Coupling mechanical motion of a single atom to a micromechanical cantilever

Wenjie Nie; Aixi Chen; Yueheng Lan

doi:10.1364/OE.25.032931

1. Introduction

Recently a lot of attention has been drawn to the coherent manipulation of the dynamics of micromechanical oscillators with cavity optomechanics [1–3]. The general setup consists of movable parts with small effective mass and high quality factor, such as oscillating cavity wall [4–7], levitated nanoparticle or nanodiamond [8–13], membrane [14–17], microscale cantilever [18–20] and a cavity field so that the interaction between the electromagnetic fields and a macroscale mechanical oscillator can be enabled by shining directly a light on the mechanical system. Various quantum characteristics of mechanical motion in the optomechanical systems, i.e., ground state cooling of mechanical motion [4, 15, 21–24], non-classical state preparation in mechanical resonators [25–27], squeezing of mechanical mode [28, 29] and optomechanical normal-mode splitting (NMS) [30, 31], have been explored extensively, where the optomechanical coupling via radiation pressure of light plays a determined role.

On the other hand, the light propagation in an optomechanical system can be manipulated by tailoring the interaction between the mechanical oscillator and the light field. A special example of controlling light is optomechanically induced transparency (OMIT) [6, 32–41], where the width of transparency window is related directly to the strength of the optomechanical coupling. Further, upon application of the cavity quantum electrodynamics (QED) [42] to the optomechanical systems, the effective optomechanical coupling between the mechanical motion and the cavity field can be promoted further by the nonlinearity of atomic ensemble [43–49], which changes the quantum characteristics of mechanical system and the light propagation properties. The influence of a single atom or qubit on the response properties of light has been investigated in detail [50, 51]. In addition, the optomechanical coupling between a light field and a collective mode of cold atomic ensemble inside a cavity is explored theoretically [52–55].

Apart from the direct optomechanical coupling via radiation pressure, the internal state of a two-level system, such as a single atom or an ultracold atomic ensemble [56–58], a Josephson-junction qubit [59, 60] or an electron spin accumulated in a carbon nanotube (CNT) [61], is used for mediating the optomechancial coupling between a light field and a mechanical motion. In particular, a strong coupling between a mechanical membrane and the mechanical motion of a single atom trapped inside an optical cavity can be achieved by two quantized fields [56], where the quantum cavity fields are coupled simultaneously to the internal state of the single atom and its mechanical motion. In this setup, the mass ratio of the atom M and the mechanical oscillator m can be maneuvered not to enter the effective coupling and therefore the strong coupling can be realized by adjusting the detunings of cavity fields. It is noted that the internal state of a two-level atom or emitter can be coupled to a nearby plane via the vacuum fluctuation force when they are very close [62–64]. This also enables a direct coupling between the internal state of a single atom and a mechanical membrane, i.e., suspended graphene sheet [63, 64].

In this work we consider a hybrid optomechanical setup consisting of a micromechanical cantilever and a single trapped atom, which are both placed in a Fabry-Pérot cavity. In particular, the micromechanical cantilever plane approaches the atom and therefore the cantilever plane shifts the energy level of the atom via vacuum effect between them, which leads to couple the mechanical motions of the single atom and the oscillating cantilever despite the smallness of the scale factor $\sqrt{M / m} ~ 10^{- 6}$ [20]. Moreover, an effective optomechanical coupling between the micromechanical cantilever and the cavity field enabled via the internal state of the single atom. We analyze the occurrence of NMS in the displacement spectrum of the micromechanical cantilever and discuss in detail influences of the atom-cantilever coupling strength and the steady-state position of atom on the NMS. Further, we investigate the response of the system to a weak probe beam and show that a double OMIT behavior is generated in the presence of the atom-cantilever coupling. These results can be used for demonstrating the effective coupling between the mechanical motions of a single atom and a massive mechanical cantilever despite the fact that the mass of atom is much less than the effective mass of the micromechanical cantilever.

2. Optomechanical model

The atom-cantilever optomechanical system studied here is depicted in Fig. 1, where we consider an atom of mass M, which is confined inside a standing-wave cavity by a external harmonic potential of frequency ω_M₀ [65]. The atom’s center-of-mass (c.m.) motion is dominated along the cavity axis x, while the other transverse motions, i.e., y-direction or z-direction in Fig. 1, are confined tightly so that the corresponding degrees of freedom of motion have been neglected. The relevant internal degrees of freedom of the atom are the ground state |g〉 and the excited state |e〉 with the bare transition frequency ω_a₀ and the free-space spontaneous emission rate Γ₀. Further, the two-level atom interacts strongly with a cavity mode of frequency ω_c ≫ ω_a₀. The system is coherently driven by a classical control field with frequency ω_L and driving amplitude ε_L.

Fig. 1 Schematic of the setup studied in the paper. The optomechanical system consists of a micromechanical cantilever and a single two-level atom both placed inside a Fabry-Pérot cavity. The system is driven by an external field with frequency ω_L and the cavity mode with frequency ω_c is probed by a weak probe field with frequency ω_p. The distance between the atom and the cantilever plane is so small that the internal state of the atom is influenced by the nearby plane due to the vacuum energy fluctuation.

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In addition, we assume that the atom approaches a micromechanical cantilever with an effective mass m and oscillation frequency ω_m₀ placed in the cavity (x − z plane). That is, the distance between the atom and the cantilever is very small (i.e., the order of the nanometes) so that the level shift of the atom induced by the zero-point energy fluctuation should be considered [62, 63]. The vacuum fluctuation leads to a direct coupling between the internal states of the two-level atom and the oscillation cantilever. It is noted that in the presence of the cantilever, the vacuum force between the atom and the cantilever is along the y-direction and therefore does not affect the mechanical motion and the oscillating frequency of the atom along the x-direction. That is, the transverse vacuum potential or force created by the nearby dielectric material is unnecessary for studying the dynamics of the atom in the x-direction. Furthermore, the micromechanical cantilever is placed in the x − z plane and geometrically different from a movable mirror [4] or dielectric membrane [14]. Therefore, it is not necessary to take into account the light radiation pressure on the cantilever because light in the cavity is reflected back inefficiently [66]. Despite of the fact, an effective optomechanical coupling between the cavity field and the oscillation cantilever can be mediated by the internal states of the single atom. In particular, the effective coupling between the mechanical motions of a single atom and a micromechanical cantilever can be generated despite the scale factor $\sqrt{M / m} ≪ 1$ .

The Hamiltonian of the system can be written as [63, 65, 67, 68]

\begin{array}{l} H = ℏ ω_{c} a^{†} a + \frac{p_{y}^{2}}{2 m} + \frac{1}{2} m ω_{m 0}^{2} y^{2} + \frac{p_{x}^{2}}{2 M} + \frac{1}{2} M ω_{M 0}^{2} x^{2} \\ + ℏ [ω_{a 0} + Δ ω_{a} (d + y)] σ_{+} σ_{-} - i ℏ g (x) (σ_{+} a - σ_{-} a^{†}) + i ℏ (ε_{L} a^{†} e^{- i ω_{L} t} - H . c .), \end{array}

where the first term in the first line is the free Hamiltonian of the driven cavity field and a the photon annihilation operator in the cavity mode of frequency ω_c = kc, satisfying the commutation relation [a, a^†] = 1. c is the speed of light in vacuum. The other terms in the first line describe the kinetic and potential energies of the cantilever and the atom, x(y) and p_x (p_y) are the position and momentum operators for the atom (cantilever) with commutation relation [x, p_x] = iℏ ([y, p_y] = iℏ). The first term in the second line is the free Hamiltonian of the atom, where Δω_a(d + y) is the frequency shift due to vacuum fluctuation [24, 62, 63] and related to the distance between the atom and the cantilever. Here d is the distance between the two-level atom and the cantilever in the static atom-plane geometry, and y the position of the cantilever. σ = |g〉〈e| is the lowering operator of the atom, σ^† its adjoint. The second term in the second line describes the interaction of the atom with the driven cavity field, where g (x) represents the cavity atom coupling strength at x, with g(x) = g₀cos(kx) [68, 69]. Finally, the last terms describe the interaction of the cavity field with the control field, with amplitude

| ε_{L} | = \sqrt{2 P_{c} κ / ℏ ω_{L}}

. κ is the decay rate of the cavity field and P_c is the laser power.

In a rotating frame with the laser frequency ω_L, the Hamiltonian (1) of the atom-cantilever optomechanical system is

\begin{array}{l} H = - ℏ Δ_{c 0} a^{†} a + \frac{p_{y}^{2}}{2 m} + \frac{1}{2} m ω_{m 0}^{2} y^{2} + \frac{p_{x}^{2}}{2 M} + \frac{1}{2} M ω_{M 0}^{2} x^{2} \\ + ℏ Δ_{a 0} (d + y) σ_{+} σ_{-} - i ℏ g_{0} \cos (k x) (σ_{+} a - σ_{-} a^{†}) + i ℏ (ε_{L} a^{†} - H . c .), \end{array}

where Δ_c₀ = ω_L − ω_c is the detuning of the pump laser from the cavity and Δ_a₀ (d + y) = ω_L − ω_a₀ −∆ω_a(d + y) is the detuning from the atomic resonance. The corresponding Heisenberg equation for the atomic polarization and the cavity field operator can be written as [69–71]

{\dot{σ}}_{-} = - (γ_{a} - i Δ_{a 0}) σ_{-} + g (x) a σ_{z} + σ_{z} \sqrt{2 γ_{a}} Γ_{i n},

\dot{a} = - (κ - i Δ_{c 0}) a + g (x) a σ_{-} + ε_{L} + \sqrt{2 κ} a_{i n} .

It is noted that a nearby boundary can modify the spontaneous emission rate of the two-level system, i.e. γ_a = γ_a(d + y). In the present model we consider a large atom-pump detuning and low saturation, i.e., ω_L ≫ ω_a and Δ_a₀(d + y) ≫ γ_a, so that the excited atomic state can be eliminated adiabatically [68, 69]. This leads to

σ_{-}^{s} = \frac{- g (x) a}{γ_{a} - i Δ_{a 0}}

. Then, the effective Hamiltonian of the atom-membrane optomechanical system become

\begin{matrix} H = \frac{p_{y}^{2}}{2 m} + \frac{1}{2} m ω_{m 0}^{2} y^{2} + \frac{p_{x}^{2}}{2 M} + \frac{1}{2} M ω_{M 0}^{2} x^{2} - ℏ Δ_{c 0} a^{†} a \\ + ℏ \frac{g_{0}^{2} a^{†} a \cos^{2} (k x)}{Δ_{a} (d) - λ_{0} y} + i ℏ (ε_{L} a^{†} - H . c .), \end{matrix}

where Δ_a(d) = ω_L − ω_a₀ − Δω_a(d) is the effective detuning of the pump laser from the atomic resonance in the static atom-plane geometry. In the derivation of Eq. (5) we used a first-order approximation of the frequency shift Δω_a(d + y) because y is very small compared to the distance d, i.e., Δω_a(d + y) = Δω_a(d) + λ₀y, with

{λ_{0} (d) = \frac{\partial Δ ω_{a} (d + y)}{\partial y} |}_{y = 0}

being the vacuum induced coupling between the single atom and the nearby plane, which is related to the level shifts of the atom in its ground and excited states [62, 63, 72, 73].

In order to calculate the vacuum induced coupling λ₀, we consider an effective isotropic two-level system at position r near a plane. The shift of the transition rate due to the vacuum effect is written as Δω_a(r) = δω_ae(r) − δω_ag(r), where δω_ag (r) and δω_ae (r) are, respectively, the frequency shifts of the two-level system in its ground state and excited state. Using the classical dyadic Green’s function G(r,r,iu) evaluated at the imaginary frequency ω = iu [62, 63], the two frequency shifts read

δ ω_{a g} (r) = \frac{3 c Γ_{0}}{ω_{a 0}^{2}} \int_{0}^{\infty} d u \frac{u^{2}}{ω_{a 0}^{2} + u^{2}} Tr {G (r, r, i u)},

and

δ ω_{a e} (r) = - \frac{δ ω_{a g} (r)}{3} - \frac{π c Γ_{0}}{ω_{a 0}} Tr Re {G (r, r, ω_{a 0})},

respectively. Further, we assume that the single atom sits near the center of the cantilever. In this case, the Green’s function of the cantilever can be approximated by that of an infinite plane. We mainly care a reflection component from a plane located at z = 0, which determines the value of the Green’s function describing the interaction of a two-level system with its own field reflected by this plane. In the vacuum regime of z > 0, the trace of this reflected component is

Tr G (z, z, ω) = \frac{{ic}^{2}}{4 π ω^{2}} \int_{0}^{\infty} d k_{| |} \frac{k_{| |}}{K_{0}} e^{2 i z K_{0}} [{(\frac{ω}{c})}^{2} r_{s} + (k_{| |}^{2} - K_{0}^{2}) r_{p}]

, where

K_{0} = \sqrt{{(\frac{ω}{c})}^{2} - k_{| |}^{2}}

and k_‖ is the parallel wavevector components; r_s and r_p are the Fresnel refection coefficients for the s and p-polarized waves. Using the specific expression of the shift of the transition rate, the coupling strength λ₀ between a two-level atom and a plane can be calculated as

\begin{array}{l} λ_{0} (d) = \frac{- 2 c^{3} Γ_{0}}{π ω_{a 0}^{2}} \int_{0}^{\infty} \frac{d u}{ω_{a 0}^{2} + u^{2}} \int_{0}^{\infty} d k_{| |} k_{| |} e^{2 i d K_{0}} \times [{(\frac{ω}{c})}^{2} r_{s} + (k_{| |}^{2} - K_{0}^{2}) r_{p}] \\ + Re \frac{c^{3} Γ_{0}}{2 ω_{a 0}^{3}} \int_{0}^{\infty} d k_{| |} k_{| |} e^{2 i d^{\sqrt{{(\frac{ω_{a 0}}{c})}^{2} - k_{| |}^{2}}}} \times [{(\frac{ω_{a 0}}{c})}^{2} r_{s} + (2 k_{| |}^{2} - {(\frac{ω_{a 0}}{c})}^{2}) r_{p}] . \end{array}

We can illustrate numerically the vacuum coupling strength λ₀ by assuming that the mechanical oscillator is a dielectric (SiN) cantilever coated with a thin copper [24]. The Fresnel reflection coefficients in the case of vacuum-metal interface are given by $r_{s} (i u, k_{| |}) = \frac{K_{3} - \sqrt{k_{| |}^{2} + ε (i u) u^{2} / c^{2}}}{K_{3} + \sqrt{k_{| |}^{2} + ε (i u) u^{2} / c^{2}}}$ and $r_{p} (i u, k_{| |}) = \frac{ε (i u) K_{3} - \sqrt{k_{| |}^{2} + ε (i u) u^{2} / c^{2}}}{ε (i u) K_{3} + \sqrt{k_{| |}^{2} + ε (i u) u^{2} / c^{2}}}$ respectively, where $K_{3} = \sqrt{k_{| |}^{2} + u^{2} / c^{2}}$ and ε is the dielectric constant describing the electromagnetic response of the metallic material. In general, the electromagnetic response of copper is described by a Drude model through the dielectric constant $ε (i u) = 1 + \frac{ω_{P}^{2}}{u^{2} + u γ}$ , where ω_P is the plasma frequency proportional to the density of conducting electrons in the metal, and γ is a damping parameter satisfying γ ≪ ω_P and therefore their contributions to the vacuum coupling strength and the frequency shift are marginal. In the calculation, we use the plasma wavelength (λ_P = 136nm) of copper corresponding to the plasma frequency ω_P = 2πc/λ_P and γ = 0.0033ω_P [74]. We choose the parameters of the atom, i.e., the transition wavelength λ_a₀ = 780 nm and the free-space spontaneous emission rate Γ₀ = 2π × 6.1 MHz, which corresponds to the decay of ⁸⁷Rb atom from its excited state |5P_3/2〉 to the ground state |5S_1/2〉 [75]. Figure 2 plots the vacuum coupling strength λ₀ as a function of the dimensionless distance ω_a₀d/c. From Fig. 2 we see that the vacuum induced coupling can be evaluated to be the order of 10¹² Hz/nm with a small distance.

Fig. 2 The vacuum coupling strength λ₀ as a function of the dimensionless distance ω_a₀d/c. The free-space spontaneous emission rate and the transition wavelength of the atom are Γ₀ = 2π × 6.1 MHz and λ_a₀ = 780 nm, respectively. The plasma frequency of copper λ_P = 136nm and γ = 0.0033ω_P.

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3. Dynamical equation and quantum fluctuations

Using the Hamiltonian (5) and taking into account the decay of the cavity mode and thermal noises affecting the atom and the micromechanical cantilever [76], the quantum dynamics of the system can be described by the following Heisenberg-Langevin equation:

\begin{array}{l} \dot{x} = p_{x} / M, \\ {\dot{p}}_{x} = - M ω_{M 0}^{2} x + \frac{ℏ k g_{0}^{2} a^{†} a \sin (2 k x)}{Δ_{a} - λ_{0} y} - γ_{M} p_{x} + ξ_{M}, \\ \dot{y} = p_{y} / m, \\ {\dot{p}}_{y} = - m ω_{m 0}^{2} y - \frac{ℏ λ_{0} g_{0}^{2} a^{†} a \cos^{2} (k x)}{{(Δ_{a} - λ_{0} y)}^{2}} - γ_{m} p_{y} + ξ_{m}, \\ \dot{a} = (i Δ_{c 0} + κ) a - \frac{i g_{0}^{2} \cos^{2} (k x) a}{Δ_{a} - λ_{0} y} + ε_{L} + \sqrt{2 κ} a_{i n}, \end{array}

where γ_M and γ_m are the damping rate of the atom and the cantilever, respectively. ξ_j(t)(j = m, M) is the Brownian noise term with zero mean value, which is characterized by the correlation function [77]

〈 ξ_{j} (t) ξ_{j} (t^{'}) 〉 = \frac{ℏ γ_{j} j}{2 π} \int d ω e^{- i ω (t - t^{'})} ω [1 + \coth (\frac{ℏ ω}{2 k_{B} T_{j}})]

, where k_B is the Boltzmann constant and T_j is the temperature surrounding the atom or the cantilever. a_in is the environmental input-noise corresponding to the operator a, which is fully characterized by the nonzero correlation function

〈 a_{i n} (t) a_{i n}^{†} (t^{'}) 〉 = δ (t - t^{'})

.

The steady-state expectation values of the atom, the cantilever and the cavity field can be obtained by setting the time derivatives to 0 in Eq. (9) as

\begin{array}{l} p_{x, y}^{s} = 0, a_{s} = \frac{ε_{L}}{κ - i Δ_{c}}, \\ x_{s} = \frac{ℏ k g_{0}^{2} {| a_{s} |}^{2} \sin (2 k x_{s})}{M ω_{M 0}^{2} Δ_{a}^{'}}, \\ y_{s} = \frac{ℏ λ_{0} g_{0}^{2} {| a_{s} |}^{2} \cos^{2} (k x_{s})}{m ω_{m 0}^{2} Δ_{a}^{'}^{2}}, \end{array}

where

Δ_{a}^{'} = Δ_{a} - λ_{0} y_{s}

and

Δ_{c} = Δ_{c 0} - g_{0}^{2} \cos^{2} (k x_{s}) / Δ_{a}^{'}

. In general, for a large coherent driving of the system, the photon number in the cavity a_s ≫ 1. Thus, we linearize the problem by considering the case with large photon number. For example, one can split the operators in Eq. (9) into their steady-state values and quantum fluctuations O = O_s + δO(O = x, p_x, y, p_y, a). By inserting the ansatz O = O_s + δO into Eq. (9) and neglecting all the terms higher than linear order in the fluctuations, i.e. δOδO, the quantum Langevin equations for the fluctuations are given by

\begin{array}{l} δ \dot{x} = ω_{M} δ p_{x}, \\ δ {\dot{p}}_{x} = - ω_{M} δ x + G_{12} δ y + G_{2 a} (δ a + δ a^{†}) - γ_{M} δ p_{x} + ξ_{M}, \\ δ \dot{y} = ω_{m} δ p_{y}, \\ δ {\dot{p}}_{y} = - ω_{m} δ y + G_{12} δ x + G_{1 a} (δ a + δ a^{†}) - γ_{m} δ p_{y} + ξ_{m}, \\ δ \dot{a} = (i Δ_{c} - κ) δ a - i G_{1 a} δ y + i G_{2 a} δ x + \sqrt{2 κ} a_{i n}, \end{array}

where

ω_{m} = \sqrt{ω_{m 0}^{2} + 2 ℏ λ_{0}^{2} g_{0}^{2} {| a_{s} |}^{2} \cos^{2} (k x_{s}) / (m Δ_{a}^{'}^{3})}

and

ω_{M} = \sqrt{ω_{M 0}^{2} - 2 ℏ k^{2} g_{0}^{2} {| a_{s} |}^{2} \cos (2 k x_{s}) / (M Δ_{a}^{'})}

are the effective frequencies associated with c.m. oscillations of the cantilever and the atom, respectively.

G_{12} = λ K g_{0}^{2} {| a_{s} |}^{2} \sin (2 k x_{s}) / Δ_{a}^{'}^{2}

is the effective coupling between the mechanical motion of the atom and the cantilever;

G_{1 a} = λ g_{0}^{2} a_{s} \cos^{2} (k x_{s}) / Δ_{a}^{'}^{2}

is the effective coupling between the cantilever and the cavity field;

G_{2 a} = K g_{0}^{2} \sin (2 k x_{s}) a_{s} / Δ_{a}^{'}

the effective coupling between the atomical motion and the cavity field. It is found that the coupling G₁₂ is independent of the mass ratio M/m and therefore the atom will affect significantly the dynamics of the cantilever.

λ = λ_{0} \sqrt{\frac{ℏ}{m ω_{m}}}

and

K = k \sqrt{\frac{ℏ}{M ω_{M}}}

. In deriving Eq. (11), we redefined the dimensionless position and momentum fluctuations δx (δy) and δp_x (δp_y) as

\sqrt{\frac{M ω_{M}}{ℏ}} δ x \to δ x (\sqrt{\frac{m ω_{m}}{ℏ}} δ y \to δ y)

and

\sqrt{\frac{1}{ℏ M ω_{M}}} δ p_{x} \to δ p_{x} (\sqrt{\frac{1}{ℏ m ω_{m}}} δ p_{y} \to δ p_{y})

1. In addition, the phase of the cavity field is chosen so that a_s is real. Correspondingly, the Brownian noises with zero mean ξ_j are also renormalized, and have the correlation function

〈 ξ_{j} (t) ξ_{j} (t^{'}) 〉 = \frac{γ_{j}}{2 π ω_{j}} \int d ω e^{- i ω (t - t^{'})} ω [1 + \coth (\frac{ℏ ω}{2 k_{B} T_{j}})]

.

The coupling strengths G₁₂ and G₁_a can be demonstrated through the normal mode splitting (NMS) of the mechanical motion. To this end, we transform to the quadratures: $δ X = (δ a^{†} + δ a) / \sqrt{2}$ , $δ Y = i (δ a^{†} - δ a) / \sqrt{2}$ , $δ X_{i n} = (δ a_{i n}^{†} + δ a_{i n}) / \sqrt{2}$ and $δ Y_{i n} = i (δ a_{i n}^{†} - δ a_{i n}) / \sqrt{2}$ . Further, introducing the column vector of fluctuation operators f(t) with f^T(t) = (δy, δp_y, δx, δp_x, δX, δY) and the corresponding column vector of noise sources n(t) with $n^{T} (t) = (0, ξ_{m}, 0, ξ_{M}, \sqrt{2 κ} δ X_{i n}, \sqrt{2 κ} δ Y_{i n})$ , Eq. (11) can be written in the following compact form:

\dot{f} (t) = J f (t) + n (t),

where the matrix J is given by

J = (\begin{matrix} 0 & ω_{m} & 0 & 0 & 0 & 0 \\ - ω_{m} & - γ_{m} & G_{12} & 0 & - \sqrt{2} G_{1 a} & 0 \\ 0 & 0 & 0 & ω_{M} & 0 & 0 \\ G_{12} & 0 & - ω_{M} & - γ_{M} & \sqrt{2} G_{2 a} & 0 \\ 0 & 0 & 0 & 0 & - κ & Δ_{c} \\ - \sqrt{2} G_{1 a} & 0 & \sqrt{2} G_{2 a} & 0 & Δ_{c} & - κ \end{matrix}) .

The stability conditions for the system demands that the real parts of all the eigenvalues of the matrix J are negative. We can use the Routh-Hurwitz criteria [78] to derive these stability conditions. Here, the explicit inequalities are quite cumbersome and therefore we always resort to numerics. Hereafter, we restrict the selected external parameters to the stable regime.

The expression for the position fluctuation of the oscillating cantilever in Fourier space is given by

\begin{array}{l} δ y (ω) = \frac{ω_{m}}{d (ω)} [A_{1} (ω) ξ_{m} (ω) + A_{2} (ω) ξ_{M} (ω)] \\ + \frac{\sqrt{2 κ} ω_{m}}{d (ω)} [A_{3} (ω) δ X_{i n} (ω) + A_{4} (ω) δ Y_{i n} (ω)], \end{array}

where

\begin{array}{l} A_{1} (ω) = {(κ + i ω)}^{2} (- ω^{2} + i γ_{M} ω + ω_{M}^{2}) - Δ_{c} (ω^{2} Δ_{c} - 2 ω_{M} G_{2 a}^{2} - Δ_{c} ω_{M}^{2} - i ω γ_{M} Δ_{c}), \\ A_{2} (ω) = ω_{M} [G_{12} {(κ + i ω)}^{2} + 2 G_{1 a} G_{2 a} Δ_{c} + G_{12} Δ_{c}^{2}], \\ A_{3} (ω) = \sqrt{2} (κ + i ω) [G_{1 a} (ω^{2} - ω_{M}^{2} - i ω γ_{M}) + G_{12} G_{2 a} ω_{M}], \\ A_{4} (ω) = \sqrt{2} Δ_{c} [G_{1 a} (ω_{M}^{2} - ω^{2} + i ω γ_{M}) - G_{12} G_{2 a} ω_{M}], \\ d (ω) = {(κ + i ω)}^{2} B_{0} - Δ_{c} B_{1}, \end{array}

with

B_{0} = ω^{4} - ω^{2} ω_{m}^{2} - i γ_{M} (ω^{3} - ω ω_{m}^{2}) - G_{12}^{2} ω_{m} ω^{2} - ω^{2} ω_{M}^{2} - ω_{m}^{2} ω_{M}^{2} - i ω γ_{m} (ω^{2} - i ω γ_{M} - ω_{M}^{2})

and

B_{1} = 2 G_{2 a}^{2} ω (ω - i γ_{m}) ω_{M} - 2 G_{2 a}^{2} ω_{m}^{2} ω_{M} + \sqrt{2} G_{1 a} ω_{m} (\sqrt{2} G_{1 a} ω^{2} - \sqrt{2} i G_{1 a} ω γ_{M} + 2 \sqrt{2} G_{2 a} G_{12} ω_{M} - \sqrt{2} G_{1 a} ω_{M}^{2}) - Δ_{c} B_{0}

. In Eq. (13) for δy(ω), the terms proportional to ξ_m and ξ_M arise from thermal noises, and the terms proportional to δX_in and δY_in originate from radiation pressure. The spectrum of fluctuation in the position of the oscillating cantilever is obtained from

S_{y} (ω) = \int \frac{d Ω}{2 π} e^{- i (ω + Ω) t} \frac{〈 δ y (ω) δ (Ω) + δ y (Ω) δ (ω) 〉}{2},

together with the nonzero correlation functions of the noise sources in the frequency domain:

\begin{array}{l} 〈 ξ_{j} (ω) ξ_{j} (Ω) 〉 = 2 π \frac{γ_{j}}{ω_{j}} ω [1 + \coth (\frac{ℏ ω}{2 k_{B} T_{j}})] δ (ω + Ω), \\ 〈 δ X_{i n} (ω) δ X_{i n} (Ω) 〉 = 〈 δ Y_{i n} (ω) δ Y_{i n} (Ω) 〉 = π δ (ω + Ω) . \end{array}

The spectrum of fluctuation in the cantilever position in Fourier space is finally obtained as

\begin{array}{l} S_{y} (ω) = {| χ (ω) |}^{2} [\frac{γ_{m} ω}{ω_{m}} β_{m} + \frac{γ_{M} ω}{ω_{M}} {| \frac{A_{2} (ω)}{A_{1} (ω)} |}^{2} β_{M}] \\ + {| χ (ω) |}^{2} κ [{| \frac{A_{3} (ω)}{A_{1} (ω)} |}^{2} + {| \frac{A_{4} (ω)}{A_{1} (ω)} |}^{2}], \end{array}

where

β_{j} = 1 + \coth \frac{ℏ ω}{2 k_{B} T_{j}}

and the effective mechanical coefficient for cantilever is written as

χ (ω) = ω_{m} / [{(ω_{m}^{e f f})}^{2} - ω^{2} + i ω γ_{m}^{e f f}]

;

ω_{m}^{e f f} = \sqrt{Re [d (ω) / A_{1} (ω) + ω^{2}]}

and

γ_{m}^{e f f} = Im [d (ω) / A_{1} (ω)] / ω

are the effective resonance frequency and effective damping rate of the cantilever, respectively. This spectrum S_y(ω) is characterized by the effective mechanical coefficient χ(ω) and any information about the modified motion of the cantilever can be obtained from the study of S_y(ω). It is noted that the modification of mechanical frequency due to radiation pressure in Eq. (17) is the so-called optical spring effect [21], which is negligibly small for a mechanical oscillator with high frequency (~ MHz), as shown in Fig. 3(a), where the normalized effective oscillation frequency

ω_{m}^{e f f} / ω_{m}

is plotted as a function of the normalized frequency ω/ω_m. Here we select the accessible parameters in optomechanical systems. For example, the wavelength of the driving field λ_L ≃ 780 nm, the total cavity length L = 1 cm and the cavity decay rate κ = 2π × 10⁵ Hz. The mass of a single ⁸⁷Rb atom M ≈ 1.42 × 10⁻²⁵ Kg, the effective oscillation frequency of atom ω_M = 2π ×1.9 MHz and the decay rate γ_M = 2π ×1 Hz due to the collisions with the background gas [76]. The single-photon Rabi frequency g₀ = 2π × 10.9 MHz and the effective pump-atom detuning

Δ_{a}^{^{'}} = 2 π \times 30 GHz

[52]. The effective mass of the cantilever m = 0.05 × 10⁻¹²Kg, the effective frequency of the cantilever ω_m = ω_M and the damping rate γ_M = 2π × 300 Hz [20, 79]. The temperatures surrounding the atom and the cantilever are same, i.e., T_m = T_M = 0.1K. The steady-stat position of atom satisfies sin(2kx_s) = 0.3 and the vacuum coupling between the atom and the cantilever λ = 2π × 9.5 MHz. We also consider that the cavity is driven on its red sideband, i.e., Δ_c = −ω_m. The driving strength ε_L and the steady-state photon number can be calculated in terms of the steady-state position x_s. In contrast to the optical spring effect, the increase of the effective mechanical damping rate plays an important role in the cooling of the mechanical motion [21]. Figure 3(b) depicts the normalized effective damping rate

γ_{m}^{e f f} / ω_{m}

as a function of the normalized frequency ω/ω_m. It is clearly seen that the effective mechanical damping rate increases significantly at ω ≈ ω_m and therefore helps drive the cantilever close to the quantum ground state.

Fig. 3 The normalized effective oscillation frequency $ω_{m}^{e f f} / ω_{m}$ (a) and the normalized effective damping rate $γ_{m}^{e f f} / ω_{m}$ (b) as functions of the normalized frequency ω/ω_m. We select atomic parameters, i.e. the free-space spontaneous emission rate Γ₀ = 2π × 6.1 MHz, the transition wavelength λ_a₀ = 780 nm, the atomic mass M ≈ 1.42 × 10⁻²⁵ Kg, the effective mass of the cantilever m = 0.05 × 10⁻¹² Kg, the effective oscillation frequency of atom ω_M = 2π × 1.9 MHz, the decay rate γ_M = 2π × 1 Hz, the Rabi frequency g₀ = 2π × 10.9 MHz, the effective pump-atom detuning $Δ_{a}^{'} = 2 π \times 30 GHz$ , the steady-state position sin(2kx_s) = 0.3 and λ = 2π × 9.5 MHz. The other parameters are λ_L ≃ 780nm, L = 1 cm, κ = 2π × 10⁵ Hz, ω_m = ω_M, γ_m = 2π × 300 Hz, T_m=T_M = 0.1K and Δ_c = −ω_m.

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In Fig. 4, we show the plot of S_y(ω) as a function of normalized frequency ω/ω_m with different effective coupling λ’s at fixed position sin(2kx_s) = 0.3 [Fig. 4(a)] and different x_s’s at fixed effective coupling λ = 2π × 3.8 MHz [Fig. 4(b)]. We used the detuning Δ_c = 0. Other parameter values are the same as those in Fig. 3. In the absence of the vacuum induced coupling λ = 0 and therefore G₁₂ = 0 and G₁_a = 0, the effective coupling G₂_a between the atomic motion and the cavity field does not lead to NMS. This situation is changed when the vacuum interaction between the atom and the nearby cantilever is included. From Fig. 4(a) and 4(b), we observe that NMS appears with sufficiently large vacuum coupling λ. This is because that the vacuum effect in the hybrid system not only induces a directed effective coupling G₁₂ between the cantilever and the atomic motion, but also a non-directed effective coupling G₁_a between the cantilever and the cavity field, which both contribute to the splitting of spectrum lines. Therefore, the NMS in the system is associated to the mixing between the mechanical mode and the atomic motion and the fluctuation of the cavity field around the steady state. Furthermore, with increasing vacuum coupling strength, the width of the splitting peaks increases. In addition, the position of the atom x_s also has an important impact on NMS. From Fig. 4(b), we observe that the width between two peaks increases with decreasing x_s. The gradual increase in the width of the splitting peaks results from the increase in the amplitudes of the effective couplings G₁₂ and G₁_a, which leads to an intriguing mixing of the mechanical mode and the atomical motion and the cavity field. As we know, NMS indicates strong coupling. In particular, the displacement spectrum of the cantilever and the NMS can be obtained by homodyne detection scheme [5, 80]. Thus, the present characteristics may be used for demonstrating the strong coupling between the mechanical motions of a single atom and a massive cantilever despite the fact that the mass of atom is much less than the effective mass of the cantilever, i.e., $\sqrt{M / m} ≪ 1$ .

Fig. 4 The displacement spectrum of the cantilever as a function of normalized frequency ω/ω_m with different λ’s for sin(2kx_s) = 0.3 (a) and different x_s’s for λ = 2π × 3.8 MHz (b). Δ_c = 0 and other parameter values are the same as those in Fig. 3.

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4. OMIT in the output field

In order to study the optical response of the hybrid system, we consider that a weak probe field with frequency ω_p and amplitude ε_p is sent into the cavity (see Fig. 1). Thus, the Hamiltonian of the whole system should include the terms related to the interaction between the driven field and the probe field, i.e., $i ℏ (ε_{p} a^{†} e^{i ω_{p} t} - H . c .)$ . Correspondingly, the dynamics of the cavity field in the quantum Langevin equations [Eqs. (9) and (11)] should include the role of the probe field, i.e., $δ \dot{a} (i Δ_{c} - κ) δ a - i G_{1 a} δ y + i G_{2 a} δ x + ε_{p} e^{- i δ t} + \sqrt{2 κ} a_{i n}$ , where δ = ω_p − ω_L is the detuning of the probe laser from the cavity. Then, using the dynamics of the quantum fluctuations around the steady-state, we can study the optical properties of the output field in the atom-cantilever optomechanical system, i.e., optomechanically induced transparency, which also demonstrates the coupling between the mechanical oscillator and the atom. Here we use the ansatz δO = O₊e⁻^iδt + O₋e^iδ to solve Eq. (11). After substituting this ansatz into Eq. (11), the following solution of interest for the response of the optomechanical system is attained,

a_{+} = ε_{p} {[β_{3} - i G_{1 a} - i G_{2 a} - \frac{{(G_{1 a} α_{2} + G_{2 a} α_{1})}^{2}}{β_{4} + i G_{1 a} α_{2} + i G_{2 a} α_{1}}]}^{- 1},

where

α_{1} = \frac{G_{2 a} β_{1} - G_{12} G_{1 a}}{β_{1} β_{2} - G_{12}^{2}}

, α₂ = (G₁_a− G₁₂α₁)/β1, β₁ = ω_m²−δ²−iδγ_m, β₂ = ωM² −δ²−iδγ_M, β₃ = κ − i(∆_c + δ), β₄ = κ + i(∆_c−δ).

Here we investigate mainly the component of the output field oscillating at the probe frequency ω_p, and therefore the expression of a₋ is not necessary to describe the four-wave mixing with frequency ω_p−2ω_f for the driving field and the weak probe field. Using the standard input-output theory [50] $〈 a_{o u t} (t) 〉 + \frac{ε_{L} + ε_{p} e^{- i δ t}}{\sqrt{2 κ}} = \sqrt{2 κ} 〈 a 〉$ and the ansatz δa = a₊e⁻^iδt + a_eⁱ, we can express the mean value of the output field as

\sqrt{2 κ} 〈 a_{o u t} (t) 〉 = 2 κ α_{s} - ε_{L} + (2 κ a_{+} - ε_{p}) e^{- i δ t} + 2 κ (a_{-}) e^{i δ t} .

It is stressed that the term related to a₊ in the above expression corresponds directly to the output field at the probe frequency ω_p via the detung δ. In order to examine the total output field at the frequency ω_p, we define an amplitude of the rescaled output field:

ε_{o u t} = 2 κ a_{+} / ε_{p},

where the constant term has been omitted. The real and imaginary parts of the amplitude of this term, Re(ε_out) and Im(ε_out), respectively, account for in-phase and out-phase quadratures of the output field spectrum and describe the absorption and dispersion by the whole system of the weak probe field, which can be measured by homodyne detections [80].

In the present hybrid optomechanical system, the mass of the atom is much less than the effective mass of the cantilever. However, the mechanical motions of a single atom and a massive cantilever oscillator can be coupled effectively by the effective coupling coefficient G₁₂, which is independent of their mass ratio M/m. Thus, the motion of a single atom may affect significantly the optical response of the system. In this section, we investigate in detail the generation of dips in the absorption spectrum induced by the vacuum-assisted atom-cantilever coupling G₁₂ and optomechanical coupling G₁_a.

We now numerically evaluate the phase quadratures Re(ε_out) and Im(ε_out)through the corresponding output field a₊. Further, we analyze in detail the OMIT phenomena in the system, induced by the mechanical motion of the atom and the cantilever. In Fig. 5, we show the absorption Re(ε_out) and dispersion Im(ε_out) in the output field, as a function of δ/ω_m with different effective coupling strength λ. In the absence of the vacuum action, i.e., λ = 0 in Fig. 5(a), we found that the system is transparent and the single-mode OMIT behavior appears when δ = ω_m. In this case the effective coupling coefficients G₁₂ = 0 and G₁_a = 0 due to the disappeared λ and the generation of the single absorption dip is induced directly by the effective coupling G₂_a between the atomic motion and the cavity field. Physically, This can be understood since the probe beam interferes destructively with the anti-Stokes field generated by the atom [6, 32], which plays a role of mechanical oscillator. In particular, we may find from Fig. 5(b)–5(d) that the double absorption dips for the output light of the probe beam appear when the vacuum induced couplings, i.e., G₁₂ and G₁_a, are included. Indeed, in the presence of the vacuum action between the atom and the nearby plane, the oscillating cantilever is coupled to the atom directly via the effective coupling G₁₂ and to the cavity field indirectly via the effective coupling G₁_a. These couplings leads to more interference channels and therefore break down the symmetry in the single OMIT interference [41]. Furthermore, the two absorption dips become more and more separated with increasing the effective coupling λ, which determines the amplitude of the vacuum induced couplings, G₁₂ and G₁_a. We see from Fig. 5 that Im(ε_out) exhibits abnormal dispersion, induced by the atom-field and the optomechanical couplings. In Fig. 6, we show the interval δ₁₂/ω_m between the two absorption dips as a function of normalized coupling λ/ω_m, where δ₁₂ = δ₁ − δ₂ and δ₁ (δ₂) is the detuning corresponding to left (right) absorption dips in Fig. 5 and can be evaluated by $d R e (ε_{o u t}) / {d δ |}_{δ = δ_{1, 2}} = 0$ . It is clearly seen from Fig. 6 that δ₁₂ increases linearly with increasing the coupling λ, which can be used to detect the atom-plane vacuum force with the hybrid optomechanical system.

Fig. 5 The absorption Re(ε_out) (solid line) and dispersion Im ε_out (dash line) in the output field are plotted as a function of δ/ω_m with different λ’s. Other parameter values are the same as those in Fig. 3.

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Fig. 6 The interval δ₁₂/ω_m between the two absorption dips is plotted as a function of the effective coupling strength λ. Other parameter values are the same as those in Fig. 3.

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In the presence of the vacuum force, the coupling coefficients G₁₂ and G₁_a depend strongly on the steady-state position of atom x_s. Fig. 7 presents the variation of the absorption Re(ε_out) and dispersion Im(ε_out) of the output field with respect to δ/ω_m with different x_s’s at given λ = 2π × 3.8 MHz. From Fig. 7, we can observe that the positions of the two absorption dips separate gradually with decreasing x_s. Correspondingly, the left and right transparency windows become more and more separated. This leads to that the interval δ₁₂ between the two absorption dips increases monotonously with decreasing x_s due to the increased effective coupling coefficients G₁₂ and G₁. In Fig. 8, we show the number of the excited atom ${η = g_{0}^{2} \cos^{2} (k x_{s}) {| a_{s} |}^{2} / (Δ_{a}^{^{'}})}^{2}$ as a function of sin(2kx_s). From Fig. 8, we see clearly that the smaller the position x_s, the larger the excited number of atom. This means that in the regime of a small value x_s, the low-excitation limit of the atom will be broken. Thus, in order to keep away from the regime and distinguish the double absorption dips as much as possible, we should select optimally the position of the trapped atom.

Fig. 7 The absorption Re(ε_out) (solid line) and dispersion Im(ε_out) (dash line) of the output field are plotted as a function of δ/ω_m with different x_s’s at given λ = 2π × 3.8 MHz. Other parameter values are the same as those in Fig. 3.

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Fig. 8 The number of the excited atom η are plotted as a function of sin (2kx_s). Other parameter values are the same as those in Fig. 7.

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The general case is considered with different frequencies for the trapped atom and the oscillating cantilever, i.e., ω_m ≠ ω_M. In Fig. 9, we show the absorption Re(ε_out) in the output field, as a function of δ/ω_m with identical [Fig. 9(b)] and different [Fig. 9(a)) and 9(c)] oscillation frequencies. The effective coupling strength λ = 2π × 3.8 MHz and the steady-state position of atom sin(2kx_s) = 0.3. In Fig. 9, we observe that with respect to the case of identical frequency ω_m = ω_M, the additional absorption peak induced by the vacuum effect moves rightward (leftward) in the case of ω_M < ω_m (ω_M > ω_m). In particular, in the case of different frequencies, the absorption spectrum profile becomes more asymmetric and the dips in absorption become shallower when the two frequencies are further apart. The increase of the interval δ₁₂ in the different frequencies enhances its sensitivity to the vacuum interaction.

Fig. 9 The absorption Re(ε_out) of the output field is plotted as a function of δ/ω_m with identical [Fig. 9(b)] and different [Fig. 9(a) and 9(c)] oscillation frequencies. The effective coupling strength λ = 2π×3.8 MHz and the steady-state position of the atom sin(2kx_s)= 0.3. Other parameter values are the same as those in Fig. 7.

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Discussion

In this work, we have designed a hybrid optomechanical system for the realization of vacuum-assisted coupling among a macromechanical cantilever, a single atom trapped inside a cavity and a single mode of a Fabry-Pérot cavity. In the presence of atom-vacuum interaction, the effective mechanical coupling between a massive cantilever and a single atom as well as an effective optomechanical coupling between the massive cantilever and the cavity field are studied with the quantum Langevin equation and its linearized dynamics around semiclassical steady states. Furthermore, we have analyzed the occurrence of NMS in the displacement spectrum of the oscillating cantilever, which is associated to the mixing between the mechanical mode and the atomic motion and the fluctuation of the cavity field. We discuss in detail numerically influences of the vacuum induced coupling and the position of the atom on the NMS, which indicates the effective strength of coupling between the mechanical motion of a single atom and a massive cantilever. In addition, we also focused our attention to the optical-response properties of the hybrid system and showed that the double-OMIT behavior in the absorption spectrum of the probe field can be observed when the effective optomechanical coupling and mechanical coupling are large enough. In particular, the separation between the two absorption dips is related directly to the vacuum coupling between the atom and the cantilever as well as the position of the trapped atom. These results have a potential application to the realization of the effective coupling between the mechanical motions of a mechanical oscillator and a single atom.

Funding

National Natural Science Foundation of China (NSFC) (11565014, 11775190, 11375093); Natural Science Foundation of Jiangxi Province (20161BAB211023 and 20171BAB201015).

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Coupling mechanical motion of a single atom to a micromechanical cantilever

Abstract

1. Introduction

2. Optomechanical model

3. Dynamical equation and quantum fluctuations

4. OMIT in the output field

Discussion

Funding

References and links

Cited By

Figures (9)

Equations (23)

Optics Express