On tests of the quantum nature of gravitational interactions in presence of non-linear corrections to quantum mechanics

When two particles interact primarily through gravity and follow the laws of quantum mechanics, the generation of entanglement is considered a hallmark of the quantum nature of the gravitational interaction. However, we demonstrate that entanglement dynamics can also occur in the presence of a weak quantum interaction and non-linear corrections to local quantum mechanics, even if the gravitational interaction is classical or absent at short distances. This highlights the importance of going beyond entanglement detection to conclusively test the quantum character of gravity, and it requires a thorough examination of the strength of other quantum forces and potential non-linear corrections to quantum mechanics in the realm of large masses.

sidered a massive particle to be placed in a coherent superposition of its spatial degrees of freedom by first preparing an internal spin degreeof-freedom of this particle in coherent superposition and allowing this particle to pass through a Stern-Gerlach apparatus.Then, crucially, he assumed this particle to interact purely gravitationally with yet another massive particle.Indeed, treating the gravitational interaction as either classical or quantum mechanical results in very different quantum states of the two particles and thus experimental results of subsequent measurements.The particles would emerge with correlated positions that are described either by an incoherent mixture or by a coherent superposition, respectively.The latter, in modern quantum information parlance referred to as an entangled state, let Feynman to conclude that "We would then have to analyze through the channel provided by the gravitational field itself via the quantum mechanical amplitudes.Therefore, there must be an amplitude for the gravitational field".
Hence, as discussed further in [2,3,4,5] the observation of the generation of entanglement between two massive particles would certify the quantum mechanical character of their gravitational interaction.If this interaction is assumed to be mediated by a gravitational field and the local dynamics follows the laws of quantum mechanics, then this would falsify the assumption of a classical force carrier and thereby imply the non-classical nature of the gravitational field [6,7,8,9,10].Note that in this argument the existence of a gravitational field is taken as an assumption that is not directly verified by the experiment [11].
Nevertheless, one must exercise caution when embarking upon an argument of this nature based on experimental data as the strength of the conclusion is inherently tied to the validation of the assumptions that underlie the very foundation of the logical inference.Indeed, recently models have been proposed in which, even when starting from product states, entanglement may be generated with semiclassical models of gravity.These models include hybrid quantum-classical ensembles [12] and variations of Bohmian quantum mechanics [13].Both models violate fundamental assumptions that enter the proofs that a classical force carrier cannot create entanglement and their validity would have to be established experimentally.
Here we add another facet to this discussion with models where quantum mechanics exhibits weak non-linear corrections to the local1 dynamics of massive particles.Non-linearities may, for example, emerge in models that couple classical gravity to quantum dynamics [14,15] -the alternative model that we wish to exclude experimentally with the observation of entanglement.In the absence of any quantum mechanical interaction the models that we will consider map product states to product states.However, in any realistic experiment there will be residual, albeit very weak, quantum interactions (e.g., electromagnetic forces -Casimir forces, dipolar interactions, etc.) [16].As we will proceed to show, under such models the entangling effect of these quantum forces -even when they are much weaker in strength than gravity at the same distance -can be amplified by non-linear corrections to quantum mechanics.This, in turn, can lead to an entanglement evolution closely resembling that expected under the influence of quantum mechanical gravity.
As a result, the observation of entanglement can be held to witness the quantum mechanical nature of the gravitational interaction without loopholes only if additional measurements limit the strength of both, non-gravitational forces and possible non-linear corrections to the local quantum dynamics at the length, time and mass scales in question.We discuss possible tests of such nonlinear corrections to quantum mechanics.
Non-linear corrections to quantum mechanics may, at first sight, appear a daring assumption for a variety of reasons.First, it has been demonstrated that specific examples of non-linear quantum mechanics that are, for example, consistent with the framework put forward by Weinberg [17,18], allow for signalling [19,20,21], an observation that would place considerable strain on notions of causality in physics.It should be pointed out, however, that proposals have been put forward that enable non-linear quantum evolutions to become consistent with Minkowski causality [22,42].Secondly, one may argue that in atomic physics the presence of non-linear corrections has been the subject of stringent experimental tests that have resulted in tight bounds on possible non-linearities in specific models [23].However, such tests have been carried out only on the spin degrees of freedom of the constituents of atomic nuclei.Under reasonable experimental parameters, tests of the quantum character of the gravitational interaction require particles that contain of order 10 12 or more nucleons.This puts them significantly beyond the mass range where exhaustive examinations of non-linear corrections have been conducted up to this point.It is perfectly conceivable that non-linear corrections to quantum mechanics scale with mass or energy of the test-masses.This could lead to considerable enhancements of non-linear effects for a testmass with 10 12 or more nucleons over those experienced in experiments with single atoms that have been tested so far [23].As a result, the existence of such non-linearities for massive particles retains the status of a plausible hypothesis, especially since, as we stressed earlier, such nonlinear corrections to quantum mechanics appear in theoretical models that couple classical gravity to quantum mechanical matter [14,15], the very alternative that the observation of entanglement generation is aiming to exclude.This motivates the following examination of the consequences of non-linear corrections to quantum mechanics in the context of experimental tests of the quantum character of gravity in interferometry with massive particles.This work represents a proof of principle, i.e., we construct a particular model where nonlinearities boost entanglement generation, while a general analysis of the effect is postponed to future work.
Outline -The paper is organised as follows: in the first section we introduce Weinberg's framework for non-linear extensions to quantum mechanics.We then proceed with a brief description of typical experimental set-ups in tests of the quantum character of gravitational interactions, and introduce a specific class of non-linear models that are used throughout the paper.After computing the entanglement dynamics and considering the implications of these results for experimental tests, we conclude by discussing how to detect non-linearities in these models via frequency measurements.
Weinberg's framework -There is no unique manner in which to incorporate non-linear terms into quantum mechanics.In this section we introduce an elegant (and rather general) framework due to Weinberg that encompasses a wide variety of possible non-linear extensions to quantum mechanics [17,18].It was introduced with the express purpose in mind of providing non-linear extensions of quantum mechanics that can then be subjected to experimental test [23].
In standard quantum mechanics, we associate with every physical system a Hilbert space H, whose elements |ψ⟩ represent its states.
This prescription ensures that each of the two subsystems will evolve under the action of its free Hamiltonian alone, which is what we expect from two non-interacting systems.Furthermore, it guarantees that the dynamics generated by h AB map product states to product states.In other words, when two non-interacting systems are initialized in a product state, their non-linear local evolution will always coincide with each subsystem's free dynamics, and the global dynamics will never entangle them spontaneously.
If we consider bilinear Hamiltonian functions of A and B, i.e., h A,B = ⟨ψ| H A,B |ψ⟩ for some Hermitian operators H A , H B , the prescription above reduces to the standard composition rule The typical experimental set-up -A possible test of the quantum character of the gravitational interaction that probes its capacity to generate entanglement prepares two test-masses at a distance r each in a state with a high degree of delocalisation ∆x.This can be achieved with the preparation of a Schrödinger cat state [24,25,16,26,27,28] or of a squeezed state of motion [29,30,31].In what follows we adopt the former as it allows us to reduce the mathematical description in the limit of large delocalisation to that of a two-dimensional Hilbert space which, in turn, allows for a straightforward application of Weinbergs formalism.Hence, we have in mind the situation depicted in Fig. 1 where the wave-function of each test-mass is prepared in a Schrödinger cat state normalisation constant N and |α| large enough so that the overlap between |α⟩ and |−α⟩, given by | ⟨α| − α⟩ | = e −2|α| 2 , is negligible.Furthermore, we assume that the spatial dynamics of the coherent states is negligible, e.g., because the par-ticles are in freefall and the experiment is sufficiently short to ensure that the dominant effect is the build-up of a relative phase.This allows us to introduce a qubit representation for each test-mass, i.e., |0⟩ ≡ |α⟩, |1⟩ ≡ |−α⟩.The initial state is then |ψ(0 Under quantum mechanics, by the symmetry of the set-up and irrespective of the precise origin of the force, a distance-dependent interaction between the two test-masses will evolve |ψ(0)⟩ into |ψ(t)⟩ = |00⟩ + e igt (|01⟩ + |10⟩) + |11⟩, i.e., the components |01⟩ and |10⟩ will accumulate a phase relative to |00⟩ and |11⟩.This dynamics is described by an interaction term of the form gH int = g (|01⟩ ⟨01| + |10⟩ ⟨10|), where g determines the coupling strength.We now add a nonlinear perturbation of the form to the local quantum dynamics of each subsystem, where can then be composed into a global Hamiltonian function h Y on the bipartite system 1+2, as mandated in Weinberg's approach [17,18].We stress that this prescription ensures that the dynamics due to h Y maps product states to product states.Then the total Hamiltonian function reads It is important to stress that, in order to obtain the numerical results presented in this paper no explicit composition of the Hamiltonian functions h 1 Y , h2 Y has been computed.Instead, we are making use of the Trotter decomposition and choose a time step dt small enough, such that we can simulate the global evolution from t to t + dt via the factorization of the global dynamics as a composition of: a) free nonlinear evolution of A under its associated Hamiltonian function h 1 Y , b) free nonlinear evolution of B under its associated Hamiltonian function h 2 Y , and c) linear evolution of A and B under the interaction term H int .Thus, in the numerical simulation we only use local nonlinear evolutions and we can avoid computing h Y explicitly.Naturally, the linear part of the Hamiltonian function, H int , alone will lead to an oscillatory entanglement dynamics with angular frequency g.In what follows we will demonstrate, perhaps surprisingly, that the non-linear contribution to the local Hamiltonian, ϵh Y , can enhance the rate of oscillation of the entanglement between the two test-masses as compared to that obtained under standard quantum mechanics.
As we are dealing with noise-free pure state dynamics, we quantify the entanglement of the evolved state |ψ(t)⟩ by the von Neumann entropy of the reduced density matrix ρ [32,33].Let us denote by E (g,ϵ) (t) the entanglement dynamics obtained for a coupling strength g of the linear part H int and a strength ϵ of the non-linear correction h Y .We chose the entanglement dynamics E (1,0) (t) corresponding to linear quantum mechanics with unit coupling strength g as the reference.As one observes from the inset of Fig. 2, with increasing ϵ the non-linear correction ϵh Y is able to boost the oscillation frequency rate of the entanglement dynamics between the two test-masses while maintaining the same shape with maximal contrast.It is then natural to ask whether for any g < 1 there is a choice ϵ = ϵ * (g) such that the frequency of oscillation of entanglement of E (g,ϵ * (g)) matches that of our reference E (1,0) (t).As it proves challenging to obtain analytical expressions for ϵ * (g), Fig. 2 shows the numerically determined relationship which is excellently fitted by an invertible function with inverse g * (ϵ).
This relationship shows that, in principle, the rate of oscillation of the entanglement can be amplified arbitrarily by the presence of the nonlinear correction to the local dynamics while maintaining a functional profile closely following that of the linear case E (1,0) (t).This statement can be made more quantitative by computing how well the non-linear entanglement dynamics E (g,ϵ * ) (t) approximates E (1,0) (t).We measure this with the mean squared deviation and plot the results in Fig. 3.The expression in Eq. ( 6) allows us to discuss the implications of these findings for realistic experimental parameters.To this end we compare the gravitational force between the two oscillators with the Casimir force between them 2 .For two dielectric spherical particles of mass M 1 and M 2 at distance r the gravitational interaction will lead to the accumulation of a relative phase at the rate while the phase due to the Casimir force will accumulate at a rate [34] where R 1 and R 2 are the radii of the two particles, ϵ d is the dielectric constant and c is the speed of light.For definiteness, assume the case of a diamond which has ϵ d = 5.7 with a mass density of ρ = 3.51 × 10 3 kg/m 3 .If we take R 1 = R 2 = 1.307 µm, r = ∆x = 200 µm, for the arrangement in Fig. 1, we find with ω g = 1 Hz and ω c = 0.096 Hz.Normally, the latter would be considered negligible compared to the former.However, from Eq. ( 6) we find that a non-linear correction h Y with a strength of the order of ϵ ∼ 2.26 Hz will be sufficient to boost the frequency of oscillation of E (ωc,ϵ) (t) purely due to Casimir interaction under a non-linear quantum mechanics to equal that of E (ωg,0) (t) expected from a quantum mechanical gravity and no Casimir force under linear quantum mechanics.
Probing non-linear corrections to QM in a harmonic oscillator -As we have seen, the presence of non-linear extensions of quantum mechanics can have significant effects on the dynamics and hence on the conclusion that can be drawn from such experiments.It is thus natural to examine whether non-linearities of the magnitude discussed above would already lead to observable consequences in current or soon-realisable experiments on a single massive particle subject to a harmonic potential.As the preparation of highly non-classical states is very challenging experimentally, we consider a harmonic oscillator initially prepared in a thermal state.We displace the trap center suddenly in space such that relative to the new coordinates the harmonic oscillator is in a displaced thermal state whose mean position and momentum are expected to oscillate periodically in time.It is reasonable to expect that the oscillation frequency will be a function of the strength of the non-linear corrections to quantum mechanics.Such frequency changes will be detectable more easily for a harmonic oscillator that is cooled to the ground state which is what we assume for the following.This has the added benefit of simplifying the analytical treatments for important limiting cases.
For the experimental set-up shown in Fig. 1 we were able to neglect the spatial dynamics and thus assumed the coherent state amplitude α to be time-independent.Now, we are interested in the full spatial dynamics of the harmonic oscillator including the non-linearities for arbitrary α.For its mathematical description we need to define three operators in such a manner that for large α they reduce to the Pauli-operators in Eq. (4) for the subspace spanned by | ± α⟩.To this end we define the parity operator and the projector P α onto the subspace spanned by | ± α⟩ which yields by lengthy but direct calculation where ∥•∥ 1 is the trace norm and σ x := |α⟩⟨−α|+ | − α⟩⟨α|.Furthermore, using the displacement operator D(β) = e βa † −β * a , we define where β ∈ C is for now a free parameter of the model.The requirement that the operator Y defined above becomes a σ y in the two-dimensional subspace spanned by |±α⟩ when |α| ≫ 1 defines the parameter β in terms of α as β = −i π 8α , as for such value of β one gets obtained by making use of the relation Thus for large α the operator Y reduces to the σ y -operator in the subspace spanned by |±α⟩.In order to be as general as possible, we can study this problem by keeping β as a free parameter, knowing that this model reduces to the qubit model studied earlier only when β satisfies the constraint above.
In keeping with the mathematical description of the experiment in Fig. 1 we proceed to choose the non-linear correction to quantum mechanics as The resulting Hamiltonian function is then which yields the non-linear Schrödinger equation We proceed by solving the resulting dynamics by making use of the formalism of generalized coherent states (GCS) [35].These are states whose evolution is described by classical Hamilton equations on a manifold that plays the role of a classical phase space.As presented in Appendix A, the GCS for this system are, to first order in ϵ, of the form |α, θ⟩ where is a generalization of the usual displacement operator D(α), parametrized by the complex coordinate α and by a set of additional complex coordinates θ := {θ µ |µ ∈ N}.The operator Π(α, θ) can be computed explicitly and its expression is found in Appendix A. The states |α, θ⟩ live on a very large manifold, and they coincide with the standard coherent states for θ µ = 0.The dynamics |α(t), θ(t)⟩ on the manifold is specified by a set of Hamilton equations, as outlined in [35].In the limit of large amplitudes, |α| ≫ 1, we find all corrections due to the non-linearity h Y to be suppressed exponentially in |α| 2 (as described in Appendix A) and the dynamics is extremely well approximated by a trajectory on the complex plane (Re(α), Im(α)) obeying α(t) = −iω 0 α(t).Thus an initial state |α, 0⟩ with |α| ≫ 1 will evolve as |α(t), 0⟩ with α(t) = αe −iω 0 t (i.e., as a linear oscillator), irrespective of the size of the non-linear strength ϵ.
In the regime of small displacement amplitudes, smaller or of the order of the spatial extent of the ground state wave function, we find in a first order amplitude expansion (see Appendix B) that the spatial dynamics still displays a periodic oscillatory behaviour, but with a frequency ω that deviates from that of the linear harmonic oscillator, ω 0 .In order to determine these frequency shifts, one needs to account for the fact that the unperturbed ground state α = 0 does not approximate well the perturbed ground state 0 0.01 0.1 Figure 4: Deviations of the oscillation frequency of the system from the frequency ω 0 of a linear oscillator.The corrected frequency is computed by initializing the system in the exact vacuum (numerically determined) and then displacing it in position by an amount α 0 .The deviation from the frequency of the linear oscillator scales approximately as ϵ/(2ω 0 ) for small amplitudes, while for large amplitudes it is always negative and vanishes exponentially as −4(ϵ/ω 0 )e −4|α0| 2 (see Appendix A for details).
(which is the state with the lowest energy expectation value).In leading order in the nonlinearity an approximation to the ground state is given by coherent state with amplitude The construction of GCS for the non-linear oscillator allows for the determination of the frequency shift (see Appendix A), which for β as large as |β| = π/4 gives This is confirmed in Fig. 4 where the numerically determined exact ground state has been displaced.The frequency shift is largest for very small displacement amplitudes α 0 .In this regime the accurate determination of the small amplitude of oscillation will however require a large number of measurements that scales approximately as α −2 0 .Indeed, given a Gaussian state with a width σ, a position measurement will have a resulting variance ∆x 2 = σ 2 .A sequence of N position measurements will reduce the variance by a factor of N , so that in order to appreciate a displacement α 0 it must be ∆x 2 /N = α 2 0 , which implies N = σ 2 /α 2 0 .For a thermal state with an average number of phonons n th , we have σ ∼ n th + 1 and therefore the number of required measurements will scale as (n th + 1) 2 /α 2 0 .Thus we need a large α 0 in order to minimize the number of required measurements.On the other hand the deviation ω/ω 0 rapidly approaches unity for α 0 > 1 which rapidly diminishes the benefits of increasing α 0 .Thus, Fig. 4 suggests that a displacement of the order of α 0 = 1 provides a good balance between size of the frequency shift and amplitude of oscillation.
As a result the experimental verification of non-linearities of order ϵ ∼ 2 Hz requires harmonic oscillators of very low frequency ω 0 whose initial state is cooled close to the ground state and displacement amplitudes of order α 0 ∼ = 1.A careful design and analysis of such an experiment beyond these simple estimates is interesting but goes beyond the scope of this work.
Discussion -We have demonstrated that the observation of entanglement generation between two massive particles, whose interaction is dominated by gravity, witnesses its quantum mechanical character only under additional assumptions that remain to be tested experimentally.Notably, the entangling capacity of very weak quantum forces, much weaker than gravity at the same distance, may be amplified by non-linear corrections to local quantum dynamics predicted in certain models that couple classical gravity to quantum matter and may closely mimic expected gravitational dynamics.It is worth mentioning that a complete theoretical analysis of the relation between entanglement and Weinberg nonlinearities is an interesting possibility for future work, as the current paper represents rather a proof of principle than a general proof.We also note that the specific model presented here ignores the effect of noise that typically accompanies classical gravity coupling to quantum matter [3,4,37], has parameters that are fine tuned and also suffers in its present formulation from the possibility of signalling.However, it is certainly possible to construct non-linear extensions to quantum mechanics that do not exhibit signalling.In particular, in [22] the author shows that any nonlinear model, where the nonlinearity arises as a dependence on local states, can be emulated by using standard quantum mechanics only, and thus cannot allow signalling.Another example is provided in [42] where the authors construct a general class of nonlinear models that do not allow signalling.These models have the property that their corresponding dynamical maps are convex quasilinear as opposed to linear.These finding suggest that other non-linear models might be found that still display the desired entanglement amplification, while not allowing for superluminal effects.Nevertheless, this model demonstrates that a conclusive test of the quantumness of gravitational interaction via the observation of entanglement generation may eventually require, at the very least, additional tests that limit non-linear corrections to quantum mechanics.A deeper analysis may reveal additional assumptions that may need to be tested independently.
This situation is reminiscent of the long journey towards establishing conclusive experimental evidence of the non-locality of nature by tests of Bell inequalities.There, increasingly sophisticated experimental tests were devised and realised to close possible loopholes -that is, establish the correctness of the assumption that are underlying the logical argument -until only quantum mechanics and its inherent non-locality remained the only reasonable explanation of the observations [38].We expect that experiments to witness quantum properties of gravity, due to their indirect nature, will experience a similar development.
Numerical simulations and figures -The data used for Figures 2, 3, and 4 has been generated via Python codes available at the GitHub repository https://github.com/gspaventa/nonlinear_oscillators in the form of two Jupyter notebooks.

A Generalized coherent states for the non-linear oscillator
The generalized coherent states (GCS) [35] framework provides a recipe for constructing coherent states for a wide class of physical systems.These are states whose evolution can be mapped into trajectories on a symplectic manifold.Furthermore, such trajectories are obtained as solutions of Hamilton equations as in classical mechanics.The simplest case being that of a standard harmonic oscillator with Hamiltonian H = ωa † a, for which the GCS are the well-known coherent states |α⟩ with α ∈ C. Indeed, the time evolution of such states can be described as classical orbits α(t) ∈ C induced by the classical Hamiltonian H(α, α * ) = ω|α| 2 .In addition to being a fascinating topic in mathematical physics, the GCS framework can be applied to various contexts, such as the study of classical limits of quantum theories [41], or the the analysis of open quantum systems in terms of their parametric representations [39,40].In this section we construct the GCS system for a single harmonic oscillator with a non-linear perturbation, and we study the impact of these non-linear corrections to its dynamics.Here, we find that even at first order in ϵ the GCS dynamics for this system is extremely convoluted.However, it is possible to get information about the dynamics from the regimes of very large and very small coherent state amplitudes.In particular, we show that the non-linear oscillator behaves as a linear one for large amplitudes, and we analyze the dynamics around the ground state to find the correction to the bare frequency ω 0 .The (non-linear) Hamiltonian that we have to analyze is where Y = D † (β)P D(β), and P = e −πa † a .The expression above defines for any state |ϕ⟩ a linear combination of the operators {1, a, a † , a † a, ϵY }.Given that the Heisenberg-Weyl algebra h 4 = Span(1, a, a † , a † a) of the quantum harmonic oscillator gives rise to the (standard) Glauber coherent states |α⟩, α ∈ C, we want to see whether it is possible to perturbatively construct the GCS for the non-linear system at hand, and write them as a correction to the unperturbed ones, to first order in ϵ.The natural first step would be to include ϵY as a new generator in h 4 but, unfortunately, the operators {1 , a , a † , a † a , ϵY } do not form a closed algebra under commutation, since for example This means that additional generators must be added in order to form a closed algebra.To this end, let us define for j, k ≥ 0 the operators which obey the following commutation relations: As per the commutator [Y jk , Y rs ], it can be written as a combination of operators N jk = a † j a k , thanks to the fact that Y 2 = 1 and [a, a † ] = 1.However, every time such a term is generated, it carries a coefficient ϵ 2 in front of it and is thus negligible if we are interested in a first order treatment.Therefore, even though the algebra would only close with the additional generators N jk , we can safely ignore this complication and consider, to first order in ϵ The maximal isotropy subgroup H (with respect to the unperturbed vacuum |0⟩) is obtained by exponentiation of the subalgebra spanned by {1, a † a}, as usual.This means that the coset G/H of displacement operators consists of elements of the form where α is a complex number, {θ µ } is a (finite) sequence of complex numbers, and Y µ := Y µ,0 are the only operators Y jk that do not annihilate the unperturbed vacuum |0⟩.An expansion in powers of ϵ to first order yields which, by defining can be rewritten as The expression above tells us that, to first order in ϵ, the GCS of the non-linear oscillator can always be constructed as The Hamiltonian function, evaluated on the coherent states |α, θ⟩, has the expression Using the fact that ⟨α| Y |α⟩ = e −2|α+β| 2 , together with the definition the equations of motion will have the form Therefore, we need to compute Ω and its derivatives.First, let us consider the resolution of the identity where the integration measure is given by By making use of Eq.34 we can rewrite Ω(α, θ) as we get where we have defined The family of integrals above turn out to be exactly solvable, by deploying techniques of complex contour integration and the Cauchy's residue theorem, as described in the following Proof.By making use of the definition of the measure dµ(α) we can turn I ν (b, c) from an integral over the complex plane to an integral over R 2 as following: (40) Now, by introducing polar coordinates ρ ∈ [0, ∞) and θ ∈ [0, 2π] we can write where and C α 0 (R) denotes the circle in the complex plane with radius R centered at α = α 0 .Now, the integral above can be easily computed with complex contour integration techniques.In particular, by invoking Cauchy's residue theorem we can compute I ν (ρ; b, c) as a sum of residues at its poles, the existence of which depends on the value of ν.For ν = 0 the integrand has a pole of order 2 in α = 0, for ν = 1 a simple pole, and for ν > 1 the integrand is an entire function and thus has no poles.Therefore we are guaranteed that I ν (ρ; b, c) = 0 for all ν > 1.We have Now, the residue at α = 0 can be computed as In this case the pole at α 0 is simple and therefore we have We then have By inserting this expression in Eq.( 41) we obtain corresponding to the frequency shift We can then conclude that for large amplitudes the frequency shift is negative, as confirmed by the numerical simulations in Fig. 4. Furthermore, the frequency shift vanishes exponentially in the initial amplitude |α 0 |.

A.2 Small amplitudes
In order to study the opposite regime, i.e. when |α| ≪ 1, we perform the following approximation.We take a first order expansion of Eq.( 52) in α and α * , allowing us to write The equations above have the following implications: first, all variables θ µ with µ ̸ = 0, 1 are cyclic for small amplitudes and therefore when starting from a coherent state with θ µ = 0, µ ̸ = 0, 1, we have θ µ (t) ≡ 0 ∀t.As per θ 0 and θ 1 , the corresponding equations of motion contains α * in a term that is proportional to ϵ.By expanding α in powers of ϵ we have the expression where α l (t) = α 0 e −iω 0 t is the solution for a linear (ϵ = 0) harmonic oscillator.Therefore in order to be consistent with a first order treatment in ϵ, we should write In order to find an expression for the new ground state and the corrections to the frequency of oscillations near the ground state, we study the spatial dynamics of the states |α, θ⟩.From now on, since for small amplitude there are only two non-cyclic theta variables, we introduce the notation θ ≡ {θ 0 , θ Surprisingly this tells us that, up to second order terms in ϵ, the spatial dynamics of the oscillator is unaffected by the time evolution of θ, and only depends on the dynamics in the complex plane α.By using the solution found in the previous section, we can write the spatial dynamics for small amplitudes as ⟨x(t)⟩ = α 0 cos (ω 0 − wϵ)t − ϵk ω 0 sin (ω 0 − wϵ)t (68) We can thus compute the new equilibrium position α v by setting α = α * = 0, which gives

A.3 Ground state
From the discussion above we can conclude that the coherent state α = 0 is not a fixed point of the dynamics anymore, and therefore the state |0⟩ is not the ground state of the system.However, we have found an expression for the new coherent state amplitude corresponding to the new spatial fixed point, and the state |α v ⟩ ≡ |α v , 0, 0⟩ turns out to be the perturbed ground state up to second order terms in ϵ, as its time evolution is where, from Eq.60 we know that θ v 0 (t) = Now, from the expressions above we note that both θ v 0 and θ v 1 are proportional to α * v , which is proportional to ϵ meaning that to first order in ϵ we can write This means that the frequency of spatial oscillations of a coherent state |α 0 ⟩ for small amplitudes will be ω 0 − w, regardless of θ(t), and this result reproduces exactly the shifts observed in numerical simulations for small displacements of the true ground state (see Fig. 4).

Figure 2 :Figure 3 :
Figure2: Relation between g and the value ϵ * that maximises the overlap with the curve corresponding to g = 1, ϵ = 0. Equivalently, given a certain non-linear strength ϵ, the value g * (ϵ) is the strength of the residual interaction that would reproduce the entanglement dynamics associated with a linear model with g = 1.(Inset): Entanglement of the composite state as a function of time for three different values of ϵ.The non-linear effects are responsible for boosting the rate at which entanglement is generated.