Entanglement of two rotating mirrors coupled to a single Laguerre-Gaussian cavity mode

Zheng Chen; Jia-Xin Peng; Jing-Jing Fu; Xun-Li Feng

doi:10.1364/OE.27.029479

1. Introduction

Quantum entanglement [1] is a very important trait in quantum physics and has become a key resource in various practical applications such as quantum information processing [2], quantum algorithms [3], quantum teleportation [4], and quantum cryptography [5]. In the past decades, tremendous efforts have been devoted to the generation of microscopic entanglement [6–10], the entangled states for photons, atoms, ions and so on in microscopic systems have been extensively investigated theoretically and prepared experimentally [11,12]. More features about entanglement in multi-body systems systems were also widely studied in [13].

On the other hand, the study of quantum entanglement for mesoscopic or macroscopic systems becomes an attractive topic and it is of great significance for the exploration of quantum-to-classical boundary [14]. The development of optomechanical cavity systems has prompted researchers to explore mechanical quantum features of macroscopic objects [15,16]. Vitali et al. investigated the entanglement between a single cavity mode and a vibrating mirror [17,18]. Afterwards, Paternostro et al. studied the macroscopic entanglement by extending to the case of two cavity modes coupled to a movable mirror [19] and other researchers extensively investigated the macroscopic entanglement of multiple cavity modes coupled to movable mirrors with various mediums inside the cavity [20–26]. The macroscopic entanglement of two vibrating objects in different cavity systems were also widely studied [27–33].

Recently, a rotation-based cavity optomechanical system was investigated. Bhattacharya and Meystre proposed a Laguerre-Gaussian cavity optomechanical system (L-GCOS) with a rotational mirror [34]. Later they studied the entanglement in such a system which is generated by exchanging orbital angular momentum between the L-G cavity mode and the rotating mirror [35]. They also investigated the entanglement between two qualitatively different degrees of freedom (vibration and rotation) for the same macroscopic object [36]. Very recently, based on the theory of ray transfer matrix, Eggleston et al. investigated the stability of a cavity consisting of two spiral phase plates in the case of incident rays far from the center of the cavity [37]. Later, a scheme of cooling a macroscopic rotating mirror in a double L-GCOS was achieved [38] and the optomechanically induced transparency in the L-GCOS was suggested to detect the orbital angular momentum of light fields [39]. Now, more and more attention has been paid to the rotation-based cavity optomechanical system, it is worth exploring how to realize the entanglement between macroscopic objects in such system.

In this paper, we investigate the entanglement between two macroscopic objects in one L-G cavity which consists of two mirrors rotating along the cavity axis z. The coupling between the two rotating mirrors is established by exchanging the orbital angular momentum with the cavity mode and the entanglement of two rotating mirrors can thus be built. We find that only in some range where the angular frequencies of the two rotating mirrors are not very close to each other, the entanglement will occur efficiently. The entanglement in our system possesses strong robustness and can be maintained at high temperature. We also study the influence of the effective cavity detuning on the entanglement and examine the minimum orbital angular momentum of the L-G cavity mode required to generate entanglement.

The paper is structured as follow. In Section II, we describe the model and introduce the Hamiltonian of the system. Then we derive the quantum Langevin equations and discuss the dynamical stability of the system in Section III. In Section IV, we study the entanglement between two rotating mirrors, and discuss its dependence on rotation frequency of the mirrors, the detuning, the ambient temperature and the orbital angular momentum of the cavity mode. In Section V, we make a brief conclusion.

2. The model and Hamiltonian of the system

The L-GCOS is illustrated in Fig. 1, which consists of two rotating mirrors. The input mirror RM1 is transparent and has no change to topological charge of the beam passed through. The two mirrors are spiral phase element [34,35] and the reflective surfaces of them are assumed to be perfect.

Fig. 1 The Laguerre-Gaussian cavity consists of two rotating mirrors (RM1 and RM2) which are made of perfectly reflecting spiral phase element. The two mirrors are mounted, respectively, on supports S1 and S2. We assume both supports are sufficiently small so that the effect of support S1 on incident light can be neglected. We assume both mirrors have the same initial phase and rotate in the same direction along the cavity axis z. From their equilibrium position (ϕ₀ = 0) of the mirrors, their angular deflections are indicated by the angle ϕ_i (i = 1, 2) and the orbital angular momentum of the beams at various points can be also indicated.

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We assume a Gaussian beam with the zero topological charge is input into the cavity, the azimuthal structure of the rotating mirrors removes the charges from the beam by reflecting. Exactly, the incident Gaussian beam first meets the mirror RM1, and the reflection gets a charge 2l while the most component is transparent and gets charge 0. Then the charge 0 beam gets charge 2l from reflecting at mirror RM2. After returning to the mirror RM1, reflection gives the beam with −2l charge, meanwhile transmission gives the beam with zero charge. With the exchange of orbital angular momentum, the coupling between the two rotating mirrors and the cavity mode is established and the entanglement of the two mirrors can be built.

As assumed above, the beam is perfectly reflected for intracavity and the loss of incident light is thus ignored. In the dimensionless units [35], the Hamiltonian of this system takes the form

\begin{matrix} H = ℏ ω_{c} a^{†} a + \frac{1}{2} \sum_{i}^{2} ℏ ω_{ϕ_{i}} (L_{z_{i}}^{2} + ϕ_{i}^{2}) + ℏ g_{1} a^{†} a ϕ_{1} \\ - ℏ g_{2} a^{†} a ϕ_{2} + i ℏ E (e^{- i ω_{L} t} a^{†} - e^{i ω_{L} t} a), \end{matrix}

where a and a^† ([a, a^†]= 1) are the annihilation and creation operators of the cavity mode with frequency ω_c, respectively, L_{z_i} and ϕ_i (i = 1, 2) the angular momentum and angular displacement of the rotating mirror, respectively, with [L_{z_i}, ϕ_i] = −i. The mass M and the radius R of the two rotating mirrors are assumed to be the same, ω_{ϕ_i} (i = 1, 2) are the angular frequencies of the two rotating mirrors. In Eq. (1), the first term is the free Hamiltonian of the cavity mode, the second term the free Hamiltonian of the two rotating mirrors, the next two terms stand for, respectively, the interaction Hamiltonian between the cavity mode and the two rotating mirrors with coupling parameters g_i, the last term is the driving Hamiltonian to the cavity mode by an external laser field with the frequency ω_L and the power

| E | = \sqrt{2 P κ / ℏ ω_{L}}

with κ being the cavity damping rate. Following in. [36], the coupling parameters are given by

g_{i} = \frac{c l}{L} \sqrt{\frac{ℏ}{I ω_{ϕ_{i}}}} (i = 1, 2),

where c is the light velocity and L is the length of the Laguerre-Gaussian cavity, l is the quantum number of the orbital angular momentum (topological charge), I = MR²/2 is the moment of inertia of the two rotational mirrors about cavity axis z.

3. Dynamical analysis

The evolution of the operators describing the dynamics of the system is given by the quantum Langevin equations [40] which includes the noise and damping due to the Brownian noise acting on the rotating mirrors as well as the vacuum fluctuation entering the cavity, In the Heisenberg picture, the corresponding quantum Langevin equations can be written as:

\begin{array}{l} \dot{a} = - i [Δ_{0} + (g_{1} ϕ_{1} - g_{2} ϕ_{2})] a - κ a + E + \sqrt{2 κ} a^{in}, \\ \dot{ϕ_{1}} = ω_{ϕ_{1}} L_{z_{1}}, \\ {\dot{L}}_{z_{1}} = - ω_{ϕ_{1}} ϕ_{1} - g_{1} a^{†} a - γ_{m_{1}} L_{z_{1}} + ε^{in}, \\ \dot{ϕ_{2}} = ω_{ϕ_{2}} L_{z_{2}}, \\ \dot{L_{z_{2}}} = - ω_{ϕ_{2}} ϕ_{2} + g_{2} a^{†} a - γ_{m_{2}} L_{z_{2}} + ε^{in}, \end{array}

where Δ₀ = ω_c − ω_L and γ_{m_i} (i = 1, 2) are the intrinsic damping rate of rotating mirrors. aⁱⁿ is a noise operator describing the vacuum radiation to input the cavity, whose delta-correlated fluctuations are [35]

〈 a^{in} (t) a^{in, †} (t) 〉 = δ (t - t^{'}),

the Brownian noise operator εⁱⁿ accounts for the mechanical noise coupling to the rotating mirrors from the thermal environment. Its mean value is zero and its fluctuations are relevant to temperature T as [40]

〈 δ ε^{in} (t) δ ε^{in} (t^{'}) 〉 = \frac{γ_{m_{i}}}{ω_{ϕ_{i}}} \int_{- \infty}^{\infty} \frac{d ω}{2 π} e^{- i ω (t - t^{'})} ω [1 + coth (\frac{ℏ ω}{2 k_{B} T})],

where k_B is the Boltzmann’s constant and T is the temperature of the environment.

By considering the quantum fluctuation of each Heisenberg operator about its steady state, we can linearize the operator around their steady-state values in the usual way that add an additional fluctuation operator to zero-mean value, a = α_s + δa.. Here δa is quantum fluctuation, other operators do the same treatment. Inserting these statements into the quantum Langevin Eq. (3), then they can be decoupled into a set of equations for steady-state values and fluctuation operators [41,42].

The steady-state values of the dynamical variables can be obtained from Eq. (3), they are

\begin{array}{l} α_{s} = \frac{| E |}{{(κ^{2} + Δ^{2})}^{\frac{1}{2}}}, \\ ϕ_{1 s} = - \frac{g_{1} {| α_{s} |}^{2}}{ω_{ϕ_{1}}}, \\ L_{z_{1}, s} = 0, \\ ϕ_{2 s} = \frac{g_{2} {| α_{s} |}^{2}}{ω_{ϕ_{2}}}, \\ L_{z_{2}, s} = 0, \end{array}

where α_s is the field amplitude of stationary cavity,

Δ = Δ_{0} - (\frac{g_{1}^{2}}{ω_{ϕ_{1}}} + \frac{g_{2}^{2}}{ω_{ϕ_{2}}})

|α_s|² is the effective cavity detuning. In general, the mean value of each dynamical variable considering here is much larger than its corresponding fluctuation, e.g. |α_s| ≫ δa. In this condition, one can safely neglect the nonlinear terms, e.g. δaδa^†. Further we can get the following linearized quantum Langevin equations:

\begin{array}{l} \dot{δ ϕ_{1}} = ω_{ϕ_{1}} δ L_{z_{1}}, \\ \dot{δ L_{z_{1}}} = - ω_{ϕ_{1}} δ ϕ_{1} - γ_{m_{1}} L_{z_{1}} - G_{1} δ X + δ ε^{in}, \\ \dot{δ ϕ_{2}} = ω_{ϕ_{2}} δ L_{z_{2}}, \\ \dot{δ L_{z_{1}}} = - ω_{ϕ_{2}} δ ϕ_{2} - γ_{m_{2}} L_{z_{2}} + G_{2} δ X + δ ε^{in}, \\ \dot{δ X} = - κ δ X + Δ δ Y + \sqrt{2 κ} X^{in}, \\ \dot{δ Y} = - κ δ Y - Δ δ X - G_{1} δ ϕ_{1} + G_{2} δ ϕ_{2} + \sqrt{2 κ} Y^{in}, \end{array}

where

G_{1} = g_{1} α_{s} \sqrt{2}

and

G_{2} = g_{2} α_{s} \sqrt{2}

are the effective optomechanical coupling parameters. In a simple situation where the input field is kept constant throughout, it is possible to choose the phase reference of the cavity field, such that the steady-state field amplitude α_s is real. Here we have defined the cavity field quadratures as

δ X = (δ a + δ a^{†}) / \sqrt{2}

and

δ Y = (δ a - δ a^{†}) / i \sqrt{2}

, the corresponding Hermitian input noise operators

X^{in} = (a^{in} + a^{in, †}) / \sqrt{2}

and

Y^{in} = (a^{in} - a^{in, †}) / i \sqrt{2}

[24].

Therefore, the Eq. (7) can be simply written as

\dot{u (t)} = Au (t) + n (t),

where u^T(t) = (δϕ₁, δL_z₁, δϕ₂, δL_z₂, δX, δY) is the vector of continuous variable fluctuation operators, n^T(t) = (0, δεⁱⁿ, 0, δεⁱⁿ,

\sqrt{2 κ} X^{in}

,

\sqrt{2 κ} Y^{in}

) is the vector of noise, and the matrix

A = (\begin{matrix} 0 & ω_{ϕ_{1}} & 0 & 0 & 0 & 0 \\ - ω_{ϕ_{1}} & - γ_{m_{1}} & 0 & 0 & - G_{1} & 0 \\ 0 & 0 & 0 & ω_{ϕ_{2}} & 0 & 0 \\ 0 & 0 & - ω_{ϕ_{2}} & - γ_{m_{2}} & G_{2} & 0 \\ 0 & 0 & 0 & 0 & - κ & Δ \\ - G_{1} & 0 & G_{2} & 0 & - Δ & - κ \end{matrix})

determines the dynamic stability of the optomechanical system. According to the Routh-Hurwitz criterion, the stability of steady-state solution can be guaranteed to exist if the real parts of the eigenvalues of A are negative. The detailed conditions for this situation to occur are derived by a simple stability-analysis test [43].

4. Simulation and analysis of the entanglement

In this section let us quantify the entanglement between the two rotating mirrors based on linearized Langevin Eq. (8) in terms of the logarithmic negativity E_N [44]. The logarithmic negativity E_N has been proposed to measure the entanglement of continuous Gaussian variables [47]. When the steady-state conditions are satisfied, the Lyapunov equation about covariance matrix V is given by

AV + {AV}^{T} = - D,

where D = Diag[0, γ_m₁(2n₁ + 1), 0, γ_m₂(2n₂ + 1), κ, κ] is the matrix of the noise correlation [9], n_i = [exp (ħω_{ϕ_i,eff}/k_BT) − 1]⁻¹ (i = 1, 2) are the mean number of thermal phonons available at the rotating mirrors of frequency ω_{ϕ_i} (i = 1, 2) and temperature T.

V is a 6 × 6 covariance matrix which can be described the entanglement structure of the system, its form is

V = (\begin{matrix} M_{1} & N_{12} & N_{1 m} \\ N_{12}^{T} & M_{2} & N_{2 m} \\ N_{1 m}^{T} & N_{2 m}^{T} & M_{M} \end{matrix}),

in which V_ij = [〈u_i(∞)u_j(∞)〉 + 〈u_j(∞)u_i(∞)〉]/2 is the element, where

u^{T} (\infty) = (δ ϕ_{1} (\infty), δ L_{z_{1}} (\infty), δ ϕ_{2} (\infty), δ L_{z_{2}} (\infty), δ X (\infty), δ Y (\infty))

is the vector of the fluctuation operators at the steady state. In the matrix V, M_j (j = 1, 2, m) is a 2 × 2 matrix describing the local properties of the rotating mirrors and cavity mode, N_ij (i, j = 1, 2, m) is a 2 × 2 matrix expressing the correlations between mirrors and cavity mode [19]. Therefore, E_N can be defined as [45]

E_{N} = max [0, - ln (2 η^{-})],

where η⁻ = 2^−1/2 [σ − (σ² − 4 |Λ|)^1/2]^1/2 and the σ = det A + det B − 2 det C can be obtained from the correlation matrix

Λ = (\begin{matrix} A & C \\ C^{T} & B \end{matrix}) .

The element of covariance matrix Λ_ij is a 4 × 4 matrix from V by taking the block in Eq. (11), the logarithmic negativity E_N about the two rotating mirrors in this system can thus be described and calculated [44–46]. In the following numerical simulation, the parameters taken are listed in Table. 1. The value of E_N is related to the parameters of the system under study. If we choose the perfect parameters that can be realized experimentally, the corresponding value of EN will also be improved. In other words, The bounds of E_N depend on the parameters achievable in the experiments.

Table 1. Definitions and approximate values of the parameters used in the text

View Table

In the following subsections we examine the entanglement between the two rotating mirrors related respectively to angular frequencies, effective detuning, temperature and the orbital angular momentum by using numerical simulation.

4.1. The rotation frequencies of the mirrors

In Fig. 2, the logarithmic negativity E_N is plotted as a function of the angular frequency ratio ω_ϕ₂/ω_ϕ₁ for different quality factor of the cavity, here the angular frequency ω_ϕ₁ of the first rotating mirror is taken as 12π × 10⁷Hz. The value of quality factors here are slightly larger than those in [27,35], but still within the reachable range. Three curves show that for a certain quality factor Q the entanglement drops sharply to zero when ω_ϕ₂/ω_ϕ₁ increases to a value d − and keeps unchanged until equals to another value d+, then it rises sharply when ω_ϕ₂/ω_ϕ₁ > d +. Apparently, the range that E_N =0 becomes narrower with the increase of the quality factor Q. Moreover, for a given ratio ω_ϕ₂/ω_ϕ₁ the entanglement E_N becomes larger with the increase of the quality factor Q. This can be easily understood because the larger quality factor Q the cavity has, the smaller the cavity decay rate is, and thus the cavity decay rate has less influence on the entanglement between two rotating mirrors.

Fig. 2 The logarithmic negativity E_N is plotted as a function of the angular frequency ratio ω_ϕ₂/ω_ϕ₁ of the rotating mirrors for different quality factors Q = 1.0 × 10⁷ (solid line), Q = 3.0 × 10⁷ (dash line) and Q = 3.0 × 10⁹ (dot line). Here the angular frequency ω_ϕ₁ of the first rotating mirror is 12π × 10⁷Hz, the other parameters are referred to Table. 1.

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The range of E_N = 0 will approach to a point ω_ϕ₂/ω_ϕ₁ = 1.0 from both sides with the increase of the quality factor Q, it indicates that for the Gaussian light input the cavity and the two rotating mirrors have the same initial phase, although there is an exchange of orbital angular momentum between light and two rotating mirrors when light is reflected by them, the exchange of total orbital angular momentum between light and two mirrors is zero for each reflection period, this is why there is no entanglement when the angular frequencies of two rotating mirrors are equal.

4.2. Effective detuning

In Fig. 3, the logarithmic negativity E_N is plotted as a function of the detuning Δ/ω_ϕ₁ for different mass of the mirrors, here ω_ϕ₁ = 12π × 10⁷Hz. With the parameters selected in Table. 1, we find that to make the stable entanglement between the two rotating mirrors appear in a steady state the effective detuning should meet the condition Δ < 0, which is similar to the case in [19].

Fig. 3 The logarithmic negativity E_N is plotted as a function of the normal optical detuning ω_ϕ₁ = 12π × 10⁷Hz for different mass of the rotating mirrors M = 35ng (solid line), M = 55ng (dashed line) and M = 75ng (dotted line). Here the angular frequency ω_ϕ₂ of the second rotating mirror is 6π × 10⁷Hz, other parameters are referred to Table. 1.

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All three curves show that the entanglement occurs from |Δ| /ω_ϕ₁ = 0.35, and the entanglement value E_N increases as the effective detuning increases and reaches its maximal value at about |Δ| /ω_ϕ₁ ≈ 1.0, then E_N decreases with the increase of the effective detuning for |Δ| /ω_ϕ₁ > 1.0 until E_N = 0. In addition, as the mass of the rotating mirrors decreases, the maximum value of entanglement increases and the entanglement can also persist for a larger region. Physically, the condition |Δ| /ω_ϕ₁ ≈ 1.0 corresponds to the optimum detuning for optomechanical cooling [39,47] and can be illustrated in terms of cavity enhanced scattering of the anti-Stokes photons by the rotating mirrors [48], it’s not surprising that the entanglement reaches a maximum at such a detuning.

4.3. Temperature

In Fig. 4, the logarithmic negativity E _N is plotted as a function of the environment temperature for different mass of the mirrors. As an example, let us first analyze the solid curve which corresponds to M = 50ng. Obviously, the solid curve shows that, with increase of the temperature, the entanglement of two rotating mirrors decreases from its maximal value E_N = 0.36 at T = 0K to E_N = 0 at T = 75K, we refer to T = 75K as the terminating temperature and denote it by T_c. The presence of the terminating temperature T_c can be explained as the harmful effect of thermal noise will eventually make the entanglement disappear with the increase of temperature..

Fig. 4 The logarithmic negativity E_N is plotted as a function of the ambient temperature for different mass of the rotating mirrors M = 50ng (solid line), M = 100ng (dashed line), M = 210ng (dotted line) and M = 300ng (dot-dashed line). Here the angular frequency ω_ϕ₂ of the second rotating mirror is 6π × 10⁷Hz, other parameters are referred to Table. 1.

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The other three curves with larger mass of the mirrors have the similar tendency, that is, the entanglement in each curve decreases with the increase of the temperature and there is a maximal entanglement at T = 0K and a terminating temperature T_c for E_N = 0. Comparing the four curves in Fig. 4 one can find with the increase of the mass, the maximal entanglement at T = 0K and the absolute value of the slope of curves become smaller, with the latter implying that the rotating mirrors with larger mass possess stronger ability to resist decoherence of the thermal environment.

Interestingly, the terminating temperature T_c increases with mass until it reaches about 210ng (dotted line), while when the mass is larger than 210ng the terminating temperature T_c becomes smaller. This phenomenon can be understood if one considering the mass has two inverse effects on the entanglement, on the one hand, the larger mass makes the entanglement become smaller, on the other hand, the larger mass makes the rotating mirrors possess robustness to resist decoherence, the competition of the two inverse effects yields such a result.

4.4. Orbital angular momentum

In Figs. 5 and 6, we plot the logarithmic negativity E_N as a function of the orbital angular momentum of the L-G cavity mode for different mass of the rotating mirrors and for different temperature, respectively. The solid curve in Fig. 5 shows that for certain mass M = 100ng and temperature T = 5K the entanglement emerges only when the orbital angular momentum l is larger than certain value 28, denoted by l_c, corresponding to a threshold orbital angular momentum of the L-G cavity mode, that is, in such a case if l < l_c the entanglement of the two rotating mirrors cannot be built. For l > l_c the entanglement increases with l. The other two curves in Fig. 5 have the similar trend, but comparing the three curves, the threshold l_c will become smaller with the decrease of the mass.

Fig. 5 The logarithmic negativity E_N is plotted as a function of the orbital angular momentum carried by the L-G mode for different mass of the rotating mirrors M = 100ng (solid line), M = 75ng (dashed line) and M = 50ng (dotted line) at the same temperature T = 5K; other parameters are referred to Table. 1.

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In fact, this result can be explained by Eq. (2) and Eq. (3), the Eqs. (3) indicate the influence of the mass and orbital angular momentum of the L-G cavity mode on the entanglement is reflected by the coupling parameters g_i (i = 1, 2) expressed in Eq. (2). From Eq. (2) we can see that one can reduce the threshold l_c by decreasing the mass. Obviously other parameters such as the cavity length, the rotating frequencies in Eq. (2) that are able to enhance coupling strength can reduce threshold l_c.

In Fig. 6 the three curves show that the threshold l_c will increase with the temperature. When M = 50ng and T = 5K, the threshold l_c is about 20, and when the mass remains unchanged and the temperature continues to increase, the value of l_c will also increase, which indicates the larger orbital angular momentum is required to overcome the thermal noise at higher temperature so as to generate entanglement.

Fig. 6 The logarithmic negativity E_N is plotted as a function of the orbital angular momentum carried by the L-G mode for different environment temperatures T = 5K (solid line), T = 15K (dashed line) and T = 25K (dotted line) at the same mass M = 50ng. Here the angular frequency ω_ϕ₂ of the second rotating mirror is 6π × 10⁷Hz, other parameters are referred to Table. 1.

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On the other hand, as shown in Figs. 5 and 6, to get a lower threshold l_c, one has to further reduce the mass or the temperature. As an example, for an experimentally-reached situation the mass and temperature are taken as M = 50ng and T = 1K, then the threshold l_c is about 19. While for the smaller mass and lower temperature such as M = 10ng and T = 300mK, a lower threshold of l_c can be achieved l_c ≈ 9. Naturally, if one wants to get a much lower threshold for l_c, the temperature and the mass are required to further decrease. Thus it theoretically indicates that a condition for small orbital angular momentum to generate entanglement in our cavity system is the smaller mass of the mirrors and the less influence of the thermal noise (that is, the lower temperature).

5. Conclusion

In summary, we have investigated a stationary macroscopic entanglement of two rotating mirrors by coupling them to a single Laguerre-Gaussian cavity mode. On the one hand, the entanglement, quantified by the logarithmic negativity E_N, is numerically calculated by using some realistic parameters. The main results obtained in this work include: (i) in some range for the angular frequencies ω_ϕ₁ and ω_ϕ₂ of the two rotating mirrors where ω_ϕ₁ and ω_ϕ₂ are not very close to each other the entanglement can occur efficiently; (ii) the maximal value of entanglement appears at the effective cavity detuning |Δ| ≈ ω_ϕ₁, and with the decrease of the mass of mirrors the entanglement increases; (iii) the entanglement has strong robustness to resist the thermal noise and it can persist to high temperature; (iv) the threshold angular momentum l_c to generate entanglement is determined by the rotational parameters strength and the temperature. On the other hand, let us compare the entanglement examined here with that in a system of a normal Fabry-Perot cavity with two linear oscillatory mirrors studied in [33]. Similarly, the steady-state entanglement between the two oscillatory mirrors also occurs when the cavity detuning is close to the mechanical frequency, but differently the robustness of the entanglement in the latter case is quite fragile, and it only exists at micro-Kelvin temperature [33]. In addition, the steady-state entanglement obtained in the latter case is quite small, which is several orders of magnitude smaller than our value of E_N. Obviously, our rotational system takes apparent advantages over the system of two linear oscillatory mirrors. Our results are meaningful to the research of the entanglement of macroscopic objects at the system of rotational cavity and have the potential applications in quantum information.

Funding

National Natural Science Foundation of China (61475168, 11674231, 11074079 and 11804225); Ministry of Science and Technology of the People’s Republic of China (2018YFA0404803).

Acknowledgments

Chen thanks the helpful discussion with M. Bhattacharya. XLF is sponsored by Shanghai Gaofeng & Gaoyuan Project for University Academic Program Development.

References

1. E. Schrodinger, “Discussion of Probability Relations between Separated Systems,” Proc. Cambridge Philos. Soc. 31(4), 555–563 (1935). [CrossRef]

2. M. A. Nielson and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000), pp. 558–559.

3. A. M. Childs and W. Van Dam, “Quantum algorithms for algebraic problems,” Rev. Mod. Phys. 82, 1–48 (2010). [CrossRef]

4. X.-M. Jin, J.-G. Ren, B. Yang, Z.-H. Yi, F. Zhou, X.-F. Xu, S.-K. Wang, D. Yang, Y.-F. Hu, S. Jiang, T. Yang, H. Yin, K. Chen, C.-Z. Peng, and J.-W. Pan, “Experimental free-space quantum teleportation,” Nat. Photon. 4(6), 376–381 (2010). [CrossRef]

5. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–190 (2002). [CrossRef]

6. B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413(6854), 400–403 (2001). [CrossRef] [PubMed]

7. M. Wallquist, K. Hammerer, P. Zoller, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, J. Ye, and H. J. Kimble, “Single-atom cavity QED and optomicromechanics,” Phys. Rev. A 81, 023816 (2010). [CrossRef]

8. K. Borkje, A. Nunnenkamp, and S. M. Girvin, “Proposal for Entangling Remote Micromechanical Oscillators via Optical Measurements,” Phys. Rev. Lett. 107, 123601 (2011). [CrossRef] [PubMed]

9. C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Ohberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012). [CrossRef]

10. A Xuereb, C Genes, and A Dantan, “Strong Coupling and Long-Range Collective Interactions in Optomechanical Arrays,” Phys. Rev. Lett. 109, 223601 (2012). [CrossRef]

11. D. Esteve, J. M. Raimond, and J. Dalibard, Quantum Entanglement and Information Processing: lecture notes of the Les Houches Summer School (Elsevier, 2004).

12. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009). [CrossRef]

13. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, “Entanglement in many-body systems,” Rev. Mod. Phys. 80(2), 517–570 (2008). [CrossRef]

14. B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev A 86, 042323 (2012). [CrossRef]

15. C. K. Law, “Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation,” Phys. Rev. A 51(3), 2537–2541 (1995). [CrossRef] [PubMed]

16. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1426 (2014). [CrossRef]

17. D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical Entanglement between a Movable Mirror and a Cavity Field,” Phys. Rev. Lett. 98, 030405 (2007). [CrossRef] [PubMed]

18. C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008). [CrossRef]

19. M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and Probing Multipartite Macroscopic Entanglement with Light,” Phys. Rev. Lett. 99, 250401 (2007). [CrossRef]

20. H. Xiong, M. O. Scully, and M. S. Zubairy, “Correlated Spontaneous Emission Laser as an Entanglement Amplifier,” Phys. Rev. Lett. 94, 023601 (2005). [CrossRef] [PubMed]

21. M. Kiffner, M. S. Zubairy, J. Evers, and C. H. Keitel, “Two-mode single-atom laser as a source of entangled light,” Phys. Rev. A 75, 033816 (2007). [CrossRef]

22. S. Qamar, M. Al-Amri, S. Qamar, and M. S. Zubairy, “Entangled radiation via a Raman-driven quantum-beat laser,” Phys. Rev. A 80, 033818 (2009). [CrossRef]

23. S. Qamar, M. Al-Amri, and M. S. Zubairy, “Entanglement in a bright light source via Raman-driven coherence,” Phys. Rev. A 79, 013831 (2009). [CrossRef]

24. L. Zhou, Y. Han, J. Jing, and W. Zhang, Phys. “Entanglement of nanomechanical oscillators and two-mode fields induced by atomic coherence,” Phys. Rev. A 83, 052117 (2011). [CrossRef]

25. W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, “Entanglement of movable mirrors in a correlated-emission laser,” Phys. Rev. A 88, 022338 (2013). [CrossRef]

26. W. Ge, M. Al-Amri, H. Nha, and M. S. Zubairy, “Entanglement of movable mirrors in a correlated emission laser via cascade-driven coherence,” Phys. Rev. A 88, 052301 (2013). [CrossRef]

27. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling Macroscopic Oscillators Exploiting Radiation Pressure,” Phys. Rev. Lett. 88, 120401 (2002). [CrossRef] [PubMed]

28. C.-J. Yang, J.-H. An, W. Yang, and Y. Li, Phys. “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92, 062311 (2015). [CrossRef]

29. J. Li, Gang Li, S. Zippilli, D. Vitali, and Tiancai Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95, 043819 (2017). [CrossRef]

30. M. J. Hartmann and M. B. Plenio, “Steady State Entanglement in the Mechanical Vibrations of Two Dielectric Membranes,” Phys. Rev. Lett. 101, 200503 (2008). [CrossRef] [PubMed]

31. J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014). [CrossRef]

32. M. Wang, X.-Y. Lü, Y.-D. Wang, J.-Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94, 053807 (2016). [CrossRef]

33. D. Vitali, S. Mancini, and P. Tombesi, “Stationary entanglement between two movable mirrors in a classically driven Fabry–Perot cavity,” J. Phys. A: Math. Theor. 40(28), 8055–8068 (2007). [CrossRef]

34. M. Bhattacharya and P. Meystre, “Using a Laguerre-Gaussian Beam to Trap and Cool the Rotational Motion of a Mirror,” Phys. Rev. Lett. 99, 153603 (2007). [CrossRef] [PubMed]

35. M. Bhattacharya, P.-L. Giscard, and P. Meystre, “Entanglement of a Laguerre-Gaussian cavity mode with a rotating mirror,” Phys. Rev. A 77, 013827 (2008). [CrossRef]

36. M. Bhattacharya, P.-L. Giscard, and P. Meystre, “Entangling the rovibrational modes of a macroscopic mirror using radiation pressure,” Phys. Rev. A 77, 030303(R) (2008). [CrossRef]

37. M. Eggleston, T. Godat, E. Munro, M. A. Alonso, H. Shi, and M. Bhattacharya, “Ray transfer matrix for a spiral phase plate,” J. Opt. Soc. Am. A 30(12), 2526–2530 (2013). [CrossRef]

38. Y.-M. Liu, C.-H. Bai, D.-Y. Wang, T. Wang, M.-H. Zheng, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-Laguerre-Gaussian-cavity with atomic ensemble,” Opt. Express , 26(5), 6143–6157 (2018) [CrossRef] [PubMed]

39. J.-X. Peng, Z. Chen, Q.-Z. Yuan, and X.-L. Feng, “Optomechanically induced transparency in a Laguerre-Gaussian rotational-cavity system and its application to the detection of orbital angular momentum of light fields,” Phys. Rev. A 99, 043817 (2019). [CrossRef]

40. C. Gardiner and P. Zoeller, Quantum Noise (Springer, 1991). [CrossRef]

41. C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49(2), 1337–1343 (1994). [CrossRef] [PubMed]

42. S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055 (1994). [CrossRef] [PubMed]

43. E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987). [CrossRef]

44. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002). [CrossRef]

45. G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004). [CrossRef]

46. R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems,” Phys. Rev. Lett. 84, 2726 (2000). [CrossRef] [PubMed]

47. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum Theory of Cavity-Assisted Sideband Cooling of Mechanical Motion,” Phys. Rev. Lett. 99, 093902 (2007). [CrossRef] [PubMed]

48. A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation Pressure Cooling of a Micromechanical Oscillator Using Dynamical Backaction,” Phys. Rev. Lett. 97, 243905 (2006). [CrossRef]

L = 1mm,	Cavity length
λ = 810mm,	Laser wavelength
ω_c = 7.4π × 10¹⁴Hz,	Cavity mode’s frequency
M = 50ng,	Rotating mirror mass
R = 10μm,	Rotating mirror radius
Q = 2.0 × 10⁷	Quality factor
ω_ϕ₁ = 12π × 10⁷Hz,	Rotating mirror angular frequency
F = 5 × 10³,	Optical finesse
l = 100ħ,	Orbital angular momentum
P = 50mW,	Input laser power
T = 1K,	Temperature
\|Δ\| /ω_ϕ₁ = 1.0,	Normalized laser detuning

Entanglement of two rotating mirrors coupled to a single Laguerre-Gaussian cavity mode

Abstract

1. Introduction

2. The model and Hamiltonian of the system

3. Dynamical analysis

4. Simulation and analysis of the entanglement

4.1. The rotation frequencies of the mirrors

4.2. Effective detuning

4.3. Temperature

4.4. Orbital angular momentum

5. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (6)

Tables (1)

Equations (14)

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