Proposal for a Quantum Delayed-Choice Experiment with Massive Mechanical Resonators

We present and analyze an experimentally-feasible implementation of a macroscopic quantum delayed-choice experiment to test a quantum wave-particle superposition on massive mechanical resonators. In our approach, the electronic spin 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 phonon can be in a coherent superposition of wave and particle, thus exhibiting both behaviors at the same time. Furthermore, we 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 environment-induced fluctuations.

Wave-particle 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. 1(a). An incident photon is split, at an input beam splitter 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 beam splitter 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 hidden-variable 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][12]. Recently, a quantum delayed-choice experiment, where BS 2 is engineered to be in a quantum superposition of being present and absent, has been proposed [13]. 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 [14,15]. This extends the conventional boundary of Bohr's complementarity principle. The quantum delayed-choice experiment has already been implemented in nuclear magnetic resonance [16][17][18], optics [19][20][21][22][23][24], and superconducting circuits [25,26]. However, all these ex-periments were performed essentially at the microscopic scale.
Here, as a first step in the macroscopic test for a coherent wave-particle superposition on massive objects, we propose and analyze a novel approach for a mechanical quantum delayed-choice experiment. Mechanical systems are not only being explored now for potential quantum technologies [27][28][29][30][31][32][33][34][35][36][37][38], but they also have been considered as a promising candidate to test fundamental principles in quantum theory [39], including, e.g., quantum superposition [40][41][42][43], wave-function collapse [44,45], quantum entanglement [46][47][48], and Bell's nonlocality [49][50][51][52]. 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 nitrogen-vacancy (NV) center in diamond to control the coherent coupling between two separated carbon nanotubes (CNTs) [53]. We focus on the electronic ground state of the NV center, which is a spin S = 1 triplet with a zero-field splitting D 2π × 2.87 GHz between spin states |0 and | ± 1 [see Fig. 1(b)]. 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.
Physical model.-We consider a hybrid system [54,55] consisting of two (labelled as k = 1, 2) parallel CNTs and an NV electronic spin, as illustrated in Fig. 1(c). The CNTs, both suspended along thex-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 vi- (a) Demonstration of the wave-particle duality using a Mach-Zehnder interferometer. A single photon is first split at the input beam splitter BS1, then undergoes a phase shift φ and finally is observed at detectors D1 and D2. The photon behaves as a wave if the output beam splitter BS2 is inserted, or as a particle if BS2 is removed. In quantum delayed-choice experiments, BS2 is set in a quantum superposition of being present and absent, and consequently, the photon can simultaneously exhibit its wave and particle nature. (b) Level structure of the driven NV spin in the electronic ground state. Here we have assumed that the Zeeman splitting between the spin states | ± 1 is eliminated by applying an external field. (c) Schematic representation of a mechanical quantum delayed-choice experiment with an NV electronic spin and two CNTs. The mechanical vibrations of the CNTs are completely decoupled or coherently coupled, depending, respectively, on whether or not the intermediate spin is in the spin state |0 , with the dc current I k through the kth CNT, and the distance d k between the spin and the kth CNT.
brating along theŷ-direction, the CNTs can parametrically modulate the Zeeman splitting of the intermediate spin through the magnetic field, yielding a magnetic coupling to the spin [56][57][58][59][60]. 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 denotes the phonon annihilation (creation) operator. The Hamiltonian characterizing the coupling of the mechanical modes to the spin is k represents the canonical phonon position operator, and g k = µ B g s y zp G k / refers to the Zeeman shift corresponding to the zero-point motion y zp = [ / (2mω m )] 1/2 . Here, µ B is the Bohr magneton, g s 2 is the Landé factor, and G k = µ 0 I k / 2πd 2 k is the magnetic-field 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 time-dependent magnetic field B x (t) = B 0 cos (ω 0 t) (with amplitude B 0 and frequency ω 0 ) along thexdirection to drive the |0 → | ± 1 transitions with Rabi frequency Ω = µ B g s B 0 / 2 √ 2 . We apply a static magnetic field B z = k=1,2 (−1) k d k G k along theẑdirection to eliminate the Zeeman splitting between the spin states |±1 [59]. This causes the same Zeeman shift, ∆ = ∆ − +3Ω 2 /∆ + , where ∆ ± = D ±ω 0 , to be imprinted on | ± 1 , and a coherent coupling, of strength Ω 2 /∆ + , between them, as shown in Fig. 1(b). We can, thus, introduce a dark state |D = (| + 1 − | − 1 ) / √ 2 and a bright state |B = (| + 1 + | − 1 ) / √ 2, with an energy splitting 2Ω 2 /∆ + . In this case, the spin state |0 is decoupled from the dark state, and is dressed by the bright state. Under the assumption of Ω/∆ 1, the dressing will only increase the energy splitting between the dark and bright states to ω q 2Ω 2 (1/∆ + 1/∆ + ). This yields a spin qubit with |D as the ground state and |B as the exited state. The spin-CNT coupling Hamiltonian is accordingly transformed to H int k=1,2 g k σ x q k , where σ x = σ + + σ − , with σ − = |D B| and σ + = σ † − . When we further restrict our discussion to a dispersive regime ω q ± ω m |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 [61][62][63], the unitary dynamics of the system is then described by an effective Hamiltonian [64], and σ z = |B B| − |D D|. The Hamiltonian H cnt includes a coherent spin-mediated CNT-CNT coupling in the beam-splitter form, which is conditioned on the spin state. Here, we neglect the direct CNT-CNT coupling much smaller than the spin-mediated coupling [64]. Furthermore, we find that the decoupling of one CNT from the spin gives rise to a spin-induced 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 delayed-choice 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 beam-splitter coupling of strength J = 2g 2 ω q / ω 2 q − ω 2 m . Unitary evolution for a time τ 0 = π/ (4J) then leads to b 1 ( 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 [64]. We now turn to the quantum delayed-choice experiment with the macroscopic CNTs. We assume that the hybrid system is initially prepared in the state |Ψ i = b † 1 ⊗ I 2 |vac ⊗ |D , where |vac refers to the phonon vacuum and I 2 is the identity operator for the second 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 single-qubit rotation |D → cos (ϕ) |0 + sin (ϕ) |D [65][66][67] 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 beam splitter, the phase shifter and the quantum output beam splitter acting in sequence on a single photon in the Mach-Zehnder interferometer, as shown in Fig. 1(a). The final state of the system therefore becomes |vac /2 describe the particle and wave behaviors, respectively. We find from Eq. (2) that the mechanical phonon is in a quantum superposition of both a wave and a particle, and thus can exhibit both characteristics simultaneously. 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 (ϕ) |0 0| + sin 2 (ϕ) |D D| is capable of reproducing the same measured statistics as in Eq. (3) [12,[68][69][70]. Thus, in order to exclude the classical interpretation and prove the existence of the coherent wave-particle superposition, the quantum coherence between the states |0 and |D should be verified [20,21,25,26]. Experimentally, such a verification can be implemented by performing quantum state tomography to show all elements of the density matrix of the spin [67]. Here, in addition to γs/2π = 200γm/2π = 80 Hz, we assume that g/2π = 100 kHz, ωm/2π = 2 MHz, Ω = 10ωm, and ∆− = 142ωm, resulting in ωq 1.5ωm and then J/2π ×12 kHz, and that n th = 100, corresponding to an environmental temperature of 10 mK.
Next, we consider how to initialize and measure the mechanical system. In order to excite the left CNT to a single-phonon state, we make the spin qubit transition frequency ω q close to the mechanical frequency ω m , after the CNT is cooled into the quantum ground state [42,[71][72][73][74][75]. The spin-CNT coupling Hamiltonian is then approximately given by a Jaynes-Cummings-like Hamiltonian When acting for a time of = π/ (2g), such a Hamiltonian can, with the spin qubit in the excited state |B , transfer a mechanical excitation to the left CNT [76]. 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 [76][77][78][79][80]. Thus by directly measuring the occupation probability of |B , the phonon number in each CNT can be obtained.
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 delayed-choice experiment. Note that during τ T , we have neglected the spin single-qubit operation time due to the driving pulse length ∼ ns [81,82]. Since 0 ≤ τ 1 ≤ 2π/J, we focus on the maximum τ T : τ max T = 5π/ (2J). A modest spin-CNT coupling g/2π = 100 kHz, which can be obtained by tuning the current I and the distance d [64], is able to mediate an effective CNT-CNT coupling J/2π 12 kHz, thus giving τ max T 0.1 ms. The relaxation time T 1 of a single NV spin at low temperatures can reach up to a few minutes. Moreover, a single spin in an ultra-pure diamond example typically has a dephasing time T 2 2 ms even at room temperature [83,84], corresponding to a dephasing rate γ s /2π 80 Hz. 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, is the equilibrium phonon occupation at temperature T , Here, H (t) is a binary Hamiltonian of the form, In Eq. (5), 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. (4) drives the phonon occupation of the kth CNT to be For a realistic CNT, we can set the mechanical linewidth to be γ m /2π = 0.4 Hz [85], leading to a single-phonon lifetime of τ m = 1/γ m 400 ms. In this situation, τ m is much longer than the total operation time τ T , γ m τ T 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. (6). 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 γ m τ T 1, the fluctuation noise δn noise k in the phonon occupation n k is expressed, according to the analysis in the Supplemental Material [64], as δn noise 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 signal-to-noise ratio defined as R k = P k /δn noise k . The signal-resolved regime often requires 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 R k < 1, in particular, at finite temperatures. Nevertheless, we find that the total fluctuation noise S 2 = δn noise 1 2 + δn noise 2 2 is kept below an upper bound B 2 = γ m τ max T + 4n th γ m τ max T , and further that assuming B 2 < 1/2 can make either or both of R 1 and 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 signal-to-noise ratio R k . The ratio 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 R 1. However, at finite temperatures, n th increases as T , causing a decrease in R, as shown in Fig. 3. Therefore, the requirement of 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. 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.
Conclusions.-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 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 spin-nanomechanical setup, the present method could be directly extended to other hybrid systems, for example, mechanical devices coupled to a superconducting atom [55,76,86]. We believe that this proposed quantum delayed-choice 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 wave-particle nature, as an additional degree of freedom of phonons, may be widely exploited for quantum information applications [14,23].

SUPPLEMENTAL MATERIAL
Here, we, first, in Sec. S1 present more details of how to obtain the spin-controlled coherent coupling between separated mechanical resonators. Second, in Sec. S2, we show the detailed implementation of the controlled Hadamard gate, the phase gate, and the mechanical quantum delayed-choice experiment. Next, in Sec. S3, we derive in detail the phonon occupation of each CNT at finite temperatures. Then, Sec. S4 describes the detailed derivation of the fluctuation noise and the detailed analysis of the requirement of resolving the coherent signal from the environment-induced fluctuation. Finally, in Sec. S5 we show the method of the numerical simulation used in this work. S1. Spin-controlled coherent coupling between separated mechanical resonators The effective Hamiltonian H eff in the article describes a spin-mediated CNT-CNT coupling conditioned on the NV spin state. This is the basic element underlying our proposal. To understand more explicitly the spin-controlled coupling between the CNTs, in this section we derive in detail the effective Hamiltonian. We consider a hybrid quantum system consisting of two parallel CNTs and an NV electronic spin (a qutrit), as depicted in Fig. S1(a). Here, for convenience, illustrations in Figs. 1(b) and 1(c) in the article are reproduced in Figs. S1(b) and S1(a), respectively. The CNTs, respectively, carry dc currents I 1 and I 2 , both along the +x-direction. A spin is placed between them, at a distance d 1 (d 2 ) from the first (second) CNT. According to the Biot-Savart law, the CNTs can, at the position of the spin, generate a magnetic field B ( = x, y, z) is a unit vector in theˆ -direction, µ 0 is the vacuum permeability, and the subscript "cnt" refers to the CNTs. When the CNTs vibrate along theŷ-direction, the magnetic field is parametrically modulated by their mechanical displacements y 1 and y 2 , and then is reexpressed, up to first order, as B cnt = B cntẑ is a first-order modification, and where B (1) cnt = k=1,2 G k y k , with a magnetic-field gradient, Note that, here, y 1 > 0 (y 2 < 0) indicates a decrease in d 1 (d 2 ). Therefore, the sign, (−1) k−1 , in Eq. (S1) does not appear in Eq. (S2). Furthermore, an external magnetic field, B ext = B x (t)x + B zẑ , is applied to the NV spin. We have assumed, as required below, that B x (t) is a time-dependent component but B z is a dc component. The Hamiltonian governing the NV spin is therefore given by where g s 2 is the Landé factor, µ B the Bohr magneton, D 2π × 2.87 GHz the zero-field splitting, and S the -component of the spin operator S ( = x, y, z). In terms of the eigenstates, {|m s , m s = 0, ±1}, of S z , the operator S is expanded as and accordingly, the Hamiltonian H NV is transformed to We find that the magnetic field along theẑ-direction causes different Zeeman shifts to be imposed, respectively, on the spin states | ± 1 , and also that the magnetic field along thex-direction drives the transition between the spin states |0 and| ± 1 . The quantum treatment of the mechanical motion demonstrates that the mechanical vibrations of the CNTs can be modelled by two single-mode harmonic oscillators with a Hamiltonian where ω k is the phonon frequency and b k (b † k ) is the phonon annihilation (creation) operator. Here, we have subtracted the constant zero-point energy ω k /2. The mechanical displacement y k is accordingly expressed as where q k is the canonical phonon position operator, and y The last line in Eq. (S8) describes a magnetic coupling between the spin and the mechanical modes. In order to realize a tunable detuning between them, B x (t) is chosen to be B x (t) = B 0 cos (ω 0 t) with amplitude B 0 and frequency ω 0 . In a frame rotating at H rot = ω 0 (| − 1 −1| + | + 1 +1|), the full Hamiltonian can be divided into two parts, account for the low-and high-frequency components, respectively. Here, we have defined Roughly, having δ ± = δ ± + 2ω 0 Ω allows one to make the rotating-wave approximation (RWA), and to straightforwardly remove H high . However, as demonstrated in Sec. S5, the accumulated error increases during the evolution, causing the dynamics driven by H low to deviate largely from that driven by H F . Thus, we are not using the RWA here. In order to suppress the error accumulation, we need to analyze the effects of H high in the limit δ ± Ω. In such a limit, we can employ a time-averaging treatment for the high-frequency component H high [S1-S3], and as a result, its effective behavior is described by the following time-averaged Hamiltonian, where the first line corresponds to the energy shifts of the spin states | ± 1 , and the second line describes a coherent coupling between these. Accordingly, the full Hamiltonian H F is approximated to be a time-independent form, As seen in Sec. S5, the error accumulation is strongly suppressed when H high is included. Tuning B (0) cnt + B z = 0 yields δ + = δ − = ∆ − and δ + = δ − = ∆ + , implying that the spin states | ± 1 have the same Zeeman shift of ∆ = ∆ − + 3Ω 2 /∆ + , as shown in Fig. S1(b). Therefore, we can define a bright state, |B = (| + 1 + | − 1 ) / √ 2, which is dressed by the spin state |0 , and a dark state, |D = (| + 1 − | − 1 ) √ 2, which decouples from the spin state |0 . In terms of the states |B and |D , the full Hamiltonian becomes The dressing mechanism allows us to introduce two dressed states, where tan (2θ) = 2 √ 2Ω/∆. Upon substituting them back into the full Hamiltonian in Eq. (S14) and then using the identity operator I = |D D| + |Φ − Φ − | + |Φ + Φ + |, we can straightforwardly obtain Here, Under the assumption of ∆ Ω, we have θ 0, such that sin (θ) sin (2θ) 0, cos (θ) cos 2 (θ) cos (2θ) 1, ω + ∆ + 4Ω 2 /∆, ω D ∆ + 2Ω 2 /∆, and |Φ + |B . In this limit, the coupling between |0 and |B only causes an energy splitting, of 2Ω 2 /∆, between the states |B and |D , so |B and |D can be used to define a spin qubit. Correspondingly, the full Hamiltonian is approximated as where ω q = 2Ω 2 /∆ + 2Ω 2 /∆ + , σ z = |B B| − |D D|, and σ x = σ + + σ − with σ − = |D B| and σ + = σ † − . Modest parameters [S4-S9], m k = 1.0 × 10 −22 kg, ω k /2π = 2 MHz, d k 2 nm, and I k 380 nA, could result in a spin-CNT coupling of up to g k /2π 100 kHz.
Furthermore, from Eq. (S22) it is found that the sequential actions of the terms σ + b 1 and σ − b † 2 , as well as of the counter-rotating terms σ − b 1 and σ + b † 2 , can transfer a mechanical phonon from the left to the right CNT, and the reverse process is caused by their Hermitian conjugates. When restricting our discussion to a dispersive regime, this phonon transfer becomes dominant. Hence, in the dispersive regime the dynamics described by H F in Eq. (S22) enables a spin quantum bus for the mechanical phonons and can be used to realize a coherent CNT-CNT coupling.
In order to show more explicitly, we rewrite H F in the interaction picture as The condition in Eq. (S23) justifies to use a time-averaging treatment of the Hamiltonian H F [S1, S2]. In the timeaveraging treatment, all terms in Eq. (S24) are considered as high-frequency components and exhibit time-averaged behaviors. Based on this, the dynamics of the system can be determined by an effective Hamiltonian Here, we have assumed that ω k = ω m . As expected, Eq. (S25) shows a coherent spin-mediated CNT-CNT coupling, corresponding to the standard linear coupler transformation, which can give rise to a direct phonon exchange. Thus in this case, the spin qubit works as a quantum bus. At the same time, it also shows that the CNT-CNT coupling can be turned off if the intermediate spin is in the state |0 . This is because the NV spin in the state |0 is decoupled from the CNTs, and the mechanical phonons can no longer be transferred from one CNT to another. Specifically, if the spin is in the state |D or |B , the CNTs are coupled; however, if the spin is instead in the state |0 , they are decoupled. Note that in Eq. (S25) ac Stark shifts caused to be imposed on the qubit have been excluded because we focus only on the quantum states of the CNTs. In the last part of this section, we evaluate the direct coupling between the CNTs. For simplicity, we assume that I k = I, d k = d, and that the CNTs have the same length L. The attractive force acting on the kth CNT is where is the force size. The work done by the force is given straightforwardly by After applying a perturbation expansion and then a quantization, this direct CNT-CNT coupling is found to be where For a modest setup [S4-S9], m = 1.0 × 10 −22 kg, ω m = 2π × 2 MHz, L = 10 nm, d = 2 nm, and I = 380 nA, we have which is much smaller than the mechanical resonance frequency ω m , and also have which is much smaller than the spin-mediated CNT-CNT coupling, for example, 2π × 12 kHz, as shown in the section below. Therefore, the direct CNT-CNT coupling can be neglected in our setup.

S2. Controlled Hadamard gate, phase gate, and mechanical quantum delayed-choice experiment
In order to implement a quantum delayed-choice experiment with macroscopic CNT mechanical resonators, we need a controlled Hadamard gate and a phase gate to act on the CNT mechanical modes. Below, we demonstrate how the effective Hamiltonian in Eq. (S25) can be used to make all required gates. Let us first consider the controlled Hadamard gate. Tuning the currents to be I k = I and, at the same time, the distances to be d k = d results in a symmetric coupling g k = g. The effective Hamiltonian H eff is accordingly reduced to H eff = H cnt ⊗ σ z , where is a beam-splitter-type interaction, and where is an effective CNT-CNT coupling strength. In our discussion, the NV spin is restricted to a subspace spanned by {|0 , |D }, where the spin is a control qubit of a Hadamard gate. The spin in the state |D mediates the coherent coupling between the separated CNTs, and causes them to evolve under the Hamiltonian H cnt in Eq. (S34). According to the Heisenberg equation of motion, b k (t) = exp (iH cnt t/ ) b k exp (−iH cnt t/ ), the unitary evolution for a time t = τ 0 ≡ π/ (4J) corresponds to a Hadamard-like gate, However, when the spin state is |0 , the two CNTs decouple from each other. In this case, their quantum states remain unchanged under the unitary evolution, yielding We have therefore achieved a spin-controlled Hadamard gate between the CNTs. That is, if the NV spin is in the state |D , then the Hadamard operation is applied to the CNTs, and if the NV spin is in the state |0 , then the states of the CNTs are unchanged. We next consider the phase gate. For the phase gate, we tune the currents to be I 1 = 0 and I 2 = 0, such that g 1 = g and g 2 = 0, causing the effective Hamiltonian in Eq. (S25) to become We find from Eq. (S40) that there exists a spin-induced shift, J, of the mechanical resonance. This dispersive shift can, in turn, introduce a dynamical phase, φ (t) = Jt, onto the first CNT. With the spin being in the state |D , we solve the Heisenberg equations of motion for the CNTs, and then obtain a phase gate, In fact, similar to the controlled Hadamard gate discussed above, the phase gate can also be controlled by the spin according to Eq. (S40).
Having achieved all required gates, we now turn to the detailed description of the macroscopic quantum delayedchoice experiment with CNT resonators. The hybrid system is initially prepared in the state |Ψ i ≡ |Ψ (0) = b † 1 ⊗ I 2 |vac ⊗ |D , where |vac refers to the phonon vacuum of the CNTs and I 2 is the identity operator on the second CNT. First, we turn on the currents of the CNTs and ensure I k = I. After a time τ 0 , a Hadamard operation is applied to the CNTs and accordingly, |Ψ i becomes Then, we turn off the current of the second CNT for a phase accumulation for a time τ 1 . As a consequence, the system further evolves to While achieving the desired phase φ, we make a spin single-qubit rotation |D → cos (ϕ) |0 + sin (ϕ) |D , and have Here, note that, we have ignored the length of the driving pulse of the spin rotation as being of the order of ns, and thus assumed that the state of the CNTs remains unchanged. At the end of the driving pulse, we turn on the current of the second CNT again and hold for another τ 0 to perform a Hadamard gate. This gate is in a quantum superposition of being present and absent. The three operations on the mechanical phonon correspond to the actions, on a single photon, of the input beam splitter, the phase shifter, and the output beam splitter, respectively, in quantum delayed-choice experiments with a Mach-Zehnder interferometer. The final state is therefore given by  P1 and (b) P2 as a function of the rotation angle ϕ and the relative phase φ. This represents a continuous transition between a particle-type behavior (ϕ = 0) and a wave-type behavior (ϕ = π/2).
describe particle and wave behaviors, respectively. This reveals that the CNT mechanical phonon is in a quantum superposition of both a particle and a wave. The probability of finding a single phonon in the kth CNT is expressed as according to Eq. (S46). In Fig. S2, we have plotted the probabilities P 1 and P 2 versus the rotation angle ϕ and the relative phase φ. In this figure we find that the mechanical phonon shows a morphing behavior between particle (ϕ = 0) and wave (ϕ = π/2). Note that the spin, in a classical mixed state of the form cos 2 (ϕ) |0 0| + sin 2 (ϕ) |D D|, would lead to the same measured statistics in Eq. (S49), that is, a local hidden variable model is capable of reproducing the quantum predictions. This is a loophole [S10-S13]. However, as discussed in Refs. [S14-S17], this loophole can be avoided as long as the second Hadamard operation is ensured to be in a truly quantum superposition of being present and absent. In our proposal, the second Hadamard operation is conditioned on the spin state. If the spin is in the |0 state, then the Hadamard operation is absent; if the spin is in the |D state, then the Hadamard operation is present; if the spin is in a quantum superposition of the |0 and |D states, then the Hadamard operation is in a quantum superposition of being present and absent. To confirm such a quantum superposition, in Fig. (S3) we numerically calculate the fidelity, F = f Ψ|ρ actual (τ T ) |Ψ f , between the desired state |Ψ f in Eq. (S46) and the actual state ρ actual (τ T ) obtained from the exact master equation in Eq. (S120). From this figure, we find that the fidelity is very close to unity even for the finite temperature of T 10 mK. Furthermore, in experiments, in order to exclude the classical interpretation and prove the existence of the coherent wave-particle superposition, the quantum coherence between the states |0 and |D should be verified. Experimentally, this coherence can be prepared by a spin single-qubit operation [S18-S20], and can be verified by performing quantum state tomography to show all the elements of the density matrix of the spin [S20]. (S120). Here, in addition to γs/2π = 200γm/2π = 80 Hz, we have assumed that g/2π = 100 kHz, ωm/2π = 2 MHz, Ω = 10ωm, and ∆− = 142ωm, resulting in ωq 1.5ωm and then J/2π 12 kHz. We have also assumed that n th = 100, which corresponds to an environmental temperature of 10 mK.

S3. Phonon occupation at finite temperatures
We begin by considering the total operation time, which is given by τ T = 2τ 0 + τ 1 , as discussed in Sec. S2. Here, τ 0 = π/ (4J) is the time for the Hadamard gate and τ 1 ∈ [0, 2π/J] is the time for the phase gate. In a realistic setup, we can assume ω m /2π 2 MHz, ω q /2π 3 MHz, and g/2π = 100 kHz, such that J/2π 12 kHz, yielding a maximum total time τ max T = 2τ 0 + τ max 1 0.1 ms, where τ max 1 = 2π/J is the maximum phase gate time. Note that, the operation time τ T depends inversely on the CNT-CNT coupling strength J, but the enhancement in J is limited by the validity of the effective Hamiltonian H eff .
The total decoherence in our setup can be divided into two parts, one from the spin and the other from the CNTs. The spin decoherence in general includes the relaxation and the dephasing. For an NV electronic spin, the relaxation time T 1 can reach up to several minutes at low temperatures and the dephasing time can be T 2 2 ms even at room temperature [S21, S22]. These justify neglecting the spin decoherence. For the mechanical decoherence, despite a long phonon life, the low mechanical frequency makes the CNT mechanical modes very sensitive to the environmental temperature. In this section and in Sec. S4, we discuss the effects of the mechanical noise on our quantum delayedchoice experiment, and demonstrate that the morphing between wave and particle can still be effectively observed even at finite temperatures.
As a result, the dissipative processes, in the hybrid system considered here, are induced only by the mechanical decoherence, which arises from the vacuum fluctuation and thermal noise. The full dynamics of the system can then be governed by the following master equatioṅ where ρ is the density operator of the system, γ m is the mechanical decay rate, Here, H (t) is a binary Hamiltonian of the form, with The three time intervals in Eq. (S51) correspond to the first Hadamard gate, the phase gate and the second Hadamard gate, respectively. Note that in Eq. (S51), we did not include the spin single-qubit rotation before the third interval because the length of the driving pulse is of the order of ns. We can derive the system evolution step by step.
Let us now consider the first evolution interval 0 < t ≤ τ 0 . During this interval, the coupling of the CNT mechanical modes introduces two delocalized phononic modes, such that H 0 is diagonalized to be and the master equation in Eq. (S50) is reexpressed, in terms of the modes c ± , aṡ In order to calculate the phonon occupations at the end of the first interval, we need to obtain the equations of motion for c † ± c ± , c † + c − , c † + c − σ z , and c † + c − σ 2 z . Here, O represents the expectation value of the operator O. Following the master equation in Eq. (S56), we have which, in turn, gives which is the phonon occupation of the kth at the end of the third interval. For a realistic CNT, the mechanical linewidth can be set to γ m /2π = 0.4 Hz [S23], and then we obtain a phonon lifetime of 400 ms, which is much longer than the maximum total time τ max T 0.1 ms. This ensures γ m τ T 1, which results in and as a result, with It is seen that on the right-hand side of Eq. (S106), the first term arises from the particle behavior of a phonon and the second term arises from its wave behavior. , respectively, and symbols correspond to numerical simulations. These analytical and numerical results exhibit an exact agreement. For all plots, in addition to γs/2π = 200γm/2π = 80 Hz, we have assumed that g/2π = 100 kHz, ωm/2π = 2 MHz, Ω = 10ωm, and ∆− = 142ωm, resulting in ωq 1.5ωm and then J/2π 12 kHz. We have also assumed that n th = 100, corresponding to an environmental temperature of 10 mK.
By substituting Eq. (S106) into Eq. (S75), the fluctuation δn k in the occupation n k is given by (δn k ) 2 = n 2 th − 2P k n th − P 2 k exp (−2γ m τ T ) − (2n th + 1) (n th − P k ) exp (−γ m τ T ) + n th (n th + 1) . (S107) Since γ m τ T 1, we have where δn signal Here, δn signal k , the quantum fluctuation induced by the Heisenberg uncertainty principle, accounts for the coherent signal, and δn noise k represents the fluctuation noise, including the vacuum (the first term) and thermal (the second term) fluctuations. To confirm the predictions of Eq. (S108), we perform numerics, as shown in Fig. S4. Specifically, we plot the fluctuation noises δn noise 1 and δn noise 2 versus the relative phase φ. The analytical expression is in excellent agreement with our numerical simulations. Furthermore, the respective CNT signal-to-noise ratios can be defined as (S111) Note that, here, we did not use δn k to define R k because δn signal k in δn k results from quantum fluctuations of the desired signal, as mentioned previously; and therefore this is not the environmental noise. In order to resolve a signal from the fluctuation noise, the ratio R k is required to be R k > 1. However, Eq. (S111) demonstrates that this criterion is not always met for all values of P k , in particular, at finite temperatures. For example, P k = 0 leads directly to R k = 0. To address this problem, we now consider the total fluctuation noise, (S112) We further assume that S 2 < P 2 1 + P 2 2 . (S113) Under this assumption, if R k < 1, then R 3−k > 1 for k = 1, 2; otherwise R 1 > 1, R 2 > 1. This means that at least one of the signals, P 1 or P 2 , is resolved for each measurement. Because the coherent phonon number equal to 1 is conserved, and therefore the signals in the two CNTs are complementary, the unresolved signal can be completely deduced from the resolved one. Thus, the criterion in Eq. (S113) ensures that the morphing behavior between wave and particle can be observed from the environment-induced fluctuation noise. In fact, for any value of P k , the total noise S is limited by an upper bound, which is independent of P k . Meanwhile, P 2 1 + P 2 2 is also limited by a lower bound √ 2/2. Thereby, in order to meet the criterion given in Eq. (S113), it is required that (S115) equation given byρ where σ z = |D D| − |0 0|, and H F is the full Hamiltonian of Eq. (S8). Here, we use the Python framework QuTiP [S24, S25] to set up this problem. However, the full Hamiltonian is time-dependent, and it takes a long time to integrate the corresponding Schrödinger equation or the master equation, in particular, for our case, where all quantum gates result from the deterministic time evolution of the system. Thus, in our numerical simulations, we replace H F with H low + H high , as in Eq. (S13). This is a reasonable replacement because in our proposal Ω (tens of MHz) is required to be much smaller than ∆ (up to ∼ GHz). In Fig. S6, we plot the unitary evolution of the phonon occupations, b † 1 b 1 and b † 2 b 2 , of the CNTs. Symbols are the exact results from the full Hamiltonian H F and solid curves are given by the approximate Hamiltonian H low + H high . We find an excellent agreement for a very long evolution time, and thus H F can be very well approximated by H low + H high . For additional comparison, we also plot the phonon occupation evolution driven only by the low-frequency component H low , corresponding to dotted curves. As seen in Fig. S6, owing to the error accumulation, the dynamics of H low deviates largely from the full dynamics of H F , even within one oscillation cycle. With the above replacement, we obtain the numerical simulations plotted in Fig. 2 of the article, and also in Fig. S4 of the Supplemental Material.