Strong mechanical squeezing in an electromechanical system

The mechanical squeezing can be used to explore quantum behavior in macroscopic system and realize precision measurement. Here we present a potentially practical method for generating strong squeezing of the mechanical oscillator in an electromechanical system. Through the Coulomb interaction between a charged mechanical oscillator and two fixed charged bodies, we engineer a quadratic electromechanical Hamiltonian for the vibration mode of mechanical oscillator. We show that the strong position squeezing would be obtained on the currently available experimental technologies.


Results
As shown schematically in Fig. 1, our model consists of a charged mechanical oscillator in the middle which is coupled to two fixed charged bodies on the left and right sides. The charged mechanical oscillator is subject to the Coulomb force due to the nearby charged bodies. The Hamiltonian describing the vibration of the charged mechanical oscillator is given by here, H 0 is the free Hamiltonian with x and p being the position and momentum operators for the vibration of the charged mechanical oscillator, with frequency ω m and mass m, and H l int ( ) describes the Coulomb interaction between the charged mechanical oscillator and the lth (l = 1, 2) charged body. C 0 U 0 is the positive charge on the charged mechanical oscillator, with C 0 and U 0 being the equilibrium capacitance and the voltage of the bias gate. r l represents the equilibrium distance between the charged mechanical oscillator and the lth charged body.
In the case of  x r r , 1 2 , the Coulomb interaction Hamiltonian H int (1) and H int (2) can be expanded to the second After defining dimensionless annihilation and creation operators for the vibration mode of mechanical oscillator using the position and momentum operators of the oscillator, is the effective mechanical coupling constant. In Eq. (4), we have neglected the zero-point energy from the first term. The second term is quadratic in position quadrature x of mechanical oscillator, which can produce quadrature squeezing through a unitary evolution on any initial state of mechanical mode. Note that the quadratic Hamiltonian H′ is similar to that of optomechanical systems for generating squeezing of the mechanical oscillator 23-30 . However in the quadratic optomechanical systems, the optomechanical coupling depends on average photons in the optical cavity. As the large photon numbers in the cavity tend to faster decay out of the cavity, the squeezing of the mechanical oscillator in refs 23-30 is limited by the cavity decay.
Next, we consider the mechanical squeezing in the different temperatures of the environment. The state of the mechanical oscillator in thermal equilibrium with an environmental temperature T is described by means of the density matrix ρ = ∑ | 〉〈 | p n n n n , where represents the population in phonon number state |n〉 with k B being the Boltzmann constant. In order to extract the squeezing properties of the mechanical mode, we need to calculate the mean square fluctuations 〈ΔQ(t)〉 2 and 〈ΔP(t)〉 2 20,30 in the position and momentum of the mechanical oscillator. Let 〈ΔQ(t)〉 2 = 〈Q(t) 2 〉 − 〈Q(t)〉 2 and 〈ΔP(t)   . Clearly, the squeezing is primarily controlled through the coupling constant g, the mechanical frequency ω m , and the temperature T.

Discussion
The degree of the squeezing S as functions of the mechanical frequency (ω m = 5 MHz-5 GHz) and the voltage of the bias gate (U 0 ) for different temperatures of the environment T = 0 K, 1 mK, 0.1 K, 1 K when C 0 = 5 nF, r 0 = 4 μm, and the elastic coefficient N/m is shown in Fig. 2. It is observed that at low temperature, the adjustment of the voltage of the bias gate could yield the strong squeezing in the high frequency. For example, we choose the realistic parameters corresponding to the experiment ω m = 5.6 MHz and m = 0.7 ng 44 . Using C 0 = 5 nF, U 0 = 10 V, and r 0 = 4 μm, we obtain S ≈ 14.8 dB at the equilibrium temperature T = 1 mK and the evolution time t = π/(2q) ≈ 5.2 ns. According to ref. 13 , there exists a critical time tot , here γ is the mechanical damping rate and ξ ≈ + n n sinh (2 ) tot t h 2 is the total phonon number with ξ being the squeeze parameter. Choosing the damping rate γ = 204 Hz 44 , we get t diss ≈ 0.16 ms, which satisfies the condition  t t diss , and thus the decoherence of the mechanical oscillator could be negligible.

Conclusion
In summary, we have proposed an effective method to generate the mechanical squeezing in the electromechanical system. This is realized through the Coulomb interaction acting on the charged mechanical oscillator and two charged bodies, implementing the strong squeezing of the mechanical oscillator. It is found that at low temperature, the squeezing can be enhanced by moderately increasing the voltage of the bias gate. Our proposed scheme would contribute to the experimental study of fundamental aspects in the macroscopic quantum effects and the precision of quantum measurements with mechanical oscillators.