Hierarchy of quantum non-Gaussian conservative motion

Mechanical quantum systems embedded in an external nonlinear potential currently offer the first deep excursion into quantum non-Gaussian motion. The Gaussian statistics of the motion of a linear mechanical quantum system, characterised by its mass and a linear-and-quadratic potential, possess a limited capacity to reduce noise in nonlinear variables. This limitation induces thresholds for noise reduction in nonlinear variables beyond which linear mechanical oscillators cannot pass. Squeezing below the thresholds for such variables is relevant for the implementation of nonlinear mechanical devices, such as sensors, processors or engines. First however, quantum non-Gaussian conservative motion must be identified in experiments with diverse nonlinear potentials. For this purpose, we provide sufficient criteria for quantum non-Gaussian motional states in conservative systems based on the observation of squeezing in nonlinear variables. We further extend these criteria to a hierarchy able to recognise the quantum non-Gaussian motion induced via diverse nonlinear potentials through their various capacities to produce nonlinear squeezing. Gaussian systems are useful for many quantum technologies but new applications will require control over nonlinear systems generating quantum non-Gaussian states. The authors present a method for the detection of quantum non-Gaussian states of mechanical particles which may be applied for future experiments in optomechanics and levitated nanoparticles in the quantum regime.

T he quantum mechanics of linear continuous variable (CV) systems is extremely well-established theoretically 1 . Linear systems can be efficiently described by linear transformations of the first and second-order statistical moments; unitary transformations are captured by the symplectic formalism. Moreover, linear transformations preserve Gaussian states, which are fully characterised by their first and second statistical moments. Due to their relative ease in preparation and manipulation in a laboratory, such linear systems are also wellestablished experimentally across several platforms including trapped ions 2 , quantum optics 3,4 , and electro-mechanics [5][6][7] . While such Gaussian systems are useful for many quantum technologies 8 , new applications will require control over highly nonlinear CV quantum devices 9 generating quantum non-Gaussian (QNG) states 10 , states which cannot be expressed as mixtures of Gaussian states.
States with a negative Wigner function form a subset of the general set of QNG states. The Hudson theorem 11,12 , states that a necessary and sufficient condition for a pure state to have a positive Wigner function is that the state be Gaussian. It follows that if the state is non-Gaussian then it cannot be both pure and positive. However, there is no such Hudson theorem for general mixed states 13 although the states in the set of mixtures of Gaussian states have positive Wigner functions. However, this set of mixtures is a proper subset of the Wigner positive states, meaning that there are Wigner positive states which are QNG. Both concepts, namely QNG and Wigner negativity, can be used to define resources for nonlinear quantum technologies 14 . The larger set of more robust QNG states is advantageous as they already contain valuable information regarding the nonlinear features of the state.
The full description of CV systems presents a duality consisting of the discretely infinite-dimensional system instantiated in photons or phonons (Fock basis methods) and the continuous infinite-dimensional system characterised by position and momentum-like quadratures (phase space methods) from which CV systems derive their name. Nonlinearity in CV systems emerges differently from these distinct paths. Photons/phonons and finite superpositions thereof can exhibit quantum non-Gaussian behaviour and can be observed in trapped ion oscillators, and microwave and optical fields by arranging for their discrete particle-like nature to be displayed [15][16][17] . The detection of quantum non-Gaussianity for photonic/phononic systems was achieved by developing quantum non-Gaussianity criteria for light based on direct photon measurements 18 , while photoncounting statistics can be used to construct loss-resilient criteria 19 and the first hierarchy of QNG features for modes of radiation or linear oscillators 20,21 . Phase space methods can also capture Fock basis quantum non-Gaussianity via the Wigner function 22 and Q-function 23,24 . Note that, while the kurtosis is a reliable indicator of non-Gaussianity 25 , it does not detect quantum non-Gaussianity.
In contrast, nanomechanical oscillators can be continuously coupled to discrete variable quantum systems in order to induce basic continuous variable nonlinear dynamics 26,27 . In addition, discrete superconducting elements 28,29 and multiple trapped ions 30,31 have been used to achieve three-wave and four-wave nonlinear couplings. More broadly, the possibility of imposing arbitrary nonlinear potentials on quantum particles is opened by the recent achievement of ground state cooling for a levitated nanoparticle [32][33][34] . While such nonlinear potentials are difficult to engineer for optical or microwave systems, mechanical systems have already been controlled in nonlinear traps [35][36][37] . Here, QNG motion can be induced by nonlinear potentials added directly to the free motion kinetic energy, an arrangement which is a current target both for trapped ions in segmented traps 35,38 and levitated objects 36,[39][40][41][42][43][44] . The potential trap of such experiments has the capacity to be rapidly reshaped between very different nonlinearities. Such systems are associated with a continuous variable readout, and position and momentum measurements to detect QNG behaviour.
In anticipation of likely developments in conservative mechanical systems exposed to externally applied nonlinear potentials, which are justifiably expected to be diverse, we propose a varied set of criteria for quantum non-Gaussian motional states constructed around the experimentally accessible statistics of nonlinear quadrature variables. At their most basic level, these criteria allow the conclusive discrimination of nonlinear from upto-quadratic potentials, but by advancing stricter conditions by excluding higher-order potentials a hierarchy of criteria for nonlinear features of the motion can be constructed. These criteria are aimed at QNG motional states generated by a nonlinear potential added to the free motion of a mechanical particle, distinct from traditional analyses established in quantum optics and other fields analysing QNG states of linear oscillators under dynamics that can also be non-conservative.

Results and discussion
Quantum non-Gaussian conservative motion. To begin this new approach consider a one-dimensional conservative mechanical system described entirely by canonical position and momentum variablesq andp satisfying ½q;p ¼ i_, with the total energyĤ given by the sum of the free motion kinetic energy,p 2 2m , and the potential energy, VðqÞ, where m is the mass and VðqÞ necessarily depends only onq. In contrast with oscillator or field modes, a generic mechanical system in the absence of external forces is described only by the kinetic energy term i.e. VðqÞ ¼ 0. While linear oscillators are represented by the specific case V 2 ðqÞ ¼ γ 1q þ γ 2 2q 2 , for arbitrary γ 1 and γ 2 ≥ 0 (however note we also include γ 2 < 0 as legitimate linear motion), mechanical systems may be described by a diverse class of potentials VðqÞ, even excluding theq 2 term. Simultaneously, conservative systems evolve only through Eq. (1). Therefore the potential energy of such linear systems does not contain terms proportional top,p 2 ,qp þpq or iðqp ÀpqÞ. Such terms do not reflect conservative mechanical motion as they are not compatible with a potential VðqÞ; this contrasts with optical systems in which similar terms, such as the phase shift and squeezing, may be more readily implemented in a controllable way.
The free Hamiltonian of systems such those considered in quantum optics is that of a harmonic oscillator. Thus, the ground and thermal states of oscillator Hamiltonians are mathematically convenient objects from which to derive the set of Gaussian states by considering all linear transformations on such states. The convex hull of Gaussian states needed for QNG analysis readily follows. However, to define the convex hull of linear mechanical states, the free motion Hamiltonianp 2 2m is insufficient, as it does not have a ground state and its eigenstates, including the zeromomentum state p ¼ 0 , are unphysical. To formulate a convex hull of motional Gaussian states we need a physical state, therefore we consider a fixed yet arbitrary mass M and a fixed non-zero quadratic potential stiffness resulting in a frequency Ω, and select the oscillator ground state of Eq. (1), with VðqÞ ¼ 1 2 MΩ 2q2 , as a fiducial state ρ 0 in order to mathematically define the Gaussian motional states generated by linear conservative dynamics. We do not need to consider a linear potential term in the definition of the fiducial state since Eq. (1) with VðqÞ ¼ V 2 ðqÞ will generate the required displacement. However since Eq. (1) with V 2 ðqÞ is not the most general upto-quadratic polynomial inq andp the set of motional Gaussian states derived from conservative motion (defined below) is a proper subset of the conventional set of Gaussian states. We stress that M and Ω are arbitrary in the sense that they provide the seed from which to grow the set of linear motion states and do not need to reflect a particular experimental setting. The selected fiducial state ρ 0 is a zero-mean Gaussian state with covariance matrix At this point one could rescale in terms of the natural length and energy scales induced by the oscillator to retrieve the conventional notation of continuous variable quantum mechanics, while remembering that the fiducial state is always defined in terms of a particular reference oscillator.
The unitary operator generated by Eq. (1) with VðqÞ ¼ V 2 ðqÞ, parameterised by time t ≥ 0, allows us to define the pure motional Gaussian conservative states as the set V 2 G ¼ fÛρ 0Û y :Û ¼ e ÀiĤ t _ g generated from the fiducial state ρ 0 , for arbitrary times t ≥ 0 and masses m > 0, and arbitrary potential coefficients γ 1 and γ 2 . The convex hull of G is denoted conv ðGÞ. For conservative motion, any state not described by the action of any mixture of such up-to-quadratic potentials on a pure Gaussian fiducial state ρ 0 is outside the set conv ðGÞ. Such states embody QNG motion for conservative systems. Differences to the general set G of Gaussian states are addressed in Supplementary Note 1.
With sufficiently large mass, the transient conservative dynamics of VðqÞ involves almost no change in the position of the particle and any nonlinear effects initially accumulate in the particle's momentum. Motivated by the momentum evolution p !p À ∂qVðqÞt induced by the infinite inertia unitary dynamics of basic monomial potentials VðqÞ ¼ γ n nþ1q nþ1 , we consider the set of variances which, over λ and n, form an infinite list of statistical tests of QNG motion, with increasingly higher order position moments contributing to the analysis. Variances of this form decompose directly into statistical quantities derived from position and momentum measurements in the form where Cov ðA; BÞ ¼ 1 2 hAB þ BAi À hAihBi is the covariance of A and B. We stress that λ is not a physical parameter here, but rather an arbitrary parameter analogous to the quadrature angle used in conventional squeezing analyses. Different values of λ indicate that different statistical quantities are being considered. Arbitrary noise reduction in linear variables (n = 1) can always be explained by a Gaussian state. To see this it is sufficient to note that the wavefunctions of a pair of canonical variables form a Fourier pair, and that the uncertainty principle between Fourier pairs is saturated by Gaussian functions. However, conservative mechanical motion does not possess this capacity. Nonconservative terms proportional to p and qp + pq in Eq. (1) are required to reach arbitrarily small S n (λ) for n = 1 (see Supplementary Note 6).
In contrast, Gaussian states from conservative linear motion and, moreover, general Gaussian states including nonconservative forces are restricted in their capacity to reduce noise in nonlinear variables. This provides an experimentally simple method to detect the QNG motion induced by conservative nonlinear potentials, as well as non-conservative linear dynamics. These tests can be easily generalised by considering a wider range of nonlinear dynamics, incorporating, instead of Eq. (3), a large variety of correlations in nonlinear combinations ofq andp, to target specific applications.
If the noise reduction in a nonlinear variable is beyond the capacity of any ρ 2 G then it is sufficient to also exclude any ρ 2 conv ðGÞ and thus detects QNG conservative motion. That is, the threshold on S n for the motional Gaussian states is also a threshold for any convex mixture of such states. This is not simple to show directly since the variance is not a convex function. However, we may take advantage of an equivalence relation induced by the structure of S n . Variances of the form we have given are independent of displacements in momentum, so that which can be seen from where p is the displacement inp. The displacement transformation maintains the Gaussianity of the state. Due to this relation there is a class of motional Gaussian states with the same value of nonlinear variance, and at least one representative of the class has zero first moment in the nonlinear variable. For this representative the variance is equivalent to the second moment and the threshold can be calculated over the set of these representatives without loss of generality. Now it is possible to see that the motional Gaussian state threshold also holds for any convex mixture of such states. Suppose that the threshold exists and is denoted Then multiplying by a probability distribution p(x) and integrating we have Thus, the threshold for the pure states is also the threshold for the mixed states. This means that once a state can be excluded from the class G it is also excluded from conv ðGÞ. For this reason we now focus the discussion on pure states.
Criteria for QNG conservative motion. As said, for a mechanical object to exhibit QNG conservative motion it is sufficient for the noise reduction in S n to exceed the capacity of the class of motional states G. The solid blue lines in Fig. 1 provide examples of the quantum non-Gaussian conservative motion thresholds T G n which bound S n (λ), n = 2, 3, as a function of the threshold parameter λ and are described by the equations T G 2 ðλÞ ¼ Variances below these thresholds unambiguously represent QNG conservative motion. The thresholds in (8) are easily extended to larger n, however, we focus on the lowest moments as they do not require as many experimental runs to accurately estimate the high order moments. Remarkably, the thresholds (8) remain valid also for the set G of general Gaussian states which includes variable mass and non-conservative termsp andqp þpq in the Hamiltonian (see Supplementary Note 2). Such thresholds safely exclude all Gaussian states, but they cannot distinguish linear non-conservative effects.
Examples of QNG motion. To illustrate the feasibility of our criteria we analyse the motion of a particle, for simplicity cooled to the ground state, corresponding to the fiducial state 0 j i, under the short-time action of purely cubic and purely quartic potentials. For short-time dynamics, such potentials can be wellapproximated locally 36 where the coefficient ofQ can simply be incorporated into the definition of the potential coefficients. To navigate toward future experiment tests, we start with the one currently existing experiment 34 and analyse the required parameters to detect QNG motion and, later, available strategies to achieve it. We leave a detailed proposal for further development and experimental analysis. With this formulation all that is required is the fiducial frequency, which we set to Ω 2π ¼ 305 kHz, following ref. 34 , a recent levitating particle experiment. Figure 1 shows the statistics for a quantum particle under two nonlinear potentials À Ák 2 , and various durations of exposure to the nonlinear potential. The large effective mass m eff = (ℏΩ) −1 indicates that for very short interaction times the particle is indeed well-approximated by large inertia, where the transient dynamics of VðqÞ are most significant. In fact, ref. 34 notes a coherence time of the levitated nanoparticle of 7.6 μs, while we consider times on the order of nanoseconds. During this extremely short interaction time QNG motion becomes visible through the thresholds in Eq (8). Subsequently the kinetic and nonlinear potential terms, as in Eq. (1), combine to produce QNG statistics for larger values of λ but with larger absolute value in S n (λ). Notably, reduction of the variance below the thresholds occurs for very weak values of the nonlinearity as well as for extremely short times t compared to the coherence time of a levitated particle. The evolution time t has to be short as the kinetic term destroys the visibility of the QNG motion. It is expected that single order of magnitude variations in the mass of levitated particles will not prevent ground state cooling with coherent scattering techniques. Keeping all other parameters constant, increasing the mass tends to further reduce the variance S n (λ) below the threshold while decreasing the mass has the opposite effect (see Supplementary Note 4). Detection of QNG motion then, is best achieved with large particle mass and weak nonlinear potentials. Such conditions are stimulating for further development and optimisation of the experimental platforms, including control the of potential and detection. While we focused on the pioneering experiment, small changes in parameters can have drastic effects on the detection of QNG motion. For example, increasing the mass tends to require smaller cubic nonlinearities for QNG motion to be visible, however, this visibility persists for orders of magnitude greater time intervals than demonstrated in Fig. 1 (see Supplementary Note 4). However further optimisation may be possible by starting from an arbitrary Gaussian state in conv ðGÞ with noise properties tuned to enhance the nonlinear squeezing. For example, if the initial state is a momentum squeezed state the cubic potential produces greater nonlinear squeezing in S 3 (λ) 46 . A recent proposal suggests that squeezed coherent states can be prepared for use in levitated systems 47 . The preparation times must be optimised to maximise the distance from the thresholds before the kinetic term impacts the noise. This seems to hold generally across S 2 and S 3 .
As said, the form of the quantum non-Gaussianity tests is motivated by the short-term dynamics of nonlinear potentials. The examples in Fig. 1 demonstrate that the effects associated with such nonlinear potentials are clearly detectable via their corresponding sufficient conditions T G n . However, we also find that S 2 (S 3 ) is insensitive to quartic (cubic) potentials. Similarly, while S 2 and S 3 are sensitive toq 5 andq 6 potentials respectively, the same insensitivity occurs when the roles are swapped. This strongly suggests a parity effect in the detection of QNG motion. This is visible in the limit of large inertia (see Supplementary Note 3) where parity effects dictate that, for a fixed n, S n does not detect QNG effects from potentials whose parity matches that of the test.
While mixtures of nonlinear terms add further complications to the analysis, we also note that the q 2 term has negative effects on the detection of QNG motion. The addition of this oscillator- . QNG motion is already visible for short evolution times t and various strengths, γ k , of the nonlinear dynamics. Clearly, S 2 efficiently detects QNG motion for the cubic potential (k = 3) and S 3 the quartic potential (k = 4). However, S 2 does not detect quartic potentials and S 3 does not detect cubic potentials. Analysis of γ k and preparation time t shows that detection of QNG motion is best achieved with large masses and weak nonlinearities, with the preparation times optimised to maximise the lateral shift from the thresholds which precedes the increase in noise deriving from the kinetic energy term.
like term tends to exacerbate the limitations induced by the kinetic energy term (see Supplementary Note 5).
Depth of QNG conservative motion. In generating the example quantum states in Fig. 1 we have assumed that the particle does not experience heating from the environment during the evolution time. This is justified by the fact that typical heating rates are on the order of milliseconds rather than nanoseconds 48 . Nevertheless the nonlinear variances are susceptible to thermalisation from noisy processes and so we investigate the resilience of such states to environmental heating by examining their capacity to remain visible to our criteria under such processes (c.f. similar operational concepts applied to nonclassicality 49 , entanglement 50,51 and Fock state quantum non-Gaussianity 18,20 ).
In Table 1 we examine this capacity under the term depth. The particle is prepared in a QNG state following Fig. 1 and subjected to thermalisation processes (see Table 1 for numbers) under free motion. We find that states prepared with stronger nonlinearities have shorter intervals for the preparation of a significant QNG state but also in survival against thermalisation. Thermalisation increases the noise in S 2 and S 3 , effectively causing the curves in Fig. 1 to rise. After the particle is heated the curves move toward a central, symmetrical position, indicating that any higher-order correlations between q and p are being erased.
Hierarchies of QNG conservative motion. Following Eq. (1), the QNG thresholds are generated by restricting VðqÞ to linear-andquadratic potentials. Lifting this restriction to consider potentials of the form V N ðqÞ ¼ ∑ N k¼0 γ k kq k generates the classes of motional The thresholds considered above are the most elementary thresholds for QNG motion for n = 2, 3. By construction V N V Nþ1 (recall that V 2 G) so that the examined set of states form a hierarchy for the conservative motion. We leave more the complex discussion required to incorporate non-conservative effects for future investigation.
Let T V N n denote a threshold in S n (λ) for the higher order set V N . If S n < T V N n then the motion evolving the fiducial state ρ 0 to the states in V N cannot produce the observed value of S n and therefore such motion is insufficient as an explanation of the observed noise reduction. This induces a hierarchy of QNG states of motion from nonlinear potentials detected by variances S n (λ). To illustrate, if S 3 ðλÞ < T V 3 3 for some λ then the state is not within the set V 3 generated by the action of up-to-cubic potentials on ρ 0 . To briefly demonstrate an example for such a hierarchy, we search for fixed λ and n such that the value S n (λ) cannot be achieved by any state in V N .
For this example again select the initial state to be simply the fiducial state ρ 0 ¼ 0 j i 0 h j, n = 3, N = 3 and the minimal squeezing parameter found at λ min ¼ À Cov ðp;q 3 Þ Var ðq 3 Þ . Then the quartic potential example from the right panel of Fig. 1, with the blue dashed line (t = 7.5 ns), produces S 3 (λ min ) ≈ 0.65. With the same mass and time, the minimum achievable at λ min with arbitrary potentials of the form V 3 is characterised by η ¼ indicates that the threshold has been passed. Additionally, this minimum under V 3 is achieved with γ 1 = γ 3 = 0 and γ 2 2π % 1:506 10 À25 J, i.e. using only the quadratic term. This again suggests a certain importance for parity in nonlinear effects measured using S n .
Beyond the QNG hierarchy. The dynamics produced by nonlinear potentials are extremely diverse, and so there is no single method to capture all nonlinear effects. Rather the implementation of the framework we present here is contingent upon the particular scenario encountered. Different choices of potential VðqÞ, and thus diverse sets V, will induce new thresholds that provide different information about observed nonlinear effects. Such choices include monomial potentialsṼ N ðqÞ ¼ γ Nq N or odd (even)-powered potentials V OðEÞ ðqÞ ¼ ∑ j γ 2jÀ1ð2jÞq 2jÀ1ð2jÞ . A concrete example is that of the tilted double well V DW ¼ γ 1q þ γ 2q 2 þ γ 4q 4 , often used to mimic the bounded cubic potential V C ¼ γ 3q 3 þ γ 4q 4 . However, using the test S 2 and following the procedure described for the hierarchy it is also possible to generate examples of QNG motion inaccessible to V DW (η > 1) using the bounded cubic potential. Using the same mass and time, a bounded cubic potential with γ 3 2π ¼ 10 À25 J and γ 4 2π ¼ 10 À26 J produces noise fluctuations at λ min not available to the V DW . Again the best approximation occurred with a purely quadratic potential ( γ 2 2π % 0:22 10 À25 J), suggesting that, despite the visual similarity, the quartic term does not accurately replicate the QNG behaviour of the cubic potential.

Conclusion
We have introduced criteria to detect quantum non-Gaussian motion for mechanical systems immersed in nonlinear potentials based upon the reduction in noise of nonlinear variables. Contrary to linear variables, the noise in nonlinear variables cannot be arbitrarily reduced using Gaussian states. These criteria rely only on appropriate measurements of the position and momentum variables, rather than full tomographic reconstructions of the quantum state. Additionally, the readout of these mechanical quadratures can be accomplished via optomechanical coupling to light 46 or through discrete variable coupling to the mechanical system 52 . For example, if a levitated particle is coupled to a microcavity then the position and momentum quadratures necessary for the reconstruction of the nonlinear squeezing can be directly recovered by the transfer of such information to the light field through the optomechanical QND coupling. Subsequently, this information can be read out via homodyne detection. The nonlinear states visible from these criteria are likely to be achievable in the near future with state-of-the-art developments in tunable nonlinear potentials for mechanical objects.
Building on these criteria, we stratify nonlinear effects by constructing a hierarchy of tests of nonlinearity. These collections of sufficient criteria serve to exclude classes of potentials as explanations for observed statistics of the motional state for conservative systems. An extension to non-conservative motion may follow a similar methodology. We emphasise that the nonlinear world is exceedingly diverse and thus stress the adaptability of our methodology to the contingency of the scenario examined. The techniques for constructing these thresholds for nonlinear Table 1 The depth of various QNG motional states. These states are as prepared in Fig. 1. The depth τ indicates approximately how long under free motion the QNG property detectable by S 2 or S 3 survives. We assume a total heating rate of Γ 2π ¼ 20:6 kHz and a thermal occupation of the particle of -n ¼ 1. The mass is as in Fig. 1 potentials are very general and imply the distinguishability, on the level of position and momentum statistics, of various nonlinear potentials which may otherwise appear similar. This provides a powerful means for crystallising our understanding of the nonlinear dynamics of mechanical bodies in quantum theory.

Data availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Code availability
The computer programmes to generate the simulations are available upon reasonable request.