We investigate the tunable bistable behavior of a hybrid nano-electro-optomechanical system (NEOMS) composed of S-shaped in the presence of two-level atoms, trapped inside a Fabry–Pérot cavity, and driven by a strong driving field and a weak probe field. The bistable behavior of the steady-state photon number and the mechanical steady-state positions are discussed. Further, we tune bistability by tuning all the coupling frequencies involved in the system and amplitude of the driving field. The present study provides the possibility of realization of a controllable optical switch depending on atom-field coupling, optomechanical coupling, electrostatic Coulomb coupling, and threshold power. In addition, we discuss that the non-linear effect of the hybrid NEOMS generates the four-wave mixing (FWM) process. Moreover, we show that the FWM process can be suppressed by the atom-field detuning and cavity-field detuning, which exhibits low photon transmission.
Nowadays in optomechanics, the micro/nano-mechanical resonators (NMRs) provide a suitable platform to study quantum coherence in a macroscopic system in various ways. The combination of a micro/nano-mechanical resonator with other miniatures like electron spin,[1] mechanical membrane,[2,3] nitrogen vacancy (NV),[4] optical cavities,[5–7] and Bose–Einstein condensate (BEC),[8,9] has a pathway in manufacturing hybrid systems.[10–12] The hybrid systems have a lot of aspiring applications, such as, dipole blockade approach,[13,14] and simulating many-body quantum systems,[15] respectively. In an optomechanical system, the interaction of laser light with a moving mirror generates the radiation pressure force[16,17] which depends on the displacement of the movable mirror. In the presence of strong driving power, the field inside the cavity is sufficiently strong to trip nonlinear behavior of the system. So far, the investigation of atom-assisted optomechanics composed of movable cavities as well as ultra-cold atoms leads to various configurations by quantum cavity field and classical pump field.[18–20] Many important works have been done in this interesting field like ground-state cooling of mechanical resonators,[21–24] electromagnetically induced transparency (EIT),[25–27] quantum dot,[28] quantum squeezing,[29–31] qubits,[32] and atomic ensemble.[33–35]
Optical bistability arises due to the nonlinear characteristics of the optical field, and therefore, exhibits hysteresis.[36] In simple optomechanical system, the optical bistability with Kerr effect was studied in Refs. [37] and [38], and in quantum degenerate Fermi gas[39] on the one hand. While in hybrid optomechanical systems composed of ultra-cold trapped atoms such as BEC,[40–43] two-level atoms[44] on the other hand. Besides, the optical bistability has also been discussed in double cavity systems[45,46] as well as in a composite photonic molecule cavity optomechanical system consisting of two whispering gallery mode microcavities.[47] Optical bistability has many practical applications in the field of nonlinear optics, for example, signal processing,[48] all-optical switching,[49] all-optical transistors,[50] and all-optical sensing elements.[51]
Like optical bistability, the four-wave mixing (FWM) intensity process is also a nonlinear effect of the quantum-optical field based on quantum coherence and quantum interference which is frequently studied in the nonlinear optics. Consequently, we have a large number of different applications of quantum coherence and interferences in optomechanical systems, names of few, frequency conversion,[52] the observation of entanglement,[53] stopped light,[54] and fast light.[55] The experimental aspect of the non-degenerate FWM in a three-level system was studied in Ref. [56]. After that, the enhanced FWM process in ultra-slow pump waves was discussed in Ref. [57]. Recently, it has been theoretically observed that the FWM can be instigated in oscillatory tunneling antisymmetric quantum wells[58] which is analogous to electromagnetically induced transparency.[59] More recently, the FWM process in optomechanical system has been studied in Ref. [60]. Moreover, the quantum mechanical behavior of the FWM was not only explained in BEC[61] but also discussed in quantum degenerate Fermi gas.[62,63] In addition, the controllable FWM has not only been studied in a two-mode optomechanical system[64] but also discussed in a composite photonic-molecule cavity optomechanical system.[65]
In the present work, we suggest a hybrid nano-electro-optomechanical system (NEOMS) for detail analysis of bistability and FWM intensity process. The suggested NEOMS comprises two charged nano-mechanical resonators NMR1 and NMR2 that are connected to an external bias gate V1 (−V2). The NMR1 (NMR2) is charged through V1 (−V2) and produces Coulomb coupling between NMR1 and NMR2. The optical cavity is loaded by N number of two-level atoms. In this work, we show that the hybrid NEOMS can be coupled with two-level atoms through the cavity field of the device. Experimental progress is made on the coupling between mechanical resonators and two-level system, e.g., superconducting qubits,[66] cascaded optomechanical devices,[67–69] Ising model,[70] and electro-optomechanical system.[71–74]
In this work: (i) we present that our numerical study is focused on resonant interactions between the low-frequency of NMR1 (NMR2) and two-level atom; (ii) the obtained results provide the threshold driving field for the bistable region that can be controlled by N and cavity field, respectively; (iii) keep in mind that the previous studies on bistability are composed of simple optomechanical systems where the bistability is a function of optomechanical coupling or a Kerr medium.[37,38] In case of hybrid optomechanical systems composed of ultra-cold atoms,[40–43] the bistability can be controlled by two coupling frequencies, i.e., atom-field coupling and optomechanical coupling. These systems work in certain parametric regime but has no external controllable source. Therefore, we consider a hybrid optomechanical system in which the optical bistability is not only controlled by new systematic parameters such as frequency ω1 (ω2) and mass m1 (m2) of the resonator, but also controlled electrostatically through Coulomb coupling gc via external bias gate V1 (−V2). By doing so, we not only control the Coulomb coupling between NMR1 and NMR2 but also control the mirror field coupling.
The rest of the paper is as follows. In Section 2, we describe the basic theory of the proposed model and derive the Heisenberg Langevin equations of motion, respectively. In Section 3, we discuss the steady-state solution of the system and analyze the bistable behavior of the steady-state photon number and the steady-state position of NMR1 (NMR2). In Section 4, we study the nonlinear behavior of the system in term of four-wave mixing process. In Section 5, we present the conclusion.
2. Theoretical model and formalism
The suggested model comprises an ensemble of two-level atoms inside a Fabry–Pérot cavity of length L, a fixed mirror, and end moving coupled charged nano-mechanical resonators NMR1 and NMR2 as shown in Fig. 1. The NMR1 is not only coupled with cavity field and sharing optomechanical coupling g0 but also coupled to NMR2 electrostatically via Coulomb coupling gc. The cavity mode is driven by a strong laser field and a weak field of amplitudes EL and εp, respectively. The charge on NMR1 is e1 = C1V1 due to V1, whereas the charge on NMR2 is e2 = −C2V2 due to −V2; here, C1 (C2) is capacitor bank. In addition, the Coulomb coupling gc is tuned by bias gate V1 (−V2). We consider the Jayen’s cumming type interaction[75] to the cavity field, and the Hamiltonian of the systems can be read as
where H0 and HI represent the free Hamiltonian and interaction Hamiltonian of the system, respectively. The free Hamiltonian is
And the interaction Hamiltonian is
where in Eq. (2), c† (c) is the creation (annihilation) operator of the cavity field, , and . Here, ℘l and ℘p correspond to laser and probe field powers. The term is the atomic transition operator between the two levels and oscillates with frequency ωa. The third term of Eq. (2) represents the energy of NMR1 (NMR2), and mj and ωj are the effective mass and mechanical frequency of the resonators. Moreover, () is the position (momentum) operator for NMRj with j = 1,2. In Eq. (3), the first term represents the atom-field interaction whereas the second term represents the mirror field interaction and expressed as g0 = ℏωc/L. Moreover, the last term belongs to Coulomb interaction between NMR1 and NMR2 and related to ,[72,76] where r0 is the relax distance between NMR1 and NMR2. Here, and are respectively the spin matrices that satisfy the commutation relation and . Here, we introduce new bosonic annihilation (creation) operator such as , , with bosonic commutation relations and .[77] At ground state, atoms satisfy the relation . By rotating frame according to laser frequency ωl, the Hamiltonian of the system is modified as
In Eq. (4), Δc = ωc – ωl is the cavity detuning whereas Δa = ωa – ωl represents the atom-field detuning corresponding to laser frequency. Moreover, δ is the probe field detuning corresponding to ωp – ωl.
Fig. 1. The demonstration of suggested NEOMS composed of N number of two-level atoms follows by moving-end two charged nano-mechanical resonators NMR1 and NMR2. The first mirror is coupled to the cavity field and MMR2, respectively. The NMR1 (NMR2) is charged by a bias voltage V1 (−V2), where gc is the Coulomb coupling and r0 is the relax position of NMR1 (NMR2). Here, q1 (q2) represents the small displacement of NMR1 (NMR2) from their relax position. Moreover, L is the cavity length and corresponds to transition operator between the two levels.
Here, we study the steady-state solution of the system, therefore, we introduce the mean value approximation where all the noise terms have been vanished. By considering all fluctuations and decay terms at the mean field approximation, i.e., ,[25] the mean value equations of Eq. (4) can be found as
Here, γ1 (γ2) is the decay term corresponding to NMR1 (NMR2), whereas γa represents the decay of the atomic oscillation. We take ℏ = 1, and represents the total coupling strength between atoms and cavity field which depends on the number of atoms. Moreover, κ is the decay term of the cavity field. Further, equation (5) represents a set of non-linear equations corresponding to the mean values. To derive the steady-state solution, we introduce new operators as Ref. [78]. Therefore, . Here, As represents the steady-state of cs, qis, pis (i = 1,2), while is considered as perturbations with frequencies ωl, ωp, 2ωL – ωp, respectively. The zero order solutions for the steady-state operator can be written as
3. Tunable bistability
We use the conventional approach of the optomechanical system to perform the bistable behavior of the steady-state photon number and mechanical steady-state position of the resonators. The set of steady-state equations (Eq. (6)) can be written as
By rearranging the terms of Eq. (7), and then equating to zero, we obtain a third order polynomial of the steady-state photon number, i.e.,
where
Here, ns = |cs|2. It is clear that equation (8) has three roots, two are for stable branches and the third one is for the unstable branch of the steady-state photon number. We have found the corresponding regimes of the bistable curve by using the cubic roots equation such that ax3 + bx2 + cx + d = 0 with inflection and critical points, i.e., , xinflection = −b/3a. The inflection point of x is that where the curve changes the concavity. So, we can write
where ns− (ns+) and ns* are the critical points of lower (upper) stable and unstable branches of the bistable curve. These critical points describe the limit range of bistability window, i.e., Δns = ns+ – ns− and have a corresponding window on amplitude of the driving field, i.e., Δ(EL) = EL (ns+) – EL (ns−). Usually, an optomechanical system works in resolved-sideband regime, i.e., κ ≪ ωm. Moreover, the strength of the driving field directly affects the resolve-sideband regime; as a result, the photon intensity increases by increasing the strength of the driven field. So we can say that the steady-state photon number exhibits the bistability phenomenon for some specific value of the driving field. The cavity field detuning reaches a certain value with the increase of the driving field, called critical detuning. Hence, at this value, the threshold bistability in the system can be seen. From Eq. (8), one can find the critical value of the cavity field detuning like this
Equation (10) represents the condition for the critical value of the cavity field detuning in the absence of atom-field coupling as well as Coulomb coupling. However, this condition is modified by considering these couplings. Therefore, we have
To analyze the optical bistability and FWM in NEOMS, we select the parametric values from the available experiments[26,79,80] as well as from theoretical work closer to experiment.[72,73] We have considered that the nano-mechanical resonator NMR1 (NMR2) oscillates with frequency ω1(ω2) = 2π × 947 kHz. Other parameters are: field frequency ωc = 2π × 15.3 × 1014 Hz, field decay rate κ/2π = 215 kHz, quality factor Q = 6700, decay rate of NMR1 (NMR2) , mass of NMR1 (NMR2) m1 = m2 = 145 ng, and the detuning frequency δ = ω1(ω2). Typically N = 10587Rb atoms are coupled to the laser field, and the wavelength of the driving field λL = 1064 nm, laser power ℘l = 12 mW, and cavity length L = 25 mm. The strength of the pump field EL/2π = 20 MHz, atom-field frequency kHz, mirror-field frequency g0/2π = 8 kHz, Coulomb frequency gc/2π = 100 kHz, atom-field detuning Δa/2π = 30 kHz with decay rate γa/2π = 0.1 kHz, and cavity-field detuning Δc = 3κ.
The steady-state solution of Eq. (7) is sketched in Fig. 2 (bifurcation diagram) by taking cavity field detuning Δc = 3.5κ against the variation of input driving field. The critical points ns− and ns+ correspond to lower and upper stable branches and the inflection point ns* corresponds to the unstable branch of the bistable curve. If we start scanning with a low driving field EL, and gradually increase the driving field frequency, the intra-cavity intensity initially follows the lower stable branch S1 of the cubic, which extends from the origin to the first critical point P1(ns*). When the amplitude of the driving field reaches to 2π × 14 MHz, the lower stable branch jumps to the upper stable branch S3, which extends from the critical point P2 (ns*) to infinity. Keep continuing to increase the value of EL, the lower stable branch will continue to follow the upper stable branch. The center branch denoted by S2 is unstable and extends between the two inflection points P1(ns*) and P2(ns*). Note that the unstable branch S2 should be negative and cannot be experimentally observed. Further, it is noteworthy that the condition is compatible with the fact that the branches with negative slope are always unstable. On the other hand, if EL decreases from 2π × 20 MHz, the intra-cavity photon number will be found decreasing by following the upper branch at first; however, whenever it reaches the second critical point P2(ns*), it will jump down to the lower stable branch and continue to decrease further along that branch.
Fig. 2. The steady-state photon number |cs|2 is demonstrated against the driving field of amplitude EL. The parameter values in this particular case are taken as kHz, g0/2π = 3.5 kHz, gc/2π = 100 kHz, Δc = 3.5κ, γ1,2 = ω1,2/Q, Δa/2π = 20 kHz, and γa/2π = 0.1 kHz.
Next, we provide a detailed analysis of electro-optomechanical switch based on tunable bistability that depends on system parameters. Figure 3(a) represents the S-shaped (bifurcation) curve of the steady-state intra-cavity photon number |cs|2 and shows its variation against the amplitude EL for different atom-field coupling frequencies. Keep in mind that the curves overlap for different values of atom-field coupling. For g/2π = 20 kHz (black solid curve), the bistable curve lies at its upper stable branch and exhibits maximum photon number of the cavity field. But, the higher stable branch jumps to its lower stable branch once we increase the atom field coupling to 2π × 30 kHz (blue dotted curve); the width of the curve also decreases and critical points of the stable branch come closer, which exhibits low photon number. Further, by increasing the strength of the atom field coupling, the bistable branch jumps step by step to its lower branch for g/2π = 40 kHz (red dashed curve) and g/2π = 50 kHz (green thick dashed curve). The width of the bistable curve decreases and becomes narrow and narrow for higher values of atom-field coupling. On the other hand, if we consider the minimum values of atom-field coupling, the lower stable branch will follow the upper stable branch. Thus figure 3(a) elucidates that the atom-field coupling has greater influence on bistability and allows us to design an optical switch that can be controlled by atom-field coupling.
Fig. 3. (a) The variation in the intra-cavity photon intensity |cs|2 is shown against the amplitude of the driving field EL for different cases of g, gc, g0, and κ, respectively. The black solid, blue dotted, red dashed, and green longer dashed curves represent the bistable information for four different values of (a) g/2π = (20,30,40,50) kHz, (b) gc/2π = (100,150,200,250) kHz, and (c) g0/2π = (2,2.5,3,3.5) kHz, while in panel (d), black solid, blue dotted, and red dashed curves exhibit the bistable behavior for three different values of κ/2π = (0.5,1.0,1.5)215 kHz. We take g0 = 2π × 4 kHz in panels (a,b,d), g/2π = 20 kHz in panels (b,c,d), while γ1,2 = ω1,2/Q, Δa/2π = 200 kHz, and γa/2π = 0.1 kHz remain the same in each panel.
Similarly, we can also think an optical switch that can be tuned by Coulomb coupling. For this purpose, we plot the bistable curve of the intra-cavity photon number |cs|2 versus driving field for different frequencies of gc as given in Fig. 3(b). We have tuned the system for several values of gc through V1 and −V2. When the potential difference across each mechanical resonator is low, the Coulomb coupling between resonators is not strong enough, i.e., gc/2π = 100 kHz (black solid curve), we have maximum photon number and the system lies in the upper stable branch. When the voltage difference across each resonator increases, the charge also increases on each resonator. It enhances the Coulomb interaction as well as the radiation pressure force between the resonators. The non-linearity in the system increases and the intra-cavity photon number |cs|2 is suppressed inside the cavity. We observe this behavior in Fig. 3(b) for gc/2π = 150 kHz (blue dotted curve), gc/2π = 200 kHz (red dashed curve), and gc/2π = 250 kHz (green solid curve). This analysis reveals that Coulomb coupling has a remarkable effect on bistable behavior of the steady-state photon number along with the driving field.
In Fig. 3(c), we show the bistable behavior of the photon intensity |cs|2 versus the variation of the driving field for several coupling frequencies of g0. In the present scenario, the radiation pressure increases corresponding to the input field, as a result, strengthening the mirror-field coupling frequency. The interaction of NMR1 with cavity field raises the radiation pressure force, which causes the scattering of photon intensity in the cavity. At lower value of optomechanical coupling, i.e., g0/2π = 2.0 kHz (black solid curve), the bistable curve follows the upper stable branch. When the frequency of the optomechanical coupling gradually increases, the upper stable path follows the lower stable path for g0/2π = 2.5 kHz (blue dashed curve). The width of the bistable curve also decreases at the same time by increasing the frequency of g0. Further, strengthening the frequency of the optomechanical coupling, the second lower stable branch follows the third lower stable branch and vice versa. So far, there is a continuous variation in the bistable curve as well as in photon number with the variation of g0. The variation in photon number is also called flip-flop phenomenon. It means that the control of the intra-cavity photon intensity is a function of optomechanical coupling. Moreover, this control parameter provides a possibility of a realizable controllable optical switch.
Now, we examine the effect of the cavity decay rate κ on optical bistability. Therefore, we display the bistable behavior of the intra-cavity photon number against the driving field for various decay rates of the cavity field, as sketched in Fig. 3(d). Here, we have found that by increasing the decay rate, as a result, the width of the bistability window decreases. The black curve represents smaller cavity field decay κ, and maximum steady-state photon number is seen for κ/2π = 0.5 × 215 kHz. When decay rate increases (the loss in cavity field increases), the photon number decreases, therefore, the stable branch goes to lower stable branch for κ/2π = 215 kHz (blue dashed curve). At this time, the photon number is low and branch points come closer. Further, by strengthening the cavity decay rate, the cavity detuning field becomes larger and generates bifurcation phenomenon in the cavity. This suppresses the steady-state photon number, as a result, we have low photon number and shift the upper stable branch to lower stable branch. Hence, from the discussion of Fig. 3(d), it is concluded that increasing the decay rate of the cavity field, the maximum photon number is suppressed, and thus bistable curve keeps continuing jumping from upper stable branch to a lower stable in successive steps.
Next, we explain the optical bistability of NEOMS by considering different oscillators and show whether the photon bistable behavior increases or decreases. In Figs. 4(a) and 4(b), we display the bistable behavior of the steady-state photon number for different oscillators against the driving field EL. Figure 4(a) indicates that, if the mass of NMR1 is gradually increased corresponding to NMR2, the bistable curve of the NEOMS will move from lower stable branch to upper stable branch. But in contrast, the masses of the resonators are decreased, and the bistable behavior of the NEOMS will exhibit reverse behavior. Further, in Fig. 4(b), we sketch the bistable behavior of the NEOMS against the amplitude of the driving field EL for different frequencies of NMR1 corresponding to NMR2. We see at lower frequency of NMR1, i.e., ω1 = 0.9ω2, the bistable curve lies in its lower stable regime as labeled by black solid curve. However, operating higher frequencies of NMR1 corresponding to NMR2, i.e., ω1 = 1.0ω2, 1.2ω2, the bistable behavior increases and the thickness of the curve becomes broadened. This means that considering different mechanical resonators corresponding to the higher frequencies, the bistable behavior of the NEOMS increases and lies at its upper branch. Moreover, the same bistable behavior for steady-state position can be observed by considering different masses and frequencies of NMRj.
Fig. 4. The bistable behavior of the intra-cavity photon number |cs|2 is shown versus the amplitude of the driving field EL for several masses of NMR1 (NMR2). (a) The black, blue, orange curves represent the bistable information for m1 = m2, m1 = 1.2m2, and m1 = 1.5m2, respectively. (b) The several colors represent the variation in photon intensity for various frequencies, i.e., ω1 = 0.9ω2, 1.0ω2, and 1.2ω2. Other parameters are counted as ω1,2/2π = 947 kHz, κ/2π = 215 kHz, gc/2π = 100 kHz, g/2π = 10 kHz, Δa/2π = 20 kHz, and Δc = 3κ.
In Figs. 5(a)–5(d), the bistable behavior of the steady-state position q1s (q2s) of the mechanical mode NMR1 (NMR2), corresponding to the steady-state component in Eq. (4), is plotted as a function of the driving field. We observe that the bistable behavior of the first mechanical mode NMR1 occurs at a certain regime for different values of optomechanical coupling and atom-field coupling, respectively. Moreover, figures 5(a) and 5(b) show the bistable behavior of the mechanical modes of the hybrid NEOMS and describe their collective influence on atom-field coupling as well as on optomechanical coupling. In Fig. 5(a), there is a slight change in the bistable curve from g/2π = 10 kHz (black solid curve) to g/2π = 20 kHz (blue dashed curve). Moreover, when the atom-field coupling g is smaller than ω1(Δa), it barely changes the bistable curve of q1s. However, multistable behavior can be observed, when the atom-field coupling and the mechanical modes become very strong or ultra-strong.
Fig. 5. The bistable behavior of the steady-state position q1s is plotted as a function of driving field for different cases: (a) g/2π = 10 kHz, 20 kHz, and 30 kHz; (b) g0/2π = (2, 2.5, 3.0, 3.5) kHz. The variation in the steady-state position q2s is sketched as a function of driving field for several values of cavity field detuning, i.e., Δc, and Coulomb coupling gc in panels (c) and (d), respectively. Other parameters are considered as Δa/2π = 10 kHz, γ1,2 = ω1,2/Q, γa/2π = 0.1 kHz, Δc = 3κ, and gc/2π = 100 kHz.
Similarly, figure 5(b) represents the bistable behavior of the steady-state position of NMR1. We know that optomechanical coupling is more sensitive as compared to atom-field coupling. If we slightly change the coupling frequency of NMR1, the bistable curve shows a prominent change in the steady-state position q1s of NMR1. The solid black, blue dotted, red dashed, and purple curves show variation in the mechanical steady-state position q1s in a descending order by selecting different coupling frequencies g0/2π = (2, 2.5, 3, 3.5) kHz. Moreover, dealing with higher optomechanical coupling frequencies, the nonlinearity increases in the system, as a result, bistable curve jumps from upper stable branch to the lower branch. In this way, an optical switch can be realized by using the pump field as input and optomechanical coupling as the control.
Next, we investigate the bistable behavior of the mechanical steady-state position q2s by operating different frequencies of the cavity field detuning and Coulomb coupling, respectively. Therefore, in Figs. 5(c) and 5(d), we illustrate the bistable behavior of the mechanical steady-state position q2s as a function of pump field for several values of Δc and gc. The stable and unstable points are changed by changing the cavity field detuning and Coulomb coupling. It is clear that from Figs. 5(c) and 5(d), the S-shaped shows reverse behavior of Figs. 5(a) and 5(b). We see that the upper branch of the bistable curve jumps to next upper stable branch for higher values of cavity field detuning and Coulomb coupling. In this case, the bistable behavior of the steady-state position q2s increases and follows higher stable branches by operating Δc = (2.0, 2.5, 3.0)κ and gc/2π = (100, 120, 140) kHz, respectively. A similar transition occurs from the upper stable branch to the lower stable branch with the decreasing strength of the driving field. These observations realize the fact that the origin of the bistable behavior is the inherent non-linearities of Δc, EL, and gc, respectively. This shows that we can develop an optical switch that can be controlled by cavity field detuning and amplitude of the driving field.
Here, we are going to introduce other possibilities of an optical switch in electro-optomechanical system with the necessity of using different scanned parameters. The first possibility is shown in Fig. 6(a) by tuning optomechanical coupling for its different frequencies. We see that the bistability curve bends low towards right at g0/2π = 10 kHz (black solid curve) and lies at its near stable regime and has low photon number. But for g0/2π = 12 kHz, the curve becomes more bended and pushed the system to its lower stable branch, see blue solid line in Fig. 6(a). Further, increasing the values of optomechanical coupling to g0/2π = (14 kHz, 16 kHz), see the red and yellow lines, the bistable curve further moves to lower and lower stable regime. Hence, in this way, we can enable or disable an optical switch by using different values of g0.
Fig. 6. A sketch of the steady-state photon number |cs|2 against the normalized cavity field detuning Δc/κ for the cases (a) g0/2π = (10, 12, 14, 16) kHz; (b) gc/2π = (100, 150, 200, 250) kHz. (c) The steady state mean intra-cavity photon numbers |cs|2 against the normalized atomic detuning Δa/ω1 for different values of atom field coupling . (d) The steady state mean photon number |cs|2 against the dimensionless detuning Δc/κ for four different frequencies of NMR1. The remaining parameters are the same as used in Fig. 9.
The other possibility of an optical switch is by tuning the Coulomb coupling through different potential difference on NMR1 and NMR2. It is clear that from Fig. 6(b), the stability of the NEOMS moves from higher stable point to lower stable point at higher value of gc. The stability curve becomes more and more bended as we increase the values of gc and produces a great change in the photon number. This shows that we can develop an optical switch which is controlled by the potential difference across each resonator. Similarly, figure 6(c) displays the bistability of the intra-cavity photon intensity versus normalized atom-field detuning Δa/ω1. The stability curve is quite straight and there is no significant difference between the width of the peak versus Δa/ω1 for different values of atom-field coupling. The system shows a slight change in stable points of the bistable curve by operating different values of atom-field coupling. The bistability is maximum at g/2π = 2 kHz (black curve), and g/2π = 3 kHz (red curve) and the photon number has maximum intensity. Increasing the values of atom-field coupling, the stability curve shows small change from stable to the unstable point for g/2π = 4 kHz, 5 kHz (blue and yellow curves).
In Fig. 6(d), the bistable behavior of the steady state photon number |cs|2 is plotted against the normalized cavity field detuning Δc/κ for four different values ω1 = 1.0ω2, 1.1ω2, 1.2ω2, and 1.3ω2. In this case, the bistability curve has a reverse character of Fig. 6(c). The width of the bistability curve and photon number increases with the increase of the frequencies of NMR1 corresponding to NMR2. The curve becomes much straighter for ω1 = 1.3ω2 which exhibits maximum photon number. We observe that the obtained threshold bistability in an optomechanical system consists of double end moving coupled-charged resonators, the frequency of each resonator as to control.
To demonstrate the effect of laser power ℘l on bistable behavior of the steady-state positions q1s and q2s, therefore, in Figs. 7(a) and 7(b), we plot q1s and q2s versus normalized cavity field detuning Δc/κ for different values of pump power ℘l. The behavior of the curve in Fig. 7(a) is near Lorentzian at ℘l = 3 mW, while in Fig. 7(b), the curve exhibits complete Lorentzian shape at ℘l = 2 mW. This means that the bistable curve of the steady-positions q1s and q2s lies at its lower stable branch and we have a symmetrical Lorentzian curve centered at Δc = 0. The Lorentzian curve in Fig. 7(b) remains unchanged by increasing the input power ℘l and only gains the height, but in Fig. 7(a), the curve changes direction and shifts toward right as we increase the value of ℘l. It becomes more and more symmetric as we further strengthen the input power. The peak gets height and becomes wider at higher values of laser power; however, no changes are observed in its direction. Moreover, dealing with different higher laser powers, the bistable behavior of the steady-state position q2s reaches a threshold bistability at critical power. However, by selecting weak laser power, the peak height becomes shorter and shorter. In this case, the system will shift towards lower stable branch, as a result, the steady-state position q1s (q2s) exhibits lower bistable behavior. From the above discussions, we can say that the occurred bistability in the nano-electro-optomechanical system enables a controlled bistable switching of the photon intensity and mechanical steady-state positions by appropriately adjusting the input power and cavity field detuning frequency.
Fig. 7. The steady-state positions q1s and q2s of the mechanical modes NMR1 and NMR2 are illustrated against the cavity field detuning Δc/κ for four different input powers ℘l. Other parameters are Δa/2π = 200 kHz, γ1,2 = ω1,2/Q, γa/2π = 0.1 kHz, g0 = 8π kHz, g/2π = 20 kHz, and gc/2π = 100 kHz.
4. Four-wave mixing intensity
Here, we investigate the FWM intensity in the hybrid nano-electro-optomechanical system. First, we discuss the analytic approach of the FWM intensity by calling input-output theory[80]
Here, δ = ωp – ωl. In Eq. (12), the first term corresponds to the output driving field with amplitude EL and frequency ωl. The second and third terms appear due to probe field having frequencies ωp and 2ωL – ωp, respectively. Moreover, the frequency 2ωL – ωp in the third term of the output field corresponds to the Stoke field. In addition, the second term is related to electro-mechanical induced transparency (EMIT), whereas the third term generates the nonlinear effect in the system which is called four-wave mixing. In four-wave mixing process, the two photons of the driving field interact with a single photon of the probe field each with frequencies ωl and ωp, generating a new photon of frequency 2ωL – ωp. In this process, the two input frequencies are selected in such a way that ωL – ωp is near a transition of a dielectric medium, such as coherent Raman scattering. The four-wave mixing intensity enhances the non-linear effect in the system, especially in nano-electro-optomechanical system which behaves as a three-mode coupled system. The coherent driving field frequency as well as the probe field frequency remains invariant during reflection, transmission, and refraction in a nonlinear medium via elastic scattering. Hence, the energy and momentum are both conserved during the interaction of field frequencies. In another perspective, the energy transfer is a coherent process and the four-wave must retain a constant phase, relative to the others, in order to avoid any destructive interference.[82] Using the steady state operator in Eq. (5) and after some simplification, we obtain the FWM term like this
where
Here, ζ′(δ) = ζ(−δ), η′(δ) = η(−δ), and X′(δ) = X(−δ). The normalized FWM intensity in term of probe field can be written as
In Fig. 8(a), we display the FWM intensity against the normalized probe field detuning δ/ω1 for several values of optomechanical coupling g0, while keeping the absence of the atom-field coupling g as well as the electrostatic Coulomb coupling gc. When optomechanical coupling g0 is insufficiently strong, the FWM intensity is far away from resonant regime. The curve has a narrow width at g0/2π = 5 kHz (black curve) and consists of symmetric single peak centered at δ/ω1 = 1. This represents the single photon resonant process. Such behavior was previously studied in a two-mode optomechanical system,[64] where different powers were operated for photon resonance process. Here, we discuss the four-wave mixing process in a tripartite system operated by a single mode cavity field along with different optomechanical couplings. Figure 8(a) indicates that increasing the value of optomechanical coupling to g0/2π = 6 kHz (blue curve), the FWM intensity moves toward the near-resonant regime and the width of the curve becomes broadened. Further, increasing the optomechanical coupling frequency to its critical values, i.e., g0/2π = (7,8) kHz, the FWM intensity reaches the resonant regime. As a result, the red and orange curves are apparently split into two asymmetric peaks centered at δ/ω1 = 0.89,0.8 on the left and 1.1, 1.15 on the right side with single dip centered at δ/ω1 = 1. Thus we have observed two-photon resonant process. Also, the magnitudes of the sideband and interior peaks are increased with the increase of g0. In addition, the FWM is not suppressed at resonance regime, i.e., δ = ω1, it is because of constructive interferences of the reflected and transmitted fields to the coupling field of NMR1. Therefore, the FWM intensity has apparent normal mode splitting in Fig. 8(a). In this case, not only the probe field but also the driving field is stored inside the cavity. It is kept in mind that when the optomechanical coupling is switched off (g0 = 0), there will be no FWM intensity that can be seen amongst reflected, transmitted, and coupling fields. Therefore, optomechanical coupling plays a vital role in generating the FWM intensity.
Fig. 8. The display of the FWM intensity versus the dimensionless frequency δ/ω1 for different values of optomechanical coupling g0 for the cases: (a) both atom field coupling and Coulomb coupling are zero, i.e., g = 0 and gc = 0; (b) g = 0 and gc ≠ 0; (c) g ≠ 0 and gc ≠ 0. Other parameters are γ1 = ω1/Q, γa/2π = 0.1 kHz, gc/2π = 100 kHz, κ/2π = 215 kHz, g/2π = 10 kHz, and EL = 2π × 20 MHz.
Figure 8(b) displays the FWM intensity against the normalized probe field detuning for several values of optomechanical coupling g0 in the presence of Coulomb coupling gc, but the atom-field coupling is kept absent (g = 0). Here, we examine the impact of Coulomb coupling on FWM intensity with the variation of g0. In the present scenario, figure 8(b) has larger peak separation as compared to Fig. 8(a) for the same value of optomechanical coupling g0, however, no additional peak appears. This shows the verification of Eq. (8) that Coulomb coupling is independent of the cavity field and only depends on optomechanical coupling. Therefore, if we switch off g0, Coulomb coupling will automatically switch off (gc = 0). Here, we see that in Fig. 8(b), the black curve appears double peaks instead of single peak as appeared in Fig. 8(a) for g0/2π = 5 kHz. The results of Fig. 8(b) reflect the previous results of FWM in a composite photonic molecule optomechanics,[65] which was previously discussed at different detunings, i.e., red detuning and blue detuning. Moreover, in Ref. [65], it has been observed that the FWM intensity decreases with the increase of pump power at red detuning while increases with the increase of pump power at blue detuning. Since our system works at red sideband regime (red detuning), therefore, we explain the FWM intensity with a different approach and hence, photon intensity increases rather than decreases like Ref. [65]. Figure 8(b) indicates that the combined coupling frequency of g0 and gc has a greater radiation pressure than from the optomechanical coupling frequency alone. In other words, the Coulomb coupling is responsible to enhance the strength of optomechanical coupling g0. The result presented in Fig. 8(b) shows that the full width at half maximum (FWHM) increases and the amplitude of the FWM also increases by turning the NMR1 coupling from weak coupling (g0/2π = 5 kHz) to strong coupling (g0/2π = 8 kHz). Moreover, when the value of the optomechanical coupling g0 lies in a certain range, bistable behavior of intra-cavity photon number can exist, where intra-cavity photon number ns determined by Eq. (8) has three possible real values, respectively, the largest and smallest are stable, and the middle one is unstable as discussed in the previous section. It is to be noted that the FWM intensity decreases with the increase of the probe field detuning and becomes zero at δ = ω1. At this frequency, the photons are completely suppressed inside the cavity and destructive interference has occurred. Therefore, the FWM intensity process is completely suppressed at resonance regime and only the probe field is stored inside the system.
At last, we examine the behavior of FWM intensity for different optomechanical couplings by introducing the atom-field coupling. We plot the FWM intensity in Fig. 8(c), defined in Eq. (14), against the normalized probe field detuning for several values of optomechanical coupling. The infix of Fig. 8(c) shows the origin of this triple-peaked structure which is appropriated to a three-photon resonance process. It is clear that once we introduce the atom–field coupling, an additional peak appears. The peaks become broadened as we increase the optomechanical coupling. Moreover, in Fig. 8(c), the amplitudes and separation of the peaks are increased to higher values for larger values of g0. We note the shifted detuning centered at δ/ω1 = 0.9, 0.8, 0.75 on the left side and 1.12, 1.17, 1.22 on the right. One can see that in Fig. 8(c), the amplitude of atom-field frequency has only one peak centered at δ/ω1 = 0.4.
From the above discussion, we conclude that in the presence of atom-field and Coulomb couplings, NEOMS faces with a high degree of non-linearity and most of the intra-cavity photons are stimulated; as a result, the FWM intensity decreases which is clearly indicated in Figs. 8(b) and 8(c). The FWM intensity can be enhanced by increasing the optomechanical coupling or increasing the input driven power or decreasing the probe field detuning. In other words, the FWM process of the hybrid NEOMS can be controlled by adjusting the optomechanical coupling corresponds to the input driving field and probe field detuning.
5. Conclusion
We present an experimentally realizable scheme to realize an electro-optomechanical switch based on optical bistability and FWM intensity in a hybrid nano-electro-optomechanical system in the presence of two-level atoms trapped inside a Fabry–Pérot cavity. We report the existence of optical bistability in steady-state photon number as a function of mirror field coupling, atom-field coupling, and Coulomb coupling along with different resonators. In the studied system, the parameters allow a considerable threshold value of the cavity detuning and power which can be useful for the tremendous applications in developing optical switches at manageable intensities. Moreover, we have explained that the FWM intensity process is the function of optomechanical coupling and no FWM can be observed when the optomechanical coupling is switched off. Further, we have observed that the FWM intensity has one peak at low resonant regime when both electrostatic Coulomb coupling and atom-field coupling are absent. In the presence of atom-field coupling and Coulomb coupling, the FWM intensity exhibits three peaks due to the additional frequency in the system. Hopefully, our reported results can be applied in designing all-optical switch, storage of optical pulse signals, optical transistors, and the memory devices.
