1. Introduction

In the emerging field of optomechanical systems, numerous rapid progresses with the radiation pressure have recently been obtained during measuring and cooling the mechanical motion [1,2]. It is reported that the quantum control of mechanical motion and the control of light or microwave field in cavity optomechanical systems can be achieved to some extent [3, 4]. The cooling of a mechanical resonator (MR) to its ground state of motional degree of freedom is an important procedure [5, 6]. Experimentally, it has been realized that micro- and nanomechanical resonator can be cooled to or near its quantum ground state in various mechanical systems either through direct cryogenic cooling [7], or laser cooling using microwave [8, 9] and optical cavity fields [10]. When the MR is cooled to the ground state, it is possible to generate and detect the non-classical states of the MR, such as the squeezed state [3, 11, 12] and the entangled state [13]. The squeezed state of a movable mirror can be obtained via feeding broadband squeezed vacuum light into the cavity in dissipative optomechanics [11]. By driving an optomechanical cavity with two controllable lasers with different amplitudes, the dissipative mechanism with the driven cavity acts as an engineered reservoir, the squeezed state of a MR can also be obtained [14]. Recently, a scheme for generation of the non-classical state of the MR is proposed, in which the MR is nonlinearly coupled to a two-level system [15]. The non-classical states of the MR can also be produced by applying the generation of motional dark states [16].

The nonlinear coherent state (NCS) as another concerned non-classical states has also attracted many interests in view of its promising application in quantum algebras [17–19], which is based on noncommutative space rather than the usual space [20]. In the description of the motion of a trapped ion, Vogel et al. put forward the definition of the NCS as the eigenstate of the non-Hermitian operator f(n̂)a [21,22], where the deformation function f(n̂) is an operator-valued function of number operator n̂ = a^†a. In this case, the commutator [f(n)a, a^† f(n)] is not a constant or a linear function of the generators of the Lie algebra but nonlinear in the generators. It is found that the NCS exhibits non-classical features such as amplitude squeezing and self-splitting, which are accompanied by pronounced quantum interference effects [21,23]. The generation schemes of the NCS are explored in some other literatures, e.g. a single-atom laser [17], a micromaser under the intensity-dependent Jaynes-Cummings model [24], an exciton in a wide quantum dot [25], and a trapped ion [21,26]. The stationary state of a single-atom laser is shown to be a phase-averaged NCS and the interaction-induced transition probabilities of the single-atom laser effectively depend on the field intensity [17]. The quantized motion of an exciton in a wide quantum dot interacting with two laser beams can be prepared in a NCS and the exciton states can be utilized for quantum information processes [25]. The NCS is employed to discuss the optimization of quantum information which is important in binary (or multibinary) communication [27]. Usually, the discussion of NCS is related with the non-Gaussian quantum states. The occurrence of the negativity of the Wigner function (WF) for some non-Gaussian states implies that they possess the non-classical properties. The non-Gaussian state plays a significant role in entanglement distillation [28, 29], quantum computation with cluster states [30], loophole-free Bell tests with the continuous variables [31, 32], and quantum error correction [33]. Besides, the non-Gaussian state can be exploited to improve the quantum communication protocols in quantum information processing [34] and improve the optimal estimation of losses at the ultimate quantum limit [35]. Recent years, it is found that the optomechanical systems possess the intrinsic nonlinearity thus hold promising applications in quantum information processing [36] and ultra-precision measurement [37]. So how to prepare the NCS of a MR in the optomechanical system is a topic of great significance.

The generation of a non-classical light source is one of the major requirements of quantum information processor associated with modern semiconductor fabrication technology [38]. For this purpose, employing photon blockade phenomenon is one of methods. This phenomenon is first proposed by Imamoḡlu [39], which occurs when the absorption of a first input photon by an optical device blocks the transmission of a second one, thereby leading to non-classical output photon statistics. The inherent non-classical property at single photon level provides a way to control single photon, which is commonly used in quantum key distribution system [40] and plays a fundamental role in proposed devices for quantum information processing [38]. The photon blockade is first observed in cavity quantum electrodynamics(QED) systems with the strong atom-cavity coupling [41], and then has been demonstrated in different solid-state schemes, e.g. in cavity QED systems with a quantum dot (QD) in photonic crystal cavities [42], with a QD strongly coupled to a photonic crystal resonator [43] and other circuit-QED systems [44, 45]. Recently, much attention has been focused on multi-photon blockade [46–48]. The photon pairs generated by the blockade are often required for generation of the entangled light. Two-photon blockade in the strong-coupling qubit-cavity system is investigated using a modified Lindblad master equation [47]. Two- and three- photon blockade phenomena observed respectively in the standard cavity QED and circuit QED systems are attributed to photon-induced tunneling [46]. It is demonstrated that multi-photon blockade may inhibit additional photon being absorbed, which is caused by the nonlinearity of the Jaynes-Cummings (JC) model [48]. In a word, it makes sense to propose a scheme in the optomechanical system for preparing a non-classical state of the MR via the multi-photon blockade.

In this paper, we propose a scheme for generating the NCS of the phonons via the two-photon blockade and the dark state of the system, then investigate its non-classical properties. A two-level QD is driven by a strong laser and the microcavity is engineered within a photonic band-gap (PBG) material. In the dressed state basis induced by the strong field, resonant excitation of two-photon can be implemented by the weak driving field. In this case, the two-photon blockade can be realized in the optomechanical system by tuning the frequency of the weak driving field. The optomechanical system can evolve into a dark state due to the damping of the microcavity and the decay of the dressed QD at selected frequencies in the PBG material. As a result, the phonon mode of the MR can be prepared in a NCS, which is a non-classical state and possesses the properties of the sub-Poisson statistics. We also demonstrate the Wigner functions (WF) of NCS, which negativity implies its non-classicality.

The paper is organized as follows. In section 2, we show the physical model for the driven QD inside an optomechanical microcavity coupled to a weak laser. The whole system is transformed into the polaron frame, and the master equation of the optomechanical system is obtained using the rotating wave, secular and Born-Markov approximations. The preparation of the NCS and its properties are investigated in section 3. Finally, we draw our conclusions in section 4.

2. System and master equations

2.1. Description of the system

The system under consideration consists of a high-Q optomechanical microcavity, which is technically engineered inside a PBG material and coupled with an excitonic two-level QD. The optomechanical microcavity with the frequency ω_c can be coupled with the MR of frequency ω_m via a combination of radiation pressure and photostriction [49]. The cavity is also driven by a weak laser field with frequency ω_p and amplitude α_p, as shown in Fig. 1. The two-level QD is represented by the usual Pauli spin- $\frac{1}{2}$ operators σ₊ and σ₋, which satisfy the commutation relations [σ₊, σ₋] = σ_z and [σ_z, σ_±] = ±2σ_±. It is placed inside the microcavity and coupled to another strong laser field with frequency ω_L and Rabi frequency Ω. This model can be realized by a optomechanical microcaviy, which includes a defective photonic crystal nanobeam and a MR with the “breathing” mode [50], and a QD ironing the surface of the beam (see Fig. 1). The motion of the MR manifests as a successive of contractions and expansions along the x axis and is driven by a strong laser. The microcavity is driven another weak laser light. In the rotating frame at the frequency of ω_L, the total Hamiltonian of the system is given by (setting h̄ = 1 throughout the paper)

 

Fig. 1 Schematic of the considered system. A photonic crystal nanobeam is clamped at both ends. The optomechanical microcavity with the frequency ω_c can be coupled with the MR of frequency ω_m via a combination of radiation pressure and photostriction [49]. The MR can travel fast and alternating expansions and contractions along the x axis, which we refer to as a “breathing” mode [50]. A QD (the small red dot in Fig) driven by a strong laser is within the photonic crystal microcavity.

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In Eq. (3), the information about the frequency dependent of the photonic crystal is stored in the coupling strength g_λ (ω_λ) which can be written as g_λ (ω_λ) = g_λD(ω_λ) with g_λ being a constant proportional to the dipole moment of the QD and D(ω_λ) corresponding to the transfer function of the photonic crystal reservoir. Experimentally, when the length of the waveguide in a 3D heterostructure is 12μm, the local photon density of states (LDOS) is demonstrated to have a jump of an order of 100 within a range of relative frequencies about 10⁻⁴ ω_b, where ω_b is the cutoff frequency of the waveguide mode [52]. The large discontinuity with step-shaped LDOS can be provided by a waveguide cutoff mode within a 3D-2D PGB heterostructure [52,53]. Besides, a forklike shape LDOS can also be engineered using a trimodal waveguide architecture in a PBG microchip using suitable cutoff frequencies for two of the three waveguide modes [52]. In this paper, the large discontinuity in the LDOS and the relative positions of the relevant frequencies considered in our study are presented in Fig 2, such that |D(ω_λ)|² = u(ω_λ − ω_b), u(ω_λ − ω_b) being a unit step function. One valuable feature of waveguide structure is that near the waveguide cutoff frequency, the field is almost the same for rods along the waveguide direction. Therefore, the QD which is close to rods and along the waveguide direction will experience the same field. The cavity resonant frequency can be engineered to be within a PBG or high-LDOS region and near the LDOS discontinuity [52, 54]. Since the QD is driven by a strong laser field, three Mollow bands appear in the fluorescent scattering spectrum [55]. When the Mollow components at different frequencies (especially at lower and higher frequencies) fall within the high LDOS region or the low LDOS region, then they will experience strongly different mode densities [53]. It will strongly influence the dynamics of QD and results in new emission effect, i.e. |D(ω_λ)|² = 1 for ω_λ > ω_b and |D(ω_λ)|² = 0 for ω_λ < ω_b. This technique supplies a way for us to eliminate spontaneous transitions at selected frequencies of the QD-cavity system. In our scheme, this property of the PBG material is very important to generate the NCS of the MR, because the generation of the dark state of the system requires the elimination of the spontaneous transitions at both lower and central frequencies of the QD-cavity system, as will be discussed later.

 

Fig. 2 Schematic representation of the LDOS and the relative position of the relevant frequencies considered in our paper. ω_b, ω_c andω_L are the photonic density-of-states bandedge frequency, the cavity frequency and the coherent driving frequency respectively. The vertical arrows represent the emission lines of the Mollow spectrum of the QD resonance fluorescence which central band occurs at the frequency ω_L, and the side bands components at ω_L ± 2Ω̄.

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The strong driving field can be viewed as a dressing field for the QD. Therefore, we diagonalize the first two terms of the Hamiltonian H₀, i.e., the Hamiltonian of the QD and the coherent interaction with the strong driving field, to find the eigenstates of the coherently driven QD [56]

In order to conveniently study the dynamics of the system including QD, microcavity and the MR, we transform the total system into the polaron frame [57] and introduce a renormalized MR-microcavity coupling strength. The polaron transformation of the total system can be written as

2.2. Derivation of the effective master equation

By use of the Baker-Campbell-Hausdorff Relation [58, 59], the exponential e^{−η(b−b†)} can be expanded as

As mentioned above, in order to explore the dynamics of the system with a three-particle direct interaction, we find that there is one time-independent Hamiltonian in H̃_I1(t), i.e., H_J = g′₁c²(a^†R₋₊ + aR₊₋), which corresponds to the Jaynes-Cummings(JC) interaction Hamiltonian describing the interaction between the dressed QD and the microcavity. The JC model has been known to describe an extensive range of coherent phenomena including the spontaneous collapse, the periodic Rabi-flopping and revivals of a two-level atom in a resonant cavity [60]. The non-classical character is investigated by the transmitted light in a driven cavity-atom system both in theoretical [61] and experimental [41] literatures. It is found that remarkable nonlinearities occurs when the coupling between the cavity mode and the atom is strong. H_J can be diagonalized, and the eigenstates consist of a ground state |0〉 = |−, 0〉 with energy E₀ = 0 and excite states $|E_{\pm }^{(n)}〉=\frac{1}{\sqrt{2}}\left(|-,n〉\pm |+,n-1〉\right)$ (n = 1, 2,···) with corresponding energies $E_{\pm }^{(n)}=\pm \sqrt{n}{g}_1^{\prime }{c}^2$. |+(−), n〉 denotes the higher- (lower-) energy eigenstate with n excitations. The splitting between the energy eigenstates in each manifold has a nonlinear dependence on n. We assume that the coupling coefficient g′₁c² is much larger than the damping rate κ of the microcavity. By cooperatively modulating the frequencies ω_p, ω_L of the driven fields and the strength Ω, two-photon transition processes can be tuned to occur resonantly and the other excitations are far off resonance. In this case, one can truncate the Hilbert space and describe the QD-cavity subsystem with the following four states

For the convenience of expression, in Eq. (19) we have denoted |0〉, $|E_{-}^{(1)}〉$, $|E_{+}^{(1)}〉$ and $|E_{-}^{(2)}〉$ as |0〉, |1〉, |2〉 and |3〉 respectively, with corresponding energies E₀ = 0, E₁ = −g′₁c², E₂ = g′₁c² and $E_3=-\sqrt{2}{g}_1^{\prime }{c}^2$. In our scheme, the decay rate κ and γ of the coupled QD-cavity system are much larger than that of the MR. We declare here that the four states concerned in Eq. (19) are established in the subspace of the reduced QD-cavity subsystem, which involves the strong driving field ω_L. Moreover, the resonant frequency of the cavity ω_c is at least a factor of 10³ with the frequency of ω_m [8–10,50,62–66]. In this situation, the energy interval between the subspaces of the reduced QD-cavity system is much larger than the energy interval of the MR Fock space. Therefore, we suppose that these above four states are enough to describe the considered system even when taking 15 Fock states of the MR into account as shall be discussed in section 3. The energy diagram of the above states of QD-cavity subsystem is shown in Fig 3. As a result, the time-dependent Hamiltonian H_J are diagonalized as

 

Fig. 3 Schematic of the four-level model. The two-photon transition from the ground state |0〉 to the higher excited state |3〉 is resonant.

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In this new basis of Eq. (19), we can rewrite the photon creation operator of the cavity and the transition operator of the QD as

Taking the similar processes, we now calculate the dissipative terms. For the dissipative contribution originating from the interaction of the QD with the photonic crystal reservoir as depicted by H̃_I3 in Eq. (18), we assume that the coupling between the QD and photonic crystal vacuum reservoir is weak and its change on the reservoir is negligible. Also, a fast time scale for the decay of the reservoir correlation is assumed such that the secular [67] and Born-Markov [58,59] approximations can be used. In quantum optics, these standard approximations have been applied with considerable success to a great deal of atom-field interaction problems in photonic crystals [55, 68]. A similar method with details has been introduced in other literature [69]. We apply the second-order perturbation theory to trace over the reservoir degrees of freedom from the following equation

The decay of the cavity ℒ_cρ = κ𝒟[a] is phenomenologically added with the definition $𝒟[𝒪]=\frac{1}{2}\left(2𝒪\rho {𝒪}^{†}-\rho {𝒪}^{†}𝒪-{𝒪}^{†}𝒪\rho \right)$. κ is the decay rate of the microcavity which may lead to a trend towards vacuum mode of the cavity. In fact, ℒ_cρ = κ𝒟[a] can also be obtained following above procedures of ℒ_λρ. After the polaron transformation of Eq. (7), the decay of the cavity is of the form

The purpose of this paper is to prepare the NCS of the MR’s phonon mode. In the next section, we will demonstrate the generation of the NCS applying the dark state of the total system. The properties of this non-classical state is also explored.

3. Preparation of NCS using dark state

Up to now, we have obtained the effective master equation which is able to describe the dynamics of three-particle system including the MR, the microcavity and the dressed QD. We also mentioned the occurrence of two-photon blockade. However, another significant object of this paper is to prepare the NCS of the phonon. In the following, on the basis of the effective Hamiltonian, we investigate the dark state of the system and propose a scheme to generate the NCS of the MR. Some properties of the NCS are also discussed.

We first consider the case that the interaction Hamiltonian V₁ and V′₂ can be neglected, i.e., V₁ = V′₂ = 0. In this case, the dynamic of the whole system is determined by the dissipation term ℒρ = ℒ_λρ + ℒ_cρ. The steady-state behavior of the system can be obtained via the steady equation ℒρ(∞) = 0 with $\rho (\infty )={\sum }_{j=0}^3{\rho }_{jj}|j〉\hspace{0.17em}〈j|\otimes {\rho }_{bj}(\infty )$, where ρ_bj(∞) is the reduced density operator of the MR in the steady state when the QD-cavity subsystem is in state |j〉. ρ_jj = 〈j|ρ| j〉 are elements of the reduced density matrix of the QD-cavity subsystem. In the steady-state solution, we find that ρ₀₀ = [γ₊/(γ₊ + γ₋)]². It should be stressed here that if the spontaneous emission rate can be eliminated γ₋ = 0, the reduced QD-cavity subsystem evolves into the state |0〉 = |−, 0〉 as depicted in Eq. (19), i.e., ρ₀₀ = 1, ρ_jj = 0, (j = 1, 2, 3). This means that the microcavity is in the vacuum state and the QD is in the lower dressed state |−〉, corresponding to ρ(∞) = |0〉 〈0| ⊗ ρ_b(∞). In fact, this condition γ₋ = 0 can be realized by technically engineering the microcavity into the PBG material. As mention before, by properly engineering the system topology, the QD coupled to the microcavity can experiences a discontinuity of the LDOS. It strongly influences the dynamics of QD and results in new emission effects, i.e. |D(ω_λ)|² = 0 for ω_λ < ω_b and |D(ω_λ)|² = 1 for ω_λ > ω_b. In this way, the spontaneous emissions occurring at frequencies ω₋ and ω_L are selectively eliminated, i.e., both the spontaneous emission rates γ₋ and γ₀ are zero. Only the spontaneous emission occurring at the frequency ω₊ = ω_L + 2Ω̄ is allowed to happen with the spontaneous emission rate γ₊. In our scheme, a nonlinear coherent state of the MR mode is generated based on the presence of a dark state. Additionally, if the PBG material were to be moved such that none of the spontaneous transitions are eliminated, the steady state of the system can not evolve into such a dark state, then the scheme for generation of the target state may be the other project which is out of our interest.

When taking the couplings of the microcavity with the QD and the weak laser field into account, i.e., V₁ ≠ 0, V′₂ ≠ 0, we devote to finding the solution which simultaneously satisfy the equations −i[V₁ + V′₂, ρ] = 0 and ℒρ = 0. Because the solution of the latter one is that in which the QD-cavity subsystem is in the state |0〉 as discussed above. Therefore the steady state of the whole system can be described by |Ψ〉 = |0, φ_b〉, (ρ = |0, φ_b〉 〈0, φ_b|), where |φ_b〉 is the motional state of the mechanical mode. When the total system is in such a state, the QD-microcavity system stops to fluoresce and the motional state |φ_b〉 can be obtained. Therefore, it can be said that the system is in a motional dark state [21], which has various types. This target state can be obtained from equation

In order to explore the properties of |φ_b〉, we expand the state |φ_b〉 as

So far, by designing a suitable coherent Hamiltonian and the interaction between the phonon bath and the coupled QD-cavity system, the mechanical mode can be prepared to the NCS |φ_b〉 which is a non-classical motional dark state. It should be highlighted here that the prepared state |φ_b〉 of the mechanical mode is completely decoupled from the QD and the cavity. This can be understood in the polaron frame described by Eq. (7), because the whole system can be described by |Ψ〉 = |0, φ_b〉, where the state |φ_b〉 of mechanical mode is decoupled from the QD and the cavity. However, when going back from this polaron frame, we obtain

Recently, when a two-level system and a mechanical mode are both strongly coupled to a cavity, the preparation of the mechanical modes in a non-Gaussian dark state has also been studied [70]. Employing additional auxiliary cavity mode to construct two Liouvillians, an approximated entangled state are prepared in the weak coupling limit, i.e., ζ = (g_q/g_m)² ≤ 1, where g_q and g_m are coupling strengthes of the cavity with a two-level system and a mechanical mode respectively. When ignoring the spontaneous emission of the two-level system and measuring the two-level system, the mechanical oscillator can be prepared in a superposition of Fock states with a fidelity of F ≈ 0.83. However, it should be emphasized that the literature [70] is distinct from this paper. In this paper, when properly tuning the frequency of the weak driving field, two-photon blockade phenomenon occurs. The QD-cavity subsystem can evolve into a dark state due to the damping of the microcavity and the elimination of the decay rate of the QD at selected frequencies in the photonic band-gap (PBG) material. By exploiting the interaction between the phonon bath and the coupled QD-cavity system, a suitable coherent Hamiltonian is designed so that the mechanical mode can naturally be prepared into a NCS. The prepared NCS of the MR in this paper is a definite pure state. Besides, the prepared NCS is completely decoupled from the QD and the cavity. The intrinsic non-linearity and non-classical behavior of the state will be analysed via phonon number statistics, second-order correlation function, and Wigner function.

3.1. Phonon number statistics

The phonon statistics of a MR are the “finger-print” of its quantum state, from which a number of useful measures of non-classicality may be inferred. The phonon number distribution (PND) of the dark states is complicated by analytical ways with the correlated polynomials. In this paper, we obtain the results by numerically solving the equation involving Laguerre polynomial. For simplicity, we factitiously choose the parameter η = g₂/ω_m = 0.305463 to satisfy the equation

We should note that η = g₂/ω_m = 0.305463 means the system approaches the single-photon strong coupling regime. If the single-photon optomechanical coupling strength g₀ is an appreciable fraction of the mechanical resonance frequency ω_m and the combined nonlinearity parameter satisfies the relation $g_0^2/\kappa {\omega }_m\ge 1$, the system is in the single-photon ultra-strong coupling regime characterized by cavity photon blockade [71, 72]. This challenging idea has inspired emerging attentions from experimental field. The nonlinearity of the phonon mode of the MR originating from the strong coupling between the microcavity and the MR has been discussed based on capacitive [73] or magnetic [74] coupling in the microwave-regime optomechanical devices. By enhancing the effect of a single-photon in the optical cavity, the strong-coupling regime is reached with the auxiliary of the squeezing of a cavity mode [4]. A single-photon ultra-strong coupling regime is proposed to be reached in an optomechanical system [75]. For a system with a quantum dot strongly coupled to a photonic-crystal nanocavity, J. Vučković et. al have studied non-classical higher-order photon correlations [76]. The strong coupling of a single two-level solid-state system with a photon is realized with a single quantum dot in a semiconductor microcavity [77]. Here, we supply some main parameters to implement our scheme with ω_c/2π = 195THz and ω_m/2π = 3.68GHz, κ/2π = 150MHz, g₁ = 5κ, which are similar to that in the literature [10]. The condition η = g₂/ω_m = 0.3 indicates that a value of g₂/2π = 1.1GHz is needed to be achieved. At present, the optomechanical coupling strength g₂ has realized g₂/2π = 11.5MHz, and is predicted to approach g₂/2π = 26MHz [65]. It is still inadequate for realizing our proposal. However, very recently, it has been proposed that applying a large-amplitude, strongly detuned mechanical parametric drive to amplify mechanical zero-point fluctuations, the radiation pressure interaction is enhanced and thus the overall coupling strength is dramatically increased, allowing us to completely enter the quantum-enabled, strong-coupling regime [78]. The strong radiation pressure forces from squeezed light on a mechanical oscillator is observed [79]. These inspiring and promising progresses enable us to believe that our scheme can be achieved in the near future.

In Eq. (37), the fact c_n+1 = 0 for n > N = 15 indicates that the MR is trapped into the Fock superposition state via the cavity damping and selectively elimination the spontaneous emission γ₋ in the PBG material. The specific expression of c_n is

 

Fig. 4 The PND P_n of the MR state in the steady-state regime versus n for different values of χ: the blue corresponds to α_p/g₁ = 0.11; the red corresponds to α_p/g₁ = 0.15. The other parameters are as follows: N = 15, Δ_a = 0, η = 0.305463.

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3.2. Second-order correlation function(SOCF)

In order to extract more information about the NCS of the MR, we calculate the corresponding second-order correlation function(SOCF) which is correlated with the coherence and statistical properties a field. The normalized SOCF of the phonon mode of the MR in the steady state can be defined as [80]

In order to get the insight into the non-classical properties of the phonon, we numerically calculate the SOCF in the Fock state representation with the following expression

 

Fig. 5 The second-order correlation function P_n of the MR state in the steady-state regime versus α_p/g₁. The other parameters are the same as those in figure 4.

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3.3. Wigner function

The Wigner function (WF) [83] was first introduced by Wigner in 1932 to calculate quantum correction to a classic distribution function of a quantum-mechanical system. It is a quasi-probability distribution which fully describes the states of a quantum system in phase space, (either the position-momentum space for an harmonic oscillator or, equivalently, the space spanned by two orthogonal quadratures of the electromagnetic fields for a single-mode state of light), in the same fashion as a probability distribution (non-negative by definition) characterizes a classical system. It has now become a useful tool to study the non-classical properties of quantum states, because the partial negativity of the WF [83,84] is indeed a good indication of the highly non-classical character of the state. In order to further unveil the non-classicality of the NCS of phonon, in this subsection, we supply the expression of WF for the phonon in Fock state representation and numerically illustrate the corresponding property.

For a single-mode system, the WF of the density operator ρ in Fock state representation |n〉 is expressed as

 

Fig. 6 The WF of the phonon in the steady-state regime with α_p/g₁ = 0.15. The other parameters are the same as those in Figure 4.

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As mentioned at the beginning of this subsection, the partial negativity of the WF [83, 84] is a good indication of the highly non-classical character of a quantum state. In order to quantitatively explore the non-classicality of the state, we study the absolute value P_W of the total negative probability of the WF, which can be defined by [84, 86]

We illustrate P_W of the total negative probability of the WF as a function of the parameter α_p/g₁ in figure 7. It can be seen from figure 7 that with the increasing of α_p/g₁, P_W takes a definite value when the parameter α_p/g₁ approaches the region α_p/g₁ > 0.09. The peak of P_W occurs at α_p/g₁ ≈ 0.145. Whereas, as found from figure 5 that g⁽²⁾(0) takes a smaller value than 1 in the region α_p/g₁ ≥ 0.1225. In this region 0.09 < α_p/g₁ < 0.1225, g⁽²⁾(0) takes a larger value than 1, being unable to tell whether or not the non-classical effect appears. But in the same region P_W takes a definite value, which is the indication of non-classicality. The region for g⁽²⁾(0) < 1 is narrower than that for P_W > 0. In other words, the WF of the phonon state has a negative region and the state possesses the non-classicality, but it does not show the sub-poissonian phonon statistics [87]. Therefore, we emphasize here that for the prepared NCS of the phonon mode in this paper, the absolute value P_W may be a better choice for quantifying the non-classicality than the SOCF.

 

Fig. 7 The absolute value P_W of the total negative probability of the Wigner function as a function of the parameter α_p/g₁. The other parameters are the same as those in figure 4.

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Up to now, we have prepared the NCS of the phonon mode of MR by using the dark state of the system. Both the facts that g⁽²⁾(0) < 1 and the WF approaches a negative region are convincing indications of the sub-Poissonian behaviour and non-classicality of the NCS. However, we should point out here that in all the calculations, the damping of the MR is neglected. Because for a MR with high quality factor, the mechanical damping γ_m could be much smaller than the cavity dissipation rate κ. The mechanical damping can be neglected during the time needed to achieve the steady states. As mentioned before, in our scheme, the four states concerned in Eq. (19) are established in the subspace of the reduced QD-cavity subsystem, which involves the strong driving field ω_L. The resonant frequency of the cavity ω_c is at least 10³ times the frequency of ω_m [8–10, 50, 62–66]. In this situation, the energy interval between the subspaces of the reduced QD-cavity system is much larger than the energy interval of the MR Fock space. Therefore, the four states of the QD-cavity subsystem and 15 Fock states of the MR are valid to describe the considered system. Here, we should mention that the increasing of ω_m corresponds to a larger quality factor, which is estimated to 1.1 × 10⁶ when ω_m = 2π × 5.3GHz at cryogenic temperature [50]. A larger ω_m than the decay of the cavity κ means the appearance of the sideband excitations. Then, the sideband cooling method can be employed. For a given cavity matched with the QD, the increasing of ω_m may dissatisfy the condition $g_0^2/\kappa {\omega }_m\ge 1$ thus lead to difficult achievement of strong coupling regime, where g₀ is the optomechanical coupling strength. In this case, as discussed before, it has been proposed recently that applying a large-amplitude, strongly detuned mechanical parametric drive to amplify mechanical zero-point fluctuations, the radiation pressure interaction is enhanced and thus the overall coupling strength is dramatically increased, allowing one to completely enter the quantum-enabled, strong-coupling regime [78]. The strong radiation pressure forces from squeezed light on a mechanical oscillator is observed [79]. These inspiring and promising progresses enable us to believe that our scheme can be achieved in the near future.

As is known that non-Gaussian states might be more appropriate than Gaussian state in the sense of the robustness in the applications of teleportation and the cloning [88–90] of quantum states. The nonlinear coherent state generated in the solid-state optomechanical system may benefit for the optomechanical quantum information processing with photons and phonons [36].

4. Conclusion

In this paper, we have proposed a scheme to prepare a NCS for the MR by using the dark state of the optomechanical system, in which a two-level QD located inside a microscopic cavity is engineered in a PBG material. The effective master equation of the optomechanical system is obtained in the polaron frame. Two-photon transition can be realized in the dressed states of the QD-microcavity subsystem via adjusting the discrepancy of the frequencies between the micro-cavity and the weak laser field. By exploiting the interaction between the phonon bath and the coupled QD-cavity system, a suitable coherent Hamiltonian is designed so that the mechanical mode can naturally be prepared into a NCS, which is a definite pure state and is completely decoupled from the QD and the cavity. The NCS of the phonon exhibits strong non-classicality since its SOCF shows the sub-Poissonian distribution and its Wigner function approaches the negative region. Our proposal provides a way to prepare a NCS with nonclassicality in an optomechanical crystal system. We hope this could stimulate discussions about the applications of non-Gaussian non-classical states in the quantum information processing [36] and quantum teleportation [91] in the optomechanical systems. The NCS could contribute to the application in optimizing quantum information in the near future, which is important in binary (or multibinary) communication [27].