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Abstract
Spontaneous wavefunction collapse theories provide the possibility to resolve the measurement problem of quantum mechanics. However, the best experimental tests have been limited by thermal fluctuations and have operated at frequencies far below those conjectured to allow the proposed cosmological origin of collapse to be identified. Here we propose to use high-frequency nanomechanical resonators to surpass these limitations. We consider a specific implementation that uses a breathing mode of a quantum optomechanical system cooled to near its motional ground state. The scheme combines phonon counting with efficient mitigation of technical noise, including nonlinear photon conversion and photon coincidence counting. It can resolve the exquisitely small phonon fluxes required for a conclusive test of collapse models as well as testing the hypothesis of a cosmological origin of the collapse noise.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Quantum mechanics is one of the most transformative physical theories of the 20th century. However, while the evolution of the quantum wave function is deterministically described by Schrödinger’s equation, the outcome of a measurement is probabilistic, given by Born’s rule. Despite recent progress [1–3], there is no consensus on how to reconcile these two viewpoints, as illustrated by the measurement paradox [4]. There are two conceptually distinct approaches: Either the interpretative postulates must be modified [5–9], or quantum mechanics approximates a deeper theory yet to be discovered. The later approach gives rise to collapse models [10,11], postulating a stochastic nonlinear modification to Schrödinger’s equation. Irrespective of whether they successfully allow the reconciliation of quantum evolution and measurement theory, these collapse models are considered the only mathematically consistent, phenomenological modifications against which quantum theory can be tested [5,12].
The most universal, well-studied collapse model is continuous spontaneous localization (CSL) [13,14], which serves as a framework to describe a variety of collapse mechanisms [10,11,15–20]. In CSL, a collapse noise field is introduced that couples nonlinearly to the local mass density. In its simplest form, this noise is white, and the model has two parameters—the collapse rate ${\lambda _c}$, which determines the interaction strength with the collapse noise field, and the correlation length ${r_c}$, which determines the spatial resolution of the collapse process [5,14]. The correlation length is expected to be ${\sim}100\;{\rm nm} $ [21], since the behavior of larger systems is generally adequately described by classical theories, whereas quantum mechanics appears to apply on smaller scales. Refined dissipative and colored models introduce two additional parameters, associating a temperature and high-frequency cutoff to the collapse noise field to ensure energy conservation and permit an identifiable physical origin of collapse [16–20]. Based on the assumption that the origin is of cosmological nature, and thermalized to the photon, neutrino, or gravitational wave background, the high-frequency cutoff is estimated to occur at ${\Omega _{{\rm csl}}}/2\pi \sim {10^{10}} - {10^{11}}\; {\rm Hz}$ [20].
To date, the most stringent unambiguous upper bounds on the collapse rate at the expected correlation length are based on mechanical resonators, with signatures of spontaneous collapse expected to manifest as an anomalous temperature increase [22–26]. However, the suggested lower bounds to the collapse-induced heating are lower than one phonon per day [20,21,27]. The challenge of resolving these exquisitely small collapse signatures over a large thermal noise background has precluded conclusive tests of CSL, and has also introduced significant challenges for data interpretation [25]. Even if these issues were resolved, quantum back-action heating [22,28] would remain orders of magnitude larger than the predicted collapse signatures. Moreover, with sizes from microns [24,25] to meters [26,29], the resonators employed to date are larger than the anticipated correlation length and have frequencies far below the expected high-frequency cutoff. As such, they are unable to provide insight into the physical origin of collapse [16–20].
In this work, we propose to test collapse theories with high-frequency nanomechanical resonators. This approach offers the advantages of miniaturization to match the expected collapse correlation length, resonance frequency around the expected high-frequency cutoff, and the ability to both exponentially suppress thermal phonons via passive cryogenic cooling and apply quantum measurement techniques to improve performance. To assess the approach, we explicitly derive the CSL decoherence for mechanical breathing modes beyond the center-of-mass (CoM) motion considered previously, and develop a specific experimental implementation that uses phonon counting in a nanoscale mechanical resonator. Our proposal includes new mitigation strategies for optical, thermal, electrical, and quantum back-action noise that provide, for the first time, to the best of our knowledge, a way to bring each of these noise sources below the expected lower bounds for collapse-induced heating. We conclude that, with challenging but plausible improvements in the state of the art, our approach could conclusively test CSL, closing the gap between measured upper bounds and predicted lower bounds on the collapse rate, and could also test the hypothesis of a cosmological origin of the collapse noise [16,17,20]. It provides an experimental pathway to answer one of the longest standing questions in physics, and also opens up possibilities for laboratory tests of astrophysical models of dark matter [30,31] and other exotic particles [32].
2. BASIC PROTOCOL
Our protocol, illustrated in Fig. 1, is based on a gigahertz nanomechanical resonator, or an array thereof, within a millikelvin environment. As opposed to standard optomechanical measurement, consisting of an optical cavity linearly coupled to a mechanical resonator [33], we propose to perform phonon counting in a three-mode optomechanical system where two optical modes are coupled via a mechanical resonator with resonance frequency $\Omega$. This approach allows collapse signatures to be spectrally distinguished from most noise sources. One mode, the probe mode, is excited by a continuous weak laser at its resonance frequency ${\omega _p}$. In the ideal case, the other, the signal mode at frequency ${\omega _s} = {\omega _p} + \Omega$, is only excited by resonant anti-Stokes Raman scattering between collapse-induced phonons and probe photons. A single-photon readout scheme minimizes both absorption heating [34] and quantum back-action heating. Signal photons are spectrally separated from probe photons by a filter cavity, while dark counts are suppressed by nonlinearly downconverting signal photons to pairs and performing coincidence detection.
Fig. 1. Illustration of protocol. Top left: Array of optomechanical cavities. Top right: Nonlinear pair production from a signal and pump photon (frequency ${\omega _{{\rm pump}}}$). Bottom: Energy-level diagram for scattering of a probe photon with a phonon. ${n_b}$, phonon number.
As a concrete example, we consider using a three-mode photonic–phononic crystal optomechanical system, such as proposed in [35–38]. We choose most parameters based on those achieved in [39], with a mechanical resonance frequency $\Omega /2\pi = 5.3\;{\rm GHz} $, a mechanical damping rate $\Gamma /2\pi = 108 \;{\rm mHz}$, an effective mass ${m_{{\rm eff}}} = 136 \; {\rm fg}$, and thermalization to the base temperature of a dilution refrigerator ($T = 10\;{\rm mK} $). We use the theoretical scattering-limited intrinsic decay rate of ${\kappa _{p,0}} = {\kappa _{s,0}} = 2\pi \cdot 9.2\;{\rm MHz} $ calculated for these devices [40] for both optical modes, where the subscripts ‘$p$’ and ‘$s$’ distinguish the probe and signal mode throughout [33]. Finally, we assume a tenfold improvement of the single-photon optomechanical coupling rate of ${g_0}/2\pi = 11.5\;{\rm MHz} $, as predicted to be feasible with optimized designs [41].
3. PHONON FLUX INDUCED BY CSL
The CSL phonon flux is ${\dot n_c} = {\lambda _c}D$, where $D$ is a geometrical factor that quantifies the susceptibility of the resonator to spontaneous collapse. The requirement that CSL should resolve the measurement problem introduces lower bounds on ${\lambda _c}$, and therefore on the phonon flux. Adler proposed ${\lambda _c} \ge {10^{- 8 \pm 2}}\;{{\rm s}^{- 1}}$ from the postulate that collapse should account for latent image formation in photography [21], while Bassi et al. proposed ${\lambda _c} \ge {10^{- 10 \pm 2}}\;{{\rm s}^{- 1}}$ from the presumption that collapse should occur in the human eye [20]. Previous experiments have focused on the effect of CSL on the CoM motion of an oscillator. Here, in contrast, we consider the effect of CSL on the fundamental breathing mode of the proposed mechanical resonator. It has the advantages of an orders-of-magnitude enhanced decoupling from the environment as well as enhanced optomechanical coupling [39]. We find that in the limit where the resonator size $R$ is much larger than ${r_c}$, the effects of CSL on the breathing motion and CoM motion are equal. However, the effect of CSL on the breathing mode is significantly more suppressed for $R \ll {r_c}$. In the regime of our work, $R \sim {r_c}$, the effect is slightly reduced (see Supplement 1 for details and derivation). Based on these results, we estimate $D = 3.6 \cdot {10^5}$ for our proposed device, which combined with the above-mentioned bounds implies minimum CSL-induced phonon fluxes of ${\dot n_c} = 3.6 \cdot {10^{- 3 \pm 2}}\;{{\rm s}^{- 1}}$ and ${\dot n_c} = 3.6 \cdot {10^{- 5 \pm 2}}\;{{\rm s}^{- 1}}$, for Adler’s and Bassi et al.’s bounds, respectively.
4. OPTOMECHANICAL DYNAMICS AND CONVERSION EFFICIENCY
To model the dynamics of the three-mode optomechanical system we employ the Born–Markov framework for open quantum systems [42–44]. (For a discussion of the applicability of this approximation, see Supplement 1.) The interaction picture Hamiltonian for our system is [35,36,44]
If the oscillator is initially in its ground state, with one photon in the probe mode, a phonon introduced by spontaneous collapse prepares the state $|{n_b}{n_p}{n_s}\rangle = |110\rangle$, where ${n_b}$ is the phonon number in the mechanical resonator, while ${n_p}$ and ${n_s}$ are the photon numbers in the probe mode and signal mode, respectively. The optomechanical conversion efficiency ${\eta _{{\rm om}}}$ for this state to emit a signal photon at frequency ${\omega _s}$ is obtained by numerically solving Eq. (2). We choose the external probe decay rate ${\kappa _{p,{\rm ex}}}/2\pi = 2.2\;{\rm MHz} $ [33], allowing operation at the threshold of strong coupling with ${g_0} \approx {\kappa _p}$. It is advantageous for efficient conversion of collapse-induced phonons to signal photons and ensures low occupancy, minimizing noise, as discussed later. We choose the signal mode to be significantly overcoupled (${\kappa _{s,{\rm ex}}}/2\pi = 0.7{\kappa _s} = 21\;{\rm MHz} $) in a trade-off between optimizing the conversion efficiency and suppressing noise from direct occupancy of the signal mode (see Section 5.C). Together, these external decay rates result in a relatively high conversion efficiency of ${\eta _{{\rm om}}} = 0.32$.
5. SOURCES OF NOISE
Four classes of noise can potentially imitate a collapse signal: thermal phonons, probe photons that leak through the system, phonons introduced by the measurement process, detector dark counts, and environmental vibrational noise.
A. Thermal Limit to Collapse Detection
A collapse signature is resolvable in a thermal noise background if ${\dot n_c}/{\dot n_{{\rm th}}} \gt 1$, where ${\dot n_{{\rm th}}} = \Gamma {({e^{\hbar \Omega /{k_B}T}} - 1)^{- 1}}$ is the thermal phonon flux. It gives a minimum testable collapse rate ${\lambda _{c,{\rm th}}} = {\dot n_{{\rm th}}}/D$. Existing experiments have operated with comparatively low-frequency oscillators in the high temperature limit ${k_B}T \gg \hbar \Omega$ [24–26,29], with thermal phonon flux significantly larger than Bassi et al.’s lower bound, and have sought to resolve small collapse signatures on top of this large thermal noise background. A significant advantage of our approach is that miniaturization and cryogenic cooling allow access to the regime where ${k_B}T \ll \hbar \Omega$. The average thermal phonon occupation is then exponentially suppressed due to Bose–Einstein statistics ${\dot n_{{\rm th}}} \approx \Gamma {e^{- \hbar \Omega /{k_B}T}}$. Figure 2 shows this exponential suppression as a function of resonator size, and compared to the CSL signal, for the simple example of the fundamental breathing mode of a silica sphere (see Supplement 1 for calculation). As can be seen, for gigahertz resonators at millikelvin temperatures, the exponential suppression allows thermal phonon fluxes beneath both Adler and Bassi et al.’s lower bounds. For the proposed photonic crystal device, we find ${\lambda _{c,{\rm th}}} = 1.7 \cdot {10^{- 17}}\;{{\rm s}^{- 1}}$, also well beneath both bounds. This provides the potential for unambiguous tests of collapse models.
Fig. 2. Heating rates of a $Q={10^7}$ silica sphere resonator versus mechanical frequency and diameter. Red traces: Heating due to the thermal environment at temperatures 300,1, and 0.01 K. Solid black traces & gray areas: Lower bounds on CSL heating rates for the breathing mode of a sphere, according to Adler [21], Bassi et al. [20], and GRW [27], assuming a resonance frequency $\Omega = c/R$ with radius $R$ and speed of sound $c = 3000 \; {\rm m/s}$. Dashed black: Heating expected from CoM motion of the sphere, with displacement equal to the surface displacement of the breathing mode (see Supplement 1). CSL heating rates drop as the resonator becomes smaller than ${r_c}{= 10^{- 7}} \;{\rm m}$. Green: Lower bound predicted from classical channel gravity [45–47]. At high frequencies and low temperatures, collapse signatures exceed the thermal heating. Blue shaded: Proposed range of ${\Omega _{{\rm csl}}}$.
B. Probe Photons Leaking through the System
Probe photons passing directly from the laser through the optomechanical system, without a scattering event, could in principle imitate a signal, obfuscating collapse signatures. We find that, using a standard laser stabilization reference cavity as a filter [48], this noise can be suppressed well below both Adler’s and Bassi et al.’s lower bounds. Similarly, if a photon is created in an optomechanical conversion process and subsequently outcoupled into the signal mode, due to energy conservation it either remains at frequency ${\omega _p}$, or has a frequency reduced by integer multiples $n$ of the mechanical resonance frequency ${\omega _p} - n\Omega$. In both cases, this noise is doubly suppressed—first by the suppression of the direct occupation pathway, and second by the filter. It makes probe photons that leak through the system a negligible source of noise, with a predicted noise rate of $1.4 \cdot {10^{- 12}}\;{{\rm s}^{- 1}}$ (see Supplement 1 and Table 1).
Table 1. Comparison of Noise Sources and Respective Testable CSL Parameter ${\lambda _c}$a
C. Measurement-Induced Phonons
Phonons introduced by the optomechanical measurement can imitate collapse signatures. These phonons are created by nonresonant scattering processes between the signal and probe modes, the three lowest order of which are shown in Fig. 3. We calculate the probability of phonon occupancy due to these processes numerically by solving Eq. (2).
Fig. 3. Signal pathways due to measurement-induced phonons. (a) Phonon created after direct excitation of signal mode. (b) Phonon due to counter-rotating transition. (c) Two phonons created by counter-rotating transition followed by resonant transition.
A probe photon can create a noise phonon by coupling directly into the signal mode instead of the probe mode [Fig. 3(a)]. This process is suppressed by the square of the resolved-sideband ratio $\Omega /{\kappa _s}$. The corresponding occupancy is shown by the dashed blue line in Fig. 4(a).
Fig. 4. Numerical calculations of noise magnitude. (a) Blue (orange): Occupancy of density matrix elements containing one (two) phonon(s); dashed blue: Contribution from direct signal mode excitation. Fast oscillations on timescale ${\Omega ^{- 1}}$ correspond to the counter-rotating transition $|010\rangle \leftrightarrow |101\rangle$; slow oscillations to the resonant process $|101\rangle \leftrightarrow |210\rangle$, with period $g_0^{- 1} = \kappa _p^{- 1}$. Dotted lines: Asymptotic values for $\kappa _p^{- 1},\kappa _s^{- 1} \ll t \ll {\Gamma ^{- 1}}$. (b) Same as (a) for $t \sim {\Gamma ^{- 1}} \gg \kappa _p^{- 1},\kappa _s^{- 1}$. (c) Cumulative probability ${p_{{\rm om}}}(t)$ of a probe photon creating a photon at frequency ${\omega _s}$. Dotted line: Asymptotic value for $t \gg {\Gamma ^{- 1}}$.
A photon that does enter the probe mode, corresponding to the state $|{n_b}{n_p}{n_s}\rangle = |010\rangle$, can introduce noise by undergoing the nonresonant phonon-creating transition $|010\rangle \to |101\rangle$, as shown in Fig. 3(b). The resulting state can also resonantly transition to a two-phonon state $|101\rangle \to |210\rangle$, as shown in Fig. 3(c). Similar to what has been mentioned above, noise phonons from this process are suppressed by ${\sim}{(\Omega /{\kappa _p})^2}$. Predicted phonon occupancies are shown in Figs. 4(a) and 4(b).
We also estimate the phonon occupancy due to optical absorption heating based on previous analyses in cryogenically cooled photonic–phononic crystals [34,40]. It only adds a small contribution to the measurement-induced phonon occupancy, as shown in Table 1 and Supplement 1.
A measurement-induced phonon can only be converted to a collapse-imitating photon at frequency ${\omega _s}$ if it scatters with a second photon entering the probe mode within the lifetime ${\Gamma ^{- 1}}$ of the mechanical resonator, as shown in Fig. 3. It is therefore possible to suppress these phonons by operating with a low average photon occupancy ${\bar n_p}$. Here, we choose the photon occupancy so that the probability of a photon entering the probe mode during one mechanical oscillation lifetime is ${\eta _p} = {\bar n_p}{\kappa _p}/\Gamma \sim 1\%$. It reduces the rate of measurement-induced photons by a factor of 100. The cumulative probability of a probe photon generating a phonon, and a second probe photon then causing emission of a photon at frequency ${\omega _s}$, is shown in Fig. 4(c). The asymptotic probability is ${p_{{\rm om}}}(t \to \infty) = 8.4 \cdot {10^{- 8}}$ (see Supplement 1).
D. Coincidence Dark Counts
Detecting collapse induced phonons at the predicted rate of less than one per day necessitates very high suppression of photon dark counts, which typically occur at hertz to kilohertz rates. One way to achieve this is to nonlinearly downconvert signal photons to pairs (bottom right inset, Fig. 1) using a bright pump beam in a third-order nonlinear medium. It has been shown that this process can convert single photons to pairs with near-unit efficiency ${\eta _\chi}$ [50] (see Supplement 1). A signal is recorded only if a coincidence detection event is registered. The coincidence dark count rate is suppressed as the square of the single-detector dark count rate ${R_{d,1}}$, ${R_{d,2}} = R_{d,1}^2 \cdot {\tau _c}$, where ${\tau _c}$ is the coincidence timing resolution. For commercially available photon counters with ${R_{d,1}} = 3.5\;{{\rm s}^{- 1}}$ and ${\tau _c} = 30\;{\rm ps} $ [51], we predict ${R_{d,2}} \sim 3.7 \cdot {10^{- 10}}\;{{\rm s}^{- 1}}$.
E. Environmental Vibrational Noise
Environmental vibrational noise, such as from seismic activity or cryogenic pumps, has proved to be one of the main noise contributions in previous lower frequency mechanical tests of spontaneous collapse [52,53]. However, we expect it not to be a significant noise source for our proposal. The magnitude of vibrational noise is greatly suppressed by moving to gigahertz frequencies. Isolation techniques can then be used to further reduce their influence. For instance, the isolation employed in gravitational wave interferometers allows the suppression of environmental vibrational noise that scales faster than ${\omega ^{- 8}}$ [54]. It would provide a suppression of more than 40 orders of magnitude for our proposal compared to low frequency cantilever experiments [25].
6. MINIMUM TESTABLE COLLAPSE RATE
For ${r_c}{= 10^{- 7}} \;{\rm m}$, the rate of coincidence counts attributed to collapse is ${R_c} = {\lambda _c}D\eta = 3.9 \cdot {10^2}{\lambda _c}$, where the efficiency $\eta = {\eta _p}{\eta _{{\rm om}}}{\eta _\chi}{\eta _d}{\eta _{\rm f}} = 1.1 \cdot {10^{- 3}}$ quantifies the fraction of phonons in the mechanical resonator that result in a coincidence count, ${\eta _\chi} = 0.95$, and ${\eta _d} = 0.64$ is the coincidence detection efficiency (see Supplement 1). This rate must exceed the sum of the noise rates, setting the limit to the minimum observable collapse rate ${\lambda _c}$, for optomechanically induced phonons, probe photons leaking through the system and thermal phonons, ${R_{{\rm om}}} = {\kappa _{p,{\rm ex}}}{\bar n_p}{p_{{\rm om}}}(t \to \infty){\eta _f}{\eta _\chi}{\eta _d}$, ${R_{{\rm phot}}} = {\kappa _{p,{\rm ex}}}{\bar n_p}{p_f}{\eta _\chi}{\eta _d}$, and ${R_{{\rm th}}} = \eta {\dot n_{{\rm th}}}$, respectively, where ${\eta _f} = 0.56$ is the transduction efficiency through the filter and ${p_f} = 3.5 \cdot {10^{- 10}}$ the probability of a probe photon leaking through the filter (see Supplement 1). The numerical values and corresponding minimum testable collapse rates are given in Table 1 and are also shown in Supplement 1. Optomechanical measurement-induced phonons and coincidence dark counts set comparable limits on ${\lambda _c}$, with negligible contributions from leaked probe photons, photoabsorption and thermally excited phonons. The minimum testable collapse rate limited by all noise sources is ${\lambda _c} = \sum\nolimits_i {\lambda _{{\rm c,i}}} = 1.4 \cdot {10^{- 12}}\;{{\rm s}^{- 1}}$.
7. SIGNAL RATE AND MEASUREMENT TIME
The predicted average time required to observe one signal due to CSL collapse is ${t_{{\rm meas}}} = ({\lambda _c}D\eta {)^{- 1}}$. Fully probing the minimum testable collapse rate of ${\lambda _c} = 1.4 \cdot {10^{- 12}}\;{{\rm s}^{- 1}}$ would require ${t_{{\rm meas}}} \gt 57 {\rm years}$. Fabricating an array of $N$ optomechanical cavities on a silicon wafer [55–58], as shown in Fig. 1, coupled to a single filter cavity, nonlinear medium and detector (or a small number of such elements) could significantly reduce this time to $t_{{\rm meas}}^{(N)} = {t_{{\rm meas}}}/N$, and also the dark count-limited testable collapse rate to $\lambda _{c,{\rm det}}^{(N)} = {\lambda _{c,{\rm det}}}/N$. While $N = 2$ optomechanical cavities would already suppress dark counts sufficiently to fully probe both Bassi et al.’s and Adler’s proposals, we estimate that $N \sim {10^4}$ may be feasible (see Supplement 1). It would essentially eliminate detector dark counts as a limit, and allow a reduction of the measurement time to about two days.
8. SCOPE AND FEASIBILITY
Many theoretical proposals based on a range of experimental systems exist to improve the resolution of collapse signatures [59–65], and to explore their noise performance [53]. However, to the best of our knowledge, this is the first proposal to offer a comprehensive noise analysis in the context of optomechanical measurement, a scalable fabrication to allow a drastic reduction of measurement time as well as to provide a method to test the hypothesis of a cosmological origin of collapse. The proposed method relies on the use of single photons to probe the phonon occupancy of the mechanical resonator, which allows a strong suppression of measurement-induced noise. This necessitates the system to operate in the single-photon strong coupling regime [33]. The only optomechanical parameters that must be improved from the current state of the art [39] to realize the protocol are a reduced optical linewidth (by a factor of ${\sim}50$), as predicted by theoretical modeling based on the device realized in [40], and an enhanced single-photon coupling rate (by a factor of ${\sim}10$), based on theoretical modeling in [41]. Alternatively, effective enhanced single-photon coupling could be achieved by coupling to a qubit or other highly nonlinear system (e.g., as demonstrated in [66]). Given the trajectory of the field, we estimate these requirements to be likely achievable in the intermediate future. Nevertheless, it is also useful to consider alternative realizations of the method.
9. QUADRATIC COUPLING
While we consider phonon counting via an optomechanical Raman interaction here, in principle the method could be implemented with any low-noise phonon-counting method applied to a high-frequency oscillator [67–71]. One promising approach may be quantum nondemolition measurement of a phonon number using nonlinear optomechanics [72]. In the regime of quadratic optomechanical coupling and resolved mechanical sidebands [33], a collapse-induced phonon imparts a frequency shift $2\bar n_{{\rm cav}}^{1/2}g_0^{(2)}$ on the optical resonance at frequency $\omega$, where ${\bar n_{{\rm cav}}} = \langle {a^\dagger}a\rangle$ is the average intracavity photon number with $a$ the annihilation operator for the optical cavity field, and $g_0^{(2)}$ the zero-point quadratic coupling rate [72–74]. The shift is detectable if it is larger than the significant noise sources, which are random fluctuations in the probe frequency, absorption heating and quantum back action from spurious linear coupling.
The linearized quadratic part of the optomechanical interaction Hamiltonian is $H_{{\rm int}}^{(2)} = \bar n_{{\rm cav}}^{1/2}g_0^{(2)}({a^\dagger} + a)(2{b^\dagger}b + {b^\dagger}{b^\dagger} + bb)$ [74]. The term proportional to ${b^\dagger}b$ yields a per-phonon optical resonance frequency shift of $2\bar n_{{\rm cav}}^{1/2}g_0^{(2)}$, which is the signature of a collapse-induced phonon. A random fluctuation $\delta \omega$ in the frequency of the probe can imitate a signal if it is larger or equal to this frequency shift, and sustained over a time comparable to the phonon lifetime ${\Gamma ^{- 1}}$. For a shot-noise limited probe, the probability of a fluctuation larger than $2\bar n_{{\rm cav}}^{1/2}g_0^{(2)}$ is given by an error function of a Gaussian distribution, so
The term proportional to ${b^\dagger}{b^\dagger}a$ converts a probe photon to two phonons, potentially imitating a collapse signature. However, the shift induced by two phonons is $4\bar n_{{\rm cav}}^{1/2}g_0^{(2)}$ and can be clearly distinguished from the collapse-induced shift caused by one phonon, and hence only amounts to a negligible source of noise (see Supplement 1).
Perhaps the most significant challenge in this approach would be to engineer a strong suppression of linear optomechanical coupling, so that the phonon flux due to quantum back action does not exceed the predicted CSL signature. If using standard architectures, there is a fundamental limit to this suppression of linear coupling [75]. Hence, either a different architecture would have to be employed [68,74,76], or the substantially more stringent condition $g_0^{(2)} \ge \kappa$ would have to be realized. The phonon flux due to quantum back action is given by ${\dot n_{{\rm ba}}} = 4g_0^2{\bar n_{{\rm cav}}}/\kappa = 4{g^2}/\kappa$. To resolve a potential CSL signature, ${\dot n_{\rm c}}$ must be greater than ${\dot n_{{\rm ba}}}$. As a result, the linear optomechanical coupling would need to be suppressed to $g \le \sqrt {{\lambda _c}D\kappa /4}$. To test ${\lambda _c}{= 10^{- 12}}$, we find the condition $g/2\pi \mathbin{\lower.3ex\hbox{$\buildrel \lt \over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} {10^{- 1}} \;{\rm Hz}$, about seven orders of magnitude lower than typical linear coupling rates in photonic–phononic crystal structures [39]. While some architectures may in principle allow for vanishing linear coupling $g$, achieving the required suppression in practice may be challenging [68,76,77]. In continuous operation, with currently available technology [39], absorption heating would exceed the expected heating from collapse by about seven orders of magnitude (see Supplement 1). Even with very large heating in the continuous domain, it may be possible to resolve the problem by operating in a pulsed regime, so that each optomechanical measurement process is completed in a timescale much shorter than the time required for absorption events to create phonons. In this case, the measurements would need to be sufficiently temporally spaced to allow for phonons to fully dissipate.
10. BOUNDS ON CSL
Figure 5 compares the predicted upper bound on the collapse rate ${\lambda _c}$ from our protocol to those of existing experiments, together with Adler’s and Bassi et al.’s lower bounds and their uncertainties for white-noise CSL, as well as colored CSL with the most common choice of the noise correlation function, as shown below. Existing upper bounds are provided by the motional stability of gravitational wave interferometers [26,29,78] (yellow region); the thermalization of ultracold cantilevers [24,25,79] (blue region); Kapitza–Dirac–Talbot–Lau (KDTL) interferometry [80–82] (dashed black); spontaneous X-ray emission from Germanium [83,84] (dotted black) and the observed temperature of neutron stars [85,86] (dashed black), which are valid however only for white noise CSL; and the non-interferometric detection of the expansion of a BEC cloud [87] (gray region). For dissipative CSL, levitated nano-oscillators provide significant bounds [55].
Fig. 5. Parameter diagram for CSL model. Excluded upper bounds for simple CSL, as well as colored CSL with ${\Omega _{{\rm csl}}}/2\pi \approx {10^{10}} - {10^{11}}\; {\rm Hz}$ : Gravitational wave detectors (yellow shaded); Cold atoms (gray shaded); Microcantilevers (blue shaded); KDTL interferometry (dashed black). Excluded for simple CSL only: Neutron stars (dashed black) and X-ray (dotted black). Proposed lower bounds: Adler (vertical blue bars and dotted blue line) and Bassi et al. (vertical black bar). Red: Predicted testable parameter space using our protocol.
The red shaded region in Fig. 5 could be tested by our protocol as discussed above. (For $N = 2$ optomechanical cavities, see Section 7.) In the case of white-noise collapse, the protocol could, for the first time, fully test Bassi et al.’s proposal. For colored CSL [16,17], we follow the common assumptions of an exponentially decaying noise correlation function [88] and that collapse noise has one of the proposed cosmological origins. In this case, the signature of colored CSL would be a high-frequency cutoff on the noise spectrum around ${\Omega _{{\rm csl}}}/2\pi \approx {10^{10}} - {10^{11}} \;{\rm Hz}$ [18–20] (see Supplement 1). The envisaged protocol would then also, for the first time, probe Adler’s prediction, since colored CSL with the above-mentioned cutoff is not tested by X-ray emission (black dotted line in Fig. 5). The mechanical resonance frequency of the proposed device is close to the frequency range in which a drastic frequency-dependent reduction of the collapse noise stemming from a cosmological origin is expected [20]. By employing a number of mechanical resonators of different frequencies (e.g., in [40] a similar resonator with a resonance frequency of ${\sim}10 \; {\rm GHz}$), or one frequency-tunable resonator [89], it could therefore be possible to identify a potential cosmological origin of collapse and differentiate collapse-induced signals from technical and environmental noise [61] (see Supplement 1).
We also evaluate the capability of the protocol to constrain parameters in gravitational collapse models. For the Diósi–Penrose model [90–92], we find that it cannot exceed existing bounds; however, for the classical channel gravity model in a typical parameter range [45–47], we predict about a one order of magnitude stronger bound than previously achieved [93] (see Supplement 1).
11. CONCLUSION
In summary, we have proposed the concept of testing quantum linearity using high-frequency mechanical oscillators. This approach offers the advantages of thermal noise suppression to well below expected collapse signatures, and the capability to test the hypothesis of a cosmological origin of collapse. As a possible implementation, we suggest a protocol based on a dual-cavity, high-frequency optomechanical device passively ground-state-cooled and operating in the strong coupling regime. We believe that this design, combined with nonlinear optical techniques to reduce dark counts, will allow measurement of the minuscule phonon flux generated by collapse-induced heating. While challenging, we believe the protocol has the potential to conclusively test CSL, and thus whether collapse mechanisms can be invoked to resolve the measurement paradox.
Funding
Australian Research Council (CE110001013, DE180101443); European Union’s Horizon 2020 research and innovation programme under the Maria Skłodowska-Curie grant agreement (663830).
Acknowledgment
The authors thank Gerard Milburn, Nathan McMahon, and James Bennett for helpful discussions, and Nicolas Mauranyapin for preparing Figure 1.
Disclosures
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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