# Mechanical Signatures of the Current Blockade Instability in Suspended Carbon Nanotubes

G. Micchi, R. Avriller, F. Pistolesi

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G. Micchi, R. Avriller, F. Pistolesi. Mechanical Signatures of the Current Blockade Instability in Suspended Carbon Nanotubes. Physical Review Letters, 2015, 115 (20), pp.206802 (1-5). 10.1103/Phys-RevLett.115.206802. hal-01178253

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Supplemental material to the paper : “Signatures of the current blockade instability in suspended carbon nanotubes”

G. Micchi, R. Avriller, and F. Pistolesi¹

¹Univ. Bordeaux, LOMA, UMR 5798, F-33400 Talence, France
CNRS, LOMA, UMR 5798, F-33400 Talence, France

(Dated: July 17, 2015)

We provide details about the analytical derivations of the displacement spectral density $S_{xx}(\omega)$ and dephasing time $\gamma_{\phi}^{-1}$ of the carbon nanotube mechanical oscillator.

# I. EVALUATION OF THE DISPLACEMENT SPECTRAL DENSITY $S_{xx}(\omega)$

In this section, we discuss how to obtain analytically the displacement spectral density $S_{xx}(\omega)$ of the carbon nanotube mechanical resonator. The effective potential $U(x)$ for the mechanical oscillator at low voltage can be expanded in Taylor series, and only the two first orders, the quadratic and quartic in $x$ are relevant for effective temperatures $T_{eff} \ll \Gamma$ (in the following, we set both the Planck constant $\hbar$ and the Boltzmann constant $k_B$ to 1). We thus derive in the following analytical expressions for $S_{xx}(\omega)$ for the two limiting cases (i) where the quartic term is a small perturbation to the quadratic one, and (ii) when the quadratic term vanishes (for $\epsilon_P = \epsilon_c$) and the potential is purely quartic in $x$.

Following Dykman et al. in Ref. [1], in the regime of very small damping rate $A/m \ll \Delta\omega$, with $\Delta\omega$ the width of the main peak in the spectral density induced by the non-linearity, the autocorrelation function for the oscillator position can be well approximated by:

$$S_{xx}(t) = \int dx_0 dy_0 P_{\text{st}}(x_0, p_0) \tilde{x}_{x_0 p_0}(t) \tilde{x}_{x_0 p_0}(0) \quad (1)$$

where $x(t)$ is the periodic function that satisfies the equation of motion $m\ddot{x} = F(x)$ with initial conditions $x(0) = x_0$, and $\dot{x}(0) = p_0/m$ and $\tilde{x}_{x_0 p_0}(t) = x_{x_0 p_0}(t) - \langle x_{x_0 p_0}(t) \rangle$. The integration is taken over the whole phase space for the classical oscillator and $P_{\text{st}}(x, p)$ is the stationary distribution that, as discussed in the main text, for $eV \ll \Gamma$ has the Gibbs form $P_{\text{st}}(x, p) = N e^{-E(x, p)/T_{\text{eff}}}$, with $E(x, p) = p^2/2m + U(x)$, and $N$ is such that the distribution is normalized $\int dx dp P_{\text{st}}(x, p) = 1$. The effective temperature $T_{\text{eff}} = eV/4$ is induced by the charge fluctuations on the dot due to the finite voltage across the leads. The damping $A/m$ and the fluctuation $D$ enter this expression in defining the form of $P_{\text{st}}$, but not in the dynamics of $x(t)$.

The expression (1) can be rewritten as follows:

$$\int_{0}^{+\infty} dE \int_{0}^{T(E)} d\tau P_{\text{st}}(E) \tilde{x}_{E}(t + \tau) \tilde{x}_{E}(\tau), \quad (2)$$

where $T(E)$ is the period of the closed trajectory $x_E(t)$ of fixed energy $E$ and $P_{\text{st}}(E) = N e^{-E/T_{\text{eff}}}$. Since $x(t)$ is periodic with a period $T(E) = 2\pi/\omega(E)$, we can introduce

its Fourier series:

$$\tilde{x}_E(t) = \sum_n e^{in\omega(E)t} x_n(E). \quad (3)$$

The spectrum then takes the form:

$$S_{xx}(\omega) = \int_{0}^{+\infty} dE P(E) \sum_{n} 2\pi\delta(\omega - n\omega(E))x_{n}^{2}(E), \quad (4)$$

with $P(E) = P_{\text{st}}(E)T(E)$. Finally, introducing the energies $E_n$ which satisfy the equation $\omega = n\omega(E_n)$, we obtain the expression for the spectral function (with $\omega > 0$) as:

$$S_{xx}(\omega) = \mathcal{N} \frac{(2\pi)^2}{\omega} \sum_{n=0}^{+\infty} \frac{e^{-E_n/T_{\text{eff}}}}{|\omega'(E_n)|} x_n^2(E), \quad (5)$$

with $\omega'(E_n) = (d\omega/dE)|_{E_n}$. The computation of $S_{xx}$ is now reduced to the computation of $\omega(E)$ and $x_n^2(E)$.

# A. Computation of $\omega(E)$

The expression for the effective force $F$ is given in the main text for the case $\epsilon_0 = (\mu_L + \mu_R + \epsilon_P)/2$:

$$F(y) = -ky + (F_0/2\pi) \sum_{a=\pm 1} \arctan[(F_0y + aeV/2)/\Gamma], \quad (6)$$

where $y = x - F_0/2k$. For $\epsilon_P < \epsilon_c$ there is only a single stable equilibrium position for $y = 0$. Expanding $F(y)$ around this point for $V = 0$ we obtain for the potential $U(y) = -\int_0^y dy' F(y')$:

$$U(y) = \frac{m\omega_0^2}{2} \left(1 - \frac{\epsilon_P}{\epsilon_c}\right) y^2 + \frac{\Gamma}{12\pi} \left(\frac{yF_0}{\Gamma}\right)^4, \quad (7)$$

where $\epsilon_c = \pi\Gamma$. The period in terms of $U$ reads:

$$T(E) = (m/2)^{1/2} \oint [E - U(y)]^{-1/2} dy. \quad (8)$$

that can be evaluated in terms of elliptic integrals:

$$\frac{\omega(E)}{\omega_0} = \frac{\pi}{2K[-m(E)]} \left(1 - \frac{\epsilon_P}{\epsilon_c}\right)^{1/2} \left(\frac{1 + C(E)}{2}\right)^{1/2}, \quad (9)$$
# Signatures of the Current Blockade Instability in Suspended Carbon Nanotubes

G. Micchi,¹ R. Avriller,¹ and F. Pistolesi¹

¹Univ. Bordeaux, LOMA, UMR 5798, Talence, France.

CNRS, LOMA, UMR 5798, F-33400 Talence, France.

(Dated: July 17, 2015)

Transport measurements allow sensitive detection of nanomechanical motion of suspended carbon nanotubes. It has been predicted that when the electro-mechanical coupling is sufficiently large a bistability with a current blockade appears. Unambiguous observation of this transition by current measurements may be difficult. Instead, we investigate the mechanical response of the system, namely the displacement spectral function; the linear response to a driving; and the ring-down behavior. We find that by increasing the electro-mechanical coupling the peak in the spectral function broadens and shifts at low frequencies while the oscillator dephasing time shortens. These effects are maximum at the transition where non-linearities dominate the dynamics. These strong signatures open the way to detect the blockade transition in devices currently studied by several groups.

PACS numbers: 73.23.Hk, 73.63-b, 85.85.+j

Recently enormous progress has been achieved in the detection of carbon nanotubes (CNT) bending modes by electronic transport measurements [1–13]. Since nanotube oscillators have remarkable mechanical properties, devices with record mass [7] and force [10] sensitivity have been realized. Transport experiments allow information to be obtained on the mechanical mode by measuring different quantities. The main ones are the oscillation amplitude in response to an external drive [1], the oscillator displacement spectral density, $S_{xx}(\omega)$ [10], and, more recently, the ring-down time of the oscillator [13].

The recent experimental advances allow one to view the behavior of such systems in the strong coupling limit from a new perspective. Defining $F_0$ as the difference of electrostatic force acting on the nanotube when one electron is added to the suspended part and $k$ as the spring constant for its displacement, one can introduce a polaronic energy scale $\epsilon_P = F_0^2/k$. For a classical system resonating at pulsation $\omega_0$ (much smaller than the bias voltage eV or the temperature T) it has been predicted [14–17] that, if $\epsilon_P \gg eV$ and T, the current in the device can be blocked and a bistability can appear (e is the electron charge and we set both the Planck and Boltzmann constant to 1). The energy $\epsilon_P$ can be estimated for actual experiments: For a CNT of 1 µm length, 1 nm radius, and suspended at a distance $d = 500$ nm from a gate one finds $\omega_0/2\pi \approx 50$ MHz, $F_0 = 10^{-14}$ N, $k = 4 \cdot 10^{-4}$ N/m, and thus $\epsilon_P \approx 16$ mK. One should thus work at very low temperatures $\ll 16$ mK: This is probably why the current blockade transition has not yet been observed for mechanical bending modes. While a (Franck-Condon) blockade [18, 19] has been observed [20] for breathing modes in the regime of incoherent transport. To increase $\epsilon_P$ one can reduce the distance of the CNT from the gate electrode or operate the system close to the Euler buckling instability [21, 22]. The energy $\epsilon_P$ scales quadratically with $d$, it is thus realistic to increase this energy up to the Kelvin range by reducing the distance $d$ to 100 nm. In any case, a clear observation of the transition

will require temperatures of the order 100 mK. At such low temperatures the typical tunnelling rate $\Gamma$ becomes larger than $T$, leading to coherent transport through the CNT. In this regime the current blockade can take place only if $\epsilon_P$ is larger than a critical value $\epsilon_c$ of the order of $\Gamma$ [14, 15]. Since one of the main experimental difficulties is to reach large values of $\epsilon_P$, the case $\epsilon_P \sim \Gamma$ is particularly interesting. The transition could then be investigated at fixed and low V and T by varying $\epsilon_P$, that can be tuned with the gate voltage. One drawback of this limit is the large width $\Gamma$ of the electronic level: The conductance dependence on $\epsilon_P$ is smooth on the scale $\Gamma$ failing to provide a clear indication of the transition.

In this Letter we show that, similar to critical phenomena, the transition can be better investigated by looking at the behavior of the phonon mode that becomes soft for $\epsilon_P = \epsilon_c$. We study $S_{xx}(\omega)$, the driving response function, and the dephasing and ring-down time as a function of the coupling constant $\epsilon_P$. We find that all of these quantities have a very peculiar behavior at the transition. The dynamics of the mechanical mode is dominated by nonlinear terms leading to a separation of time scales that is maximal at the transition. The theory presented gives clear indication on how to unambiguously observe the transition using available methods of measurement.

**The model.** We consider a suspended CNT. We assume that a single electronic level is relevant for transport. We neglect the spin degrees of freedom. The Hamiltonian reads:

$$H = H_L + H_R + H_T + (\epsilon_0 - xF_0)d^\dagger d + \frac{p^2}{2m} + \frac{k}{2}x^2, \quad (1)$$

where $d$ is the destruction operator for the electronic level on the dot, $x$ is the displacement of the relevant mechanical mode, $p$ the conjugated momentum, $m$ the mode effective mass, $k$ the spring constant (giving a pulsation $\omega_0 = \sqrt{k/m}$) and $F_0$ the electrostatic force acting on the dot when an electron is added. The first three terms describe the leads and their coupling:
with

$$C(E) = \left( 1 + \frac{4\pi^2}{3} \frac{\varepsilon_P^2 E}{\varepsilon_c^3 (1 - \varepsilon_P / \varepsilon_c)^2} \right)^{1/2}, \quad (10)$$

and $m(E) = \frac{C(E)-1}{C(E)+1}$. $K[-m(E)]$ is the complete elliptic integral of the first kind with parameter $-m(E)$ [see Ref. [2]]. We will need in the following the derivative of $\omega(E)$ for vanishing energy:

$$\omega'(0) \equiv \left. \frac{d\omega}{dE} \right|_{E=0} = \frac{\pi^2 \omega_0}{4 \varepsilon_c} \left( \frac{\varepsilon_P}{\varepsilon_c} \right)^2 \left( 1 - \frac{\varepsilon_P}{\varepsilon_c} \right)^{-3/2} . \quad (11)$$

## B. Case (i) weakly non linear oscillator

<figure><img src="image_1.png" /><figcaption>FIG. 1. Normalized displacement spectral density $S_{xx}(\omega)/S_{xx}^{max}$ as a function of the rescaled frequency $(\omega - \omega_m)/T_{eff}\omega'(0)$, in the case of weak electro-mechanical coupling $(\varepsilon_P \ll \varepsilon_c)$.</figcaption></figure>

Sufficiently far from the transition $(\varepsilon_P \ll \varepsilon_c)$, the oscillator is weakly non-linear and we can treat the quartic part of the potential as a small perturbation. We can thus calculate $S_{xx}$ by expanding the expression (5) to leading order in the non-linearity. The main contribution comes from the first harmonic whose amplitude can be approximated with the harmonic expression $x_1^2(E) \approx E/2m\omega_0^2$. The energy dependent resonating frequency is approximated by the expression:

$$\omega(E) = \omega_m + \omega'(0)E + \dots \quad (12)$$

where $\omega_m/\omega_0 = (1 - \varepsilon_P/\varepsilon_c)^{1/2}$. We obtain then:

$$S_{xx}(\omega) = \frac{\pi \omega_m \varepsilon_c}{m \omega_0^2 (\varepsilon_c - \varepsilon_P) \omega |\omega'(0)|} \frac{\omega - \omega_m}{\omega'(0) T_{\text{eff}}} e^{-\frac{\omega - \omega_m}{\omega'(0) T_{\text{eff}}} \omega} . \quad (13)$$

From Eq. (13) and Fig. 1 we see that the spectral density is defined for $\omega > \omega_m$ (there is actually an upper bound of the order of $\omega_0$, but the effective temperature being very low this limit is not visible) and has a maximum at

$\omega = \omega_m + \omega'(0)T_{eff}$. The full width at half maximum (FWHM) of the spectral line $\Delta\omega$ reads

$$\Delta\omega = \Delta_2\omega'(0)T_{\text{eff}}, \quad (14)$$

where $\Delta_2 \approx 2.446$ is a numerical coefficient corresponding to the FWHM in Fig. 1.

We thus find that the effect of the non-linearity on the width of the resonance is linear with the bias voltage $eV$ and controlled by $\omega'(0)$, at least as far as the quartic terms does not become dominant. We will estimate at the end of the next subsection the limit of validity of this approach by comparing with the result of the purely quartic term.

## C. Case (ii) purely quartic oscillator

<figure><img src="image_2.png" /><figcaption>FIG. 2. Normalized displacement spectral density $S_{xx}(\omega)/S_{xx}^{max}$ as a function of the rescaled frequency $\omega/\omega_{max}$, at the critical point $(\varepsilon_P = \varepsilon_c)$. The spectral line maximum is located at frequency $\omega_{max} = B\omega_0 (eV/4\Gamma)^{1/4}$.</figcaption></figure>

At the critical point $(\varepsilon_p = \varepsilon_c)$, the quadratic part of the potential vanishes and the mechanical oscillator becomes purely quartic. In this regime, we obtain for the oscillator frequency:

$$\frac{\omega(E)}{\omega_0} = B \left(\frac{E}{\Gamma}\right)^{1/4}, \quad (15)$$

with $B = \frac{1}{2} \left(\frac{\pi^3}{3}\right)^{1/4} \frac{\Gamma[3/4]}{\Gamma[5/4]} \approx 1.212$ a numerical constant and $\Gamma[x]$ the Euler gamma function [2]. The oscillator frequency has thus a scaling in energy $\propto (E/\Gamma)^{1/4}$.

Remarkably, the displacement spectral density of the quartic oscillator also has a simple analytical expression:

$$S_{xx}(\omega) = \tilde{B} \frac{\Gamma^2}{\omega_0 F_0^2} \left(\frac{\Gamma}{T_{\text{eff}}}\right)^{3/4 + \infty} \sum_{n=1}^{+\infty} e^{-E_n/T_{\text{eff}}} \frac{\alpha_n^2 E_n}{n\Gamma}, \quad (16)$$
$H_α = \sum_k (\epsilon_{\alpha k} - \mu_\alpha) c_{\alpha k}^\dagger c_{\alpha k}$, with $\alpha = L$ and $R$, for left and right lead, $\epsilon_{\alpha k}$ the electronic spectrum, $\mu_\alpha$ the chemical potential; and $H_T = \sum_k t_\alpha c_{\alpha k}^\dagger d + \text{h.c.}$ the tunnelling Hamiltonian. From these quantities one can define the single-level width $\Gamma_\alpha \equiv \pi t_\alpha^2 \rho_\alpha$ with $\rho_\alpha$ the density of states and $\Gamma = \Gamma_L + \Gamma_R$.

In the Born-Oppenheimer limit ($\Gamma_\alpha \gg \omega_0$) the displacement of the mechanical mode can be described by a Langevin equation:

$$m\ddot{x} + A(x)\dot{x} + m\omega_0^2 x = F_e(x) + \xi(t), \quad (2)$$

where the dissipation $A(x)$, the average force $F_e(x) = F_0 \langle d^\dagger d \rangle$, and the stochastic force $\xi(t)$ are due to the electrons tunneling through the quantum dot [15, 16]. The explicit expressions for $A$, $F_e$, and $\langle \xi(t) \xi(t') \rangle = D(x) \delta(t - t')$ have been obtained in Ref. [15]:

$$F_e(x) = F_0 \left[ \frac{1}{2} + \frac{1}{\pi} \sum_{\alpha} \frac{\Gamma_{\alpha}}{\Gamma} \arctan \frac{\mu_{\alpha} - \epsilon_0 - F_0 x}{\Gamma} \right], \quad (3)$$

$A(x) = (F_0^2 \Gamma / \pi) \sum_\alpha \Gamma_\alpha / [(\mu_\alpha - \epsilon_0 - F_0 x)^2 + \Gamma^2]^2$, and $D(x) = F_0^2 \Gamma_L \Gamma_R / (\pi \Gamma^3) [h(\mu_L) - h(\mu_R)]$, where $h(\mu) = \arctan z + z/(z^2 + 1)$ with $z = (\mu - \epsilon_0 - F_0 x)/\Gamma$. In the same limit a Fokker-Planck equation for the probability $P(x, p, t)$ can be derived [23, 24]:

$$\partial_t P = \frac{p}{m} \partial_x P - F \partial_p P + \frac{A}{m} \partial_p (pP) + \frac{D}{2} \partial_p^2 P, \quad (4)$$

with $F(x) = F_e(x) - kx$.

**Softening of the mechanical mode.** We assume that the device is symmetric: $\Gamma_\alpha = \Gamma/2$. The presence of the mechanical coupling modifies the electron-hole symmetry point for $\epsilon_0$ to the value $\epsilon_0 = (\mu_L + \mu_R)/2 + \epsilon_P/2$. We will always assume this value from this point on. Defining $y = x - F_0/2k$, one determines $F(y) = -ky + (F_0/2\pi) \sum_{a=\pm 1} \arctan[(F_0 y + a eV/2)/\Gamma]$ which depends only on the bias voltage $eV = \mu_L - \mu_R$ and is anti-symmetric in $y$. The equilibrium positions are defined by the solutions of the equation $F(y) = 0$. The line $\epsilon_P = \epsilon_c(V) \equiv \pi \Gamma [1 + (eV/2\Gamma)^2]$ [for $eV/\Gamma < 2/\sqrt{3}$] separates the monostable region from the bistable region (see inset of Fig. (1)).

Let us now define $\omega_m$ at a stable point $y_\beta$ as $m\omega_m^2 = -(dF/dy)_{y_\beta}$. It goes smoothly from $\omega_0$ to 0 for $\epsilon_P < \epsilon_c$ with the analytic form $\omega_m^2/\omega_0^2 = (\epsilon_c - \epsilon_P)/\epsilon_c$, while for $\epsilon_P \gtrsim \epsilon_c$ it reads $2(\epsilon_P - \epsilon_c)/\epsilon_c$ [see Fig. (1) orange dashed lines].

The vanishing of $\omega_m$ suggests that its direct measurement should allow the detection of the transition with great accuracy. As in phase transitions, this mode becomes soft, leading to a strong response at the transition. The dip in the gate voltage dependence of $\omega_m$ observed by four different groups [3, 4, 9, 12] is the precursor of this softening. Nevertheless, one should be cautious since the definition of $\omega_m$ only takes into account the first derivative of the force at the minimum of the potential. When

<figure><img src="image_2.png" /><figcaption>FIG. 1. Density plot of $S_{xx}$ as a function of $\omega$ and $\epsilon_P$. The values of $\omega_m$ (orange dashed line), $2\omega_m$ (blue dot-dashed line), and $3\omega_m$ (red dotted line) are shown. The units of $S_{xx}$ are $x_{zpm}^2/\omega_0 = (m\omega_0^2)^{-1}$, where $x_{zpm} = (m\omega_0)^{-1/2}$ is the zero-point motion displacement. The symmetry of the potential implies that only odd harmonics are present for $\epsilon_P < \epsilon_c$. Inset: phase diagram in the plane $eV-\epsilon_P$ for the stability of the effective potential.</figcaption></figure>

this term vanishes the next order terms in $y$ become important and the response of the system can no longer be predicted simply by the value of $\omega_m$. Therefore, in the following we calculate the typical measurable quantities and study their behavior when $\epsilon_P$ is swept through the transition.

**Fluctuation spectrum.** We define the displacement fluctuation spectrum $S_{xx}(\omega) = \int e^{i\omega t} dt \langle \tilde{x}(t) \tilde{x}(0) \rangle$, with $\tilde{x}(t) = x(t) - \langle x \rangle$. This quantity has been measured recently in Ref. [10]. We can obtain $S_{xx}$ numerically from the Fokker-Planck description following the method used in Ref. [17]. Writing Eq. (4) as $\partial_t P = \mathcal{L}_0 P$ the spectrum takes the form:

$$S_{xx}(\omega) = -2\mathrm{Tr} \left[ \hat{x} \frac{\mathcal{L}_0}{\omega^2 + \mathcal{L}_0^2} \hat{x} P_{\mathrm{st}} \right] \quad (5)$$

where $P_{\mathrm{st}}$ is the stationary solution of the problem, satisfying both $\mathcal{L}_0 P_{\mathrm{st}} = 0$ and the normalization condition $\mathrm{Tr} P_{\mathrm{st}} = 1$. The operator $\hat{x}$ is defined as $\hat{x}P \equiv \tilde{x}P(x, p)$.

Let's begin by discussing the stationary solution of Eq. (4): $P_{\mathrm{st}}$. In agreement with similar models [22, 25] we find that for sufficiently small $eV$, even if the system is out of equilibrium, the stationary distribution function takes the simple Gibbs form $P_{\mathrm{st}}(x, p) = N \exp\{-E(x, p)/T_{\mathrm{eff}}\}$, with $T_{\mathrm{eff}} = eV/4$, $N$ a normalization factor, $E(x, p) = p^2/2m + U(x)$, $U(x) = -dF/dx$, and $U(x) = 0$ at its minimum. This result is due to the smooth dependence on $x$ of both $D(x)$ and $A(x)$ on the scale of the spread of the probability distribution $P(x, p)$ for $eV \ll \Gamma$.

We come now to the displacement spectrum obtained
where $\dot{B} = 16 \cdot 3^{3/4} \pi^{1/4} f(1)/\Gamma[3/4]$ is a numerical constant. $\alpha_n = \int_{-1}^{1} du \frac{u}{f(1)\sqrt{1-u^4}} \cos\left(n\pi \frac{f(u)}{f(1)}\right)$ is a parameter depending on the harmonic index $n$ and involving the integral function $f(u) = \int_{-1}^{u} dv/\sqrt{1-v^4}$. The energy $E_n = (3^{1/4} f(1)\omega/\pi^{5/4} n\omega_0)^4 \Gamma$ satisfies the equation $\omega = n\omega(E_n)$. We evaluate numerically the values of $\alpha_n$ for the first harmonics: $\alpha_1 \approx -0.477$, $\alpha_3 \approx -0.021$ and $\alpha_5 \approx -9.3 \cdot 10^{-4}$. In Eq. (16), the main contribution to $S_{xx}(\omega)$ is given by the first harmonic $n=1$, the other harmonics $n \ge 3$ having a smaller weight. The normalized line shape of the displacement spectral density can be further approximated, retaining only the contribution of the first harmonic by:

$$ \frac{S_{xx}(\omega)}{S_{xx}(\omega_{max})} \approx \left( \frac{\omega}{\omega_{max}} \right)^4 e^{-\left[ \left( \frac{\omega}{\omega_{max}} \right)^4 - 1 \right]}, \quad (17) $$

where $\omega_{max} = B\omega_0 (T_{eff}/\Gamma)^{1/4}$ is the position of the maximum of the spectral density. From Eq. (17) and Fig. 2, we see that the spectral density has a different line shape compared to the weak non linear oscillator in Fig. 1. Its FWHM is given by:

$$ \Delta\omega = \Delta_4\omega_{max}. \quad (18) $$

where $\Delta_4 \approx 0.585$ is a numerical coefficient corresponding to the FWHM in Fig. 2. Finally, we get $\Delta\omega \approx 0.709\omega_0 (T_{eff}/\Gamma)^{1/4}$.

In contrast to the quasi-harmonic oscillator [see Eq. (14)], the resonance width of the quartic oscillator at the critical point does not scale linearly with the effective temperature, but with a scaling law $\propto (T_{eff}/\Gamma)^{1/4}$. Let's finally find the range of validity of the approximation used in Sec. IB. We first remark that $\Delta\omega$ in Eq. (14) diverges close to the transition, signalizing the breakdown of the quasi-harmonic approximation. At the transition, $\Delta\omega$ expressed in Eq. (18) is finite. We estimate the crossover between both regions to happen when both estimations of the resonance width $\Delta\omega$ in Eq. (14) and in Eq. (18) are equal. We find that the crossover between the two regimes takes place for $1 - \varepsilon_p/\varepsilon_c \approx 1.71(eV/\varepsilon_c)^{1/2}$.

## II. EVALUATION OF THE OSCILLATOR DEPHASING TIME $\gamma_\phi^{-1}$

### A. Linear response of the oscillator under weak driving

We now consider that the mechanical oscillator is under the influence of a weak driving force of frequency $\omega_D$ and amplitude $F_D$, so that the total force applied on the oscillator is $F(x) + F_D \cos(\omega_D t)$. The evolution equation for the probability distribution is then given by:

$$ \partial_t P = \{\mathcal{L}_0 + 2\mathcal{L}_D \cos(\omega_D t)\} P, \quad (19) $$

where $\mathcal{L}_D = -F_D \partial_p/2$. One can show that for weak driving, an approximate solution of Eq. (19) has the form $P(x_0, p_0, t) = P_{st}(x_0 - x_i(t), p_0 - p_i(t))$, where $x_i(t)$ and $p_i(t)$ are solutions of the equations of motion $\dot{x}_i(t) = p_i(t)/m$ and $\dot{p}_i(t) = F(x_i) + F_D \cos(\omega_D t) - A(x_i)p_i(t)/m$. Namely, the probability distribution for the oscillator in presence of driving is obtained by moving rigidly the centre of the stationary Gibbs distribution along the deterministic trajectory of the damped oscillator. If the driving is weak compared to the effective temperature, one can expand $P(x_0, p_0, t)$ to linear order in the driving strength:

$$ P(x_0, p_0, t) \approx P_{st}(x_0, p_0) \left\{ 1 - \frac{x_i(t)F(x_0) - p_i(t)p_0/m}{T_{eff}} \right\}. \quad (20) $$

### B. Ring-down response of the oscillator

<figure><img src="image_3.png" /><figcaption>FIG. 3. Normalized ring-down displacement of the oscillator $\tilde{x}(t)/\tilde{x}(0)$ (green curve) and its envelope (blue curve) at the critical point $\varepsilon_p = \varepsilon_c$.</figcaption></figure>

In the following, we suppose that the driving has been imposed far in the past. The linear response to this driving generates at time $t=0$ an initial condition that we choose for simplicity to be $x_i(0) = x_i$ and $p_i(0) = 0$. At this same time $t=0$, the driving is suddenly switched off and the system is let relaxing toward equilibrium. The ring-down dynamics at time $t > 0$ is encoded into the average response of the oscillator displacement $x(t) = \int \int dx_0 dp_0 P(x_0, p_0, 0) x_{x_0, p_0}(t)$.

Introducing $\tilde{x}_{x_0 p_0}(t) = x_{x_0 p_0}(t) - \langle x_{x_0 p_0}(t) \rangle$, we obtain from Eq. (20):

$$ \tilde{x}(t) \approx -\frac{x_i}{T_{eff}} \iint dx_0 dp_0 P_{st}(x_0, p_0) F(\tilde{x}_0) \tilde{x}_{x_0, p_0}(t), \quad (21) $$
from Eq. (5) that we show for $eV/\Gamma = 5 \cdot 10^{-3}$ in Fig. 1 and 2. As anticipated, as coupling increases the resonance broadens and shifts at low frequency. More surprisingly, the peak position shows a minimum at a small but finite value of $\omega$ at the transition with a maximal broadening (cf. Fig. 2-b). The position of the minimum and its width depend on the bias voltage. A strong telegraph noise appears for $\varepsilon_P \gtrsim \varepsilon_c$ (dark region at $\omega \to 0$), signaling the hopping of the systems between the two minima in the potential [15, 17]. This is also a strong indication of the transition [see Fig. (2)-d]. In the bistable phase ($\varepsilon_P > \varepsilon_c$) a double peak in the spectral function is visible.

In order to understand this behavior we take advantage of the separation of time scales of the problem. The damping $A(x)$ and the fluctuations $D(x)$ are both generated by the non-equilibrium electronic transport and, by hypothesis, are parametrically smaller (as $\omega_0/\Gamma$) than the Hamiltonian terms in Eq. (4). This implies that the system performs many oscillations on the closed trajectory in the phase space that satisfies $E(x(t), p(t)) = E$, before drifting to a nearby trajectory on the slow time scale $\gamma_E^{-1}$, where $\gamma_E = \int dx P_{st}(x, p) A(x)/m$ is the average dissipation coefficient [16, 23]. For each energy $E$ one can then calculate the pulsation of the closed trajectory $\omega(E) = 1/[2\pi(m/2)^{1/2} \oint [E - U(x)]^{-1/2} dx]$. The non-linearities present in $U(x)$ induce dispersion in $\omega(E)$. Then, from our definition, $\omega_m$ indicates only $\omega(0)$. Energies up to $eV$ are populated as an effect of the stochastic fluctuations. They all contribute to the fluctuation spectrum leading to an inhomogeneous broadening that spans the frequencies between $\omega_m$ and $\omega(eV)$.

In order to provide a quantitative verification of this interpretation we calculate the spectrum by neglecting the effect of the dissipation and considering only the interference of the different trajectories populated according to $P_{st}$ [26, 27]. This gives $S_{xx}(t) = \int dx_0 dy_0 P_{st}(x_0, p_0) \tilde{x}(t) \tilde{x}(0)$, where $x(t)$ satisfies the equation of motion $m\ddot{x} = F(x)$, with the initial conditions $x(0) = x_0$, and $\dot{x}(0) = p_0/m$. In terms of the Fourier coefficients of fixed energy periodic trajectories $[\tilde{x}_E(t) = \sum_n e^{in\omega(E)t} x_n(E)]$ the spectrum takes the form:

$$S_{xx}(\omega) = \int_{0}^{\infty} \mathcal{P}(E) dE \sum_{n} 2\pi \delta(\omega - n\omega(E)) x_{n}^{2}(E), \quad (6)$$

$$\text{with } \mathcal{P}(E) = N e^{-E/T_{\text{eff}}} 2\pi/\omega(E).$$

Two limits can be analyzed [28]: When the quartic term is much smaller than the quadratic one, the full width at half height of the resonance is $\Delta\omega \approx 1.5\omega_0(eV/\varepsilon_c)(\varepsilon_P/\varepsilon_c)^2(1 - \varepsilon_P/\varepsilon_c)^{-3/2}$ with a small positive shift of the maximum from $\omega_m$ of the same order. In the opposite limit of vanishing harmonic term ($\varepsilon_P = \varepsilon_c$) the potential can be approximated as quartic. In this case $\Delta\omega/\omega_0 = 0.50(eV/\Gamma)^{1/4}$ with a peak position $\omega_M$ at $0.85\omega_0(eV/\Gamma)^{1/4}$. This voltage dependence can be related to the dispersion of $\omega(E)$ that vanishes as $E^{1/4}$ for

<figure><img src="image_3.png" /><figcaption>FIG. 2. Comparison of the full numerical solution of the Fokker-Planck equation for $S_{xx}$ (blue solid lines) with the one obtained with Eq. (6) (dots) for $\varepsilon_P/\varepsilon_c = 0.87$, 1, 1.11 (a, b, and c panel, respectively) in units of $(m\omega_0^2)^{-1}$. The numbers label the order of the harmonic, while the letters h and l in the c panel indicate the high- and low-frequency contributions. Panel d: $S_{xx}(0)$ as a function of $\varepsilon_P/\Gamma$ indicating the onset of the telegraph noise at the transition.</figcaption></figure>

$\varepsilon_P = \varepsilon_c$. Remarkably, at criticality the Q-factor of the oscillator defined as $\omega_M/\Delta\omega$ takes the universal value 1.71, independently of $V$ or $\Gamma$. The crossover between the two regimes takes place for $1 - \varepsilon_P/\varepsilon_c \approx 1.71(eV/\varepsilon_c)^{1/2}$, thus for the values considered in Fig. 1 the quartic region is restricted to $1 - \varepsilon_P/\varepsilon_c < 0.03$. Finally, the double peak of the spectral function for $\varepsilon_P > \varepsilon_c$ can be explained by the contributions of the low-energy high-frequency trajectories around each single minimum, and those, at higher energy and lower frequency, revolving around both minima. The comparison with the numerical result presented in Fig. (2) shows a very good agreement.

Driving. Let us consider the other main tool used to detect mechanical motion: The response to a driving force of frequency $\omega_D$. We can find the linear response of the system by letting $F(x) \to F(x) + F_D \cos(\omega_D t)$ in Eq. (4). The evolution operator becomes $\mathcal{L}(t) = \mathcal{L}_0 + 2\mathcal{L}_D \cos(\omega_D t)$, with $\mathcal{L}_D = -F_D \partial_p/2$. After a transient time the solution can be written as a Fourier series $P(t) = \sum_n e^{in\omega_D t} P_n$ where each Fourier component can in turn be expanded as a power series of the driving parameter $F_D$: $P_n = \sum_{k=0}^{\infty} P_{n,k}$, with $P_{n,k}$ of order $F_D^k$. This leads to the equation for each component $(in\omega_D - \mathcal{L}_0)P_{n,0} = 0$ and

$$(in\omega_D - \mathcal{L}_0)P_{n,k+1} = \mathcal{L}_D(P_{n+1,k} + P_{n-1,k}), \quad (7)$$

with the condition $\text{Tr}P(t) = 1$. Eq. (7) can be solved by recursion. The time dependence of the displacement then reads $\tilde{x}(t) \equiv \text{Tr}[\hat{x}P(t)] = F_D\chi(\omega_D)e^{i\omega_D t} + \text{c.c.}$, where

$$\chi(\omega) = \text{Tr}[\hat{x}P_1(t)] = \text{Tr}[\hat{x}(i\omega - \mathcal{L}_0)^{-1}\partial_p P_{st}]. \quad (8)$$
which can be rewritten as:

$$ \tilde{x}(t) \approx -\frac{x_i}{T_{\text{eff}}} \int_{0}^{+\infty} dE \int_{0}^{T(E)} dt P_{st}(E) F[\tilde{x}_E(0)] \tilde{x}_E(t). \quad (22) $$

Similarly to Sec.I, we perform the Fourier expansion of the periodic trajectories $\tilde{x}_E(t) = \sum_n e^{in\omega(E)t}x_n(E)$. In the case where $\varepsilon_P < \varepsilon_c$, the electronic force acting upon the oscillator is given by Eq. (7). We can further perform the average over energy and phase of the orbit in Eq. (22):

$$ \tilde{x}(t) \approx \frac{x_i}{T_{\text{eff}}} \int_{0}^{+\infty} dE P(E) \left\{ m\omega_m^2 \sum_n x_n^2(E) e^{in\omega(E)t} + \sum_{\{n_i\}} \frac{F_0^4 x_{n_1}(E) x_{n_2}(E) x_{n_3}(E) x_{n_4}(E)}{3\pi m \Gamma^3} e^{in_4\omega(E)t} \delta_{n_4, \sum_{j=1}^3 n_j} \right\}. \quad (23) $$

Eq. (23) enables to compute the ring-down dynamics of the oscillator displacement $\tilde{x}(t)$ as well as to extract a characteristic dephasing time $\gamma_{\varphi}^{-1}$ of the mechanical oscillator.

In the following, we derive analytical expressions for $\tilde{x}(t)$ for the two limiting cases (i) where the quartic term is a small perturbation to the quadratic one, and (ii) when the quadratic term vanishes (for $\varepsilon_P = \varepsilon_c$) and the potential is purely quartic in $x$.

### 1. Case (i) weakly non linear oscillator

Far from the transition ($\varepsilon_P < \varepsilon_c$), the oscillator is weakly non-linear, and we can use the same approximations as in Sec.IB to derive from Eq. (23) an analytical expression for the oscillator displacement:

$$ \tilde{x}(t) \approx x_i \frac{(1 - (\gamma_{\varphi} t)^2) \cos(\omega_m t) - 2\gamma_{\varphi} t \sin(\omega_m t)}{(1 + (\gamma_{\varphi} t)^2)^2}. \quad (24) $$

where the dephasing time $\gamma_{\varphi}^{-1}$ of the oscillator is given by:

$$ \gamma_{\varphi}^{-1} = \frac{1}{\omega'(0)T_{\text{eff}}}. \quad (25) $$

Eq. (25) shows that after the driving has been switched off, the oscillator follows an oscillating behaviour given by the natural frequency of the vibration $\omega_m = \omega(0)$. This fast oscillation decays as a power law.

As expected, we find that the typical decay or dephasing time $\gamma_{\varphi}^{-1}$ of the oscillator is inversely proportional to the broadening of the displacement spectral density $\Delta\omega$ [see Eq. (14)], namely:

$$ \Delta\omega = \Delta_2\gamma_{\varphi}. \quad (26) $$

### C. Case (ii) purely quartic oscillator

At the critical point ($\varepsilon_P = \varepsilon_c$), the quadratic part of the potential vanishes and the mechanical oscillator becomes purely quartic. We adopt the same approximations as in Sec.IC to derive from Eq. (23) an analytical expression for the oscillator displacement:

$$ \frac{\tilde{x}(t)}{\tilde{x}(0)} \approx \frac{1}{\Gamma[7/4]} \int_{0}^{+\infty} dy y^{3/4} e^{-y} \cos(\omega_{\text{max}} y^{1/4} t), \quad (27) $$

where the frequency $\omega_{\text{max}} = B\omega_0 (T_{\text{eff}}/\Gamma)^{1/4}$ is the same as in Sec.IC. The oscillator displacement in Eq. (27) is plotted in Fig. 3.

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[2] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, 55 (Courier Corporation, 1964).
<figure><img src="image_4.png" /><figcaption>FIG. 3. Time dependence of $\langle \tilde{x}(t) \rangle$ (green solid line) and its envelope (blue solid line) for $\epsilon_P/\epsilon_c = 0.32, 1$, and $1.11$ (a, b, and c panel, respectively). The exponential decay on the scale $2/\gamma_E$ is shown red dashed. In panel a and b the orange dotted line gives the result of the analytical expressions discussed in the text. Panel d: comparison of the $\epsilon_P$-dependence of $\Delta\omega, 2\gamma_\varphi$, and $\gamma_E$.</figcaption></figure>

Naturally, the relation between $\chi(\omega)$ and $S_{xx}(\omega)$ comes into question. If $P_{\text{st}}$ has a Gibbs form then $F_D \mathcal{L}_0 \hat{x} P_{\text{st}} = -2T_{\text{eff}} \mathcal{L}_D P_{\text{st}}$. This leads to a fluctuation-dissipation relation:

$$ \mathrm{Im}[\chi(\omega)] = \frac{\omega}{2T_{\mathrm{eff}}} S_{xx}(\omega). \qquad (9) $$

Thus, for $eV \ll \Gamma$, $\chi$ and $S_{xx}$ give access to the same information in two independent ways. For larger voltages expression (8) always holds while Eq. (9) will be violated.

**Ring-down behavior.** Finally, let us consider the response for time $t > 0$ of the oscillator when the coherent drive is switched off at $t = 0$. A damped harmonic oscillator relaxes exponentially on a time scale (the ring-down time) given by the same dissipation coefficient that also determines the width of the resonance of the response function. For nano-mechanical oscillators it has recently been shown [13] that this may not be the case. Non-linearities induce frequency noise, which in turn is responsible for phase fluctuations of $\tilde{x}(t)$. The average over many realizations of $\tilde{x}(t)$ decays then on the time scale

$\gamma_\varphi^{-1}$, the value for which phase fluctuations become the order of $2\pi$. Since the energy is insensitive to the phase, its average decays on the same time scale as the single realization $\gamma_E^{-1}$.

With respect to our problem, we use the solution of the Fokker-Planck equation with driving [Eq. (7)] as the initial condition and then find $\tilde{x}(t)$ and $E(t)$ from the evolution of the probability with $\mathcal{L}_0$. The result as a Laplace transform reads:

$$ \langle \tilde{x}(z) \rangle = \mathrm{Tr} \left[ \hat{\tilde{x}}(z - \mathcal{L}_0)^{-1} P(t=0) \right]. \qquad (10) $$

In a similar way we can calculate also the Laplace transform of the evolution of the total energy by letting $\hat{\tilde{x}} \to E(\hat{x}, \hat{p})$ in Eq. (10). One can then obtain the time dependence by numerically implementing the Cauchy theorem $\langle \tilde{x}(t) \rangle = \oint_C \langle \tilde{x}(z) \rangle e^{-zt} dz/(2\pi i)$, where $C$ is a contour that encloses the poles of $\langle \tilde{x}(z) \rangle$ for $\text{Re}z < 0$.

We find that the energy exponentially decays on the scale $\gamma_E^{-1}$, even at the transition. On the other hand, as shown in Fig. (3), $\langle \tilde{x}(t) \rangle$ decays on a much shorter scale that we define $\gamma_\varphi^{-1}$. Fig. (3)-d shows the $\epsilon_P$ dependence of $\Delta\omega, 2\gamma_\varphi$, and $\gamma_E$. The width $\Delta\omega$, obtained from the form of $S_{xx}$, coincides within the numerical accuracy with $2\gamma_\varphi$, proving that frequency noise is the responsible of the faster decay of $\langle \tilde{x}(t) \rangle$. Both present a pronounced maximum at $\epsilon_P = \epsilon_c$, indicating the transition. Using the approach presented for the analytical calculation of $S_{xx}(\omega)$ we find that $\langle \tilde{x}(t) \rangle$ decays as $1/(1+t^2\gamma_\varphi^2)^2$, where $\gamma_\varphi = 0.41\Delta\omega$ [29]. Similarly for $\epsilon_P = \epsilon_c$ the decay scale is proportional to $\Delta\omega$ with the analytical form given by the dotted line of Fig. (3).

**Conclusions.** We found that the study of the mechanical properties of the suspended carbon nanotube open new perspectives for the observation of the current-blockade transition occurring at low temperature and voltage for $\epsilon_P = \pi\Gamma$. Indeed, for that value the quadratic part of the effective potential vanishes, leading to strong frequency and phase fluctuations that remarkably modify the typically measured response functions ($S_{xx}, \chi$, and ring-down behavior). The $Q$-factor of the resonator takes the minimum and universal value 1.71 at criticality, where the separation of time scales is also maximal. These results can lead to the observation of the current-blockade transition by mechanical measurements in devices currently investigated by several experimental groups.

**Acknowledgements.** We acknowledge support from ANR-10-BLANC-0404 QNM.

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