

Regular Articles Twomode Squeezing in an Electromechanical ResonatorAbstractA mechanical resonator integrated with piezoelectric transducers enables mechanical nonlinearities to be dynamically engineered to emulate nondegenerate parametric downconversion. In this configuration, millions of phonons are simultaneously generated in pairs in two macroscopic vibration modes, which results in the amplification of their motion by more than 20 dB. Mechanical twomode squeezed states are also created in parallel, which exhibit fluctuations 5 dB below the thermal level of their constituent modes and they harbor correlations between the modes that become perfect as their amplification is increased. This remarkable observation of correlations between two massive phonon ensembles establishes the means to create an entangled macroscopic mechanical system at the single phonon level. Keywords: electromechanical resonator, parametric downconversion, squeezedstates, phonons 1. IntroductionElectromechanical systems consist of a single degree of mechanical freedom hosting a spectrally pure resonance that is embedded in an electrical transduction circuit, and they have emerged as a versatile platform for a range of technological applications and to study fundamental science [1, 2]. For instance, their small inertial mass and high quality resonance have enabled the development of ultraprecise sensors that can detect a single electron spin and even the mass of a proton leading to a new class of medicinal diagnostic technology [3–5]. The electrical transduction circuit can also enable the underlying potential energy landscape of the mechanical resonator to be dynamically engineered, yielding a range of nonlinear motional dynamics that can be harnessed for both information storage and processing, which brings forth the concept of mechanical computation offering both ultralow power consumption and the capacity for unprecedented parallel data processing [6–9]. Most tantalizingly, a high frequency mechanical resonator operated at sufficiently low temperatures can even condense into its quantum ground state, where its low mass and spectral purity yield quantum zero point fluctuations that are large enough to be observed, enabling a macroscopic quantum system to be studied [10–12]. In this instance, although the mechanical resonator is composed of billions of atoms, only the phonons sustained by the fundamental resonance mode are quantized, and it corresponds to a tangible vibration of the mechanical element. This phonon picture, namely the collective excitation of atoms in the fundamental vibration mode, enables concepts from atomic molecular optical (AMO) physics developed for photons, that is, a quanta of electromagnetic radiation, to be exploited. Indeed, it is the laser cooling techniques pioneered in AMO physics that have been utilized most successfully to cool the macroscopic mechanical oscillator so that on average, its fundamental mode is occupied by much less than one phonon [13, 14]. Although the notion of a macroscopic mechanical resonator composed of billions of atoms being considered only in terms of the number of phonons sustained by its fundamental mode might seem counterintuitive, this paradigm has successfully been exploited to demonstrate the signature feature of photonics, namely lasing but with the localized phonons in the mechanical resonator [15–18]. Consequently, the ability to cool a solidstate macroscopic mechanical resonator into its ground state opens up the possibility of generating an allmechanical macroscopic entangled state [10, 19], which would enable foundational aspects of quantum mechanics to be queried such as the nature of the quantum to classical divide and the absence of quantum phenomena in our everyday classical world [20, 21]. 2. Twomode squeezed statesEntangled photons in the guise of twomode squeezed states were among the first nonclassical states of light to be generated in the lab and nowadays are routinely siphoned from spontaneous parametric downconversion [22]. This typically involves a χ^{(2)} nonlinear crystal that is exposed to a strong laser pump beam whose dielectric polarization responds nonlinearly to the pump’s electric field. In this process, a pair of photons—the signal and the idler—are generated at the expense of the pump photons, which conserve both energy and momentum, as depicted in Fig. 1(a). Remarkably, the quantum fluctuations in the amplitude and phase of the generated photon pair are correlated, as encapsulated by the Einstein, Podolsky, and Rosen (EPR) paradox, where their individual fluctuations are amplified, while their relative fluctuations are reduced below the vacuum noise level [23–26]. The EPR paradox recast in the Bell inequality has been routinely violated via the entangled signal and idler photons whose fluctuations remain correlated even when they are separated over distances of kilometers [27, 28]. The nonclassical light generated from spontaneous parametric downconversion has subsequently engendered a whole host of quantumenabled technologies including quantum communication [29], optical quantum computing [30], quantum teleportation [31], quantum enhanced measurements and metrology [32, 33]. This unprecedented success has led to twomode squeezed states being translated to other parts of the electromagnetic spectrum, for instance microwaves utilizing superconducting circuits [34]. Specifically, a superconducting resonator terminated with a nonlinear element such as a superconducting quantum interference device (SQUID), as generically depicted in Fig. 1(b), can enable parametric downconversion [35, 36]. Pumping the SQUID with flux in these socalled Josephson parametric amplifiers (JPAs) yields a nonlinear variation in the boundary conditions of the microwave resonator which creates a frequency modulation in the standing waves it hosts. If the pump frequency coincides with the sum of two modes (say ω_{i} and ω_{ii} as depicted in Fig. 1(b)) in the microwave resonator, it nondegenerately amplifies their vacuum noise, generating signal and idler photons in a process reminiscent to that described above where these photons are entangled [37, 38]. Indeed, JPAlike devices have now opened the door to many of the concepts pioneered with optical twomode squeezed states but in an onchip microwave circuit that could even be yoked into a quantum computer. A variation on these microwave circuits has recently emerged in the form of cavity electromechanical systems where the cavity, namely the microwave resonator, is composed of an inductor and capacitor in series, and the mechanical resonator forms a mechanically compliant element of the parallel plate capacitor, thus capacitively coupling the mechanics to the microwaves as generically depicted in Fig. 1(c) [39–41]. Electric fields in the forms of microwaves can be pumped into this hybrid circuit via a capacitively coupled transmission line that nonlinearly modulates the energy stored in the capacitor and hence the frequencies of the mechanics and the cavity. If the pump coincides with the sum frequency of the subsystems (i.e. ω_{i} and ω_{1} as depicted in Fig. 1(c)), it behaves like a nondegenerate parametric amplifier and it creates photons and phonons in pairs that are correlated in a twomode squeezed state [42, 43]. This remarkable demonstration illustrates that two vastly dissimilar systems, namely light and sound, can be entangled in a macroscopic context, providing a new regime in which quantum physics can be explored [14]. The cavity electromechanical system detailed in Fig. 1(c) straddles the two extremes of light and sound, and it naturally suggests the possibility of a purely mechanical analogue of a twomode squeezed state in a phononcavity electromechanical system as depicted in Fig. 1(d). As in the microwave case, the mechanical resonator can also sustain multiple modes, say ω_{1} and ω_{2}, as shown in Fig. 1(d), which if nonlinearly coupled, could yield nondegenerate parametric amplification. In contrast to the microwave circuit that needs to be engineered with a nonlinear element, namely the SQUID, to enable the modes to couple, the modes in the mechanical resonator can dispersively couple naturally via the strain induced from the motion of a given mode that modifies the restoring potential, that is, the spring constant and hence, the natural frequency of the other modes [44]. Consequently, if the strain in the mechanical element is sinusoidally pumped, it will nonlinearly modulate the frequencies of all the localized modes, and once this modulation coincides with a pair of modes, it will nondegenerately amplify their motion and in the process generate a twomode squeezed state [45]. Critically, however, an entanglement will only be generated if the constituent modes are initialized in their quantum ground state [10–12], but the ability to harness this interaction, even if the modes in question are thermalized, would lay a pivotal marker on the road to generating an allmechanical macroscopic entanglement [46].
3. Electromechanical resonatorAlthough degenerate parametric amplification was demonstrated in the early nineties, it could not be used to disentangle correlations between the signal and idler phonons due to their having identical frequencies and being spatially localized to the same mode in the mechanical resonator [47]. Alternatively, a nondegenerate variant of this process in the first two modes of a beam resonator yielded only a modest phonon generation rate, which was insufficient to significantly amplify the thermomechanical fluctuations before they were dissipated, resulting in statistically insignificant correlations [45]. To that end, the electromechanical resonator shown in Fig. 2(a) was developed, which consists of two mechanical elements that are strongly coupled via the exaggerated overhangs between them [48]. This results in the two lowestorder vibration modes shown in Figs. 2(b) and 2(c) that are extracted from a finite element calculation and henceforth labeled symmetric (S) and asymmetric (A), which are closely spaced in frequency. In addition, the spatial profiles of both modes have a large overlap yielding greater strainmediated dispersive coupling between them. This combination of enhanced dispersive coupling and a smaller frequency separation, almost tending towards a degenerate configuration, is designed to yield larger nondegenerate parametric amplification when the strain in this system is modulated at the sum frequency of both modes. The piezoelectric effect is utilized in order to modulate the strain in the mechanical modes. By design, the electromechanical system is fabricated from gallium arsenide (GaAs), which is piezoelectrically active [6, 49]. The piezoelectric transducer is formed from a GaAs conducting layer and a gold gate electrode sandwiching an insulating GaAs layer, which is integrated directly into the mechanical element, as shown in Fig. 2(a). Application of an electric field across the transducer generates inplane strain that can nonlinearly modulate the spring constants and hence, the frequencies of the modes supported by the mechanical system [6]. The Hamiltonian for the electromechanical system in this configuration can then be expressed as: Here, P and Q are the canonical momentum and conjugate position of the symmetric and asymmetric modes with mass m and natural frequency ω_{n}, where the summation expresses their kinetic and potential energies. The last term describes the piezoelectrically activated nondegenerate parametric downconversion with amplitude Λ and frequency ω_{p} = ω_{S} + ω_{A}, that is, at the sum frequency of the modes in question. Ostensibly, this Hamiltonian is classical, which superficially suggests the unavailability of twomode squeezing. However, the last term is analogous to (ab + a^{†}b^{†}) for nondegenerate parametric downconversion in cavity electromechanical systems in the frame rotating at the pump frequency, where a, b and a^{†}, b^{†} are the annihilation and creation operators for the mechanical and microwave resonators, respectively, and is the intrinsic coupling rate between them that is enhanced by n pump photons [41–43]. Consequently, Λ encapsulates the intrinsic coupling rate between the symmetric and asymmetric modes that is enhanced by the piezoelectric pump, which results in phonon pairs being simultaneously generated in them [50]. Remarkably, these phonons should still be correlated, even if the two modes are thermalized and far from their quantum ground state, as long as their generation rate exceeds the rate at which they are dissipated from both modes [46]. 4. Results4.1 Nondegenerate parametric amplificationTo ascertain if this latter expectation is satisfied, the thermomechanical fluctuations of both modes are spectrally measured via optical interferometry with a 3µW helium neon laser probing the right element of the electromechanical system at room temperature and in a high vacuum [46]. This measurement reveals the modes with natural frequencies ω_{S }/ 2π ≈ 246 kHz and ω_{A }/ 2π ≈ 262 kHz with quality factors Q_{n} = ω_{n }/ жд ω_{n} of 1300 and 2200, respectively, from their spectral power bandwidth жд ω_{n}, and it corresponds to both modes sustaining > 10^{7} phonons. Next nondegenerate parametric downconversion is activated, and the thermal fluctuations from both modes are monitored. The results of this measurement shown in Fig. 2(d) indicate that as the piezoelectric pump voltage is increased, the thermomechanical fluctuations of both modes are enhanced. This amplification can be referenced to the bare thermal fluctuations, and it yields gains of more than 20 dB that are accompanied by a narrowing of the power spectral bandwidth from both modes, as detailed in Fig. 2(e). At the largest pump amplitudes (>0.275 V_{rms}) жд ω_{n} → 0, resulting in both modes undergoing nondegenerate parametric resonance [51, 52]. To confirm these experimental observations, the equations of motion for both modes are extracted from the Hamiltonian in eq. (1) and reformulated in their rotating frames with Q_{n} = X_{n} cos(ω_{n}t) + Y_{n} sin(ω_{n}t), where X_{n} and Y_{n} are the slowly varying inphase and quadrature components. The resultant equations are then numerically solved, as detailed in a previous study [46], and they faithfully reproduce the experimental response as shown in Fig. 2(f), thus verifying that the amplification can be ascribed to nondegenerate parametric downconversion.
The temporal dynamics of this amplification can also be acquired by mixing the output from the optical interferometer with local oscillators locked onto the resonances of both modes and then demodulated in a phase sensitive detector (PSD). The PSD samples the random displacement fluctuations of the mechanical modes driven by the thermal Langevin force at a rate of 50 ms over a period of 300 s. This yields four time series for the inphase and quadrature components of both modes, enabling their phase portraits to be constructed as shown in Figs. 3(a) and 3(b). This measurement reveals that the thermal fluctuations of both modes are random and uncorrelated as evidenced by their circular distribution in phase space, indicating all vibration phases are equally likely (red and blue points in Figs. 3(a) and 3(b)). Repeating this measurement with a pump amplitude of 0.25 V_{rms} confirms that the fluctuations in both modes are enhanced via the nondegenerate parametric downconversion while retaining their random nature; namely, this amplification is phase insensitive [53]. The observed amplification arises from phonons generated in both modes from the same process; hence, their fluctuations should be correlated. To that end, the cross quadratures are constructed in phase space from the inphase component of the symmetric mode and the quadrature component of the asymmetric mode and vice versa, as shown in Figs. 3(c) and 3(d). This unveils squeezed distributions where a particular phase orientation is amplified while the perpendicular phase is deamplified. The narrowness of this distribution implies the existence of correlations between the symmetric and asymmetric modes (if no correlations between the modes existed, the crossquadratures would yield circular and therefore random distributions) and is the signature feature of a twomode squeezed state [54, 55]. However, in order to quantitatively verify the existence of correlations between the modes, two criteria need to be satisfied. First, the twomode squeezed distributions should exhibit smaller fluctuations than the bare distributions; otherwise, the correlations would be washed out by the thermomechanical noise from both modes. Second, the correlations should be statistically evident from the temporal data used to construct the phase portraits.
4.2 Squeezing below the thermal noiseThe former condition can in fact be visibly confirmed by examining the data in Figs. 3(c) and 3(d), which indicate that the deamplified phases encompass a narrower distribution than the bare thermal motion from both modes. Quantitatively, new axes x^{+} and y^{−} are introduced, as shown in the inset to Fig. 3(c), onto which the counts in the squeezed distributions are projected, as shown in Figs. 4(b) and 4(c) [46]. This reveals Gaussian profiles with a zero mean displacement, as expected from random fluctuations driven by the thermal Langevin force, whose full width at half maximum can be used to extract their standard deviation σ. This analysis is repeated as a function of nonzero pump amplitude for the cross quadratures X_{S}: Y_{A} and X_{A}: Y_{S}, where the standard deviations for both the amplified and deamplified phases corresponding to the x^{+} and y^{−} axes are determined as shown in Fig. 4(a). Naturally, as the pump amplitude is increased, the fluctuations along the x^{+} axis are amplified, while concurrently they are reduced along the y^{−} axis where entropy is conserved in this process. The gain can then be extracted by normalizing with the narrowest standard deviation from the bare thermal distribution (corresponding to the quadrature component of the asymmetric mode). While this slightly overestimates the amplification, it is consistent with the spectral response detailed in Fig. 2(e). However, this normalizing reference underestimates the deamplification, yielding a conservative 5dB suppression of the mechanical fluctuations below the thermal level where further reduction is inhibited by noise from the piezotransducers at large pump voltages [46].
To confirm these observations, the thermal Langevin force, in the equations of motion extracted from the Hamiltonian in eq. (1), is decomposed into inphase and quadrature components [16, 45, 46]. This leads to a new set of composite variables for the squeezed distributions given by X_{S} ± Y_{A} and X_{A} ± Y_{S} that naturally correspond to the rotated axes x^{+} and y^{−} introduced in Fig. 3(c) [37, 46]. The pumpinduced gain in the fluctuations of these composite variables can then be extracted from the Langevin correlation function, yielding the solid lines in Fig. 4(a). This verifies that the noise reduction below the thermal level arises from the nondegenerate parametric downconversion. 4.3 Correlation coefficientThe statistical correlations between the time series for the inphase and quadrature components from both modes can be analyzed via the absolute correlation coefficient cov(Z_{i}Z_{i}) /σ_{Zi}σ_{Zi}, where Z_{i} ∈ {X_{S}, Y_{S}, X_{A}, Y_{A}} and the numerator describes the covariance. Analyzing the bare thermal fluctuations (shown by the red and blue points in Figs. 3(a) and 3(b)) enables a correlation coefficient matrix to be constructed, yielding 16 permutations. However, only 10 combinations are relevant due to their symmetry, as shown in Fig. 5(a). This reveals that the diagonal elements corresponding to the autocorrelations of Z_{i} yield a coefficient of 1, which indicates that they are perfectly correlated with themselves, as one would expect. However, all the offdiagonal elements yield a coefficient of 0, indicating an absence of correlations; for instance cov(X_{S}Y_{S}) /σ_{XS}σ_{YS} ≈ 0, which is unsurprising, as this maps the circular distribution in Fig. 3(a) corresponding to the random uncorrelated fluctuations driven by the thermal Langevin force. Repeating this analysis as a function of pump amplitude ranging from 0.1, 0.2, and 0.25 V_{rms} as shown in Figs. 5(b)–(d) indicates that the autocorrelations in the diagonal elements remain perfect. However, the offdiagonal elements corresponding to the crossquadrature phase portraits in Figs. 3(c) and 3(d), namely, X_{S}: Y_{A} and X_{A}: Y_{S}, emerge and converge towards 1, which implies that the two modes have become perfectly correlated from the simultaneous generation of phonon pairs via nondegenerate parametric downconversion.
5. Implications and outlookAlthough both mechanical modes sustain more than 10 million phonons at room temperature, the simultaneous generation of phonon pairs from the nondegenerate parametric downconversion occurs at a rate that renders their fluctuations indistinguishable. Thus, remarkably, these thermalized macroscopic mechanical vibration modes become perfectly entwined in a twomode squeezed state. Ultimately, if the modes can be operated with phonon populations of much less than one, then these correlations will manifest themselves in a macroscopic allmechanical entanglement [10, 42, 56]. Access to such a state would be extremely tantalizing, as it would provide an invaluable platform to investigate the absence of quantum mechanical phenomena in our everyday classical world [20]. From a technological point of view, these results herald the emergence of quantum optics to phononics, and thus, concepts such as quantum cryptography and optical quantum computing could potentially be harnessed with sound in a microchip. More immediately, the possibility of even greater squeezing becomes available via more strongly coupled modes in conjunction with a more efficient piezoelectric pump [57]. This enhanced squeezing would not only offer detectors that could operate below the quantum limit, yielding unprecedented sensitivities for metrological applications [33] and fundamental science [58], but it could even be utilized to create room temperature entanglements, thus bringing quantum sound into a more technologically accessible regime [59]. References
