Nonlinear optical effect and nonlinear optical mass sensor based on graphene optomechanical system

Chen Hua-Jun

doi:10.7498/aps.69.20191745

Abstract

Graphene, atomically thin two-dimensional (2D) nanomaterial consisting of a single layer of carbon atoms, has received tremendous attention in the past few decades. Graphene may be considered as an excellent nanomaterial for fabricating nanomechanical resonator systems to investigate the quantum behavior of the motion of micromechanical resonators because of its unique properties of low mass density, high frequency, high quality-factor, and intrinsically small size. Additionally, graphene optomechanics based on a bilayer graphene resonator coupled to a microwave on-chip cavity, where light and micromechanical motion interact via the radiation pressure, has been demonstrtated experimentally recently. In this work, we demonstrate theoretically the nonlinear optical effect including optical bistability and four-wave mixing under the regimes woth different parameters and detunings in a graphene resonator-microwave cavity system. When the graphene optomechanics is driven by one strong pump laser beam, we find that the optical bistability can be controlled by tuning the power and the frequency of the pump beam. The four-wave mixing (FWM) phenomenon is also investigated and we find that sharp peaks in the FWM spectrum exactly are located at the resonant frequency of graphene resonator. Therefore, a straight nonlinear optical means for determining the resonant frequency of the graphene resonator is presented. Setting the cavity field resonating with pump field, and then scanning the probe frequency across the cavity frequency, one can easily and exactly obtain the resonant frequency of the resonator from the FWM spectrum. We further theoretically propose a mass sensor based on the graphene optomechanical system. The mass of external nanoparticles deposited onto the graphene resonator can be measured conveniently by tracking the shift of resonant frequency due to mass changing in the FWM spectrum. Compared with optomechanical mass sensors in linear regime, the nonlinear optical mass sensor may be immune to the detection noise. The system may have potential applications in communication networks for frequency conversion and provide a new platform for high sensitive sensing devices.

Keywords:

PACS:

42.50.Wk	(Mechanical effects of light on material media, microstructures and particles)
42.65.Pc	(Optical bistability, multistability, and switching, including local field effects)
78.47.nj	(Four-wave mixing spectroscopy)

Authors and contacts

Chen Hua-Jun

School of Mechanics and Photoelectric Physics, Anhui University of Science and Technology, Huainan 232001, China

Corresponding author: Chen Hua-Jun, chenphysics@126.com

FullText HTML

1. 引　言

石墨烯是由单层碳原子构成的具有原子尺寸厚度的二维纳米材料, 由于二维石墨烯材料具有一些显著的特征, 包括超低质量密度、超高频率、超高机械品质因子、以及本质的小尺寸等特性, 在过去十几年中吸引了人们的广泛兴趣. 基于这些特性, 石墨烯常被当作制备纳米机械振子系统和研究纳米机械振子量子运动行为的常用材料^[1-4]. 基于以上石墨烯独特的性质, 采用石墨烯编制的微纳机械振子系统将在微弱力的探测^[5-7]和纳米颗粒质量传感^[8,9]等方面有着重要的应用. 但是我们注意到, 目前基于石墨烯的应用依然停留在经典领域. 虽然现在石墨烯振子的制备和探测采用光学和高频技术^[10]已经完全可以实现, 但该方案依然停留在经典物理学领域, 实现量子领域的途径依然尚未发现.

采用纳米机械振子系统如何实现由经典物理学领域过渡到量子领域受到人们的广泛关注^[11], 最近已经有三个研究组分别通过压电晶体谐振器^[12]、超导微波振子^[13]和光力学晶体^[14]论证了机械振子的振动量达到小于1(即所说的基态冷却). 基于石墨烯编制的机械振子系统通过光-热相互作用在实验上实现了机械模式与光学模式的耦合^[15], 该方案中石墨烯材料覆盖于芯片尺度上的沟槽上形成光学谐振腔. 遗憾的是由于石墨烯材料编制的末端腔镜几乎是透明的(将近98%的光被透射出去), 导致由石墨烯振子构成的光腔品质因子很低(远小于1), 因此, 典型腔光力系统中的辐射压力^[11,16-18]现象很难在石墨烯末端腔镜构成的微腔中观测到^[15].

另一方面, 用于研究由辐射压力诱导的相干光声相互作用的光力系统^[11], 由于其在声子激光^[19,20]、压缩光产生^[21-23]、基态冷却^[12-14]等方面的潜在应用而受到研究者的广泛关注. 光力诱导透明^[24-28]作为光力系统中的著名现象也在光力系统中被观测到, 将为基于固态器件的光存储指明新的方向^[29,30]. 基于光力诱导透明的机械调制的慢光和快光现象在光波和微波领域相继被观测到^[27,29,31], 也将为芯片尺度上的存储、过滤或同步光传播的固态架构提供新的前景. 此外, 最近一些新奇的现象及应用也相继在光力系统中被证实, 如声子激光^[32,33]、光力放大现象^[34,35]、透明与高阶边带产生^[36,37], 奇异点光力学效应等^[38,39], 以及光力系统中的非互易^[40]. 这一系列些新奇现象的发现, 不仅拓展了对光力系统的理论研究, 而且也为光力系统的潜在应用指明了方向.

由于石墨烯振子系统中的辐射压力远小于腔光力系统中的辐射压力^[11], 为了放大石墨烯振子系统中的辐射压力, 利用石墨烯振子和高品质因子的超导微波腔来制造腔光力系统是一种很自然的选择. 最近, 将具有高品质因子的多层石墨烯振子与超导微波腔通过电容耦合构成复合石墨烯光力系统在实验上被证实^[41,42]. 采用多层石墨烯的原因是尽管多层石墨烯的质量远大于单层, 但多层石墨烯的电阻较低, 而低电阻特性有益于与超导微波腔耦合. 前不久, 实验上也制备出了基于双层石墨烯机械振子与微波腔耦合而构成的复合石墨烯光力系统, 该方案实现了通过辐射压力诱导的机械运动与微波腔的相互作用^[43], 为通往量子领域打开了一扇大门.

此外, 光力系统也提供了一个研究非线性光与物质相互作用的平台. 在光力系统中的非线性现象中, 光学双稳态和四波混频作为典型的非线性现象而受到研究者的广泛关注. 近些年, 光学双稳态现象在许多光力系统中被观测到, 如波色爱因斯坦光力系统^[44,45]、量子阱光力系统^[46]、超冷原子系统^[47,48]以及复合光力系统^[49,50]. 当用一束频率为 ${\omega _{\rm{p}}}$ 较强的泵浦光和一束频率为 ${\omega _{\rm{s}}}$ 较弱的探测光同时驱动光力系统时, 探测透射场中不仅有频率为 ${\omega _{\rm{p}}}$ 和 ${\omega _{\rm{s}}}$ 的光场, 也将会出现一个频率为 $2{\omega _{\rm{p}}} - {\omega _{\rm{s}}}$ 的四波混频光场^[51,52]. 然而, 很少有研究者关注石墨烯光力系统中的光学双稳态和四波混频现象.

本文研究由石墨烯纳米机械振子与微波腔耦合而形成的石墨烯光力系统以及该系统中的非线性现象. 首先, 研究了该系统中的光学双稳态行为, 并且双稳态可由泵浦场的功率和频率所控制. 其次, 当同时用一束频率为 ${\omega _{\rm{p}}}$ 的较强泵浦场和一束频率为 ${\omega _{\rm{s}}}$ 较弱的信号光来驱动该系统时, 透射场中将出现四波混频场. 通过对该系统中的四波混频现象进行研究, 给出一种由四波混频谱测量机械振动频率的方法. 最后, 提出一种测量纳米颗粒质量的非线性光学方案. 当石墨烯振子上沉积纳米颗粒时, 由四波混频谱中的共振频移可测量出纳米颗粒的质量.

2. 模型与理论

石墨烯光力系统模型如图1所示^[41-43], 其中石墨烯机械振子与高品质因子的微波腔通过电容耦合, 在图1中C代表等效电容(带下标的是相应的电容器), L代表等效电感(带下标的是相应的电感). 我们用一个等效的电容 ${C_{\rm{g}}}(x) = \displaystyle\int {\dfrac{{{\varepsilon _0}h}}{{d - U(z)}}} {\rm{d}}z$ 来描述有效电容, 该表达式中h代表石墨烯振子的宽度, $U(z)$ 代表模式的形状, d是真空带隙, x是石墨烯振子的最大偏差位移. 该石墨烯耦合系统的总电容 ${C_{{\rm{tot}}}}(x) = {C_{\rm{g}}}(x) + C$ , 因此该微波腔的腔频为 ${\omega _{\rm{c}}} = {1 / {\sqrt {L{C_{{\rm{tot}}}}(x)} }}$ . 分别用频率为 ${\omega _{\rm{p}}}$ 和 ${\omega _{\rm{s}}}$ 的强泵浦光和弱探测光驱动该石墨烯光力系统, 并且以 ${\omega _{\rm{p}}}$ 为旋转参考系, 得到系统的哈密顿量如下^{[41-43,53-57]}:

$图 1 石墨烯光力系统与非线性质量传感示意图, 其中该系统由一束频率为${\omega _{\rm{p}}}$的泵浦光和一束频率${\omega _{\rm{s}}}$的信号光驱动\r\nFig. 1. Schematic of graphene optomechanical system and nonlinear optical mass sensor driven by a strong pump field ${\omega _{\rm{p}}}$ and a weak signal field ${\omega _{\rm{s}}}$.$

图 1 石墨烯光力系统与非线性质量传感示意图, 其中该系统由一束频率为

${\omega _{\rm{p}}}$

的泵浦光和一束频率

${\omega _{\rm{s}}}$

的信号光驱动

Fig. 1. Schematic of graphene optomechanical system and nonlinear optical mass sensor driven by a strong pump field

${\omega _{\rm{p}}}$

and a weak signal field

${\omega _{\rm{s}}}$

下载: 全尺寸图片幻灯片

$\begin{split} H =\; & \hbar {\varDelta _{\rm{p}}}{a^\dagger }a + \hbar {\omega _{\rm{m}}}{b^\dagger }b - hg{a^\dagger }a({b^\dagger } + b)\\ & + {\rm{i}}\hbar \sqrt {{k_{\rm{e}}}} ({E_{\rm{p}}}{a^\dagger } - E_{\rm{p}}^*a) \\ &+ {\rm{i}}\hbar \sqrt {{k_{\rm{e}}}} \Big({E_{\rm{s}}}{a^ + }{{\rm{e}}^{ - {\rm{i}}\varOmega t}} - E_{\rm{s}}^*a{{\rm{e}}^{{\rm{i}}\varOmega t}}\Big), \end{split}$

(1)

式中第一项和第二项是自由哈密顿量, 这里引入了二次量子化的产生算符 ${a^\dagger }({b^\dagger })$ 和湮灭算符 $a(b)$ 来分别描述腔场和石墨烯振子, 其中 ${\varDelta _{\rm{p}}} = {\omega _{\rm{c}}} - {\omega _{\rm{p}}}$ 代表腔频与泵浦频率失谐, ${\omega _{\rm{m}}}$ 为石墨烯振子频率. 第三项是微波腔场与石墨烯振子通过辐射压力耦合的相互作用项, g是耦合强度. 最后两项分别是微波腔场与泵浦光场和探测光场之间的相互作用, 探测光场与泵浦光场之间的失谐为 $\varOmega = {\omega _{\rm{s}}} - {\omega _{\rm{p}}}$ . ${E_{\rm{p}}}$ 和 ${E_{\rm{s}}}$ 分别是泵浦光场和信号光场的振幅, 与相应的功率之间的关系为 $\left| {{E_{\rm{p}}}} \right| = \sqrt {{P / {\hbar {\omega _{\rm{p}}}}}}$ 和 $\left| {{E_{\rm{s}}}} \right| = \sqrt {{{{P_{\rm{s}}}} / {\hbar {\omega _{\rm{s}}}}}}$ , ${k_{\rm{e}}}$ 描述腔场损耗率.

通过朗之万运动方程, 同时加入影响腔场和石墨烯振子的噪声和阻尼, 得到如下量子郎之万方程:

$\begin{split}\dot a(t) =\; & - ({\rm{i}}{\varDelta _{\rm{p}}} + \kappa )a(t) + {\rm{i}}ga(t)X(t)\\ &- \sqrt {{\kappa _{\rm{e}}}} ({E_{\rm{p}}} + {E_{\rm{s}}}{{\rm{e}}^{ - {\rm{i}}\varOmega t}}) + {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over a} _{{\rm{in}}}}(t),\end{split}$

(2)

$\ddot X(t) + {\gamma _{\rm{m}}}\dot X(t) + \omega _{\rm{m}}^{\rm{2}}X(t) = 2g{\omega _{\rm{m}}}{a^ + }(t)a(t) + \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \xi } (t),$

(3)

其中 $X = {b^\dagger } + b$ 是位置算符, ${\gamma _{\rm{m}}}$ 是石墨烯振子的衰减率, $\kappa$ 是微波腔模的衰减率.

${{\overset{\frown} a} _{{\rm{in}}}}(t)$ 是 $\delta$ 关联的郎之万噪声算符, 它的平均值 $\Big\langle {{{\overset{\frown} a}_{{\rm{in}}}}(t)} \Big\rangle = 0$ , 其关联函数为

$\left\langle {{{\overset{\frown} a}_{{\rm{in}}}}(t){{\overset{\frown} a}_{{\rm{in}}}}(t')} \right\rangle \sim \delta (t - t').$

(4)

石墨烯振子受一个平均值为零的布朗随机力影响, 其关联函数如下

$\begin{split} & \Big\langle {{{\overset{\frown} {\xi }}^\dagger }(t)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\xi } (t')} \Big\rangle \\=\; & \frac{{{\gamma _{\rm{m}}}}}{{{\omega _{\rm{m}}}}}\int {\frac{{{\rm d}\omega }}{{2{\text{π}}}}\omega {{\rm{e}}^{ - {\rm{i}}\omega (t - t')}}\left[ {1 + \coth \left(\frac{{\hbar \omega }}{{2{k_{\rm{B}}}T}}\right)} \right]},\end{split}$

(5)

式中 ${k_{\rm{B}}}$ 为玻尔兹曼常数, T是该石墨烯光力系统所处环境温度.

由于泵浦光场强度远大于探测光场强度, 可把腔场算符和石墨烯振子算符写成稳态值和小的涨落之和^[58], 即 $a = {a_{\rm{s}}} + {\rm{\delta }}a$ 和 $X = {X_{\rm{s}}} + {\rm{\delta }}X$ , 然后代入方程(2)和(3)中, 同时忽略高阶非线性项, 得到两组方程: 分别为稳态平均值方程组和小的涨落方程组. 稳态平均值方程组由 ${a_{\rm{s}}} = {\sqrt {{\kappa _{\rm{e}}}} {E_{\rm{p}}}} / {\left[ {{\rm{i}}\left( {{\varDelta _{\rm{p}}} + g{X_{\rm{s}}}} \right) + \kappa } \right]}$ 和 ${X_{\rm{s}}} = 2 g{\left| {{a_{\rm{s}}}} \right|^2}/{\omega _{\rm{m}}}$ 组成, 它们决定了腔内光子数( ${n_{\rm{c}}} = {\left| {{a_{\rm{s}}}} \right|^2}$ ):

${n_{\rm{c}}}[{\kappa ^2} + {({\varDelta _{\rm{p}}} - 2{g^2}{n_{\rm{c}}}/{\omega _{\rm{m}}})^2}] = {\kappa _{\rm{e}}}{\left| {{E_{\rm{p}}}} \right|^2}.$

(6)

对于与小的涨落有关的方程, 做变换^[58]

$\left\langle {\delta O} \right\rangle = {O_ + }{{\rm{e}}^{ - {\rm{i}}\varOmega t}} + {O_ - }{{\rm{e}}^{{\rm{i}}\varOmega t}},$

(7)

解方程得到

${a_ - } = \frac{{{\rm{i}}ga_{\rm{s}}^2{\chi ^2}\sqrt {{\kappa _{\rm{e}}}} {E_{\rm{s}}}}}{{[\kappa + {\rm{i}}(\bar \varDelta + {\rm{\varOmega }})][\kappa - {\rm{i}}(\bar \varDelta - \varOmega )] - {g^2}{\chi ^2}n_{\rm{c}}^2}},$

(8)

其中 $\chi = {{2 g{\omega _{\rm{m}}}} / {(\omega _{\rm{m}}^2 - {\varOmega ^2} + {\rm{i}}\varOmega {\gamma _{\rm{m}}})}}$ , $\bar \varDelta = {\varDelta _{\rm{p}}} - g{X_{\rm{s}}} - g\chi {n_0}$ .

引入标准的输入-输出关系^[58] ${a_{{\rm{out}}}}(t) = {a_{{\rm{in}}}}(t) - \sqrt {{\kappa _{\rm{e}}}} a(t)$ , 得到

$\begin{split} & \left\langle {{a_{{\rm{out}}}}(t)} \right\rangle \\ =\;& ({E_{\rm{p}}} - \sqrt {{\kappa _{\rm{e}}}} {a_0}){{\rm{e}}^{ - {\rm{i}}{\omega _{\rm{p}}}t}} + ({E_{\rm{s}}} - \sqrt {{\kappa _{\rm{e}}}} {a_ + }){{\rm{e}}^{ - {\rm{i}}(\varOmega + {\omega _{\rm{p}}})t}} \\ &- \sqrt {{\kappa _{\rm{e}}}} {a_ - }{{\rm{e}}^{ - {\rm{i}}(\varOmega - {\omega _{\rm{p}}})t}} \\ =\;& ({E_{\rm{p}}} - \sqrt {{\kappa _{\rm{e}}}} {a_0}){{\rm{e}}^{ - {\rm{i}}{\omega _{\rm{p}}}t}} + ({E_{\rm{s}}} - \sqrt {{\kappa _{\rm{e}}}} {a_ + }){{\rm{e}}^{ - {\rm{i}}{\omega _{\rm{s}}}t}} \\ &- \sqrt {{\kappa _{\rm{e}}}} {a_ - }{{\rm{e}}^{ - {\rm{i}}(2{\omega _{\rm{p}}} - {\omega _{\rm{s}}})t}},\\[-12pt] \end{split}$

(9)

其中 ${a_{{\rm{out}}}}(t)$ 是输出场算符. 方程(9)中的输出场包含了三个部分: 第一项对应于振幅为 ${E_{\rm{p}}}$ 频率为 ${\omega _{\rm{p}}}$ 的驱动场; 第二项对应于频率为 ${\omega _{\rm{s}}}$ 的探测场(反斯托克斯场)导致光力诱导透明产生^{[17,24-29,44]}; 第三项对应于频率为 $2{\omega _{\rm{p}}} - {\omega _{\rm{s}}}$ 的四波混频场(斯托克斯场)^[52]. 四波混频场的强度可定义如下:

${\rm{FWM}} = \left| {\frac{{\sqrt {{\kappa _{\rm{e}}}} {a_ - }}}{{{E_{\rm{s}}}}}} \right|,$

(10)

其取决于光力耦合强度g和泵浦功率P.

通过对四波混频的研究, 本文进一步呈现一种基于石墨烯光力系统的非线性光学质量传感方案. 当石墨烯振子表面吸附额外纳米颗粒时(如金属纳米颗粒、病毒、细胞、抗体等), 石墨烯振子质量的变化引起灵敏的共振频移. 质量传感的基本原理便是探测由沉积在石墨烯振子上的额外纳米颗粒所产生的共振频移 ${\rm{\delta }}f$ . 当质量为 ${\rm{\delta }}m$ 的待测纳米粒子吸附在有效质量为 ${m_{{\rm{eff}}}}$ 石墨烯振子表面上时(这里 ${m_{{\rm{eff}}}} \gg{\rm{\delta }}m$ ), 石墨烯振子的振动频率便会移动到 ${\omega _{\rm{m}}} + {\rm{\delta }}f$ . 在四波混频谱中监测微小频移 ${\rm{\delta }}f$ , 这样沉积在石墨烯振子表面上的纳米颗粒的质量就可被确定. 微小频移与沉积在石墨烯振子表面上的纳米颗粒质量遵循如下关系式:

${\rm{\delta }}m = {\Re ^{ - 1}}{\rm{\delta }}f,$

(11)

其中 ${\Re ^{ - 1}} = {( - 2 M/{\omega _{\rm{m}}})^{ - 1}}$ 是质量响应率^[59].

3. 数值结果与讨论

在22 mK的温度下, 石墨烯纳米振子耦合到微波腔系统的参数如下^[43]: m = 10 fg, m_eff = 1.89 × 10^–13 g, C = 45 fF, C_g = 0.4 fF, ω_c/2π = 7.82 GHz, ω_m/2π = 24 MHz, κ/2π = 5.6 MHz, g/2π = 35 Hz, 石墨烯机械振子的机械品质因子Q_m = 15 × 10³, γ_m/2π = ω_m/Q_m = 1.6 kHz.

(6)式描述腔内光子数 ${n_{\rm{c}}}$ 的立方方程, 将呈现出光学双稳态行为. 图2画出了在三个不同泵浦功率驱动下腔内光子数 ${n_{\rm{c}}}$ 作为腔场与泵浦场频率失谐 ${\varDelta _{\rm{p}}} = {\omega _{\rm{c}}} - {\omega _{\rm{p}}}$ 的函数. 当泵浦功率比较小时, 如 $P < 1.0\;\text{μ}{\rm{W}}$ ( $P = 0.2\;{\rm{ \text{μ} W}}$ ), 虽然图2中没有给出, 但腔内光子数 ${n_{\rm{c}}}$ 的曲线呈现洛伦兹线形. 随着进一步增加泵浦功率P, 当泵浦功率达到临界值时, 双稳态现象产生. 图2给出泵浦功率从 $P = 1.0\;{\rm{ \text{μ} W}}$ 到 $P = 4.0\;{\rm{\text{μ} W}}$ 腔内光子数 ${n_{\rm{c}}}$ 的演化过程. 显然腔内光子数曲线由小泵浦功率下的洛伦兹线形转换到较大功率下的不对称曲线. 由(6)式可知腔内光子数 ${n_{\rm{c}}}$ 有三个实根, 其中最大和最小的根是稳定的, 而中间的一个根是不稳定的. 腔内光子数 ${n_{\rm{c}}}$ 取决于泵浦功率P, 并且较大的腔场-泵浦场失谐 ${\varDelta _{\rm{p}}}$ 也有益于双稳态现象的观测.

$图 2 在三个不同泵浦功率条件下石墨烯光力腔内光子数作为腔-泵浦失谐${\varDelta _{\rm{p}}}$的函数\r\nFig. 2. Mean intracavity photon number of graphene optomechanical cavity as a function of the cavity-pump detuning ${\varDelta _{\rm{p}}}$ with four pump powers.$

图 2 在三个不同泵浦功率条件下石墨烯光力腔内光子数作为腔-泵浦失谐

${\varDelta _{\rm{p}}}$

的函数

Fig. 2. Mean intracavity photon number of graphene optomechanical cavity as a function of the cavity-pump detuning

${\varDelta _{\rm{p}}}$

with four pump powers.

下载: 全尺寸图片幻灯片

此外, 本文进一步讨论了光学双稳态行为分别在红边带失谐( ${\varDelta _{\rm{p}}} = {\omega _{\rm{m}}}$ )和蓝边带失谐( ${\varDelta _{\rm{p}}} = - {\omega _{\rm{m}}}$ )条件下的演化过程. 图3(a)和图3(b)分别给出在 ${\varDelta _{\rm{p}}} = {\omega _{\rm{m}}}$ 和 ${\varDelta _{\rm{p}}} = - {\omega _{\rm{m}}}$ 条件下, 腔内光子数 ${n_{\rm{c}}}$ 作为泵浦功率P的函数. 显然, 光学双稳态呈现类似磁滞回线的行为. 不同的是, 在两种不同的失谐条件下, 腔内光子数 ${n_{\rm{c}}}$ 随着泵浦功率P的演化过程呈现镜像反对称.

$图 3 (a)在失谐${\varDelta _{\rm{p}}} = {\omega _{\rm{m}}}$时, 腔内光子数${n_{\rm{c}}}$作为泵浦功率P的函数; (b) 在失谐${\varDelta _{\rm{p}}} = - {\omega _{\rm{m}}}$时, 腔内光子数${n_{\rm{c}}}$作为泵浦功率P的函数\r\nFig. 3. (a) The mean intracavity photon number ${n_{\rm{c}}}$ as a function of P for ${\varDelta _{\rm{p}}} = {\omega _{\rm{m}}}$; (b) mean intracavity photon number ${n_{\rm{c}}}$ as a function of P for ${\varDelta _{\rm{p}}} = - {\omega _{\rm{m}}}$.$

图 3 (a)在失谐

${\varDelta _{\rm{p}}} = {\omega _{\rm{m}}}$

时, 腔内光子数

${n_{\rm{c}}}$

作为泵浦功率P的函数; (b) 在失谐

${\varDelta _{\rm{p}}} = - {\omega _{\rm{m}}}$

时, 腔内光子数

${n_{\rm{c}}}$

作为泵浦功率P的函数

Fig. 3. (a) The mean intracavity photon number

${n_{\rm{c}}}$

as a function of P for

${\varDelta _{\rm{p}}} = {\omega _{\rm{m}}}$

; (b) mean intracavity photon number

${n_{\rm{c}}}$

as a function of P for

${\varDelta _{\rm{p}}} = - {\omega _{\rm{m}}}$

下载: 全尺寸图片幻灯片

以上现象的物理本质来自于光相干干涉效应: 当用泵浦光场和探测光场同时驱动该石墨烯光力系统时, 导致辐射压力在拍频 $\varOmega$ 共振相干, 诱导石墨烯振子在其共振频率处振动, 进而诱导出斯托克斯( ${\omega _{\rm{S}}} = {\omega _{\rm{p}}} - {\omega _{\rm{m}}}$ )和反斯托克斯散射光( ${\omega _{{\rm{AS}}}} = {\omega _{\rm{p}}} + {\omega _{\rm{m}}}$ ). 当微波腔分别处于红边带失谐( ${\varDelta _{\rm{p}}} = {\omega _{\rm{m}}}$ )和蓝边带失谐( ${\varDelta _{\rm{p}}} = - {\omega _{\rm{m}}}$ )时, 处于频率为 ${\omega _{\rm{p}}} - {\omega _{\rm{m}}}$ 斯托克斯散射光和频率为 ${\omega _{\rm{p}}} + {\omega _{\rm{m}}}$ 的反斯托克斯散射光将分别被抑制, 而频率为 ${\omega _{\rm{p}}} + {\omega _{\rm{m}}}$ 的反斯托克斯散射光和频率为 ${\omega _{\rm{p}}} - {\omega _{\rm{m}}}$ 的斯托克斯散射光将重新建立腔肠, 诱导光学双稳态产生.

另一方面, 四波混频现象是腔光力系统中的另一种非线性行为. 在图4中通过(10)式画出了四波混频强度作为探测场-微波腔场失谐 ${\varDelta _{\rm{s}}} = {\omega _{\rm{s}}} - {\omega _{\rm{c}}}$ 的函数. 石墨烯光力系统的参数中石墨烯振子频率为 ${\omega _{\rm{m}}} = 24\;{\rm{ MHz}}$ , 图4中进一步考虑其他的振子频率, 如ω_m= 22 MHz, ω_m= 26 MHz和ω_m = 28 MHz. 从图4(a)发现, 虽然随着石墨烯振子频率的增加, 四波混频强度呈现减弱的趋势, 但四波混频谱线中尖峰的位置正对应于 ${\varDelta _{\rm{s}}} = \pm {\omega _{\rm{m}}}$ , 即尖峰的位置正对应于石墨烯振子的频率, 如图4(b)和图4(c)所示. 该现象的物理本质来自于机械振动模式与两束光场通过微波腔相互作用而产生的量子相干效应. 当探测场与泵浦场之间的失谐 $\varOmega = {\omega _{\rm{s}}} - {\omega _{\rm{p}}}$ 相接近时, 石墨烯振子开始振动并产生斯托克斯散射光( ${\omega _{\rm{s}}} = {\omega _{\rm{p}}} - {\omega _{\rm{m}}}$ ). 该过程中, 电子在 ${\omega _{\rm{p}}} - {\omega _{\rm{m}}}$ 频率附近吸收两个光子, 放出一个光子, 诱导四波混频产生.

$图 4 (a) 在四个不同石墨烯振子频率时, 四波混频谱FWM作为探测-腔失谐${\varDelta _{\rm{s}}}$的函数; (b) 和 (c)分别是左边和右边尖峰的放大\r\nFig. 4. (a) The four-wave mixing (FWM) spectrum as a function of probe-cavity detuning ${\varDelta _{\rm{s}}}$ under four different graphene resonator frequencies; (b) and (c) are the amplifications of the left and right peaks.$

图 4 (a) 在四个不同石墨烯振子频率时, 四波混频谱FWM作为探测-腔失谐

${\varDelta _{\rm{s}}}$

的函数; (b) 和 (c)分别是左边和右边尖峰的放大

Fig. 4. (a) The four-wave mixing (FWM) spectrum as a function of probe-cavity detuning

${\varDelta _{\rm{s}}}$

under four different graphene resonator frequencies; (b) and (c) are the amplifications of the left and right peaks.

下载: 全尺寸图片幻灯片

图4同时也指明一种测量石墨烯振子频率的非线性光学方法. 石墨烯振子频率的确定包括两步骤: 第一步, 使泵浦场频率与微波腔场频率相等, 即 ${\varDelta _{\rm{p}}} = 0$ ; 第二步, 探测场扫描石墨烯光力系统, 四波混频谱中尖峰的位置便对应石墨烯振子的振动频率. 这样通过探测四波混频谱便可以较容易而且直接地测得石墨烯振子的振动频率.

石墨烯振子的频率确定后, 接下来给出一种测量纳米颗粒质量的非线性光学方法. 质量传感的原理在于监测由吸附在石墨烯振子上的微小纳米颗粒所引起的微小频移 ${\rm{\delta }}f$ , 然后通过(11)式便可确定纳米颗粒的质量. 本文以测量纳米颗粒(如金属纳米颗粒、生物分子等)的质量为例, 纳米颗粒质量的数量级为飞克( $1\;{\rm{ fg}} = {10^{ - 15}}\;{\rm{ g}}$ ). 把纳米颗粒沉积到石墨烯振子的表面(如图1所示), 然后通过观测四波混频谱中的频移, 这里假定纳米颗粒均匀地分布在石墨烯振子的表面, 并且纳米颗粒的质量不影响石墨烯振子的弹性系数. 图5给出了石墨烯振子表面没有吸附(黑色曲线)和吸附(彩色曲线)纳米颗粒时四波混频谱作为探测失谐 ${\varDelta _{\rm{s}}}$ 的函数. 当石墨烯振子表面没有沉积纳米颗粒时(黑色曲线), 四波混频谱线中尖峰的位置正对应于石墨烯振子的频率. 当石墨烯振子表面吸附纳米颗粒后, 随着沉积纳米颗粒质量的增加, 频率呈现显著的移动, 如图5中彩色曲线所示. 通过测量频率的移动, 便可确定纳米颗粒的质量. 图5中的插图显示了频移与沉积到石墨烯振子表面上纳米颗粒质量之间的线性关系, 斜率表示石墨烯振子的质量灵敏度.

$图 5 当把纳米颗粒沉积到石墨烯振子表面上时, 四波混频谱的频移. 插图是纳米颗粒的质量与频移之间的线性关系\r\nFig. 5. The four-wave mixing (FWM) spectrum after landing the nanoparticles on the surface of graphene resonator and the color curves shows the mechanical frequency-shifts. The inset shows the linear relationship between the frequency-shifts and the mass of the nanoparticles.$

图 5 当把纳米颗粒沉积到石墨烯振子表面上时, 四波混频谱的频移. 插图是纳米颗粒的质量与频移之间的线性关系

Fig. 5. The four-wave mixing (FWM) spectrum after landing the nanoparticles on the surface of graphene resonator and the color curves shows the mechanical frequency-shifts. The inset shows the linear relationship between the frequency-shifts and the mass of the nanoparticles.

下载: 全尺寸图片幻灯片

质量响应率 $\Re = {{\partial {\omega _{\rm{m}}}}/ {\partial M}} = - {\omega _{\rm{m}}}/2 M$ 是评估机械振子质量传感性能的重要参数. 显然振子的质量密度越低、频率越高、机械品质因子越高其灵敏度就越高. 本文中 $\left| \Re \right| = 6.35 \times {10^{19}}\;{\rm{Hz}}/{\rm{g}}$ . 在质量传感过程中各种噪声源, 包括本质噪声和外在噪声^[59,60], 将会影响石墨烯振子的质量传感性能和灵敏度. 振子机械运动的本质热噪声是主要的噪声源, 将会影响传感灵敏度, 由于石墨烯光力系统工作在较低的温度条件下^[34] ( $T = 22\;{\rm{ mK}}$ ), 因此热噪声可被有效降低. 外在的噪声源主要体现在设备的探测噪声. 但是对于一些复杂精密的信号读出器件来说, 探测噪声将是主要的噪声源, 而且将变得非常显著. 目前的理论和实验都论证了非线性效应能有效增强质量传感的灵敏度^[61,62]. 非线性质量传感能产生较大的共振振幅和输出信号, 但不会同时放大噪声信号, 因此非线性现象有益于抵消探测噪声的影响和提高信噪比. 此外, 有研究表明非线性质量传感可能对探测噪声免疫.

另外, 与传统测量微小颗粒质量的质谱仪相比, 本文所提出的非线性光学质量传感方案不需要待测粒子被电离, 也能有效地克服电学测量过程中所产生的热效应. 同时由于采用泵浦-探测双光驱动石墨烯光力系统产生了一个拍频, 因此对于高频和低频的机械振子都适用. 当机械振子的寿命较长时, 质量传感的精确度将显著提高^[63-65].

4. 结　论

本文研究了基于石墨烯光力系统中的非线性光学现象, 分别着重讨论了非线性光学效应中的光学双稳态和四波混频现象. 考虑同时用一束较强的泵浦光和一束较弱的探测光来驱动石墨烯光力系统. 通过对泵浦光光腔和失谐的控制, 实现了对光学双稳态的有效操控. 此外, 也讨论了共振条件下的四波混频, 研究不同石墨烯振子频率下的四波混频谱, 发现四波混频谱中尖峰的位置对应于石墨烯振子的频率, 这样给出了一种测量石墨烯振子频率的非线性光学方法. 基于该系统进一步提出一种非线性光学质量传感方案, 并且该非线性方案将对探测噪声免疫. 石墨烯光力学系统在实验上的证实也为质量传感的实施提供了基础.

References

[1]	Chen C, Rosenblatt S, Bolotin K I, Kalb W, Kim P, Kymissis I, Stormer H L, Heinz T F, Hone J 2009 Nat. Nanotechnol. 4 861 Google Scholar
[2]	Eichler A, Moser J, Chaste J, Zdrojek M, Wilson-Rae I, Bachtold A 2011 Nat. Nanotechnol. 6 339 Google Scholar
[3]	Song X, Oksanen M, Sillanpää M A, Craighead H G, Parpia J M, Hakonen P J 2012 Nano. Lett. 12 198 Google Scholar
[4]	Chen C, Lee S, Deshpande V V, Lee G H, Lekas M, Shepard K, Hone J 2013 Nat. Nanotechnol. 8 923 Google Scholar
[5]	Bunch J S, van der Zande A M, Verbridge S S, Frank I W, Tanenbaum D M, Parpia J M, Craighead H G, McEuen P L 2007 Science 315 490 Google Scholar
[6]	Moser J, Güttinger J, Eichler A, Esplandiu M J, Liu D E, Dykman M I, Bachtold A 2013 Nat. Nanotechnol. 8 493 Google Scholar
[7]	Stapfner S, Ost L, Hunger D, Reichel J, Favero I, Weig E M 2013 Appl. Phys. Lett. 102 151910 Google Scholar
[8]	Chiu H Y, Hung P, Postma H W C 2008 Nano. Lett. 8 4342 Google Scholar
[9]	Chaste J, Eichler A, Moser J, Ceballos G, Rurali R, Bachtold A A 2012 Nat. Nanotechnol. 7 301 Google Scholar
[10]	Singh V, Sengupta S, Solanki H S, Dhall R, Allain A, Dhara S, Pant P, Deshmukh M M 2010 Nanotechnology 211 65204
[11]	Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391 Google Scholar
[12]	O’Connell A D, Hofheinz M, Ansmann M, Bialczak R C, Lenander M, Lucero E, Neeley M, Sank D, Wang H, Weides M, Wenner J, Martinis J M, Cleland A N 2010 Nature 464 697 Google Scholar
[13]	Teufel J D, Donner T, Li D, Harlow J W, Allman M S, Cicak K, Sirois A J, Whittaker J D, Lehnert K W, Simmonds R W 2011 Nature 475 359 Google Scholar
[14]	Chan J, Alegre T P M, Safavi-Naeini A H, Hill J T, Krause A, Gröblacher S, Aspelmeyer M, Painter O 2011 Nature 478 89 Google Scholar
[15]	Barton R A, Storch I R, Adiga V P, Sakakibara R, Cipriany B R, Ilic B, Wang S P, Ong P, McEuen P L, Parpia J M, Craighead F G 2012 Nano. Lett. 12 4681 Google Scholar
[16]	Peterson R W, Purdy T P, Kampel N S, Andrews R W, Yu P L, Lehnert K W, Regal C A 2016 Phys. Rev. Lett. 116 063601 Google Scholar
[17]	Chen H J 2018 J. Appl. Phys. 124 153102 Google Scholar
[18]	Rossi M, Mason D, Chen J, Tsaturyan Y, Schliesser A 2018 Nature 563 53 Google Scholar
[19]	Grudinin I S, Lee H, Painter O, Vahala K J, 2010 Phys. Rev. Lett. 104 083901 Google Scholar
[20]	Jing H, Özdemir S K, Lü X Y, Zhang J, Yang L, Nori F 2014 Phys. Rev. Lett. 113 053604 Google Scholar
[21]	Brooks D W C, Botter T, Schreppler S, Purdy T P, Brahms N, Stamper-Kurn D M 2012 Nature 488 476 Google Scholar
[22]	Safavi-Naeini A H, Gröblacher S, Hill J T, Chan J, Aspelmeyer M, Painter O 2013 Nature 500 185 Google Scholar
[23]	Purdy T P, Yu P L, Peterson R W, Kampel N S, Regal C A 2013 Phys. Rev. X 3 031012
[24]	Agarwal G S, Huang S M 2010 Phys. Rev. A 81 041803 Google Scholar
[25]	Weis S, Riviere R, Deleglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Science 330 1520 Google Scholar
[26]	Teufel J D, Li D, Allman M S, Cicak K, Sirois A J, Whittaker J D, Simmonds R W 2011 Nature 471 204 Google Scholar
[27]	Safavi-Naeini A H, Alegre T P M, Chan J, Eichenfield M, Winger M, Lin Q, Hill J T, Chang D E, Painter O 2011 Nature 472 69 Google Scholar
[28]	Karuza M, Biancofiore C, Bawaj M, Molinelli C, Galassi M, Natali R, Tombesi P, Di Giuseppe G, Vitali D 2013 Phys. Rev. A 88 013804 Google Scholar
[29]	Zhou X, Hocke F, Schliesser A, Marx A, Huebl H, Gross R, Kippenberg T J 2013 Nat. Phys. 9 179 Google Scholar
[30]	Fan L, Fong KY, Poot M, Tang H X 2015 Nat. Commun. 6 5850 Google Scholar
[31]	Massel F, Heikkilä T T, Pirkkalainen J M, Cho S U, Saloniemi H, Hakonen P J, Sillanpää M A 2011 Nature 480 351 Google Scholar
[32]	Jing H, Özdemir S K, Lü X Y, Zhang J, Yang L, Nori F 2014 Phys. Rev. Lett. 113 053604
[33]	Jiang Y, Maayani S, Carmon T, Nori F, Jing H 2018 Phys. Rev. Appl. 10 064037 Google Scholar
[34]	Jiao Y, Lü H, Qian J, Li Y, Jing H 2016 New J. Phys. 18 083034 Google Scholar
[35]	Lu T X, Jiao Y F, Zhang H L, Saif F, Jing H 2019 Phys. Rev. A 100 013813 Google Scholar
[36]	Zhang H, Saif F, Jiao Y, Jing H 2018 Opt. Express 26 25199 Google Scholar
[37]	Jiao Y F, Lu T X, Jing H 2018 Phys. Rev. A 97 013843 Google Scholar
[38]	Lü H, Özdemir S K, Kuang L M, Nori F, Jing H 2017 Phys. Rev. Appl. 8 044020 Google Scholar
[39]	Lü H, Wang C, Yang L, Jing H 2018 Phys. Rev. Appl. 10 014006 Google Scholar
[40]	Li B, Huang R, Xu X, Miranowicz A, Jing H 2019 Photonics Res. 7 630 Google Scholar
[41]	Weber P, Guttinger J, Tsioutsios I, Chang D E, Bachtold A 2014 Nano. Lett. 14 2854 Google Scholar
[42]	Singh V, Bosman S J, Schneider B H, Blanter Y M, Castellanos-Gomez A, Steele G A 2014 Nat. Nanotechnol. 9 820 Google Scholar
[43]	Song X, Oksanen M, Li J, Hakonen P J, Sillanpää M A 2014 Phys. Rev. Lett. 113 027404 Google Scholar
[44]	Chen B, Jiang C, Zhu K D 2011 Phys. Rev. A 83 055803 Google Scholar
[45]	Chen B, Jiang C, Li J J, Zhu K D 2011 Phys. Rev. A 84 055802 Google Scholar
[46]	Sete E A, Eleuch H 2012 Phys. Rev. A 85 043824 Google Scholar
[47]	Kanamoto R, Meystre P 2010 Phys. Rev. Lett. 104 063601 Google Scholar
[48]	Purdy T P, Brooks. D W C, Botter T, Brahms N, Ma Z Y, Stamper-Kurn D M 2010 Phys. Rev. Lett. 105 133602 Google Scholar
[49]	Yan D, Wang Z H, Ren C N, Gao H, Li Y, Wu J H 2015 Phys. Rev. A 91 023813 Google Scholar
[50]	Xiong W, Jin D Y, Qiu Y, Lam C H, You J Q 2016 Phys. Rev. A 93 023844 Google Scholar
[51]	Huang S, Agarwal G S 2010 Phys. Rev. A 81 033830 Google Scholar
[52]	Jiang C, Cui Y, Liu H 2013 Europhys. Lett. 104 34004 Google Scholar
[53]	严晓波, 杨柳, 田雪冬, 刘一谋, 张岩 2014 物理学报 63 204201 Google Scholar Yan X B, Yang L, Tian X D, Liu Y M, Zhang Y 2014 Acta Phys. Sin. 63 204201 Google Scholar
[54]	陈雪, 刘晓威, 张可烨, 袁春华, 张卫平 2015 物理学报 64 164211 Google Scholar Chen X, Liu X W, Zhang K Y, Yuan C H, Zhang W P 2015 Acta Phys. Sin. 64 164211 Google Scholar
[55]	Liu Y C, Hu Y W, Wong C W, Xiao Y F 2013 Chin. Phys. B 22 114213 Google Scholar
[56]	Liu Y L, Wang C, Zhang J, Liu Y X 2018 Chin. Phys. B 27 024204 Google Scholar
[57]	Dobrindt J M, Kippenberg T J 2010 Phys. Rev. Lett. 104 033901 Google Scholar
[58]	Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391
[59]	Ekinci K L, Yang Y T, Roukes M L 2004 J. Appl. Phys. 95 2682 Google Scholar
[60]	Yie Z, Zielke M A, Burgner C B, Turner K L 2011 J. Micromech. Microeng. 21 025027 Google Scholar
[61]	Ramos D, Mertens J, Calleja M, Tamayo J 2008 Appl. Phys. Lett. 92 173108 Google Scholar
[62]	Dai M D, Eom K, Kim C W 2009 Appl. Phys. Lett. 95 203104 Google Scholar
[63]	Li J J, Zhu K.D 2013 Phys. Rep. 525 223 Google Scholar
[64]	陈华俊, 方贤文, 陈昌兆, 李洋 2016 物理学报 65 194205 Google Scholar Chen H J, Fang X W, Chen C Z, Li Y 2016 Acta Phys. Sin. 65 194205 Google Scholar
[65]	Chen H J, Chen C Z, Li Y, Fang X W, Tang X D 2017 Opt. Commun. 382 73 Google Scholar

Cited By

期刊类型引用(4)

1.	李涛，张莹. 基于光线光学的强激光非线性效应仿真与分析. 激光杂志. 2023(05): 252-256 . 百度学术
2.	廖庆洪，邱海燕，喻富，肖敏. 三腔复合光力系统的透射谱和四波混频现象研究. 中国激光. 2023(14): 234-241 . 百度学术
3.	谢宝豪，陈华俊，孙轶. 多模光力系统中光力诱导透明引起的慢光效应. 物理学报. 2023(15): 71-80 . 百度学术
4.	喻富，肖添，何高倩，廖庆洪. 超导量子比特耦合微波腔和机械谐振器系统的探测场吸收特性研究. 激光与光电子学进展. 2022(03): 294-300 . 百度学术

其他类型引用(5)

图 1 石墨烯光力系统与非线性质量传感示意图, 其中该系统由一束频率为 ${\omega _{\rm{p}}}$ 的泵浦光和一束频率 ${\omega _{\rm{s}}}$ 的信号光驱动

Figure 1. Schematic of graphene optomechanical system and nonlinear optical mass sensor driven by a strong pump field ${\omega _{\rm{p}}}$ and a weak signal field ${\omega _{\rm{s}}}$ .

DownLoad: Full-Size Img PowerPoint

图 2 在三个不同泵浦功率条件下石墨烯光力腔内光子数作为腔-泵浦失谐 ${\varDelta _{\rm{p}}}$ 的函数

Figure 2. Mean intracavity photon number of graphene optomechanical cavity as a function of the cavity-pump detuning ${\varDelta _{\rm{p}}}$ with four pump powers.

DownLoad: Full-Size Img PowerPoint

图 3 (a)在失谐 ${\varDelta _{\rm{p}}} = {\omega _{\rm{m}}}$ 时, 腔内光子数 ${n_{\rm{c}}}$ 作为泵浦功率P的函数; (b) 在失谐 ${\varDelta _{\rm{p}}} = - {\omega _{\rm{m}}}$ 时, 腔内光子数 ${n_{\rm{c}}}$ 作为泵浦功率P的函数

Figure 3. (a) The mean intracavity photon number ${n_{\rm{c}}}$ as a function of P for ${\varDelta _{\rm{p}}} = {\omega _{\rm{m}}}$ ; (b) mean intracavity photon number ${n_{\rm{c}}}$ as a function of P for ${\varDelta _{\rm{p}}} = - {\omega _{\rm{m}}}$ .

DownLoad: Full-Size Img PowerPoint

图 4 (a) 在四个不同石墨烯振子频率时, 四波混频谱FWM作为探测-腔失谐 ${\varDelta _{\rm{s}}}$ 的函数; (b) 和 (c)分别是左边和右边尖峰的放大

Figure 4. (a) The four-wave mixing (FWM) spectrum as a function of probe-cavity detuning ${\varDelta _{\rm{s}}}$ under four different graphene resonator frequencies; (b) and (c) are the amplifications of the left and right peaks.

DownLoad: Full-Size Img PowerPoint

图 5 当把纳米颗粒沉积到石墨烯振子表面上时, 四波混频谱的频移. 插图是纳米颗粒的质量与频移之间的线性关系

Figure 5. The four-wave mixing (FWM) spectrum after landing the nanoparticles on the surface of graphene resonator and the color curves shows the mechanical frequency-shifts. The inset shows the linear relationship between the frequency-shifts and the mass of the nanoparticles.

DownLoad: Full-Size Img PowerPoint

[1]	Chen C, Rosenblatt S, Bolotin K I, Kalb W, Kim P, Kymissis I, Stormer H L, Heinz T F, Hone J 2009 Nat. Nanotechnol. 4 861 Google Scholar
[2]	Eichler A, Moser J, Chaste J, Zdrojek M, Wilson-Rae I, Bachtold A 2011 Nat. Nanotechnol. 6 339 Google Scholar
[3]	Song X, Oksanen M, Sillanpää M A, Craighead H G, Parpia J M, Hakonen P J 2012 Nano. Lett. 12 198 Google Scholar
[4]	Chen C, Lee S, Deshpande V V, Lee G H, Lekas M, Shepard K, Hone J 2013 Nat. Nanotechnol. 8 923 Google Scholar
[5]	Bunch J S, van der Zande A M, Verbridge S S, Frank I W, Tanenbaum D M, Parpia J M, Craighead H G, McEuen P L 2007 Science 315 490 Google Scholar
[6]	Moser J, Güttinger J, Eichler A, Esplandiu M J, Liu D E, Dykman M I, Bachtold A 2013 Nat. Nanotechnol. 8 493 Google Scholar
[7]	Stapfner S, Ost L, Hunger D, Reichel J, Favero I, Weig E M 2013 Appl. Phys. Lett. 102 151910 Google Scholar
[8]	Chiu H Y, Hung P, Postma H W C 2008 Nano. Lett. 8 4342 Google Scholar
[9]	Chaste J, Eichler A, Moser J, Ceballos G, Rurali R, Bachtold A A 2012 Nat. Nanotechnol. 7 301 Google Scholar
[10]	Singh V, Sengupta S, Solanki H S, Dhall R, Allain A, Dhara S, Pant P, Deshmukh M M 2010 Nanotechnology 211 65204
[11]	Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391 Google Scholar
[12]	O’Connell A D, Hofheinz M, Ansmann M, Bialczak R C, Lenander M, Lucero E, Neeley M, Sank D, Wang H, Weides M, Wenner J, Martinis J M, Cleland A N 2010 Nature 464 697 Google Scholar
[13]	Teufel J D, Donner T, Li D, Harlow J W, Allman M S, Cicak K, Sirois A J, Whittaker J D, Lehnert K W, Simmonds R W 2011 Nature 475 359 Google Scholar
[14]	Chan J, Alegre T P M, Safavi-Naeini A H, Hill J T, Krause A, Gröblacher S, Aspelmeyer M, Painter O 2011 Nature 478 89 Google Scholar
[15]	Barton R A, Storch I R, Adiga V P, Sakakibara R, Cipriany B R, Ilic B, Wang S P, Ong P, McEuen P L, Parpia J M, Craighead F G 2012 Nano. Lett. 12 4681 Google Scholar
[16]	Peterson R W, Purdy T P, Kampel N S, Andrews R W, Yu P L, Lehnert K W, Regal C A 2016 Phys. Rev. Lett. 116 063601 Google Scholar
[17]	Chen H J 2018 J. Appl. Phys. 124 153102 Google Scholar
[18]	Rossi M, Mason D, Chen J, Tsaturyan Y, Schliesser A 2018 Nature 563 53 Google Scholar
[19]	Grudinin I S, Lee H, Painter O, Vahala K J, 2010 Phys. Rev. Lett. 104 083901 Google Scholar
[20]	Jing H, Özdemir S K, Lü X Y, Zhang J, Yang L, Nori F 2014 Phys. Rev. Lett. 113 053604 Google Scholar
[21]	Brooks D W C, Botter T, Schreppler S, Purdy T P, Brahms N, Stamper-Kurn D M 2012 Nature 488 476 Google Scholar
[22]	Safavi-Naeini A H, Gröblacher S, Hill J T, Chan J, Aspelmeyer M, Painter O 2013 Nature 500 185 Google Scholar
[23]	Purdy T P, Yu P L, Peterson R W, Kampel N S, Regal C A 2013 Phys. Rev. X 3 031012
[24]	Agarwal G S, Huang S M 2010 Phys. Rev. A 81 041803 Google Scholar
[25]	Weis S, Riviere R, Deleglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Science 330 1520 Google Scholar
[26]	Teufel J D, Li D, Allman M S, Cicak K, Sirois A J, Whittaker J D, Simmonds R W 2011 Nature 471 204 Google Scholar
[27]	Safavi-Naeini A H, Alegre T P M, Chan J, Eichenfield M, Winger M, Lin Q, Hill J T, Chang D E, Painter O 2011 Nature 472 69 Google Scholar
[28]	Karuza M, Biancofiore C, Bawaj M, Molinelli C, Galassi M, Natali R, Tombesi P, Di Giuseppe G, Vitali D 2013 Phys. Rev. A 88 013804 Google Scholar
[29]	Zhou X, Hocke F, Schliesser A, Marx A, Huebl H, Gross R, Kippenberg T J 2013 Nat. Phys. 9 179 Google Scholar
[30]	Fan L, Fong KY, Poot M, Tang H X 2015 Nat. Commun. 6 5850 Google Scholar
[31]	Massel F, Heikkilä T T, Pirkkalainen J M, Cho S U, Saloniemi H, Hakonen P J, Sillanpää M A 2011 Nature 480 351 Google Scholar
[32]	Jing H, Özdemir S K, Lü X Y, Zhang J, Yang L, Nori F 2014 Phys. Rev. Lett. 113 053604
[33]	Jiang Y, Maayani S, Carmon T, Nori F, Jing H 2018 Phys. Rev. Appl. 10 064037 Google Scholar
[34]	Jiao Y, Lü H, Qian J, Li Y, Jing H 2016 New J. Phys. 18 083034 Google Scholar
[35]	Lu T X, Jiao Y F, Zhang H L, Saif F, Jing H 2019 Phys. Rev. A 100 013813 Google Scholar
[36]	Zhang H, Saif F, Jiao Y, Jing H 2018 Opt. Express 26 25199 Google Scholar
[37]	Jiao Y F, Lu T X, Jing H 2018 Phys. Rev. A 97 013843 Google Scholar
[38]	Lü H, Özdemir S K, Kuang L M, Nori F, Jing H 2017 Phys. Rev. Appl. 8 044020 Google Scholar
[39]	Lü H, Wang C, Yang L, Jing H 2018 Phys. Rev. Appl. 10 014006 Google Scholar
[40]	Li B, Huang R, Xu X, Miranowicz A, Jing H 2019 Photonics Res. 7 630 Google Scholar
[41]	Weber P, Guttinger J, Tsioutsios I, Chang D E, Bachtold A 2014 Nano. Lett. 14 2854 Google Scholar
[42]	Singh V, Bosman S J, Schneider B H, Blanter Y M, Castellanos-Gomez A, Steele G A 2014 Nat. Nanotechnol. 9 820 Google Scholar
[43]	Song X, Oksanen M, Li J, Hakonen P J, Sillanpää M A 2014 Phys. Rev. Lett. 113 027404 Google Scholar
[44]	Chen B, Jiang C, Zhu K D 2011 Phys. Rev. A 83 055803 Google Scholar
[45]	Chen B, Jiang C, Li J J, Zhu K D 2011 Phys. Rev. A 84 055802 Google Scholar
[46]	Sete E A, Eleuch H 2012 Phys. Rev. A 85 043824 Google Scholar
[47]	Kanamoto R, Meystre P 2010 Phys. Rev. Lett. 104 063601 Google Scholar
[48]	Purdy T P, Brooks. D W C, Botter T, Brahms N, Ma Z Y, Stamper-Kurn D M 2010 Phys. Rev. Lett. 105 133602 Google Scholar
[49]	Yan D, Wang Z H, Ren C N, Gao H, Li Y, Wu J H 2015 Phys. Rev. A 91 023813 Google Scholar
[50]	Xiong W, Jin D Y, Qiu Y, Lam C H, You J Q 2016 Phys. Rev. A 93 023844 Google Scholar
[51]	Huang S, Agarwal G S 2010 Phys. Rev. A 81 033830 Google Scholar
[52]	Jiang C, Cui Y, Liu H 2013 Europhys. Lett. 104 34004 Google Scholar
[53]	严晓波, 杨柳, 田雪冬, 刘一谋, 张岩 2014 物理学报 63 204201 Google Scholar Yan X B, Yang L, Tian X D, Liu Y M, Zhang Y 2014 Acta Phys. Sin. 63 204201 Google Scholar
[54]	陈雪, 刘晓威, 张可烨, 袁春华, 张卫平 2015 物理学报 64 164211 Google Scholar Chen X, Liu X W, Zhang K Y, Yuan C H, Zhang W P 2015 Acta Phys. Sin. 64 164211 Google Scholar
[55]	Liu Y C, Hu Y W, Wong C W, Xiao Y F 2013 Chin. Phys. B 22 114213 Google Scholar
[56]	Liu Y L, Wang C, Zhang J, Liu Y X 2018 Chin. Phys. B 27 024204 Google Scholar
[57]	Dobrindt J M, Kippenberg T J 2010 Phys. Rev. Lett. 104 033901 Google Scholar
[58]	Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391
[59]	Ekinci K L, Yang Y T, Roukes M L 2004 J. Appl. Phys. 95 2682 Google Scholar
[60]	Yie Z, Zielke M A, Burgner C B, Turner K L 2011 J. Micromech. Microeng. 21 025027 Google Scholar
[61]	Ramos D, Mertens J, Calleja M, Tamayo J 2008 Appl. Phys. Lett. 92 173108 Google Scholar
[62]	Dai M D, Eom K, Kim C W 2009 Appl. Phys. Lett. 95 203104 Google Scholar
[63]	Li J J, Zhu K.D 2013 Phys. Rep. 525 223 Google Scholar
[64]	陈华俊, 方贤文, 陈昌兆, 李洋 2016 物理学报 65 194205 Google Scholar Chen H J, Fang X W, Chen C Z, Li Y 2016 Acta Phys. Sin. 65 194205 Google Scholar
[65]	Chen H J, Chen C Z, Li Y, Fang X W, Tang X D 2017 Opt. Commun. 382 73 Google Scholar

[1]	Hu Sheng-Run, Ji Xue-Qiang, Wang Jin-Jin, Yan Jie-Yun, Zhang Tian-Yue, Li Pei-Gang. Ultralow switching threshold optical bistable devices based on epsilon-near-zero Ga₂O₃-SiC-Ag multilayer structures. Acta Physica Sinica, 2024, 73(5): 054201. doi: 10.7498/aps.73.20231534
[2]	Ge Yun-Ran, Zheng Kang, Ding Chun-Ling, Hao Xiang-Ying, Jin Rui-Bo. Efficient optical nonreciprocity based on four-wave mixing effect in semiconductor quantum well. Acta Physica Sinica, 2024, 73(1): 014201. doi: 10.7498/aps.73.20231212
[3]	Li Yun-Heng, Yu Ke, Zhu Tian-Yu, Yu Tong, Shan Si-Chao, Gu Ya-Zhou, Li Zhi-Tong. Optical bistability in topological layered structure and its application in photonic neural network. Acta Physica Sinica, 2024, 73(16): 164208. doi: 10.7498/aps.73.20240569
[4]	Jiang Yang-Yang, Xia Xiao-Shuang, Li Jian-Bo. Four-wave mixing properties in bilayer graphene nanosystem. Acta Physica Sinica, 2023, 72(12): 126801. doi: 10.7498/aps.72.20230012
[5]	Cao Lei-Ming, Du Jin-Jian, Zhang Kai, Liu Sheng-Shuai, Jing Jie-Tai. Experimental generation of multimode quantum correlations between a conical probe and a conical conjugate based on a four-wave mixing process. Acta Physica Sinica, 2022, 71(16): 160306. doi: 10.7498/aps.71.20220081
[6]	Xu Xiao-Yin, Liu Sheng-Shuai, Jing Jie-Tai. Amplification of entangled beam based on four-wave mixing process. Acta Physica Sinica, 2022, 71(5): 050301. doi: 10.7498/aps.71.20211324
[7]	. Amplification of entangled beam based on four-wave mixing process. Acta Physica Sinica, 2021, (): . doi: 10.7498/aps.70.20211324
[8]	Zhong Yin-Yin, Pan Xiao-Zhou, Jing Jie-Tai. Quantum entanglement in coherent feedback system based on the cascaded four wave mixing processes. Acta Physica Sinica, 2020, 69(13): 130301. doi: 10.7498/aps.69.20200042
[9]	Yang Rong-Guo, Zhang Chao-Xia, Li Ni, Zhang Jing, Gao Jiang-Rui. Quantum manipulation of entanglement enhancement in cascaded four-wave-mixing process. Acta Physica Sinica, 2019, 68(9): 094205. doi: 10.7498/aps.68.20181837
[10]	Yang Jian-Yong, Chen Hua-Jun. All-optical mass sensing based on ultra-strong coupling quantum dot-nanomechanical resonator system. Acta Physica Sinica, 2019, 68(24): 246302. doi: 10.7498/aps.68.20190607
[11]	Cao Ya-Min, Wu Bao-Jian, Wan Feng, Qiu Kun. Principle and noise performance of optical phase arithmetic devices using four wave mixing. Acta Physica Sinica, 2018, 67(9): 094208. doi: 10.7498/aps.67.20172638
[12]	Dong Li-Juan, Xue Chun-Hua, Sun Yong, Deng Fu-Sheng, Shi Yun-Long. Loss-induced localized field enhancement and optical bistable state in heterostructure containing single-negative materials. Acta Physica Sinica, 2016, 65(11): 114207. doi: 10.7498/aps.65.114207
[13]	Chen Hua-Jun, Fang Xian-Wen, Chen Chang-Zhao, Li Yang. Coherent optical propagation properties and ultrahigh resolution mass sensing based on double whispering gallery modes cavity optomechanics. Acta Physica Sinica, 2016, 65(19): 194205. doi: 10.7498/aps.65.194205
[14]	Yang Jin-Jin, Li Hui-Jun, Wen Wen, Huang Guo-Xiang. Optical bistability via active Raman gain in an n-type atomic medium. Acta Physica Sinica, 2012, 61(22): 224204. doi: 10.7498/aps.61.224204
[15]	Liu Jun-Hai, Wan Yong, Han Wen-Juan, Yang Hong-Wei, Zhang Huai-Jin, Wang Ji-Yang, Petrov Valentin. Study on the optical bistability in the laser oscillation of Yb doped vanadate crystals. Acta Physica Sinica, 2010, 59(1): 293-299. doi: 10.7498/aps.59.293
[16]	Yin Jing-Chan, Xiao Xiao-Sheng, Yang Chang-Xi. Experimental study of slow light based on four-wave mixing wavelength conversion and dispersion in optical fibers. Acta Physica Sinica, 2010, 59(6): 3986-3991. doi: 10.7498/aps.59.3986
[17]	Yang Yong-Ming, Xu Qi-Ming, Zhang Yan-Peng. Repeatedly dressed four-wave mixing in N5B five-level atomic system. Acta Physica Sinica, 2009, 58(1): 290-297. doi: 10.7498/aps.58.290
[18]	Li Pei-Li, Huang De-Xiu, Zhang Xin-Liang. Ultrahigh-speed all-optical encoder based on four-wave mixing in semiconductor optical amplifiers with PolSK modulated signals. Acta Physica Sinica, 2009, 58(3): 1785-1792. doi: 10.7498/aps.58.1785
[19]	Mao Qing-He, Feng Su-Juan, Jiang Jian, Zhu Zong-Jiu, Liu Wen-Qing. Bistability-mapping of L-band dual-wavelength erbium-doped fiber laser based on fiber loop mirror. Acta Physica Sinica, 2007, 56(1): 296-300. doi: 10.7498/aps.56.296
[20]	Chen Jun, Liu Zheng-Dong, You Su-Ping. The hole burning and optical bistability of quasi-Λ-type four-level atom system. Acta Physica Sinica, 2006, 55(12): 6410-6413. doi: 10.7498/aps.55.6410

期刊类型引用(4)

1.	李涛，张莹. 基于光线光学的强激光非线性效应仿真与分析. 激光杂志. 2023(05): 252-256 . 百度学术
2.	廖庆洪，邱海燕，喻富，肖敏. 三腔复合光力系统的透射谱和四波混频现象研究. 中国激光. 2023(14): 234-241 . 百度学术
3.	谢宝豪，陈华俊，孙轶. 多模光力系统中光力诱导透明引起的慢光效应. 物理学报. 2023(15): 71-80 . 百度学术
4.	喻富，肖添，何高倩，廖庆洪. 超导量子比特耦合微波腔和机械谐振器系统的探测场吸收特性研究. 激光与光电子学进展. 2022(03): 294-300 . 百度学术

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板