Nonreciprocal transmission and fast-slow light effects in a cavity optomechanical system

Jun-Hao Liu; Ya-Fei Yu; Zhi-Ming Zhang

doi:10.1364/OE.27.015382

1. Introduction

In recent years, optical nonreciprocity has got a lot of attentions for its important applications in photonic network, signal processing, and one-way optical communication protocols. In the common nonreciprocal devices, such as, isolator, circulator, nonreciprocal phase shifter, the transmission of the information is not symmetric. At present, the researches about the optical nonreciprocity mainly focused on two aspects: one is the transmission properties of the signal fields, another is the photonic statistical properties of the signal fields.

For the first aspect, scientists have demonstrated that many physical effects and physical systems, such as Faraday rotation effect in the magneto-optical crystals [1–3], optical nonlinearity [4], spatial-symmetry-breaking structures [5,6], optoacoustic effects [7,8], the parity-time-symmetric structures [9–13], can be used to realize the optical nonreciprocal transmission. Efforts have also been made to study the nonreciprocal transmission in cavity optomechanical systems [14–18]. Manipatruni et al. demonstrated that the optical nonreciprocal transmission was based on the momentum difference between the forward and backward-moving light beams in a Fabry-Perot cavity with one moveable mirror [19]. Hafezi et al. proposed a scheme to achieve the nonreciprocal transmission in a microring resonator by using an unidirectional optical pump [20]. Metelmann and Clerk discussed a general method for nonreciprocal photon transmission and amplification via reservoir engineering [21]. Shen et al. experimentally demonstrated the non-magnetic nonreciprocity in a whispering gallery microresonator [22]. Ruesink et al. studied the nonreciprocity and magnetic-free isolation in a compact system based on optomechanical interactions [23]. Peterson et al. demonstrated an efficient frequency-converting microwave isolator based on the optomechanical interactions [24]. Mirza et al. studied the optical nonreciprocity and slow light propagation in coupled spinning optomechanical resonators [25]. More recently, Lépinay et al. experimentally realized the nonreciprocal amplification in a microwave optomechanical device [26].

For the second aspect, the researches on the photonic statistic properties of the transmitted fields in nonreciprocal devices are fewer. At present, the relevant theoretical works include the nonreciprocal photon blockade [27], the authors discussed how to create and manipulate nonclassical light via photon blockade in rotating nonlinear devices. They found that the light with sub-Poissonian or super-Poissonian photon-number statistics can emerge when driving the resonator from its left or right side. Subsequently, Xu et al. proposed a scheme to manipulate the statistic properties of the photons transport nonreciprocally via quadratic optomechanical coupling [28].

In this paper, we study the nonreciprocal transmission and the fast-slow light effects in a cavity optomechanical system, as shown in Fig. 1. We show that, when the intrinsic photon loss of the cavity equals the external coupling loss of the cavity, we can achieve the nonreciprocal transmission of the signal fields with the red-sideband pumping or the blue-sideband pumping. We also show that, when the intrinsic photon loss is much less than the external coupling loss, the nonreciprocity of the system about the optical transmission properties almost disappears, now the system exhibits a nonreciprocal fast-slow light propagation phenomenon, i.e., the group velocity of the right-moving signal field will be sped up (fast light), while the group velocity of the left-moving signal field will be slowed down (slow light), or vice versa.

Fig. 1 (a) Schematic diagram of our proposed model. An optomechanical microtoroid cavity supports a clockwise circulating mode (â) and a counter-clockwise circulating mode (ĉ), both the two cavity modes couple with the mechanical mode (b̂) via the radiation pressure. The pump fields (ε_ap, ε_cp) and signal fields (ε_as, ε_cs) couple with the cavity modes by an optical fiber. (b) The nonreciprocal transmission: the right-moving signal field is completely transmitted (T_a = 1), while the left-moving signal field is blocking-up (T_c = 0). (c) The nonreciprocal fast-slow light: both the right-moving field and the left-moving signal field are transmitted (T_a = T_c = 1). However, the group delay of the right-moving signal field is negative (τ_a < 0), that corresponds to the fast light. The group delay of the left-moving signal field is positive (τ_c > 0), that corresponds to the slow light.

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2. Model

Our system model is shown in Fig. 1(a). We consider an optomechanical microtoroid cavity, which supports a clockwise circulating mode (â) and a counter-clockwise circulating mode (ĉ), both the two cavity modes couple with the mechanical mode (b̂) via the radiation pressure. The cavity mode â (ĉ) is driven simultaneously by a strong pump field ε_ap (ε_cp) and a weak signal field ε_as (ε_cs). The total Hamiltonian of the system can be expressed as

H_{total} = H_{om} + H_{aps} + H_{cps} + H_{ac},

where H_om = ħω₀â^†â + ħω₀ĉ^†ĉ + ħω_mb̂^†b̂ + ħg(â^†â + ĉ^†ĉ)(b̂^† + b̂) is the Hamiltonian of the cavity optomechanical system. â(ĉ) and b̂ are the annihilation operators of the clockwise (counter clockwise) circulating cavity mode and the mechanical mode with frequency ω₀ and ω_m, respectively. g is the optomechanical coupling strength between the cavity modes and the mechanical mode. H_aps = iħε_ap(â^†e^−iω_apt − H.c.) + iħε_as(â^†e^−iω_ast − H.c.) describes the interactions of the cavity mode â with the pump field of amplitude

ε_{ap} = \sqrt{2 κ P_{ap} / ℏ ω_{ap}}

and the signal field of amplitude

ε_{as} = \sqrt{2 κ P_{as} / ℏ ω_{as}}

, respectively, in which κ is the coupling decay rate of the cavity, and P_ap (P_as) is the laser power. Similarly, H_cps = iħε_cp(ĉ^†e^−iω_cpt − H.c.) + iħε_cs(ĉ^†e^−iω_cst − H.c.) describes the interaction Hamiltonian of cavity mode ĉ with the pump field of amplitude

ε_{cp} = \sqrt{2 κ P_{cp} / ℏ ω_{cp}}

and the signal field of amplitude

ε_{cs} = \sqrt{2 κ P_{cs} / ℏ ω_{cs}}

, respectively. The last term H_ac = ħJ(â^†ĉ+ ĉ^†â) represents the interaction between the two cavity modes with the strength J.

For simplicity, we assume that the two pump fields have the same frequency, i.e., ω_ap = ω_cp = ω_p. In the rotation frame with H_r = ω_p(â^†â + ĉ^†ĉ), the system Hamiltonian can be written as

\begin{array}{l} H & = & ℏ Δ {\hat{a}}^{†} \hat{a} + ℏ Δ {\hat{c}}^{†} \hat{c} + ℏ ω_{m} {\hat{b}}^{†} \hat{b} + ℏ g ({\hat{a}}^{†} \hat{a} + {\hat{c}}^{†} \hat{c}) ({\hat{b}}^{†} + \hat{b}) \\ + i ℏ ε_{ap} ({\hat{a}}^{†} - H . c .) + i ℏ ε_{as} ({\hat{a}}^{†} e^{- i δ_{as} t} - H . c .) \\ + i ℏ ε_{cp} ({\hat{c}}^{†} - H . c .) + i ℏ ε_{cs} ({\hat{c}}^{†} e^{- i δ_{cs} t} - H . c .) \\ + ℏ J ({\hat{a}}^{†} \hat{c} + {\hat{c}}^{†} \hat{a}), \end{array}

where Δ = ω_c − ω_p is the frequency detuning between the cavity field (â, ĉ) and the pump field (ε_ap, ε_cp), and δ_as = ω_as − ω_p (δ_cs = ω_cs − ω_p) is the frequency detuning between the signal field ε_as (ε_cs) and the pump field ε_ap (ε_cp). The system dynamics is fully described by the set of quantum Heisenberg-Langevin equations

\begin{array}{l} \frac{d \hat{a}}{d t} & = & - (i Δ + κ_{t}) \hat{a} - i g \hat{a} ({\hat{b}}^{†} + \hat{b}) - i J \hat{c} + ε_{ap} + ε_{as} e^{- i δ_{as} t} + \sqrt{2 κ} {\hat{a}}_{in}, \\ \frac{d \hat{c}}{d t} & = & - (i Δ + κ_{t}) \hat{c} - i g \hat{c} ({\hat{b}}^{†} + \hat{b}) - i J \hat{a} + ε_{cp} + ε_{c s} e^{- i δ_{cs} t} + \sqrt{2 κ} {\hat{c}}_{in}, \\ \frac{d \hat{b}}{d t} & = & - (i ω_{m} + γ) \hat{b} - i g ({\hat{a}}^{†} \hat{a} + {\hat{c}}^{†} \hat{c}) + \sqrt{2 γ} {\hat{b}}_{in}, \end{array}

where the cavity has the damping rate κ_t = κ_in + κ, which are assumed to be due to the intrinsic photon loss and external coupling loss, respectively, and the mechanical mode has the damping rate γ. â_in (ĉ_in) and b̂_in are the δ-correlated operators of the input noises for the cavity mode â (ĉ) and the mechanical mode b̂, respectively. These noise operators satisfy 〈â_in〉 = 〈ĉ_in〉 = 〈b̂_in〉 = 0.

In this model, we are interested in the mean response of the system. Thus, in the following, we turn to calculate the evolutions of the expectation values of â, ĉ, b̂, and we denote 〈â〉 ≡ A, 〈ĉ〉 ≡ C, 〈b̂〉 ≡ B, 〈â^†〉 ≡ A^*, 〈ĉ^†〉 ≡ C^*, 〈b̂^†〉 ≡ B^*. By using the mean-field assumption 〈âb̂ĉ〉 = 〈â〉〈b̂〉〈ĉ〉, we can write the equations for the mean values as

\begin{array}{l} \frac{d A}{d t} & = & - (i Δ + κ_{t}) A - i g A (B + B^{*}) - i J C + ε_{ap} + ε_{as} e^{- i δ_{as} t}, \\ \frac{d C}{d t} & = & - (i Δ + κ_{t}) C - i g C (B + B^{*}) - i J A + ε_{cp} + ε_{cs} e^{- i δ_{cs} t}, \\ \frac{d B}{d t} & = & - (i ω_{m} + γ) B - i g ({| A |}^{2} + {| C |}^{2}) . \end{array}

Equations (4) can be solved by using the perturbation method in the limit of the strong pump fields, while taking the signal fields to be weak. Using the linearization approximation, we make the following ansatz [29]

X = X_{0} + X_{a +} e^{- i δ_{as} t} + X_{a -} e^{i δ_{as} t} + X_{c +} e^{- i δ_{c s} t} + X_{c -} e^{i δ_{c s} t},

where X can be any one of the quantities A, B, C, or their complex conjugates A^*, C^*, B^*. X₀ represents the steady-state mean value of the corresponding system mode, and X_a+, X_a−, X_c+, X_c− are the additional fluctuations. By substituting Eq. (5) into Eqs. (4), and keeping only the first-order in the small quantities and neglecting the nonlinear terms like A_a+C_a+, A_a+B_c−, B_c−C_a+, · · · , we can obtain the steady-state mean value equations, and the fluctuation equations for the cavity mode components A_a+ and C_c+ (see the appendix). By solving these equations, we find that A_a+ = η(δ_as)ε_as, C_c+ = ξ(δ_cs)ε_cs, the concrete forms of the coefficients η(δ_as) and ξ(δ_cs) are tediously long, and we will not write them out here.

The relation among the input, internal, and output fields is given as [30] X^out = Xⁱⁿ − 2κX. By using the ansatz again, we write the output field X^out as $X_{0}^{out} + X_{a +}^{out} e^{- i δ_{as} t} + X_{a -}^{out} e^{i δ_{as} t} + X_{c +}^{out} e^{- i δ_{cs} t} + X_{c -}^{out} e^{i δ_{cs} t}$ . Then we can obtain the output field components $A_{a +}^{out} = ε_{as} - 2 κ A_{a +}$ and $C_{c +}^{out} = ε_{cs} - 2 κ C_{c +}$ . The transmissivities can be written as $t_{a} (δ_{as}) = A_{a +}^{out} / ε_{as}$ , $t_{c} (δ_{cs}) = C_{c +}^{out} / ε_{cs}$ . The nonreciprocal transmission is then described by the normalized transmissivities (transmission spectra)

\begin{array}{l} T_{a} & = & {| t_{a} (δ_{as}) |}^{2} = {| 1 - 2 κ η (δ_{as}) |}^{2}, \\ T_{c} & = & {| t_{c} (δ_{cs}) |}^{2} = {| 1 - 2 κ ξ (δ_{cs}) |}^{2} . \end{array}

What’s more, in the resonant region of the transmission spectra, the output signal fields have the phase dispersions ϕ_a(ω_as) = arg[T_a(ω_as)] and ϕ_c(ω_cs) = arg[T_c(ω_cs)], which can cause the group delay [31]

τ_{a} = \frac{d ϕ_{a} (ω_{as})}{d ω_{as}}, τ_{c} = \frac{d ϕ_{c} (ω_{cs})}{d ω_{cs}} .

The group delay τ_a (τ_c) > 0 corresponds to the slow light propagation of the signal field, and the group delay τ_a (τ_c) < 0 corresponds to the fast light propagation of the signal field. In the following, we will discuss the nonreciprocal transmission (T_a = 1, T_c = 0 or T_a = 0, T_c = 1) and the nonreciprocal fast-slow light effects (τ_a > 0, τ_c < 0 or τ_a < 0, τ_c > 0), respectively.

In this paper, the parameters are chosen based on the recently experiment [32, 33]: ω_m = 2π × 10 MHz and γ = 2π × 10² Hz (quality factor Q_m = 10⁵), the equivalent mass of the mechanical resonance m = 5 ng, and the equivalent cavity length l = 1 mm. The damping rate of the optical cavity κ = 2π × 1 MHz, the wavelength of the pump laser λ = 1064 nm. The other parameters are J = 2π × 10³ Hz, κ_in = 2π × 1 MHz.

3. Nonreciprocal transmission

In this section, we numerically evaluate the transmission spectra T_a and T_c to show the possibility of achieving the nonreciprocal transmission of the signal fields.

Firstly, we assume that the system works near the red sideband (Δ = ω_m). In Fig. 2 we plot T_a and T_c as a function of δ_as/ω_m and δ_cs/ω_m, respectively. Here we hold the pump power P_c constant, P_c = 100nW [34]. We can see that the transmission of the left-moving signal field is simply that of a bare resonator, and T_c exhibits a dip near the point δ_cs/ω_m = 1 (T_c ≈ 0).

Fig. 2 The transmission spectra T_a (red solid lines) and T_c (blue dashed lines) as a function of δ_as/ω_m and δ_cs/ω_m, respectively. The system works near the red sideband (Δ = ω_m). The parameters are: (a) P_a = P_c, (b) P_a = 10²P_c, (c) P_a = 10⁴P_c, (d) P_a = 10⁵P_c. The other parameters are stated in the text.

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By adjusting the pump power P_a, we find that the transmission of the right-moving signal field can be obviously modified. Near the point δ_as/ω_m = 1, T_a will exhibit a very narrow peak and gradually increase with the increase of P_a. When P_a = 10⁴P_c, we have T_a ≈ 1. If we continue to increase P_a, the spectrum will exhibit a split, and this is associated with the normal mode splitting [35]. The above features can result in a nonreciprocal transmission of the signal fields in our system, i.e., the right-moving signal field is completely transmitted, while the left-moving signal field is blocking-up.

Then we consider that the system works near the blue sideband (Δ = −ω_m), and we also choose P_c = 100nW. In Fig. 3, we can see that T_c exhibits a dip near the point δ_cs/ω_m = −1 (T_c ≈ 0). By adjusting the pump power P_a, T_a will exhibit a very narrow peak near the point δ_as/ω_m = −1, with the increase of P_a, the peak value will first increase and then decrease. When P_a = 9.5P_c, we have T_a > 1, T_c < 1, now the right-moving signal field can be amplified, while the left-moving signal field cannot be amplified. When P_a = 10⁴P_c, we have T_a ≈ 1, T_c ≈ 0. Now the system can also be used to realize the nonreciprocal transmission of the signal fields.

Fig. 3 The transmission spectra T_a (red solid lines) and T_c (blue dashed lines) as a function of δ_as/ω_m and δ_cs/ω_m, respectively. The system works near the blue sideband (Δ = −ω_m). The parameters are: (a) P_a = P_c, (b) P_a = 6P_c, (c) P_a = 8.5P_c, (d) P_a = 9.5P_c, (e) P_a = 5 × 10²P_c, (f) P_a = 10⁴P_c. The other parameters are stated in the text.

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In addition, in our system, the transmissive direction of the signal field can be changed by adjusting the ratio of P_a and P_c. For example, when Δ = ω_m, P_a = 100nW and P_c = 10⁴P_a, now the left-moving signal field is completely transmitted while the right-moving signal field is blocking-up.

4. Nonreciprocal fast-slow light effects

In this section, we show how to realize the nonreciprocal fast-slow light propagation of the signal fields, i.e., both the right-moving and left-moving signal fields can be completely transmitted, while the group velocity of the right-moving signal field will be sped up and the left-moving signal field will be slowed down, or vice versa.

In Fig. 4, we plot T_a (T_c) as a function of δ_as/ω_m (δ_cs/ω_m) for different intrinsic photon loss rate κ_in under the unbalanced-pumping conditions (P_a = 10⁵P_c, P_c = 100nW). We find that near the point δ_as/ω_m = −1 the transmission T_a of the right-moving signal field is always approximately equal to 1 for different κ_in. While the transmission T_c of the left-moving signal field can be changed from 0 to 1 with the decrease of κ_in near the point δ_cs/ω_m = −1. In other words, the nonreciprocity in the transmission is weakened with the decrease of κ_in, and when κ_in is small enough the nonreciprocity about the transmission properties will disappear.

Fig. 4 The transmission spectra T_a (red solid lines) and T_c (blue dashed lines) as a function of δ_as/ω_m and δ_cs/ω_m, respectively, for different intrinsic photon loss rate κ_in. The system works near the blue sideband (Δ = −ω_m). The parameters are: (a) κ_in = κ, (b) κ_in = 10⁻¹κ„ (c) κ_in = 10⁻²κ, (d) κ_in = 10⁻³κ. The other parameters are stated in the text.

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However, in this situation (κ_in ≪ κ), the nonreciprocity of system is shown in the group delay properties of the signal fields. In Fig. 5, we plot τ_a and τ_c as a function of δ_as/ω_m and δ_cs/ω_m, respectively. We can see that, in the range of the parameters we considered (we have plotted the transmission spectra T_a and T_c and we can guarantee that T_a ≈ T_c ≈ 1 for all the parameters used in Fig. 5), the group delay of the right-moving signal field is negative near the point δ_as/ω_m = −1 (the group velocity will be sped up), that corresponds to the fast light propagation. While the group delay of the left-moving signal field is positive near the point δ_cs/ω_m = −1 (the group velocity will be slowed down), that corresponds to the slow light propagation. This shows that the system can exhibit a nonreciprocal fast-slow light propagation of the signal fields with the blue-sideband pumping.

Fig. 5 The group delay τ_a (red solid lines) and τ_c (blue dashed lines) as a function of δ_as/ω_m and δ_cs/ω_m, respectively. The system works near the blue sideband (Δ = −ω_m). The parameters are: (a) P_a = 5 × 10⁴P_c, (b) P_a = 1 × 10⁵P_c, (c) P_a = 2 × 10⁵P_c, (d) P_a = 5 × 10⁵P_c. The other parameters are stated in the text.

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Furthermore, we can change the propagation direction of the fast-slow light by adjusting the ratio of of P_a and P_c. For example, in Fig. 5(b), we have τ_a ≈ −0.3μs and τ_c ≈ 0.3μs. However, if we choose P_a = 100nW and P_c = 10⁵P_a, then we have τ_a ≈ 0.3μs and τ_c ≈ −0.3μs, now the right-moving signal field is slow light and the left-moving signal field is fast light.

In addition, we have also considered the nonreciprocal fast-slow light propagation of the signal fields when the system works near the red-sideband, and we plot the transmission spectrum T_a (T_c) and group delay τ_c (τ_c) as a function of δ_as/ω_m(δ_cs/ω_m), respectively (not shown in this paper). We find that for the parameters we used in this system, the nonreciprocal fast-slow light propagation phenomenon will not happen with the red-sideband pumping.

5. Conclusion

In summary, we have studied the nonreciprocal transmission and the fast-slow light effects in a cavity optomechanical system. We have shown that, for both the red-sideband pumping or the blue-sideband pumping, the system can act as an optical unidirectional isolator. We have also shown that, if the intrinsic photon loss is much less than the external coupling loss, the nonreciprocity of the system about the optical transmission almost disappears, now the system reveals an interesting nonreciprocal fast-slow light propagation phenomenon with the blue-sideband pumping. Our proposed model might have applications in the photonic network.

Appendix

By substituting Eq. (5) into Eqs. (4), we can obtain the the steady-state mean value equations

\begin{array}{l} 0 = - (i Δ + κ_{t}) A_{0} - i g A_{0} (B_{0} + B_{0}^{*}) - i J C_{0} + ε_{ap}, \\ 0 = - (i Δ + κ_{t}) C_{0} - i g C_{0} (B_{0} + B_{0}^{*}) - i J A_{0} + ε_{cp}, \\ 0 = - (i ω_{m} + γ) B_{0} - i g ({| A_{0} |}^{2} + {| C_{0} |}^{2}) . \end{array}

In this system, we are interested on the dynamics of the cavity mode components A_a+e^−iδ_ast and C_c+e^−iδ_cst which are resonance with the corresponding signal fields ε_ase^−iδ_ast and ε_cse^−iδ_cst, respectively. We can obtain

\begin{array}{l} Φ_{a} B_{a +} = - i g (A_{0}^{*} A_{a +} + A_{a +}^{*} A_{0} + C_{0}^{*} C_{a +} + C_{a +}^{*} C_{0}), \\ Ω_{a} A_{a +} = - i g A_{0} (B_{a +} + B_{a +}^{*}) - i J C_{a +} + ε_{as}, \\ Ω_{a} C_{a +} = - i g C_{0} (B_{a +} + B_{a +}^{*}) - i J A_{a +}, \end{array}

and

\begin{array}{l} Φ_{c} B_{c +} = - i g (A_{0}^{*} A_{c +} + A_{c +}^{*} A_{0} + C_{0}^{*} C_{c +} + C_{c +}^{*} C_{0}), \\ Ω_{c} C_{c +} = - i g C_{0} (B_{c +} + B_{c +}^{*}) - i J A_{c +} + ε_{cs}, \\ Ω_{c} A_{c +} = - i g A_{0} (B_{c +} + B_{c +}^{*}) - i J C_{c +}, \end{array}

where Φ_k = i(ω_m − δ_ks) + γ and

Ω_{k} = i [Δ + g (B_{0} + B_{0}^{*}) - δ_{ks}] + κ_{t}

, k = a, c.

Funding

National Natural Science Foundation of China (11574092, 61775062, 61378012, 91121023); National Basic Research Program of China (2013CB921804).

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Nonreciprocal transmission and fast-slow light effects in a cavity optomechanical system

Abstract

1. Introduction

2. Model

3. Nonreciprocal transmission

4. Nonreciprocal fast-slow light effects

5. Conclusion

Appendix

Funding

References

Cited By

Figures (5)

Equations (10)

Optics Express