Hybrid systems for the generation of non-classical mechanical states via quadratic interactions

Hybrid systems for the generation of non-classical mechanical states via quadratic interactions

Carlos Sánchez Muñoz Present address: Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, UK carlossmwolff@gmail.com Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Saitama, 351-0198, Japan    Antonio Lara Dpto. Física Materia Condensada C03, Instituto Nicolas Cabrera (INC), Condensed Matter Physics Institute (IFIMAC), Universidad Autónoma de Madrid, 28049, Madrid, Spain    Jorge Puebla Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Saitama, 351-0198, Japan    Franco Nori Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Saitama, 351-0198, Japan Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA
August 23, 2019
Abstract

We present a method to implement two-phonon interactions between mechanical resonators and spin qubits in hybrid setups, and show that these systems can be applied for the generation of nonclassical mechanical states even in the presence of dissipation. In particular, we demonstrate that the implementation of a two-phonon Jaynes-Cummings Hamiltonian under coherent driving of the qubit yields a dissipative phase transition with similarities to the one predicted in the model of the degenerate parametric oscillator: beyond a certain threshold in the driving amplitude, the driven-dissipative system sustains a mixed steady state consisting of a “jumping cat”, i.e., a cat state undergoing random jumps between two phases. We consider realistic setups and show that, in samples within reach of current technology, the system features non-classical transient states, characterized by a negative Wigner function, that persist during timescales of fractions of a second.

Introduction.—Over the past decades, technological developments have allowed to implement new classes of extremely sensitive nanomechanical oscillators, such as membranes or microcantilevers, that are finding applications in a wide variety of areas, from biological detection Calleja et al. (2012) to ultrasensitive mass sensing Spletzer et al. (2008); Gil-Santos et al. (2009); Chaste et al. (2012) or NMR imaging Mamin et al. (2007); Poggio and Degen (2010); Mamin et al. (2012). There has been a growing interest in studying hybrid systems in which these mechanical elements are coupled to some other quantum actor, allowing to explore the quantum limits of mechanical motion Wallquist et al. (2009); Xiang et al. (2013); Treutlein et al. (2014), with prominent examples such as cavity optomechanics setups Gigan et al. (2006); Thompson et al. (2008); Kippenberg and Vahala (2008); Marquardt and Girvin (2009); Aspelmeyer et al. (2014). Many of these works aim to explore the quantum limit of mesoscopic objects consisting of billions of atoms by cooling them close to the ground state Blencowe (2004); Chan et al. (2011); Poot and van der Zant (2012) and generating inherently quantum states, such as squeezed states Jähne et al. (2009) or quantum superpositions Hoff et al. (2016).

In this work, we present hybrid setups that are able to achieve a two-phonon coherent coupling between a mechanical mode and a spin qubit, described as a two-level system (TLS). It is known for systems involving some kind of two-particle interaction plus a nonlinearity Drummond et al. (1980, 1981); Gilles et al. (1994); Hach III and Gerry (1994); Benito et al. (2016); Krippner et al. (1994); Hu and Nori (1996, 1997, 1999); Nation et al. (2012); Tan et al. (2013); Everitt et al. (2014); Mirrahimi et al. (2014); Leghtas et al. (2015); Minganti et al. (2016); Bartolo et al. (2016), that the mechanical system can evolve into motional cat states. Although these states are ultimately washed out by decoherence, our proposed setup features non-classical transient states, characterized by a negative Wigner function, during timescales that can extend up to seconds. After this, the system reaches a mixed steady state that has been understood  as a cat state flipping its phase at random times Minganti et al. (2016). This offers an attractive platform both for the study of fundamental questions in quantum mechanics—such as decoherence, spontaneous symmetry breaking and ergodicity in dissipative quantum systems Zurek (2003); Mølmer (1997); Cresser (2001); Benito et al. (2016)—and for practical applications where nonclassical mechanical states can be envisaged as a technological resource Ralph et al. (2003); Lund et al. (2008); Joo et al. (2011); Facon et al. (2016); Albert et al. (2016); Mirrahimi et al. (2014) .

Our proposal is based on hybrid devices in which a single spin qubit embedded in a magnetic field gradient couples to a mechanical oscillator through the position-dependent Zeeman shift Rabl et al. (2009, 2010); Arcizet et al. (2011); Kolkowitz et al. (2012); Pigeau et al. (2015); Wei et al. (2015); Li et al. (2016). We consider the qubit to be given by the electronic spin of nitrogen-vacancy (NV) centers, which are excellent candidates due to their outstanding coherence and control properties Childress et al. (2006); Gaebel et al. (2006); Hanson et al. (2006); Hanson and Awschalom (2008); Buluta et al. (2011); Doherty et al. (2013). Several works Ma et al. (2016); Cai et al. (2017) have analyzed particular geometries in which the equilibrium position of the system leaves the spin in a point of null magnetic gradient, leading to a quadratic dependence of the coupling with the position. These works proposed to use this dependence to couple two different modes of the oscillator in order to effectively enhance the linear coupling between one of these modes and the TLS. In contrast, we propose here to use these geometries to achieve degenerate, two-phonon exchange between one mode of the resonator and the TLS, which gives rise to physical phenomena with no analogue in linearly-coupled systems Wang et al. (2017).

Figure 1: (a) Proposed configuration: an NV center in a diamond film (red) is placed on top of a mechanical oscillator (e.g., a thin membrane) and placed between two magnets with aligned magnetization. Dimensions have been altered for visual clarity. (b) Magnetic field lines, computed for two Dy magnetized cylinders of 30 nm diameter separated by a gap of  nm. Red (blue)-shaded areas mark the regions where . Both first derivatives are zero at the position of the NV center—represented by a circle—when the oscillator is at rest. (c) Magnetic field along the axis for .

Setup proposal.—We consider an NV center placed on top of a mechanical oscillator at a position and surrounded by a magnetic field . An NV center consists of a nitrogen atom and an adjacent vacancy in diamond, and its electronic ground state can be described as a spin triplet with states , with . The Hamiltonian of the system reads (we set hereafter) , where stands for the Hamiltonian of the NV center, for the mechanical mode, and the last term describes a perturbation on the NV center due to the external magnetic field, where is the Bohr magneton, is the Landé factor of the NV center, and is its spin operator. The last term provides the mechanism that couples the qubit and the mechanical degree of freedom. We will assume that the mechanical mode oscillates only along the axis, so that the position of the NV center is given by , being the displacement of the oscillator with respect to the equilibrium point. Setting , we can expand the Hamiltonian in terms of up to second order, . Our proposal relies on considering a magnetic field with an extremum at the position of the NV center, which will cancel the first derivative in the expansion and provide a second-order coupling to the mechanical mode. For simplicity, we will consider that the field has also null second derivatives along the and axis, so that the mechanical mode only couples to . This latter assumption is not necessary, but we show here a particular proposal in which this is indeed the case. By writing the position operator of the mechanical oscillator as , with the zero-point fluctuation amplitude, the resonant mechanical frequency, and the annihilation operator, the resulting Hamiltonian becomes:

(1)

where the two-phonon coupling is given by , and . The critical parameters here in order to maximize this coupling are the second gradient of the magnetic field and the zero-point motion of the oscillator, which in both cases should be as high as possible.

For practical purposes, we will focus on cases where the magnetic field is generated by nanomagnets, since they provide high gradients at short distances Mamin et al. (2007); Poggio and Degen (2010); Mamin et al. (2012). In Fig. 1, we propose a particular arrangement of magnets that yields the required spatial magnetic field profile. An NV center injected in a diamond film Ohashi et al. (2013) is placed on top of a resonator of nanometer-scale thickness that oscillates along the direction; the extension of the diamond film should be much smaller than that of the oscillator to minimize any possible impact on its properties. Diamond films can be compatible, for instance, with silicon nitride substrates Almeida et al. (2011); Baudrillart et al. (2016); Skoog et al. (2012). The resonator is positioned in the gap between two cylindrical nanomagnets with saturated magnetization along the axis. The size of the gap is considered to be of the order of tens of nanometers. In the region between the magnets, this geometry yields a strong magnetic field in the direction and a negligible field in the and directions, as we show in Fig 1(b-c). Moreover, every component of the field has a null derivative with respect to at the middle point. This gives rise to the quadratic coupling between the NV center and the oscillator.

Figure 2: (a) Two-photon coupling rate versus zero-point fluctuation amplitude of the oscillator, for two magnets separated by  nm. (b) Two-photon coupling rate versus the magnet separation for three different magnetic materials, for a resonator with   fm. (c) Cooperativity versus the oscillator quality factor and the zero-point fluctuations, for a magnet separation of  nm, oscillator frequency  , temperature  , and pure-dephasing rate  . The white line marks the onset of quantum effects. Point A corresponds to a feasible point for state-of-the-art technology at mK temperatures Ghadimi et al. (2018), with   fm and  ; B is the point taken in most part of the text for clarity of results:   fm and  

Two-phonon coupling rates.—In order to estimate the achievable two-photon coupling rate in realistic setups, we simulated the magnetic field generated by two cylinders of nanometer size with saturated magnetization for three different materials (, and ) (see Appendix .1). stands as the best choice due to its high saturation magnetization Mamin et al. (2012); Scheunert et al. (2014). Figure 2(b-c) is an example of the simulated magnetic field for two cylinders of with  nm of diameter,  nm of height and separated by a gap of  nm. In this configuration, one can obtain values of  . To obtain the corresponding two-phonon coupling rate , one must consider a specific implementation of the mechanical oscillator. The relevant parameter is the zero-point fluctuation amplitude, , which ranges from tens of femtometers in systems such as oscillators Norte et al. (2016); Ghadimi et al. (2018) to hundreds of femtometers in systems such as carbon nanotubes Huttel et al. (2009), graphene resonators Singh et al. (2014); Weber et al. (2016), wires Arcizet et al. (2011) or cantilevers Poggio et al. (2007). Figure 2(a) shows versus for a fixed separation between magnets of  nm. As an example, for a value of   fm, the resulting coupling rate is  . The dependence of this value on the gap between the magnets is shown on Fig. 2(b). Linear couplings induced by non-zero first-order gradients due to imperfect alignment can be disregarded at the two-phonon resonance condition. Alternatively, they could be used to calibrate the device and measure the state of the oscillator by properly tuning the qubit energy (see Appendix .4-.5).

Quantum effects.—To address the possibility of observing quantum effects, a relevant figure of merit is the cooperativity  Kolkowitz et al. (2012); Schuetz et al. (2015), where is the dephasing rate of the qubit, is the oscillator decay rate ( being the quality factor), and is the average number of thermal phonons at the oscillator at the temperature . Values of the cooperativity mark the onset of quantum effects. The impact of spin relaxation is not relevant here, since relaxation times can reach hundreds of seconds at low temperatures Jarmola et al. (2012). Regarding pure dephasing rates, can achieve room temperature values  Bar-Gill et al. (2013) using dynamical decoupling techniques, which have already been used in very similar setups Kolkowitz et al. (2012). Once , and are established, the cooperativity is fully determined by the oscillator parameters, , and . Figure 2(c) shows versus and for (typical of systems such as wires Arcizet et al. (2011) or nanobeams Ghadimi et al. (2018)),   and  . As an example, an oscillator with  ,  fm Ghadimi et al. (2018) and   (point A in Fig. 2(c)) yields at these conditions, and can show quantum effects for dephasing rates  , which have already been achieved experimentally Bar-Gill et al. (2013); Ovartchaiyapong et al. (2014). Recently, room-temperature values have been demonstrated in oscillators fabricated via soft-clamping and strain engineering techniques Tsaturyan et al. (2017); Ghadimi et al. (2018), with values expected at dilution refrigerator temperatures (Tsaturyan et al. (2017). Therefore, although demanding, these conditions are withing reach of state-of-the-art technology. For the sake of clarity of results, we will consider hereafter a slightly more optimistic value of   fm (giving  ), and set   and   as in Ref. Ghadimi et al. (2018) (this choice is shown as point B in Fig. 2(c)). We take   and , giving and . While the proximity of the NV center to the surface in a diamond film might render longer dephasing rates than in the bulk, we note that we are also considering cryogenic temperatures, which is known to enhance coherence times by several orders of magnitude Jarmola et al. (2012). At these low temperatures, several techniques exist in order to minimize the influence of heat induced by, e.g., RF voltage; most of these solutions are related to the design of heat sinks, cooling fins, etc., and the selection of proper materials for heat dissipation Savin et al. (2006).

Figure 3: Signatures of a dissipative phase transition in the steady-state. (a) Phonon number and fluctuations of the position operator versus the driving amplitude . (b) Wigner function for three values of the driving amplitude. Here,  ,  ,  ,  ,  .
Figure 4: Dynamics of the density matrix and of a single quantum trajectory. (a) Phonon population versus time. Vertical lines depict times when a phonon emission process takes place. (b) Wigner function of the mechanical mode given by the density matrix (upper) and the wavefunction of a single trajectory (lower), at the times marked with arrows on the top of panel (a). The last two columns of the bottom row depict a “jumping cat”, at times before and after a phonon emission process. Parameters are those of Fig. 3.

Dissipative dynamics of the driven, two-phonon Jaynes-Cummings Hamiltonian.—By adding two oscillating magnetic fields, one in the axis with frequency in the MW regime; and another in the axis with frequency , we obtain (see Appendix .3 and Refs. Rabl et al. (2009); Li et al. (2016)) an effective, coherently driven two-phonon Jaynes-Cummings Hamiltonian:

(2)

where is the lowering operator of the effective TLS, and denotes the amplitude of the driving. We will consider the resonant situation . In order to describe the dynamics of the system under dissipation, this Hamiltonian needs to be supplemented with the usual Lindblad terms Carmichael (2002), giving the master equation for the dynamics of the density matrix, , where . We consider the system to be actively cooled to a thermal phonon population close to zero, which can be done, for instance, by means of laser cooling Chan et al. (2011); Teufel et al. (2011); Peterson et al. (2016) or using another spin qubit Rabl et al. (2009). We therefore exclude incoherent pumping terms of the kind from the master equation, at the expense of using an increased resonator linewidth , with the natural linewidth, and is the number of thermal phonons in the oscillator in the absence of cooling Kolkowitz et al. (2012).

The two-phonon Hamiltonian (2) is reminiscent of quantum optical systems with two-photon interactions that have attracted considerable interest Gilles et al. (1994); Hu and Nori (1996, 1997, 1999); Everitt et al. (2014); Mirrahimi et al. (2014); Leghtas et al. (2015). Different systems with two-particle interactions and some kind of nonlinearity—e.g., two-photon losses in the case of the degenerate parametric oscillator (DPO) Drummond et al. (1980, 1981); Gilles et al. (1994); Hach III and Gerry (1994); Benito et al. (2016); Nation et al. (2012), a Kerr nonlinearity Goto (2016) or, as in the present case, a TLS Wang et al. (2017)—, have been shown to develop transient cat states Krippner et al. (1994); Tan et al. (2013); Everitt et al. (2014); Leghtas et al. (2015); Goto (2016); Wang et al. (2017) that, through unavoidable single-photon losses, tend to a steady state characterized by a Wigner function with phase bimodality Gilles et al. (1994); Hach III and Gerry (1994); Benito et al. (2016); Bartolo et al. (2016) and no interference fringes. Research on the DPO has shown that such steady state corresponds to a succession of random jumps between cat states of opposite phase when single trajectories are considered Minganti et al. (2016), i.e., a sustained “jumping cat” Garraway and Knight (1994). In the following, we discuss the appearance of analogue nonclassical effects in our system.

Figure 5: Cattiness versus time for an initial vacuum state. (a) Parameters are those of Fig. 3, with   (solid, blue), (dashed, yellow) and (dotted, red). Inset: Wigner function at the time of maximum ,  ms. (b)   and parameters of point A in Fig. 2(c), consistent with Ref. Ghadimi et al. (2018) at  :  fm ( ),   (estimated), and  , giving .

Figure 3(a) depicts the phonon population and the variance of the position operator in the steady state versus the driving amplitude . In close similarity to the DPO Drummond et al. (1980, 1981); Gilles et al. (1994); Hach III and Gerry (1994); Benito et al. (2016), we observe a phase transition characterized by the development of two lobes in the Wigner function, preceded by some degree of squeezing. This occurs when the phonon population is , a point where its dependence with changes from to . Note that here, phase bimodality does not originate from the two-level nature of the driven TLS Alsing and Carmichael (1991), but is rather a consequence of the phase symmetry of the master equation, which is invariant under the change  Benito et al. (2016). Figure 4 shows the transient dynamics of the oscillator towards the steady state, computed for the density matrix and for a single quantum trajectory Plenio and Knight (1998) for a system initially in the ground state. The Wigner function of the oscillator shows an initial squeezing along two directions that is eventually confined in phase space due to the TLS nonlinearity (see Appendix .6). Individual quantum trajectories reveal that the bimodal steady state consists of a cat state undergoing random phase flips due to single-phonon losses Minganti et al. (2016), as shown in the last two columns of Fig. 4(b), that capture two times, before and after a single-phonon emission event. Each of these cat states has an extremely long lifetime, surviving with fidelities for times longer than a millisecond (see Appendix .7).

Transient non-classical states.— The high quality factors of state-of-the-art nanoresonators Ghadimi et al. (2018) allows for non-classical states to develop and evolve in timescales of tenth of a second before every trace of coherence is washed out. We show this by plotting the evolution of the “cattiness” , defined by dividing the integrated negative parts of the Wigner function of the state by that of a reference cat state  Everitt et al. (2014), so that only for nonclassical states and for cat states. The results shown in Fig. 5 demonstrate that we can observe unambiguous nonclassical features lasting up to seconds even with state-of-the-art setups Ghadimi et al. (2018). Several routes to detect these quantum states are discussed in Appendix .5. Once in the steady state, a feedback protocol has been proposed Minganti et al. (2016) in order to enhance the decay rate only when the system is in one of the two possible cat states, and therefore stabilize the system in the other. We note that the combination of recently developed single-phonon detectors Cohen et al. (2015); Riedinger et al. (2016) and the optical control of decay via active cooling makes the system proposed here an attractive platform to implement such feedback protocols, e.g., switching between two effective quality factors—by changing the driving amplitude of the cooling laser—whenever a single phonon is detected.

Acknowledgements.
The authors kindly acknowledge P. B. Li, C. Navarrete-Benlloch, X. Hu and A. Miranowicz for useful discussions and comments. C.S.M acknowledges funding from a Short-Term Grant from the Japanese Society for the Promotion of Science (JSPS) and from the Marie Sklodowska-Curie Fellowship QUSON (Project No. 752180). J.P. was supported by the RIKEN Incentive Research Project Grant No. FY2016. F.N. is supported in part by the MURI Center for Dynamic Magneto-Optics via the Air Force Office of Scientific Research (AFOSR) (FA9550-14-1-0040), Army Research Office (ARO) (Grant No. 73315PH), Asian Office of Aerospace Research and Development (AOARD) (Grant No. FA2386-18-1-4045), Japan Science and Technology Agency (JST) (the ImPACT program and CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (JSPS-RFBR Grant No. 17-52-50023), RIKEN-AIST Challenge Research Fund, and the John Templeton Foundation.

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Appendix

.1 Magnetic field simulations

For the case of , we have considered a saturation magnetization of  Scheunert et al. (2014), exchange stiffness    Saraiva et al. (2010), and a magnetic damping   Woltersdorf et al. (2009); for , we take  Cantu-Valle et al. (2015),   Manfred et al. (2003) and   Schoen et al. (2016); and for ,  Schoen et al. (2016),   Liu and Morisako (2008) and   Schoen et al. (2016). Note that, regarding the fabrication of these nanopillars, outstanding control on the size and shape can be achieved by using a variety of techniques, such as molecular beam epitaxy Pigeau et al. (2012) or focused electron-beam-induced deposition (FEBID) Gavagnin et al. (2014). We calculated the magnetic field generated by the cylinders using MuMax3 Vansteenkiste et al. (2014), a finite differences, open-source solver of the Landau-Lifshitz-Gilbert equation Gilbert (2004).

Figure 6: Procedure to obtain an effective TLS. (a) Bare energy levels defined in the basis of bright and dark states and in the frame rotating with , see Eq. (6). is coupled to through the microwave magnetic field in the axis, and to through the magnetic field in the axis. (b) The microwave field dresses the states and . In the basis of dressed energy levels, the states and define an effective TLS, driven by the field in axis. If , .

.2 Master equation and quantum trajectory simulations

Simulation of the system dynamics in a dissipative environment has been performed following two different techniques: master equation simulations and the method of quantum trajectories.

For the master equation simulations, we solved numerically the differential equations that govern the evolution of the density matrix:

(3)

where . This was done by truncating the Hilbert space setting a maximum number of phonons in the oscillator. For the calculations done in this manuscript, was enough to guarantee convergence for the highest values of driving considered.

The method of quantum trajectories yields a stochastic evolution of a pure wavefunction, which averaged over many different realizations provides the same predictions as the master equation for the density matrix. At every finite time step , for each element of the type in the master equation, the wavefunction can randomly undergo a quantum jump with probability that transforms the system as

(4)

(under proper normalization). The occurrence of a jump is determined by generating a random number at each time step, so that the jump occurs whenever ( must be chosen small enough so that, at every time step, ). When no jump occurs, the wavefunction evolves as

(5)

where is a non-Hermitian Hamiltonian, .

.3 Derivation of the two-phonon, driven Jaynes-Cummings Hamiltonian

Here we define an effective TLS in a way very similar to the one outlined in Refs. Rabl et al. (2009); Li et al. (2016). Our starting point is the Hamiltonian:

(6)

By working in the basis of bright and dark states

(7)
(8)

and assuming, , we can perform a rotating wave approximation and apply a unitary transformation to Eq. (6) in order to move to a rotating frame where the time dependence with is eliminated:

(9)

with . The driving term with can be removed by working in the dressed basis of states and :

(10)
(11)

with , , and , where . In the limit , we have . This makes it evident that we should take the opposite limit , since this will increase the difference between and , allowing us to spectrally isolate one of these transitions as an effective TLS. In that case, , and , so that , and . By defining an effective TLS with lowering operator and transition energy , and assuming , we can make a rotating wave approximation to eliminate fast-rotating terms and perform a final unitary transformation to remove the remaining time dependence, yielding the driven, two-phonon Jaynes-Cummings Hamiltonian of Eq. (2) in the main text. Here and in the main text, we define to lighten the notation. The scheme presented here is sketched in Fig. 6. The derivation will be valid for , , and , with . Since  , a sensible choice of parameters is  , giving  . Since we want  , this implies a value  , which fulfills the conditions above and sets the maximum limit for the driving .

.4 First order magnetic gradient effects due to imperfect alignment

In the main text, we considered a geometry in which the equilibrium point of the oscillator lies exactly at the center of the gap between two nanomagnets, yielding a null first-order gradient of the magnetic field and therefore a pure quadratic coupling. However, it is important to address to which extent unavoidable deviations from a perfectly aligned situation might render noticeable effects due to the induced coupling through first-order gradients. In the simulation depicted in Fig. 7, we observe that a misalignment of  nm is able to induce first-order couplings in the range of . Taking into account that we are considering mechanical modes with frequencies   and fixing the two-phonon resonant condition , we see that first-order gradient terms of the kind will rotate as and can therefore be neglected beside unimportant frequency shifts (e.g., even for as high as , one can still achieve full two-phonon Rabi oscillations governed by by tuning the TLS frequency to , with ).

By tuning the TLS in resonance with the mechanical mode, , the first-order coupling can be used as well to stabilize the resonator close to the middle point. The variation in as the oscillator is moved would yield different responses of the TLS, which could be used in a feedback loop to correct the position of the resonator. It has been demonstrated that picometer stability can be achieved by adding feedback control to piezo-actuators via spectroscopy arrangements, which can be readily obtained by monitoring the NV center light emission Walder et al. (2015); Hümmer et al. (2016).

Figure 7: First and second-order coupling rates as a function of the equilibrium position of the oscillator with respect to the center of the gap between the magnets.

.5 Detection of mechanical non-classical states

In the main text, we have focused on the generation of non-classical states of motion. In an experimental implementation, it is vital do to have a scheme to detect and reconstruct such states. There is a great body of work regarding the reconstruction of mechanical states Vanner et al. (2015); here, we comment on the route consisting of a displacement and a phonon number measurement. One possible way to see that this technique allows to reconstruct the quantum state is to note that the Wigner function can be written as

(12)

with the displacement operator and the parity operator. Through phonon number measurements, we can obtain the expected value of the parity operator for different displaced states and reconstruct the Wigner function.

Figure 8: Simulation of a state-reconstruction measurement. The phonon number distribution is measured mechanical state displaced by a fixed amplitude for a different set of displacement angles . The states used here are: (a) a non-classical mechanical state generated by our system (the same as in the inset of Fig. 5 in the main text); (b) a cat state with the same phonon population as in (a); (c) a coherent state with the same phonon population as in (a). This data can be used to retrieve the quantum state of the system via reconstruction algorithms.

We can also picture the number measurement on a displaced state as an homodyne measurement; by considering the displaced annihilation operator , with , the resulting number operator is

(13)

For , we obtain that the number measurement of the displaced state minus an offset measures the quadrature amplitude :

(14)

Measured over one complete cycle in , the quadrature amplitudes provide tomographically complete information about the quantum system Lvovsky and Raymer (2009); Vogel and Risken (1989). Figure 8 shows a simulation of the proposed measurement; the phonon number distribution of a mechanical state displaced by a fixed amplitude is measured for a set displacement angles . The difference between the resulting data for the distinct states considered is apparent even to the naked eye; this information can be used to infer the quantum state through multiple reconstruction algorithms, like maximum likelihood or entropy maximization Vanner et al. (2015).

The qubit-resonator coupling can also be used in order to perform state tomography of the mechanical oscillator. First order effects allow us to go from a resonant two-phonon coupling regime to a resonant or dispersive one-phonon coupling regime by tuning the TLS energy out of the two-phonon resonance. This would allow, for instance, to measure the state of the oscillator generated by the two-phonon interaction by suddenly switching the TLS energy to a regime of dispersive interaction governed by , which can be used to employ techniques of state reconstruction via displacement and number measurement through Ramsey interferometry of the qubit Haroche and Raimond (2006); Deleglise et al. (2008); Vanner et al. (2015).

.6 Confined dynamics in phase space

The role of nonlinearity brought by the TLS is to limit the dynamics to a region of the phase space of the oscillator. This is clearer if we represent Eq. (2) in the basis that diagonalizes the driven-TLS Hamiltonian Sánchez Muñoz et al. (2018), considering, for simplicity, the resonant case :

(15)

At high driving, , the terms proportional to are counter-rotating and do not contribute to the dynamics provided that , where is the phonon population in the oscillator. In this case, the evolution under the terms is decoupled for the two eigenstates of , , and takes the form of a squeezing operation along the angle . Due to this squeezing operation, the phonon population of the oscillator grows. Once the population reaches a value such that , the counter-rotating terms enter into action and distort the evolution. They act as a barrier in phase space, preventing a small initial population from growing past a given threshold, in close similarity the physics of confined quantum Zeno dynamics Raimond et al. (2010, 2012); Signoles et al. (2014). This is shown in Fig. 4 in the main text, where we depict the evolution, from the moment the driving is turned on, of a system initially in its ground state. In order to gain insight in the driven-dissipative nature of the dynamics, we show the phonon population and Wigner functions of the reduced cavity system, computed both from the density matrix and from the wavefunction of a single quantum trajectory Plenio and Knight (1998). Initially, the TLS is in its ground state, which is described in the dressed basis as a linear superposition . Therefore, the mechanical mode evolves in a superposition of being squeezed along the and axes, yielding a cross-like pattern in phase space, as shown in the first column of Fig. 4(b). After that, the counter-rotating terms enter into action and the squeezing is distorted, yielding a ribbon-like pattern in a confined region of phase space. Finally, the interplay between the coherent evolution and dissipation in the resonator yields a steady Wigner function with two lobes associated to the coherent states .

.7 Long-lived cats in a quantum trajectory

When considering a single quantum trajectory, any cat state in which the system is found to be remains stable for very long times, even in the absence of any feedback protocol. The reason is that they are only affected by the random quantum jumps that flip their phase. At any time, the probability to undergo a jump during a small time interval is . Therefore, if the system is initialized in one of the two cat states that compose the mixed steady state, this state remains stable with a fidelity:

(16)

where we considered time intervals shorter than the phonon lifetime, i.e., . Since phonon lifetimes can reach hundreds of seconds in oscillators with high quality factors Ghadimi et al. (2018), a cat state in this system can in fact be extremely long lived. This is shown in Fig. 9, where we selected a pure state of the quantum trajectory at a random time (once the evolution is stationary) that is very close to a cat state, let it evolve as a mixed state under the master equation, and computed the fidelity to a cat state with the same population as the initial state. This shows that the cat state can be maintained stable in this system with a fidelity for times  ms, during which it can be used as a resource for quantum applications Ralph et al. (2003); Lund et al. (2008); Joo et al. (2011); Facon et al. (2016); Albert et al. (2016).

Figure 9: Fidelity to a cat state versus time for an initial state chosen from a random time () in the quantum trajectory, once the steady-state limit is reached. The initial state resembles a cat with fidelity . The blue, dashed line corresponds to the expression for , which is valid for short times. For long times, the fidelity tends to . Simulation parameter, those of Figs. 3-4.
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