Unconditional measurement-based quantum computation with optomechanical continuous variables

Unconditional measurement-based quantum computation
with optomechanical continuous variables

Oussama Houhou o.houhou@qub.ac.uk School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, UK Laboratory of Physics of Experimental Techniques and Applications, University of Médéa, Médéa 26000, Algeria    Darren W. Moore dmoore32@qub.ac.uk Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, UK    Sougato Bose s.bose@ucl.ac.uk Department of Physics and Astronomy, University College London, London WC1E 6BT, UK    Alessandro Ferraro a.ferraro@qub.ac.uk School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, UK o.houhou@qub.ac.uk School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, UK Laboratory of Physics of Experimental Techniques and Applications, University of Médéa, Médéa 26000, Algeria dmoore32@qub.ac.uk School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, UK s.bose@ucl.ac.uk Department of Physics and Astronomy, University College London, London WC1E 6BT, UK a.ferraro@qub.ac.uk School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, UK
Abstract

Universal quantum computation encoded over continuous variables can be achieved via Gaussian measurements acting on entangled non-Gaussian states. However, due to the weakness of available nonlinearities, generally these states can only be prepared conditionally, potentially with low probability. Here we show how universal quantum computation could be implemented unconditionally using an integrated platform able to sustain both linear and quadratic optomechanical-like interactions. Specifically, considering cavity opto- and electro-mechanical systems, we propose a realisation of a driven-dissipative dynamics that deterministically prepares the required non-Gaussian cluster states — entangled squeezed states of multiple mechanical oscillators suitably interspersed with cubic-phase states. We next demonstrate how arbitrary Gaussian measurements on the cluster nodes can be performed by continuously monitoring the output cavity field. Finally, the feasibility requirements of this approach are analysed in detail, suggesting that its building blocks are within reach of current technology.

Introduction.— Measurement-based quantum computation (MBQC) is a powerful approach to process information encoded in quantum systems Briegel et al. (2009), which requires solely local measurements on an entangled state (cluster state) Raussendorf and Briegel (2001); Raussendorf et al. (2003). This approach gives significant theoretical insights into fundamental questions about the origin of the power of quantum computing Van den Nest et al. (2006); Gross et al. (2009); Bremner et al. (2009); Anders and Browne (2009); Raussendorf (2013); Bermejo-Vega et al. (2017), and it offers promising applicative opportunities provided large enough clusters can be built, including the demonstration of quantum computational supremacy Hangleiter et al. (2018); Bermejo-Vega et al. (2018) and the realisation, in condensed matter systems Brennen and Miyake (2008); Cai et al. (2010); Li et al. (2011); Wei et al. (2011); Aolita et al. (2011); Else et al. (2012); Wei and Raussendorf (2015); Miller and Miyake (2015), of fault-tolerant processors with high resilience thresholds Raussendorf and Harrington (2007); Raussendorf et al. (2007).

In view of the relevance of MBQC, major efforts have been devoted to its experimental implementation. In the setting of finite-dimensional (discrete-variable) quantum systems, various experimental demonstrations of small-size MBQC have been reported Walther et al. (2005); Kiesel et al. (2005); Prevedel et al. (2007); Tame et al. (2007); Lu et al. (2007); Vallone et al. (2008); Barz et al. (2012); Bell et al. (2014); Lanyon et al. (2013). However, the largest clusters to date have been generated in the context of continuous-variable (CV) systems Braunstein and Van Loock (2005); Serafini (2017), with photonic clusters composed of up to one million modes Yokoyama et al. (2013); Chen et al. (2014); Roslund et al. (2014); ichi Yoshikawa et al. (2016); Cai et al. (2017); Rigas et al. (2012). Such achievements stem from the fact that these clusters belong to the class of Gaussian states Ferraro et al. (2005); Weedbrook et al. (2012); Adesso et al. (2014); Genoni et al. (2016), and CV Gaussian entanglement is generally available unconditionally (deterministically), contrary to discrete-variable systems whose entanglement typically relies on post-selection 111See Refs. Lindner and Rudolph (2009); Schwartz et al. (2016); Pichler et al. (2017) for progresses towards deterministic generation of discrete variable clusters and Refs. Spiller et al. (2006); Anders et al. (2010); Roncaglia et al. (2011); Proctor et al. (2017); Gallagher and Ferraro (2018) for alternative measurement-based approaches.. Despite such remarkable progress, in order to realize universal computation Lloyd and Braunstein (1999), these photonic clusters have to be either equipped with non-Gaussian measurements Menicucci et al. (2006); Gu et al. (2009) or interspersed with non-Gaussian states Menicucci (2014). Unfortunately, both strategies require high order non-linearities, which are hard to implement deterministically in optics and in fact stand as a major roadblock 222Notice that much effort has been devoted to counteract this issue, leading to proposals in which the necessary non-Gaussian elements can be obtained on-demand Marek et al. (2011); Yukawa et al. (2013); Marshall et al. (2015); Miyata et al. (2016); Marek et al. (2017); Takeda and Furusawa (2017); however these still require the use of quantum memories which are hard to realize Lvovsky et al. (2009); Yoshikawa et al. (2013) and that we avoid here.. As a remedy, we propose here to use non-linear optomechanical systems, with the aim of providing a feasible path to unlock the full potential of unconditional MBQC.

Our approach is motivated by recent experimental breakthroughs in cavity optomechanics Milburn and Woolley (2011); Aspelmeyer et al. (2014), which lends itself as a disruptive new platform for CVs in which the information carrier is embodied in the centre of mass motion of a mechanical oscillator. Indeed ground state cooling Teufel et al. (2011); O’Connell et al. (2010); Noguchi et al. (2016a), squeezing beyond the parametric limit Lei et al. (2016); Wollman et al. (2015); Pirkkalainen et al. (2015), two-oscillator entanglement Ockeloen-Korppi et al. (2018); Riedinger et al. (2018) and non-locality Marinkovic et al. (2018) have been achieved experimentally, with further scalability and integrability within reach Massel et al. (2012); Damskägg et al. (2016); Grass et al. (2016); Nielsen et al. (2017); Noguchi et al. (2016b). Crucially, optomechanics has a significant advantage to photonics in the unconditional non-linearity embedded in the radiation pressure dynamics Bhattacharya et al. (2008); Thompson et al. (2008). For driven systems this manifests primarily as a quadratic coupling in the position of the oscillator Thompson et al. (2008); Woolley et al. (2008); Hertzberg et al. (2010); Rocheleau et al. (2010); Nunnenkamp et al. (2010); Purdy et al. (2010); Sankey et al. (2010); Hill et al. (2011); Karuza et al. (2013); Flowers-Jacobs et al. (2012); Massel et al. (2012); Li et al. (2012); Hill (2013); Doolin et al. (2014); Kaviani et al. (2015); Paraïso et al. (2015); Lee et al. (2015); Kim et al. (2015); Abdi and Hartmann (2015); Kalaee et al. (2016); Fonseca et al. (2016); Brawley et al. (2016); Leijssen et al. (2017); Zhang et al. (2017); Dellantonio et al. (2018).

Here we consider a driven-dissipative opto-mechanical system. By taking advantage of the control over the mechanical state granted by externally driving the cavity, and arranging for either dissipative engineering Kronwald et al. (2013); Ikeda and Yamamoto (2013); Houhou et al. (2015) or continuous monitoring Clerk et al. (2008); Genoni et al. (2016); Moore et al. (2017), we are able to provide schemes for the deterministic preparation of non-Gaussian cluster states and local measurements sufficient to achieve computational universality. The integration of these schemes into a single experimental platform constitutes, as far as we know, the first proposal for universal MBQC with CVs that can be implemented unconditionally.

Measurement based computation with CVs.— As said, MBQC is predicated on the existence of a highly entangled multipartite resource state known as the cluster state. For our purposes, a cluster state is associated with a mathematical lattice graph of vertices , and edges that define the adjacency matrix with entries if (with ). Consider an -oscillator system, with each oscillator characterised by the canonical position and momentum operators, being their respective annihilation operator. The CV cluster state Zhang and Braunstein (2006); Menicucci et al. (2006) is operationally defined by first preparing all vertices (embodied by the oscillators) in a product state of momentum-squeezed vacua , where , , , and is a shorthand for the degree of squeezing. Then, controlled-phase operations are applied for any edge . These can be compactly written defining the multi-oscillator operator , with . Consequently, the resulting standard cluster state is given by (see Fig. 1); this is a Gaussian state and the degree of squeezing (with ) determines its quality for computational purposes Gu et al. (2009).

Figure 1: Circuit representing the preparation of a cluster state with squeezing and adjacency matrix . In the absence (presence) of cubic operations —given in the dashed box— the standard (non-Gaussian) cluster is obtained.

The computation proceeds via a series of local projective measurements on the cluster nodes. These measurements implement the gates of the program to be computed, whose output is embodied in the state of the non-measured nodes. The Lloyd-Braunstein criterion Lloyd and Braunstein (1999), first developed for circuit-based computation, allows a distinction to be drawn between Gaussian and non-Gaussian gates. A finite set of Gaussian gates is sufficient to perform any multimode Gaussian operation. However, it is only when an additional non-Gaussian gate is at disposal that universality is unlocked, in the sense that any Hamiltonian can be simulated to arbitrary precision. In MBQC, Gaussian measurements on the cluster are sufficient to implement arbitrary Gaussian gates Gu et al. (2009), including in extremely compact ways Ferrini et al. (2013). On the other hand, as mentioned, several proposals for implementing non-Gaussian gates are extant in the literature Ghose and Sanders (2007); Gu et al. (2009); Marek et al. (2011); Yukawa et al. (2013); Marshall et al. (2015); Miyata et al. (2016); Marek et al. (2017); Takeda and Furusawa (2017). Here we focus on a method in which the standard cluster is modified using non-Gaussian resources — called cubic-phase states Gottesman et al. (2001). This modified non-Gaussian cluster is particularly advantageous for scaling to large numbers of operations since it allows for the measurement strategy to remain Gaussian Gu et al. (2009); Gottesman et al. (2001).

We will first present a general exposition of the optomechanics model we wish to base our proposal on, and then introduce two complementary schemes allowing us to prepare the modified non-Gaussian cluster and perform on it arbitrary Gaussian measurements.

Optomechanics implementation.— Consider an array of mechanical resonators, each with distinct frequency , immersed in a cavity field of frequency and driven by an external field . The Hamiltonian for such a system is . Due to radiation pressure the cavity frequency becomes dependent on the mechanical positions Aspelmeyer et al. (2014). Expanding in powers of these latter, we write: , with and . In addition, we consider the case of a multi-tone drive, with the complex driving amplitudes and the driving frequencies. The standard linearisation procedure for an externally driven cavity Aspelmeyer et al. (2014) yields, in the frame rotating with the free terms, the following Hamiltonian PRL ()

(1)

where the are the detunings of the field with the cavity, and are the amplifications of the single phonon-photon couplings due to the external driving. Then, we consider four driving fields per each mechanical resonator with detunings and amplitudes (). Moreover, we consider an additional drive that is resonant with the cavity (), with amplitude . Hamiltonian (1), in the rotating wave approximation (RWA), becomes PRL ()

(2)

with , (; ) and . Notice that independent control over each term in the Hamiltonian (2) is possible 333Note that the parameters , , can be tuned by varying both the bare optomechanical couplings and the driving amplitudes, while is set by changing only., which is in turn crucial for our purposes. The aforementioned RWA holds in a regime satisfying and (, , ), given that the frequencies do not overlap PRL ().

As said, dissipation is central for our aims. We model the evolution of the system by a master equation in which the cavity mode dissipates at a rate and the mechanical oscillators are in contact with a thermal bath Carmichael (1993); Gardiner and Zoller (2000):

(3)

where and denote the mechanical damping rate and mean phonon number for the -th mechanical oscillator. The standard super-operator for Markovian dissipation is denoted as ().

The cubic phase state.— The finitely-squeezed cubic phase state of a single system is defined as Gottesman et al. (2001)

(4)

A core result of our proposal is that the cubic phase state of a single mechanical oscillator can be unconditionally generated as the steady state of the dynamics given in Eq. (3) (with ), applying suitable drive amplitudes and phases. The coefficients of the linear terms, and , are associated only with Gaussian steady states Houhou et al. (2015). Indeed, the ratio of the amplitudes of these determines the degree of squeezing 444We describe the level of squeezing as  dB Gu et al. (2009). of the steady state Houhou et al. (2015); Kronwald et al. (2013). Non-Gaussianity at the steady state derives instead from the remaining coefficients as follows. By choosing the driving strengths as , with , we obtain the Hamiltonian

(5)

It can be proven analytically PRL () that, neglecting the mechanical thermal noise, the master equation (3) has the steady state where is the vacuum state of the cavity and is the mechanical finitely squeezed cubic phase state defined in Eq. (4) with . The stability condition of the system’s dynamics is inherited from the linear system: . Notice that for this Hamiltonian, the cubic phase state is the only steady state regardless of the initial state PRL ().

In order to consider the effect of non-zero mechanical noise, we numerically find the steady state of Eq. (3) and then we calculate the fidelity between the latter and the state in Eq. (4). This is shown in Fig. 2 where we plot the fidelity as a function of the mean phonon number of the bath and the mechanical damping rate. As expected, the mechanical noise has a noxious effect on the target cubic phase state; the higher the temperature (quantified by ) or mechanical damping rate () the lower the fidelity.

Figure 2: Fidelity of the noisy cubic phase state with the noiseless one as a function of the mean phonon number () and mechanical damping rate (). Each point of the plot was obtained with the following parameters: (), , and .

This analysis shows that a cubic phase state can be generated in the massive mechanical oscillator of an optomechanics experiment. This is a result of interest in its own, given the highly non classical character of such a state — which displays a non-positive Wigner function and a high degree of quantum non-Gaussianity Takagi and Zhuang (2018); Albarelli et al. (2018) — and its deterministic attainability. As mentioned, for this state to be considered as a resource for computing, we must also show that it can be embedded in a standard Gaussian cluster state 555A seminal proposal to realize, via optomechanical-like interactions, states potentially useful for MBQC is given in Ref. Pirandola et al. (2006); however, there a non-linearized and probabilistic approach is considered. We also notice that the recent deterministic proposal in Ref. Brunelli et al. (2018) use similar techniques to the ones presented here but the states obtained there are not useful for MBQC (see also Brunelli and Houhou (2018) for further analysis)..

Non-Gaussian cluster states.— We aim to generate a modified non-Gaussian cluster state sufficient to perform universal computation by interspersing the standard state with cubic-phase states. In particular we will now show how the dissipative dynamics described by Eq. (3) can be adapted to generate the state , where denotes the cubic non-linearities, and we have defined and (see Fig. 1). The state allows the implementation of universal computation since it can be composed of nodes with zero non-linearity, as the standard Gaussian one, and nodes with . For any given computation, Gaussian measurements will then “tailor” this non-Gaussian cluster accordingly to the program to be implemented. In this way, cubic gates can be implemented only when needed 666Similar tailoring techniques have been considered in the context of both continuous Menicucci (2014) and discrete variables Bermejo-Vega et al. (2017). In the latter, Gaussian and non-Gaussian gates are substituted with Clifford and non-Clifford ones..

Adapting the Hamiltonian switching scheme considered in Refs. Ikeda and Yamamoto (2013); Houhou et al. (2015), one can generate the state via dissipation engineering. The switching scheme involves steps such that at each one the driving fields are tuned to implement the transformation . This implies that, at the step, the Hamiltonian is where is a positive parameter. At each step, the system is allowed to reach its steady state (i.e., the vacuum of the collective mode ) and then the Hamiltonian is switched, by modifying the driving fields, for the next step to begin (see Ref. PRL () for details). Therefore, if the system is initially in vacuum (and neglecting the mechanical damping), after the steps the mechanical state is given by the target cluster state, in the basis of the local modes 777We should mention that our switching scheme introduced here is not only a generalisation of a previous protocol Houhou et al. (2015) for the generation of Gaussian cluster states, but also conforms with the canonical preparation of Gaussian cluster states if we restrict our selves to first sideband drivings only..

Fig. 3 demonstrates the effectiveness of the switching scheme for generating a two-node non-Gaussian cluster. In the absence of mechanical noise (solid red line), the fidelity with the target state increases monotonically in each step and it reaches unit fidelity at the steady state (at the end of the second step, provided longer evolution time is allowed). When the mechanical environment is considered (dot dashed line), the fidelity reaches a maximum (during the second step) before the noise starts to negatively affect the quality of the target cluster state. As already seen in Fig. 2, the thermal noise has a detrimental effect on the performance of the switching scheme, however high degrees of fidelities can still be achieved. Part of this negative effect is due to the fact that the oscillators are assumed to be initialized in thermal equilibrium with their environment (with mean phonon numbers and and mechanical damping ), rather than in the ground state. This effect can then be circumvented to a large degree by first independently cooling the oscillators (red detuned sideband cooling) Marquardt et al. (2007); Wilson-Rae et al. (2007). This can be seen in the dashed blue curve of Fig. 3, which in fact closely approximates the noiseless scenario.

Figure 3: The fidelity of the preparation of a two-node non-Gaussian cluster state. The nodes of the cluster consist of a squeezed state and a cubic phase state with same amount of squeezing. We used the following parameters: (), , , and evolution duration . Pre-cooling (dashed line) the oscillators close to the ground state greatly increases the (maximum) achievable fidelity of the scheme.

Local Gaussian measurements— For the non-Gaussian cluster state above to be useful in quantum computation, one finally requires the capacity to perform Gaussian measurements on individual nodes. Unfortunately, the mechanical modes embodying our cluster state are not directly accessible to measurement and must be probed instead using the cavity field as a detector. Conveniently, since the resonators are assumed to have distinct and well-spaced frequencies PRL (), we may address each oscillator individually. In particular, by properly driving the system, it is possible to engineer a quantum non-demolition (QND) interaction between the cavity position quadrature and an arbitrary quadrature of any given oscillator Clerk et al. (2008); Moore et al. (2017).

Consider again Hamiltonian (2) with (), i.e. addressing only the first sidebands. Let and . In this case one has a sum of QND interactions, , where is the cavity position quadrature and is an arbitrary quadrature of the mechanical mode . Each oscillator can be addressed in turn by setting all but the amplitude of interest to zero. In this case we have an -step process with each step described by . Let us stress that the mechanical quadrature to be measured, which in turn depends on the program to be implemented, is simply selected by the phase of the external driving. Continuously monitoring, via homodyne detection, the output cavity field’s position quadrature drives the mechanical system towards an eigenstate of the chosen quadrature (represented by a vacuum state squeezed along an appropriate axis defined by ). For the purposes of computation, this is equivalent to performing a projective quadrature measurement directly onto the cluster state Moore et al. (2017). As said, the latter are in turn sufficient to perform any multimode operation, when operating on the non-Gaussian cluster .

In PRL () we provide an example of how to implement the minimal building block of universal MBQC by using the tools introduced so far. In particular, we consider the universal non-Gaussian gate defined as the operator Weedbrook et al. (2012) — called cubic phase gate — and show that it can be reliably implemented on a squeezed state via local Gaussian measurements on the two-node non-Gaussian cluster of Fig. 3.

Experimental feasibility.— The protocol proposed above to prepare non-Gaussian cluster states requires physical platforms exhibiting linear and quadratic position coupling with the cavity field. Moreover, the system needs to operate in the resolved sideband regime and the conditions and (, , ) must be met to ensure the validity of the RWA used in our derivation of the dynamics. These requirements may be realised in current and near future experiments. In fact, there are many platforms that can be used to implement our scheme, including membrane-in-the-middle configurations Thompson et al. (2008); Sankey et al. (2010); Flowers-Jacobs et al. (2012); Karuza et al. (2013); Lee et al. (2015), ultracold atoms inside a cavity Purdy et al. (2010), photonic crystals Paraïso et al. (2015); Kalaee et al. (2016); Leijssen et al. (2017); Kaviani et al. (2015), circuit-QED Kim et al. (2015), electro-mechanical systems Woolley et al. (2008); Hertzberg et al. (2010); Rocheleau et al. (2010); Massel et al. (2012); Dellantonio et al. (2018), micro-disks Li et al. (2012); Hill et al. (2011); Hill (2013); Doolin et al. (2014), and optically levitated particles Chang et al. (2010); Abdi and Hartmann (2015); Fonseca et al. (2016). In particular, very large quadratic couplings are within reach of current experiments Paraïso et al. (2015); Kalaee et al. (2016); Hill et al. (2011). Also we mention that linear to quadratic ratios of up to may be obtained Zhang et al. (2017); Brawley et al. (2016).

Furthermore, the linear to quadratic couplings ratio can be improved by optimising the experimental design. For instance, one may exploit the membrane tilting in membrane-in-the-middle setups Thompson et al. (2008); Sankey et al. (2010) or fine positioning the microdisc in microtorid optomechanical systems Li et al. (2012). Also, our protocols can be implemented in electrical circuits by controlling the bias flux and coupling capacitance as proposed in Kim et al. (2015), or considering magnetically or optically levitated particles as suggested in Pino et al. (2018); Chang et al. (2010).

Conclusions and outlook.— Continuous-variable systems are convenient for fault-tolerant computation since they naturally offer high-dimensional spaces in which the discrete units of quantum information can be resiliently encoded Chuang et al. (1997); Gottesman et al. (2001); Lund et al. (2008); Menicucci (2014); Michael et al. (2016); Fukui et al. (2017), as recently proven experimentally in the context of circuit-based quantum computation Ofek et al. (2016); Rosenblum et al. (2018); Flühmann et al. (2018). In this respect, the alternative measurement-based approach considered here is promising, thanks to the availability of high threshold schemes Raussendorf and Harrington (2007); Raussendorf et al. (2007). In particular, we have shown that a setting where mechanical oscillators act as the information carriers, rather than photons, provides the advantage that the core ingredients for universal computation —non-Gaussian cluster states and Gaussian operations— can be realized unconditionally. This opens the way to deterministic fault-tolerant quantum computation in integrable platforms where linear and quadratic optomechanics-like interactions can be simultaneously achieved.

We thank M. Brunelli, M. Paternostro, and M. Sillanpää for helpful discussions. A.F. and O.H. acknowledge support from the EPSRC project EP/P00282X/1 and the EU Horizon2020 Collaborative Project TEQ (grant agreement nr. 766900). O.H. acknowledges support from the SFI-DfE Investigator programme (grant 15/IA/2864). D.M. acknowledges the Coordinator Support funds from Queen’s University Belfast.

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  • (105) See Supplemental Material at [URL will be inserted by publisher] for more details.
  • (106) Note that the parameters , , can be tuned by varying both the bare optomechanical couplings and the driving amplitudes, while is set by changing only.
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  • (112) A seminal proposal to realize, via optomechanical-like interactions, states potentially useful for MBQC is given in Ref. Pirandola et al. (2006); however, there a non-linearized and probabilistic approach is considered. We also notice that the recent deterministic proposal in Ref. Brunelli et al. (2018) use similar techniques to the ones presented here but the states obtained there are not useful for MBQC (see also Brunelli and Houhou (2018) for further analysis).
  • (113) Similar tailoring techniques have been considered in the context of both continuous Menicucci (2014) and discrete variables Bermejo-Vega et al. (2017). In the latter, Gaussian and non-Gaussian gates are substituted with Clifford and non-Clifford ones.
  • (114) We should mention that our switching scheme introduced here is not only a generalisation of a previous protocol Houhou et al. (2015) for the generation of Gaussian cluster states, but also conforms with the canonical preparation of Gaussian cluster states if we restrict our selves to first sideband drivings only.
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Supplementary Material:

Unconditional measurement-based quantum computation

with optomechanical continuous variables

Supplemental Material

Oussama Houhou

Darren W. Moore

Sougato Bose

Alessandro Ferraro

Appendix A Hamiltonian derivation

Our optomechanical system consists of mechanical oscillators interacting with a cavity mode via radiation pressure, and the cavity is pumped by a multi-tone field. The full Hamiltonian of the system is given by Aspelmeyer et al. (2014)

(S1)

where and are, respectively, the annihilation operator and frequency of the cavity field, , and are the frequency and (dimensionless) position and momentum quadratures of the mechanical oscillator, and is the (classical) pump of the cavity.

The cavity’s frequency becomes dependent on the mechanical positions Aspelmeyer et al. (2014). We expand in powers of up to the second order:

(S2)

with and are the the position and position-squared couplings of the mechanical oscillator with the cavity field. The Hamiltonian becomes

(S3)

Allowing our system to be in contact with vacuum reservoir for the cavity and thermal bath for the mechanical oscillator, leads to the following Heisenberg - Langevin equations Gardiner and Zoller (2000) for the system operators:

(S4)
(S5)
(S6)

where and are the damping rates for the cavity mode and the mechanical oscillator, and and are the input noise operators for the cavity and mechanical oscillator respectively, satisfying the correlation relations:

(S7)
(S8)
(S9)
(S10)

with denoting the mean phonon number.

We aim to derive an effective Hamiltonian for the system involving quantum fluctuations around the (classical) fields steady states. Replacing the system operators, in equations (S4)-(S6), by their mean-fields: , and , the classical equations of motion become:

(S11)
(S12)
(S13)

The multi-tone driving field can be written as :

(S14)

where is the frequency of the pump field, and is its complex amplitude. We consider the following ansatz for the intra-cavity field at the steady state Milburn and Woolley (2011):

(S15)

where the constants are the complex amplitudes of the cavity at the steady state. By substituting expression (S15) in Eq. (S12) we find:

(S16)

If we assume weak coupling such that for we have

(S17)

the time dependent terms in Eq. (S16) can be neglected. And if we denote by and the values of position and momentum at the steady state, it is easy to find the following:

(S18)
(S19)
(S20)

where is the detuning of the drive with respect to the cavity.

Having obtained the steady state for all fields, we can derive a Hamiltonian of the system in terms of the quantum fluctuations around the classical steady values. First, we split the system operators into classical part and quantum fluctuation,

(S21)

then we substitute (S21) in equations(S4)-(S6). Assuming a strong drive, , we find:

(S22)
(S23)
(S24)

Equations (S22)–(S24) correspond to the following effective Hamiltonian:

(S25)

where we defined and .

The explicit time-dependence of Hamiltonian (S25) can be removed by, first, going to a frame rotating with the free terms of the system where the Hamiltonian transforms to

(S26)

Then, we consider a resonant drive with the cavity, with amplitude and null detuning, and four driving fields per each mechanical oscillator, with detunings and amplitudes (). By invoking the rotating wave approximation (RWA), we may write the Hamiltonian as

(S27)

where we have defined and .

Appendix B Validity of the rotating wave approximation

The validity of the RWA introduced in Sec. A will be justified here. Hamiltonian (S27) is obtained from Eq. (S26) by discarding all time-dependent (counter-rotating) terms and keeping only the resonant ones. The counter-rotating terms may be written as

(S28)

with the following expressions:

(S29)
(S30)
(S31)
(S32)
(S33)

Now we can state the necessary conditions to safely neglect the counter-rotating terms. For the RWA to be valid, the following constraints must be met:

(S34)

We study the validity of the RWA in more details for the interesting case of the preparation of the cubic phase state of a mechanical oscillator. The system consists of a cavity and one mechanical oscillator (). The full Hamiltonian of the system is again with

(S35)