Cooling mechanical motion via vacuum effect of an ensemble of quantum emitters

Wenjie Nie; Aixi Chen; Yueheng Lan

doi:10.1364/OE.23.030970

1. Introduction

Recently, Cavity optomechanical system with movable mechanical oscillator has received extensive attention, in which a significant interaction between the light and the mechanical system via radiation pressure can be generated [1–3]. The optomechancial system can be used for demonstrating the quantum-mechanical effect of macroscopic resonator, such as the generation of the continuous-variable optomechanical entanglement [4–11], quantum squeezing and optical cooling of mechanical mode [12–20], and non-classical state preparation in mechanical resonators [21–24]. The classical dynamics in an optomechanical system [25,26] and the relation between the classical and quantum characteristics of a mechanical motion are discussed [27]. Further, quantum-classical transition and the quantum information processing based on the optomechanical system have been investigated in detail [28,29], where the ground-state cooling of mechanical resonator plays a role of the precondition.

So far, many theoretical and experimental efforts are devoted to achieve the ground-state cooling of mechanical resonator using various cooling schemes [30–37]. In general, the ground-state cooling of a mechanical resonator demands that the variances for the quantum fluctuation of the mechanical resonator’s position and momentum should both quickly tend to 1/2. In order to reach this goal, the effective mechanical damping rate under the perturbation of the radiation pressure should increase significantly [36], which can be achieved by enhancing the cooling anti-Stokes process and suppressing the heating Stokes’ in the resolved sideband limit, that is, the frequency of the mechanical resonator is greater than the decay rate of the optical field [30,31]. However, it is noted that the resolved sideband condition is a bit tight except for a few special optomechanical systems [38,39], which is hard to fulfill in experiment. In this regard, one way to circumvent the limitation is to use electromagnetically-induced-transparency (EIT) cooling scheme for a mechanical resonator coupled to a superconducting flux qubit, where the quantum interference can cancel unwanted heating effect [40]. The EIT ground-state cooling scheme in another hybrid optomechanical systems with a three-level atomic medium or a single electronic spin qubit are discussed in detail [41,42]. The ground-state cooling in the regime of bad cavity is achieved by adding a good cavity with a small cavity decay rate, which can vary the effective optical response of the whole cavity system and thus satisfy the effective resolved sideband condition for the ground-state cooling of a mechanical resonator [43–45].

In this work, we investigate the ground-state cooling of a mechanical resonator in a hybrid optomechanical system composed of a standard optomechanical cavity and an ensemble of two-level quantum emitters trapped inside the cavity. We consider that the ensemble is positioned near the cavity mirror and therefore the motion of the cavity mirror changes the transition rate of each quantum emitter via vacuum fluctuation, which induces a direct coupling between the internal states of the quantum emitters and the mechanical mode [46] which changes significantly the dynamics of the whole system and its steady-state characteristics compared with those in the absence of the vacuum force. We investigated the influences of the vacuum coupling strength on the effective frequency and the effective damping rate of the movable mirror by studying the dynamics of the quantum fluctuation of the system. Further, we focus on the effect of the vacuum fluctuation on the ground-state cooling of the mechanical resonator, in the presence of which we found that the ground-state cooling of the mechanical motion can be achieved in the bad cavity limit. Further, we also analyze in detail the effect of the cavity and atomical detuning, the coupling strength between the optical field and the quantum emitter and the decay rate of the quantum emitter on the motional ground-state cooling.

2. Model and Hamiltonian

The hybrid optomechanical system investigated here is shown in Fig. 1, in which a thin ensemble of N quantum emitters is positioned inside a standard optomechanical cavity consisting of one fixed mirror and one movable mirror. Further, the ensemble of the quantum emitters (position q₀) is modeled by a two-level system with the excited-state |e〉 and ground-state |g〉, which is driven by an external field with frequency ω_f. Thus, the quantum emitters can mediate a linear driving of the single-mode field inside the optical cavity. Correspondingly, the movable mirror interacts with the driven field via the optical radiation pressure. In general, the movable mirror in an optomechanical system is treated as a quantum-mechanical harmonic oscillator with mass m, frequency ω_m, and damping rate γ_m. In order to realize the direct coupling between the two-level ensemble and the mechanical mirror, we assume that the ensemble is positioned near the right movable mirror, i.e. the distance between the emitter and movable mirror is of the order of nanometers. In this case, the vacuum energy fluctuation around the quantum emitters leads to a modification of its electronic state. Consequently, the transition rate of each emitter depends strongly on the position x of the movable mirror in the direction of the cavity axis, i.e. ω_a = ω_a(d + x), where d (d ≫ x) is the distance between the quantum emitter and the mirror in a static emitter-plane geometry [47,48]. The coupling strength between the quantum emitter and the cavity mirror can be calculated as $λ_{0} = {\frac{\partial ω_{a} (d + x)}{\partial x} |}_{x = 0}$ . The total Hamiltonian of the hybrid optomechanical system can be written as [49–51]

\begin{array}{l} H & = & \bar{h} ω_{c} a^{†} a + \frac{p^{2}}{2 m} + \frac{1}{2} m ω_{m}^{2} x^{2} + \frac{1}{2} \bar{h} ω_{a} (d + x) \sum_{i = 1}^{N} σ_{z}^{(i)} + \bar{h} g (a \sum_{i = 1}^{N} σ_{+}^{(i)} + a^{†} \sum_{i = 1}^{N} σ_{-}^{(i)}) \\ - \bar{h} χ_{c} a^{†} a x + \bar{h} (Ω \sum_{i = 1}^{N} σ_{+}^{(i)} e^{- i ω_{f} t} + Ω^{*} \sum_{i = 1}^{N} σ_{-}^{(i)} e^{i ω_{f} t}) \end{array}

where a is the photon annihilation operator of the cavity mode with frequency ω_c and decay rate κ, which satisfies the commutation relation [a, a^†] = 1. x and p are the position and momentum operator of the movable mirror with commutation relation [x, p] = ih̄. The two-level system can be described by the pseudospin-1/2 operators σ_z, σ₊ and σ₋, which satisfy the commutation relations [σ₊, σ₋] = σ_z and [σ_z, σ_±] = ±2σ_±. g (χ_c = ω_c/L) is the coupling strength between the cavity field and the quantum emitter (movable mirror). Ω is the driving strength of the ensemble and L is the cavity length. In particular, we consider that the number of the quantum emitters is very large, i.e. N ≫ 1, but the excitation probability of a single quantum emitter is very small. In this low excitation limit, the collective operators can be defined as

\sum_{i = 1}^{N} σ_{+}^{(i)} = \sqrt{N} S_{+}

and

\sum_{i = 1}^{N} σ_{-}^{(i)} = \sqrt{N} S_{-}

, where the operators S₊ and S₋ satisfy the fundamental commutation relation [S₋,S₊] ≃ 1 [41,52]. In addition, the condition of low excitation limit should not be appreciably altered by the driving field and the interaction with the cavity, so that

〈 \sum_{i = 1}^{N} σ_{z}^{(i)} 〉 = 〈 2 S_{+} S_{-} - N 〉 ≃ - N

.

Fig. 1 A hybrid optomechanical system model. An ensemble of quantum emitters is positioned inside a standard optomechanical cavity, which is modeled by a two-level system with the excited-state |e〉 and ground-state |g〉 and driven by an external field with frequency ω_f. The motion of the mirror nearby the quantum emitter changes its transition rate via vacuum fluctuation and therefore leads to the coupling between the quantum emitter and the mechanical mode.

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In order to derive the coupling strength between the quantum emitter and the movable mirror, the specific expression of the transition rate of the quantum emitter is needed. We first decompose the transition rate of the emitter ω_a(d + x) into two parts, the bare frequency ω₀ plus the total frequency shift Δω_a(d + x). Thus, in the linear approximation, we have ω_a(d + x) = ω₀ + Δω_a(d) + λ₀x. In the following, we derive the coupling strength λ₀ in terms of the frequency shift. The frequency shift of an effective isotropic two-level system at position r near a plane can be expressed as [46,50,51]

Δ ω_{a} (r) = δ ω_{ae} (r) - δ ω_{ag} (r),

where δω_ag(r) and δω_ae(r) are, respectively, the frequency shifts of the emitter at positon r in its ground state and excited state. The two frequency shifts can be calculated straightforwardly in terms of the classical dyadic Green’s function G(r, r, iu) evaluated at the imaginary frequency ω = iu [46,50,51], i.e.

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

and

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

where c is the speed of light and Γ₀ is the free-space spontaneous emission rate of the two-level system. Further, we assume that the quantum emitters sit near the center of the movable mirror. In this case, the Green’s function of the movable mirror can be approximated by that of an infinite plane. Moreover, a reflection component from a plane located at z = 0 determines the value of the Green’s function, which describes the interaction of a two-level system with its own field reflected by this surface. In the vacuum side with z > 0, the trace of this reflected component is

Tr G (z, z, ω) = \frac{i c^{2}}{4 π ω^{2}} \int_{0}^{\infty} d k_{‖} \frac{k_{‖}}{K_{0}} e^{2 iz 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 s and p-polarized waves. Using Eqs. (2)–(4), the coupling strength λ₀ can be calculated as

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

It is noted that the influence of the resonant cavity on the Casimir-Polder interaction of a two-level system nearby a cavity wall has been neglected when the cavity length L is large enough. This is because that the change of the density of states of the electromagnetic field inside the cavity does not significantly alter the Casimir-Polder force [53]. Furthermore, the dynamical Casimir-Polder effect due to the dissipation of the electromagnetic fields inside the cavity can be neglected safely with a large optical cavity [54,55].

In a rotating frame at the laser frequency ω_f, the total Hamiltonian of the hybrid optomechanical system becomes

\begin{array}{l} H & = & \bar{h} Δ_{c 0} a^{†} a + \frac{1}{2} \bar{h} ω_{m} (p^{2} + q^{2}) + \bar{h} (Δ_{a 0} + λ q) S_{+} S_{-} \\ + \bar{h} g \sqrt{N} (a S_{+} + a^{†} S_{-}) - \bar{h} χ a^{†} a q + \bar{h} \sqrt{N} (Ω S_{+} + Ω^{*} S_{-}), \end{array}

where Δ_c0 = ω_c − ω_f is the detuning between the cavity mode and the driving field of the emitter, and Δ_a0 = ω₀ − ω_f + Δω_a(d) is the detuning between the quantum emitter and its driving field. The position and momentum operators x and p have been nondimensionalized as

\sqrt{\frac{m ω_{m}}{\bar{h}}} x \to q

and

\sqrt{\frac{1}{\bar{h} m ω_{m}}} p \to p

in Eq. (6), and therefore the coupling strengths become

λ = λ_{0} \sqrt{\bar{h} / m ω_{m}}

and

χ = χ_{c} \sqrt{\bar{h} / m ω_{m}}

. During the derivation of Eq. (6) we neglected a constant term of N since it does not influences the system dynamics.

3. Dynamical evolution of the system

Here we focus mainly on the quantum dynamics of the nonlinear optomechanical system and investigate further the cooling characteristics of the mechanical oscillator in the presence of the vacuum effect. In particular, the ground-state cooling of the mechanical motion is expected to be possible even in the unresolved sideband regime i.e. k > ω_m by adjusting the decay rate of the quantum emitter. Using the Hamiltonian (6) and taking into account the effects of the noises and damping, the full dynamics of the system can be described by the following Heisenberg-Langevin equation of motion:

\begin{array}{l} \dot{q} & = & ω_{m} p, \\ \dot{p} & = & - ω_{m} q - λ N / 2 + χ a^{†} a - γ_{m} p + ξ (t), \\ \dot{a} & = & - (κ + i Δ_{c 0}) a + i χ aq - i {GS}_{-} + \sqrt{2 κ} a_{in} (t), \\ {\dot{S}}_{-} & = & - (Γ_{a} + i Δ_{a 0}) S_{-} - i Ga - i λ q S_{-} - i η + \sqrt{2 Γ_{a}} Γ_{-} (t), \end{array}

where

G = g \sqrt{N}

and

η = Ω \sqrt{N}

are the collective coupling strength of the two-level ensemble. The term −γ_mp depicts a damping force acting on the movable mirror with damping rate γ_m, which results mainly from the contact with the thermal bath through the external support. Further, we consider a high-frequency oscillator operating in the MHz range and thus its damping rate (γ_m = ω_m/Q) can be evaluated to be of the order of 10² Hertz with a mechanical quality factor Q ∼ 10⁵ [56]. ξ(t) is the Brownian noise force with zero mean acting on the movable mirror, which satisfies the correlation [57]

〈 ξ (t) ξ (t^{'}) 〉 = \frac{γ_{m}}{ω_{m}} \int \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 constant and T is the thermal bath temperature related to the movable mirror. a_in is the optical input-noise operator characterized fully by its correlation, which is given in the Markovian approximation as

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

. Γ₋(t) is the environmental noises corresponding to the operator S₋, which obeys a similar correlation,

〈 Γ_{-} (t) Γ_{-}^{†} (t^{'}) 〉 = δ (t - t^{'})

. Note that a nearby boundary can modify the emission rate of the two-level system, i.e.

Γ_{a} (z) = Γ_{0} + \frac{2 π c Γ_{0}}{ω_{0}} Tr Im {G (z, z, ω_{0})}

[46,50,51]. In the linear approximation, we have Γ_a(d + x) ≃ Γ_a(d) + Λ₀x, where

Λ_{0} = {\frac{\partial Γ_{a} (d + x)}{\partial x} |}_{x = 0}

. However, the coupling term Λ₀x is always much smaller than the zeroth order term Γ_a(d) and the coupling term λ₀x. Therefore the term Γ_a(d + x) ≃ Γ_a(d).

We first solve the steady-state expectation of the system, which can be obtained by replacing the operators by their averages in Eq. (7) and then setting all the time derivatives to zero. Further, using the mean field approximation (factorization assumption) 〈aq〉 = 〈a〉〈q〉 [58], the steady-state values can be derived as follows:

\begin{array}{l} p_{s} & = & 0, q_{s} = \frac{χ {| α_{s} |}^{2} - λ N / 2}{ω_{m}}, \\ α_{s} & = & \frac{- i {GS}_{s}}{κ + i Δ_{c}}, \\ S_{s} & = & \frac{- i η}{Γ_{a} + \frac{G^{2} κ}{κ^{2} + Δ_{c}^{2}} + i (Δ_{a} - \frac{G^{2} Δ_{c}}{κ^{2} + Δ_{c}^{2}})}, \end{array}

where Δ_c = Δ_c0 − χq_s and Δ_a = Δ_a0 + λq_s are, respectively, the effective detunings of the cavity mode and the quantum emitter. In the following, we focus on the dynamical evolution of the quantum fluctuation around these steady-state expectation, which is described by a set of linearized equations of motion for the fluctuation operators δq, δp, δa and δS₋. This can be derived by spliting the each operator in Eq. (7) into the steady-state value and corresponding quantum fluctuation, e.g. q = q_s + δq, p = p_s + δp, a = α_s + δa and S₋ = S_s + δS₋ as

\begin{array}{l} δ \dot{q} & = & ω_{m} δ p, \\ δ \dot{p} & = & - ω_{m} δ q + χ (α_{s}^{*} δ a + α_{s} δ a^{†}) - γ_{m} p + ξ (t), \\ δ \dot{a} & = & - (κ + i Δ_{c}) δ a + i χ α_{s} δ q - i G δ S_{-} + \sqrt{2 κ} a_{in} (t), \\ δ {\dot{S}}_{-} & = & - (Γ_{a} + i Δ_{a}) δ S_{-} - i G δ a - i λ S_{s} δ q + \sqrt{2 Γ_{a}} Γ_{-} (t), \end{array}

where we have considered the case |α_s| ≫ 1 and therefore all the terms higher than linear order in the fluctuations δq, δp, δa, δσ₋ and δσ_z in Eq. (7) are neglected safely.

Defining new quadrature fluctuation operators for the nonlinear optomechanical system $δ X = (δ a + δ a^{†}) / \sqrt{2}$ , $δ Y = (δ a - δ a^{†}) / \sqrt{2} i$ , $δ U = (δ S_{-} + δ S_{+}) / \sqrt{2}$ , $δ V = (δ S_{-} - δ S_{+}) / \sqrt{2} i$ and the corresponding Hermitian input noise operators $δ X^{in} = (δ a^{in} + δ a^{in, †}) / \sqrt{2}$ , $δ Y^{in} = (δ a^{in} - δ a^{in, †}) / \sqrt{2} i$ , $δ u = (δ Γ_{-} + δ Γ_{-}^{†}) / \sqrt{2}$ , $δ v = (δ Γ_{-} - δ Γ_{-}^{†}) / \sqrt{2} i$ , the Eq. (9) can be written in a more compact form, ḟ(t) = Jf(t) + n(t), with the column vector of fluctuation operator f^T(t) = (δp(t), δq(t), δX(t), δY(t), δU(t), δV(t)) and the corresponding column vector of noise $n^{T} (t) = (0, ξ, \sqrt{2 κ} δ X_{in}, \sqrt{2 κ} δ Y_{in}, \sqrt{2 Γ_{a}} δ u, \sqrt{2 Γ_{a}} δ v)$ . J is the drift matrix, which is given by

J = (\begin{matrix} 0 & ω_{m} & 0 & 0 & 0 & 0 \\ - ω_{m} & - γ_{m} & χ_{p} & χ_{n} & 0 & 0 \\ - χ_{n} & 0 & - κ & Δ_{c} & 0 & G \\ χ_{p} & 0 & - Δ_{c} & - κ & - G & 0 \\ G_{n} & 0 & 0 & G & - Γ_{a} & Δ_{a} \\ - G_{p} & 0 & - G & 0 & - Δ_{a} & - Γ_{a} \end{matrix}),

with

χ_{p} = χ (α_{s}^{*} + α_{s}) / \sqrt{2}

,

χ_{n} = χ (α_{s} - α_{s}^{*}) / \sqrt{2} i

,

G_{p} = λ (S_{s}^{*} + S_{s}) / \sqrt{2}

,

G_{n} = λ (S_{s} - S_{s}^{*}) / \sqrt{2} i

. For simplify, we choose properly the phase of the quantum emitter so that S_s is real. Further, the stability of the system demands that all the eigenvalues of the drift matrix J have negative real parts. These stability conditions for the hybrid optomechanical system can be obtained by using the Routh-Hurwitz criteria [59]. In the following section, We always resort to numerical calculation to check the stability of the steady-state solutions because the explicit inequalities satisfying stability condition are quite cumbersome. Correspondingly, all the external parameters used by us are also chosen to satisfy the stability condition.

4. Cooling of the mechanical motion

We are interested in the cooling characteristics of the mechanical resonator in the hybrid optomechanical system, where the two-level ensemble is driven by an external laser. In particular, we investigate the important role of the vacuum coupling in the ground-state cooling of the mechanical motion. In order to achieve this goal, we solve Eq. (9) by taking its Fourier transform to obtain the position fluctuation of the movable mirror around the steady state as δq(ω) = χ(ω)F(ω), where F is the Fourier transform of the total force acting on the movable mirror. The effective mechanical susceptibility of the movable mirror introduced here is defined as $χ (ω) = ω_{m} / (ω_{eff}^{2} - ω^{2} + i ω γ_{eff})$ , where the effective resonance frequency and effective damping rate of the movable mirror are given by

ω_{eff} = \sqrt{ω_{m} Re [\frac{A (ω) + B (ω) + C (ω)}{D (ω)}]}

and

γ_{eff} = γ_{m} + \frac{ω_{m}}{ω} Im [\frac{A (ω) + B (ω) + C (ω)}{D (ω)}],

respectively, in which we define

\begin{array}{l} D (ω) & = & {[G^{2} - Γ_{a} κ - i (Γ_{a} + κ) ω + ω^{2}]}^{2} + 2 G^{2} Δ_{a} Δ_{c} + {(Γ_{a} + i ω)}^{2} Δ_{c}^{2} + Δ_{a}^{2} β_{1}, \\ A (ω) & = & - G^{2} β_{3} (Γ_{a} + i ω) + {(Γ_{a} + i ω)}^{2} [β_{1} ω_{m} - Δ_{c} (χ_{n}^{2} + χ_{p}^{2})], \\ B (ω) & = & Δ_{a} [(G^{2} Δ_{c} + β_{1} Δ_{a}) ω_{m} - (G^{2} + Δ_{a} Δ_{c}) χ_{n}^{2} - Δ_{c} Δ_{c} χ_{p}^{2} - G G_{p} β_{2}], \\ C (ω) & = & G (Γ_{a} + i ω) (G β_{4} + G_{p} β_{5}) + G^{2} [G^{2} ω_{m} - G G_{p} χ_{p} + Δ_{a} (Δ_{c} ω_{m} - χ_{p}^{2})], \\ β_{1} (ω) & = & {(κ + i ω)}^{2} + Δ_{c}^{2}, \\ β_{2} (ω) & = & (κ + i ω) χ_{n} + Δ_{c} χ_{p}, \\ β_{3} (ω) & = & (κ + i ω) ω_{m} + χ_{n} χ_{p}, \\ β_{4} (ω) & = & χ_{n} χ_{p} - (κ + i ω) ω_{m}, \\ β_{5} (ω) & = & (κ + i ω) χ_{p} - χ_{n} Δ_{c} . \end{array}

In order to show the influences of the vacuum effect on the cooling of the movable mirror and how the ground-state cooling of the movable mirror is realized, we can count the effective phonon number in the mirror’s motion,

N_{ph} = (〈 δ p^{2} 〉 + (δ q^{2}) - 1) / 2,

where 〈δp²〉 and 〈δq²〉 are the momentum and displacement variances of the movable mirror in the steady state, and expressed as

〈 δ p^{2} 〉 = \frac{1}{2 π} \int_{- \infty}^{\infty} S_{p} (ω) d ω

and

〈 δ q^{2} 〉 = \frac{1}{2 π} \int_{- \infty}^{\infty} S_{q} (ω) d ω

, respectively [36]. In view of definition of the spectrum S_q(ω) of fluctuation in position of the movable mirror 〈δq(ω′)δq(ω)〉_s = S_q(ω)δ(ω′ + ω), we have

S_{q} (ω) = {| χ (ω) |}^{2} [S_{t} (ω) + S_{f} (ω) + S_{a} (ω)],

where

\begin{array}{l} S_{t} (ω) & = & \frac{γ_{m} ω}{ω_{m}} (1 + coth \frac{\bar{h} ω}{2 k_{B} T}), \\ S_{f} (ω) & = & \frac{2 κ}{{| D (ω) |}^{2}} [{| h_{1} (ω) |}^{2} + {| h_{2} (ω) |}^{2}], \\ S_{a} (ω) & = & \frac{2 Γ_{a}}{{| D (ω) |}^{2}} [{| h_{3} (ω) |}^{2} + {| h_{4} (ω) |}^{2}], \\ h_{1} (ω) & = & (G^{2} Δ_{a} + β_{6} Δ_{c}) χ_{n} + [(Γ_{1} + i ω) β_{7} + (κ + i ω) Δ_{a}^{2}] χ_{p}, \\ h_{2} (ω) & = & (G^{2} Δ_{a} + β_{6} Δ_{c}) χ_{p} + [(Γ_{a} + i ω) β_{7} - (κ + i ω) Δ_{a}^{2}] χ_{n}, \\ h_{3} (ω) & = & G β_{8} χ_{n} - G χ_{p} [(κ + i ω) Δ_{a} + (Γ_{a} + i ω) Δ_{c}], \\ h_{4} (ω) & = & G [(Γ_{a} + i ω) β_{5} - G^{2} χ_{p} - Δ_{a} β_{2}], \\ β_{6} (ω) & = & {(κ + i ω)}^{2} + Δ_{a}^{2}, \\ β_{7} (ω) & = & G^{2} + (ω - i Γ_{a}) (ω - i κ), \\ β_{8} (ω) & = & G^{2} - Γ_{a} κ - i ω (Γ_{a} + κ) + ω^{2} + Δ_{a} Δ_{c} . \end{array}

The spectrum S_p(ω) of the momentum fluctuation of the movable mirror can be written as S_p(ω) = (ω/ω_m)² S_x₁(ω). We next investigate the cooling of the movable mirror in the absence or the presence of the vacuum coupling. The dependence of the effective frequency and the damping rate of the movable mirror on the vacuum coupling strength is also studied.

To illustrate numerically the effective oscillation frequency of the movable mirror, we need to calculate the frequency shift and emission decay rate of the two-level system near the movable mirror and then derive the vacuum coupling strength λ₀. We assume the movable mirror is a thick dielectric (SiN) membrane coated with copper. Then, the reflection coefficient characterizing copper-vacuum interface is given by the standard Fresnel coefficients $r_{s} (i u, k_{‖}) = \frac{K_{3} - \sqrt{k_{‖}^{2} + ε (i u) u^{2} / c^{2}}}{K_{3} + \sqrt{k_{‖}^{2} + ε (i u) u^{2} / c^{2}}}$ and $r_{p} (i u, k_{‖}) = \frac{ε (i u) K_{3} - \sqrt{k_{‖}^{2} + ε (i u) u^{2} / c^{2}}}{ε (i u) K_{3} + \sqrt{k_{‖}^{2} + ε (i u) u^{2} / c^{2}}}$ , where $K_{3} = \sqrt{k_{‖}^{2} + u^{2} / c^{2}}$ . Further, the frequency shift Δω_a(d) can be calculated with Green’s function. In addition, we need the electromagnetic response of copper, which 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.

Figure 2(a) plots the normalized effective oscillation frequency ω_eff/ω_m as a function of the normalized frequency ω/ω_m with different distances d. For the explicit calculation of the vacuum coupling strength, 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 [60]. In addition, we select the accessible parameters in experiment, i.e. the intrinsic frequency of the movable mirror ω_m = 2π × 2 × 10⁶ Hz, the mass of the movable mirror m = 5 ng and the decay rate of the movable mirror γ_m = 2π × 100 Hz. Other parameters i.e. the effective detunings Δ_c = −ω_m and Δ_a = −ω_m; the cavity decay rate κ = 5ω_m and the free-space spontaneous emission rate Γ₀ = 0.1ω_m; The cavity length is L = 0.01 m and the collective coupling strength G = ω_m. We consider that the two-level ensemble is driven by a laser with wavelength λ_L = 1064 nm and the collective coupling strength |η| = 10¹² Hz. We see from Fig. 2(a) that in the high-frequency regime of the mechanical oscillator, the frequency of the movable mirror is not altered significantly and therefore the optical spring effect [36,61] of the system is very small. However, the change of the effective frequency becomes significant with decreasing distance d, which means that the stronger the vacuum coupling, the larger the optical spring effect. In particular, the frequency shift of the movable mirror due to the optical spring effect becomes relevant with a small mechanical frequency [61] and therefore contributes to the measurement of the CP force by the optomechanical device.

Fig. 2 The normalized effective oscillation frequency ω_eff/ω_m and the normalized effective damping rate γ_eff/ω_m are plotted as a function of the normalized frequency ω/ω_m with different distances. We use the plasma wavelength λ_P = 136 nm of copper corresponding to the plasma frequency ω_P = 2πc/λ_P and γ = 0.0033 ω_P to calculate the coupling strength between the quantum emitter and the nearby mirror. Other parameters are, respectively, ω_m = 2π × 2 × 10⁶ Hz, m = 5 ng, γ_m = 2π × 100 Hz, Δ_c = −ω_m, Δ_a = −ω_m, κ = 5ω_m, Γ₀ = 0.1ω_m and L = 0.01 m. In addition, the quantum emitter is driven by a laser with wavelength λ_L = 1064 nm and the collective coupling strength |η| = 10¹² Hz. The collective coupling strength G = ω_m.

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Figure 2(b) depicts the normalized effective damping rate γ_eff/ω_m as a function of the normalized frequency ω/ω_m with different distances d. We can see from Fig. 2(b) that in the presence of the vacuum coupling, the effective damping rate increases significantly with decreasing d. When the distance is very large, i.e. ω₀d/c → ∞, the vacuum coupling disappears and the effective damping rate is minimal. Indeed, in the presence of the vacuum coupling, the decay rate of the two-level ensemble Γ_a and the coupling strength λ increase with decreasing d. Thus, the effective damping rate [see Eq. (12)] depends strongly on the distance, which can be made very large by decreasing d in the chosen parameter regime. The significant increase in the effective damping rate is essential for cooling mechanical motion which drives the movable mirror close to the quantum ground state. Physically, the increase of the decay rate Γ_a helps dissipate the phonon in the mechanical motion via the decay channel of the atom ensemble, which enables the ground-state cooling of the movable mirror (see below). The ground state of a mechanical system is approached as the effective phonon number n_ph < 1, which corresponds generally to the situation of the energy equipartition with both variances 〈δq²〉 ≃ 1/2 and 〈δp²〉 ≃ 1/2. This demands that the initial mean-thermal excitation quantum number n = [exp(h̄ω_m/k_BT) − 1]⁻¹ is not excessively large, which is possible when the temperature is low enough, e.g., T = 0.1 K. We focus mainly on the ground-state cooling of the mechanical part in the bad cavity limit, i.e. κ = 5ω_m, in which the condition of the effective resolved sideband is satisfied by adjusting the bare decay rate of the quantum emitter, i.e. Γ₀ ≪ ω_m.

In Fig. 3, we plot the effective phonon number as a function of the dimensionless effective cavity detuning Δ_c/ω_m with different distances d. It is clear that the effective phonon number is always larger than one in the absence of the vacuum coupling i.e. ω₀d/c → ∞. Therefore, the ground state of the mechanical resonator can not be achieved. In the presence of the vacuum coupling, the effective phonon number n_ph < 1 in a finite interval of Δ_c, which shrinks gradually and vanishes with increasing distance d. Correspondingly, the minimum of the effective phonon number $n_{ph}^{min}$ increases with increasing distance d. The cooling mechanism can be understood from the interference and the level configuration in Fig. 4. In general, in order to cool the mechanical mode, the heating process in Fig. 4 should be suppressed as much as possible [30,62,63]. In the absence of vacuum coupling, the destructive quantum interference exists between the two different excitation pathways, from |n, n_a, m〉 → |n + 1, n_a, m + 1〉 directly and from |n, n_a, m〉 → |n + 1, n_a, m + 1〉 → |n, n_a + 1, m + 1〉 → |n + 1, n_a, m + 1〉 indirectly, which leads to the suppression of the heating process and therefore the cooling of the mechanical mode. However, when the vacuum coupling is included, the additional excitation pathway, i.e. from |n, n_a, m〉 → |n, n_a +1, m+1〉, will also take place. Then, the two excitation pathways, from |n, n_a, m〉 → |n + 1, n_a, m + 1〉 directly and from |n, n_a, m〉 → |n, n_a + 1, m + 1〉 → |n + 1, n_a, m + 1〉 indirectly, destructively interfere. In addition, the destructive interference also takes place between two indirect pathways, from |n, n_a, m〉 → |n + 1, n_a, m + 1〉 → |n, n_a + 1, m + 1〉 → |n + 1, n_a, m + 1〉 through the radiation pressure coupling and from |n, n_a, m〉 → |n, n_a + 1, m + 1〉 → |n + 1, n_a, m + 1〉 through the vacuum coupling. Therefore, increase of the interference channels in contrast to the case with only optomechanical coupling helps suppress the heating process in the hybrid system and enhance ground state cooling of the mechanical motion. Consequently, the effective phonon number n_ph is related to the strength of vacuum coupling and increases with increasing distance d. Then, the vacuum coupling may be used for controlling the cooling of the mechanical motion.

Fig. 3 The effective phonon number N_ph is plotted as a function of the dimensionless effective cavity detuning Δ_c/ω_m with different distances d. The collective coupling strength |η| = 0.8 × 10¹² Hz and the temperature of the movable mirror is selected as T = 0.1 K. Other parameter values are the same as those in Fig. 2.

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Fig. 4 Plot the energy level diagram in the hybrid optomechanical system, where |n, n_a, m〉 denotes the state of n photons in optical mode, n_a atomic excitations in collective excitation mode and m phonons in mechanical mode. The black double arrow denotes the coupling between states |n, n_a + 1, m + 1〉 and |n + 1, n_a, m + 1〉 through the atom-field coupling. The black solid (dashed) arrow denotes the cooling (heating) process of the mechanical mode through the radiation pressure coupling and the red solid (dashed) arrow denotes the cooling (heating) process of the mechanical mode through vacuum coupling.

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Figure 5 shows the effective phonon number as a function of the dimensionless effective detuning of the emitter Δ_a/ω_m with different distances d. The effective detuning Δ_c = −ω_m and therefore the cavity is tuned to the Stokes sideband. We can see that the curve for the effective phonon number has two minima, which are, respectively, reached at the Anti-Stokes sideband resonance, i.e. Δ_a ≃ ω_m and Stokes sideband resonance, i.e. Δ_a ≃ −ω_m. In the absence of the vacuum coupling, the effective phonon number is always larger than one. Things changed when the vacuum coupling between the quantum emitter and the movable mirror is included. For example, when ω₀d/c = 0.10, the two minimal values of the effective phonon number are n_ph = 0.07 and n_ph = 0.23, respectively. Moreover, the minimum of the effective phonon number also increases with increasing distance d. This is because in the presence of vacuum coupling, the destructive quantum interference appearing between the additional excitation pathways through vacuum coupling further suppresses the heating transition in the system, where one pair of pathways are |n, n_a, m〉 → |n + 1, n_a, m + 1〉 and |n, n_a, m〉 → |n, n_a + 1, m + 1〉 → |n + 1, n_a, m + 1〉, and the other pair of pathways are |n, n_a, m〉 → |n + 1, n_a, m + 1〉 → |n, n_a + 1, m + 1〉 → |n + 1, n_a, m + 1〉 and |n, n_a, m〉 → |n, n_a + 1, m + 1〉 → |n + 1, n_a, m + 1〉. These results suggest that a strong vacuum coupling is preferable for achieving the ground-state cooling of a mechanical oscillator.

Fig. 5 The effective phonon number N_ph is plotted as a function of the dimensionless effective detuning of the emitter Δ_a/ω_m with different distances d. The effective detuning Δ_c = −ω_m. Other parameter values are the same as those in Fig. 3.

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The collective coupling strength G will influence significantly the effective phonon number and therefore the ground-state cooling of the mechanical motion. In the following, we focus on the two cases in the absence and the presence (ω₀d/c = 0.1) of the vacuum coupling. The effective phonon number is shown in Fig. 6 as a function of the coupling strength G. We see from Fig. 6 that the curve for the effective phonon number has only one minimum in the absence of vacuum coupling. The optimal coupling strength G for cooling mechanical motion results from a pair of excitation pathways (|n, n_a, m〉 → |n + 1, n_a, m+ 1〉 and |n, n_a, m〉 → |n + 1, n_a, m + 1〉 → |n, n_a + 1, m + 1〉 → |n + 1, n_a, m + 1〉) through the radiation pressure. In particular, in this case the ground-state cooling of the mechanical motion can not be obtained with the selected parameters. When the vacuum coupling is included, i.e. ω₀d/c = 0.1, there exist two minima of the effective phonon number, which are smaller than one and therefore the ground-state cooling of the mechanical motion can be achieved with a proper coupling strength G. Similarly, these optimal coupling strengths G for the cooling of mechanical motion result from the transition |n, n_a, m〉 → |n + 1, n_a, m + 1〉 through radiation pressure coupling and the transition |n, n_a, m〉 → |n, n_a + 1, m + 1〉 through vacuum coupling, both of which contribute to the suppression of the heating process in the mechanical mode.

Fig. 6 The effective phonon number N_ph is plotted as a function of the collective coupling strength g in absence and presence of the vacuum coupling (ω₀d/c = 0.1). The effective detuning Δ_c = −ω_m. Other parameter values are the same as those in Fig. 3.

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The effective phonon number is shown in Fig. 7 as a function of the free-space spontaneous emission rate of the quantum emitter Γ₀ in the absence and presence of the vacuum coupling. The aim of Fig. 7 is to present clearly the important role of the quantum emitter for the cooling of the movable mirror in the bad cavity limit. We see from Fig. 7 that the effective phonon number is smaller than one only in a finite region near Γ₀ = 0.1ω_m when the vacuum coupling is included, which means that the decay rate Γ₀ of the quantum emitter helps satisfy the effective resolved sideband condition and therefore enhances the cooling process of the mechanical motion in the bad cavity limit, which is similar to the ground-state cooling of a mechanical resonator in a coupled cavity optomechanical system via an electromagnetically-induced- transparency (EIT)-like mechanism [43]. In the absence of the vacuum coupling, the effective phonon number is also always larger than one and therefore the ground state of the movable mirror can not be approached. These results suggest that we should select proper coupling strength G and decay rate Γ₀ in order to get optimal cooling for the movable mirror.

Fig. 7 The effective phonon number N_ph is plotted as a function of the free-space spontaneous emission rate of the quantum emitter Γ₀ in the absence and presence of the vacuum coupling (ω₀d/c = 0.1). Other parameter values are the same as those in Fig. 6.

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5. Conclusion

In conclusion,we studied the cooling of a mechanical resonator in a hybrid optomechanical system with an ensemble of two-level quantum emitters, which is coupled to the nearby movable mirror via vacuum force. The hybrid optomechanical cavity is driven along with the atom ensemble. We adopt the quantum Langevin equation to deal with the hybrid system and focus on the linearized dynamics around the steady state of the system. Further, by applying the Fourier transform method, the effective resonance frequency and effective damping rate of the movable mirror are derived. We also investigated the cooling characteristics of the mechanical motion by counting the effective phonon number in the absence and presence of the vacuum effect. We showed that the ground-state cooling of the mechanical motion can be approached in the bad cavity regime, when the vacuum coupling between the emitter ensemble and the movable mechanical resonator is included. The influences of the parameters of the cavity and the quantum emitter on the cooling of the mechanical motion are investigated in detail numerically. The results obtained here are not only useful for realizing the quantum control and manipulation of macroscopic mechanical motion but also able to provide one possible utilization of the vacuum force in an optomechanical device.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11304010, No. 11565014, No. 11365009 and No. 11375093, the startup Foundation for Doctors of East China Jiaotong University, under Grant No. 26541001.

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Cooling mechanical motion via vacuum effect of an ensemble of quantum emitters

Abstract

1. Introduction

2. Model and Hamiltonian

3. Dynamical evolution of the system

4. Cooling of the mechanical motion

5. Conclusion

Acknowledgments

References and links

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Figures (7)

Equations (16)

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