be exploited for storing quantum information encoded in the waveparticle degree of freedom of photons. In addition, optomechanical interactions can couple a mechanical mode to optical modes at different frequencies.^{74} Thus, the mechanical waveparticle degree of freedom may be employed to map quantum information encoded in the waveparticle degree of freedom from photons at a given frequency to photons at any desired frequency. The mechanical waveparticle nature, as a new degree of freedom, may find various applications in quantum information.
We believe that the macroscopicity of our singlephonon waveparticle superposition is highly counterintuitive, as based on a refined version of the quantum paradox, even if the mechanical resonators are in the singlephononexcitation regime. Indeed, we analyzed a “nested” kitten state, as given in Eq. (16), where the particle and wave states, given in Eqs. (17) and (18), are purely mechanical kitten states for ϕ ≠ ±π/2. Moreover, we have described a method, based on mechanicalmode squeezing, which enables the amplification of smallexcitation Schrödinger kitten states, given in Eqs. (17) and (18), to largeexcitation Schrödinger cat states of the massive mechanical resonators. For these reasons, an experimental realization of our proposal can be a fundamental test of a coherent waveparticle superposition of massive objects with phonon excitations, which can be increased exponentially by squeezing. Hence, this proposed quantum delayedchoice experiment of massive mechanical resonators not only leads to a better understanding of quantum theory at the macroscopic scale, but also indicates that, like the vertical and horizontal polarizations of photons, the mechanical waveparticle nature, as an additional degree of freedom of phonons, may be widely exploited for quantum information applications.
Data availability
The data that supports the findings of this study are available in the Supplementary Information file. Additional data are also available from the corresponding authors upon reasonable request.
References
 1.
Bohr, N. in Quantum Theory and Measurement, (eds Wheeler, J. A. & Zurek, W. H.) 9–49 (Princeton University Press, Princeton, 1984).
 2.
Wheeler, J. A. in Mathematical Foundations of Quantum Theory, (eds Marlow, A. R.) 9–48 (Academic Press, Cambridge, 1978).
 3.
Ma, X.s, Kofler, J. & Zeilinger, A. Delayedchoice gedanken experiments and their realizations. Rev. Mod. Phys. 88, 015005 (2016).
 4.
Hellmuth, T., Walther, H., Zajonc, A. & Schleich, W. Delayedchoice experiments in quantum interference. Phys. Rev. A 35, 2532 (1987).
 5.
Kim, Y.H., Yu, R., Kulik, S. P., Shih, Y. & Scully, M. O. Delayed “Choice” Quantum Eraser. Phys. Rev. Lett. 84, 1 (2000).
 6.
Jacques, V. et al. Experimental realization of Wheeler’s delayedchoice gedanken experiment. Science 315, 966–968 (2007).
 7.
Jacques, V. et al. Delayedchoice test of quantum complementarity with interfering single photons. Phys. Rev. Lett. 100, 220402 (2008).
 8.
Manning, A. G., Khakimov, R. I., Dall, R. G. & Truscott, A. G. Wheeler’s delayedchoice gedanken experiment with a single atom. Nat. Phys. 11, 539 (2015).
 9.
Liu, Y., Lu, J. & Zhou, L. Information gain versus interference in Bohr’s principle of complementarity. Opt. Express 25, 202–211 (2017).
 10.
Vedovato, F. et al. Extending Wheeler’s delayedchoice experiment to space. Sci. Adv. 3, e1701180 (2017).
 11.
Chaves, R., Lemos, G. B. & Pienaar, J. Causal modeling the delayedchoice experiment. Phys. Rev. Lett. 120, 190401 (2018).
 12.
Ionicioiu, R. & Terno, D. R. Proposal for a quantum delayedchoice experiment. Phys. Rev. Lett. 107, 230406 (2011).
 13.
Adesso, G. & Girolami, D. Quantum optics: waveparticle superposition. Nat. Photon. 6, 579 (2012).
 14.
Shadbolt, P., Mathews, J. C. F., Laing, A. & O’Brien, J. L. Testing foundations of quantum mechanics with photons. Nat. Phys. 10, 278 (2014).
 15.
Roy, S. S., Shukla, A. & Mahesh, T. S. NMR implementation of a quantum delayedchoice experiment. Phys. Rev. A 85, 022109 (2012).
 16.
Auccaise, R. et al. Experimental analysis of the quantum complementarity principle. Phys. Rev. A 85, 032121 (2012).
 17.
Xin, T., Li, H., Wang, B.X. & Long, G.L. Realization of an entanglementassisted quantum delayedchoice experiment. Phys. Rev. A 92, 022126 (2015).
 18.
Tang, J.S. et al. Realization of quantum Wheeler’s delayedchoice experiment. Nat. Photon. 6, 600–604 (2012).
 19.
Peruzzo, A., Shadbolt, P., Brunner, N., Popescu, S. & O’Brien, J. L. A quantum delayedchoice experiment. Science 338, 634–637 (2012).
 20.
Kaiser, F., Coudreau, T., Milman, P., Ostrowsky, D. B. & Tanzilli, S. Entanglementenabled delayedchoice experiment. Science 338, 637–640 (2012).
 21.
Yan, H. et al. Experimental observation of simultaneous wave and particle behavior in a narrowband singlephoton wave packet. Phys. Rev. A 91, 042132 (2015).
 22.
Rab, A. S. et al. Entanglement of photons in their dual waveparticle nature. Nat. Commun. 8, 915 (2017).
 23.
Long, G.L., Qin, W., Yang, Z. & Li, J.L. Realistic interpretation of quantum mechanics and encounterdelayedchoice experiment. Sci. China Phys. Mech. Astron. 61, 030311 (2018).
 24.
Zheng, S.B. et al. Quantum delayedchoice experiment with a beam splitter in a quantum superposition. Phys. Rev. Lett. 115, 260403 (2015).
 25.
Liu, K. et al. A twofold quantum delayedchoice experiment in a superconducting circuit. Sci. Adv. 3, e1603159 (2017).
 26.
Blencowe, M. Quantum electromechanical systems. Phys. Rep. 395, 159–222 (2004).
 27.
Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. 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).
 29.
Iijima, S. Helical microtubules of graphitic carbon. Nature 354, 56 (1991).
 30.
Liu, D. E. Sensing Kondo correlations in a suspended carbon nanotube mechanical resonator with spinorbit coupling. Quantum Eng. 1, e10 (2019).
 31.
Buluta, I., Ashhab, S. & Nori, F. Natural and artificial atoms for quantum computation. Rep. Prog. Phys. 74, 104401 (2011).
 32.
Xiang, Z. L., Ashhab, S., You, J. Q. & Nori, F. Hybrid quantum circuits: superconducting circuits interacting with other quantum systems. Rev. Mod. Phys. 85, 623 (2013).
 33.
Rabl, P. et al. Strong magnetic coupling between an electronic spin qubit and a mechanical resonator. Phys. Rev. B 79, 041302 (2009).
 34.
Rabl, P. et al. A quantum spin transducer based on nanoelectromechanical resonator arrays. Nat. Phys. 6, 602 (2010).
 35.
Kolkowitz, S. et al. Coherent sensing of a mechanical resonator with a singlespin qubit. Science 335, 1603–1606 (2012).
 36.
Li, P.B., Xiang, Z.L., Rabl, P. & Nori, F. Hybrid quantum device with nitrogenvacancy centers in diamond coupled to carbon nanotubes. Phys. Rev. Lett. 117, 015502 (2016).
 37.
Cao, P., Betzholz, R., Zhang, S. & Cai, J. Entangling distant solidstate spins via thermal phonons. Phys. Rev. B 96, 245418 (2017).
 38.
Gamel, O. & James, D. F. V. Timeaveraged quantum dynamics and the validity of the effective Hamiltonian model. Phys. Rev. A 82, 052106 (2010).
 39.
Qin, W. et al. Exponentially enhanced lightmatter interaction, cooperativities, and steadystate entanglement using parametric amplification. Phys. Rev. Lett. 120, 093601 (2018).
 40.
Huang, P. et al. Observation of an anomalous decoherence effect in a quantum bath at room temperature. Nat. Commun. 2, 570 (2011).
 41.
Lillie, S. E. et al. Environmentally mediated coherent control of a spin qubit in diamond. Phys. Rev. Lett. 118, 167204 (2017).
 42.
Xing, J. et al. Experimental investigation of quantum entropic uncertainty relations for multiple measurements in pure diamond. Sci. Rep. 7, 2563 (2017).
 43.
Xue, F., Wang, Y. D., Liu, Y.x & Nori, F. Cooling a micromechanical beam by coupling it to a transmission line. Phys. Rev. B 76, 205302 (2007).
 44.
You, J. Q., Liu, Y.x & Nori, F. Simultaneous cooling of an artificial atom and its neighboring quantum system. Phys. Rev. Lett. 100, 047001 (2008).
 45.
Grajcar, M., Ashhab, S., Johansson, J. R. & Nori, F. Lower limit on the achievable temperature in resonatorbased sideband cooling. Phys. Rev. B 78, 035406 (2008).
 46.
Ma, Y., Yin, Z.q, Huang, P., Yang, W. L. & Du, J. F. Cooling a mechanical resonator to the quantum regime by heating it. Phys. Rev. A 94, 053836 (2016).
 47.
Clark, J. B., Lecocq, F., Simmonds, R. W., Aumentado, J. & Teufel, J. D. Sideband cooling beyond the quantum backaction limit with squeezed light. Nature 541, 191–195 (2017).
 48.
Wang, X., Miranowicz, A., Li, H.R. & Nori, F. Hybrid quantum device with a carbon nanotube and a flux qubit for dissipative quantum engineering. Phys. Rev. B 95, 205415 (2017).
 49.
O’Connell, A. D. et al. Quantum ground state and singlephonon control of a mechanical resonator. Nature 464, 697 (2010).
 50.
Scully, M. O. & Zubairy, M. S. Quantum Optics. (Cambridge University Press, Cambridge, 1997).
 51.
Liu, Y.x, Wei, L. F. & Nori, F. Generation of nonclassical photon states using a superconducting qubit in a microcavity. Europhys. Lett. 67, 941 (2004).
 52.
Hofheinz, M. et al. Generation of Fock states in a superconducting quantum circuit. Nature 454, 310 (2008).
 53.
Hofheinz, M. et al. Synthesizing arbitrary quantum states in a superconducting resonator. Nature 459, 546 (2009).
 54.
Doherty, M. W. et al. The nitrogenvacancy colour centre in diamond. Phys. Rep. 528, 1–45 (2013).
 55.
Jelezko, F., Gaebel, T., Popa, I., Gruber, A. & Wrachtrup, J. Observation of coherent oscillations in a single electron spin. Phys. Rev. Lett. 92, 076401 (2004).
 56.
Jiang, L. et al. Repetitive readout of a single electronic spin via quantum logic with nuclear spin ancillae. Science 326, 267–272 (2009).
 57.
Liu, G.Q. et al. Demonstration of entanglementenhanced phase estimation in solid. Nat. Commun. 6, 6726 (2015).
 58.
Liu, G.Q. et al. Singleshot readout of a nuclear spin weakly coupled to a nitrogenvacancy center at room temperature. Phys. Rev. Lett. 118, 150504 (2017).
 59.
Balasubramanian, G. et al. Ultralong spin coherence time in isotopically engineered diamond. Nat. Mater. 8, 383–387 (2009).
 60.
BarGill, N., Pham, L. M., Jarmola, A., Budker, D. & Walsworth, R. L. Solidstate electronic spin coherence time approaching one second. Nat. Commun. 4, 1743 (2013).
 61.
Moser, J., Eichler, A., Güttinger, J., Dykman, M. I. & Bachtold, A. Nanotube mechanical resonators with quality factors of up to 5 million. Nat. Nanotechnol. 9, 1007–1011 (2014).
 62.
Leggett, A. J. Testing the limits of quantum mechanics: motivation, state of play, prospects. J. Phys. 14, R415 (2002).
 63.
Korsbakken, J. I., Whaley, K. B., Dubois, J. & Cirac, J. I. Measurementbased measure of the size of macroscopic quantum superpositions. Physi. Rev. A 75, 042106 (2007).
 64.
Gerry, C. & Knight, P. Introductory Quantum Optics. (Cambridge University Press, Cambridge, 2005).
 65.
Agarwal, G. S. Quantum Optics. (Cambridge University Press, Cambridge, 2013).
 66.
Ourjoumtsev, A., TualleBrouri, R., Laurat, J. & Grangier, P. Generating optical Schrödinger kittens for quantum information processing. Science 312, 83–86 (2006).
 67.
Hofer, S. G., Lehnert, K. W. & Hammerer, K. Proposal to Test Bell’s Inequality in Electromechanics. Phys. Rev. Lett. 116, 070406 (2016).
 68.
Marinković, I. et al. Optomechanical bell test. Phys. Rev. Lett. 121, 220404 (2018).
 69.
OckeloenKorppi, C. F. et al. Stabilized entanglement of massive mechanical oscillators. Nature 556, 478 (2018).
 70.
Sekatski, P., Brunner, N., Branciard, C., Gisin, N. & Simon, C. Towards quantum experiments with human eyes as detectors based on cloning via stimulated emission. Phys. Rev. Lett. 103, 113601 (2009).
 71.
Lund, A. P., Jeong, H., Ralph, T. C. & Kim, M. S. Conditional production of superpositions of coherent states with inefficient photon detection. Phys. Rev. A 70, 020101(R) (2004).
 72.
Gu, X., Kockum, A. F., Miranowicz, A., Liu, Y.x & Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep. 718–719, 1–102 (2017).
 73.
Fiore, V. et al. Storing optical information as a mechanical excitation in a silica optomechanical resonator. Phys. Rev. Lett. 107, 133601 (2011).
 74.
Dong, C., Fiore, V., Kuzyk, M. C. & Wang, H. Optomechanical dark mode. Science 338, 1609–1613 (2012).
Acknowledgements
W.Q. thanks PengBo Li for valuable discussions. W.Q. and J.Q.Y. were supported in part by the National Key Research and Development Program of China (Grant No. 2016YFA0301200), the China Postdoctoral Science Foundation (Grant No. 2017M610752), and the NSFC (Grant No. 11774022). G.L.L. is supported in part by National Key Research and Development Program of China (2017YFA0303700); Beijing Advanced Innovation Center for Future Chip (ICFC). A.M. and F.N. acknowledge the support of a grant from the John Templeton Foundation. F.N. is supported in part by the: MURI Center for Dynamic MagnetoOptics via the Air Force Office of Scientific Research (AFOSR) (FA95501410040), Army Research Office (ARO) (Grant No. Grant No. W911NF1810358), Asian Office of Aerospace Research and Development (AOARD) (Grant No. FA23861814045), Japan Science and Technology Agency (JST) (QLEAP program, ImPACT program, and CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (JSPSRFBR Grant No. 175250023, and JSPSFWO Grant No. VS.059.18N), and RIKENAIST Challenge Research Fund.
Author information
Affiliations
Contributions
W.Q. and A.M. developed the theory and performed the calculations. G.L.L., J.Q.Y., and F. N. supervised the project. All authors researched, collated, and wrote this paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Qin, W., Miranowicz, A., Long, G. et al. Proposal to test quantum waveparticle superposition on massive mechanical resonators. npj Quantum Inf 5, 58 (2019). https://doi.org/10.1038/s4153401901729
Received:
Accepted:
Published:
Further reading

Generation of Strong Mechanical–Mechanical Entanglement by Pump Modulation
Advanced Quantum Technologies (2021)

Multimode Interference Induced Optical Routing in an Optical Microcavity
Annalen der Physik (2021)

Tunable optomechanically induced transparency by controlling the darkmode effect
Physical Review A (2020)

Quantumenhanced interferometry with asymmetric beam splitters
Science China Physics, Mechanics & Astronomy (2020)

Nanoassembly technique of carbon nanotubes for hybrid circuitQED
Applied Physics Letters (2020)