Stationary quantum entanglement between a massive mechanical membrane and a low frequency LC circuit

We study electro-mechanical entanglement in a system where a massive membrane is capacitively coupled to a {\it low frequency} LC resonator. In opto- and electro-mechanics, the entanglement between a megahertz (MHz) mechanical resonator and a gigahertz (GHz) microwave LC resonator has been widely and well explored, and recently experimentally demonstrated. Typically, coupling is realized through a radiation pressure-like interaction, and entanglement is generated by adopting an appropriate microwave drive. Through this approach it is however not evident how to create entanglement in the case where both the mechanical and LC oscillators are of low frequency, e.g., around 1 MHz. Here we provide an effective approach to entangling two low-frequency resonators by further coupling the membrane to an optical cavity. The cavity is strongly driven by a red-detuned laser, sequentially cooling the mechanical and electrical modes, which results in stationary electro-mechanical entanglement at experimentally achievable temperatures. The entanglement directly originates from the electro-mechanical coupling itself and due to its quantum nature will allow testing quantum theories at a more macroscopic scale than currently possible.


I. INTRODUCTION
In optomechanics, an optical field can couple to a massive mechanical oscillator (MO) via the radiation pressure force [1]. This approach provides the possibility to prepare quantum states of macroscopic systems by manipulating optical degrees of freedom. Over the past decade, significant experimental progress has been achieved in observing quantum effects in massive mechanical systems, including reaching the quantum ground state [2,3], quantum squeezing of the mechanical motion [4][5][6], quantum entanglement between two MOs [7,8], and between a MO and an electromagnetic field [9,10], among many others. Such quantum states of massive objects have important implications for both quantum technologies, e.g., quantum sensing [11], quantum transducers [12], as well as foundational studies of decoherence theories at the macro scale and the boundary between the quantum and classical worlds [13].
In this paper, we provide a scheme to entangle a massive MO with a macroscopic low-frequency LC oscillator. Specifically, we consider a tripartite system where a mechanical membrane is capacitively coupled to an LC resonator and further optomechanically coupled to an optical cavity. Unlike most other approaches with GHz resonators [2,9,14,15], the LC resonator we consider here is in the radio frequency domain, around 1 MHz [16,17], and close to the mechanical frequency. Such a low-frequency LC resonator means a much larger product L × C (L-inductance; C-capacitance) than that at microwave frequency (10 6 larger for frequency of 1 MHz compared to 1 GHz), which typically implies a much larger number of charges and a much bigger LC circuit. The membrane-LC interaction takes a nonlinear form H int = g 0 xq 2 [18][19][20], where x is the mechanical position, q the charge and g 0 the bare electro-mechanical coupling rate, which is a radiation pressure-like interaction. We note that the entanglement between a nanomechanical resonator and an LC resonator of microwave frequency has been well stud-ied [18,19], where the electro-mechanical interaction is a radiation pressure type ∝ g 0 xb † b. Here b is the annihilation operator of the LC field and q 2. The interaction is derived by taking the rotating wave approximation (RWA) and neglecting fast oscillating terms, which is valid only for the LC frequency much larger than the mechanical frequency, ω LC ω m . Such a radiation pressure interaction predicts the generation of electro-mechanical entanglement if an appropriate microwave drive is adopted [18]. However, when the LC frequency is approaching the mechanical frequency, like in Ref. [16,17] and as considered here, it is not clear how to apply an appropriate driving field such that (stationary) electromechanical entanglement can be produced. In other words, it is not apparent how to apply the mechanism of Ref. [18] to two nearly resonant low-frequency oscillators.
Inspired by recent experiments [16,17], we apply a DC drive for the LC circuit, which significantly enhances the effective electro-mechanical coupling rate. The linearized interaction takes the form ∝ gδxδq [16,20], where g is the effective coupling rate. Based on this coupling, electro-mechanical entanglement can indeed be created, but only at unrealistic extremely low temperature (typically below 0.1 mK for 1 MHz oscillators), and the entanglement completely vanishes at typical cryogenic temperatures of a few tens of millikelvin, because of the low resonant frequencies and thus large thermal occupations. In our approach, we overcome this limitation and show that by further coupling the mechanical membrane to an optical cavity field via radiation pressure, and by driving the cavity with a red-detuned laser, the mechanical and electrical modes get significantly cooled, which leads to the emergence of electro-mechanical entanglement. This entanglement is in the steady state regime and is robust against temperature. Here, the red-detuned cavity cools the mechanical mode, which then acts as a cold bath for the electrical mode [21]. We show that this is equivalent to assuming that the MO-LC system operates at extremely low temperatures without the need for an optical cavity. The remainder of the paper is organized as follows: in Sec. II, we introduce our tripartite opto-electro-mechanical system, provide its Hamiltonian and the corresponding Langevin equations, and in Sec. III we show how to obtain the steady-state solutions of the system and quantify the entanglement. In Sec. IV, we present the results of electro-mechanical entanglement and discuss optimal parameter regimes for obtaining the entanglement and its detection. Finally, we draw the conclusions in Sec. V.

II. THE SYSTEM
We consider a tripartite opto-electro-mechanical system, as shown in Fig. 1, which consists of an LC electrical circuit, a MO, and an optical cavity. An experimental realization of a suitable MO could be a metal coated nanomembrane [16,17], which is capacitively coupled to an LC resonator and further coupled to an optical cavity field via radiation pressure. Specifically, the radiation pressure of the cavity field causes a mechanical displacement which further changes the capacitance of the LC circuit, and conversely, the voltage fluctuation in the LC circuit leads to an optical phase shift via the mediation of the MO. The Hamiltonian of the system reads where a (a † ) is the annihilation (creation) operator of the cavity mode, x and p (q and φ) are the dimensionless position and momentum (charge and flux) quadratures of the mechanical (LC) resonator, and therefore [a, a † ] = 1 and [x, p] = [q, φ] = i. The resonance frequencies ω c , ω m , and ω LC = 1 √ LC are of the cavity, mechanical, and LC resonators, respectively, where L (C) is the inductance (capacitance) of the LC circuit. The capacitance C(x) is a function of the mechanical position x, which characterizes the capacitive coupling to the MO. The single-photon optomechanical coupling rate G 0 =− dω c dx | x s , and the electro-mechanical coupling rate g = /mω m and q 0 = √ /Lω LC are the zero-point fluctuations of the mechanical and LC oscillators, respectively, with the effective mass m of the MO, and x s and q s are the average of the position x and charge q. The last two terms in the Hamiltonian denote the electric driving for the LC circuit and the laser driving for the cavity, respectively, where V is a DC bias voltage (see Fig. 1 (a)), and E = √ 2P l κ/ ω l is the coupling between the cavity field with decay rate κ and the driving laser with frequency ω l and power P l .
In the frame rotating at the drive frequency ω l , the quantum Langevin equations (QLEs) governing the system dynamics are given bẏ where ∆ 0 = ω c − ω l , γ m and γ LC = 2R/L (with R the resistance of the circuit) are the mechanical and electrical damping rates, respectively, a in is the input noise operator for the cavity, whose mean value is zero and the only non-zero correlation is The Langevin force operator ξ accounts for the Brownian motion of the MO and is autocorrelated as where we have made a Markovian approximation valid for large mechanical quality factors Q m = ω m /γ m 1 [22], and n m k B T ω m is the equilibrium mean thermal phonon number in the high temperature limit, with k B the Boltzmann constant and T the environmental temperature.
We work with a strongly driven cavity, which leads to a large amplitude of the cavity field a 1. This allows us to linearize the system dynamics (essentially the nonlinear optomechanical interaction) around the semiclassical averages by writing any operator as O = O + δO (O = a, x, p, q, φ) and neglecting small second-order fluctuation terms. Therefore, the QLEs Eq. (2) are separated into two sets of equations: one is for averages O s ≡ O and the other for zero-mean quantum fluctuations δO. The steady-state averages can be obtained by setting the derivatives to zero and solving the following equa-tions a s = E κ + i∆ , where ∆ = ∆ 0 − G 0 x s . The expression of q s shows a linear dependence of q s (and thus of the coupling g) on the bias voltage V, which means that the electro-mechanical coupling strength can be significantly improved by increasing the bias voltage [16]. The linearized QLEs describing the quadrature fluctuations (δX, δY, δx, δp, δq, δφ), with δX = (δa + δa † )/ where G = √ 2G 0 a s is the effective optomechanical coupling, and 2 are the quadratures of the cavity input noise. Note that in deriving the above QLEs, we have chosen a phase reference such that a s is real and positive. The fluctuation of the bias voltage δV ≡ V −V can be considered as the input noise for the flux, and is autocorrelated as which corresponds to the quantum version of the Johnson-Nyquist noise correlation [23] for a resistor R = γ LC 2 L at temperature T by including the vacuum fluctuation. In such a way, the noise correlation for the operator δV(t) ≡ q 0 δV(t) can be written in the form withn LC k B T ω LC being the thermal occupancy of the LC oscillator, which takes a consistent form as that for the Langevin force operator ξ. This is the reason why we defined the damping rate γ LC as twice its conventional definition γ LC = R/L.

III. STEADY-STATE SOLUTIONS AND QUANTIFICATION OF GAUSSIAN ENTANGLEMENT
We are interested in the quantum correlation between the mechanical and LC oscillators in the stationary state. Owing to the fact that the dynamics are linearized and all input noises are Gaussian, the Gaussian nature of the state will be preserved for all times. The steady state of the quantum fluctuations of the system is therefore a threemode Gaussian state and is completely characterized by a 6 × 6 covariance matrix (CM) C, which is defined as C i j = 1 2 u i (t)u j (t ) + u j (t )u i (t) (i, j = 1, 2, ..., 6), where u(t) = δX(t), δY(t), δx(t), δp(t), δq(t), δφ(t) T . The stationary CM C can be obtained by solving the Lyapunov equation [24] AC + CA T = −D, where A is the drift matrix determined by the QLEs (6), given by and D = diag κ, κ, 0, γ m (2n m + 1), 0, γ LC (2n LC + 1) is the diffusion matrix, which is defined by n i (t)n j (t ) + n j (t )n i (t) /2 = D i j δ(t − t ), with the vector of input noises To quantify the Gaussian entanglement, we adopt the logarithmic negativity [25], which is a full entanglement monotone under local operations and classical communication [26] and sets an upper bound for the distillable entanglement [25]. The logarithmic negativity is defined as [27] whereν − = min eig|iΩ 2C4 | (with the symplectic matrix Ω 2 = ⊕ 2 j=1 iσ y and the y-Pauli matrix σ y ) is the minimum symplectic eigenvalue of the partially transposed CMC 4 = P 1|2 C 4 P 1|2 , with C 4 being the 4 × 4 CM of the mechanical and electrical modes, obtained by removing in C the rows and columns related to the cavity field, and P 1|2 = diag(1, −1, 1, 1) being the matrix that performs partial transposition on CM [28].

IV. ELECTRO-MECHANICAL ENTANGLEMENT IN THE STEADY STATE
In this section, we present the results of the entanglement between the mechanical and LC oscillators. All results are in the steady state guaranteed by the negative eigenvalues (real parts) of the drift matrix A. We adopt experimentally feasible parameters [16,17]: ω c /2π = 200 THz, ω m /2π = 1 MHz, κ = 0.1ω m , γ m = 10 −6 ω m , γ LC = 10 −5 ω LC , and consider the LC frequency ω LC as a variable which is tuned around ω m . To avoid additional low-frequency electronic noises, LC frequencies much below 1 MHz will not be considered. We work in the resolved sideband limit, κ ω m , and assume a relatively large Q factor of the LC oscillator compared to those typically demonstrated at room temperature [16,17] as we place the system at cryogenic temperatures where superconductivity can significantly improve the Q factor [29]. At a few tens of millikelvin, the mechanical and LC oscillators still exhibit significant thermal excitations because of their low frequencies. Therefore, we use a red-detuned laser to drive the cavity and stimulate the optomechanical anti-Stokes process, which results in cooling of the mechanical mode [3], and owing to the linearized MO-LC coupling, the electrical mode also gets cooled. In such a system, it is even possible to cool a 1 MHz LC resonator into its quantum ground state from temperature of a few tens of millikelvin [21]. The cooling process in this hybrid system can be considered as the transport of thermal excitations from the electrical mode to the mechanical mode, and then to the cavity mode, which eventually dissipates the heat via cavity photon leakage to the environment. The low effective temperatures of the mechanical and electrical modes are a precondition for observing their entanglement if strong coupling rates are used. This is veri- fied numerically and shown in Fig. 2, where the entanglement is maximal for a cavity-laser detuning ∆ ω LC . We assume both the optomechanical and the electro-mechanical coupling to be strong G, g > κ, in order to significantly cool both the mechanical and electrical modes [21] and to ultimately create the desired electro-mechanical entanglement. Figure 2 also shows that in our system two nearly resonant oscillators are preferred to maximize the entanglement. If the couplings are further increased (cf. Fig. 2(b)) the system becomes unstable for ∆ < ∼ 0.4ω m .
We further show the stationary MO-LC entanglement as a function of the two coupling rates g and G for the resonant case [16,17] in Fig. 3. It is clear that the entanglement grows with increasing coupling strengths and strong couplings G, g > κ are generally required to obtain considerable entanglement. The coupling strengths are restricted by the stability condition and the system starts to be unstable when g > 8κ; G > 6κ. It would in principle be interesting to compare the mechanism of the entanglement generation in the present system with other tripartite systems, such as atom-lightmirror [30] and magnon-photon-phonon [31] system, which also contain linear coupling (atom-light; magnon-photon) and nonlinear coupling (light-mirror; magnon-phonon). However, in contrast to these approaches where the two linearly coupled subsystems oscillate at relatively high frequency and thus the RWA g(m + m † )(c + c † ) → g(mc † + m † c), where m and c are two mode operators, is always satisfied, here the linearly coupled mechanical and LC oscillators are both at low frequencies and the optimal coupling rate g is comparable to their frequencies (see Fig. 3). Therefore, the RWA is not valid in our case and one has to consider the full interaction and the effect of counter-rotating terms g(mc + m † c † ). In fact, we verified that no electro-mechanical entanglement can be found in the tripartite system if the MO-LC coupling is assumed to be a beamsplitter type g(mc † + m † c). This type of interaction is equivalent to assuming weak coupling g ω m/LC in the original full interaction g(m + m † )(c + c † ), and thus it is possible to use the RWA. As clearly visible from Fig. 3, this situation of weak coupling g < κ ω m/LC does not produce any en- tanglement, and hence the counter-rotating terms must be considered for entanglement. In contrast, the two linearly coupled (via beamsplitter interaction) subsystems in Refs. [30,31] get entangled because they are respectively resonant with the two entangled mechanical sidebands of the driving field.
We further investigate the entanglement as a function of temperature and the data in Figure 4 shows that it is robust against temperature, surviving up to ∼100 mK, based on realistic parameters. Even though the electro-mechanical coupling rate g and the Q factor of the LC resonator we use are larger than the demonstrated values [16,17], it is realistic to assume that they can be achieved at low temperature and by properly designing the system [29].
Finally, we would like to discuss how to detect the electromechanical entanglement. The task requires to essentially measure the four quadratures of the mechanical and electrical modes, (x, p, q, φ), based on which the CM can be reconstructed and the logarithmic negativity can then be computed according to the definition in Eq. (11). To measure the mechanical quadratures, we adopt the strategy used in Refs. [9,10,32], i.e., sending a weak red-detuned probe field with detuning equal to the mechanical frequency ∆ p ω m into the cavity, which maps the mechanical state onto the anti-Stokes sideband of the probe field at cavity resonance. Thus, by homodyning the probe output field, the two mechanical quadratures are measured. The quadratures of the electrical mode can also be measured by employing a homodyne scheme at radio frequency.
In order to avoid measuring the whole 4 × 4 CM for quan-tifying entanglement, alternatively, one can also verify the entanglement by using the Duan criterion [33], which requires the measurement of only two collective quadratures, X + = x + q, and Y − = p − φ. A sufficient condition for entanglement is that the two collective quadratures should satisfy the following inequality δX 2 + + δY 2 − < 2.
(12) Figure 5 shows that in moderate ranges of temperature and LC Q factor the inequality is fulfilled, indicating the presence of electro-mechanical entanglement. The entanglement survives up to 19 mK for Q LC = 2 × 10 4 , and 86 mK for Q LC = 10 5 .

V. CONCLUSIONS
We have provided a straightforward but effective approach to preparing entangled states of low-frequency mechanical and LC resonators. At typical cryogenic temperatures, the two resonators still contain significant thermal excitations, which effectively destroy their joint quantum correlations. In order to solve this, we couple the mechanical element to an optical cavity via the radiation pressure force, which can act as an additional cold bath: by drving the cavity with a red-detuned laser, both the mechanical and electrical modes are sequentially cooled, resulting in remarkable electro-mechanical entanglement emerging from thermal noise. The entanglement originates from the electro-mechanical coupling and can be in the stationary state and robust against temperature.
The present work can be considered as a complementary study to the widely explored situation where the LC frequency, typically in microwave domain [2,9,14,15], is much larger than the mechanical frequency, and in this case electro-mechanical entanglement can be directly generated by adopting an appropriate microwave drive [18]. The entanglement generated in this work, however, uses a different mechanism and is of low-frequency resonators (both around 1 MHz), which implies its macroscopic quantum nature, and would allow us to test quantum theories at a more macroscopic level [34][35][36].