Cross-Kerr effect in a parity-time symmetric optomechanical system

X. H. Zhao; B. P. Hou; L. Liu; Y. H. Zhao; M. M. Zhao

doi:10.1364/OE.26.018043

1. Introduction

An optomechanical system (OMS), which is one representative model to describe the interaction between the cavity intensity and the mechanical movement, is composed by a fixed mirror and a movable mirror that is coupled to the optical field in the cavity by radiation pressure [1–3]. It is well known that the OMS has many important applications in the fields of the quantum mechanics, quantum information and quantum optics, such as in the optomechanical quantum information processing [4], the entanglement between the macroscopic oscillator and the cavity field [5–8], the cooling of mechanical oscillators to their quantum ground states [9–11], detection of ultrahigh precision [12,13], the demonstration of the quantum nonlinearities [14–17], and the generation of the macroscopic quantum superposition [18].

In particular, the optomechanically induced transparency phenomenon (OMIT), one of the important absorption properties in the optomechanical system has attracted much attention [19–21]. Recently, many schemes to modulate the OMIT have been proposed to present many interesting optical phenomena. For example, in a two-mode optomechanical system composed by two cavity modes coupling to a common mechanical oscillator, the mechanical-mode splitting of the movable mirror as well as the OMIT in the splitting region were investigated [22]. In such an optomechanical system, the optomechanically induced absorption (OMIA) was predicted [23] and the OMIT and slow light were theoretically demonstrated [24]. Also, three-pathway OMIT in the coupled-cavity optomechanical system was investigated [25]. Recently, we have considered the OMIT and OMIA accompanied by the normal-mode splitting (NMS) in strongly tunnel-coupled optomechanical cavities [26]. Additionally, the OMIT with higher-order sidebands in a generic optomechanical system or in an optomechanical system coupled to a charged object was discussed [27,28].

Besides the optical manipulations of the OMIT in above discussions, the mechanical modulations of the OMIT by using external coherent forces to drive the optomechanical system were well studied. The OMIT in the charged optomechanical system driven by Coulomb force was used to precisely measure the charge number of small charged objects [29]. The OMIT in the optomechanical system driven by an external time-dependent force was studied [30]. Also, the phase-dependent OMIT in an coherently driven optomechanical system was theoretically investigated [31]. Recently, we considered the local modulation of the double OMIT and optomechanically induced amplification by using two time-dependent forces to drive two coupled nanomechanical oscillators, respectively [32].

In general, the real energy eigenvalues in quantum mechanics are accompanied by the Hermitian Hamiltonian. However, it was shown that certain non-Hermitian Hamiltonians that respect parity and time-reversal symmetries (PT symmetry) can exhibit real spectra [33]. Non-Hermitian-based quantum mechanics respecting PT symmetry attracted considerable attention because of its plentiful applications, such as in nonreciprocal light propagation [34–36], loss induced optical transparency [37], unidirectional reflectionless resonances [38,39], enhanced nonlinear optics [40], PT-symmetric phonon laser [41], and PT-symmetry-breaking chaos [42].

In the beginning, non-Hermitian-based complex quantum mechanics is debated on its experimental realization. Fortunately, due to the formal equivalence between the quantum-mechanical Schrödinger equation and the paraxial optical diffraction equation, it was recognized that the complex PT-symmetric potentials can be realized in optics by spatially modulating the refractive index with properly placed gain and loss in a balanced manner [37]. Subsequently, the PT symmetry has been experimentally realized in waveguides [34, 36], microwave billiard [43], microcavities [35], and two coupled LRC circuits [44]. Recently, the PT symmetry has been explored on the platform of the optomechanical system [41, 42, 45–48]. For example, it was theoretically demonstrated that the coupled optomechanical systems can be used to observe PT symmetry for the mechanical degrees of freedom [45]. The Optomechanically induced absorption was enhanced by the PT-symmetry [46]. And the OMIT in the optomechanical system with PT symmetry was well studied [47,48].

Recently, Heikkilä et al have proposed that the cross-Kerr interaction mediated by the Josephson effect can boost the optomechanical radiation-pressure coupling by several orders of magnitude [49,50]. Subsequently, the influences of the cross-Kerr coupling as well as optical parametric amplifier on the effective frequency, damping, normal mode splitting, ground state cooling, and steady state entanglement of an optomechanical system were studied [51]. And the effects of the cross-Kerr coupling on the red and blue sidebands as well as the effective mechanical damping were demonstrated [52]. Also, the bistability and the probe absorption in the presence of the cross-Kerr interaction were studied [53].

As is presented in above discussions, the optomechanical interaction induced phenomena have been extensively investigated in the context of the PT-symmetry [41, 42, 45–48]. To our knowledge, the physics related to the cross-Kerr interaction between the cavity field and the mechanical oscillator in the PT-symmetric optomechanical system have not been investigated. In the present paper, we shall add an active cavity to couple the optomechanical cavity with the cross-Kerr interaction, and investigate the OMIT influnced by the cross-kerr effect under PT symmetry. By investigating the effects of the tunnel-coupling and the cross-Kerr interaction in the passive-active case, we find some distinctive properties which are different from those in the double-passive case. The paper is organized as follows: In Sec. II we describe the model and dynamical equations. OMIT in double-passive cross-Kerr optomechanical system is investigated in Sec.III. The distinctive absorption properties in passive-active cross-Kerr optomechanical system are discussed in Sec.IV. Finally, in Sec. V we summarize our main results.

2. Model and dynamical equations

The optomechanical system considered here is shown in Fig. 1. It consists of a passive cavity with a loss rate κ₁ and an active cavity with a gain rate κ₂, which are directly connected each other with a coupling constant J. In the passive cavity, the cavity field is coupled to the mechanical oscillator with the optomechanical interaction induced by the radiation pressure, simultaneously with the cross-Kerr coupling between the passive cavity and the mechanical oscillator, which is mediated by a superconducting charge qubit [49, 50]. Here â_i( ${\hat{a}}_{i}^{+}$ ) denotes the annihilation (creation) operator for the ith cavity field with frequency ω_ci (i = 1, 2), and b̂(b̂⁺) for the annihilation (creation) operator of the mechanical mode with frequency ω_m. The left fixed mirror in the passive cavity is coherently driven by a strong coupling field $ε_{L} = \sqrt{(2 κ_{1} P_{L}) / ℏ ω_{L})}$ with frequency ω_L, and a probe field $ε_{p} = \sqrt{(2 κ_{1} P_{p}) / ℏ ω_{p})}$ with frequency ω_P, in which P_L and P_p are the powers of the coupling and probe fields, respectively.

Fig. 1 Schematic diagram of the two-cavity-coupled optomechanical system. In the passive cavity with a loss rate κ₁, the cavity field is coupled to the mechanical oscillator not only by optomechanical interaction but by the cross-Kerr coupling mediated by a superconducting charge qubit. Additionally, the left passive cavity is directly coupled to the right active cavity with a gain rate κ₂. The passive cavity is driven by a coupling field ∊_L and a probe field ε_P, respectively.

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The Hamiltonian of the system in the rotating frame at the frequency ω_L of the coupling field reads

\begin{array}{l} H & = & ℏ Δ_{1} {\hat{a}}_{1}^{+} {\hat{a}}_{1} + ℏ Δ_{2} {\hat{a}}_{2}^{+} {\hat{a}}_{2} + ℏ ω_{m} {\hat{b}}^{+} \hat{b} + ℏ g {\hat{a}}_{1}^{+} {\hat{a}}_{1} ({\hat{b}}^{+} + \hat{b}) \\ + ℏ G {\hat{a}}_{1}^{+} {\hat{a}}_{1} {\hat{b}}^{+} \hat{b} + ℏ J ({\hat{a}}_{1}^{+} {\hat{a}}_{2} + {\hat{a}}_{1} {\hat{a}}_{2}^{+}) + i ℏ ε_{L} ({\hat{a}}_{1}^{+} - {\hat{a}}_{1}) \\ + i ℏ ε_{p} ({\hat{a}}_{1}^{+} e^{- i δ t} - {\hat{a}}_{1} e^{i δ t}) . \end{array}

with δ = ω_p − ω_L and Δ_i = ω_ci − ω_L (i = 1, 2) is detuning of the coupling field from the corresponding cavity. The first three terms denote the free energies of the two cavities and the mechanical oscillator. The next two terms describe the optomechanical and the cross-Kerr couplings between the passive cavity and the mechanical oscillator, respectively. The direct coupling between the two cavities is denoted by the term with coupling coefficient J. The applications of the coupling and the probe fields on the left fixed mirror are given by the last two terms.

The quantum Langevin equations for the operators of the cavity and the movable mirror in a rotating frame with the coupling frequency ω_L are given by:

\begin{array}{l} {\dot{\hat{a}}}_{1} & = & - i Δ_{1} {\hat{a}}_{1} - i g {\hat{a}}_{1} (\hat{b} + \hat{b}) - i G {\hat{a}}_{1} {\hat{b}}^{+} \hat{b} - i J {\hat{a}}_{2} + ε_{L} \\ + ε_{p} e^{- i δ t} - κ_{1} {\hat{a}}_{1} + \sqrt{2 κ_{1}} {\hat{a}}_{in, 1} . \end{array}

{\dot{\hat{a}}}_{2} = - i Δ_{2} {\hat{a}}_{2} - i J {\hat{a}}_{1} + κ_{2} {\hat{a}}_{2} + \sqrt{2 | κ_{2} |} {\hat{a}}_{in, 2} .

\dot{\hat{b}} = - i ω_{m} \hat{b} - i g {\hat{a}}_{1}^{+} {\hat{a}}_{1} - i G {\hat{a}}_{1}^{+} {\hat{a}}_{1} \hat{b} - γ_{m} \hat{b} + \hat{ξ} .

in which γ_m and ξ̂ are the damping rate and the noise operator associated to the Brownian motion of the mirror. And, â_in,k (k = 1, 2) is the input vacuum in the ith cavity with zero mean value.

In an optomechanical system, the Heisenberg-Langevin equations are nonlinear due to the optomechanical interaction, and it is difficult to get an analytic solution to these equations. The probe field is weaker than the coupling field in the optomechanically induced transparency, we could use the linearization approach of quantum optics to get an analytical understanding. Correspondingly, the cross-Kerr coupling between the passive cavity and the mechanical oscillator should be linearized because of its smaller strength than that of the optomechanical interaction [49]. Then, we have linearized the cross-Kerr interaction following the linearization procedure used in linearizing optomechanical interaction. Specifically, the variables of the mirror and the cavities can be divided into the steady parts and the fluctuation ones as â_i = a_si + δâ_i, ${\hat{a}}_{i}^{+} = a_{si}^{*} + δ {\hat{a}}_{i}^{+}$ (i = 1, 2), b̂ = b_s + δb̂, ${\hat{b}}^{+} = b_{s}^{*} + δ {\hat{b}}^{+}$ . The linearized Heisenberg-Langevin equations for the fluctuation operators are obtained by neglecting all the nonlinear terms. The dynamical behavior of the system can be obtained by solving the linearized equation of motion for the fluctuations (δâ₁, δâ₂, δb̂) around their steady-state parts (a_s1, a_s2, b_s), which are mainly determined by the coupling field. The solutions of the steady parts (a_s1, a_s2, b_s) are obtained as

\begin{array}{l} a_{s 1} = \frac{ε_{L} (κ_{2} - i Δ_{2})}{(κ_{1} + i Δ_{111}) (κ_{2} - i Δ_{2}) - J^{2}}, \\ a_{s 2} = \frac{i J ε_{L}}{(κ_{1} + i Δ_{111}) (κ_{2} - i Δ_{2}) - J^{2}}, \\ b_{s} = \frac{- i g a_{s 1}^{*} a_{s 1}}{γ_{m} + i Ω_{m}} . \end{array}

where

Δ_{11} = Δ_{1} + g (b_{s} + b_{s}^{*})

,

Δ_{111} = Δ_{11} + G b_{s} b_{s}^{*}

,

Ω_{m} = ω_{m} + G a_{s 1}^{*} a_{s 1}

. Correspondingly, the Langevin equations for the expectation values (δa₁, δa₂, δb) of the fluctuations (δâ₁, δâ₂, δb̂) are given by

\begin{array}{l} δ {\dot{a}}_{1} & = & - (i Δ_{111} + κ_{1}) δ a_{1} - i g a_{s 1} (δ b + δ b^{*}) \\ - i G a_{s 1} (b_{s}^{*} δ b + b_{s} δ b^{*}) - i J δ a_{2} + ε_{p} e^{- i δ t} . \end{array}

δ {\dot{a}}_{2} = - (i Δ_{2} - κ_{2}) δ a_{2} - i J δ a_{1} .

\begin{array}{l} δ \dot{b} & = & - (i ω_{m} + γ_{m}) δ b - i g (a_{s 1}^{*} δ a_{1} + a_{s 1} δ a_{1}^{*}) \\ - i G b_{s} (a_{s 1}^{*} δ a_{1} + a_{s 1} δ a_{1}^{*}) - i G a_{s 1}^{*} a_{s 1} δ b . \end{array}

Using the following ansatz (in the rotating frame)

\begin{array}{l} δ a_{1} = a_{1 +} e^{- i δ t} + a_{1 -} e^{i δ t}, \\ δ a_{2} = a_{2 +} e^{- i δ t} + a_{2 -} e^{i δ t}, \\ δ b = b_{+} e^{- i δ t} + b_{-} e^{i δ t} . \end{array}

with δ being redefined by δ = σ * ω_m + ω_m, we get

a_{1 +} = \frac{ε_{p} (Q_{21} - M_{+})}{{| Q_{1} Q_{2} |}^{2} N_{12}^{2} + (M_{-} + Q_{12}) (M_{+} - Q_{21})},

where

\begin{array}{l} N_{12} = \frac{1}{N_{-}} + \frac{1}{N_{+}}, \\ Q_{12} = \frac{{| Q_{1} |}^{2}}{N_{-}} - \frac{{| Q_{2} |}^{2}}{N_{+}} + \frac{J^{2}}{H_{-}}, \\ Q_{21} = \frac{{| Q_{2} |}^{2}}{N_{-}} - \frac{{| Q_{1} |}^{2}}{N_{+}} - \frac{J^{2}}{H_{+}} . \end{array}

and

\begin{array}{l} H_{\pm} = κ_{2} \pm i Δ_{2} + i δ, \\ Q_{1} = i g a_{s 1} + i G a_{s 1} b_{s}^{*}, \\ Q_{2} = i g a_{s 1} + i G a_{s 1} b_{s}, \\ M_{\pm} = - κ_{1} \pm i Δ_{111} + i δ, \\ N_{\pm} = \pm i ω_{m} \pm i G a_{s 1} a_{s}^{*} - γ_{m} + i δ . \end{array}

3. OMIT in doulbe-passive cross-Kerr optomechanical system

Now we investigate the effects of the optical tunnel coupling on the probe absorption properties in the double-passive optomechanical system with κ₁ > 0 and κ₂ < 0. In the case of the weak tunnel coupling for J < |κ₂ + κ₁|/2, which is shown in Fig. 2(a), it is shown that the absorption doublet decreases with the tunnel strength J and leads to a shallower transparency window. When increasing the tunnel coupling into the strong region with J > |κ₂ + κ₁|/2, we can see that the two absorption peaks still decrease with J. If the tunnel coupling is increased further, e.g. J = 5κ₁ and J = 7κ₁, which are shown respectively by dashed and dotted curves in Fig. 2(b), the absorption peaks are distorted outside and separated more widely from each other. This distorted absorption spectrum is superposed by the OMIT spectrum induced by the optomechanical interaction and the normal-mode splitting resulted from the strong tunnel coupling [23].

Fig. 2 The absorption ε_R as a function of the redefined detuning σ = (δ − ω_m)/ω_m for different values of the weak tunnel coupling (J < |κ₂ + κ₁|/2) (a): J = 0.5κ₁(black, solid curve); J = κ₁(red, dashed curve); J = 2κ₁(blue, dotted curve). In the case of the strong tunnel coupling (J > |κ₂ + κ₁|/2), the tunnel coupling J is given by different values as (b): J = 3κ₁(black, solid curve); J = 5κ₁(red, dashed curve); J = 7κ₁(blue, dotted curve). The other values of the parameters are given by: P_L = 3mW, g = 250, ω_c1/2π = 1.3GHz, κ₁/2π = 0.1MHz, κ₂ = −5κ₁, ω_m/2π = 6.3MHz, γ_m = 40Hz and G = 0.

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Now we consider the cross-Kerr effects on the probe absorption in the double-passive optomechanical system. At this time, the probe absorption spectrum in Fig. 3 is changed into an asymmetric OMIT spectrum, in which one absorption peak with broader width is located around the resonant point σ = 0 (δ = ω_m) and another absorption line lies on the right side of δ = ω_m. Also, it is shown that the right absorption line becomes thinner and moves father from the resonant point with the increase of the cross-Kerr parameter G.

Fig. 3 The absorption ε_R as a function of the redefined detuning σ = (δ − ω_m)/ω_m for different values of the cross-Kerr parameter G : G = 0.0005g (black, solid curve); G = 0.001g (red, dashed curve); G = 0.002g (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2 except for J = 0.

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The asymmetry of the absorption spectrum is the distinctive feature induced by the cross-Kerr effect, which can be confirmed by the quantitative findings. In the resolved sideband regime ω_m ≫ κ₁, we get the reduced form of the Eq. (10) as

δ a_{1} = \frac{ε_{p}}{κ_{1} - i σ - \frac{{| g a_{s 1} - G b_{s}^{*} a_{s 1} |}^{2}}{i (σ - G a_{s 1}^{*} a_{s 1}) - γ_{m}} .}

From Eq. (13), we can get the following characteristic parameters: (i) the optomechanically induced transparency (OMIT) is located at ω_K = G|a_s1|², which is induced by the cross-Kerr interaction. This can be seen by the term with G in Eq. (1), which provides a mechanical frequency shift by the cavity photons. Therefore, the resonance excitation leads the OMIT at the shifted frequency. This can be confirmed by the quantitative findings that the normalized frequency shift ω_K/ω_m are given by 0.0407 in solid curve, 0.0814 in dashed curve and 0.1627 in dotted curve, which are well coincided with their locations shown in Fig. 3, respectively. This implies that we can use the location of the OMIT induced by the cross-Kerr effect in the probe absorption spectrum to accurately measure the cross-Kerr strength G. On the other hand, it is seen from the term with G in Eq. (1) that the cross-Kerr effect can lead to the cavity frequency shift by the quanta number of the mechanical resonator. Thus the location of the OMIT in Fig. 3 can also be used to detect the quanta in the mechanical resonator or to prepare the mechanical Fock states. (ii) We can get the normalized frequencies σ_L and σ_R centered by the left resonance peak and the right line, which are given by

\begin{array}{l} σ_{L} = - \frac{G a_{s 1}^{*} a_{s 1} {| g a_{s 1} - G b_{s}^{*} a_{s 1} |}^{2}}{{| G a_{s 1}^{*} a_{s 1} |}^{2} + {(κ_{1} - γ_{m})}^{2}}, \\ σ_{R} = G a_{s 1}^{*} a_{s 1} + \frac{G a_{s 1}^{*} a_{s 1} {| g a_{s 1} - G b_{s}^{*} a_{s 1} |}^{2}}{{| G a_{s 1}^{*} a_{s 1} |}^{2} + {(κ_{1} - γ_{m})}^{2}} . \end{array}

It is shown in Eq. (14) that the right absorption line is located symmetrically about the resonant point and is successively shifted by

G a_{s 1}^{*} a_{s 1}

. Due to the smaller value of the fraction part in Eq. (14), the distance between two absorption peaks is mainly determined by the cross-Kerr

G a_{s 1}^{*} a_{s 1}

. This is why the right absorption line is remarkably shifted to the right side while the left peak is not explicitly moved with the cross-Kerr coupling. For the given values in Fig. 3, we can get the the locations normalized by the mechanical frequency

ω_{m} : (\frac{σ_{L}}{ω_{m}}, \frac{σ_{R}}{ω_{m}}) \approx (- 0.0118, 0.053)

, (−0.007, 0.088), (−0.004, 0.167), which are well coincided with the locations shown in the solid, dashed and dotted curves in Fig. 3.

(iii) Similarly, we get the widths D_L and D_R for the left resonance peak and the right line in the probe absorption spectrum, which are given by

\begin{array}{l} D_{L} = κ_{1} - \frac{(κ_{1} - γ_{m}) {| g a_{s 1} - G b_{s}^{*} a_{s 1} |}^{2}}{{| G a_{s 1}^{*} a_{s 1} |}^{2} + {(κ_{1} - γ_{m})}^{2}}, \\ D_{R} = γ_{m} + \frac{(κ_{1} - γ_{m}) {| g a_{s 1} - G b_{s}^{*} a_{s 1} |}^{2}}{{| G a_{s 1}^{*} a_{s 1} |}^{2} + {(κ_{1} - γ_{m})}^{2}} . \end{array}

It is shown from Eq. (15) that the widths of the left resonance peak and the right line are mainly determined by κ₁ and γ_m due to the smaller value of the fraction part, respectively. To make the interference more obvious in the optomechanical system, the cavity decay κ₁ is assumed to be quite larger than the mechanical decay γ_m [21]. This is why the resonance peak around the resonant point is much wider than the right line. Additionally, we can see that the cross-Kerr effect reduces the width for the right absorption line while increases that for the left absorption peak. This can be confirmed by the quantitative findings:

(\frac{D_{L}}{ω_{m}}, \frac{D_{R}}{ω_{m}}) \approx (0.0112, 0.0046)

, (0.0145, 0.0014), (0.0155, 0.0004), which are well coincided with the locations shown in the solid, dashed and dotted curves in Fig. 3.

In above discussions, we consider how the probe absorption is influenced by the optical tunnel or the cross-Kerr interaction individually. Here we display the optical properties when both the optical tunnel coupling and the cross-Kerr interaction are simultaneously applied on the system. First, we display the variation of the optical properties with the tunnel coupling for fixed cross-Kerr parameter with G = 0.001g. In the weak tunnel coupling between two cavities J < |κ₂ + κ₁|/2, it is shown in Fig. 4(a) that the broader peak around resonance and the absorption line induced by the cross-Kerr effect are not shifted while their heights are decreased with increase of J. When the tunnel coupling is tuned into strong region of J > |κ₂ + κ₁|/2, shown in Fig. 4(b), the right absorption line becomes broader and the peak around the resonant point is split into two ones due to the normal-mode splitting induced by the strong tunnel coupling between the two cavities. Second, we consider the probe absorption properties varying with the cross-Kerr parameter G when the two cavities are tunnel coupled, which is not plotted here due to the length limit of the paper. The absorption spectrum exhibits the similar behavior to that in the absence of the tunnel coupling shown in Fig. 3 except that the absorption peak around the resonant point becomes broader and the right absorption line turns taller with the increase of the cross Kerr parameter.

Fig. 4 The absorption ε_R as a function of the redefined detuning σ = (δ − ω_m)/ω_m with G = 0.001g and for different values of the weak tunnel coupling (J < |κ₂ + κ₁|/2) (a): J = κ₁(black, solid curve); J = 1.6κ₁(red, dashed curve); J = 2κ₁(blue, dotted curve). In the case of the strong tunnel coupling for J > |κ₂ + κ₁|/2, the tunnel coupling J is given by different values as (b): J = 3κ₁(black, solid curve); J = 3.6κ₁(red, dashed curve); J = 5κ₁(blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.

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4. OMIT in passive-active cross-Kerr optomechanical system

Following the similar pattern to that in above section, we now consider the optical properties in the passive-active cross-Kerr optomechanical system. As is well known that the PT-symmetric optical system can be realized by using two linearly coupled optical modes with a gain in one mode and a loss in another one [41,42,46–48]. To investigate the PT-symmetry in the present system, we only consider the optical modes by ignoring the driving fields and take into account the optomechanical interaction and cross-Kerr effect by replacing the mechanical variables with their steady values, which provide nonlinear frequency shift of the optical modes. The dynamical equations for the optical variables are given by

i \frac{\partial}{\partial t} (\begin{array}{l} {\hat{a}}_{1} \\ {\hat{a}}_{2} \end{array}) = (\begin{matrix} Δ_{111} - i κ_{1} & J \\ J & Δ_{2} + i κ_{2} \end{matrix}) (\begin{array}{l} {\hat{a}}_{1} \\ {\hat{a}}_{2} \end{array}) .

in which the nonlinear interactions induced frequency shift is

Δ_{111} = Δ_{1} + g (b_{s}^{*} + b_{s}) + G {| b_{s} |}^{2}

. By diagonalizing Eq. (16), we obtain the complex frequency as

ω_{\pm} = \frac{1}{2} {Δ_{111} + Δ_{2} + i (κ_{2} - κ_{1}) \pm \sqrt{{[Δ_{111} - Δ_{2} - i (κ_{2} + κ_{1})]}^{2} + 4 J^{2}}} .

The real and imaginary parts of the complex frequency in Eq. (17) denote eigenfrequencies of the two optical supermodes as well as the corresponding linewidths. In above discussions, the system is investigated under the resonant conditions: Δ₁₁₁ = Δ₂ = ω_m. Then the complex frequency in Eq. (17) reduces to the expression

ω_{\pm} = ω_{m} + \frac{i}{2} (κ_{2} - κ_{1}) \pm \sqrt{4 J^{2} - {(κ_{2} + κ_{1})}^{2}} .

We can see that when J > |κ₂ + κ₁|/2 the optical Hamiltonian has two real eigenfrequencies and two-coupled optical system is PT symmetric. If J < |κ₂ + κ₁|/2, the eigenfrequencies of the optical Hamiltonian are complex and its PT-symmetry is broken. Without loss of generality, we set the optical loss and gain have different amplitudes instead in a balanced manner.

Firstly, we display how the tunnel coupling between the cavities influences the probe absorption in the passive-active case by setting κ₁/2π = 0.1MHz, κ₂ = 5κ₁. It is shown in Fig. 5(a) that the two peaks in the OMIT spectra increase with the tunnel coupling parameter J, which take contrary behavior to that in the double-passive case shown in Fig. 2(a). In the case of strong tunnel-coupling region, the OMIT distorted by normal-mode splitting in Fig. 2(b) in the double-passive case is changed into the spectrum of optomechanically induced amplification in the whole given frequency range shown in Fig. 5(b), in which two amplification dips located symmetrically around the resonant point become wider with increase of the tunnel parameter. The spectrum in Fig. 5(b) is almost the mirror symmetry of that in the double-passive case shown in Fig. 2(b). The probe absorption influenced by the cross-Kerr coupling are shown in Fig. 6, in which the two cavities are coupled to construct the passive-active Hamiltonian. Comparing with Fig. 3, we find that in the passive-active case the broad absorption peak becomes taller and thinner, while the right line turns shorter with the increase of the cross-Kerr coupling.

Fig. 5 The absorption ε_R as a function of the redefined detuning σ = (δ − ω_m)/ω_m in the passive-active case (κ₁/2π = 0.1MHz, κ₂ = 5κ₁). The weak tunnel coupling (J < |κ₂+κ₁|/2) is given by different values (a): J = κ₁(black, solid curve); J = 1.6κ₁(red, dashed curve); J = 2κ₁(blue, dotted curve). The strong tunnel coupling (J > |κ₂ + κ₁|/2) is given by different values (b): J = 4κ₁(black, solid curve); J = 6κ₁(red, dashed curve); J = 8κ₁(blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.

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Fig. 6 The same settings are used as those in Fig. 3 except for the presence of tunnel coupling (J = 2κ₁) and in the passive-active case(κ₂ = 5κ₁).

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In the presence of the cross-Kerr effect, the variation of the probe absorption spectra with the tunnel coupling, shown in Fig. 7, tells that the absorption peaks in the passive-active case are narrower and become taller with increase of the tunnel coupling J, which is contrary to the double-passive case shown in Fig. 4(a). When increasing the tunnel coupling up into the strong region J > |κ₂ + κ₁|/2, the two peaks in the OMIT spectrum display different behaviors from those in the double-passive cross-Kerr optomechanical system. Specifically, the absorption peak around resonant point in Fig. 8(a) with J = 4κ₁ is changed into an amplification window and the right absorption line becomes taller than that in the case of the weak tunnel coupling shown in Fig. 7. When the tunnel coupling J is increased further up to J = 6κ₁, which is shown in Fig. 8(b), the absorption peak around resonance is not only turned into an amplification window, but the right absorption line is changed into an amplification line. This can be used to realize a switching from absorption to amplification by only adjusting the tunnel interaction. If the tunnel coupling J is set as J = 8κ₁ shown in Fig. 8(c), the amplification window is split into two parts due to the normal-mode splitting induced by the strong tunnel coupling between the two cavities. From Fig. 8, we can see that the two absorption peaks in the OMIT spectrum are changed into amplification dips in different steps: the absorption peak around resonance is converted into an amplification one faster than the right absorption line by adjusting the tunnel coupling strength.

Fig. 7 The same settings are used as those in Fig. 4(a) except for the presence of cross-Kerr effect (G = 0.001g).

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Fig. 8 The absorption ε_R as a function of the redefined detuning σ = (δ − ω_m)/ω_m with G = 0.001g in the passive-active case. The strong tunnel coupling (J > |κ₂ + κ₁|/2) is given by different values: (a) J = 3κ₁(black, solid curve); (b) J = 3.6κ₁(red, dashed curve); (c) J = 5κ₁(blue, dotted curve).The same settings are used as those in Fig. 4(b) except for the presence of cross-Kerr effect (G = 0.001g).

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Additionally, the comparison between the Fig. 4(b) and Fig. 8 tells that the normal-mode splitting in the double-passive case is more sensitive than that in the passive-active case. It is seen from Eq. (18) that the two eigenfrequencies have the same real value for J < |κ₂ + κ₁|/2 and lead to a single broad resonance at δ = ω_m (σ = 0). This can explain the facts that a broad absorption peak or an amplification dip are located around the resonant point in the weak tunnel-coupled region, which are displayed by Fig. 4(a) and Figs. 8(a) and 8(b), respectively. If the tunnel coupling lies in the strong region, i.e. J > |κ₂ + κ₁|/2, the eigenfrequencies bifurcate and leads to the appearance of the normal-mode splitting, which is illustrated by Figs. 4(b) and 8(c).

5. Summary

We have investigated the cross-Kerr effects in a parity and time symmetric optomechanical system, in which one active cavity with a gain is directly coupled to a passive cavity injected by the cross-Kerr coupling and the optomechanical interaction between the cavity field and the mechanical oscillator. For a full investigation, we firstly consider the cross-Kerr effects in the double-passive case. It is shown that the combined effect of the optomechanical interaction and cross Kerr effect leads to an asymmetric OMIT spectrum, in which the broad absorption peak is located around the resonant point and the sideband absorption line lies at the frequency mainly determined by the cross Kerr strength. In the passive-active case, the absorption peaks are increased with the weak tunnel coupling, which is contrary to that in the double-passive case. When the tunnel coupling is tuned into the strong regime, the broad absorption peak and the sideband absorption line are changed into the corresponding amplification dips in succession. This can be used to realize a switching from absorption to amplification by only adjusting the tunnel interaction.

Funding

National Natural Science Foundation of China (10647007); the Science Foundation of Sichuan Province of China (2018JY0180).

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Cross-Kerr effect in a parity-time symmetric optomechanical system

Abstract

1. Introduction

2. Model and dynamical equations

3. OMIT in doulbe-passive cross-Kerr optomechanical system

4. OMIT in passive-active cross-Kerr optomechanical system

5. Summary

Funding

References and links

Cited By

Figures (8)

Equations (18)

Optics Express