Effect of laser phase noise on the steady-state field-mirror entanglement and ground-state cooling in a Laguerre-Gaussian optorotational system

Yupeng Chen; Sumei Huang; Li Deng; Aixi Chen

doi:10.1364/OE.522152

1. Introduction

As the essence of quantum mechanics, quantum entanglement is a phenomenon [1] that can’t be observed in classical mechanics, and is one of the most important resources in quantum communication and quantum computing [2]. So far, quantum entanglement in the microscopic domain has been developed rapidly [3–5]. However, there has been little progress in the macroscopic domain.

In the past two decades, cavity optomechanics has gained considerable attention [6]. The foundation of cavity optomechanics is the interaction between a cavity field and a mechanical oscillator generated by radiation pressure. Because cavity optomechanical systems have the advantage of strong interaction between the cavity field and the mechanical oscillator, they are considered as one of the best platforms for studying macroscopic quantum phenomena. At present, many quantum phenomena have been studied in cavity optomechanical systems, such as quantum entanglement between various modes [7–12], quantum coherence [13], and photon anti-bunching [14].

The Laguerre-Gaussian (L-G) beam is a typical structured beam, which has a helical wavefront and a hollow center in the intensity distribution [15,16]. The L-G beam is usually generated by a spiral phase element, which can produce a L-G beam with orbital angular momentum as high as $l=1000\hbar$ [17,18]. In a L-G cavity optorotational system, a L-G cavity mode is coupled to a spiral phase element used as a cavity end mirror due to the exchange of angular momentum between them [19,20]. Recently, the L-G cavity optorotational system has attracted a great deal of interest. Many physical phenomena have been studied in such a system, such as entanglement between the L-G cavity mode and the rotating mirror [20], cooling of the rotating mirror with fewer restrictions [21], generation of sideband effects under matching conditions [22,23], observation of optomechanical-induced transparency [24] and double optomechanical-induced transparency [25], generation of Fano resonance, realization of slow-to-fast light conversion [26], preparation of entanglement between various modes [20,27,28], and so on.

The study of entanglement in cavity optomechanical systems mainly focuses on the entanglement between different modes at the steady state of the system. However, existing studies on entanglement in cavity optomechanical systems rarely consider the impact of driving laser noise. In fact, real driving lasers have both the amplitude and phase noises. It has been found the laser phase noise has a detrimental influence on optomechanical entanglement [29]. Additionally, it has been shown that laser phase noise has a negative impact on the ground-state cooling of mechanical oscillators [29–33]. Furthermore, studies have shown that laser phase noise is strongly colored, and assuming white laser phase noise overestimates its effect [29]. Besides, it has been shown that the phase noise heating can affect the resolved sideband cooling of a levitated nanoparticle and the thermal heating is not the main factor in high vacuum [34]. The low pass filters which is realized by additional filtering cavities can decrease the effect of the phase noise [35,36].

In this paper, we study the effect of laser phase noise on the steady-state entanglement between the cavity mode and the rotating mirror and on the ground-state cooling of the rotating mirror in a L-G cavity optorotational system. The results indicate that laser phase noise influences the field-mirror entanglement significantly, especially in the regime close to the unstable boundary. The field-mirror entanglement can be strongly influenced by both the strength of the laser phase noise and the band center of the laser phase noise spectrum. The maximum field-mirror entanglement decreases and the entanglement regime becomes narrower with increasing the laser phase noise strength or increasing band center of the frequency noise spectrum. In addition, we find that increasing the power of the input laser and the angular momentum of the L-G cavity mode do not always enhance the field-mirror entanglement. What’s more, in the presence of laser phase noise, ground-state cooling of the rotating mirror can still be realized at a lower input laser power.

The structure of this paper is as follows: in Section 2, we present the studied model and Hamiltonian of the system. Then we give the quantum Langevin equations of the system in the presence of the laser noise. The stationary entanglement between the cavity mode and the rotating mirror is quantified by the Logarithmic negativity. In Section 3, we show the results of the optomechanical entanglement in the steady state and analyze the effect of laser phase noise on the optomechanical entanglement. In Section 4, we derive the expression of the effective phonon number of the rotating mirror and show the results of the ground-state cooling in the presence of the laser phase noise. In Section 5, we discuss the experimental feasibility. In Section 6, we draw the conclusions.

2. Quantum Langevin equations of the system

The system under study is a typical L-G optorotational system, which is formed by two spiral phase elements used as the cavity mirrors [37], as shown in Fig. 1. The input mirror is fixed and can partially transmit the light, while the rear mirror can completely reflect the light and rotate around the $Z$ axis, which can be seen as a harmonic oscillator with the angle displacement $\phi _{0}$ and the angular momentum $L_{z0}$. When a Gaussian beam with topological charge 0 is injected into the cavity through the input mirror, the topological charge of the transmitted light is also 0. When the rear mirror reflects the light, the topological charge of the reflected light is $2l$. When the light is reflected by the input mirror, the topological charge of the reflected light is 0. The exchange $2l\hbar$ of angular momentum between the cavity mode and the rear mirror happens once in a cycle of the photons moving in the cavity, so the time interval between angular momentum exchanges is $\frac {2L}{c}$, where $L$ is the length of the cavity and $c$ is the light speed in vacuum. Thus, the torque acting on the rear mirror exerted by the cavity mode is $\frac {\hbar cl}{L}=\hbar \xi _\phi$, where $\xi _\phi$ is the optorotational coupling parameter [37].

Fig. 1. Sketch of the L-G optorotational system [20]. A Gaussian laser beam is sent to the optical cavity to drive a L-G cavity mode. IC and RM are the spiral phase elements and are the input mirror and rear mirror respectively. The RM is set on the support. It can rotate around the $z$ axis. The angular displacement of the rotating mirror from its equilibrium position $\phi _{0}=0$ is denoted by $\phi$. The topological charge $l$ on each beam at different points is also shown.

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Taking into account the noise of the driving laser, the Hamiltonian of the system has the following form [20,29]:

(1)$$H=\frac{L_{z0}^2}{2I}+\frac{1}{2}I\omega_\phi^2\phi_0^2+\hbar\omega_{c}a^{\dagger}a-\hbar \xi_\phi a^{\dagger}a\phi+i\hbar E(t)(a^{\dagger}e^{{-}i[\omega_0 t+\Phi(t)]}-a e^{i[\omega_0 t+\Phi(t)]}).$$

In Eq. (1), the first and the second term are the energy of the rotating mirror with mass $m$ and radius $R$. $I$ is the moment of inertia of the rotating mirror, given by $I=\frac {1}{2}mR^2$. The third term represents the energy of the cavity mode with frequency $\omega _c$, where $a$ and $a^{\dagger }$ represent the annihilation and creation operators of the cavity mode, respectively. The fourth term represents the interaction between the cavity mode and the rotating mirror. The last term represents the interaction between the cavity mode and the driving laser with frequency $\omega _0$. We assume that there are amplitude noise and phase noise in the input laser. The parameter $E(t)=E_0+\varepsilon (t)$ is the amplitude of the input laser, where $E_{0}$ is the mean amplitude of the input laser and $\varepsilon (t)$ is the real, zero-mean laser amplitude noise. The $E_0$ is related to the power $P$ of the input laser by $E_{0}=\sqrt {\frac {2\kappa P}{\hbar \omega _0}}$ with $\kappa$ being the cavity decay rate. The parameter $\Phi (t)$ describes the zero-mean laser phase noise. Then we replace the angular momentum $L_{z0}$ and the angular displacement $\phi _0$ by the dimensionless angular momentum operator $L_z=\frac {L_{z0}}{\sqrt {I\hbar \omega _\phi }}$ and the dimensionless angular displacement operator $\phi =\phi _0\sqrt {\frac {I\omega _\phi }{\hbar }}$, respectively. The Hamiltonian of the system can be written as:

(2)$$H=\frac{\hbar\omega_{\phi}}{2}(L_z^2+\phi^2)+\hbar\omega_{c}a^{\dagger}a-\hbar ga^{\dagger}a\phi+i\hbar E(t)(a^{\dagger}e^{{-}i[\omega_0 t+\Phi(t)]}-a e^{i[\omega_0 t+\Phi(t)]}),$$

where $g=\frac {cl}{L}\sqrt {\frac {\hbar }{I\omega _\phi }}$ is the optorotational coupling coefficient. The commutation relation between the angular displacement operator and the angular momentum operator is $[\phi,L_z]=i$.

The system dynamics can be described using the Heisenberg equations of motion. In addition, in order to describe the system dynamics more accurately, we consider the damping and noise terms for the cavity and mechanical modes. We obtain the equations of motion of the system operators as follows:

(3)$$\begin{aligned} & \dot{\phi}=\omega_\phi L_z,\\ & \dot{L_z}={-}\omega_\phi \phi+ga^\dagger a-\frac{D_\phi}{I}L_z+\xi,\\ & \dot{a}={-}(\kappa+i\omega_{c})a+iga\phi+E(t)e^{{-}i[\omega_0 t+\Phi(t)]}+\sqrt{2\kappa}a_{in}(t), \end{aligned}$$

where $D_\phi$ is the intrinsic damping constant of the rotating mirror and $a_{in}$ is the operator of the input vacuum noise, whose nonzero correlation function is [38]:

(4)$$\begin{aligned} \langle a_{in}(t)a_{in}^\dagger(t')\rangle =\delta(t-t'). \end{aligned}$$

In addition, $\xi$ is the Hermitian Brownian noise operator and its correlation function is [20]:

(5)$$\langle\xi(t)\xi(t')\rangle =\frac{D_\phi}{\omega_\phi I}\int\frac{d\omega}{2\pi}e^{{-}i\omega(t-t')}\omega[1+coth(\frac{\hbar\omega}{2k_BT})],$$

where $k_B$ is the Boltzmann constant and $T$ is the temperature of the environment. Due to the laser phase noise, we need to analyze under the rotating frame at frequency $\omega _0+\dot {\Phi }(t)$, instead of the rotating frame at frequency $\omega _0$. The cavity mode operator is transformed according to $a(t)=\tilde a(t)e^{-i\omega _{0}(t-t_{0})-i\int _{t_{0}}^{t}dt' \dot {\Phi }(t')}$, where $t_{0}\rightarrow -\infty$ is the time instant at which $E(t)$ is real. Thus, Eq. (3) becomes:

(6)$$\begin{aligned} & \dot{\phi}=\omega_\phi L_z,\\ & \dot{L_z}={-}\omega_\phi \phi+g\tilde a^\dagger \tilde a-\frac{D_\phi}{I}L_z+\xi,\\ & \dot{\tilde a}={-}\kappa \tilde a-i(\Delta_0-\dot{\Phi}-g\phi)\tilde a+E_0+\varepsilon+\sqrt{2\kappa}\tilde{a}_{in}(t), \end{aligned}$$

where $\Delta _0=\omega _c-\omega _0$ is the cavity detuning, and $\tilde {a}_{in}(t)=a_{in}(t)e^{i\omega _{0}(t-t_{0})+i\int _{t_{0}}^{t}dt' \dot {\Phi }(t')}$ is still the input vacuum noise, whose correlation function is given by $\langle \tilde {a}_{in}(t)\tilde {a}_{in}^{\dagger }(t')\rangle =\delta (t-t')$. We drop the $\tilde { }$ on $a$ for convenience. It is seen that the phase noise $\Phi$ and the amplitude noise $\varepsilon$ of the driving laser affect the time evolution of the cavity mode $a$.

Under the assumption that the L-G cavity mode is driven by an intense laser, the steady-state amplitude $|a_{s}|$ of the L-G cavity mode is much larger than $1$ ($|a_s|\gg 1$). In this situation, we write each operator in Eq. (6) as a sum of the steady-state mean value and the fluctuation operator ($\phi =\phi _{s}+\delta \phi, L_{z}=L_{zs}+\delta L_{z}, a=a_{s}+\delta a$). Since we use the rotating frame at frequency $\omega _0+\dot {\Phi }(t)$, the amplitude $a_{s}$ of the classical stationary coherent state is independent of the laser noise. We assume the classical steady state is time-independent which means the laser phase noise and laser amplitude noise don’t have any effect on the steady states [29]. The mean values of the operators are found to be:

(7)$$\begin{aligned} & L_{zs}=0,\\ & \phi_s=\frac{g\lvert a_{s}\rvert^2}{\omega_\phi},\\ & a_{s}=\frac{E_0}{\kappa+i\Delta}, \end{aligned}$$

where the subscript $s$ represents the steady-state mean value of the operator and $\Delta =\Delta _0-g\phi _s$ is the effective cavity detuning, including the effect of the radiation torque. It is seen that the steady-state amplitude $a_{s}$ of the cavity field depends on the power $P$ of the input laser, and the steady-state angular displacement $\phi _{s}$ of the rotating mirror is proportional to the intensity $|a_{s}|^{2}$ of the cavity field. When $|a_{s}|>>1$, we can neglect the high-order terms in fluctuations. Thus, the fluctuation operators of the system satisfy the linearized quantum Langevin equations:

(8)$$\begin{aligned} & \delta\dot{\phi}=\omega_\phi \delta L_z,\\ & \delta\dot{L_z}={-}\omega_\phi\delta\phi+g(a_s\delta a^\dagger+a_{s}^{{\ast}}\delta a)-\frac{D_\phi}{I}\delta L_z+\xi,\\ & \delta\dot{a}={-}(\kappa+i\Delta)\delta a+iga_{s}\delta\phi+i\dot{\Phi}a_s+\varepsilon+\sqrt{2\kappa}\tilde{a}_{in}, \end{aligned}$$

where we have ignored the term $i\dot {\Phi }\delta a$, because it is much smaller than the term $i\dot {\Phi }a_{s}$ when $|a_{s}|\gg 1$ [29]. We obtain the following linearized equations for the quadrature fluctuation operators:

(9)$$\begin{aligned} & \delta\dot{\phi}=\omega_\phi \delta L_z,\\ & \delta\dot{L_z}={-}\omega_\phi\delta\phi+G\delta X_{\Delta}-\frac{D_\phi}{I}\delta L_z+\xi,\\ & \delta\dot{X_{\Delta}}={-}\kappa\delta X_{\Delta}+\Delta\delta Y_{\Delta}+\sqrt{2}cos\theta_{\Delta}\varepsilon+\sqrt{2\kappa}X_{\Delta}^{in},\\ & \delta\dot{Y_{\Delta}}={-}\kappa\delta Y_{\Delta}-\Delta\delta X_{\Delta}+G\delta\phi+\sqrt{2}|a_s|\dot{\Phi}+\sqrt{2}sin\theta_{\Delta}\varepsilon+\sqrt{2\kappa}Y_{\Delta}^{in}, \end{aligned}$$

where $\delta X_{\Delta }=(\delta ae^{i\theta _{\Delta }}+\delta a^\dagger e^{-i\theta _{\Delta }})/\sqrt {2}$ and $\delta Y_{\Delta }=(\delta ae^{i\theta _{\Delta }}-\delta a^\dagger e^{-i\theta _{\Delta }})/(i\sqrt {2})$ are the amplitude and phase fluctuation operators of the cavity mode, respectively, and $X_{\Delta }^{in}=(\tilde {a}_{in}e^{i\theta _{\Delta }}+\tilde {a}_{in}^\dagger e^{-i\theta _{\Delta }})/\sqrt {2}$ and $Y_{\Delta }^{in}=(\tilde {a}_{in}e^{i\theta _{\Delta }}-\tilde {a}_{in}^\dagger e^{-i\theta _{\Delta }})/(i\sqrt {2})$ are the amplitude and phase of the input vacuum noise, respectively, $\theta _{\Delta }=\textrm {arctan}[\Delta /\kappa ]$, and $G=\sqrt {2}ga_s$ represents the effective optorotational coupling strength.

Next, we consider further the effect of laser noise. In currently available stabilized lasers, the amplitude noise can be neglected, while phase noise cannot be neglected and is responsible for the nonzero laser linewidth $\Gamma _{l}$ observed in every real laser system [30–33]. We can yield the laser spectrum from the correlation function of the field via the Fourier transform [29]:

(10)$$\begin{aligned} S_L(\omega)=\int d(\tau)e^{i\omega\tau}C(\tau)=\int d\tau e^{i\omega\tau}\left\langle e^{(i\Phi(t+\tau)-i\Phi(t))}\right\rangle. \end{aligned}$$

The laser frequency noise $\Phi (t)$ can be described by a Gaussian Stochastic process with zero mean. So we can give the form of $C(\tau )$ [29]:

(11)$$\begin{aligned} C(\tau)=e^{(-\frac{1}{2}\int_0^\tau ds\int_0^\tau ds'\langle \dot{\Phi}(s)\dot{\Phi}(s') \rangle)} \end{aligned}$$

If we consider $\langle \dot {\Phi }(s)\dot {\Phi }(s') \rangle =2\Gamma _l\delta (s-s')$ as the correlation function of laser frequency noise. The frequency noise spectrum is flat: $S_{\dot {\Phi }}(\omega )=2\Gamma _l$ and $C(\tau )=e^{-\Gamma _l|\tau |}$, which implies a Lorentzian laser spectrum with linewidth $\Gamma _{l}$ [29]. However, the effect of laser phase noise is overestimated by taking the flat frequency noise spectrum [32]. We consider a frequency noise spectrum with a bandpass filter form, which is closer to a realistic situation [29]:

(12)$$\begin{aligned} S_{\dot{\Phi}}(\omega)=2\Gamma_l\frac{\Omega^4}{(\Omega^2-\omega^2)^2+\omega^2\gamma^2}, \end{aligned}$$

where $\Gamma _l$ represents both the laser phase noise strength and the laser linewidth, $\Omega$ is the band center and $\gamma$ is the bandwidth of the frequency laser spectrum. We assume the frequency noise variable $\varPsi =\dot {\Phi }$ obeys the following Langevin equations [29]:

(13)$$\begin{aligned} & \dot{\varPsi}=\Omega\theta,\\ & \dot{\theta}={-}\Omega\varPsi-\gamma\theta+\Omega\sqrt{2\Gamma_l}\epsilon, \end{aligned}$$

where $\theta$ is an auxiliary variable and $\epsilon$ is the white noise whose correlation function is $\langle \epsilon (t) \epsilon (t') \rangle =\delta (t-t')$. Under this assumption, we can prove that the frequency noise spectrum $S_{\dot {\Phi }}(\omega )$ (Eq. (12)) can be reproduced (see the Appendix). The dynamics of the system in the presence of the laser phase noise can be described by the combination of Eq. (9) with Eq. (13), which can be written in the matrix form:

(14)$$\begin{aligned} & \dot{u}(t)=Au(t)+n(t), \end{aligned}$$

where $u(t)=(\delta \phi,\delta L_z,\delta X_{\Delta },\delta Y_{\Delta },\varPsi,\theta )^{T}$ is the vector of the fluctuation operators, $n(t)=(0,\xi (t),\sqrt {2\kappa }X_{\Delta }^{in}(t),\sqrt {2\kappa }Y_{\Delta }^{in}(t),0,\Omega \sqrt {2\Gamma _l}\epsilon (t))^{T}$ is the vector of the system noises, $A$ is a $6\times 6$ drift matrix:

(15)$$\begin{aligned} A=\begin{pmatrix} 0 & \omega_\phi & 0 & 0 & 0 & 0 \\ -\omega_\phi & -\frac{D_\phi}{I} & G & 0 & 0 & 0 \\ 0 & 0 & -\kappa & \Delta & 0 & 0 \\ G & 0 & -\Delta & -\kappa & \sqrt{2}|a_s| & 0 \\ 0 & 0 & 0 & 0 & 0 & \Omega \\ 0 & 0 & 0 & 0 & -\Omega & -\gamma \\ \end{pmatrix}. \end{aligned}$$

3. Stationary entanglement between the L-G cavity mode and the rotating mirror

In this section, we investigate the effect of laser phase noise on the stationary entanglement between the cavity mode and the rotating mirror. We need to ensure the system is stable when calculating the entanglement, and the stable conditions can be derived from the Routh-Hurwitz criteria [39]. We assume that all of the noises of the system are in a Gaussian state. The quantum Langevin equations have been linearized above. Hence, the steady state of the system is a Gaussian state, which can be described by $6\times 6$ covariance matrix $V$, and $V_{ij}=(\langle u_i(\infty )u_j(\infty )+u_j(\infty )u_i(\infty )\rangle )/2$. The entanglement between the cavity mode and the rotating mirror in the steady state is calculated by the covariance matrix $V$, which can be obtained using the Lyapunov equation:

(16)$$\begin{aligned} AV+VA^\top={-}D, \end{aligned}$$

where the diffusion matrix $D$ is the matrix of noise correlations, defined as: $\frac {1}{2}\langle n_i(t)n_j(t')+n_j(t')n_i(t)\rangle =D_{ij}\delta (t-t')$. In the limit of a large quality factor of the rotating mirror $Q=\frac {\omega _\phi }{D_\phi /I}\gg 1$, we can obtain the correlation function of the thermal noise $\xi$ [40]:

(17)$$\begin{aligned} \frac{1}{2}\langle \xi(t)\xi(t')+\xi(t')\xi(t)\rangle\simeq\frac{D_\phi}{I}(2\bar{n}+1)\delta(t-t'), \end{aligned}$$

where $\bar {n}=(e^{\frac {\hbar \omega _{eff}}{k_BT}}-1)^{-1}$ represents the number of mean thermal phonons, $k_B$ is the Boltzmann constant, $T$ is the temperature of the environment and $\omega _{eff}^2=\omega _{\phi }^2-\frac {2\xi _\phi ^2(2\kappa )P}{I\omega _c}(\frac {\Delta }{\Delta ^2+\kappa ^2})(\frac {\kappa ^2-(\omega ^2-\Delta ^2)}{(\kappa ^2+(\omega -\Delta )^2)(\kappa ^2+(\omega +\Delta )^2)})$ represents the effective frequency of the rotating mirror [37]. The matrix $D$ is a diagonal matrix and is found to be $diag[0,\frac {D_\phi }{I}(2\bar {n}+1),\kappa,\kappa,0,2\Gamma _l\Omega ^2]$. The logarithmic negativity $E_N$ can be used to measure the stationary entanglement between the L-G cavity mode and the rotating mirror, and is given by [41,42]:

(18)$$\begin{aligned} E_N=max[0,-ln2\eta^-], \end{aligned}$$

where $\eta ^-\equiv 2^{-\frac {1}{2}}\{\sum (V')-[\sum (V')^2-4detV']^\frac {1}{2}\}^\frac {1}{2}$. The $4\times 4$ matrix $V'$ is the submatrix of the matrix $V$ and we write the matrix $V'$ in a $2\times 2$ matrix:

(19)$$\begin{aligned} V'=\begin{pmatrix} A & C \\ C^\top & B \\ \end{pmatrix}, \end{aligned}$$

where $A$ and $B$ are the variances of the mechanical and cavity modes, and $C$ is the correlation between the mechanical and cavity modes. Besides, $\sum (V')\equiv detA+detB-2detC$.

We choose parameters as shown in Table 1 [20]. And we assume the laser phase noise spectrum has a bandpass filter form [29].

Table 1. The value of some parameters used in the text [20]

View Table

In Fig. 2, we show the contour plot of the entanglement $E_N$ between the cavity mode and the rotating mirror versus the input power $P$ and the effective cavity detuning $\Delta$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. To ensure the system is working in the resolved sideband region, the cavity decay rate is fixed at $\frac {\kappa }{\omega _\phi }=0.5$. Besides, the frequency noise spectrum is fixed at a moderate band center $\frac {\Omega }{2\pi }=50$ kHz. We see from Fig. 2 that the effect of the laser phase noise on the entanglement is significant. Without the laser phase noise((a) $\Gamma _l=0$), the maximum entanglement is over 0.45 and is generated close to the unstable regime. In the presence of the laser phase noise, as the strength $\Gamma _{l}$ of the laser phase noise increases, not only the maximum entanglement decreases and is generated at a larger cavity detuning, but also the entanglement region becomes narrower. In addition, we find the maximum entanglement appears at the input power $P=80$ mW. Moreover, increasing the power of the input laser does not always increase the entanglement. The reason is that increasing the power of the input laser makes the effective optorotational coupling strength larger, which increases the entanglement, but also makes the photon number $|a_s|^2$ larger, which amplifies the effect of the laser phase noise.

Fig. 2. Contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the input power $P$ and the effective cavity detuning $\Delta$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. The other parameters are $\frac {\kappa }{\omega _\phi }=0.5$, $\frac {\Omega }{2\pi }=50$ kHz, $\gamma =\frac {\Omega }{2}$, $l=100\hbar$.

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In Fig. 3, we show the contour plot of the entanglement $E_N$ between the cavity mode and the rotating mirror versus the input power $P$ and the cavity decay rate $\kappa$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. The frequency noise spectrum is fixed at a moderate band center $\frac {\Omega }{2\pi }=50$ kHz. We set the cavity detuning at $\frac {\Delta }{\omega _\phi }=1$. When laser phase noise is absent, the maximum entanglement appears close to the unstable regime. When laser phase noise is present, with increasing strength $\Gamma _{l}$ of the laser phase noise, the maximum entanglement decreases and the entanglement region is narrower. It is noted that the detrimental effect of the laser phase noise on the entanglement is significant close to the unstable regime. What’s more, for sufficiently high phase noise, the entanglement only exists in the resolved sideband region ($\kappa /\omega _{\phi }<1$).

Fig. 3. Contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the input power $P$ and the cavity decay rate $\kappa$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. The other parameters are $\frac {\Delta }{\omega _\phi }=1$, $\frac {\Omega }{2\pi }=50$ kHz, $\gamma =\frac {\Omega }{2}$, $l=100\hbar$.

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In Fig. 4, we show the contour plot of the entanglement $E_N$ between the cavity mode and the rotating mirror versus the input power $P$ and the effective cavity detuning $\Delta$ for different band centers of the frequency noise spectrum, (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. To ensure the system working in the resolved sideband region, the cavity decay rate is fixed at $\frac {\kappa }{\omega _\phi }=0.5$. Besides, the strength of the laser phase noise is fixed at a moderate strength $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. At a small band center($\Omega =30$ kHz) and a small bandwidth($\gamma =15$ kHz), the maximum entanglement can be larger than 0.275, while at a large band center($\Omega =140$ kHz) and a large bandwidth($\gamma =70$ kHz), the entanglement almost vanishes, so the band center $\Omega$ of the frequency noise spectrum can affect the entanglement significantly. It is noted that increasing the band center $\Omega$ of the frequency noise spectrum can decrease the maximum entanglement and make the entanglement region narrower.

Fig. 4. Contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the input power $P$ and the effective cavity detuning $\Delta$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. The other parameters are $\frac {\kappa }{\omega _\phi }=0.5$, $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $\gamma =\frac {\Omega }{2}$, $l=100\hbar$.

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In Fig. 5, we show the contour plot of the entanglement $E_N$ between the cavity mode and the rotating mirror versus the input power $P$ and the cavity decay rate $\kappa$ for different band centers of the frequency noise spectrum, (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. The frequency noise spectrum is fixed at a moderate strength $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. We set the cavity detuning at $\frac {\Delta }{\omega _\phi }=1$. Similar to Fig. 4, the maximum entanglement decreases rapidly and the entanglement region is narrower with increasing the band center $\Omega$ of the frequency noise spectrum. What’s more, the entanglement is more robust to the laser phase noise in the resolved sideband region.

Fig. 5. Contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the input power $P$ and the cavity decay rate $\kappa$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. The other parameters are $\frac {\Delta }{\omega _\phi }=1$, $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $\gamma =\frac {\Omega }{2}$, $l=100\hbar$.

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In Fig. 6, we show the contour plot of the entanglement $E_N$ between the cavity mode and the rotating mirror versus the angular momentum $l$ of the cavity mode and the effective cavity detuning $\Delta$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. Without the laser phase noise, increasing the angular momentum of the cavity mode can increase the entanglement. When laser phase noise is present, increasing the angular momentum is not always increasing the entanglement, which is similar to the effect of the power of the input laser on the entanglemnt. With increasing the angular momentum, the optorotational coupling coefficient and the intracavity photon number become larger, which increases the entanglement, but also amplifies the effect of the laser phase noise, leading to a decrease in the entanglement. What’s more, an increase in the strength $\Gamma _{l}$ of the laser phase noise leads to decreasing the maximum entanglement and narrowing the entanglement region.

Fig. 6. Contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the angular momentum $l$ of the cavity mode and the effective cavity detuning $\Delta$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. The other parameters are $\frac {\kappa }{\omega _\phi }=0.5$, $\frac {\Omega }{2\pi }=50$ kHz, $\gamma =\frac {\Omega }{2}$, $P=80$ mW.

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In Fig. 7, we show the contour plot of the entanglement $E_N$ between the cavity mode and the rotating mirror versus the angular momentum $l$ of the cavity mode and the effective cavity detuning $\Delta$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. It is seen that a larger band center $\Omega$ of the frequency noise spectrum can decrease the maximum entanglement and make the entanglement region narrower. Besides, as predicted by the numerical analysis, a large angular momentum is not always beneficial for the entanglement due to the effect of the laser phase noise. To resist the negative effect of the larger band center $\Omega$ of the frequency noise spectrum on the entanglement, the effective cavity detuning need to be increased.

Fig. 7. Contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the angular momentum $l$ of the cavity mode and the effective cavity detuning $\Delta$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. The other parameters are $\frac {\kappa }{\omega _\phi }=0.5$, $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $\gamma =\frac {\Omega }{2}$, $P=80$ mW.

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In Fig. 8, we show the contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the bandwidth $\gamma$ and band center $\Omega$ of the frequency noise spectrum. It is seen that increasing the band center $\Omega$ of the frequency noise spectrum can decrease the entanglement significantly while the changing the bandwidth $\gamma$ of the frequency laser spectrum has a little effect on the entanglement in a large range of $\Omega$. Thus the band center $\Omega$ of the frequency noise spectrum is the major factor affecting the entanglement in the presence of laser phase noise.

Fig. 8. Contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the bandwidth $\gamma$ and band center $\Omega$ of the frequency noise spectrum. The other parameters are $\frac {\Delta }{\omega _\phi }=1$, $\frac {\kappa }{\omega _\phi }=0.5$, $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $l=100\hbar$, $P=80$ mW.

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In Fig. 9, we show the contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the length $L$ of the cavity and the mass $m$ of the rotating mirror. From Fig. 9, it is seen that decreasing the length $L$ of the cavity or the mass $m$ of the rotating mirror can increase the entanglement first and then decrease the entanglement. This result is similar to the influence of increasing the angular momentum $l$ of the L-G cavity mode on the entanglement. Note that too large or too small the length $L$ of the cavity or the mass $m$ of the mirror is not beneficial for the field-mirror entanglement. Based on the optorotational coupling coefficient $g=\frac {cl}{L}\sqrt {\frac {\hbar }{I\omega _\phi }}$ and the moment of inertia of the rotating mirror $I=\frac {1}{2}mR^2$, we can get $g\propto \frac {1}{L}$ and $g\propto \frac {1}{R}$, so the influence of the radius $R$ of the rotating mirror on the entanglement is similar to that of the cavity length $L$ on the entanglement.

Fig. 9. Contour plot of entanglement $E_N$ between the cavity mode and the rotating mirror versus the length $L$ of the cavity and the mass $m$ of the rotating mirror. The other parameters are $\frac {\Delta }{\omega _\phi }=1$, $\frac {\kappa }{\omega _\phi }=0.5$, $\frac {\Gamma _l}{2\pi }=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $\frac {\Omega }{2\pi }=50$ kHz, $\gamma =\frac {\Omega }{2}$, $l=100\hbar$, $P=80$ mW.

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4. Ground-state cooling of the rotating mirror

It is next of interest to consider how the laser phase noise affects the ground-state cooling of the rotating mirror in the L-G cavity optorotational system. In the optomechanical system, the ground-state cooling of the movable mirror in the presence of laser phase noise has been studied, showing that ground-state cooling can be realized with sufficiently low frequency noise at the resonance frequency $\omega _{\phi }$ of the mechanical resonator [32]. What’s more, in the resolved sideband limit, at low input power, the mechanical resonator can still reach ground-state in the presence of laser phase noise [29].

On Fourier transforming the quantum Langevin equations (Eq. (9) and Eq. (13)), we can obtain the angular momentum fluctuations of the rotating mirror:

(20)$$\begin{aligned} \delta L_z(\omega)=\frac{1}{d(\omega)}\left(\frac{2a_s\Delta G\Omega^2\sqrt{\Gamma_l}}{(\omega^2-\Omega^2+i\omega\gamma)b(\omega)}\epsilon(\omega)+\frac{\sqrt{2\kappa}\Delta G}{b(\omega)}Y_{\Delta}^{in}(\omega)+\frac{\sqrt{2\kappa}G(\kappa-i\omega)}{b(\omega)}X_{\Delta}^{in}(\omega)+\xi(\omega)\right), \end{aligned}$$

where $d(\omega )=-i\omega +i\frac {\omega _\phi ^2}{\omega }+\frac {D_\phi }{I}-i\frac {\omega _\phi G^2\Delta }{\omega b(\omega )}$ and $b(\omega )=(\kappa -i\omega )^2+\Delta ^2$. The first term in the angular momentum fluctuations is the contribution of the laser phase noise, the second and third terms are the contributions of the input vacuum noise, and the last term is the contribution of the thermal noise of the rotating mirror. Then the spectrum of angular momentum fluctuations of the rotating mirror is defined by [43]:

(21)$$\begin{aligned} S_{L_z}(\omega)=\frac{1}{4\pi}\int d\Omega e^{{-}i(\omega+\Omega)t}\langle\delta L_z(\omega)\delta L_z(\Omega)+\delta L_z(\Omega)\delta L_z(\omega)\rangle. \end{aligned}$$

In addition, to calculate the spectrum of fluctuations, we need the noise correlation functions in the frequency domain [29,43],

(22)$$\begin{aligned} & \langle X_{\Delta}^{in}(\omega) {X_{\Delta}^{in}}^\dagger(\Omega)\rangle=\pi\delta(\omega+\Omega),\\ & \langle Y_{\Delta}^{in}(\omega) {Y_{\Delta}^{in}}^\dagger(\Omega)\rangle=\pi\delta(\omega+\Omega),\\ & \langle \xi(\omega) \xi(\Omega)\rangle=2\pi\frac{D_\phi}{I}(2\bar{n}+1)\delta(\omega+\Omega),\\ & \langle \epsilon(\omega) \epsilon(\Omega)\rangle=2\pi\delta(\omega+\Omega). \end{aligned}$$

Thus, the spectrum of angular momentum fluctuations of the rotating mirror can be obtained:

(23)$$\begin{aligned}S_{L_z}(\omega)=\frac{1}{|d(\omega)|^2}\left( \frac{4 a_s^2\Delta^2 G^2 \Omega^4 \Gamma_l}{((\omega^2-\Omega^2)^2+\omega^2\gamma^2)|b(\omega)|^2}+\frac{\kappa\Delta^2G^2}{|b(\omega)|^2}+\frac{\kappa G^2(\kappa^2+\omega^2)}{|b(\omega)|^2}+\frac{D_\phi}{I}(2\bar{n}+1)\right) . \end{aligned}$$

In Eq. (23), the first term represents the contribution of the laser phase noise, the second and third terms represent the contributions of the input vacuum noise, and the fourth term represents the contribution of the mechanical thermal noise. By Fourier transforming Eq. (9), we can obtain $\delta \phi (\omega )=i\frac {\omega _\phi }{\omega }\delta L_z(\omega )$ and the spectrum of angular displacement fluctuations of the rotating mirror $S_\phi (\omega )=\frac {\omega ^2}{\omega _\phi ^2}S_{L_z}(\omega )$. Then we can derive the effective phonon number $n_{eff}$ of the rotating mirror by the stationary mean energy of the rotating mirror [29]:

(24)$$\begin{aligned} U=\frac{\hbar\omega_\phi}{2}(\langle\delta\phi^2\rangle+\langle\delta L_z^2\rangle)=\hbar\omega_\phi(n_{eff}+\frac{1}{2}), \end{aligned}$$

where $\langle \delta \phi ^2\rangle =\frac {1}{2\pi }\int _{-\infty }^{+\infty } S_\phi (\omega )d\omega$ and $\langle \delta L_z^2\rangle =\frac {1}{2\pi }\int _{-\infty }^{+\infty } S_{L_z}(\omega )d\omega$.

In Fig. 10, we show the contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the input power $P$ and the effective cavity detuning $\Delta$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. In Fig. 11, we show the contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the input power $P$ and the cavity decay rate $\kappa$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. We fix the angular momentum of the L-G cavity mode at $l=80\hbar$. In Figs. 10 and 11, the region where $n_{eff}<1$ becomes narrower with increasing the strength $\Gamma _l$ of the laser phase noise, which means that increasing the strength $\Gamma _l$ of the laser phase noise can amplify the effect of the laser phase noise on the cooling of the rotating mirror. Note that ground-state cooling can still be realized at a lower input power. Similar to the entanglement, for a larger strength $\Gamma _l$ of the laser phase noise, ground-state cooling can be realized only in the resolved sideband limit.

Fig. 10. Contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the input power $P$ and the effective cavity detuning $\Delta$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. The other parameters are $\frac {\kappa }{\omega _\phi }=1$, $\frac {\Omega }{2\pi }=50$ kHz, $\gamma =\frac {\Omega }{2}$, $l=80\hbar$.

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Fig. 11. Contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the input power $P$ and the cavity decay rate $\kappa$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. The other parameters are $\frac {\Delta }{\omega _\phi }=1$, $\frac {\Omega }{2\pi }=50$ kHz, $\gamma =\frac {\Omega }{2}$, $l=80\hbar$.

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In Fig. 12, we show the contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the input power $P$ and the effective cavity detuning $\Delta$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. In Fig. 13, we show the contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the input power $P$ and the cavity decay rate $\kappa$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. In Figs. 12 and 13, the region where $n_{eff}<1$ becomes narrower with increasing the band center $\Omega$ of the frequency noise spectrum. In addition, ground-state cooling can be realized at a lower input power and in the resolved sideband limit for a larger band center $\Omega$ of the frequency noise spectrum. Similar to the strength $\Gamma _{l}$ of the laser phase noise, increasing the band center of frequency noise spectrum can amplify the effect of the laser phase noise on the cooling of the rotating mirror.

Fig. 12. Contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the input power $P$ and the effective cavity detuning $\Delta$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. The other parameters are $\frac {\kappa }{\omega _\phi }=1$, $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $\gamma =\frac {\Omega }{2}$, $l=80\hbar$.

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Fig. 13. Contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the input power $P$ and the cavity decay rate $\kappa$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. The other parameters are $\frac {\Delta }{\omega _\phi }=1$, $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $\gamma =\frac {\Omega }{2}$, $l=80\hbar$.

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In Fig. 14, we show the contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the angular momentum of the cavity mode $l$ and the effective cavity detuning $\Delta$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. In Fig. 15, we show the contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the angular momentum of the cavity mode $l$ and the effective cavity detuning $\Delta$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. In Figs. 14 and 15, the region where $n_{eff}<1$ becomes narrower with increasing the strength $\Gamma _{l}$ of the laser phase noise and larger band center $\Omega$ of the frequency noise spectrum. In addition, the ground-state cooling can still be realized at a larger strength of the laser phase noise and a larger band center $\Omega$ of the frequency noise spectrum at $l=50\hbar$. Too large angular momentum can amplify the effect of the laser phase noise on the cooling, while too small angular momentum is not beneficial for exchanging the orbital angular momentum between the L-G beam and the rotating mirror to release the excitation quanta of the rotating mirror for cooling.

Fig. 14. Contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the angular momentum of the cavity mode $l$ and the effective cavity detuning $\Delta$ for different strengths $\Gamma _l$ of the laser phase noise, (a) $\Gamma _l=0$, (b) $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and (c) $\frac {\Gamma _l}{2\pi }=1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. The other parameters are $\frac {\kappa }{\omega _\phi }=1$, $\frac {\Omega }{2\pi }=50$ kHz, $\gamma =\frac {\Omega }{2}$, $P=1$ mW.

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Fig. 15. Contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the angular momentum of the cavity mode $l$ and the effective cavity detuning $\Delta$ for different band centers of the frequency noise spectrum $\Omega$($\gamma =\frac {\Omega }{2}$), (a) $\frac {\Omega }{2\pi }=30$ kHz, (b) $\frac {\Omega }{2\pi }=80$ kHz, and (c) $\frac {\Omega }{2\pi }=140$ kHz. The other parameters are $\frac {\kappa }{\omega _\phi }=1$, $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $\gamma =\frac {\Omega }{2}$, $P=1$ mW.

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In Fig. 16, we show the contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the bandwidth $\gamma$ and band center $\Omega$ of the frequency noise spectrum. In Fig. 16, we find that the increase of band center $\Omega$ of the frequency noise spectrum can increase the effective phonon number $n_{eff}$ significantly for a smaller bandwidth $\gamma$ of the frequency noise spectrum. For a larger bandwidth $\gamma$ of the frequency noise spectrum, increasing the band center $\Omega$ of the frequency noise spectrum have a little effect on the effective phonon number of the rotating mirror. Moreover, we find that increasing the bandwidth $\gamma$ of the frequency noise spectrum can decrease the effective phonon number $n_{eff}$ significantly for a larger band center $\Omega$ of the frequency noise spectrum. Thus, the band center $\Omega$ of the frequency noise spectrum is the major factor affecting the ground state cooling in the presence of laser phase noise for a smaller band center $\Omega$ of the frequency noise spectrum, while the bandwidth $\gamma$ of the frequency noise spectrum is the major factor affecting the ground state cooling in the presence of laser phase noise for a larger band center $\Omega$ of the frequency noise spectrum.

Fig. 16. Contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the bandwidth $\gamma$ and band center $\Omega$ of the frequency noise spectrum. The other parameters are $\frac {\Delta }{\omega _\phi }=1$, $\frac {\kappa }{\omega _\phi }=1$, $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $l=80\hbar$, $P=1$ mW.

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In Fig. 17, we show the contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the length $L$ of the cavity and the mass $m$ of the rotating mirror. From Fig. 17, it is seen that decreasing the length $L$ of the cavity or the mass $m$ of the rotating mirror can decrease the effective phonon number first and then increase the effective phonon number. This result is similar to the influence of increasing the angular momentum $l$ of the L-G cavity mode on the effective phonon number. Note that too large or too small the length $L$ of the cavity or the mass $m$ of the mirror is not beneficial for the ground state cooling. Since $g\propto \frac {1}{L}$ and $g\propto \frac {1}{R}$, the influence of the radius $R$ of the rotating mirror is similar to that of the cavity length $L$ on the effective phonon number $n_{eff}$ of the rotating mirror.

Fig. 17. Contour plot of the effective phonon number $n_{eff}$ of the rotating mirror versus the length $L$ of the cavity and the mass $m$ of the rotating mirror. The other parameters are $\frac {\Delta }{\omega _\phi }=1$, $\frac {\kappa }{\omega _\phi }=1$, $\frac {\Gamma _l}{2\pi }=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, $\frac {\Omega }{2\pi }=50$ kHz, $\gamma =\frac {\Omega }{2}$, $l=80\hbar$, $P=1$ mW.

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5. Experimental feasibility

In this section, we discuss the feasibility of the system in the experiment. Experimentally, the high-$l$ L-G modes can be prepared by using the phase element and the topological charge in the L-G beam can be achieved as high as 1000 [18]. Moreover, a mirror with mass 25 ng, radius 15 $\mu$m, and mechanical quality factor $1.3\times 10^5$ is used in a recent experiment [44], which focuses on the cooling of the mirror. In addition, it has been demonstrated experimentally that the torsional frequency of a nanomechanical oscillator can be in the range of 3–20 MHz [45]. Therefore, with the development of the technology, the rotating mirror with mass 50 ng, radius 250 $\mu$m, frequency $2\pi \times 10$ MHz, and mechanical quality factor $10^7$ in our proposal is within reach, and it is possible to realize the experiment we have proposed.

6. Conclusions

We have studied the effects of the laser phase noise on both the steady-state field-mirror entanglement and the ground-state cooling of the rotating mirror in the L-G optorotational system. We find that the band center of the frequency noise spectrum is the major factor affecting the entanglement. Increasing the strength of laser phase noise or the band center of the frequency noise spectrum can lead to decreasing the maximum entanglement and narrowing the entanglement region. Increasing the power of input laser and the angular momentum of the L-G cavity mode don’t always increase the entanglement. In numerical simulations, the entanglement first increases and then decreases with increasing the power of input laser and the angular momentum of the L-G cavity mode. Decreasing the length of cavity and the mass and radius of rotating mirror first increase the entanglement and then decrease the entanglement. In addition, the laser phase noise has a strong destructive effect on the entanglement close to the unstable regime, and the entanglement is more robust against the laser phase noise in the resolved sideband region. When the laser phase noise is present, the system generates a large entanglement at $\Gamma _{l}/(2\pi )=0.01$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$, and the entanglement still exists at $\Gamma _{l}/(2\pi )=0.1$ k$\frac {\mathrm {Hz^2}}{\mathrm {Hz}}$. It is noted that the linewidth 80 Hz of the laser can be achieved experimentally [46]. Thus, with the laser phase noise, the field-mirror entanglement in the optorotational system can be achieved with realistic experimental parameters.

Moreover, considering the laser phase noise, the ground-state cooling can still be realized in the optorotational system. The effective phonon number of the rotating mirror increases for a larger strength and larger band center of the frequency noise spectrum, but the ground-state cooling can still be realized at a lower power of input laser. And the band center of the frequency noise spectrum is the major factor affecting the effective phonon number for a smaller band center of the frequency noise spectrum, while the bandwidth of the frequency noise spectrum is the major factor affecting the effective phonon number for a larger band center of the frequency noise spectrum. If the power of the input laser (or the angular momentum of the L-G cavity mode) is too large or too small, it is difficult to realize the ground state cooling. The effective phonon number first decreases and then increases with increasing the power of input laser and the angular momentum of the L-G cavity mode. Decreasing the length of cavity, the mass and radius of rotating mirror first decrease the effective phonon number and then increase the effective phonon number. What’s more, in the resolved sideband region, the effective phonon number is lower at a low input power.

Appendix: How to reproduce Eq. (12) from Eq. (13)

Here we show that the frequency noise spectrum $S_{\dot {\Phi }}(\omega )$ (Eq. (12)) can be reproduced by Eq. (13). On Fourier transforming Eq. (13), we obtain

(25)$$\begin{aligned} \varPsi(\omega)={-}\frac{\Omega^2\sqrt{2\Gamma_l}\epsilon(\omega)}{\omega^2-\Omega^2+i\omega\gamma}. \end{aligned}$$

The frequency noise spectrum $S_{\dot {\Phi }}(\omega )=S_{\varPsi }(\omega )$ is defined as

(26)$$\begin{aligned} S_{\varPsi}(\omega)=\frac{1}{4\pi}\int d\Omega e^{{-}i(\omega+\Omega)t}\langle \varPsi(\omega) \varPsi(\Omega)+ \varPsi(\Omega) \varPsi(\omega)\rangle. \end{aligned}$$

With the help of the correlation function of the noise $\epsilon$ in the frequency domain, $\langle \epsilon (\omega ) \epsilon (\Omega )\rangle =2\pi \delta (\omega +\Omega )$, we obtain

(27)$$\begin{aligned} S_{\varPsi}(\omega)=2\Gamma_l\frac{\Omega^4}{(\Omega^2-\omega^2)^2+\omega^2\gamma^2}, \end{aligned}$$

which is Eq. (12).

Funding

National Natural Science Foundation of China (12174344, 91636108, 12175199); Natural Science Foundation of Zhejiang Province (LY21A040007, LZ20A040002); Sci-Tech Academy, Zhejiang University(17062071-Y).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Parameter	Description	Value	Units
$L$	Length of the cavity	0.1	mm
$m$	Mass of the mirror	50	ng
$R$	Radius of the mirror	250	$μ$ m
$Q$	Mechanical quality factor	$10^{7}$
$T$	Temperature of the environment	0.4	K
$ω_{ϕ}$	Frequency of the rotating mirror	$2 π \times 10$	MHz
$ω_{c}$	Cavity resonance frequency	$2 π \times 10^{14}$	Hz

Effect of laser phase noise on the steady-state field-mirror entanglement and ground-state cooling in a Laguerre-Gaussian optorotational system

Abstract

1. Introduction

2. Quantum Langevin equations of the system

3. Stationary entanglement between the L-G cavity mode and the rotating mirror

4. Ground-state cooling of the rotating mirror

5. Experimental feasibility

6. Conclusions

Appendix: How to reproduce Eq. (12) from Eq. (13)

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (17)

Tables (1)

Equations (27)

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