Force-induced transparency and conversion between slow and fast lights in optomechanics

TL;DR

This paper theoretically explores how external forces can induce transparency and enable conversion between slow and fast light in an optomechanical cavity, offering new methods for controlling light velocity and force detection.

Contribution

It introduces a novel theoretical framework showing force-controlled transparency and light conversion in optomechanics, leveraging RWA and anti-RWA effects.

Findings

Force-induced transparency of slow/fast light demonstrated

Conversion between slow and fast lights controlled by external force

Potential applications in light velocity control and force sensing

Abstract

The optomechanics can generate fantastic effects of optics due to appropriate mechanical control. Here we theoretically study effects of slow and fast lights in a single-sided optomechanical cavity with an external force. The force-induced transparency of slow/fast light and the force-dependent conversion between the slow and fast lights are resulted from effects of the rotating-wave approximation (RWA) and the anti-RWA, which can be controlled by properly modifying the effective cavity frequency due to the external force. These force-induced phenomena can be applied to control of the light group velocity and detection of the force variation, which are feasible using current laboratory techniques.

Figures

Equations

$$\begin{array}{rl}H=& \hslash {\Delta }_c{c}^{†}c+(\frac{p^2}{2m}+\frac{1}{2}m{\omega }_m^2{q}^2)-\chi q{c}^{†}c-fq\\ & +i\hslash [({\epsilon }_d+{\epsilon }_p{e}^{-i\delta t})c^{†}-H.c.],\end{array}$$

$$⟨\dot{q}⟩$$

$$⟨\dot{p}⟩$$

$$⟨\dot{c}⟩$$

$$q_0$$

$$c_{+}$$

$$\begin{array}[]{cc}m\omega_{m}^{2}\frac{\chi^{2}}{\hbar^{2}}q_{0}^{3}-(f\frac{\chi^{2}}{\hbar^{2}}+2m\omega_{m}^{2}\frac{\chi}{\hbar}\Delta_{c})q_{0}^{2}&\\ +[m\omega_{m}^{2}(\kappa^{2}+\Delta_{c}^{2})+2f\frac{\chi}{\hbar}\Delta_{c}]q_{0}-[f(\kappa^{2}+\Delta_{c}^{2})+\chi\varepsilon_{d}^{2}]&=0,\end{array}$$

$${\epsilon }_{out}=({\epsilon }_d-2\kappa c_0)+(1-2\kappa c_{+}){\epsilon }_p{e}^{-i\delta t}-2\kappa c_{-}{\epsilon }_p^{\ast }{e}^{i\delta t}.$$

$$\begin{array}[]{lll}\varepsilon_{T}&=&2\kappa c_{+}\\ &=&\dfrac{2\kappa}{[\kappa-i(\delta-\Delta)]+\frac{2i\omega_{m}\beta}{(\delta^{2}-\omega_{m}^{2}+i\gamma_{m}\delta)-\frac{2i\omega_{m}\beta}{\kappa-i(\delta+\Delta)}}},\end{array}$$

$$\begin{array}[]{lll}\varepsilon_{T}&\simeq&\dfrac{2\kappa}{[\kappa-i(\delta-\Delta)]+\dfrac{2i\omega_{m}\beta}{\delta^{2}-\omega_{m}^{2}+i\gamma_{m}\delta}}\\ &\simeq&\dfrac{2\kappa}{[\kappa-i(\delta-\Delta)]+\dfrac{\beta}{\frac{\gamma_{m}}{2}-i(\delta-\omega_{m})}},\end{array}$$

$$\begin{array}[]{lll}\varepsilon_{T}&\simeq&\dfrac{2\kappa}{[\kappa-i(\delta-\Delta)]-\dfrac{\beta}{\frac{\gamma_{m}}{2}-i(\delta+\omega_{m})}},\end{array}$$

$$\epsilon =1-2\kappa c_{+}.$$

$$\tau =\frac{\partial \Phi }{\partial {\omega }_p}{|}_{\bar{\omega }}=\mathrm{Im}[\frac{1}{\epsilon }\frac{\partial \epsilon }{\partial {\omega }_p}]{|}_{\bar{\omega }}=\mathrm{Im}[\frac{1}{\epsilon }\frac{\partial \epsilon }{\partial \delta }]{|}_{\delta =\pm {\omega }_m},$$

Full text

I Introduction

The cavity optomechanics (COM), combining mechanical modes with optical modes via radiation pressure, has attracted considerable attention recently. Extensive research efforts have presented interesting quantum properties and nonlinear effects by optomechanics, such as entanglement Bose et al. (1997); Vitali et al. (2007); Sun et al. (2012), squeezing Liao and Law (2011); Han et al. (2013), normal mode splitting Huang and Agarwal (2009), Kerr effect Lü et al. (2013), optomechanically induced transparency (OMIT) Agarwal and Huang (2010); Huang and Agarwal (2011); Safavinaeini et al. (2011); Weis et al. (2010); Bai et al. (2016), optical solitonsGan et al. (2016), and chaosLü et al. (2015); Bakemeier et al. (2015), which are associated with potential applications in quantum information processing Agarwal and Huang (2012); Xiong et al. (2015) and precision measurements Zhang et al. (2012); Aspelmeyer et al. (2014); He (2015).

Among the above mentioned items, the OMIT, a kind of induced transparency arisen from the interference of excitation pathways in optomechanical systems is the research focus of the present paper. We have noticed that the electromagnetically induced transparency (EIT) in atoms can produce slow and fast lights Hau et al. (1999), which is the technique with appealing applications in optical storage Tucker et al. (2005); Xing et al. (2007), optical telecommunication Boyd and Gauthier (2009), signal processing Willner et al. (2008) and interferometry Shi et al. (2007). So we wonder if the OMIT, with analogy to the EIT, could also work for producing slow/fast light and even beyond. In fact, there have been publications Chen et al. (2011); Zhan et al. (2013); Tarhan (2013); Jiang et al. (2013); Tarhan et al. (2013); Akram et al. (2015a, b) for slow and fast light effects associated with the COM using similar behavior to those with multi-level atoms. As shown below, however, we will go for a further step with the COM by presenting an experimentally feasible proposal for a force-induced transparency with slow/fast light and a conversion between the slow and fast lights.

Specifically, different from the traditional transparency proposals Imamoǧlu et al. (1991); Fleischhauer et al. (2005); Li et al. (2016); Mücke et al. (2010), where the slow/fast light can only be adjusted with an external optical field, e.g. the power and frequency of the pump field, our study shows that we can control exactly by an external force the group velocity of lights with a fixed pump field. This external force employed for the control could be Coulomb-relevant Zhang et al. (2012) or magnetic effects prl-108-120801 .

Two kinds of external forces are usually employed in the optomechanics. One is the constant force nanotech-7-509 ; prl-114-093602 including electric field force Zhang et al. (2012), magnetic field force prl-108-120801 , elastic force apl-105-014108 ; pra-94-043855 and optical gradient force nphoton-4-211 ; The other is the time-harmonic-driving force pra-91-043843 ; srep-5-11278 , which could be achieved with piezoelectric coupling nphys-9-712 and Lorentz force njp-9-35 . In our scheme, we choose a constant force as the external force, which can modify the eigen-frequencies of the cavity by adjusting the cavity length. Such an external force is similar to the one in Ref.prl-114-093602 , where the force is applied to balance the effective force from the nonlinear optical effect.

Compared with the optical manipulations on the group velocity of lights Imamoǧlu et al. (1991); Fleischhauer et al. (2005); Li et al. (2016); Mücke et al. (2010), which are limited by the power and frequency ranges of the laser field, the external force in our scheme works in a larger regime. Due to the external force, the effective eigen-frequency of the optomechanical cavity is modified, and thus we may achieve the conversion between slow and fast lights in this way. This conversion is physically governed by the conditions for the rotating-wave approximation (RWA) and the anti-RWA which are present as anti-Stokes and Stokes processes in the optomechanics, respectively.

As pointed out below, the slow/fast light is originally from the effect of RWA/anti-RWA of the parameters in the system, which is a more fundamental factor than the anti-Stokes/Stokes process as mentioned in Refs. Akram et al. (2015a, b). The latter is valid only for the situation of the third-order nonlinear coupling, but the former can explain the fast/slow light effects in various physical systems including both the linear Imamoǧlu et al. (1991); Fleischhauer et al. (2005); Li et al. (2016); Mücke et al. (2010) and the nonlinear coupling systems Chen et al. (2011); Zhan et al. (2013); Tarhan (2013); Jiang et al. (2013); Tarhan et al. (2013); Akram et al. (2015a, b). Moreover, our proposal is more simplified and effective, and experimentally feasible using current techniques nanotech-7-509 since it is only required to apply an external force on the optomechanical resonator, much more easily adjusted than the idea with an additional atom Akram et al. (2015a) or nano-resonator Akram et al. (2015b). In Ref.Akram et al. (2015a), the slow/fast light is adjusted by the detuning between the optomechanical cavity and a cavity-confined atom, which is hard to manipulate experimentally. The control of the slow/fast light in Akram et al. (2015b) is made by Coulomb coupling between two nanomechanical resonators (NRs), which is also experimentally challenging.

Furthermore, the effects in our work can be observed even in room temperature since the noise is much less than the mean value of the output field Karuza et al. (2013). In particular, due to one-to-one correspondence between the external force and the group velocity of the light in some special regimes, our proposal could be used to achieve precision measurements and operations. As such, our work provides a new and effective way to control the group velocity of the light in optomechanical systems with an external force.

The rest of the paper is structured as follows. In Sec. II, we present the Hamiltonian and the steady state of the optomechanics. In Sec. III, we deduce the output of the probe field and its time delay. In Sec. IV, we give some simulations and discussions for the force-induced light transparency and the force-dependent conversion between the slow and fast lights under some experimentally available conditions. The conclusion is given in the last section.

II Hamiltonian and Steady states

As sketched in Fig. 1, an optomechanical cavity is driven by a strong pump field with frequency $\omega_{d}$ and power $P_{d}$, and by a weak probe field with frequency $\omega_{p}$ and power $P_{p}$. The oscillating mirror with mass $m$ and frequency $\omega_{m}$ couples to the Fabry-Perot cavity with frequency $\omega_{c}$ via radiation pressure force, and experiences an external force $f$. As mentioned above, this force must be a constant force, such as a Coulomb force Zhang et al. (2012) or a magnetic force with a steady electric current Ma et al. (2015).

In the rotating frame at frequency $\omega_{d}$ of the pump field, the Hamiltonian of the system is given by Zhang et al. (2012); Ma et al. (2015)

[TABLE]

with detunings $\Delta_{c}=\omega_{c}-\omega_{d}$ and $\delta=\omega_{p}-\omega_{d}$. The first term is the free Hamiltonian for the cavity with the annihilation (creation) operator $c$ $(c^{\dagger})$. The second term describes the energy for the oscillating mirror with $q$ ($p$) being the position (momentum) operator. The third term represents the radiation pressure effect between the cavity and the oscillating mirror with a coupling strength $\chi=\frac{\hbar\omega_{c}}{L}$, where $L$ is the cavity length. The forth term is associated with the external force on the oscillating mirror. The last two terms are the interactions between the cavity and two input fields with strengths $\varepsilon_{d}=\sqrt{\frac{2\kappa P_{d}}{\hbar\omega_{d}}}$ and $\varepsilon_{p}=\sqrt{\frac{2\kappa P_{p}}{\hbar\omega_{p}}}$, respectively, where $\kappa$ is cavity decay rate.

To get the mean response of the system, we employ the Heisenberg-Langevin equations and the mean-field approximation Agarwal and Huang (2010). Then the mean-value equations of our model can be written as

[TABLE]

where $\gamma_{m}$ is the decay rate of the movable mirror. The steady-state response of Eq. (2) contains many Fourier components, where we are only interested in the linear response of the system for the probe field.

To obtain steady-state solutions to Eq. (2), we assume the equation Boyd (2008) $\langle s\rangle=s_{0}+s_{+}\varepsilon_{p}e^{-i\delta t}+s_{-}\varepsilon_{p}^{\ast}e^{i\delta t}$ with $s=q,p,c$, and these three terms $s_{0,\pm}$ are associated with the frequencies $\omega_{d}$, $\omega_{p}$, $2\omega_{d}-\omega_{p}$, respectively. Inserting the three equations into Eq. (2), we obtain

[TABLE]

where $\beta=\chi^{2}|c_{0}|^{2}/(2m\hbar\omega_{m})$, and $\Delta=\Delta_{c}-\chi q_{0}/\hbar$ is the effective cavity-pump detuning, depending on the steady-state position $q_{0}$ of the mirror. Assuming that the above solutions are based on the mean value much larger than the noise, we consider that the effects resulted from those solutions could be observed at room temperature, similar to the one in Ref. Karuza et al. (2013).

With the steady-state solution in Eq. (3), the steady-state equation for the position $q_{0}$ can be rewritten as

[TABLE]

which means that the steady state for the position $q_{0}$ depends on two tunable parameters: the pump power $\varepsilon_{d}$ and the external force $f$ (see Fig. 2). In other words, the effective cavity frequency (cavity-pump detuning) $\omega_{c}^{\prime}=\omega_{c}-\chi q_{0}/\hbar$ ($\Delta=\Delta_{c}-\chi q_{0}/\hbar$) can be adjusted by controlling the pump power and the external force. It reminds us the possibility to realize some force-induced/dependent physics.

III Output light and time delay

With the application of the input-output relation Walls and Milburn (1994) $\varepsilon_{out}=\varepsilon_{in}-2\kappa\langle c\rangle,$ we have the output field

[TABLE]

For simplicity, we assume the quadrature of the output field as

[TABLE]

whose real and imaginary parts are associated with the absorption and dispersion, respectively Agarwal and Huang (2010). Moreover, the output field varies with both the pump strength $\varepsilon_{d}$ and the external force $f$, implying that the external force, in addition to the pump field, can construct the light transparency (see Eq. (4) and Fig. 2).

To follow the force-induced transparency of the probe light, we suppose the system working in the resolved-sideband regime due to $\omega_{m}\gg\kappa$. This is the condition for the normal mode splitting in optomechanics, and the strongest radiation coupling can be achieved when the system reaches the first-order red/blue sideband with $\delta=\pm\omega_{m}$ or $\delta=\pm\Delta$.

In the case of $\Delta\simeq\omega_{m}$ (i.e., the RWA case), the optomechanics works in the first-order red sideband. With the application of $\delta^{2}-\omega_{m}^{2}\simeq 2\omega_{m}(\delta-\omega_{m})$, we neglect the small term $2i\omega_{m}\beta/[\kappa-i(\delta+\Delta)]$. Thus Eq. (6) is rewritten as

[TABLE]

which is the expression for the slow light [see Figs. 3(a) and (b)].

Similarly, in the case of $\Delta\simeq-\omega_{m}$ (i.e., the case of anti-RWA), the system is governed by the first-order blue sideband, and we have $\delta^{2}-\omega_{m}^{2}\simeq-2\omega_{m}(\delta+\omega_{m})$,

[TABLE]

which is the solution for the fast light [see Figs. 3(c) and (d)]. In this situation, a very small gain can be achieved in the absorption of the output field ($Re[\varepsilon_{T}]$). This gain of the probe light originates from the anti-RWA process with both a photon and a phonon simultaneously created or annihilated. Due to the large cavity decay, however, the photon-phonon creation dominates the system evolution, which is supported by the external field. As such, we have the gain in the absorption of the output field.

The slow/fast light conversion in our scheme is very different from the previously proposed transparencies using atoms Imamoǧlu et al. (1991); Fleischhauer et al. (2005), coupled cavities Li et al. (2016) and atom-cavity hybrids Mücke et al. (2010). Compared with Eq. (7), Eq. (8) has a sign difference in the denominator, which can be understood as a switch between the effects of RWA and anti-RWA. Since the two solutions of $\varepsilon_{T}$ under RWA and anti-RWA take two fast changes in absorption/dispersion ($Re[\varepsilon_{T}]$/$Im[\varepsilon_{T}]$) with the slopes in different signs, the conversion between the slow and fast lights can be achieved by controlling the parameters to reach the RWA and anti-RWA regimes. This viewpoint is different from in Refs. Akram et al. (2015a, b), where the fast/slow light is explained as a characteristic in the anti-Stokes/Stokes process. Actually, the anti-Stokes/Stokes process Akram et al. (2015a, b) owns the fast/slow light due to the third-order nonlinear coupling as the radiation coupling. In contrast, by the fact that Eq. (6) is reduced to Eq. (7) under the RWA and to Eq. (8) under the anti-RWA, we consider that the slow and fast effects originate fundamentally from the RWA and the anti-RWA employed for the parameters. This is a more fundamental reason than the anti-Stokes/Stokes effects for the slow/fast light in general systems. For example, the slow light effect observed in the previous publications Imamoǧlu et al. (1991); Fleischhauer et al. (2005); Li et al. (2016); Mücke et al. (2010) is due to the involvement of only the RWA.

The transmission of the probe field, defined by the ratio of the output and input field amplitudes at the frequency of the probe light Weis et al. (2010); Zhan et al. (2013), is given by

[TABLE]

In the regime of the narrow transparency window, there is a rapid variation of the probe phase $\Phi(\omega_{p})=\mathrm{arg}[\varepsilon]=\frac{1}{2i}\mathrm{ln}(\frac{\varepsilon}{\varepsilon^{\ast}})$. This variation is associated with the group velocity delay as Tarhan et al. (2013); Gu et al. (2015)

[TABLE]

where $\bar{\omega}=\omega_{d}\pm\omega_{m}$, and $\delta=\pm\omega_{m}$ is the condition for the two-photon resonance. $\tau>0$ and $\tau<0$ correspond to the slow and fast light propagation, respectively.

IV Simulations and discussion

To demonstrate the force-induced light transparency and the force-dependent conversion between the slow and fast lights, we have made some simulations using following experimental parameters Gröblacher et al. (2009): $\lambda=1064$ nm, $P_{d}=0.2$ mW, $L=25$ mm, $\kappa/2\pi=215$ kHz, $m=145$ ng, $\omega_{m}/2\pi=947$ kHz, $\gamma_{m}/2\pi=141$ Hz. In what follows, to justify the feasibility of our scheme, we assume a fixed pump light which is far detuned from the cavity as $\Delta_{c}=-10\omega_{m}$.

IV.1 Force-induced light transparency

In the absence of the external force, the effective detuning between the optomechanical cavity and the pump field is $\Delta\approx-10\omega_{m}$, which is far detuned from the resonator frequency $\omega_{m}$. In this case, even if the pump power is set to some feasible values, no OMIT can be observed due to the large detuning of the pump light from the cavity field and the limitation of the work regime for the optical manipulation. However, if an external force $f=f_{1}$ is applied on the optomechanics, as shown in Fig. 3, the OMIT appears since the condition of $\Delta=\omega_{m}$ can be satisfied. This is due to the fact that the external force pushes the system into the red-sideband regime under the RWA by increasing the effective cavity frequency. As shown in Fig. 4, with the increase of the force, the central frequency for transparency remains unchanged, whereas the main peak of the output field moves from the low frequency to the higher due to the increase of the effective cavity frequency.

In contrast, with an alternative external force $f=f_{2}$ applied, the system moves into the blue-sideband regime under the anti-RWA. The fast light due to force-induced transparency is thus produced.

IV.2 Force-dependent slow/fast light conversion

As discussed above, when the external force $f=f_{1}$ is applied, there is a force-induced transparency for the slow light [see Figs. 3(b)]. In contrast, when the external force is $f=f_{2}$, the system works in the blue-sideband regime with an effective detuning $\Delta=-\omega_{m}$, and the fast light effect is available in this situation [see Figs. 3(d)]. In this context, it is natural to ask if there is a possibility to have a conversion between the slow light and the fast one.

The answer to this possibility is positive, as shown below. The physics for the control of the slow and fast lights can be understood from Eq. (6). There are fast changes in absorption/dispersion in very narrow spectral ranges [see Fig. 3(a) and (c)], which are followed by large changes in the refractive index due to Eq. (6) satisfying the Kramers-Kronig relations Shahoei and Yao (2012). As a result, if $f=f_{1}$, the system works in the red-sideband regime with a positive change in the refractive index, yielding a low group velocity [see Fig. 3(b)]. In contrast, the system turns to be in the blue-sideband regime once $f=f_{2}$ is applied, which creates a high group velocity [see Fig. 3(d)] due to a negative change in the refractive index.

Since the probe transmission $\varepsilon$ depends on both the pump power and the external force in our approach, the group velocity delay $\tau$ can be tuned by both the power of the pump light and the external force, which are different from previous ideas with $\tau$ modified only by the pump power Tarhan (2013); Jiang et al. (2013); Tarhan et al. (2013); Akram et al. (2015a). To show this, we plot $\tau$ as a function of the force around the detuning of $\delta=\pm\omega_{m}$ in Fig. 5. It implies that the external force $f$ can be used to control the group velocity of the probe light even with a fixed pump field, and also means the possibility to measure the external force using this property. In particular, the delay $\tau$ approximately linearly varies with $f$ at the point near $f=f_{1}$ ($f=f_{2}$) for which the slope is $d\tau/df\approx$ 244(242) s/N at $\Delta\approx\omega_{m}$ ($\Delta\approx-\omega_{m}$). Within the regime with one-to-one correspondence between the group velocity and the external force, we may perform precision control or measurement for the group velocity using a certain external force.

Moreover, in Fig. 5, with the increase of the external force $f$ for the slow (fast) light, the approximate and analytic results intersect at the point of $f=f_{1}$ ($f=f_{2}$) where the system meets exactly the condition for the red(blue)-sideband regime. In contrast to the previous works Chen et al. (2011); Tarhan (2013); Zhan et al. (2013); Jiang et al. (2013); Tarhan et al. (2013); Akram et al. (2015a, b) with the time delay expressed by reduced analytic solutions, we fully consider the contribution from the effects of both RWA and anti-RWA in the measurement of the group velocity using a certain external force. In this context, our work provides a further understanding of the slow and fast lights in comparison with the previous treatments Chen et al. (2011); Tarhan (2013); Zhan et al. (2013); Jiang et al. (2013); Tarhan et al. (2013); Akram et al. (2015a, b).

V Conclusion

In summary, we have studied and explained the slow and fast light effects in a single-sided optomechanical cavity under an external force. The two special characters of the optomechanical cavity, i.e., the force-induced light transparency and conversion related to the slow/fast light, can be fully controlled by the effective cavity frequency modified by the external force. In particular, we pointed out that the effect of RWA/anti-RWA of the parameters is the fundamental reason to generate the slow/fast lights. Since our proposal is feasible using current laboratory techniques, we believe that our scheme provides a new way to producing tunable fast and slow lights, which helps inspiring more potential applications for optomechanics.

Acknowledgements.

JQZ thanks Yong Li for helpful discussion. This work is supported by National Key R&D Program of China No.2017YFA0304500, by National Fundamental Research Program of China under Grant 2013CB921803, by National Natural Science Foundation of China under grants No.91636220, No.91421111, No.11674360 and No.11375136, and by the research fund of City College, Wuhan University of Science and Technology under grant No.2014CYBSKY001.