Fano profile in a novel double cavity optomechanical system with harmonically bound mirrors
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Vaibhav N Prakash1, Aranya B Bhattacherjee,2,31School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067, India 2Department of Physics, Birla Institute of Technology and Science, Pilani, Hyderabad Campus, Hyderabad—500078, India
First author contact:3Author to whom any correspondence should be addressed. Received:2020-01-16Revised:2020-05-9Accepted:2020-05-15Online:2020-08-04
Abstract We investigate a double cavity optomechanical system generating single and double Fano resonance (multi-Fano). By altering a single parameter, the tunnelling rate g of the middle mirror, we are able to switch between single and double Fano line shapes. The first spectral line shape is stronger in the case of multi-Fano than in the case of single Fano. Also the behaviour of the steady state value of the displacement of the middle mirror, with respect to g, heavily influences the behaviour of double Fano lines in our scheme. This tunability along with using a single pump and signal/probe laser has an added advantage in situations where only low power consumption is available. Keywords：optomechanics;double Fano resonance;optical switch;single Fano resonance
PDF (826KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Vaibhav N Prakash, Aranya B Bhattacherjee. Fano profile in a novel double cavity optomechanical system with harmonically bound mirrors. Communications in Theoretical Physics, 2020, 72(9): 095501- doi:10.1088/1572-9494/ab95fe
1. Introduction
Fano resonance was first understood in the context of Rydberg atoms as an interference effect between discrete and continuum states [1]. Since then it has been observed in various physical processes involving bound states inside a continuum in atom–atom scattering. Besides atomic physics, Fano resonances can be found in nuclear physics, plasmonics and cavity optomechanics [2–4]. A similar effect has been discovered in Mie scattering of small particles with negative dielectric susceptibility and weak dissipation rate [5]. Fano resonances are commonly observed when photons travelling through different paths undergo interference. In cavity optomechanics, they occur due to the constructive and destructive interference between two different pathways the photons travel to build the cavity field [6]. The interaction of the discrete optomechanical ground state with broad continuum state in a Λ-type system results in an asymmetric line shape. This asymmetric structure of Fano line shape can then be conveniently used to produce fast all optical switches in communication networks [7–10] as opposed to the conventional switching components having long Lorentzian tails. For similar reasons, Fano resonances can be used in optical sensors as it is very sensitive to changes in refractive index [11, 12]. Fano minima has also been used for state transfer and transduction between microwave and optical photons [13]. From a practical point of view it then becomes important to have greater tunability and efficiency to produce such line shapes so that they can be effortlessly incorporated into quantum devices.
On the other hand there has been a recent surge of interest in double cavity optomechanical system (OMS). This is due to the greater tunability of membrane-in-the-middle devices over single cavity Fabry–Perot systems. Double cavity OMSs have been effectively used for ground state cooling [14, 15], entanglement [16–18] and transduction [19] between microwave and optical photons. Hybrid optomechanical technologies comprising of toroidal shaped whispering gallery mode (WGM) micro-resonators have been successfully used to induce non-reciprocity between communication channels [20, 21], an essential step towards quantum communication.
Here we report on the use of a double cavity OMS with two movable mirrors in producing single and double Fano resonances depending on the value of the tunnelling rate g of the middle mirror/membrane. In our system we have used a single pump and probe to build an optical field inside a Fabry–Perot cavity. The double cavity OMS with two harmonically bound mirrors and single pump and probe lasers seems to be energy efficient from a previously proposed system [6]. Similar results have been achieved in interferometres having mirrors with different mechanical frequencies [22] controlled by mechanical pumps [23]. Another major difference between these models and ours is the dependence on a single parameter, i.e. the photon tunnelling rate g, to produce single and double Fano line-shapes. In our model of double cavity OMS, double Fano line occurs due to the simultaneous coupling of mirrors with the radiation pressure in both the cavities. We assume that this coupling occurs at the natural frequency of the mirrors and is achieved by detuning the pump field with respect to the cavity frequencies of both the cavities, simultaneously. We show that the strength of the second Fano line in the double Fano case depends on the tunnelling rate g of the middle mirror. The effect of changing g on transition between single and double Fano resonances is also considered. After giving a short description of the system Hamiltonian and establishing the Langevin equations in section 2, we plot single and double Fano resonances in section 3. In section 4 we describe the behaviour of the two Fano line-shapes in double Fano case, w.r.t. g. This is followed by a discussion and conclusion in section 5.
2. Basic model
We start with a scheme which we have previously discussed in [24], shown in Figure 1. Cavity A is driven by an intense pump/control laser of frequency ωc and has an average photon number ${\overline{n}}_{a}=\langle {a}^{\dagger }a\rangle $ , where a (a†) is the annihilation (creation) operator of optical mode confined in cavity A.
Figure 1.
New window|Download| PPT slide Figure 1.Schematic diagram of double cavity system. The pump and probe input strengths are εc and εp respectively. κ1 and κ2 are the decay rates of cavity A and B respectively. Cavity B is separated from A by the middle mirror M1 whose transmitivity can be altered. On the other hand mirror M2 has zero transmitivity. The output aout is measured on the same side as the input, the measured quantity being proportional to the reflection coefficient Tb.
The transparency of the middle mirror allows for the tunnelling of photons with rate g into cavity B which has photons with average photon number ${\overline{n}}_{b}=\langle {b}^{\dagger }b\rangle $ where b (b†) is the annihilation (creation) operator of optical mode confined in cavity B. This leads to the coupling of the two mirrors M1 and M2 via radiation pressure forces from the cavity fields. In a frame rotating with frequency ωc, the system Hamiltonian can be written as,$ \begin{eqnarray}\displaystyle \begin{array}{rcl}\hat{H} & = & \sum _{j=1}^{2}\left(\displaystyle \frac{{\hat{p}}_{j}^{2}}{2{m}_{j}}+\displaystyle \frac{1}{2}{m}_{j}{{\rm{\Omega }}}_{j}^{2}{\hat{x}}_{j}^{2}\right)-{\hslash }{{\rm{\Delta }}}_{1}{\hat{a}}^{\dagger }\hat{a}-{\hslash }{{\rm{\Delta }}}_{2}{\hat{b}}^{\dagger }\hat{b}\\ & & -{\hslash }{G}_{1}{\hat{x}}_{1}{\hat{a}}^{\dagger }\hat{a}-{\hslash }{G}_{2}({\hat{x}}_{2}-{\hat{x}}_{1}){\hat{b}}^{\dagger }\hat{b}\\ & & +{\hslash }g({\hat{a}}^{\dagger }\hat{b}+\hat{a}{\hat{b}}^{\dagger })+{\rm{i}}{\hslash }\sqrt{\eta \kappa }{\epsilon }_{c}({\hat{a}}^{\dagger }-\hat{a})\\ & & +{\rm{i}}{\hslash }\sqrt{\eta \kappa }{\epsilon }_{p}({\hat{a}}^{\dagger }{{\rm{e}}}^{-{\rm{i}}{\rm{\Omega }}t}-\hat{a}{{\rm{e}}}^{{\rm{i}}{\rm{\Omega }}t}),\end{array}\end{eqnarray}$where m1 and m2 are the ‘bare’ masses of mirrors M1 and M2 respectively. Here, ωj (j=1, 2) is the mechanical frequency of the oscillator Mj. In our scheme both the movable mirrors vibrate with the same mechanical frequency i.e. ω1=ω2=ωm. We assume that the field in cavity B couples with equal strength to both mirrors M1 and M2. Also ${\hat{x}}_{1}$, ${\hat{x}}_{2}$ and ${\hat{p}}_{1}$, ${\hat{p}}_{2}$ are the position and momentum operators following the commutation relations, $[{\hat{x}}_{j},{\hat{p}}_{j}]={\rm{i}}{\hslash }$ (j=1, 2) for mirrors M1 and M2. G1,2 denotes the cavity frequency shift per resonator displacement for M1,2. It is related to single-photon optomechanical coupling strength by, ${g}_{{0}_{\mathrm{1,2}}}={G}_{\mathrm{1,2}}{x}_{\mathrm{zp}}$. Here ${x}_{\mathrm{zp}}$ is the standard deviation of the zero point motion of the oscillator. Thus ${G}_{1}=\tfrac{{\omega }_{1}}{{L}_{1}}$ and ${G}_{2}=\tfrac{{\omega }_{2}}{{L}_{2}}$, where ω1 (ω2) are resonant frequencies of cavity A (B) and L1,2 are the corresponding cavity lengths. It is to be noted that G1,2 can be made very large as the effective length can be made very small [25]. Cavity A is driven by an external input laser field consisting of a strong control field and weak probe field denoted by ${a}_{\mathrm{in}}(t)={\epsilon }_{c}{{\rm{e}}}^{-{\rm{i}}{\omega }_{c}t}+{\epsilon }_{p}{{\rm{e}}}^{-{\rm{i}}{\omega }_{p}t}$ with field strengths εc and εp and frequencies ωc and ωp respectively. The field strengths are given as ${\epsilon }_{c}=\sqrt{{P}_{c}/{\hslash }{\omega }_{c}}$ and ${\epsilon }_{p}=\sqrt{{P}_{p}/{\hslash }{\omega }_{p}}$ where Pc and Pp are the control and probe field powers, respectively. Without loss of generality we assume that εc and εp are real. κex is the external decay rate between the i/o system and the optical cavity while κ is the total decay rate. η=κex/κ is the coupling coefficient and is fixed in the critical coupling regime (η=0.5) [26, 27]. Here ${{\rm{\Delta }}}_{j}={\omega }_{c}-{\omega }_{j}$ (j=1, 2), is the cavity detuning of cavity A (j=1) and B (j=2) while ${\rm{\Omega }}={\omega }_{p}-{\omega }_{c}$ is the detuning of the probe field with respect to the control field frequency ωc. In our semi-classical description the operators are replaced by c-numbers while the noise terms are excluded. The Hamiltonian in equation (1) gives rise to the following Heisenberg Langevin equation,$ \begin{eqnarray}\,\dot{a}={\rm{i}}({{\rm{\Delta }}}_{1}+{G}_{1}{x}_{1})a-{\rm{i}}{gb}+\sqrt{\eta \kappa }{\epsilon }_{c}+\sqrt{\eta \kappa }{\epsilon }_{p}{{\rm{e}}}^{-{\rm{i}}{\rm{\Omega }}t}-\displaystyle \frac{\kappa }{2}a,\end{eqnarray}$$ \begin{eqnarray}\dot{b}={\rm{i}}({{\rm{\Delta }}}_{2}+{G}_{2}[{x}_{2}-{x}_{1}])b-{\rm{i}}{ga}-\displaystyle \frac{\kappa }{2}b,\,\end{eqnarray}$$ \begin{eqnarray}{\dot{x}}_{j}=\displaystyle \frac{{p}_{j}}{{m}_{j}},\,\end{eqnarray}$$ \begin{eqnarray}{\dot{p}}_{1}=-{m}_{1}{{\rm{\Omega }}}_{1}^{2}{x}_{1}-{\hslash }{G}_{2}{b}^{\dagger }b+{\hslash }{G}_{1}{a}^{\dagger }a-\displaystyle \frac{{\gamma }_{1}}{2}{p}_{1},\,\end{eqnarray}$$ \begin{eqnarray}{\dot{p}}_{2}=-{m}_{2}{{\rm{\Omega }}}_{2}^{2}{x}_{2}+{\hslash }{G}_{2}{b}^{\dagger }b-\displaystyle \frac{{\gamma }_{2}}{2}{p}_{2},\,\end{eqnarray}$where γ1,2 are the decay rates of oscillations of mirror M1,2 respectively. All decay terms in the above equations can be derived using the standard master equation. We do not elaborate on the master equation approach here since it becomes important only if internal system dynamics is of interest.
3. Fano resonances
Fano resonance in a cavity OMS arises due to the interference between photons from to two coherent processes, the direct photon absorption and the indirect Raman process. For simplicity we consider a Λ-type system as shown in figure 2. Here, ω is the cavity resonance frequency and ωc is the frequency of the strong pump laser. The oscillating mirror with natural frequency ωm creates sidebands (Stokes and anti-Stokes) on the pump laser via Raman scattering. When the pump is red-detuned from the cavity frequency (${\rm{\Delta }}={\omega }_{c}-\omega \approx -{{\rm{\Omega }}}_{m}$) the anti-Stokes sideband becomes resonant with the cavity frequency. This leads to the coherent exchange between photons and phonons. The state $| f\rangle $ coherently interacts with state $| e\rangle $ as depicted in figure 2. Further when the weak probe laser is detuned from ωc such that it is near resonance with the cavity frequency ω, photons can be directly injected into the cavity. This is labelled as pathway 1 in figure 2. Some of these photons from probe laser can then be stimulated down to state $| f\rangle $ to produce phonons with frequency ωm and then back to the excited state $| e\rangle $ with the help of the strong pump laser. This is labelled as pathway 2 in figure 2. The cavity field is thus built through these two coherent processes or two different pathways. Just like the two slit experiment, since there is no way of knowing which slit the photons have come out, here there is no way of knowing which pathway the photons have traversed. This leads to the interference between cavity photons. δ in figure 2, is the difference between the resonances of the probe field, ${\omega }_{p}\approx \omega $ and ${\omega }_{p}\approx {\omega }_{c}+{{\rm{\Omega }}}_{m}$ [6]. It is a parameter which controls this interference. A complete destructive interference (δ=0) gives rise to optomechanically induced transparency (OMIT) while in general when $\delta \ne 0$, a partial constructive and destructive interference occurs between cavity photons and causes asymmetric Fano profiles.
Figure 2.
New window|Download| PPT slide Figure 2.Schematic diagram of photon pathways in a Λ-type system. Pathway 1 is the direct photon absorption at cavity frequency ω and pathway 2 is the indirect or Raman induced photon absorption due to detuning of pump laser by the mechanical frequency ωm. δ is the difference between the resonances of the probe field at frequency ωp.
3.1. Single Fano
In cavity optomechanics, a Λ-type system shown in figure 2 can be formed using either single or multiple cavities separated by dielectric membranes (or partially transmitting mirrors). It has been shown [6] that single Fano resonance can even occur in a single Fabry–Perot cavity provided that the anti-Stokes Raman field is offset from the cavity resonance frequency such that their is a difference ωL (or δ in our case) between the probe field and the anti-Stokes field. This gives rise to partial constructive and destructive interference causing asymmetric Fano line-shape. In a double cavity OMS too we can have single Fano provided that both the end mirrors are fixed. In that case the Hamiltonian in equation (1) will be modified;$ \begin{eqnarray}\begin{array}{rcl}\hat{H} & = & \left(\displaystyle \frac{{\hat{p}}_{1}^{2}}{2{m}_{1}}+\displaystyle \frac{1}{2}{m}_{1}{{\rm{\Omega }}}_{1}^{2}{\hat{x}}_{1}^{2}\right)-{\hslash }{{\rm{\Delta }}}_{1}{\hat{a}}^{\dagger }\hat{a}-{\hslash }{{\rm{\Delta }}}_{2}{\hat{b}}^{\dagger }\hat{b}\\ & & -{\hslash }{G}_{1}{\hat{x}}_{1}{\hat{a}}^{\dagger }\hat{a}+{G}_{2}{\hat{x}}_{1}{\hat{b}}^{\dagger }\hat{b}\\ & & +{\hslash }g({\hat{a}}^{\dagger }\hat{b}+\hat{a}{\hat{b}}^{\dagger })+{\rm{i}}{\hslash }\sqrt{\eta \kappa }{\epsilon }_{c}({\hat{a}}^{\dagger }-\hat{a})\\ & & +{\rm{i}}{\hslash }\sqrt{\eta \kappa }{\epsilon }_{p}({\hat{a}}^{\dagger }{{\rm{e}}}^{-{\rm{i}}{\rm{\Omega }}t}-\hat{a}{{\rm{e}}}^{{\rm{i}}{\rm{\Omega }}t}).\end{array}\end{eqnarray}$Heisenberg Langevin equations (2a)–(2e) will also be modified according to equation (3);$ \begin{eqnarray}\,\dot{a}={\rm{i}}({{\rm{\Delta }}}_{1}+{G}_{1}{x}_{1})a-{\rm{i}}{gb}+\sqrt{\eta \kappa }{\epsilon }_{c}+\sqrt{\eta \kappa }{\epsilon }_{p}{{\rm{e}}}^{-{\rm{i}}{\rm{\Omega }}t}-\displaystyle \frac{\kappa }{2}a,\end{eqnarray}$$ \begin{eqnarray}\,\dot{b}={\rm{i}}({{\rm{\Delta }}}_{2}-{G}_{2}{x}_{1})b-{\rm{i}}{ga}-\displaystyle \frac{\kappa }{2}b,\,\end{eqnarray}$$ \begin{eqnarray}{\dot{x}}_{1}=\displaystyle \frac{{p}_{1}}{{m}_{1}},\,\end{eqnarray}$$ \begin{eqnarray}{\dot{p}}_{1}=-{m}_{1}{{\rm{\Omega }}}_{1}^{2}{x}_{1}-{\hslash }{G}_{2}{b}^{\dagger }b+{\hslash }{G}_{1}{a}^{\dagger }a-\displaystyle \frac{{\gamma }_{1}}{2}{p}_{1},\,\end{eqnarray}$where x1 is the displacement of the middle mirror/membrane. The above equations are almost similar to the ones in [26], where a toroidial WGM microresonator was studied instead of a Fabry–Perot cavity. Using standard mean field and linearization approach [28] and breaking the above c-numbers (equations (4a)–(4d)) into their respective Fourier components (Y = ${\sum }_{n=\mathrm{1,2}}{S}_{n}{{\rm{e}}}^{\pm {\rm{i}}{\rm{\Omega }}t}$) (appendix), we arrive at the anti-stokes field in cavity A:$ \begin{eqnarray}{A}_{1}^{-}=-\displaystyle \frac{({{gG}}_{2}| \overline{b}| {q}_{1}+{\rm{i}}{D}_{3}{G}_{1}| \overline{a}| {q}_{1}+{D}_{3}\sqrt{\eta \kappa }{\epsilon }_{p})}{{D}_{1}{D}_{3}+{g}^{2}},\end{eqnarray}$where ${D}_{1}={{\rm{\Theta }}}_{1}+{\rm{i}}{\rm{\Omega }}$, ${D}_{2}={{\rm{\Theta }}}_{1}-{\rm{i}}{\rm{\Omega }}$, ${D}_{3}={{\rm{\Theta }}}_{2}+{\rm{i}}{\rm{\Omega }}$, ${D}_{4}\,={{\rm{\Theta }}}_{2}-{\rm{i}}{\rm{\Omega }}$ and ${{\rm{\Theta }}}_{j}={\rm{i}}{\overline{{\rm{\Delta }}}}_{j}-\kappa /2,(j=1,2)$. Since we are using a single input–output port, we can measure only the backward reflection coefficient of the input probe field. The standard input–output relation leads to;$ \begin{eqnarray}{a}_{\mathrm{out}}={C}_{\mathrm{cb}}{{\rm{e}}}^{-{\rm{i}}{\omega }_{c}t}+{C}_{\mathrm{pb}}{{\rm{e}}}^{-{\rm{i}}{\omega }_{p}t}-\sqrt{\eta \kappa }{A}_{1}^{+}{{\rm{e}}}^{-{\rm{i}}(2{\omega }_{c}-{\omega }_{p})t},\end{eqnarray}$where ${C}_{\mathrm{cb}}={\epsilon }_{c}-\sqrt{\eta \kappa }$$\overline{a}$, ${C}_{\mathrm{pb}}={\epsilon }_{p}-\sqrt{\eta \kappa }{A}_{1}^{-}$, are the complex coefficients for steady state and backward reflection, respectively. Using ${C}_{\mathrm{pb}}$ we calculate the normalised backward reflection coefficient, ${T}_{b}=| {C}_{\mathrm{pb}}/{\epsilon }_{p}{| }^{2}$ (appendix, equation (A2)). Since the pump laser is red-detuned by the mechanical frequency ωm, from equation (6), the Stokes field, with coefficient ${A}_{1}^{+}$, will be off-resonant by $2{\omega }_{c}-{\omega }_{p}$ from the cavity. Hence we do not show it here but it is straightforward to calculate. Figure 3 gives the normalised backward reflection coefficient Tb for different values of tunnelling rate g and ω/ωm ∈ [0.98, 1.02]. This is done to emphasise changes in the complete destructive interference (leading to OMIT) of the cavity field to a constructive and destructive interference (leading to asymmetric Fano line shapes). OMIT occurs at g/ωm=0 (figure 3(a)) since both the probe and the anti-Stokes Raman field completely destructively interfere at cavity resonance. For 0.0<g/ωm≤0.4 (figures 3(b)–(e)) Tb intensity changes gradually with peak around 0.9. For g/ωm∼0.6 (figure 3(f)) the peak drastically increases to around 1.0.
Figure 3.
New window|Download| PPT slide Figure 3.Backward reflection coefficient for membrane-in-middle resonator with fixed end mirrors. m1=20 ng, Pc=1 mW, G1=2π×13 GHz nm−1, G2=2π×13 GHz nm−1, γ1=γ2=2π×41 kHz, κ=2π×15 MHz, Δ2=Δ1=−ωm and ω1=ωm=2π×51.8 MHz.
3.2. Multi-Fano
Fano profiles depend on the number of dressed states and resonances of the probe laser. For example a single dressed state with single resonance gives rise to a single OMIT. On the other hand the occurrence of a double OMIT has been recently shown [29] in a piezo-mechanical system coupled with an OMS. The piezo-mechanical coupling, forms two dressed states with two resonances, changing a single OMIT into a double OMIT. Now a shift in the detuning of the coupling laser by an amount δ will give rise to more number of resonances than dressed states leading to asymmetric Fano profiles. Here we specifically show the existence of double asymmetric Fano-line shapes in a double cavity OMS. The system dynamics are defined by equations (2a)–(2e) and following the same procedure given in section 3.1, the anti-stokes field component ${A}_{1}^{-}$ is given as;$ \begin{eqnarray}{A}_{1}^{-}=\displaystyle \frac{{{gG}}_{2}({q}_{2}-{q}_{1})\overline{b}-{\rm{i}}{D}_{3}{G}_{1}{q}_{1}\overline{a}-{D}_{3}\sqrt{\eta \kappa }}{{D}_{1}{D}_{3}+{g}^{2}}.\end{eqnarray}$The normalised backward reflection coefficient ${T}_{b}=| {C}_{\mathrm{pb}}/{\epsilon }_{p}{| }^{2}$ as defined below equation (6), is given in appendix (equation (A4)). We plot Tb in figures 4(a)–(f). The x-axis range here is again ω/ωm ∈ [0.98, 1.02] so as to emphasise changes in Fano profiles. As in figure 3(a), in figure 4(a) we have a single OMIT structure at g/ωm=0.0. This system does not show double OMIT since at g/ωm=0.0 the system turns into a single cavity OMS with complete destructive interference of the Raman anti-stokes and the probe field at cavity resonance with response function (${\epsilon }_{T}=\sqrt{\eta \kappa }{A}_{1}^{-}/{\epsilon }_{p}$) of the OMS being,$ \begin{eqnarray}{\epsilon }_{T}=\displaystyle \frac{\eta \kappa }{\tfrac{\kappa }{2}-{\rm{i}}y+\tfrac{\beta }{\tfrac{{\gamma }_{m}}{2}-{\rm{i}}y}}.\end{eqnarray}$We note that instead of having a large parameter space, we have fixed all the parameters except for the tunnelling rate g. This makes the system experimentally easy to access. Comparing figures 3 and 4 we see that the Fano line-shapes are roughly similar upto around g/ωm=0.2. At around g/ωm=0.3 we observe the first signs of single Fano going into double Fano. At g/ωm=0.4 we clearly see the separation of the two distinct asymmetric Fano line-shapes. The first Fano line-shape for double Fano case is more sharp than that for the single Fano case (figure 3). This means easier switching when implemented in all-optical switches. We also note that the difference between strength of dips (y-axis) for the two Fano lines remain more or less constant with increasing g. Hence a sharper first Fano line also means a sharper second Fano line.
Figure 4.
New window|Download| PPT slide Figure 4.Backward reflection coefficient with middle and one end mirror not fixed. m1=m2=20 ng, Pc=1 mW, G1=2π×13 GHz nm−1, G2=2π×13 GHz nm−1, γ1=γ2=2π×41 kHz, κ=2π×15 MHz, Δ2=Δ1=−ωm and ω1=ω2=ωm=2π×51.8 MHz.
4. Fano profile behaviour
Next we try to model the behaviour of the double Fano profile. Using analytical tools (Mathematica), we measure the distance (in terms of ω/ωm) between the two Fano line-shapes in the backward reflection (Tb) plot. The dependency of the measured separation values between the two line-shapes is plotted with respect to g for 0.4≤g/ωm≤1. We choose this interval due to prominence of the weaker line-shape. For comparison, in figure 5(a), we also plot scaled up steady state values ${\overline{x}}_{1}$ and ${\overline{x}}_{2}$ for displacements of mirrors M1 and M2 respectively:$ \begin{eqnarray}{\overline{x}}_{1}=\displaystyle \frac{{\hslash }({G}_{1}| \overline{a}{| }^{2}-{G}_{2}| \overline{b}{| }^{2})}{{m}_{1}{{\rm{\Omega }}}_{1}^{2}},\end{eqnarray}$$ \begin{eqnarray}{\overline{x}}_{2}=\displaystyle \frac{{\hslash }{G}_{2}| \overline{b}{| }^{2}}{{m}_{2}{{\rm{\Omega }}}_{2}^{2}}.\end{eqnarray}$
Figure 5.
New window|Download| PPT slide Figure 5.(a) Calculated steady state values ${\overline{x}}_{1}$ and ${\overline{x}}_{2}$ scaled up by an order of 1011 compared with measured values (using Mathematica) of the widths (ω/ωm) between Fano line-shapes. (b) Fits of two different functions with the measured values (using Python).
It can be seen in figure 5(a) that the behaviour of measured values closely mimics that of ${\overline{x}}_{1}$. It should be noted that large separation value amounts to well-resolved Fano lines. From figure 5(a) it can be seen that the transmitivity (or tunnelling rate g) of the middle mirror M1 should be relatively high to guarantee well-resolved Fano lines. There is a rapid increase for 0.6≤g/ωm≤0.9 saturating at g/ωm=0.95 before decreasing till g/ωm=1. We use two functions to fit the data as shown in figure 5(b), a generalised logistic function and the Moffat function [30],$ \begin{eqnarray}{Y}_{\mathrm{gen}}(x)=a+\displaystyle \frac{c}{1+T{{\rm{e}}}^{-B{\left(x-M\right)}^{\tfrac{1}{T}}}},\end{eqnarray}$$ \begin{eqnarray}{Y}_{\mathrm{moff}}(x)=A{\left(1+{\left(\displaystyle \frac{x-\mu }{\sigma }\right)}^{2}\right)}^{-\beta },\end{eqnarray}$with appropriate fit parameters. A generalised logistic function is a cumulative function depicting, among many things, diffusion. On the other hand Moffat function is a point spread function used in astrophysics to fit ‘seeing-limited’ images of point sources [31], a blurring of light caused by the Earth’s atmosphere. It is a measure of diffusion of light emanating from a point source registering on a photographic plate after passing through the Earth’s atmosphere. Both the fits give low χ2/d.o.f (degree of freedom) values, of the order of 10−4, but the Moffat function gives a better fit as seen from figure 5(b).
From the fits we can see that in our scheme the separation between the two Fano lines in double a Fano profile is related to the transmission of photons through the middle mirror and mimics a diffusion process. We do not suggest that photon diffusion occurs between cavity A and B but nonetheless it might be useful in future studies to understand the effects of changing g on Fano lines in terms of diffusion.
5. Discussion and conclusion
Here we do not delve into quantitatively understanding the results given in figures 3 and 4 following equations for the normalised backward reflection coefficients, but try to intuitively understand them. We have assumed that the strong pump laser is red-detuned from resonant frequencies of cavity A and B by the same amount ωm i.e. ${{\rm{\Delta }}}_{1}={{\rm{\Delta }}}_{2}\approx -{{\rm{\Omega }}}_{m}$. Since mirror M2 has the same natural frequency ωm as M1, we get another state $| {f}_{2}\rangle $ degenerate with $| {f}_{1}\rangle $ as shown in Figure 6. For non-zero value of the tunnelling rate g, the middle mirror M1 allows photons into cavity B. Just as in the case of single Fano explained in section 3, the simultaneous coupling of photons in cavity A and B with mirrors M1 and M2 gives rise to the shifts δ1 and δ2, respectively. Single OMIT occurs when δ1=δ2=0 i.e. when field in cavity A couples to mirror M1 (g/ωm=0). Figure 7 gives the strength of fields in both the cavities. At lower values of the tunnelling rate g (i.e. g/ωm∼0.2), the field strength in cavity B is much smaller than that in cavity A. This means that it cannot substantially affect the motion of mirror M2 via radiation pressure. Thus the coupling of state $| {f}_{2}\rangle $ with state $| e\rangle $ is practically non-existent. ${\delta }_{1}\ne 0$ but δ2=0, implying only single Fano. Hence Fano profiles in figures 3 and 4 look pretty similar upto around g/ωm=0.2. Once the value of g/ωm≥0.3, the optical field in cavity B becomes strong enough to influence motion of M2 and hence ${\delta }_{2}\ne 0$. This might be the reason we start observing a shift from single to double Fano around g/ωm=0.3 in figure 4. The line-shapes become prominent for g/ωm≥0.4 since the field strengths in cavity A and B increases with g.
Although ideally cavity A and B should contain the same number of photons at $g/{{\rm{\Omega }}}_{m}=1.0$, the discrepancies in field strengths at g/ωm=1.0 as shown in figure 7 might be due to approximations (for e.g. truncation of higher order optomechanical coupling, mean field approach and rotating wave approximation) used in calculating these values. These are good approximations for large cavity field strength but they do have their short comings. For instance, in single-photon optomechanics nonlinear photon–phonon interaction becomes important and one can no longer assume linear response of oscillating mirrors to solve the equations of motion [32]. It is to be noted that in our study the single photon optomechanical coupling ${g}_{{0}_{\mathrm{1,2}}}\lt {\kappa }_{1.2}$ and hence still in the weak optomechanical regime. In the single photon strong coupling regime one has to include higher order expansion of cavity frequency ω(x), leading to higher order optomechanical coupling rate.
In conclusion, we have successfully shown the presence of double Fano profiles in our scheme. We have also shown that the same scheme can be used to generate single and double Fano lines-shapes by simply controlling the tunnelling rate g of the middle mirror M1. The behaviour of the separation between the line-shapes closely follows the behaviour of the steady state displacement of M1 and mimics a diffusion process. The first Fano line has a larger dip in the multi-Fano case than in the single Fano for the same value of g/ωm. The double Fano line-shapes also become resolved for $0.6\leqslant g/{{\rm{\Omega }}}_{m}\leqslant 0.95$. Hence our scheme can be implemented for sensitive devices in need of sharp spectral lines like in all-optical switches and quantum sensors.
,2,31School of Physical Sciences, 
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