Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay

As illustrated in figure 1(a), we study the transmission rates of the probe field and the second-order sideband in a single active optomechanical cavity [37]. The Hamiltonian of the system can be written at the simplest level as

$$\begin{eqnarray}&&{H}_{1}={\hslash }{{\rm{\Delta }}}_{{\rm{l}}}{a}^{\dagger }a+\displaystyle \frac{{p}^{2}}{2m}+\displaystyle \frac{1}{2}m{\omega }_{{\rm{m}}}^{2}{x}^{2}-{\hslash }{{ga}}^{\dagger }{ax}+{\rm{i}}{\hslash }({\varepsilon }_{{\rm{l}}}{a}^{\dagger }+{\varepsilon }_{{\rm{p}}}{a}^{\dagger }{{\rm{e}}}^{-{\rm{i}}\xi t}-{\rm{h.c.}}),\end{eqnarray} \tag{ 1 }$$

with the optical operator a and the mechanical position or momentum operator x or p, and the cavity–pump detuning and the probe–pump detuning, respectively, ${{\rm{\Delta }}}_{{\rm{l}}}\equiv {\omega }_{{\rm{c}}}-{\omega }_{{\rm{l}}},\,\,{{\rm{\Delta }}}_{{\rm{p}}}\equiv {\omega }_{{\rm{p}}}-{\omega }_{{\rm{c}}},\,\,\xi \equiv {\omega }_{{\rm{p}}}-{\omega }_{{\rm{l}}}$. The Heisenberg equations of motion of this system are then

$$\begin{eqnarray}\ddot{x} & + & {{\rm{\Gamma }}}_{{\rm{m}}}\dot{x}+{\omega }_{{\rm{m}}}^{2}x-\displaystyle \frac{{\hslash }g}{m}{a}^{\dagger }a=0,\\ \dot{a} & + & ({\rm{i}}{{\rm{\Delta }}}_{{\rm{l}}}-{\rm{i}}{gx}-\kappa )a-{\varepsilon }_{{\rm{l}}}={\varepsilon }_{{\rm{p}}}{{\rm{e}}}^{-{\rm{i}}\xi t}.\end{eqnarray} \tag{ 2 }$$

The optical or mechanical gain and damping terms are added phenomenologically into the equation (2) [7, 33]. Setting all the derivatives as zero leads to the steady-state values

$$\begin{eqnarray}&&{x}_{{\rm{s}}}=\displaystyle \frac{{\hslash }g}{m{\omega }_{{\rm{m}}}^{2}}{| {a}_{{\rm{s}}}| }^{2},\,\,\,{a}_{{\rm{s}}}=\displaystyle \frac{{\varepsilon }_{{\rm{l}}}}{-\kappa +{\rm{i}}{{\rm{\Delta }}}_{{\rm{l}}}-{\rm{i}}{{gx}}_{{\rm{s}}}}.\end{eqnarray} \tag{ 3 }$$

All the operators are then divided into their stationary values and the remaining fluctuations, i.e. $x={x}_{{\rm{s}}}+\delta x,a={a}_{{\rm{s}}}+\delta a$, and the equations satisfied by the fluctuation terms are

$$\begin{eqnarray}\delta \ddot{x} & + & {{\rm{\Gamma }}}_{{\rm{m}}}\delta \dot{x}+{\omega }_{{\rm{m}}}^{2}\delta x-\displaystyle \frac{{\hslash }g}{m}({a}_{{\rm{s}}}\delta {a}^{\dagger }+{a}_{{\rm{s}}}^{* }\delta a+\delta {a}^{\dagger }\delta a)=0,\\ \delta \dot{a} & + & ({\rm{i}}{{\rm{\Delta }}}_{{\rm{l}}}-{\rm{i}}{{gx}}_{{\rm{s}}}-\kappa )\delta a-{\rm{i}}g({a}_{{\rm{s}}}\delta x+\delta x\delta a)-{\varepsilon }_{{\rm{p}}}{{\rm{e}}}^{-{\rm{i}}\xi t}=0.\end{eqnarray} \tag{ 4 }$$

Here the second-order nonlinear terms such as $\delta x\delta a$ and $\delta {a}^{\dagger }\delta a$ are preserved in equation (4). Now we solve these equations by using the ansatz

$$\begin{eqnarray}\delta a & = & {A}_{+}^{(1)}{{\rm{e}}}^{-{\rm{i}}\xi t}+{A}_{-}^{(1)}{{\rm{e}}}^{{\rm{i}}\xi t}+{A}_{+}^{(2)}{{\rm{e}}}^{-2{\rm{i}}\xi t}+{A}_{-}^{(2)}{{\rm{e}}}^{2{\rm{i}}\xi t},\\ \delta x & = & {X}^{(1)}{{\rm{e}}}^{-{\rm{i}}\xi t}+{X}^{(1)* }{{\rm{e}}}^{{\rm{i}}\xi t}+{X}^{(2)}{{\rm{e}}}^{-2{\rm{i}}\xi t}+{X}^{(2)* }{{\rm{e}}}^{2{\rm{i}}\xi t}.\end{eqnarray} \tag{ 5 }$$

Substituting equation (5) into equation (4), and first neglecting the second-order parts in equation (5), leads to the linear response of the system

$$\begin{eqnarray}{A}_{+}^{(1)} & = & \displaystyle \frac{[\lambda (\xi ){\lambda }_{{\rm{s}}2}(\xi )+{\rm{i}}{\hslash }{g}^{2}{| {a}_{{\rm{s}}}| }^{2}]{\varepsilon }_{{\rm{p}}}}{\lambda (\xi ){\lambda }_{{\rm{s}}1}(\xi ){\lambda }_{{\rm{s}}2}(\xi )+{\rm{i}}[{\lambda }_{{\rm{s}}1}(\xi )-{\lambda }_{{\rm{s}}2}(\xi )]{\hslash }{g}^{2}{| {a}_{{\rm{s}}}| }^{2}},\\ {X}^{(1)} & = & \displaystyle \frac{{\lambda }_{{\rm{s}}2}(\xi ){\hslash }{{ga}}_{{\rm{s}}}^{* }{\varepsilon }_{{\rm{p}}}}{\lambda (\xi ){\lambda }_{{\rm{s}}1}(\xi ){\lambda }_{{\rm{s}}2}(\xi )+{\rm{i}}[{\lambda }_{{\rm{s}}1}(\xi )-{\lambda }_{{\rm{s}}2}(\xi )]{\hslash }{g}^{2}{| {a}_{{\rm{s}}}| }^{2}},\end{eqnarray} \tag{ 6 }$$

where $\lambda (\xi )=m({\omega }_{{\rm{m}}}^{2}-{\xi }^{2}-{\rm{i}}\xi {{\rm{\Gamma }}}_{{\rm{m}}}),\,{\lambda }_{{\rm{s}}1,{\rm{s}}2}(\xi )=-{\rm{i}}\xi -\kappa \pm ({\rm{i}}{{\rm{\Delta }}}_{{\rm{l}}}-{\rm{i}}{{gx}}_{{\rm{s}}}).$ Combining equations (4)–(6) gives the second-order sideband amplitude

$$\begin{eqnarray}&&{A}_{+}^{(2)}=\displaystyle \frac{{C}_{1}{X}^{(1)2}+{C}_{2}{A}_{+}^{(1)}{X}^{(1)}}{{\lambda }_{{\rm{s}}2}(\xi )[\lambda (2\xi ){\lambda }_{{\rm{s}}1}(2\xi ){\lambda }_{{\rm{s}}2}(2\xi )+{\rm{i}}({\lambda }_{{\rm{s}}1}(2\xi )-{\lambda }_{{\rm{s}}2}(2\xi )){\hslash }{g}^{2}{| {a}_{{\rm{s}}}| }^{2}]}\end{eqnarray} \tag{ 7 }$$

where ${C}_{1}=-{\rm{i}}{\hslash }{g}^{4}{a}_{{\rm{s}}}{| {a}_{{\rm{s}}}| }^{2},\,{C}_{2}={\hslash }{g}^{3}({\lambda }_{{\rm{s}}2}(2\xi )-{\lambda }_{{\rm{s}}2}(\xi )){| {a}_{{\rm{s}}}| }^{2}+{\rm{i}}g{\lambda }_{{\rm{s}}2}(\xi )\lambda (2\xi ){\lambda }_{{\rm{s}}2}(2\xi ).$ We note that ${A}_{+}^{(2)}$, being proportional to ${\varepsilon }_{{\rm{p}}}^{2}$, is indeed smaller than the first-order term ${A}_{+}^{(1)}$ for a weak probe light. The first term or the second term of ${A}_{+}^{(2)}$ denotes an upconverted-like process of the pump light or the first-order sideband, respectively [20].

By including two optical modes ${a}_{\mathrm{1,2}}$ and the mechanical mode containing now in the passive resonator (see figure 1(b)), we write the Hamiltonian of this system as

$$\begin{eqnarray}&&{H}_{2}={H}_{1}+{\hslash }{{\rm{\Delta }}}_{{\rm{l}}}{a}_{2}^{\dagger }{a}_{2}-{\hslash }J({a}_{1}^{\dagger }{a}_{2}+{a}_{2}^{\dagger }{a}_{1}),\end{eqnarray} \tag{ 8 }$$

with the replacement $a\,\to \,{a}_{1}$ for the expression of ${H}_{1}$. The Heisenberg equations of this compound system are

$$\begin{eqnarray}\ddot{x} & = & -{{\rm{\Gamma }}}_{{\rm{m}}}\dot{x}-{\omega }_{{\rm{m}}}^{2}x+\displaystyle \frac{{\hslash }g}{m}{a}_{1}^{\dagger }{a}_{1},\\ {\dot{a}}_{1} & = & (-{\rm{i}}{{\rm{\Delta }}}_{{\rm{l}}}+{\rm{i}}{gx}-\gamma ){a}_{1}+{\rm{i}}{{Ja}}_{2}+{\varepsilon }_{{\rm{l}}}+{\varepsilon }_{{\rm{p}}}{{\rm{e}}}^{-{\rm{i}}\xi t},\\ {\dot{a}}_{2} & = & (-{\rm{i}}{{\rm{\Delta }}}_{{\rm{l}}}+\kappa ){a}_{2}+{\rm{i}}{{Ja}}_{1}.\end{eqnarray} \tag{ 9 }$$

Expanding the operators as follows

$$\begin{eqnarray}{a}_{1} & = & {a}_{1,{\rm{s}}}+{A}_{1+}^{(1)}{{\rm{e}}}^{-{\rm{i}}\xi t}+{A}_{1-}^{(1)}{{\rm{e}}}^{{\rm{i}}\xi t}+{A}_{1+}^{(2)}{{\rm{e}}}^{-2{\rm{i}}\xi t}+{A}_{1-}^{(2)}{{\rm{e}}}^{2{\rm{i}}\xi t},\\ {a}_{2} & = & {a}_{2,{\rm{s}}}+{A}_{2+}^{(1)}{{\rm{e}}}^{-{\rm{i}}\xi t}+{A}_{2-}^{(1)}{{\rm{e}}}^{{\rm{i}}\xi t}+{A}_{2+}^{(2)}{{\rm{e}}}^{-2{\rm{i}}\xi t}+{A}_{2-}^{(2)}{{\rm{e}}}^{2{\rm{i}}\xi t},\\ x & = & {x}_{{\rm{s}}}+{X}^{(1)}{{\rm{e}}}^{-{\rm{i}}\xi t}+{X}^{(1)* }{{\rm{e}}}^{{\rm{i}}\xi t}+{X}^{(2)}{{\rm{e}}}^{-2{\rm{i}}\xi t}+{X}^{(2)* }{{\rm{e}}}^{2{\rm{i}}\xi t},\end{eqnarray} \tag{ 10 }$$

where ${a}_{i,{\rm{s}}}$ (i = 1, 2) and x_s denotes the steady values, leads to the amplitude of the first-order sideband ${A}_{1+}^{(1)}$ and ${X}^{(1)}$, as given in [34], and also the second-order sideband amplitude ${A}_{1+}^{(2)}$. We find that it has exactly the same form as the single-cavity result, i.e. the equation (7), but with the replacements: $a\to {a}_{1},\,{A}_{+}^{(1)}\to {A}_{1+}^{(1)},$ and

$$\begin{eqnarray*}&&{\lambda }_{{\rm{s}}1}\to {\lambda }_{{\rm{d}}1},\,\,{\lambda }_{{\rm{s}}2}\to {\lambda }_{{\rm{d}}2},\,\,{\lambda }_{{\rm{d}}1,{\rm{d}}2}(\xi )\equiv -{\rm{i}}\xi +\gamma \pm \left({\rm{i}}{{\rm{\Delta }}}_{{\rm{l}}}-{\rm{i}}{{gx}}_{{\rm{s}}}+\displaystyle \frac{{J}^{2}}{{\rm{i}}{{\rm{\Delta }}}_{{\rm{l}}}-{\rm{i}}\xi -\kappa }\right).\end{eqnarray*}$$

The output-field expectation value can be obtained by using the standard input–output relation, i.e. ${a}_{1}^{\mathrm{out}}={a}_{1}^{\mathrm{in}}-\sqrt{\gamma }{a}_{1}$, where a₁ⁱⁿ and a₁^out are the input and output field operators, respectively. The transmitted rate of the probe [34] and the efficiency of the sideband (i.e. the ratio of the amplitude of the output second-order sideband to the amplitude of the input field) [20] then read

$$\begin{eqnarray*}&&{| {t}_{{\rm{p}}}| }^{2}={\left|\displaystyle \frac{{a}_{1}^{\mathrm{out}}}{{a}_{1}^{\mathrm{in}}}\right|}^{2}={\left|1-\displaystyle \frac{\gamma }{{\varepsilon }_{{\rm{p}}}}{A}_{1+}^{(1)}\right|}^{2},\,\,\,\eta =\left|\displaystyle \frac{\gamma }{{\varepsilon }_{{\rm{p}}}}{A}_{1+}^{(2)}\right|.\end{eqnarray*}$$

Also, taking into account of the second-order sideband frequency $2\xi +{\omega }_{{\rm{l}}}$, the group delay of the probe light or the second-order sideband is given by

$$\begin{eqnarray*}&&{\tau }_{{\rm{g}}}=\displaystyle \frac{d\arg ({t}_{{\rm{p}}})}{d\xi }{| }_{\xi ={\omega }_{{\rm{m}}}},\,\,\,{\tau }_{{\rm{g}}}^{\prime }=\displaystyle \frac{d\arg ({A}_{1+}^{(2)})}{d\omega }{| }_{\xi ={\omega }_{{\rm{m}}}}=\displaystyle \frac{d\arg ({A}_{1+}^{(2)})}{2d\xi }{| }_{\xi ={\omega }_{{\rm{m}}}},\end{eqnarray*}$$

respectively. In the following simulations, experimentally accessible parameters are chosen, i.e. ${\varepsilon }_{{\rm{p}}}/{\varepsilon }_{{\rm{l}}}=0.05$, (the resonator radius) $R={\omega }_{{\rm{c}}}/g=34.5\,\mu {\rm{m}}$, m = 50 ng, ${\omega }_{{\rm{c}}}=1.93\times {10}^{5}\,\mathrm{GHz}$, ${\omega }_{{\rm{m}}}=2\pi \times 23.4\,\mathrm{MHz}$, $\gamma =6.43\,\mathrm{MHz}$, ${{\rm{\Gamma }}}_{{\rm{m}}}=2.4\times {10}^{5}\,\mathrm{Hz}$, and ${{\rm{\Delta }}}_{{\rm{l}}}={\omega }_{{\rm{c}}}-{\omega }_{{\rm{l}}}={\omega }_{{\rm{m}}}$ or ${{\rm{\Delta }}}_{{\rm{p}}}\equiv {\omega }_{{\rm{p}}}-{\omega }_{{\rm{c}}}$. These parameter values of coupled resonators have also been used in such a wide range of fields from nonreciprocal light propagation [26], loss-induced revival of lasing [31], phonon lasing [6, 7] and phonon diode [38] to low-power chaos [33] and ultra-sensitive force measurement [39].

In sharp contrast to the conventional OMIT profile in a lossy cavity, an inverted-OMIT profile appears for the probe field in an active cavity (see figure 2(a)), characterizing a non-amplifying dip in the otherwise strongly amplifying region [34, 35]. However, as shown in figure 2(b), the double-peak spectrum exists for η also in the active OMIT. Nevertheless, η is still enhanced by the gain for $\sim 60 \%$ at ${{\rm{\Delta }}}_{{\rm{p}}}=0$, for ${P}_{{\rm{l}}}=933\,\mu {\rm{W}}$. For lower values of P_l, the transmission of the probe is decreased due to the weakened OMIT effect. Correspondingly, the enhanced optical absorption leads to the enhanced two-phonon up-converted process of the pump field, i.e. the increasing of η. However, as ${P}_{{\rm{l}}}\to 0$, both of two signals are rapidly suppressed, since the whole OMIT mechanism tends to disappear [20].

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The gain can also strongly affect the dispersion of the system, hence greatly altering the phases of the two signals. Figures 2(c) and (d) show the resulting group delays of the probe and the up-converted sideband. For low-power values, both ${\tau }_{{\rm{g}}}$ and ${\tau }_{{\rm{g}}}^{\prime }$ are enhanced significantly due to the gain, i.e. ∼50 or 100 times for their maximum values in comparison with the lossy-cavity results. In particular, a remarkable slow-to-fast light transition happens at ${P}_{{\rm{l}}}\sim 0.12\,\mathrm{mW}$, for $\kappa /\gamma =1$, which is reminiscent of the gain-assisted fast light in an atomic gas [36]. Clearly, in this weak-power regime, the OMIT effect is relatively weak and thus we observe the similar gain-dominated group delays of both two signals (see the regime 1 in figure 2(d)). In contrast, for higher values of P_l, the more prominent OMIT effect leads to the reversed fast-to-slow light transition for the probe (see the regime 2 in figure 2(d)). The relevant second-order signal has no such a transition as it tends to be relatively suppressed, as in the passive OMIT [20], leading to a tunable (negative or positive) relative delay of the two signals.

As shown in figures 3(a), the second-order profile is similar to the first-order case [34]: the field amplification tends to be maximized as the gain and loss approach to the balance ($\kappa /\gamma =1$), and the reversed-gain dependence works for both processes. In fact, by taking ${\omega }_{{\rm{m}}}/{\omega }_{{\rm{c}}}\sim 0$, ${x}_{{\rm{s}}}\sim 0$, we approximately have

$$\begin{eqnarray}&&\eta \approx \left|\displaystyle \frac{{\rm{i}}{{\hslash }}^{2}{g}^{4}| {a}_{1,{\rm{s}}}{| }^{2}{a}_{1,{\rm{s}}}^{* }{\kappa }^{4}\gamma {\varepsilon }_{{\rm{p}}}}{2{m}^{2}{{\rm{\Gamma }}}_{{\rm{m}}}^{2}{\omega }_{{\rm{m}}}^{3}{({J}^{2}-\kappa \gamma )}^{3}[m{\omega }_{{\rm{m}}}{{\rm{\Gamma }}}_{{\rm{m}}}({J}^{2}-\kappa \gamma )+{\rm{i}}{\hslash }{g}^{2}{| {a}_{1,{\rm{s}}}| }^{2}({J}^{2}-{\kappa }^{2})]}\right|,\end{eqnarray} \tag{ 11 }$$

i.e., η is most preferable in the vicinity of the exceptional point (EP) at the gain–loss balance [26, 34], i.e. $\kappa /\gamma =1$, with $J/\gamma =1$. Clearly, for $\kappa /\gamma \lt 1$, the loss in the passive cavity can be compensated by the gain, leading to the sideband amplification of the transmission spectrum; after passing by the EP, the optical supermodes become spontaneously localized in either the amplifying or the lossy cavity [26], thus the loss of the passive cavity is increased again. This, of course, does not means that η is maximized for $\kappa /\gamma =1$ at all detuning values [40]. Still, at ${{\rm{\Delta }}}_{{\rm{p}}}=0$, we have ∼3 times enhancement of η by tuning $\kappa /\gamma$, in comparison with even the single-active-cavity situation (see figure 3(b)), indicating the advantage of using a compound system.

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The associated optical group delay is shown in figure 4. For the coupled passive resonators ($\kappa /\gamma =-1$), ${\tau }_{{\rm{g}}}$ and ${\tau }_{{\rm{g}}}^{\prime }$ changes within the scope of only several microseconds, as in the single-passive-cavity case (see figure 4(a)). For $\kappa /\gamma \lt 1$, due to the loss compensation by gain, similar results as in the single-active-cavity case can be observed, i.e. the gain-dominated low-power regime, with similarly giant enhancement of both ${\tau }_{{\rm{g}}}$ and ${\tau }_{{\rm{g}}}^{\prime }$ (around the slow-to-fast light transition, see figure 4(b)), which is then followed by the OMIT-dominated regime for higher values of P_l. In contrast, for $\kappa /\gamma \geqslant 1$, as shown in [34], the regime 1 as shown in figure 4(b) now disappears, since the slow-to-fast light transition in the low-power regime is reversed to the fast-to-slow light transition, a feature characterizing the unconventional PT-broken regime (see figures 4(c) and (d)) [34]. Accompanying with this reversal, the group delay of the second-order signal tends to be rapidly suppressed in the PT-broken regime. This feature does not exist at all in any single cavity case, in which the relative group delay of the probe and the second-order signal becomes significant only for higher values of the pump power (see e.g. figure 2(d)). In practice, this may provide highly sensitive low-power diagnoses of different phases of the OMIT systems.

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