Steady-state negative Wigner functions of nonlinear nanomechanical oscillators

The manipulation of mechanical degrees of freedom by light scattering, at present known as optomechanics, has attracted increasing interest in recent years [1, 2]. Current research in this direction is largely driven by substantial progress in cooling mechanical oscillators to their ground states [3–8] and carrying out displacement measurements close to the standard quantum limit [9–11]. These developments set the stage for investigations of the quantum regime of macroscopic mechanical degrees of freedom, an endeavour that could lead to experimental tests of potential limitations of quantum mechanics [12]. Here, we introduce an optoelectromechanical technique for preparing a mechanical oscillator in a nonclassical steady state with a negative Wigner function by enhancing the anharmonicity of its motion.

As has been known since the initial formulation of quantum mechanics, the dynamics of a purely harmonic quantum system is hard to distinguish from its classical counterpart [13]. As an example, electrical circuits in the superconducting regime do not display quantum behaviour unless they feature a nonlinear element such as a Josephson junction. Along these lines, nanomechanical oscillators that are coupled to nonlinear ancilla systems have been considered [6, 14, 15]. Here, in contrast, we introduce a framework that only makes use of intrinsic properties of the mechanical oscillator. This approach allows one to prepare the latter in nonclassical states that are stationary states of its dissipative motion. The lifetime of these states is thus not limited by the lifetime of the oscillator's excitations.

In fact, doubly clamped mechanical resonators such as nanobeams naturally feature an intrinsic, geometric nonlinear contribution to their elastic energy that gives rise to a Duffing nonlinearity [16–19] and typically only becomes relevant for large deflection amplitudes. To investigate quantum effects of mechanical degrees of freedom [20–22] it is thus of key importance to devise means for enhancing the nonlinearity of nanomechanical structures.

Here we consider inhomogeneous electrostatic fields to decrease the harmonic oscillation frequency of a nanomechanical oscillator [23]. This procedure enhances the amplitude of its zero point motion and therefore leads to an amplification of its nonlinearity per phonon, which eventually becomes comparable to the optical linewidth of a high-finesse cavity. When combined with an optomechanical coupling to the cavity, this approach opens up a variety of possibilities for manipulating the mechanical oscillator by driving the cavity modes with lasers. In particular, the motional sidebands split into separated lines for each phonon number n, cf figure 1. This feature allows us to selectively transfer population from the phonon Fock state |n〉 to |n + 1〉 (|n'〉 to |n' − 1〉) by driving the spectral line corresponding to n (n') phonons in the blue (red) sideband.

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To demonstrate the capabilities of our approach, we show that suitably adjusted laser fields prepare the mechanical oscillator in a stationary state close to a phonon Fock state, i.e. in a state for which the quantum behaviour of an oscillator is most obvious [24]. The highly nonclassical nature of this state manifests itself in a Wigner function that reaches the lowest possible value of −2/π. This can be verified via the power spectrum of an optical probe field [25, 26]. Here, the stationary nature of the motional states considerably simplifies their measurement as compared to phonon Fock states that might be prepared by projective measurements [27–29].

Our scheme can, for example, be implemented with state of the art nanoelectromechanical systems (NEMS) comprising carbon-based mechanical resonators [30, 31] dispersively coupled to high-finesse toroidal [32] or fibre-based [33] microcavities (see figure 2). Carbon-based oscillators are particularly well suited as they combine very small transverse dimensions with ultra-low dissipation [34].

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The remainder of the paper is organized as follows. In section 2, we discuss the Hamiltonian of the oscillating nanobeam, including its geometric nonlinearity. In section 3, we then introduce the optomechanical coupling of the beam to a high-finesse cavity and present the master equation that describes the dynamics of our system including the relevant damping mechanisms. In section 4, we provide an analysis of how driven cavity modes can drive the nanomechanical resonator into a nonclassical steady state by adiabatically eliminating the photon degrees of freedom. In what follows, we analyse important aspects of an implementation in more detail. In section 5, we describe the electrostatic technique that lowers the harmonic oscillation frequency of the beam and hence increases its nonlinearity per phonon. Section 6 describes a possible feasible experimental setup including a carbon nanotube (CNT) coupled to a microtoroidal cavity. We then present examples of states that could be prepared in this setup in section 7. Finally, we discuss a technique to measure the prepared state (section 8), and end the paper with the conclusions and an outlook.

The dynamics of the fundamental flexural mode of a thin beam can be described by the Hamiltonian [35]

$$\begin{equation} H_{\mathrm{m}}=\frac{P^2}{2m_{*}}+\frac{1}{2}m_{*}{\omega_{\mathrm{m}}}^2X^2+\frac{\beta}{4}X^4, \end{equation} \tag{ 1 }$$

where X is the deflection, P its conjugate momentum, ω_m the mode frequency and m_* the effective mass. We assume clamped–clamped boundary conditions and apply thin beam theory [36]. This yields $\omega _{\mathrm {m},0}=c_{\mathrm {s}}\tilde {\kappa }(\frac {4.73}{L})^2$ for the mode frequency without applying elastic stress or other external forces, and $\beta = 0.060\,m_{*}\omega _{\mathrm {m},0}^{2}\tilde {\kappa }^{-2}$ for the nonlinearity which results from the stretching induced by the deflection. Here, L is the suspended length, c_s is the phase velocity of compressional phonons and $\tilde {\kappa }$ is given by the relevant transverse dimension of the beam times a geometric prefactor.

The single-mode model (1) is valid for sufficiently low phonon occupancies and weak nonlinearities such that phonon–phonon interactions involving higher harmonics can be neglected. Thus, we focus on the latter regime and write H_m in terms of phonon creation (annihilation) operators b^† (b) to obtain

$$\begin{equation} H_{\mathrm{m}} = \hbar {\omega_{\mathrm{m}}}{b^{\dagger}} b+\hbar\frac{\lambda}{2}({b^{\dagger}}+b)^4 \end{equation} \tag{ 2 }$$

with nonlinearity $\lambda = \frac {\beta }{2} x_{\mathrm {ZPM}}^{4}/\hbar$, where $x_{\mathrm {ZPM}}=\sqrt {\hbar /2m_{*}{\omega _{\mathrm {m}}}}$ is the amplitude of the oscillator's zero-point motion.

Note that λ∝ω_m⁻² can be significantly enhanced by 'softening' the mode, i.e. lowering ω_m. While this can be achieved by applying compressive stress along the beam [35, 37], we follow here an alternative approach to allow for better tunability. We consider an in-plane flexural resonance (cf figure 2(b)) and use tip electrodes to apply a strong inhomogeneous static electric field that polarizes the beam. As a dielectric body is attracted towards stronger fields, this adds an inverted parabola to the potential that counteracts the harmonic contribution due to elasticity. Thereby, as will be borne out in detail in section 5, the oscillation frequency is reduced [23, 38, 39].

We consider a typical optomechanical setup where the displacement X = x_ZPM(b + b^†) of the nanomechanical oscillator dispersively couples to several laser-driven cavity resonances. In a frame rotating with the frequencies of the laser fields ω_L,j the system Hamiltonian reads

$$\begin{equation} H=\hbar \sum_{j}\left[-\Delta_ja_j^{\dagger}a_j+\frac{\Omega_j}{2}\left(a_j^{\dagger}+a_j\right)+G_{0,j}{x_{\mathrm{ZPM}}} a_j^{\dagger}a_j\left({b^{\dagger}}+b\right)\right]+H_{\mathrm{m}}, \end{equation}\noindent \tag{ 3 }$$

where Δ_j = ω_L,j − ω_j is the detuning of a given laser j from the closest cavity mode with frequency ω_j and photon annihilation operator a_j, $\Omega _j / 2 = \sqrt {P_{\mathrm {in},j}\kappa _{\mathrm {ex}}/\hbar \omega _{\mathrm {L},j}}$ is the driving strength associated with that laser with input power P_in,j and κ_ex is the corresponding external decay rate. The coupling strength $G_{0,j}=\frac {\partial \omega _{j}}{\partial X}$ corresponds to the shift of the respective cavity resonance as per the oscillator's deflection [40].

We incorporate the damping of the cavity field at rate κ and mechanical dissipation at rate γ_m in a master equation and shift the cavity normal coordinates to their steady-state value a_j → a_j + α_j, with

$$\begin{equation} \alpha_j=\frac{\Omega_j}{2 \Delta_j + {\mathrm{i}} \kappa}. \end{equation} \tag{ 4 }$$

Neglecting higher-order terms in the photon–phonon coupling for the regime where 〈a^†_ja_j〉 ≪ |α_j|² [3, 4, 40] leads to the linearized Hamiltonian

$$\begin{equation} H'=\hbar\sum_j\left[-\Delta_ja_j^{\dagger}a_j+\left(\frac{g^{*}_{\mathrm{m},j}}{2}a_j+\mathrm{H.c.}\right)\left({b^{\dagger}}+b\right)\right]+H_{\mathrm{m}}, \end{equation} \tag{ 5 }$$

where g_m,j = 2α_jx_ZPMG_0,j is the enhanced optomechanical coupling. The term ∼|α|²(b^† + b), causing a static shift of the mechanical mode due to radiation pressure, is omitted here since it can be compensated for by a suitable choice of electrostatic softening fields (see section 5). The master equation for the shifted system now reads

$$\begin{equation} \dot\rho=-\frac{{\mathrm{i}}}{\hbar}\left[H',\rho\right]+\sum_j\frac{\kappa}{2}\mathcal{D}_{\mathrm{c},j}\rho+\frac{{\gamma_{\mathrm{m}}}}{2}\mathcal{D}_{\mathrm{m}}\rho, \end{equation} \tag{ 6 }$$

where

$$\begin{equation} \mathcal{D}_{\mathrm{c},j}\rho=2a_j\rho{a_j}^{\!\dagger}-{a_j}^{\!\dagger}a_j\rho-\rho{a_j}^{\!\dagger}a_j, \end{equation} \tag{ 7 }$$

$$\begin{equation} \mathcal{D}_{\mathrm{m}}\rho=(\overline n+1)\left\{2b\rho b^{\dagger}-b^{\dagger}b\rho-\rho b^{\dagger}b\right\}+\overline n\left\{2b^{\dagger}\rho b- bb^{\dagger}\rho-\rho bb^ {\dagger}\right\}, \end{equation} \tag{ 8 }$$

with the Bose number $\overline n=1/[\exp (\hbar {\omega _{\mathrm {m}}}/k_{\mathrm {B}}T)-1]$ at the environment temperature T. In equation (8), we have applied a rotating wave approximation (RWA) that is justified for the high mechanical quality factors we are interested in [40].

Equation (6) describes the dynamics of our system. However, the behaviour of the mechanical mode becomes more apparent after adiabatically eliminating the photon degrees of freedom, as we do in the next section.

For |g_m,j| ≪ κ, we now derive a reduced master equation (RME) for the mechanical motion by adiabatically eliminating the cavity degrees of freedom [40]. We start by applying an RWA to the mechanical Hamiltonian which yields

$$\begin{equation} H_{\mathrm{m}}'= \hbar{\omega'_{\mathrm{m}}}{b^{\dagger}} b+\hbar\frac{\lambda'}{2}{b^{\dagger}}{b^{\dagger}} bb, \end{equation} \tag{ 9 }$$

with the shifted frequency ω_m' = ω_m + λ' and the nonlinearity per phonon λ' = 6λ. Here, the RWA is warranted provided the relevant phonon numbers n fulfil n²λ' ≪ 6ω_m. Within the RWA, the mechanical energy eigenstates are Fock states and the large nonlinearity, $\lambda ' \gg |{g_{\mathrm {m},j}}|^{2}/\kappa , \overline {n} {\gamma _{\mathrm {m}}}$, we are interested in naturally leads to Fock number resolved dynamics. In an interaction picture with respect to H_m' transition operators between neighbouring phonon Fock states rotate at frequencies corresponding to the respective mechanical energy level spacings, as

$$\begin{equation} {\mathrm{e}^{\mathrm{i}H_{\mathrm{m}}' t/\hbar}} b \,{\mathrm{e}^{-\mathrm{i}H_{\mathrm{m}}' t/\hbar}} = \sum_n{\mathrm{e}^{-\mathrm{i}\delta_{n}t}}b_n, \end{equation} \tag{ 10 }$$

with $b_n=\sqrt {n}|n-1\rangle \langle n|$

and

$$\begin{equation} \delta_n=(E_n-E_{n-1})/\hbar\approx{\omega'_{\mathrm{m}}}+\lambda'(n-1). \end{equation} \tag{ 11 }$$

We further focus on the resolved sideband regime for the softened frequency κ ≪ ω_m, which is suitable for efficient back action cooling, as well as $\overline {n}{\gamma _{\mathrm {m}}}\ll |{g_{\mathrm {m},j}}|^2/\kappa$, to ensure that the influence of the cavity modes on the mechanical motion overwhelms environmental heating. The resulting RME for the reduced density operator of the nanomechanical resonator, $\mu =\mathrm {Tr}_{\mathrm {c}}\left \{\rho \right \}$, reads

$$\begin{equation} \dot\mu=-\frac{\mathrm{i}}{\hbar}\left[H_{\mathrm{m}}'+\hbar\sum_{n,j}\Delta_{\mathrm{m},j}^{(n)}b_n^{\dagger}b_n,\mu\right] +\sum_{n,j} \sum_{\sigma=+,-} \frac{A_{\sigma,j}^n}{2} \, \mathcal{D}_{\sigma}^n\mu + \frac{{\gamma_{\mathrm{m}}}}{2}\mathcal{D}_{\mathrm{m}}\mu, \end{equation} \tag{ 12 }$$

where the driven cavity modes act as an additional bath for each number transition with damping terms

$$\begin{equation} \mathcal{D}_+^n\mu=2b_n^{\dagger}\mu b_n-b_nb_n^{\dagger}\mu-\mu b_nb_n^{\dagger}, \end{equation} \tag{ 13 }$$

$$\begin{equation} \mathcal{D}_-^n\mu=2b_n\mu b_n^{\dagger}-b_n^{\dagger}b_n\mu-\mu b_n^{\dagger} b_n. \end{equation} \tag{ 14 }$$

Note that the frequency shifts

$$\begin{equation} \Delta^{(n)}_{\mathrm{m},j}=g^{2}_{\mathrm{m},j}\left\{\frac{\Delta_j+\delta_n}{4(\Delta_j+\delta_n)^2+\kappa_j^2}+\frac{\Delta_j-\delta_n}{4(\Delta_j-\delta_n)^2+\kappa_j^2}\right\} \end{equation} \tag{ 15 }$$

are negligible compared to ω_m',λ'. The cavity-induced transition rates between neighbouring Fock states

$$\begin{equation} A_{\pm,j}^n=\frac{|{g_{\mathrm{m},j}}|^2 \kappa}{4(\Delta_j\mp\delta_n)^2+\kappa^2}, \end{equation} \tag{ 16 }$$

however, can exceed the mechanical damping $A_{\pm ,j}^n>{\gamma _{\mathrm {m}}} \overline n$, which leads to a significant impact on the mechanical steady state.

The steady state of the RME (12) is diagonal in the Fock basis,

$$\begin{equation} \mu=\sum_nP_n{|n\rangle}{\langle n|}. \end{equation} \tag{ 17 }$$

Hence the relations

$$\begin{equation} \frac{P_n}{P_{n-1}}=\frac{\sum_jA_{+,j}^n+{\gamma_{\mathrm{m}}}\overline{n}}{\sum_jA_{-,j}^n+{\gamma_{\mathrm{m}}}\left[\overline{n}+1\right]},\quad\sum_nP_n=1, \end{equation} \tag{ 18 }$$

determine all occupation probabilities P_n and therefore the mechanical state μ. The relations (18) show the central feature of the driven nonlinear nanoresonator. Here $|{g_{\mathrm {m},j}}|^2/\kappa \gg {\gamma _{\mathrm {m}}}\overline {n}$ implies $P_n/P_{n-1} \approx (\sum _jA_{+,j}^n)/(\sum _jA_{-,j}^n)$. For a detuning Δ_j = ± δ_n we find that Aⁿ_±,j > Aⁿ_∓,j and the corresponding laser drive therefore enhances (decreases) P_n/P_n−1.

This feature can be used to approach phonon Fock states |n〉, as illustrated in figure 1 for n = 1. Fock states are highly nonclassical and for n odd they exhibit a maximally negative Wigner function at the origin. Ensuring that each laser drive preferentially addresses only one transition |n〉 → |n ± 1〉 naturally requires λ'⪆κ. This can be achieved by electrostatic mode softening as we describe in the next section.

The harmonic frequency ω_m of the nanoresonator can be reduced by adding an electrostatic potential V_es∝ − X² to the elastic potential, where X is the deflection of the resonator. This is achieved by placing charged electrodes at both sides of the nanoresonator (see figure 2), which generate a strongly inhomogeneous electrostatic field. To estimate the potential that softens the beam, we consider its electrostatic energy line density

$$\begin{equation} \mathcal{W}(x,y)=-\textstyle\frac{1}{2}[\alpha_{\parallel}E^2_{\parallel}(x,y)+\alpha_{\perp}E^2_{\perp}(x,y)]. \end{equation} \tag{ 19 }$$

Here, y and x are the co-ordinates along the beam and the direction of its deflection, E_||,⊥ are the electrostatic field components parallel and perpendicular to the beam and α_||,⊥ the respective screened polarizabilities. The electrostatic energy as a functional of the deflection x(y) thus reads

$$\begin{equation} V_{\mathrm{es}}=\int_0^L\mathcal{W}(x(y),y)\,{\rm d}y. \end{equation} \tag{ 20 }$$

We expand $\mathcal {W}(x(y),y)$ to second order in the transverse deflection x(y) from its equilibrium configuration x(y) = 0. The leading contributions for the fundamental mode with profile ϕ₀(y) can be found by considering x(y) = ϕ₀(y)X. They read

$$\begin{equation} V_{\mathrm{es}} \approx V_{\mathrm{es},0} + V_{\mathrm{es},1} X + V_{\mathrm{es},2} X^{2}, \end{equation} \tag{ 21 }$$

where V_es,0 is an irrelevant constant and

$$\begin{equation} V_{\mathrm{es},1}=\int_{0}^{L}{\rm d}y\frac{\partial\mathcal{W}(x,y)}{\partial x}\big |_{x=0}\phi_0(y), \end{equation} \tag{ 22 }$$

$$\begin{equation} V_{\mathrm{es},2}=\frac{1}{2}\int_{0}^{L}{\rm d}y\frac{\partial^{2}\mathcal{W}(x,y)}{\partial x^{2}} \big|_{x=0}\phi^{2}_0(y). \end{equation} \tag{ 23 }$$

The linear term V_es,1X shifts the beam's equilibrium position and can be employed to compensate for shifts induced by the dispersive coupling to the cavity fields by satisfying $V_{\mathrm {es},1}=-\hbar \sum _jG_0|\alpha _j|^2$. The quadratic term V_es,2X² in turn is negative and reduces the fundamental frequency as

$$\begin{equation} \omega^{2}_{\mathrm{m}} = \omega_{\mathrm{m},0}^{2} - \frac{2}{m_{*}} |V_{\mathrm{es},2}|. \end{equation} \tag{ 24 }$$

At $|V_{\mathrm {es},2}| = \frac {m_{*}}{2}\omega _{\mathrm {m},0}^{2}$, the buckling instability [35, 37] occurs and by tuning the electrostatic fields below it, one can reduce ω_m by any desired factor ζ = ω_m,0/ω_m. Note that the softening increases the zero point motion amplitude and, hence, also the optomechanical coupling per photon G₀x_ZPM. A particularly useful feature of this dielectric approach to soften the nanobeam is that the electrostatic potential and hence ω_m, G₀x_ZPM and λ can be tuned in situ. To illustrate the feasibility of our approach, we now turn to the description of a feasible experimental setup for its implementation.

As a suitable device for realizing our scheme, we envisage a setup with a single-walled CNT as the mechanical oscillator (c_s = 21 000 ms⁻¹) that interacts with the evanescent field of a whispering gallery mode of a microtoroid cavity [32], cf figure 2. A CNT is a favourable candidate since without softening (ω_m = ω_m,0) one obtains $\lambda \propto (m_{*} \tilde {\kappa }^2)^{-1}$, indicating that a low effective mass m_* and a small transverse dimension $\propto \tilde {\kappa }$ are desirable. For a CNT of radius R, one has $\tilde {\kappa }=R/\sqrt {2}$. We consider a wide-band-gap semiconducting CNT that for the infrared laser wavelengths involved will behave as a dielectric [41]. Taking into account that the transverse optical polarizability is negligible (i.e. depolarization) and approximating the longitudinal one by its static value, the optomechanical coupling can be estimated as (cf [32])

$$\begin{equation} G_{0,j}\sim\omega_j \frac{\alpha_{\parallel} L}{\epsilon_0V_{\mathrm{c}}} \xi^2\kappa_{\perp}{\mathrm{e}}^{-2\kappa_{\perp}d}\, C, \end{equation} \tag{ 25 }$$

where V_c = πa²_cL_c is the mode volume of the whispering gallery mode with cavity length L_c and effective mode waist 2a_c. ξ is the ratio of the field at the cavity surface to the maximum field inside, d the distance between the chip carrying the nanotube and the microtoroid's surface and κ⁻¹_⊥ the decay length of the evanescent field. The correction factor C depends on the type of whispering gallery mode and the placement of the CNT. We assume for simplicity that the NEMS and photonic chips are perpendicular to each other and find for a transverse electric (TE) mode and optimal placement $C\approx 0.17/\sqrt {\kappa _{\perp }(d+a_{\mathrm {c}})}$. This optimum involves a small displacement of the CNT midpoint from the equatorial plane of the toroid to ensure coupling to the in-plane motion and an angle of ∼50° between the CNT axis and the equatorial plane of the toroid to circumvent depolarization.

In this setup, the tip electrodes used to lower the mechanical oscillation frequency need to be positioned close to the cavity surface to ensure a sufficiently inhomogeneous electrostatic field while maintaining the optomechanical coupling high enough. Hence, a concomitant degradation of the cavity's finesse due to photon scattering or absorption at the tip electrodes is a natural concern. We find that cavity losses are minimized if one uses TE whispering gallery modes and tip electrodes of sub-wavelength diameter d_el that are aligned approximately parallel to the equatorial plane of the toroid (cf figures 2(a) and (b)). For this arrangement the dominant loss mechanism is absorption. The ratio of absorbed power P_a to power circulating in the cavity P_c can be estimated to be (cf [42], chapter 8)

$$\begin{equation} \frac{P_{\mathrm{a}}}{P_{\mathrm{c}}} \lesssim \frac{\pi \sigma d_{\mathrm{el}} \xi^{2}}{c \varepsilon_{0} a_{\mathrm{c}}} \sqrt{\frac{\pi}{\kappa_{\perp} a_{\mathrm{c}}}} \mathrm{e}^{-2 k_{\perp} d} \sin \theta, \end{equation} \tag{ 26 }$$

where σ is the 2D optical conductivity of the electrodes and θ their misalignment with respect to the equatorial plane of the toroid. The resulting finesse is given by

$$\begin{equation} \mathcal{F} = \left(\frac{1}{\mathcal{F}_{\mathrm{c}}} + \frac{P_{\mathrm{a}}}{2 P_{\mathrm{c}}}\right)^{-1}, \end{equation} \tag{ 27 }$$

where $\mathcal {F}_{\mathrm {c}}$ is the bare cavity finesse. For relevant cavity parameters we find $\frac {P_{\mathrm {a}}}{2 P_{\mathrm {c}}} \ll \mathcal {F}_{\mathrm {c}}^{-1}$ if we assume θ ⩽ 1°, d_el ⩽ 10 nm and $\sigma \lesssim 2 \times 10^{-5}\,\Omega ^{-1}$. Metallic CNTs, for example, fulfil the latter two requirements provided their resonances do not match the relevant cavity frequencies ω_j [43].

In the following, we now give some examples of steady states for the mechanical oscillator that can be generated in the setup described here.

To illustrate the potential of our scheme, we show in figure 3 the results for a (10,0) CNT characterized by R = 0.39 nm, α_∥ = 142(4π₀ Å²) and α_⊥ = 10.9(4π₀ Å²) [44]. The nanotube has a length of L = 1.0 μm and is coupled to whispering gallery modes with wavelengths 2πc/ω_j ≈ 1.1 μm, a_c = 1.4 μm and ξ = 0.4 of a silica microtoroid with finesse $\mathcal {F}_{\mathrm {c}}= 3{\times }10^6$ [45] and circumference L_c = 1.35 mm. The applied voltages are chosen so that tip electrodes at a distance of 20 nm from the tube induce maximal fields at its axis of E_∥ ≈ 1.20 × 10⁷ V m⁻¹ and E_⊥ ≈ 1.778 × 10⁶ V m⁻¹ and lower its mechanical frequency by a factor ζ = 4.0. For the optomechanical couplings, we consider g_m,j/2π = 21.0 kHz for all j which can be achieved with d = 50 nm, κ_ex/κ = 0.1 and a launched power P_in,j = 1.2 W. The above parameters lead to ω_m/2π = 5.23 MHz, λ/2π = 209 kHz and κ/2π = 52.3 kHz. We assume CNT Q-values of Q_m = 5 × 10⁶ and Q_m = 5 × 10^5 2 and an environmental temperature T = 20 mK.

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For these parameters and for laser detunings Δ₁ ≈ δ₁ and Δ_2,3 ≈ −δ_2,3 a single phonon Fock state can be prepared with 91% fidelity (for Q_m = 5 × 10⁶), as shown in figure 3. This steady state of the mechanical oscillator has a Wigner function that is clearly negative at the origin, W(0,0) = −0.53 for Q_m = 5 × 10⁶ and W(0,0) = −0.30 for Q_m = 5 × 10⁵.

The results shown in figure 3 have been obtained from a numerical solution of equation (6) with numerically optimized detunings Δ_j. These numerical results do not rely on the rotating wave approximation used in equation (9). Here the fast rotating terms lead to corrections to the energy level spacings δ_n that can be comparable to κ. Hence to match the laser detunings with sufficient accuracy to the energy level spacings δ_n, the latter have been determined numerically.

In turn, for the more moderate parameters: L = 1.7 μm, $\mathcal {F}_{\mathrm {c}}=2\times 10^6$, L_c = 1.80 mm (κ/2π = 64.2 kHz), ζ = 3.3 (which imply ω_m/2π = 2.13 MHz and λ/2π = 85.5 kHz), launched laser powers P_in,1/2 = 22 mW and P_in,3 = 44 mW, Q_m = 1.5 × 10⁶ and T = 30 mK, we still find a significant negative peak of depth W_min = −0.15 (see figure 4).

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We note that the deep resolved sideband and undercoupled conditions used here, where the frequencies of the driving lasers are detuned by several cavity linewidths from the optical resonances of the cavity and losses into the coupling fibre are only a small fraction of the total losses, imply that the power absorbed inside the cavity is much smaller than the launched power (below 30 μW in the example of figure 3 and below 7 μW in the example of figure 4). Whereas an intracavity absorption ∼1 μW has already been shown to be compatible with a cryogenic environment [46], even higher finesses and lower absorption could be attained by using crystalline resonators [47]. Additionally, the CNT polarizabilities could be enhanced well above the static values we have assumed by profiting from the corresponding excitonic optical resonances [48], allowing for a substantial reduction in the laser powers required.

Furthermore, in both examples, the voltages applied to soften the frequency of the CNT nanomechanical resonator generate electric fields at the CNT that are well below its threshold for dielectric breakdown and are comparable to fields applied in recent experiments [23].

Finally, we note that our approach can be generalized to the preparation of Fock states with n > 1.

To verify whether a nonclassical steady state with a negative Wigner function W has been successfully prepared, one can measure sidebands of the output power spectrum S(ω) of an additional cavity mode weakly driven by a probe laser with frequency ω_L. Within our quantum noise approach [40] and under the same conditions for which equation (12) holds, we find for the sidebands (i.e. where ω ≠ ω_L) (figure 5)

$$\begin{equation} S(\omega)\propto\sum_n \left[\frac{n \, \Gamma_n \, A_-^n \, P_{n}}{\left(\omega-\omega_{\mathrm{L}}-\delta_n\right)^2+\frac{\Gamma_n^{2}}{4}} + \frac{n \, \Gamma_n \, A_+^n \, P_{n-1}}{\left(\omega-\omega_{\mathrm{L}}+\delta_n\right)^2+\frac{\Gamma_n^{2}}{4}}\right], \end{equation} \tag{ 28 }$$

with sideband linewidths

$$\begin{equation} \fl \Gamma_{n}=n [ A_-^n + A_+^n + {\gamma_{\mathrm{m}}} (2\overline{n}+1)]+(n-1)[A_-^{n-1} + {\gamma_{\mathrm{m}}} (\overline{n}+1)]+(n+1) [A_+^{n+1}+{\gamma_{\mathrm{m}}}\overline{n}]. \end{equation} \tag{ 29 }$$

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Since the negativity of the Wigner function is in our scheme expected to be maximal at the origin where $W(0,0) = \frac {2}{\pi } \sum _{n} (-1)^{n} P_{n}$, the nonclassicality of the state can be read off from S(ω). For a probe laser that is resonant with a cavity mode, one has Aⁿ₊ = Aⁿ₋ and the ratios of peak intensities S(ω_L + δ_n)/S(ω_L − δ_n) ≈ P_n/P_n−1 together with $1 = \sum_{n}\!P_{n}$ allow us to determine the P_n assuming that occupations above a certain Fock number are negligible.

We have shown that a dielectrically enhanced nonlinearity in nanomechanical oscillators enables controlling their motion at a Fock state resolved level. The scheme set forth opens new avenues for optomechanical manipulations of nanobeams, in particular by exploiting the in situ tunability of the electrostatic softening approach introduced.

This work is part of the DFG Emmy Noether project HA 5593/1-1 and was supported by Nanosystems Initiative Munich. The authors thank J Kotthaus, T J Kippenberg and A Schliesser for useful discussions.