Introduction

A great variety of important physical phenomena can be effectively described in a two-dimensional Hilbert space, when the system's dynamics effectively involves only two relevant states. The most common examples include oscillatory, decaying and/or relaxation effects in: elementary particles (e.g., neutrino and kaon oscillation1,2), atoms (e.g., Rabi oscillation and spontaneous emission3), molecules (e.g., tunnelling in double-well potentials, like Ammonia inversion4,5,6) and crystals (e.g., spin relaxation7,8).

In such systems, oscillations occur because the relevant states are not eigenstates of the system's Hamiltonian. To be definite and without loss of generality within the two dimensional formalism, let us take the eigenstates |+〉 and |−〉 of the operator as the relevant states and as the Hamiltonian, where ωx is the characteristic oscillating frequency (for example, for Ammonia, ωx = 24 GHz is the inversion frequency). If we start from any eigenstate of , we observe the coherent oscillation between |+〉 and |−〉 with frequency ωx. In this idealised situation, temporal oscillations remain coherent in time, with a constant amplitude. However, in practice they lose coherence and decay more or less rapidly, because the system is exposed to external noises. Such environmental effects can be effectively described by Lindblad-type master equations9,10,11.

Oscillations become of great conceptual importance when the two relevant states |+〉 and |−〉 become “macroscopically” distinct. This is the typical situation with chiral molecules, as we will see. The observation of oscillations between two such states is directly connected with the highly-debated problem (both theoretically and experimentally) of the quantum-to-classical transition: how linear quantum mechanics copes with macroscopic classical variables, where “classical” implies no superposition9,10,11,12. The fundamental question is whether such “macroscopic oscillations” persist when the system increases in size (and assuming that environmental sources of noise are kept under control) as predicted by quantum mechanics, or alternatively if they unavoidably decay in time because of intrinsic nonlinear effects in the dynamics. This second possibility is predicted by collapse models12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27.

Collapse models have been extensively studied in the literature. There has been also a rapid progress in experimental searches of nonlinear effects predicted by collapse models23, in particular by delocalizing large massive objects with matter-wave interferometry and optomechanical techniques28,29,30,31,32,33. In order to motivate further experimental searches of such nonlinear effects, here we follow a different approach by studying how collapse affects naturally oscillating quantum systems. In these cases, it is not necessary to create the superpositions in the laboratory, as they appear spontaneously from the dynamics.

Collapse models add stochastic and nonlinear terms to the Schrödinger dynamics, which induce the collapse of the wave function. In the most well-studied collapse models (CSL16, QMUPL20), a noise-field is nonlinearly coupled to the spatial degrees of freedom of any massive system, inducing the suppression of spatial coherence. These models are discussed later in the text. Here it suffices to say that, when restricting to a 2D Hilbert space, where the states |+〉 and |−〉 describe two different spatial configurations, then the collapse dynamics takes the form34:

with Wt a standard Wiener process and λ the collapse rate depending on the size of system and the nature of oscillation. The last two terms of Eq. (1) induce the collapse of the wave function either to |+〉 or |−〉, according to the Born probability rule. In experimental situations, only averages over the noise are relevant. These can be computed from the density matrix , where denotes the stochastic average. It is not difficult to prove that obeys the following Lindblad-type equation34:

Quantum linearity (manifested by the oscillatory behavior) is well preserved when ωx ≫ λ, while nonlinearity (i.e., no quantum superposition) becomes dominant when λ ≫ ωx. In this way collapse models provide a quantitative description for the transition from the microscopic quantum world to the macroscopic classical one.

For any given physical system, one has to derive λ from the full collapse dynamics, in the same way in which the characteristic frequency ωx can be deduced, at least in principle, from the complete Hamiltonian of the system. In the next sections, we will compute λ for three different types of oscillatory systems: neutrinos, neutral mesons and chiral molecules. We will show that, contrary to what previously predicted35, collapse models cannot be tested with neutrinos. The collapse effect is stronger for neutral mesons, but still beyond experimental reach. Instead, chiral molecules offer promising candidates for testing collapse models.

Eq. (2) has the same form as that describing an oscillatory system under environmental noises9,10,11. This means that, in analysing the effect of collapse models on oscillating systems, one has to consider also environmental effects, which tend to mask the collapse effects, by damping oscillations in a similar way. In each case, we will compare predictions of collapse models with decoherence effects.

Results

We present the analysis of how collapse models modify the oscillatory behavior of neutrinos, neutral mesons and chiral molecule. We will use the mass-proportional Continuous Spontaneous Localization (CSL) model23; details of CSL dynamics are explained in Methods section. The representation of the collapse dynamics in the position-basis predicts the following collapse rate for the off-diagonal elements of the density matrix of a generic system consisting of N nucleons:

where (see Methods), N is the number of nucleons in the spatial superposition, with rC = 10−5 cm and , are distinct positions of nucleons in spatial superposition.

The connection between the full characterization of λCSL, given by Eq. (3), with the two-dimensional one given by Eq. (2) is not always straightforward. It depends on the system under study and sometimes needs careful analysis and lengthy calculations, as we will show.

In applying collapse models to experiments, one has always to take decoherence effects into account, as they produce apparently similar effects. For neutrinos and chiral molecules, collisions are the dominant source of decoherence. Using collisional decoherence theory9,10,11, we exploit the phenomenological formula: λDEC ~ n v σDEC (with v the relative velocity, n the density of bath particles and σDEC the decoherence scattering cross section) in order to estimate the decoherence rate. For mesons, we provide upper bounds on λDEC using available experimental data.

Neutrino oscillation

Effective description of neutrino oscillations

The flavour eigenstates of neutrinos |να〉 (with α = e, μ, τ for electronic, muonic and tauonic neutrinos) are linear combinations of mass eigenstates: , with the unitary mixing matrix. Therefore, for a neutrino in an initial flavour eigenstate, the transition probability between different flavour eigenstates shows an oscillatory behaviour in the course of time1,2. This oscillation may be damped either by environmental interactions, or by nonlinearities in the dynamics such as those predicted by collapse models. Neutrinos are the lightest massive particles, therefore it seems unlikely that they show any spontaneous collapse effect. However, they can travel very long distances through space and there could be enough time during the flight, for collapse effects to build up appreciably. Therefore, it is not clear beforehand whether neutrinos can play any role in testing spontaneous collapses.

The collapse rate in neutrino oscillations

The effect of collapse models on neutrino oscillation was first elaborated by Christian35, using the Diósi-Penrose (DP) gravity-induced collapse model18,19,21. Gravity is fundamentally nonlinear, therefore when properly taken into account, it induces a nonlinear modification of the Schrödinger equation. According to the analysis of Christian (see also36), the predicted magnitude of the oscillation damping (λt) is between ~ 10−2 and ~ 1 for cosmogenic neutrinos. This value is strong enough to be tested with high-precision techniques. However, this strong predicted effect is questionable, for the following reason. For point-like constituents, like neutrinos, gravitational self-energy diverges, implying a divergence in DP model. To avoid this problem, Diósi18,19 originally introduced a cutoff for small lengths, equal to the nuclear size. However, Ghirardi, Grassi and Rimini37 showed that a much larger cutoff (~ 10−7 m) is needed, in order for the model to be consistent with known experimental data. On the other hand, the effective size of the neutrinos, as introduced by Christian, is ~ 10−30 m, well beyond any reasonable cutoff. Therefore the result cannot be trusted.

We compute the collapse effect on neutrino oscillations using the CSL model, which is free from the divergences contained in the DP model (apart from standard quantum field theoretical ones, which can be treated with usual renormalization techniques). The dynamics of neutrino oscillation is phenomenologically described in a 3D–Hilbert space of flavour, while the CSL collapse occurs in space. Therefore, the major task is to link the spatio-temporal description of Eq. (8) to the 3D–dynamics neutrino oscillations. Since the CSL model is a field theoretical model, this can be properly done by resorting standard quantum field theoretical techniques. Treating the noise as perturbation, for the collapse rate we obtain36:

where m0 = 1 amu, mj are the eigenvalues of mass-eigenstates and with pi the momentum. In the relativistic regime, as appropriate for neutrinos, one has . By taking the largest mass difference in Eq. (4), one finds the following upper bound:

where the energy (E) and the time of flight (t) of the neutrinos depend on the type of neutrinos under study. In Table I, this damping factor has been computed for neutrinos originating from three different sources. The CSL collapse effect is very tiny and non detectable with present-day technology, the reason being that neutrinos are too light, although they can travel long distances.

Table 1 Theoretical values of CSL collapse rate and decoherence rate for neutrinos, mesons and chiral molecules. By moving from elementary particle scale to molecular scale, the collapse rate λCSL increases significantly. The decoherence hides collapse effects in neutrinos and mesons, but it can be reduced at a negligible level compared with collapse rates of chiral molecules. Results show that quantum linearity can be in principle tested using chiral molecules. However, engineering a proper experiment is not straightforward
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Decoherence effects in neutrinos

We also analyze decoherence effects on neutrino oscillations due to the scattering with particles (mainly leptons), during their flight through space. The experimental value of the relevant scattering cross section, σDEC, are known in the literature38,39. The average density of electrons in outer space and in the atmosphere are respectively and , while the average density of neutrinos is about nν ~ 108/m3 everywhere (electrons and neutrinos are the two main sources of decoherence36). Assuming the neutrino velocity v equal to the velocity of light in vacuum, we get:

with the decoherence rate in the out-space [atmosphere]. Neutrinos travel through the atmosphere within ~ 10−4 s, the remaining time being spent in traveling through outer space. Taking both contributions from atmosphere and outer space into account and using data listed in Table I, the decoherence damping factor for cosmogenic neutrinos (CN) turns out to be: λCNt ~ 10−5. For solar neutrinos (SN) instead, one gets: λSNt ~ 10−18, which is hardly detectable, in agreement with well-known experimental results2,40.

This analysis shows that, since environmental decoherence on neutrino oscillations is much stronger than the CSL collapse effect (and comparable with that–overestimated–predicted by Christian35), even if technology were able in principle to discriminate collapse effects on neutrino oscillations, these effects would be masked by unavoidable decoherence effects.

Neutral mesons

Effective description of neutral mesons oscillations

As a second example of oscillating quantum systems, we consider neutral mesons. Differently from neutrinos, they offer the advantage that decoherence effects can be kept low, since they are produced in a very controlled environment.

A meson consists of a quark and an antiquark. For example there exists the neutral K-meson (K0, made of and d; or , made of s and ) or the neutral B-meson system (B0, made of and d; or , made of b and ). The phenomenology of these oscillating and decaying systems is usually described by a 2 × 2 non-Hermitian Hamiltonian whose stationary states are the mass eigenstates41: where m1,2 and Γ1,2 are masses and decay widths. The mass eigenstates are related to the flavour eigenstates via , if and only if we assume conservation and neglect violation. We can safely assume such a linear combination because the violation is a very small effect in our case.

The collapse rate for neutral mesons' oscillations

In computing the predictions of collapse models for the oscillations in neutral mesons, we follow the same approach as the one we did for neutrino oscillations. We perform the computation by expanding the CSL dynamics to the first significant perturbative order, in order to find the dominant effect. The calculation is long but straightforward and is fully reported in41. The final result for the collapse rate is:

We list the damping rates in Table I for distinct mesons, using the experimental values given in Ref. 48. With not much surprise, the obtained values are much larger than those for neutrinos.

The decoherence rate in mesons

Environmental decoherence effects have been investigated42 and compared with experimental data43,44,45,46,47. Bounds from experimental data of the CPLEAR experiment44 and to the more refined data of the KLOE experiment of the DAPHNE collider45,46,47 were obtained in terms of a phenomenological time-independent parameter ζ42. This parameter, first introduced by Schrödinger, quantifies the spontaneous factorization of an initially entangled wave function in a chosen basis. The best value, obtained by measuring 2-pion final states, is: ζ = 0.003 ± 0.018stat ± 0.006syst. Since this is a time averaged quantity, we can use it only for small times, when ζ ≈ λdecot. From that we may deduce an upper bound on the decoherence rate which is about 8 × 107 Hz with %90 confidence level. Thus, comparing this value with those in Table I, we see that collapse models are not directly measurable for strangeness oscillations; for other types of mesons similar considerations hold. To test collapse models for mesons one has to find observables being more sensitive to the CLS effect.

Chiral molecules

Effective description of chiral molecules

Another very relevant example of oscillating quantum system is given by chiral molecules, in which case |+〉, |−〉 represents two configurations with different macroscopic properties, e.g., optical activity. The classical example is Ammonia inversion phenomenon4. In general, non-rigid molecules and molecular complexes have at least two stable configurations that can be transformed to each other by a large-amplitude vibration4. In the zero-th order approximation, this vibration can be described by the motion of a particle of effective mass μ in a double-well potential V(q), where q is a generalized large-amplitude coordinate. The minima of the wells are positioned at q = ±q0/2, separated by a barrier V0 (see Fig. 1). Molecular configurations are described by localized states (say “chiral” states) at each minima. The tunnelling through the height barrier leads to measurable level splittings in the molecular spectra, which has been observed for a large variety of non-rigid molecules and molecular complexes4,49,50.

Figure 1
figure 1

A symmetric double-well potential.

Important parameters are the tunnelling frequency ωx = (E2 − E1)/ℏ, the height barrier V0 and the minima separation of q0. The molecular structures are associated with left and right chiral states, ψL and ψR, that are localized in each minima. The tunnelling splitting is manifested as doubling in the spectra of the molecule4.

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In the limit V0 ≫ ω0 ≫ kBT (where T is temperature, kB is Boltzmann constant and ω0 = [V″(±q0/2)/ μ]1/2 is the small-amplitude vibration in either well), the state of the molecule is effectively confined in the two-dimensional Hilbert space spanned by two chiral states5,6. Thus, the Hamiltonian becomes with ωx the level splitting due to the tunneling.

The collapse rate in chiral molecules

We consider superpositions of chiral states as spatial superpositions of an atom or group of atoms between two distinct molecular configurations. So differently from the case of neutrinos and neutral mesons, we can immediately derive the collapse rate from Eq. (3). Typical non-rigid molecules are within a range of size 1–100 Å. This implies that chiral coherence is distributed over the region whose dimension is much smaller than rC = 103 Å. We can then expand F(r) to the leading order of r in Eq. (3) and we obtain:

where mi is the mass (in amu) of i-th atom, n is the number of atoms in the spatial superposition (e.g., for Ammonia, we have three Hydrogen atoms in superposition; n = 3) and and are positions of i-th atom in the two chiral conformations where the origin is the chirality center (e.g., for Ammonia, they are positions respect to the Nitrogen atom). The mere knowledge of positions of atoms in chiral structures of the molecule is enough to compute the collapse rate using Eq. (8). However, when data about the effective mass μ and the minima separation q0 of the double-well potential is available, then one can simply use the following simpler formula for the collapse rate:

with μ the effective mass (in amu) moving in the double-well potential. For example, μ = mN(3mH)/(mN + 3mH) ≈ 3 amu and q0 = 0.8Å in the case of Ammonia4.

We apply Eq. (8) to compute the collapse rates for some pyramidal chiral sulfoxides51. The results are shown in Table I. For each sulfoxide, we obtain the enantiomeric equilibrium structures with DFT (B3LYP), using a minimum basis set by Firefly program52. As expected, the collapse rates are by many orders of magnitude stronger than those of neutrinos and mesons.

Decoherence rate in chiral molecules

If we model the chiral coherence as the spatial superposition of a quantum Brownian particle of effective mass μ over the distance q0 (see next section for more detail), then we can use the linearized quantum Brownian dynamics to compute σDEC5,6. In this way we can compute the dominant contribution to λDEC by using Eqs. (3) and (13) of Ref. 6. We consider the London dispersion potential for collisions. Then, for the density of background gas about ~ 1010 particles/m3 (the conventional ultra-high vacuum) and the background temperature of , we obtain: λDEC ~ 10−6–10−4 Hz. Considering the cryogenic vacuum where n ~ 105 particles/m353, then we get: λDEC ~ 10−11–10−9 Hz. Accordingly, for chiral molecules decoherence can be practically reduced to a negligible level compared to collapse effects (see Table I), thus quantum nonlinearities can be in principle tested using chiral coherence.

Estimates of bounds on Λ

We showed that, contrary to the cases of neutrinos and mesons, spontaneous collapse effects (quantified by λ) can be in principle tested with chiral molecules because environmental effects can be controlled in such a way that the decoherence becomes negligible. Of course, the great challenge is to find a feasible experimental scheme. We leave the question to future research. In the meantime, one can follow a different strategy and use the spectroscopic data of tunneling splittings to introduce upper bounds on Λ. Here, we discuss this strategy.

According to Eq. (2) the dipole moment shows no oscillation when λ ≥ ωx. If this is the case, then the spectra of the molecule should show no tunneling splitting. Accordingly, the experimental observation of a tunneling frequency ωx implies that λ < ωx. This places an upper bound on the collapse parameter Λ, which according to Eq. (9) can be written as follows:

The smaller the observed tunnelling frequency, the stronger the bound (see Table II). To our best knowledge, the smallest molecular tunnelling splitting that has been observed is of the order of a few Hertzs for Ru-D2 complex with NMR spectroscopy55, where μ = 2 amu and q0 ~ 1–2 Å. Accordingly, we get Λ < 105 Hz, which is 1014 times larger than the standard CSL value. This should be compared with one of the best available experimental bounds on the collapse rate: Λ < 10−5 Hz, which is obtained by quantum interfere of massive objects with a mass of 7 × 103 amu, in matter-wave interferometry experiment29. So, also this bound is very week, but could be significantly improved.

Table 2 Current bounds on the collapse constant Λ, coming from observation of tunnelling. We used available spectroscopic data about tunnelling splittings (see main text). These bounds should be compared with the best experimental bound on Λ, which is obtained by wave-matter interferometry of molecules with mass m = 7 × 103 amu29. Using a molecular spectroscopic technique with relative resolution of R ≤ 10−14, it is in principle possible to introduce bounds on Λ, which could compete with those coming from matter-wave interferometry
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According to Eq. (10), in order to obtain stronger bound on Λ, one should move toward smaller tunnelling frequencies (i.e., smaller ωx) or larger effective sizes (i.e., larger μ and q0). In molecular systems, the effective size that can be simply described by a double-well potential, is limited in the ranges q0 ~ 1–10 Å and μ ~ 1–100 amu4. According to Eq. (10), we get: Λ < αωx, with α varying in the range 4–4 × 106. Therefore, among possible strategies for testing collapse effects in molecular systems, the observation of smaller tunnelling frequencies is the most flexible.

This strategy becomes even more promising if we consider recent progresses in high-resolution spectroscopic methods49,50. As we discussed before, ωx is manifested as level splittings in rotational-vibrational spectra, where molecular modes cover the typical frequency range of ω = 109–1014 Hz (from microwave to UV4,49,50). Thus, if we use a spectroscopic method with resolution R = ωx/ω, we find: Λ < β R, where β varies in the range 4 × 109–4 × 1020 Hz. So, with a relative resolution R ~ 10−14, which is in the range of available highest resolution spectroscopy techniques49,50, one can reach a bound for Λ, comparable with that obtained in matter wave interferometry. With better resolutions, we can set stronger bounds. Molecules of the form Y-X-X-Y may serve as candidate molecules where their torsional internal rotation can be simply described by a double-well potential4,49,50, with a very tiny tunnelling splitting when Y is a heavy atom (e.g., for Cl2O2, we have the theoretical value of ωx ~ 10−11 Hz49,50).

Discussion

We computed the predictions of the Continuous Spontaneous Localization (CLS) collapse model for the damping of the oscillatory behavior of three distinct naturally oscillating quantum systems: neutrinos, mesons and chiral molecules. The numerical results are summarized in Table I. We also analysed the main decoherence effects on these systems and compared them with the predictions of the CSL collapse model. The values we obtained for the collapse rates for the first two types of systems are much smaller than the main decoherence rates; consequently, possible violations of the superposition principle cannot be directly observed in these oscillatory systems. Chiral molecules are better candidates. We suggest a new type of experiment with chiral molecules, which can serve as a test of quantum linearity and which can possibly put stronger upper bounds on the collapse parameters, than those already available from the literature.

Our formulation of chiral molecules also includes any system whose effective dynamics is described by a double-well potential. A very promising line of research is the study of systems that can be artificially prepared in a double-well potential where its parameters (V0, μ and q0) are adjustable at will. Then, by tuning them in proper ranges, e.g. larger μ, one can hope to set further bounds on the collapse parameters.

Methods

CSL model

We consider the most commonly used collapse models in the literature: the mass proportional Continuous Spontaneous Localization (CSL) model17. The CSL dynamics is:

with the standard quantum Hamiltonian, the standard quantum average (here is where nonlinearity enters the equation), m0 = 1 amu, γ > 0 the strength of the collapse process, which is a new phenomenological constant of the model, Wt(r) an ensemble of independent Wiener processes, one for each point in space, and:

where is the annihilation operator of a particle of type j at position r, and:

with rC the correlation length, the other new phenomenological constant of the model. After averaging over all possible realizations of the stochastic processes, the dynamics for the density matrix is given by:

where the second term on the right hand side is the collapse term. It tells that superpositions of states which are closer than rC are efficiently localized, while superpositions of terms which are further separated are suppressed, with a rate proportional to γ and to the size of the system27.

Values of collapse parameters

The value of the correlation length is commonly fixed to 16. For the collapse strength γ, two values have been proposed in the literature. Ghirardi, Pearle and Rimini16 set , while Adler27 sets . These values are in agreement with all known experimental data. Much larger values are ruled out because the collapse would become so strong to be detectable also for isolated microscopic systems, contrary to experimental evidence. Much smaller values are also ruled out, because in such cases the collapse would become so weak that the localization of the wave function of macroscopic objects would not be guaranteed anymore. Without this, collapse models would lose their interest. In our analysis, we consider the strongest value of γ suggested by Adler. By defining the collapse rate as: , we get: . This is the numerical value we used in the text.