Abstract
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, zeropoint displacement uncertainty and squeezing factor of strained multilayer graphene NEMS, are studied. Our research promotes the measured precision limit of graphenebased nanotransducers by reducing quantum noises through squeezed states.
Introduction
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 [3], infinitesimal displacement [4], and gravitationalwave [5] detections. When detecting very weak physical quantities, the mechanical motion of a nanoresonator 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 [68]. Thermomechanical noise squeezing has been studied by Rugar and Grutter [9], 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. [10] have successfully achieved parametric amplification and backaction noise squeezing using a qubitcoupled nanoresonator.
To study quantumsqueezing effects in mechanical systems, zeropoint displacement uncertainty, Δx_{zp}, the best achievable measurement precision, is introduced. In classical mechanics, the complex amplitudes, X = X_{1 }+ iX_{2}, where X_{1 }and X_{2 }are the real and imaginary parts of complex amplitudes respectively, can be obtained with complete precision. In quantum mechanics, X_{1 }and X_{2 }do not commute, with the commutator [X_{1}, X_{2}] = iħ/M_{eff}w, and satisfy the uncertainty relationship ΔX_{1}ΔX_{2 }≥ (ħ/2M_{eff}w)^{1/2}. Here, ħ is the Planck constant divided by 2π, M_{eff }= 0.375ρLWh/2 is the effective motional doubleclamped film mass [11,12], ρ is the volumetric mass density, L, W, and h are the length, width, and thickness of the film, respectively, and w = 2f_{0 }is the fundamental flexural mode angular frequency with
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 doubleclamped film [13]. Therefore, Δx_{zp }of the fundamental mode of a NEMS device with a doubleclamped 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, freestanding graphene membranes have been fabricated [14], providing an excellent platform to study quantumsqueezing 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 graphenebased transducers with ultrahigh sensitivity. In this study, we have studied quantumsqueezing effects of strained multilayer graphene NEMS based on experimental devices proposed by Chen et al. [15].
Results
Displacement uncertainty of graphene NEMS
A typical NEMS device with a doubleclamped freestanding graphene membrane is schematically shown in Figure 1. The substrate is doped Si with high conductivity, and the middle layer is SiO_{2 }insulator. A pump voltage can be applied between the membrane and the substrate. The experimental data of the devices are used in our simulation [15]. For graphene, we use a Young's modulus of E = 1.03 × 10^{12 }Pa, volumetric mass density of ρ = 2200 kg/m^{3}, based on previous theories and scanning tunneling microscope experiments [13,15,16].
Figure 1. Schematic of a doubleclamped graphene NEMS device.
In graphene sensors and transducers, to detect the molecular adsorbates or electrostatic forces, a strain ε will be generated in the graphene film [15,17]. When a strain exists in a graphene film, the tension T_{s }in Equation 1 can be deduced as T_{s }= ESε = EWhε. The zeropoint displacement uncertainty of the strained graphene film is given by
where ρ' represents the effective volumetric mass density of graphene film after applying strain. The typical measured strains in [15] are ε = 4 × 10^{5 }when ρ' = 4ρ and ε = 2 × 10^{4 }when ρ' = 6ρ. Based on Equation 2, measurable Δx_{zp }of the strained multilayer graphene films of various sizes are shown in Figure 2, and typical Δx_{zp }values of graphene NEMS under various ε are summarized in Table 1.
Figure 2. Δx_{zp }versus multilayer graphene film sizes with strains. (a) Monolayer graphene. (b) Bilayer graphene. (c) Trilayer graphene.
Table 1. Calculated Δx_{zp }(10^{4}nm) of monolayer (Mon), bilayer (Bi), and trilayer (Tri) graphene versus strain ε (L = 1.1 μm, W = 0.2 μm)
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.
Quantumsqueezing effects of graphene NEMS
To analyze quantumsqueezing effects in graphene NEMS devices, a backactionevading 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 [18]. The quantum capacitance C_{q }and screen capacitance C_{s }cannot be neglected [1820] owing to a graphene film thickness on the atomic scale. The quantum capacitance of monolayer graphene [21,22] is C_{q}^{monolayer }= 2e^{2}n^{1/2}/(ħv_{F}π^{1/2}), 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 [23].
Pumping the graphene membrane film from an initial thermal equilibrium state at frequency w_{m }= w, the variance of the complex amplitudes, ΔX^{2}_{1,2}(t, θ), are given by [24]
where N = [exp(ħw/k_{B}T)  1]^{1 }is the average number of quanta at absolute temperature T and frequency w, k_{B }is the Boltzmann constant, τ = Q/w is the relaxation time of the mechanical vibration, Q is the quality factor of the NEMS, and η = C_{T}V^{2}/8d^{2}M_{eff}w_{m}. When θ = 0, a maximum modulation state, namely, the best quantumsqueezed state, can be reached [9,21], and ΔX_{1 }can be simplified as ΔX_{1}(t) = [(ħ/2M_{eff}w_{a})(2N + 1)(τ^{1 }+ 2η)^{1}(τ^{1 }+ 2ηexp(τ^{1 }+ 2η)t)]^{1/2}. As t → ∞, the maximum squeezing of ΔX_{1 }is always finite, with expression of ΔX_{1}(t → ∞) ≈ [ħ(2N + 1)(1 + 2Qη)^{1}/2M_{eff}w]^{1/2}. The squeezing factor R, defined as R = ΔX_{1}/Δx_{zp }= ΔX_{1}/(ħ/2M_{eff}w)^{1/2}, can be expressed as
where ε is the strain applied on the graphene film. In order to achieve quantum squeezing, R must be less than 1. According to Equation 4, R values of monolayer and bilayer graphene films with various dimensions, strain ε, and applied voltages at T = 300 K and T = 5 K have been shown in Figure 3. Quantum squeezing is achievable in the region log R < 0 at T = 5 K. As shown in Figure 3, the applied strain increases the R values because of the increased fundamental angular frequency and the decreased Δx_{zp }caused by strain, which makes squeezing conditions more difficult to reach. Figure 4a has shown that ΔX_{1 }changes with applied voltages at T = 5 K, the red line represents the uncertainties of X_{1 }and the dashed reference line is ΔX = Δx_{zp}. As shown in Figure 4a, applying a voltage larger than 100 mV, we can obtain ΔX_{1 }< Δx_{zp}, which means that the displacement uncertainty is squeezed, and the quantum squeezing is achieved. Some typical R values of monolayer graphene film, obtained by varying the applied voltage V, such as strain ε, have been listed in Table 2 (with T = 300 K and Q = 125) and Table 3 (with T = 5 K and Q = 14000). As shown in Tables 2 and 3 and Figure 3, lowering the temperature to 5 K can dramatically decrease the R values. The lower the temperature, the larger the quality factor Q, which makes the squeezing effects stronger.
Figure 3. Log R versus applied voltages for graphene film structures at T = 300 K with Q = 125 and T = 5 K with Q = 14000. (a) Monolayer graphene and (b) bilayer graphene.
Figure 4. (a) ΔX_{1 }versus applied voltages of graphene film and the dashed reference line is ΔX = Δx_{zp}. (b) Time dependences of ΔX_{1 }and ΔX_{2}, which are expressed in units of Δx_{zp}, where time is in units of t_{ct}, θ = 0, and the dashed reference line is ΔX = Δx_{zp}. L = 1.1 μm, W = 0.2 μm, d = 0.1 μm, T = 5 K, Q = 14000, and V = 2.5V.
Table 2. 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)
Table 3. 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)
In contrast to the previous squeezing analysis proposed by Rugar and Grutter [9], in which steadystate solutions have been assumed and the minimum R is 1/2, we use timedependent 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.
Discussion
The ordering relation of Δx_{zp }for multilayer graphene is Δx_{zp}^{trilayer }< Δx_{zp}^{bilayer }< Δx_{zp}^{monolayer }shown in Figure 5a, as the zeropoint displacement uncertainty is inversely proportional to the film thickness. Squeezing factors R of multilayer graphene films follow the ordering relation; R_{trilayer }> R_{bilayer }> R_{monolayer}, as shown in Figure 5b, as R is proportional to the thickness of the graphene film. The thicker the film, the more difficult it is to achieve a quantumsqueezed state, which also explains why traditional NEMS could not achieve quantum squeezing due to their thickness of several hundred nanometers.
Figure 5. (a) Δx_{zp }versus various graphene film sizes. (b) Log R versus multilayer graphene film lengths and applied voltages at T = 5 K
For a clear view of squeezing factor R as a function of film length L, 2D curves from Figure 5b are presented in Figure 6. It is found that R approaches unity as L approaches zero, while R tends to be zero as L approaches infinity as shown in Figure 6a,b. It explains why R has some kinked regions, shown in the upper right part of Figure 5b with black circle, when the graphene film length is on the nanometer scale shown in Figure 3. To realize quantum squeezing, the graphene film length should be in the order of a few micrometers and the applied voltage V should not be as small as several mV, shown in Figure 6b. As L → 0, where the graphene film can be modeled as a quantum dot, the voltage must be as large as a few volts to modulate the film to achieve quantum squeezing. As L → ∞, where graphene films can be modeled as a 1D chain, the displacement uncertainty would be on the nanometer scale so that even a few mV of pumping voltage can modulate the film to achieve quantum squeezing easily.
Figure 6. R versus L with ε = 0.4 × 10^{5}, and V = 20 mV, 1.5 V.
By choosing the dimensions of a typical monolayer graphene NEMS device in [15] 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 radiofrequency singleelectrontransistor detection system can in principle attain such sensitivities [25]. In order to verify the quantum squeezing effects, a displacement detection scheme need be developed.
Conclusions
In conclusion, we presented systematic studies of zeropoint 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 zeropoint 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, highsensitivity graphenebased nanotransducers can be developed based on quantum squeezing.
Abbreviation
NEMS, nanoelectromechanical system.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both SY and YX designed and conducted all the works and drafted the manuscript. Both ZJ and YW have read and approved the final manuscript.
Acknowledgements
The authors gratefully acknowledge Prof. Raphael Tsu at UNCC, Prof. JeanPierre 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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