Controllable optomechanically induced transparency and ponderomotive squeezing in an optomechanical system assisted by an atomic ensemble

Yin Xiao; Ya-Fei Yu; Zhi-Ming Zhang

doi:10.1364/OE.22.017979

1. Introduction

Recently, a phenomenon called optomechanically induced transparency (OMIT) has been predicted theoretically [1] and observed experimentally [2–4]. OMIT means that the transmission of a probe light through an optomechanical cavity can be drastically influenced when introducing a second light, i.e. the coupling light. In analogy to the electromagnetically induced transparency (EIT) in atomic ensembles, the realization of OMIT can be used to manipulate the group velocity of light [5, 6]. The optomechanical systems have also been recognized as good systems for the purpose of optical memories, since the mechanical systems can have very long coherence times [7–9]. There are many schemes for realizing OMIT, for example, the light propagation in a cavity optomechanical system with a Bose-Einstein condensate (BEC) has been theoretically investigated[6]. It is also shown how the group delay and advance of the probe field can be controlled by the power of the coupling field in an optomechanical cavity with a moving nanomechanical mirror [5]. By using OMIT, optomechanical systems can be used as a single-photon router [10] which has been demonstrated experimentally with a cavity electromechanical system [11]. In [12], Shahidani et al. proposed a system to control and manipulate OMIT with a nonlinear medium. As is well known, a typical cavity optomechanical system couples a movable mirror and a cavity field by the radiation pressure [13, 14]. Besides OMIT, this coupling can also lead to many other remarkable effects, for example, the quantum ground state cooling of the nanomechanical resonators [15–17], the quantum state-transfer [18–20], entanglement [21, 22] and squeezing [23–25]. Among these phenomena, just like the interaction between electromagnetic radiation and atoms which has led to interesting quantum feature of squeezing [26, 27], the ponderomotively squeezed light can be generated in an optomechanical cavity via the interaction between cavity field and vibrating mirror [25, 28]. Achieving squeezed states experimentally is an important goal because of its applications in ultrahigh precision measurements [29–31].

In this paper, we explores two features (i.e., OMIT and ponderomotive squeezing) of the optomechanical system assisted by a low-excited two-level atomic ensemble. A hybrid system containing atomic ensemble has attracted much attention in recent years [32, 33]. Here we first propose a scheme for realizing a controllable OMIT, which means that the width of the transparency window can be adjusted flexibly. This is different from the system proposed in [12], in which the authors use a nonlinear crystal consisting of a Kerr medium and a degenerate optical parametric amplifier (OPA) to achieve this goal. Comparing with [12], we just need to adjust one parameter to realize the controllable transparency window. Besides, if we choose suitable parameters, we can make a switch for the probe field, i.e. controlling the system to be optomechaically induced transparency (OMIT) or to be optomechanically induced absorption (OMIA) for the probe field. We expect that this effect can become an applied technology in the near future. We then study the squeezing spectrum of the transmitted field without importing the probe field, since we are mainly interested in the effects of the coupling between the atomic ensemble and cavity field. Besides the driving laser coupling the cavity field, we find that the coupling strength between the atomic ensemble and the cavity field and that between the atomic ensemble and the external optical field also play an important role.

2. Model and theory

As shown in Fig. 1, the system under study consists of a generic optomechanical system and an atomic ensemble. The atomic ensemble is driven by a strong classical external field of frequency ω_c, and the cavity mode of frequency ω₀ is driven simultaneously by a strong classical external field of frequency ω_c and a weak probe field of frequency ω_p. The model Hamiltonian reads

\begin{array}{l} H & = & \bar{h} ω_{0} c^{†} c + \frac{\bar{h} ω_{a}}{2} \sum_{i = 1}^{N} σ_{z}^{i} + (\bar{h} G c \sum_{i = 1}^{N} σ_{+}^{i} + H . c) + (\bar{h} Ω e^{- i ω_{c} t} \sum_{i = 1}^{N} σ_{+}^{i} + H . c) \\ + (\frac{p^{2}}{2 m} + \frac{1}{2} m ω_{m}^{2} q^{2}) - \bar{h} g c^{†} c q + i \bar{h} ε_{c} (c^{†} e^{- i ω_{c} t} - H . c) + i \bar{h} (ε_{p} c^{†} e^{- i ω_{p} t} - H . c), \end{array}

here, the first term describes the free Hamiltonian of cavity field in which c(c^†) is the annihilation (creation) operator of the cavity field. The second term is the free Hamiltonian of the atom ensemble. The third and fourth terms are the interaction Hamiltonian describing the coupling between the atom ensemble and the cavity field and that between the atom ensemble and the external driving field, respectively.

σ_{z}^{i} = {| e 〉}^{i} 〈 e | - {| g 〉}^{i} 〈 g |

,

σ_{+}^{i} = {| e 〉}^{i} 〈 g |

, and

σ_{-}^{i} = {| g 〉}^{i} 〈 e |

are the Pauli matrices for the ith atom in the atomic ensemble. The number of atoms in the atomic ensemble is N, and we assume that all the atoms have the same excited (ground) state |e〉 (|g〉), so they have the same transition frequency ω_a. The atomic ensemble is arranged in a thin layer whose size in the direction of the cavity axis is much smaller than the wavelength of the cavity field, thus all the atoms have the same coupling strength G with the cavity field [34]. Similarly, when the wavelength of external driven field is much larger than the size of atomic ensemble in the vertical direction, we can also assume that the coupling coefficient Ω for each atom is the same. The fifth term is the Hamiltonian of the mechanical mode with resonance frequency ω_m and effective mass m. The sixth term is the interaction Hamiltonian describing the radiation pressure interaction between the cavity mode and the mechanical resonator, where g is the optomechanical coupling rate between the mechanical mode and cavity mode. The last two terms describe the interaction of the cavity field with the coupling field and that of the cavity field with the probe field, with the amplitude

ε_{c} = \sqrt{\frac{2 κ P_{c}}{\bar{h} ω_{c}}}

and

ε_{p} = \sqrt{\frac{2 κ P_{p}}{\bar{h} ω_{p}}}

, respectively. κ is the decay rate of the cavity field, and P_c and P_p are the laser powers. We have also assumed that no direct interaction exists between the atoms and the mirror, the indirect interaction between them solely relies on the cavity field.

Fig. 1 Sketch of the system. A two-level atomic ensemble driven by an external optical field couples with the cavity field in a typical optomechanical cavity. The cavity is driven by a coupling laser and a probe laser.

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Since all the atoms have the same transition frequency ω_a, we can consider the atomic ensemble in the cavity as a whole to be a Hopfield dielectric [35], thus the atoms can be completely described by a type of collective low-energy excitations. Then we introduce the following operators of the atomic collective excitation modes for the atomic ensemble to simplify the Hamiltonian [34],

A^{†} = \frac{1}{\sqrt{N}} \sum_{i = 1}^{N} σ_{+}^{i}, A = {(A^{†})}^{†} .

In the low-excitation limit with large N, the above operators satisfy the standard bosonic commutation relations

[A, A^{†}] \approx 1 .

Then we have

\sum_{i = 1}^{N} σ_{z}^{i} = 2 A^{†} A - N .

The Hamiltonian (1) can be rewritten in terms of the atomic collective operators A(A^†) as

\begin{array}{l} H & = & \bar{h} ω_{0} c^{†} c + \bar{h} ω_{a} A^{†} A + (\bar{h} G_{A} c A^{†} + \bar{h} χ A^{†} e^{- i ω_{c} t} + H . c) \\ + \frac{p^{2}}{2 m} + \frac{1}{2} m ω_{m}^{2} q^{2} - \bar{h} g c^{†} c q + i \bar{h} ε_{c} (c^{†} e^{- i ω_{c} t} - H . c) + i \bar{h} (ε_{p} c^{†} e^{- i ω_{p} t} - H . c), \end{array}

where

G_{A} = \sqrt{N} G

is the effective coupling strength between the cavity field and the atomic ensemble, and

χ = \sqrt{N} Ω

is the effective coupling strength between the atomic ensemble and the external driving field. We can see from G_A and χ that the coupling coefficients are enhanced by

\sqrt{N}

times. In the interaction picture with respect to H₀ = h̄ω_c (c^†c + A^†A), the interaction Hamiltonian is given in the form as

\begin{array}{l} H & = & \bar{h} Δ_{c} c^{†} c + \bar{h} Δ_{a} A^{†} A + (\bar{h} G_{A} c A^{†} + \bar{h} χ A^{†} + H . c) \\ + \frac{p^{2}}{2 m} + \frac{1}{2} m ω_{m}^{2} q^{2} - \bar{h} g c^{†} c q + i \bar{h} ε_{c} (c^{†} - H . c) + i \bar{h} (ε_{p} c^{†} e^{- i δ t} - H . c), \end{array}

where Δ_c = ω₀ − ω_c, Δ_a = ω_a − ω_c and δ = ω_p − ω_c are the detunings.

3. OMIT

Applying the Heisenberg equations of motion for operators c, A, p, and q, we derive the quantum Langevin equations as follows,

\dot{c} = - i Δ_{c} c - i G_{A} A + i g c q - κ c + ε_{c} + ε_{p} e^{- i δ t} + \sqrt{2 κ} c_{in},

\dot{A} = - i Δ_{a} A - i G_{A} c - i χ - γ_{a} A + \sqrt{2 κ} A_{in},

\dot{p} = - m ω_{m}^{2} q + \bar{h} g c^{†} c - γ_{m} p + ξ,

\dot{q} = \frac{p}{m},

where κ is the cavity decay rate, γ_a is the decay rate of the atomic transition |e〉 ↔ |g〉 and γ_m is the damping rate of mechanical oscillating mirror; c_in and A_in are the input vacuum noise operators with zero mean value, and their only nonzero correlation functions are

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

and

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

, respectively [36]. ξ is the Brownian stochastic force with zero mean value and its correlation function is

〈 ξ (t) ξ (t^{'}) 〉 = m \bar{h} γ_{m} \int \frac{d ω}{2 π} e^{- i ω (t - t^{'})} ω [coth (\frac{\bar{h} ω}{2 k_{B} T}) + 1]

[37]. Since we are interested in the mean response of the system, we write the Langevin equations for the mean values. And then we can obtain the steady-state mean values of c, A, p, and q by setting all the time derivatives of Eqs. (7)–(10) to zero. They are given by

A_{s} = - i f c_{s} - \frac{i χ}{i Δ_{a} + γ_{a}}, p_{s} = 0, q_{s} = \frac{\bar{h} g {| c_{s} |}^{2}}{m ω_{m}^{2}}, c_{s} = \frac{ε_{c} - f χ}{i (Δ_{c} - {| f |}^{2} Δ_{a} - g q_{s}) + (κ + {| f |}^{2} γ_{a})},

where

f = \frac{G_{A}}{i Δ_{a} + γ_{a}}

. We then linear the problem for δo ≪ o_s (o = c, A, p, q), inserting the ansatz o = o_s + δo into Eqs. (7)–(10) and retain only first order terms in the small quantities δo. Thus the quantum Langevin equations for the fluctuations are given by

δ \dot{c} = - (κ + i Δ) δ c - i G_{A} δ A + i λ δ q + ε_{p} e^{- i δ t} + \sqrt{2 κ} c_{in},

δ \dot{A} = - (γ_{a} + i Δ_{a}) δ A - i G_{A} δ c + \sqrt{2 κ} A_{in},

δ \dot{p} = - γ_{m} δ p - m ω_{m}^{2} δ q + \bar{h} λ (δ c + δ c^{†}) + ξ,

δ \dot{q} = \frac{δ p}{m},

where Δ = Δ_c − gq_s and λ = gc_s. In order to solve Eqs. (12)–(15), we make the ansatz δo = o₊e^−iδt + o₋e^iδt (o = c, A, p, q). Substituting this ansatz into Eqs. (12)–(15), we derive the following solution of interest

c_{+} = \frac{m [κ - i (Δ + δ) + α] (ω_{m}^{2} - δ^{2} - i δ γ_{m}) + i \bar{h} λ^{2}}{m [κ + i (Δ + δ) + β] [κ - i (Δ + δ) + α] (ω_{m}^{2} - δ^{2} - i δ γ_{m}) + i \bar{h} λ^{2} (2 i Δ - α + β)} ε_{p},

where

α = \frac{G_{A}^{2}}{γ_{a} - i (δ + Δ_{a})}

and

β = \frac{G_{A}^{2}}{γ_{a} - i (δ - Δ_{a})}

. Using the input-output relation c_out (t) = c_in (t) − 2κc(t) [36], one obtains,

c_{out} (t) = (ε_{c} - 2 κ c_{s}) e^{- i ω_{c} t} + (ε_{p} - 2 κ c_{+}) e^{- i (δ + ω_{c}) t} - 2 κ c_{-} e^{i (δ - ω_{c}) t},

The transmission of the probe field is then given by,

\begin{array}{l} t_{p} & = & \frac{ε_{p} - 2 κ c_{+}}{ε_{p}} \\ = & 1 - 2 κ \frac{m [κ - i (Δ + δ) + α] (ω_{m}^{2} - δ^{2} - i δ γ_{m}) + i \bar{h} λ^{2}}{m [κ + i (Δ + δ) + β] [κ - i (Δ + δ) + α] (ω_{m}^{2} - δ^{2} - i δ γ_{m}) + i \bar{h} λ^{2} (2 i Δ - α + β)} . \end{array}

Here we introduce the normalized transmission

T_{p} = \frac{t_{p} - t_{r}}{1 - t_{r}},

where t_r is resonance transmission in the absence of a coupling laser,

t_{r} = t_{p} (δ = ω_{m}, λ = 0) .

The optomechanically induced transparency is then described by,

{| T_{p} |}^{2} = {| 1 - κ \frac{m [κ - i (Δ + δ) + α] (ω_{m}^{2} - δ^{2} - i δ γ_{m}) + i \bar{h} λ^{2}}{m [κ + i (Δ + δ) + β] [κ - i (Δ + δ) + α] (ω_{m}^{2} - δ^{2} - i δ γ_{m}) + i \bar{h} λ^{2} (2 i Δ - α + β)} |}^{2} .

For numerical calculation, we choose the parameters from the recent experiment [38]: the wavelength of the laser λc = 2πc/ω_c = 1064nm, the total cavity length in absence of the coupling laser L = 25mm, the mass of the oscillating mirror m = 145ng, the cavity decay rate κ = 2π × 215 × 10³Hz, the frequency of the moving mirror ω_m = 2π × 947 × 10³Hz, the mechanical factor Q = ω_m/γ_m = 6700. The parameters of atoms are G_A = 2π × 1.59 × 10³Hz and γ_a = 2π × 400Hz which are based on [39]. The atomic-cavity field detuning Δ_a = 0 is considered here. For the sideband resolved limit ω_m ≫ κ, and G_A, γ_a,

\frac{G_{A}^{2}}{γ_{a}} ≪ κ

, the effective mechanical damping rate γ_eff can be obtained as

γ_{eff} = γ_{m} (1 + C),

where C = 2h̄λ²/mω_mκγ_m. The γ_eff is related to the width of the transparency window [1, 2] and depends on |c_s|². Thus we can control the value of c_s to achieve the goal of adjusting the width of the transparency window. From Eq. (11), one has

{| c_{s} |}^{2} \propto {| ε_{c} - f χ |}^{2} .

In Fig. 2, we show that the transparency window changes with different parameters. Like the typical optomechanical system for OMIT, increasing the driving laser strength coupling with the cavity field will increase the width of transparency window, since it increases the coupling rate λ which is equivalent to the Rabi frequency in the atomic EIT [2]. We show this in Fig. 2(a) without importing the driving field coupling with the atomic ensemble. However, we often hope to adjust the transparency window with a small power by another pump laser. Here, in this system, it is easy to achieve this goal because the pump power is enlarged by nearly 4 times ( $\frac{G_{A}}{γ_{a}} \approx 4$ ) due to its coupling with atomic ensemble. From Eq. (23), we find that when the coupling strength χ is in the range (0, 0.5ɛ_c) and P_c = 15mW, the width of window decreases compared with the case when χ = 0. Once the coupling strength χ exceeds 0.5ɛ_c, the width of window increases. We show this controllable transparency window in Fig. 2(b). This is quite different from [12], in which the authors control the transparency window only in the presence of two nonlinearities. If we choose suitable parameters to allow χ = 0.25ɛ_c, we find an interesting phenomenon, that is, we change the OMIT to OMIA for the probe field as we can see it in Fig. 2(c). This can also be understood from Eq. (23), |c_s|² = 0 at this time. This phenomenon can be used to make a switch for probe field. No matter what the value of the pump laser P_c is, we just set χ as 0.25ɛ_c to realize the switch for probe field. From the discussion above, we can find that the external field driving the atomic ensemble changes the mean photon number of the cavity field, so we can adjust the coupling strength between this field and the atomic ensemble to control the transparency window desirably. By defining ζ = 1 + T_p, whose real and imaginary parts represent the behaviour of absorption and dispersion, respectively, we show its dispersion properties in Fig. 2(d) with different coupling parameters χ. As we all know, the dispersion properties change the group velocity of light, thus the parameters χ has a distinct influence on slow light as seen in Fig. 2(d). In [40], the authors give a detailed discussion about the effect of the coupling strength between the cavity field and the atomic ensemble on the OMIT in a similar system, so we do not discuss it here again.

Fig. 2 (a) The transparency window with the laser power P_c as a parameter. (b) The transparency window with the coupling strength χ as a parameter. (c) The optical switch for the probe field. (d) The behaviour of the dispersion.

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4. Ponderomotive squeezing

In this section, we analyze the squeezing properties of the transmitted field, which is accessible to experiment and useful for practical applications [29–31]. It is more convenient to work in the frequency domain because, experimentally, fluctuations of the electric field are more convenient to be measured in the frequency domain than in the time domain. The squeezing spectrum of the transmitted field is given by [28, 41]

\begin{array}{l} S_{θ} (ω) & = & \int_{- \infty}^{\infty} 〈 δ X_{θ}^{out} (t + τ) δ X_{θ}^{out} (t) 〉 e^{- i ω τ} d τ \\ = & 〈 δ X_{θ}^{out} (ω) δ X_{θ}^{out} (ω) 〉, \end{array}

where

δ X_{θ}^{out} (ω) = e^{- i θ} δ c_{out} (ω) + e^{i θ} δ c_{out}^{†} (ω)

. Since

[X_{θ}^{out}, X_{θ + \frac{π}{2}}^{out}] = 2 i

, quadrature squeezing occurs when

〈 δ X_{θ}^{out} (ω) δ X_{θ}^{out} (ω) 〉 < 1

, i.e., S_θ (ω) < 1. Then we rewrite Eq. (24) to be

S_{θ} (ω) = 1 + 2 B_{c^{†} c} + e^{- 2 i θ} B_{c c} + e^{2 i θ} B_{c^{†} c^{†}},

where 〈δc(ω)δc(ω′)〉 = δ(ω + ω′)B_cc and other terms are defined in a similar way. As the phase angle θ is an adjustable parameter, we choose appropriate θ by solving dS_θ(ω)/dθ = 0, then we obtain

e^{2 i θ_{opt}} = \pm \frac{B_{c c} (ω)}{| B_{c c} (ω) |},

here, we choose the negative value to optimize the degree of squeezing. So Eq. (25) can be rewritten as

S_{opt} (ω) = 1 + 2 B_{c^{†} c} - 2 | B_{c c} | .

For calculating in frequency domain, we define the Fourier transform for an operator u (u = δc, δA, δp, c_in, A_in and ξ) as

u (t) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} e^{- i ω t} u (ω) d ω, u^{†} (t) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} e^{- i ω t} u^{†} (- ω) d ω,

which lead to the following nonzero correlation functions in the frequency domain [36, 37]

〈 c_{in} (Ω) c_{in}^{†} (ω) 〉 = δ (Ω - ω),

〈 A_{in} (Ω) A_{in}^{†} (ω) 〉 = δ (Ω - ω),

〈 ξ (Ω) ξ (ω) 〉 = \bar{h} γ_{m} m ω (1 + coth \frac{\bar{h} ω}{2 k_{B} T}) δ (Ω + ω) .

Then we rewrite Eqs. (12)–(15) in the frequency domain, which can be written in matrix form as

M (ω) Z (ω) = Y (ω),

where Z = (δc, δc^†, δA, δA^†, δp, δq)^T,

Y = {(\sqrt{2 κ} c_{in}, \sqrt{2 κ} c_{in}^{†}, \sqrt{2 γ_{a}} A_{in}, \sqrt{2 γ_{a}} A_{in}^{†}, ξ, 0)}^{T}

, and

M = (\begin{matrix} Λ_{1} & 0 & i G_{A} & 0 & 0 & - i λ \\ 0 & Λ_{2} & 0 & - i G_{A} & 0 & i λ \\ i G_{A} & 0 & Θ_{1} & 0 & 0 & 0 \\ 0 & - i G_{A} & 0 & Θ_{2} & 0 & 0 \\ - \bar{h} λ & - \bar{h} λ & 0 & 0 & γ_{m} - i ω & m ω_{m}^{2} \\ 0 & 0 & 0 & 0 & 1 & i m ω \end{matrix}),

in which we have defined Λ₁ = κ − i(ω − Δ), Λ₂ = κ − i(ω + Δ), Θ₁ = γ_a − i(ω − Δ_a), and Θ₂ = γ_a − i(ω + Δ_a). By solving Eq. (32), we obtain

δ c (ω) = E_{1} (ω) c_{in} E_{2} (ω) c_{in}^{†} + E_{3} (ω) A_{in} + E_{4} (ω) A_{in}^{†} + E_{5} (ω) (ξ),

δ c^{†} (ω) = E_{1}^{*} (- ω) c_{in}^{†} + E_{2}^{*} (- ω) c_{in} + E_{3}^{*} (- ω) A_{in}^{†} + E_{4}^{*} (- ω) A_{in} + E_{1}^{*} (- ω) ξ,

where

E_{1} (ω) = \frac{\sqrt{2 κ} Θ_{1} [i \bar{h} λ^{2} Θ_{2} + m (G_{A}^{2} + Θ_{2} Λ_{2}) (ω_{m}^{2} - ω^{2} - i ω γ_{m})]}{d (ω)},

E_{2} (ω) = \frac{i \sqrt{2 κ} \bar{h} λ^{2} Θ_{1} Θ_{2}}{d (ω)},

E_{3} (ω) = \frac{- i \sqrt{2 γ_{a}} G_{A} [i \bar{h} λ^{2} Θ_{2} + m (G_{A}^{2} + Θ_{2} Λ_{2}) (ω_{m}^{2} - ω^{2} - i ω γ_{m})]}{d (ω)},

E_{4} (ω) = \frac{- \sqrt{2 γ_{a}} \bar{h} λ^{2} G_{A} Θ_{1}}{d (ω)},

E_{5} (ω) = \frac{i λ Θ_{1} (G_{A}^{2} + Θ_{2} Λ_{2})}{d (ω)},

\begin{array}{l} d (ω) = & - i \bar{h} λ^{2} [Θ_{1} (G_{A}^{2} + Θ_{2} Λ_{2}) - Θ_{2} (G_{A}^{2} + Θ_{1} Λ_{1})] \\ - m (G_{A}^{2} + Θ_{2} Λ_{2}) (G_{A}^{2} + Θ_{1} Λ_{1}) (ω_{m}^{2} - ω^{2} - i ω γ_{m}) . \end{array}

By using the results above and the input-output relation, we can obtain B_c^†c and B_cc in the expression of squeezing spectrum (27),

B_{c^{†} c} = 2 κ [{| E_{2} (- ω) |}^{2} + {| E_{4} (- ω) |}^{2} + \bar{h} γ_{m} m ω (1 + coth [\frac{\bar{h} ω}{2 k_{B} T}]) {| E_{5} (- ω) |}^{2}],

\begin{array}{r} B_{c c} = 2 κ [E_{1} (ω) E_{2} (- ω) + E_{3} (ω) E_{4} (- ω) - \frac{E_{2} (- ω)}{\sqrt{2 κ}} \\ + E_{5} (ω) E_{5} (- ω) \bar{h} γ_{m} m ω (1 + coth [\frac{\bar{h} ω}{2 k_{B} T}])] . \end{array}

Now we focus on the dependence of the degree of squeezing on the coupling strength between the cavity field and the atomic ensemble. The parameters are: χ = 0, P_c = 5mW and other parameters are the same as in Fig. 2. The parameters we choose meet the stability conditions which can be obtained by the Routh-Hurwitz criterion [42]. Then we show the squeezing spectrum versus the normalized frequency in Fig. 3. In Fig. 3(a), we find that the transmitted field exhibits squeezing for both resonance frequencies in the bare cavity. However, when we set the coupling strength as G_A = 0.08κ, the degree of squeezing increases at the blue-detuned frequency while it decreases at the red-detuned frequency, but they both present the property of squeezing, as shown in Fig. 3(b). In Fig. 3(c), if we increase the coupling strength G_A to 0.1κ, the degree of squeezing at the blue-detuned frequency is increased further, yet the frequency at the red-detuned presents nearly no squeezing. But we cannot increase the degree of squeezing to the perfect squeezing corresponding to S_opt(ω) = 0, and we can find it in Fig. 3(d), in which the degree of squeezing decreases rather than increases with the coupling strength increasing to 0.8κ. This can be understood by considering the effective cavity decay rate

κ_{eff} = κ + \frac{G_{A}^{2}}{γ_{a}}

which can be obtained from Eq. (11). As we all know, the cavity decay rate will reduce the photon number drastically when it is large enough. So when we have a large coupling strength G_A, the decay will predominate the process. We have also made a numerical calculation about the effects of a larger coupling strength G_A on the properties of squeezing, and found that there were no squeezing just as the case in which the optomechanical coupling λ = 0. Thus this provides us a good methods for controlling the degree of squeezing and finding the optimal squeezing.

Fig. 3 Squeezing with different coupling strength G_A

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5. Conclusion

In conclusion, we theoretically investigate two features of an optomechanical system assisted by a low-excited two-level atomic ensemble, i.e., OMIT and pondermotive squeezing. These two features have important practical applications. In such a system, we first realize a controllable transparency window by the external driving field with a small power, and make a switch for the probe field when choosing suitable parameters. Then we study the properties of the pondermotive squeezing in the frequency domain by changing the coupling strength between the cavity-field and the atomic ensemble, and we find a meaningful phenomenon by adjusting the coupling strength, i.e, the degree of squeezing presents different characteristics at different resonance frequency with increasing the coupling strength. We expect that our results can be realized in experiment and have practical applications in the near future.

Acknowledgments

This work was supported by the Major Research Plan of the NSFC (Grant No. 91121023), the NSFC (Grant Nos. 61378012 and 60978009), the SRFDPHEC (Grant No. 20124407110009), the “973”Program (Grant Nos. 2011CBA00200 and 2013CB921804), and the PCSIRT (Grant No. IRT1243).

References and links

1. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010). [CrossRef]

2. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010). [CrossRef] [PubMed]

3. J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London) 471, 204 (2011). [CrossRef]

4. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472, 69 (2011). [CrossRef]

5. Devrim Tarhan, Sumei Huang, Ö zgür, and E. Müstecaplioğlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Phys. Rev. A 87, 013824 (2013). [CrossRef]

6. Bin Chen, Cheng Jiang, and Ka-Di Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Phys. Rev. A 83, 055803 (2011). [CrossRef]

7. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706 (2009). [CrossRef]

8. Victor Fiore, Yong Yang, Mark C. Kuzyk, Russell Barbour, Lin Tian, and Hailin Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011). [CrossRef] [PubMed]

9. E. Verhagen, S. Delalise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63 (2012). [CrossRef]

10. G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801(R) (2012). [CrossRef]

11. C. Jiang, B. Chen, and K.-D. Zhu, “Demonstration of a single-photon router with a cavity electromechanical system,” J. Appl. Phys. 112, 033113 (2012). [CrossRef]

12. S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013). [CrossRef]

13. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172 (2008). [CrossRef] [PubMed]

14. F. Marquardt and S. M. Girvin, “Trend: optomechanics,” Physics 2, 40 (2009). [CrossRef]

15. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007). [CrossRef] [PubMed]

16. M. Bhattacharya and P. Meystre, “Trapping and cooling a mirror to its quantum mechanical ground state,” Phys. Rev. Lett. 99, 073601 (2007). [CrossRef] [PubMed]

17. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008). [CrossRef]

18. L. Tian and H. L. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82, 053806 (2010). [CrossRef]

19. Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012). [CrossRef] [PubMed]

20. T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Coherent state transfer between itinerant microwave fields and a mechanical oscillator,” Nature (London) 495, 210–214 (2013). [CrossRef]

21. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, 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]

22. 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]

23. A A Clerk, F Marquardt, and K Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008). [CrossRef]

24. Jie-Qiao Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011). [CrossRef]

25. 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, 1337 (1994). [CrossRef] [PubMed]

26. N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426 (1995). [CrossRef] [PubMed]

27. E. Alebachew and K. Fessah, “Interaction of a two-level atom with squeezed light,” Opt. Commun. 271, 154 (2007). [CrossRef]

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

29. D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature (London) 448, 476 (2012). [CrossRef]

30. A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature (London) 500, 185 (2013). [CrossRef]

31. T. P. Purdy, P.-L. Yu, R.W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X , 3, 031012 (2013).

32. D. Meiser and P. Meystre, “Coupled dynamics of atoms and radiation-pressure-driven interferometers,” Phys. Rev. A 73, 033417 (2006). [CrossRef]

33. K. Hammerer, K. Stannigel, C. Genes, and P. Zoller, “Optical lattices with micromechanical mirrors,” Phys. Rev. A 82, 021803(R) (2010). [CrossRef]

34. C. P. Sun, Y. Li, and X. F. Liu, “Quasi-spin-wave quantum memories with a dynamical symmetry,” Phys. Rev. Lett. 91, 147903 (2003). [CrossRef] [PubMed]

35. J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev. 112, 1555 (1958). [CrossRef]

36. C. W. Gardiner and P. Zoller, Quantum noise (Springer) (2004).

37. V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001). [CrossRef]

38. S. Gröblacher, K. Hammerer, M. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London) 460, 724 (2009). [CrossRef]

39. C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008). [CrossRef]

40. Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” J. Phys. B: At. Mol. Opt. Phys. 44165505 (2011). [CrossRef]

41. M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32, 2887 (1985). [CrossRef] [PubMed]

42. 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 (1987). [CrossRef] [PubMed]

Controllable optomechanically induced transparency and ponderomotive squeezing in an optomechanical system assisted by an atomic ensemble

Abstract

1. Introduction

2. Model and theory

3. OMIT

4. Ponderomotive squeezing

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (43)

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