Can the displacemon device test objective collapse models?

The quantum measurement problem and the quantum-to-classical interface have been much debated since the inception of quantum mechanics. This debate has sparked numerous questions such as “Is there a limit to the extent that the Schrödinger equation may be applied?” and “What are the mechanisms that cause a coherent quantum superposition to lose its interference features and result in classical behavior?” This debate and these particular questions explore the two key dynamical processes in quantum mechanics, namely, the linear deterministic Schrödinger evolution and the non-linear stochastic wavefunction collapse owing to a measurement. Several different physical interpretations of these two processes have become popular among various groups within the physics and philosophy communities. However, it is often very difficult or even impossible to experimentally distinguish between these differing views of the world.

One resolution to the measurement problem and the quantum-to-classical interface is provided by objective collapse theories. These propose physical mechanisms, not encompassed by standard Schrödinger evolution, which cause wavefunctions to collapse without the need for collapse by a measurement.

Following early progress in modifying the Schrödinger equation,^1–4 spontaneous localization models were studied by Ghirardi, Rimini, and Weber (GRW).⁵ GRW theory satisfies the requirements of a physical collapse model with the key ingredient being that an N-particle wavefunction $ψN(x1,…,xN)$ experiences a spontaneous Gaussian localization with the effect of premultiplying it by an operator of the form $exp (−(X−xk)2/(2r2))$⁠, where r is the localization length, X is the position operator, and k = 1, …, N. These Gaussian jumps happen at randomly distributed times with a mean rate λ which is chosen to preserve unitary time evolution for microscopic systems, but importantly accounts for classical features emerging as the number of particles in an object is increased.⁶ A further refinement of this type of collapse theory combines the ideas of stochastic dynamical reduction together with GRW to obtain continuous spontaneous localization (CSL),⁷ which is now a widely studied model. In contrast to GRW theory, CSL assumes that the localization operations happen continuously in time. Like GRW, CSL behavior is characterized by two phenomenological parameters, the collapse rate $λCSL$ and the localization length $rCSL$⁠. The goal of many experimental tests of CSL is to impose bounds on these constants. To this end, $rCSL=100$ nm and $λCSL=10−8±2$ Hz have been identified as an interesting part of parameter space to explore.^8–11 

Another contender to reconcile the issues presented by the measurement problem, offered by Diósi^12,13 and later rederived through a different line of reasoning by Penrose,¹⁴ is now referred to as Diósi–Penrose (DP) collapse. This gravity-induced collapse model is argued by Penrose to arise from the incompatibility of the theories of general relativity and quantum mechanics. The essence of the theory is that for a massive particle in a superposition of two well-localised states, each state has an associated space-time geometry leading to a conflict with Einstein's theory of general covariance.¹⁵ This superposition of differing gravitational fields leads to an instability which induces wave-function collapse after a characteristic lifetime $tDP$⁠. To quantify the difference between two space-times, Penrose has suggested using the gravitational self-energy from classical physics $EG$⁠, and relating this to $tDP$ via $tDP=ℏ/EG$⁠.

Tests of objective collapse models aim to detect deviations from standard quantum theory. Non-interferometric tests focus on sensing the additional sources of decoherence which are predicted by collapse models, usually in the form of heating. Searching for this excess noise has led to experimental realizations including test masses on ultracold cantilevers,¹⁶ underground tests,¹⁷ and entangling macroscopic diamonds.^18,19 Meanwhile, a host of non-interferometric experiments have been proposed using a range of platforms including optomechanics,²⁰ superconducting devices,²¹ ultracold atoms,²² and levitated nanospheres.²³ Notably in optomechanics, building on the Bose–Jacobs–Knight proposal,²⁴ the Marshall–Simon–Penrose–Bouwmeester mirror-superposition scheme proposes how to study spatial superposition states involving 10¹⁴ atoms.²⁵ On the other hand, interferometric tests strive to create large superpositions with increasingly massive objects in order to probe the quantum-to-classical transition;²⁶ these experiments include atom interferometry,²⁷ superpositions of large molecules,^28,29 interferometry of free-falling nano-particles,³⁰ and also proposals with ultracold atomic systems.³¹ Meanwhile, electromechanical devices have been used in several experiments that, while not testing collapse models directly, nevertheless probe fundamental aspects of quantum behavior, such as coherent exchange between a mechanical resonator and a qubit,³² and entanglement between a resonator and a microwave field³³ or another resonator.³⁴ Moreover, there is growing evidence that electromechanical devices are a viable route for tests of collapse, particularly with mechanical oscillators coupled to superconducting circuits.³⁵ 

In this work, we provide an experimental scheme to probe collapse models using the displacemon platform.³⁶ This device consists of a vibrating nanobeam coupled to a superconducting qubit such that operations performed on the qubit allow us to generate non-Gaussian states of the center-of-mass of the mechanical system. We begin with a brief overview of this device and the qubit operations required to produce mechanical superpositions. A scheme is then presented that allows us to monitor the decoherence of these large-scale mechanical superpositions via qubit measurement and thereby test for sources of decoherence that could be attributed to DP or CSL collapse theories. We carefully examine the feasibility of our protocol given current and near-term experimental parameters and find that with technical improvements in isolating the device from vibrational noise, the displacemon device provides an experimentally viable route for testing DP and CSL models of objective collapse beyond existing bounds.

A matter-wave interferometer works by passing the test particle through a series of diffraction gratings and then measuring its position. A well-known example is the electron double-slit interferometer. Quantum superposition manifests itself as a series of spatial fringes, which arise from interference between different paths a particle could take through the gratings. This kind of matter-wave interferometry, now realized using increasingly massive test particles in sophisticated interferometers, allows the most stringent interferometry-based exclusions of spontaneous collapse theories.³⁷ The largest mass tested so far is 27 000 amu, surviving in a superposition for 7 ms.²⁹ However, this mass is still too small to test gravitational decoherence and some theories of spontaneous collapse.

Progress in this area continues to be rapid, but to explore the gravitational-decohererence scale using conventional matter-wave interferometers, two major challenges need to be addressed. First, as test masses become larger, it becomes increasingly difficult to create suitable diffraction gratings. Existing state-of-the-art interferometers use finely patterned membranes in the first and final step. However, the test particles interact with these gratings by van der Waals interaction, which becomes stronger as the particle diameter increases and introduces unwanted smearing of the wavefunction.³⁸ Future proposed interferometers will use all-optical gratings,³⁰ but here the large mismatch between the optical wavelength and the de Broglie wavelength of a massive test particle imposes other stringent requirements on the interferometer. The second challenge is the requirement for the particle to evolve in free-fall, so that the wavefunction has time to spread out before the final measurement.

The displacemon³⁶ provides an approach to circumvent both these challenges by using as its test particle a doubly clamped cantilever coupled to a superconducting qubit. The key idea is to create a synthetic diffraction grating, realized through a series of operations on the qubit. Figure 1(a) summarizes the principle, and how it relates to a conventional matter-wave interferometer. In a conventional interferometer, a test particle moving along the axis of the interferometer encounters a series of gratings, each of which either blocks it or allows it to pass. (In an optical grating, the particle may be ejected from the apparatus rather than blocked, but the effect is the same.) Each grating effectively performs a projective measurement of the particle's position, in a basis that contains the states {“blocked,” “passed”}.³⁹ The process of passing through the gratings and then registering the particle's position is thus equivalent to a series of projective position measurements, with the final measurement outcome recorded only if all previous measurements have projected to “passed” (i.e., the particle makes it through the interferometer unimpeded). Figures 1(a) and 1(b) illustrates this equivalence.

The Hamiltonian of the flux-coupled qubit-resonator system is

where $ωq$ is the bare qubit frequency, $σz$ is the Pauli operator operating in the qubit basis with eigenvectors $|+〉$ and $|−〉$⁠, and λ is the qubit-mechanical coupling parameter, which may depend on time t. The mechanical resonator has creation (annihilation) operator $a (a†)$ and mechanical frequency $Ω/2π$⁠. The displacement is $X≡XZP(a+a†)$⁠, where $XZP≡ℏ/2mΩ$⁠. The interaction between the mechanical oscillator and the qubit shifts the energy splitting of the qubit by an amount proportional to the mechanical displacement [Fig. 1(c)].

In the simplest implementation, the required projective measurement is realized as follows [Fig. 1(d)]:

• i.

  The qubit is prepared in an eigenstate $(|+〉+|−〉)/2$⁠.

• ii.
  The system evolves under the Hamiltonian (1) for time $τR$⁠. In the qubit's rotating frame, it precesses at a rate proportional to the mechanical displacement X. (In the parameter regime considered here, the mechanical frequency is small enough that the particle can be assumed stationary during this step. Reference 36 explains how the protocol should be modified if this assumption is violated.) At the end of this step, the accumulated qubit precession is

  $Θ=λXXZPτR.$

  (2)
• iii.
  The qubit is subjected to a $π/2$ rotation about a horizontal axis and is projectively measured in the ${|+〉,|−〉}$ basis. The probability to obtain the result $|+〉$ is

  $p+=cos2Θ.$

  (3)

Conditioning on this outcome (i.e., analyzing subsequent data only in those experimental runs for which this outcome was obtained) therefore projects the particle's displacement as shown in Fig. 1(b), thus realizing an effective grating. Conditioning on $|−〉$ results in a π phase-shifted interference fringe.

In a conventional interferometer, the particle encounters gratings that are located at different positions along its path [Fig. 1(a)]. In the displacemon, the effective grating is turned on and off in time [Fig. 1(b)]. In both cases, state preparation of the matter degree of freedom manifests itself as a diffraction pattern in the transverse coordinate X.

We now explain the proposed device in detail and obtain expressions for each of the quantities in the Hamiltonian (1).

The resonator is modeled as a superconducting beam with length $ℓ$⁠, a square cross-section, and side length D [Fig. 2(d)]. For simplicity, we assume that the beam has been fabricated without tension. Its motion is described by the Euler–Bernoulli beam equation,⁴⁰ 

where u(Z, t) is the displacement profile of the beam as a function of time t and coordinate along the beam Z, with Z = 0 being the center. Here, ρ₀ is the density of the beam and E is its Young's modulus, and Poisson's ratio has been neglected.

To solve for the frequency of the fundamental displacement mode, we write

which means that $u0(Z)$ must satisfy

where

The solution, respecting the boundary conditions $u0(±ℓ/2)=0$ and normalizing⁴⁰ so that $∫u02(Z) dZ=1$⁠, is

The additional boundary conditions $u0′(±ℓ/2)=0$ constrain k to obey

whose lowest solution is $kℓ≈4.730$⁠. Substituting this solution into Eq. (7) gives the frequency of the fundamental mode,

An ideal square beam has two degenerate fundamental modes corresponding to orthogonal directions of motion. However, only the out-of-plane mode couples strongly to the qubit.⁴¹ Furthermore, fabrication imperfections will generally break the degeneracy of the two modes. (If necessary, this can be enforced by designing the beam width to be different from its thickness.) We therefore consider only the out-of-plane mode in the rest of this work.

The qubit is based on the split concentric transmon design.⁴² Figure 2 shows how this design relates to the conventional transmon.^43,44 A conventional transmon [Fig. 2(a)] consists of two superconducting islands coupled via a capacitance C in parallel with a Josephson junction. The Josephson junction acts as an effective non-linear inductor, meaning that the transmon is an anharmonic LC resonator whose two lowest states form the qubit basis. The qubit frequency is

where $EC≡e2/2C$ is the charging energy and $EJ$ is the Josephson energy. Clearly, if $EJ$ depends on X, this realizes the qubit-mechanical coupling envisaged in Sec. II B.

One way to introduce such a coupling is to replace the single Josephson junction by a pair of junctions forming a superconducting quantum interference device (SQUID), making a “split transmon”⁴⁵ [Fig. 1(b)]. Assuming that each junction has Josephson energy $EJ0/2$⁠, the Josephson energy of the combined junctions is

where $Φ$ is the magnetic flux through the SQUID loop and $Φ0≡h/2e$ is the superconducting flux quantum. If a superconducting beam is introduced into one arm of the loop and the qubit is operated in a static magnetic field, then the displacement of the beam modulates $Φ$ and therefore $EJ$⁠. Electromechanical coupling between a vibrating beam and a transmon was recently demonstrated using this principle.⁴⁶ 

A drawback of the circuit in Fig. 2(b) is that the qubit is sensitive to magnetic field noise, which modulates $Φ$⁠. To mitigate this, the design is modified so that it contains two flux loops [Fig. 2(c)], meaning that the two islands of the transmon are now the inner and outer circles. Figure 1(d) sketches a realization of this circuit. The effective Josephson energy is⁴² 

where $ΔΦ≡Φ1−Φ2$ is the difference in enclosed flux between the two loops. From Eq. (11), the qubit frequency is therefore

where

is the qubit frequency when $ΔΦ=0$⁠. The flux difference $ΔΦ(t)$⁠, and therefore the qubit frequency, can be tuned using a small quasi-static perpendicular field gradient generated by an on-chip bias coil.

Such a qubit can be initialized, controlled, and measured using standard techniques of circuit quantum electrodynamics (cQED).⁴⁴ In summary, the simplest initialization is by thermal relaxation to the ground state $|+〉$⁠. Coherent control is achieved by applying microwave bursts resonant with $ωq$⁠. Projective readout in the $σz$ basis is achieved by coupling the qubit to an off-resonant cavity and measuring the state-dependent change to the cavity's resonant frequency.

The coupling described by Eq. (1) arises if the qubit frequency given by Eq. (14) depends on the beam's displacement. This occurs in the presence of an in-plane magnetic field $B||$⁠, applied orthogonal to the beam axis [Fig. 1(d)]. In the presence of such a field, a displacement X in the fundamental mode changes the flux difference by an amount

where

is a geometrical coupling factor that relates the displacement of the beam to the area it sweeps out.

Applying Eq. (16) to Eq. (14) gives

This allows us to calculate the electromechanical coupling rate in Eq. (1):

where the time dependence arises through the tunable flux offset $ΔΦ(t)$⁠.

In principle, Eq. (24) allows arbitrarily large $λ(t)$⁠. In practice, the inevitable asymmetry between the two Josephson junctions has the effect of limiting the useful range of $ΔΦ$⁠, meaning that the $tan πΔΦ(t)2Φ0$ factor is likely to be of order unity.

We consider two realizations of the device in Fig. 2, with different levels of ambition. The parameters of these devices are shown in Table I. Device A is designed to have the same mass as the vibrating mirror in the Marshall–Simon–Penrose–Bouwmeester proposal to test for gravitational collapse.²⁵ Device B is designed to have the Planck mass. The dimensions of devices A and B are chosen to be within the capabilities of microfabrication; obviously, other combinations of length and width could be chosen to give the same mass. For both devices, the mechanical quality factor is taken to be $Q=1.1×106$⁠, the same as in Ref. 46 and typical for aluminum nanobeams.

The qubit frequency is set by the geometry (which determines $EC$⁠) and the Josephson junction parameters (which determine $EJ$⁠). We again take parameters close to those experimentally measured in Ref. 46 by assuming $ωq0/2π=8$ GHz and $ωq/2π=6$ GHz, which requires $ΔΦ≈0.620 Φ0$⁠.

We now consider the evolution of the resonator following the conditioning step described in Sec. II B. The spatial wavefunction of the conditioned state is proportional to the initial wavefunction multiplied by a sinusoid, meaning that the effective grating performs the transformation:³⁶ 

Here, $|ψ〉M$ is the mechanical wavefunction, $ϒ+$ is a measurement operator, $R(θ)≡e−iθa†a$ is the resonator rotation operator, and

For a derivation of Eq. (25), see our previous proposal.³⁶ Multiplying the wavefunction by a sinusoid is equivalent to dividing it into two branches, one of which has received an impulse $+ℏ|α|/XZP$ and the other which has received $−ℏ|α|/XZP$⁠. After a quarter of a mechanical period, the centroids of these two branches are spatially separated by a distance

The largest separation will occur if $|α|$ is maximized. In Eq. (26), the time integral extends at most over the qubit decoherence time $T2*$⁠, since otherwise the qubit fringes required to create the effective grating are washed out. Thus, during the qubit precession step, if $λ(t)$ is set to its largest accessible value $λmax$⁠, then the largest accessible separation is

Taking $ΔΦ=0.620Φ0$ during this step, as explained in Sec. III D, gives

and therefore

and

In the standard theory of quantum mechanics, the thermal decoherence of the mechanical state ρ may be modeled by the Caldeira–Leggett master equation

which describes quantum Brownian motion.⁶ The first term in Eq. (34) describes unitary evolution of the mechanical oscillator with Hamiltonian $H0=ℏΩa†a$⁠, while the term $−iγ[X,{P,ρ}]/ℏ$ describes dissipation, which occurs over a timescale $1/γ=Q/Ω$⁠.

The final term $−Dth[X,[X,ρ]]$ in Eq. (34) describes momentum diffusion and leads to the decoherence of quantum superpositions. Namely, the effect of this term is to cause spatial superpositions of size $2ΔX$ to decohere over a timescale⁴⁸ 

Here, the thermal decoherence rate is $Dth=2mγkBTeff/ℏ2$⁠, and the thermal occupation of the oscillator is $N¯=kBTeff/ℏΩ$⁠, which is valid in the limit⁶ $kBTeff≫ℏΩ$⁠. Here, $Teff$ is the effective temperature of the environment, which incorporates the cryogenic temperature and other effects such as ambient mechanical noise, which we discuss further in Sec. VII.

A term $−DP[P,[P,ρ]]$ may be added to Eq. (34) to bring the dynamics into Lindblad form,⁴⁹ where P is the mechanical momentum operator. This term describes spatial diffusion at a rate $DP=γ/(8mkBTeff)$⁠. However, in the temperature regime considered in this work, $kBTeff≫ℏΩ$⁠, this term is negligible. We use the corresponding Langevin equations, which describe this quantum Brownian motion in the Heisenberg picture, in order to solve for the open-system dynamics of the mechanical state in Sec. VI. This enables us to properly model the effects of dissipation and quantum decoherence while avoiding any violation of positivity when solving Eq. (34) at short times with a non-negligible spatial diffusion term.⁵⁰ 

In Diósi–Penrose theory,^14,51 the collapse of a superposition arises from the difference in the mass distributions $ρ(r)$ and $ρ′(r)$ in the two branches. The lifetime of a superposition is

where¹⁵ 

parameterizes the incompatibility of the space-time generated by the two branches of the superposition, and r and $r′$ are spatial coordinates. Equation (37) is proportional to the gravitational self-energy of the difference between the two mass distributions, and $γDP$ is a constant¹⁵ which takes the value $γDP=1/8π$ if $|EG|$ is equal to the gravitational self-energy (rather than only proportional to it).

For the devices considered here, the spatial separation $2ΔX$ is much smaller than the distance between adjacent atoms a. (Compare Table I with the nearest-neighbour distance in aluminum, which is $a=2.9×10−10$ m.) This means that Eq. (37) can be broken down into a sum of self-energies created by the individual nuclei, each in a superposition

Here, $EG(1)(2ΔXi)$ is the self-energy of the superposition of a single nucleus separated by $2ΔXi$⁠, which is the separation of the entire beam at the location of the ith nucleus (see Fig. 3). Since

with $u0(Z)$ defined by Eq. (8), Eq. (38) becomes

where ρ₀ and $ma$ are the density and atomic mass of the material from which the beam is made. The factor 2 takes account of the fact that the two branches of the wavefunction correspond to beam displacements in different directions.

A full expression for $EG(1)$ is given in Ref. 52; modeling the nuclei as hard spheres with radius σ, it is

The value of σ is the size of the nuclear mass distribution that acts as the gravitational source. The simplest choice is that σ is the nuclear radius, which in aluminium⁵³ is $σnuc=3.06×10−15$ m. Since Table I shows that $ΔX≫σnuc$⁠, the first term in Eq. (41) dominates, meaning that

and therefore

where m is the mass of the beam, and the last line makes the assumption¹⁵ that $γDP=1/8π$⁠. This leads to the predicted collapse lifetime $tDP$ shown in Table I. Any superposition in device A would collapse due to gravity in approximately 18 ms, or 25 mechanical periods; for device B, the collapse time is much less than the mechanical period, so that a superposition that survives even for half a period would be enough to exclude this collapse model.

For gravitational collapse to be observable, the conventional decoherence rate needs to be reduced below the gravitational rate $1/tDP$⁠. In Sec. VI, we outline a scheme to detect this decoherence rate using the displacemon device.

Above, we assumed that σ is the same as the nuclear radius $σnuc$⁠. It has also been suggested⁵² that σ should be the radius of the ground-state wavepacket if the mechanical state is initialized in the ground state prior to superposition state preparation. There are two ways of defining this. One is to say that the wavepacket size is equal to the beam's zero-point amplitude, i.e., $σ=XZP$⁠. If we make the crude approximation that the wavepacket gravitates in the same way as a uniform sphere with this radius, then we still have $ΔXmax≫σ$ and so Eq. (45) is modified to

Alternatively, the wavepacket size may be taken as the zero-point amplitude of a single atom in the crystal, which is

where $Ωp$⁠, the maximum of the lower phonon dispersion branch, which in aluminum⁵⁴ is $Ωp≈2π×3$ THz. We now have $ΔXmax<σ$⁠, and so (in the approximation that the wavepacket still gravitates as a uniform sphere), Eq. (41) is modified to

which follows an expression from Ref. 15, keeping only the leading term in $ΔXi/XZP(1)$⁠. A full analysis should take into account that $ΔXi$ depends on both Z and t, but in the gross approximation that $ΔXi∼ΔXmax$ then we find

again assuming $γDP=1/8π$⁠.

For completeness, the gravitational energies and lifetimes that follow from Eqs. (46) and (50) are listed in Table I.

To observe the collapse predicted by the DP model, we require that $tth>tDP$⁠. Using Eqs. (35) and (36), this condition may be rewritten as

The right-hand side of Eq. (51) is a function of the separation $ΔX$⁠. Note that $ΔX$ is a dynamical variable that varies between an initial value of zero and $ΔXmax$ over one quarter of a mechanical cycle. Over this timescale, the dissipation rate is much smaller than the mechanical frequency $Ω/2π$ and the decoherence rates, so may be ignored. Furthermore, the separation $ΔXmax$ may be controlled with the magnetic field $B||$⁠. Therefore, we may maximize Eq. (51) with respect to $ΔX$ to estimate the maximum thermal occupation of the environment $N¯max$ that still allows DP collapse to be observed, and the corresponding environmental temperature $Teffmax$⁠.

To maximize the right-hand side of Eq. (51), we account for the dependence of $EG$ on $ΔX$ through Eq. (41) in the approximation $ΔXi≈ΔX$⁠, which is valid for $ℓ≫D$⁠. The term $EG/(ΔX)2$ is then a decreasing function of $ΔX$ for¹⁵ $ΔX≥σ$⁠. Hence, in the region $ΔX≥σ$ we find that Eq. (51) is maximized at $ΔX=σ$ to give $N¯<N¯max$ with

First, we take the gravitational radius to be the nuclear radius $σ=σnuc$⁠, in which case device A requires $N¯max=1.3×105$ (⁠$Teffmax=8.7 mK$⁠) while device B requires $N¯max=3.9×109$ (⁠$Teffmax=1.5 K$⁠). Second, we take σ to be the zero-point amplitude of the beam $σ=XZP$⁠. In this case, we find a more stringent requirement for device A of $N¯max=95$ (⁠$Teffmax=6.3 μK$⁠), while device B may operate at $N¯max=3.4×108$ (⁠$Teffmax=0.13 K$⁠). Device B can operate at a higher effective temperature for both choices of σ, which is attributed to the $1/Ω2$ dependence of $N¯max$⁠. These relatively high effective temperatures required by device B are encouraging given recent experimental progress in cooling Planck mass objects.⁵⁵ 

Finally, we consider the case when the gravitational radius is given by the zero-point amplitude of a single atom in the crystal $σ=XZP(1)$⁠. As this corresponds to the region $ΔX<σ$ we take $ΔX=ΔXmax$ to provide an estimate for $N¯max$⁠. Combining Eqs. (50) and (51) then gives $N¯<N¯max=GmaQ/(4Ω2(XZP(1))3)$⁠, which gives $N¯max=2.1×10−5$ (⁠$Teffmax=1.4×10−12 K$⁠) for device A and $N¯max=0.62$ (⁠$Teffmax=2.4×10−10 K$⁠) for device B. These results break our initial assumption that $kBTeff≫ℏΩ$ and do not correspond to physically realizable temperatures. Hence, the displacemon is not capable of sensing DP collapse with $σ=XZP(1)$⁠.

Treating the constituents as uniform spheres,¹⁵ the term $EG/(ΔX)2$ is maximized in the limit $ΔX→0$⁠, which seems to increase the $N¯$ our devices can tolerate in (51). However, we note that in this limit, both decoherence timescales $tDP$ and $tth$ tend to infinity and we do not expect to see any collapse as there is no spatial superposition. Furthermore, in our testing protocol—see Sec. VI—in the limit $ΔX→0$ the mechanical state is initialized in a thermal state. In this limit, there is no quantum decoherence and the thermal state simply equilibrates with its environment.

In order to model the dynamical evolution of the displacemon resonator under the assumptions of CSL, we can add the term $−DCSL[X,[X,ρ]]$ to the right-hand side of the master equation in Eq. (34), corresponding to a momentum diffusion process with rate $DCSL$⁠. This additional term is valid for regimes where the resonator has an oscillation amplitude⁵⁶ $XZP≪rCSL$⁠, and would only be detectable over conventional thermal noise if $DCSL>Dth$⁠. The geometry of the superposed macroscopic object dictates $DCSL$⁠. Assuming that the superposed displacemon beam is a cuboid of length $ℓ$ and width D such that the volume is given by $V=ℓD2$⁠, it follows that^56,57

where

and $Nn$ (not to be confused with $N¯$⁠) is the total number of nucleons that make up the resonator. Using $Nn=m/amu$⁠, we have that $Nn=3.1×1015$ for device A, and $1.3×1019$ for device B. The evolution of the resonator is governed by Eq. (34) (adapted to include CSL decoherence, as discussed) and is captured by the following quantum Langevin equations:

where $fth$ and $fCSL$ are the stochastic force terms arising from the standard theory of thermal decoherence, and CSL, respectively. These operators obey the following relations:

and $〈fth(t)fCSL(t′)〉=0$ at all times since the two noise sources are uncorrelated. By exploiting qubit measurements on the displacemon device, we can test the thermal decoherence of a mechanical resonator and deduce whether the rate agrees with conventional decoherence theory or whether there is a deviation which can be attributed to a non-zero $DCSL$⁠. The proposed procedure for this is described in detail in Sec. VI.

The displacemon device has several features which can be exploited to dynamically quantify the degree of decoherence of the resonator. From the statistics of qubit measurements taken at regular time intervals, one can infer both the extent of decoherence in the mechanical system and the rate at which this occurs. The following method should be read with CSL in mind as it exploits the dynamical description provided by the Langevin equations [Eqs. (A2a) and (A2b)]. While Penrose does not offer a dynamical description for gravity-induced collapse, we shall demonstrate that the following protocol can still be used to determine whether or not total decoherence of the mechanical state occurs on the timescale of $tDP$ or $tth$⁠.

Our method to test the decoherence rate of a mechanical superposition using the displacemon device is as follows:

1. The resonator is cooled toward its ground state using the qubit-measurement pre-cooling scheme outlined in Ref. 36. By using a sequence of pulses and selecting specific qubit outcomes, we can discard higher energy states and select the lowest-energy thermal states. Thus, the mechanical state $ρn¯$ is prepared in a thermal state with $n¯$ being the initial thermal occupation number.

2. The qubit-mechanical interaction is switched on [Eq. (1)] and a measurement of the qubit in the state $|+〉$ heralds an effective diffraction grating [see Eq. (25)] in the mechanical system. This then leads to a spatial mechanical superposition (as outlined in Sec. IV). The post-measurement state of the mechanical system is given by

   $ρin=ϒ+ρn¯ϒ+†Tr[ϒ+†ϒ+ρn¯],$

   (57)

   where the denominator is the heralding probability of creating such a state: $Tr[ϒ+†ϒ+ρn¯]=(1+e−2(1+2n¯|α|2))/2$⁠. We neglect any mechanical decoherence which may occur during the implementation of the measurement operator since $ϒ+$ is enacted on the timescale of the qubit lifetime $T2*$ which is much shorter than the mechanical period.

3. Following the qubit measurement outcome $|+〉$⁠, the mechanical state $ρ(t)$ is decoupled from the qubit and the position and momentum operators evolve according to Eqs. (A2a) and (A2b), in the Heisenberg picture.

4. Finally, after a time t_k, we apply a second effective grating operation $ϒ+$ by again switching on the qubit-mechanical interaction and measuring the qubit in the state $|+〉$⁠. The time elapsed between applying the first and second gratings is $tk=kπ/Ω$ where k is the integer number of mechanical half-periods. By waiting for mechanical half-periods we can neglect the R operator in Eq. (25). As the qubit is only coupled to the mechanical system during the fast grating operations, we are not constrained by the qubit coherence time; a key strength of our protocol. Over many experimental runs, we calculate the probability of measuring the qubit in the $|+〉$ state at time intervals t_k after it was first measured in the $|+〉$ state. This conditional probability is denoted by $P(k,D)=Tr[ϒ+†ϒ+ρ(tk)]$⁠, where $D=Dth$ according to conventional decoherence theory, or $D=Dth+DCSL$ if we incorporate the predictions from CSL theory. In the absence of any decoherence effects (⁠$D=0$⁠) this expression can be readily evaluated to give

   $P(k,0)=3+4e−2(1+2n¯)|α|2+e−8(1+2n¯)|α|24(1+e−2(1+2n¯)|α|2),$

   (58)

   and is bounded between [0.75, 1] for all $n¯$ and $|α|$⁠. Therefore, observing $P(k,D)<0.75$ necessarily implies the presence of decoherence either by conventional means, or via an additional collapse-theory mechanism. If $P(k,D)=0.5$ then the qubit is equally likely to be in the $|+〉$ or $|−〉$ state and so the mechanical state has completely decohered.

5. Alternatively, to test DP-style collapse we can measure the timescale required for $P(k,D)$ to reduce to an appropriate value (depending on the initial state) indicating that the mechanical state has undergone decoherence, and compare this with the timescales $tDP$ and $tth$⁠.

6. Note: We are interested in the probability that a second qubit measurement yields $|+〉$⁠. As this is a binary-outcome measurement, instances where the second measurement is $|−〉$ also contribute to our empirical estimate of $P(k,D)$ and so these experimental runs need not be discarded.

Using this experimental procedure, we can investigate how the probability $P(k,D)$ scales with various parameters as we tune $DCSL$ and $Dth$⁠. From this, we can identify the regions of parameter space where the predicted effects of CSL are greater than those of conventional decoherence and thus can be feasibly tested.

We can derive an analytical expression for $P(k,D)$ by recasting it in the Heisenberg picture $P(k,D)=〈ϒ+†(tk)ϒ+(tk)〉$⁠, where $ϒ+(tk)=cos (|α|X(tk)/XZP)$⁠. By inserting the solutions to the Langevin equations into the definition of $ϒ+(tk)$ [see Eq. (25)], we expect the probability to decay with k according to

For a derivation of this formula see the  Appendix. In regimes, such as ours, with high quality factors $Q=Ω/γ≫1$ we can make the following approximation:

demonstrating that the probability decays exponentially with time toward a value of 0.5. The slowly decaying envelope function $P0(k)$ [defined in Eq. (A6) of the  Appendix] determines the initial probability $P(0,D)$ and depends on k, the grating parameter $|α|$⁠, and the properties of the mechanical oscillator γ, $Ω$⁠, and $n¯$⁠.

The theoretical decay of the probabilities $P(k,Dth)$ and $P(k,Dth+DCSL)$ are plotted for devices A and B in Figs. 4(a) and 4(b). For these plots, the CSL decay rate has been calculated assuming $rCSL=10−7$ m and $λCSL=10−11$ Hz (a factor of 10³ times smaller than Adler's predicted value,⁹ thus probing a more challenging region of parameter space). The mechanical parameters and $|α|$ have been set to the values given in Table I. The probability in Eq. (60) decays exponentially with $|α|2$ and measurements are only performed after at least half a mechanical period has elapsed. Accordingly, we have used $|α|=0.1$ for device B so that CSL is properly distinguishable from purely thermal decoherence; this can be achieved by setting $B||=0.01$ mT [see Eq. (31)]. The initial thermal occupation numbers of the resonators are set to $n¯=100$⁠, assuming that both devices have been pre-cooled. The bath environment is assumed to have a thermal occupation number $N¯=kBTeff/(ℏΩ)$ with values $N¯=103$ (equivalent to an effective temperature of 0.07 mK) for device A and $N¯=5×106$ (or 1.9 mK) for device B. We have chosen these values of $N¯$ for these plots in order to most clearly visually demonstrate the exponential decay of $P(k,D)$⁠. However, we discuss below the experimental challenge of achieving such low temperatures and the interplay between the bath temperature and testable CSL parameters.

To maximally distinguish between decoherence effects in standard quantum theory versus CSL, we can investigate the percentage difference between the expected probabilities of a second $|+〉$ qubit measurement as predicted by the two theories:

For this, we use the full analytical result for $P(k,D)$ given in Eq. (59). As we scan the discrete variable k (i.e., vary the time between applying the two diffraction gratings) we can find the maximum percentage difference between the two probabilities $ΔMax$ for different values of $Dth$ and $DCSL$⁠. This quantity is plotted as a function of $λCSL$ and $N¯$ for devices A and B in Figs. 5(a) and 5(b), using the parameters in Table I. As before, we have used initial resonator occupation numbers of $n¯=100$⁠. We only consider $N¯>n¯=100$ as we expect the bath to always be warmer than the device itself.