Quantum-squeezing effects of strained multilayer graphene NEMS
© Xu et al; licensee Springer. 2011
Received: 1 March 2011
Accepted: 20 April 2011
Published: 20 April 2011
Quantum squeezing can improve the ultimate measurement precision by squeezing one desired fluctuation of the two physical quantities in Heisenberg relation. We propose a scheme to obtain squeezed states through graphene nanoelectromechanical system (NEMS) taking advantage of their thin thickness in principle. Two key criteria of achieving squeezing states, zero-point displacement uncertainty and squeezing factor of strained multilayer graphene NEMS, are studied. Our research promotes the measured precision limit of graphene-based nano-transducers by reducing quantum noises through squeezed states.
The Heisenberg uncertainty principle, or the standard quantum limit [1, 2], imposes an intrinsic limitation on the ultimate sensitivity of quantum measurement systems, such as atomic forces , infinitesimal displacement , and gravitational-wave  detections. When detecting very weak physical quantities, the mechanical motion of a nano-resonator or nanoelectromechanical system (NEMS) is comparable to the intrinsic fluctuations of the systems, including thermal and quantum fluctuations. Thermal fluctuation can be reduced by decreasing the temperature to a few mK, while quantum fluctuation, the quantum limit determined by Heisenberg relation, is not directly dependent on the temperature. Quantum squeezing is an efficient way to decrease the system quantum [6–8]. Thermomechanical noise squeezing has been studied by Rugar and Grutter , where the resonator motion in the fundamental mode was parametrically squeezed in one quadrature by periodically modulating the effective spring constant at twice its resonance frequency. Subsequently, Suh et al.  have successfully achieved parametric amplification and back-action noise squeezing using a qubit-coupled nanoresonator.
where E is the Young's modulus of the material, T s is the tension on the film, A is 0.162 for a cantilever and A is 1.03 for a double-clamped film . Therefore, Δx zp of the fundamental mode of a NEMS device with a double-clamped film can be given by Δx zp = ΔX 1 = ΔX 2 = (ħ/2M eff w)1/2. In a mechanical system, quantum squeezing can reduce the displacement uncertainty Δx zp.
Recently, free-standing graphene membranes have been fabricated , providing an excellent platform to study quantum-squeezing effects in mechanical systems. Meanwhile, a graphene membrane is sensitive to external influences, such as atomic forces or infinitesimal mass (e.g., 10-21 g) due to its atomic thickness. Although graphene films can be used to detect very infinitesimal physical quantities, the quantum fluctuation noise Δx zp of graphene NEMS devices (approx. 10-2 nm), could easily surpass the magnitudes of signals caused by external influences. Thus, quantum squeezing becomes necessary to improve the ultimate precision of graphene-based transducers with ultra-high sensitivity. In this study, we have studied quantum-squeezing effects of strained multilayer graphene NEMS based on experimental devices proposed by Chen et al. .
Displacement uncertainty of graphene NEMS
Calculated Δx zp (10-4nm) of monolayer (Mon), bilayer (Bi), and trilayer (Tri) graphene versus strain ε (L = 1.1 μm, W = 0.2 μm)
ε= 4 × 10-5
ε= 2 × 10-4
According to the results in Figure 2 and Table 1, we find Δx zp large strain < Δx zp small strain; one possible reason is that larger applied strain results in smaller fundamental angular frequency and Δx zp, therefore, the quantum noise can be reduced.
Quantum-squeezing effects of graphene NEMS
To analyze quantum-squeezing effects in graphene NEMS devices, a back-action-evading circuit model is used to suppress the direct electrostatic force acting on the film and modulate the effective spring constant k of the membrane film. Two assumptions are used, namely, the film width W is on the micrometer scale and X 1 >> d, where d is the distance between the film and the substrate. Applying a pump voltage V m(t) = V[1+ sin(2w m t + θ)], between the membrane film and the substrate, the spring constant k will have a sinusoidal modulation k m(t), which is given by k m(t) = sin(2w m t + θ)C T V 2/2d 2, where C T is the total capacitance composed of structure capacitance C 0, quantum capacitance C q, and screen capacitance C s in series . The quantum capacitance C q and screen capacitance C s cannot be neglected [18–20] owing to a graphene film thickness on the atomic scale. The quantum capacitance of monolayer graphene [21, 22] is C q monolayer = 2e2 n 1/2/(ħv Fπ1/2), where n is the carrier concentration, e is the elementary charge, and v F ≈ c/300, where c is the velocity of light, with bilayer C q bilayer = 2 × 0.037m e e 2/πħ 2, and trilayer C q trilayer = 2 × 0.052m e e 2/πħ 2, where m e is the electron mass .
R values of monolayer graphene versus various strain ε and voltage V (L = 1.1 μm, W = 0.2 μm, and T = 300 K with Q = 125)
ε= 4 × 10-5
ε= 2 × 10-4
V = 2 V
V = 10 V
R values of monolayer graphene versus various strain ε and voltage V (L = 1.1 μm, W = 0.2 μm, and T = 5 K with Q = 14000)
ε= 4 × 10-5
ε= 2 × 10-4
V = 2 V
V = 10 V
In contrast to the previous squeezing analysis proposed by Rugar and Grutter , in which steady-state solutions have been assumed and the minimum R is 1/2, we use time-dependent pumping techniques to prevent X 2 from growing without bound as t → ∞, which should be terminated after the characteristic time t ct = ln(QC T V 2/4M eff w 2 d 2)4M eff wd 2/C T V 2, when R achieves its limiting value. Therefore, we have no upper bound on R. Figure 4b has shown the time dependence of ΔX 1 and ΔX 2 in units of t ct, and the quantum squeezing of the monolayer graphene NEMS has reached the limiting value after one t ct time. Also, to make the required heat of conversion from mechanical energy negligible during the pump stage, t ct << τ must be satisfied. We find t ct/τ ≈ 1.45 × 10-5 for the monolayer graphene parameters considered in the text.
By choosing the dimensions of a typical monolayer graphene NEMS device in  with L = 1.1 μm, W = 0.2 μm, T = 5 K, Q = 14000, V = 2.5 V, and ε = 0, we obtain Δx zp = 0.0034 nm and R = 0.374. After considering quantum squeezing effects based on our simulation, Δx zp can be reduced to 0.0013 nm. With a length of 20 μm, Δx zp can be as large as 0.0145 nm, a radio-frequency single-electron-transistor detection system can in principle attain such sensitivities . In order to verify the quantum squeezing effects, a displacement detection scheme need be developed.
In conclusion, we presented systematic studies of zero-point displacement uncertainty and quantum squeezing effects in strained multilayer graphene NEMS as a function of the film dimensions L, W, h, temperature T, applied voltage V, and strain ε applied on the film. We found that zero-point displacement uncertainty Δx zp of strained graphene NEMS is inversely proportional to the thickness of graphene and the strain applied on graphene. By considering quantum capacitance, a series of squeezing factor R values have been obtained based on the model, with R monolayer < R bilayer < R trilayer and R small strain < R large strain being found. Furthermore, high-sensitivity graphene-based nano-transducers can be developed based on quantum squeezing.
The authors gratefully acknowledge Prof. Raphael Tsu at UNCC, Prof. Jean-Pierre Leburton at UIUC, Prof. Yuanbo Zhang at Fudan University, Prof. Jack Luo at University of Bolton, and Prof. Bin Yu at SUNY for fruitful discussions and comments. This study is supported by the National Science Foundation of China (Grant No. 61006077) and the National Basic Research Program of China (Grant Nos. 2007CB613405 and 2011CB309501). Dr. Y. Xu is also supported by the Excellent Young Faculty Awards Program (Zijin Plan) at Zhejiang University and the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP with Grant No. 20100101120045).
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