When a trichromatic laser field is applied to a cavity optomechanical system within the single-photon strong-coupling regime, we find that the motion of mirror can evolve into a dark state such that the cavity field mode cannot absorb energy from the external field. Via tuning three components of the pumping field to be resonant to the carrier, red-sideband and blue-sideband transitions in the displaced representation respectively, the state of mirror motion can exhibit non-classical properties, such as that in the Lamb-Dicke limit, the state evolves into a squeezed coherent state, and beyond the limit, the state can become a squeezed non-Gaussian state.
© 2014 Optical Society of America
A great surge of interest in macroscopic mechanical systems covers manipulation of the center-of-mass motion [1–4], preparation of nonclassical states of mechanical oscillators  and light fields [6, 7], etc. In special, preparations of the nonclassical states of mechanical oscillators can demonstrate the quantum control at large scales , which possess potential applications in such as precision measurement , long-distance quantum communication and networking [10–12]. Nonclassical states such as squeezed state , Schrödinger ‘cats’  and Fock states  have been already investigated in various optomechanical systems. In these studies linearized Langevin equation or master equation to quantum fluctuations around classical steady states approach is employed to find system observables, in which the position of the mechanical oscillator linearly modulates the cavity frequency while the cavity field is driven by a strong optical field [16, 17].
With the rapid technical advancement, the single-photon strong-coupling regime has been experimentally realized in ultracold atomic gas system [6, 18, 19], where the presence of a single photon displaces the mechanical oscillator by more than its zero-point uncertainty. There are several micro-optomechanical systems [20, 21] currently approaching to the regime in optical domain, and the microwave optomechanical circuit system also gives the possibility to approach the elusive single-photon strong coupling limit [22, 23]. In this regime the single-photon coupling strength becomes comparable with the frequency of mechanical oscillator and larger than the damping rate of the cavity mode, and then the inherently nonlinear interaction becomes much more significant. The nonlinearity is much helpful to create more general, possibly more interesting and useful states at the level of single photons and phonons, especially non-Gaussian quantum states with negative Wigner funcitons, which indicate strong nonclassicality and genuine quantum behavior and have advantages over Gaussian ones in quantum information protocols . For example, Ren et al.  proposed a heralded probabilistic scheme to generate mechanical NOON state via manipulating photon transport in the single-photon strong-coupling optomechanical systems, Rabl  investigated the photon blockade effect in the single-photon optomechanical system, which may provide an essential ingredient for potential applications as a quantum nonlinear device, and Akram et al.  proposed a scheme to prepare entangled mechanical ‘cat’ states via conditional single-photon optomechanics.
In this paper we study the preparation of non-classical states of mirror motion when a trichromatic field is applied to a cavity optomechanical system within the single-photon strong-coupling regime. Via utilizing the quantum interference effects , which are much useful in the optomechanical system such as manipulation of electromagnetically induced transparency (EIT) to produce desired transmission [29–32], the motion of the mirror may evolve into a dark state such that the cavity-field mode cannot absorb energy from the external field, and accordingly, the optomechanical cavity is decoupled from the external field. However, compared with  where a bichromatic laser field is applied to couple the carrier and red-sideband transitions in the displaced representation, in our system when a third frequency component couples to blue-sideband transition, we will demonstrate that more interesting dark states can be generated. We resort to variance of quadrature operator and Wigner function to analyze the non-classical properties of the states and we find that in the Lamb-Dicke limit, i.e., the single-photon coupling strength much smaller than the mechanical frequency, the obtained state can be a squeezed coherent state, while beyond the limit, i.e., in the strong-coupling regime, the obtained state can become a squeezed non-Gaussian state with the negativity of Wigner function. As a consequence, for Gaussian and non-Gaussian optomechanical system the use of an extra frequency component to drive the blue-sideband transition can lead to the squeezed state of mechanical motion [33–35]. Moreover, in contrast to earlier generation of non-classical mechanical states relying on conditional measurement  and non-Gaussian input light , we can achieve the non-Gaussian mechanical state only with use of a trichromatic laser field.
The organization of the paper is as follows. In Sec. 2, the single-photon optomechanical system is introduced. In Sec. 3 the preparation of non-classical states of mirror motion and the validity of the optomechanical state are discussed, and in the last the conclusion is drawn.
2. Description of the single-photon optomechanical system
We consider a general optomechanical system, in which the cavity field couples to a movable end mirror via radiation pressure. The cavity field is driven by a trichromatic laser field with respective frequencies ωi and amplitudes Ωi (i = 1, 2, 3). The setup is schematically shown in Fig. 1, and the Hamiltonian of the system is given by (ħ = 1)
The first two terms of Ĥ′ correspond to a Hamiltonian Ĥ0 without driving, which conserves the photon number, i.e., [â†â, Ĥ0] = 0, and it can be diagonalized by introducing a displaced oscillator basis as37]. The corresponding eigenvalue is εn,p = pωm − n2g2/ωm, where the energy of phonon state shifts g2/ωm induced by a single-photon radiation pressure.
Within the eigenbasis of Ĥ0, the cavity annihilation operator â is expressed in the form38], and are associated Laguerre polynomials. Then, from Eq. (5) the Hamiltonian Ĥ′ can be reexpressed as
3. Preparation of non-classical states for the mirror motion
Now we turn to the problem of how to effectively reduce the Hamiltonian (8) to a desired form, which can be shown to operate in a finite-dimensional Hilbert space. In the case that the frequency components ωi of trichromatic field are equally spaced, the detunings Δi satisfy the conditions:(8) can be decomposed into two parts 26], which is based on the fact that the excitation to two-photon space from one-photon states is far off-resonance, i.e., . Accordingly, the excitation to higher photon number space is much weaker and negligible.
The Hamiltonian Ĥeff describes the resonant transitions between zero- and one-photon states, which is sketched in Fig. 2. In there the frequency element ω1 resonantly drives the carrier transitions |ψ0,p〉 ↔ |ψ1,p〉 , element ω2 resonantly couples to the red-sideband transitions |ψ1,p〉 ↔ |ψ0,p+1〉 and element ω3 resonantly couples to the blue-sideband transitions |ψ0,p〉 ↔ |ψ1,p+1〉. In the system there exists a class of dark states |D〉 for the effective Hamiltonian in Eq. (11), which is the eigenvector of zero eigenvalue, i.e., Ĥeff|D〉 = 0 . The explicit expression for the dark state is given byEq. (11) becomes the form discussed in  if there is Ω3 = 0, i.e., blue-sideband transition is not driven, and the recursive relation in Eq. (15) is consistent with the result therein. However, with use of the frequency element ω3 to couple to the blue-sideband transition, we will display that the obtained dark state can exhibit squeezing in both the weak- and strong-coupling regime. Especially in the strong-coupling regime, the mechanical dark state can possess both the squeezing and non-Gaussian quantum properties.
From the Hamiltonian (11), we can see that for a given positive integer N, the Hilbert space can be confined within N phonon number states if one sets the transition coefficients to the N +1 phonon number states and zero. Thus, there will be none excitations from N phonon states to N + 1 phonon states, as shown by the dashed lines in Fig. 2. The explicit expressions for and read
With the special value of gN, the recursive relations in Eq. (15) become a group of N + 1 linear equations. With the condition β0 = 1, all the coefficients βp can be calculated from the first N recursive equations, and to obtain the self-consistent results the last equationEq. (15) into a matrix form
Because of β0 = 1, there are N terms to calculate. Therefore, in order to obtain the nontrivial solutions of the equations, the rank of coefficient matrix M should be N. The condition of rank(M) = N can be achieved via utilizing the recurrence relation of continuant , and choosing an appropriate value of Ω1 for a given group of Ω2 and Ω3 together with the coupling strength gN given from Eqs. (16).
3.1. Squeezed coherent mechanical state within the Lamb-Dikce Limit
In the following we first consider the dark state of the oscillator motion when the system is well within the Lamb-Dicke limit g ≪ ωm, i.e., ξ ≪ 1. From the expressions of the associated Laguerre polynomials, , and in the Lamb-Dicke limit becomeEq. (15) becomes (22) is simplified as 40]. To fulfill the relation g ≪ ωm, we can choose the gN = 0.1916ωm with the mirror’s motion state confined within phonon number N = 99, where the value ξ2 = 0.037 well falls in the approximation in Eq. (21). In Fig. 3, we plot the numerical result of the phonon number distribution Pm calculated from Eq. (18) compared with that of squeezed coherent state obtained from Eq. (24) with the parameters: Ω1 = 0.0038ωm, Ω2 = 0.0225ωm, Ω3 = −0.015ωm, where the populations are hardly distributed above the phonon number n = 20, and we plot the distributions within phonon number n = 20 to clearly show the similarity. The quadrature operator of phonon mode is and the variance of X̂ is determined by the relation Eq. (24) is (ΔX̂)2 = 0.5(ν − μ)2 = 0.1(7dB), and from the numerical results we can obtain the variance (ΔX̂)2 = 0.0952(7.2dB). The variance is smaller than the fluctuation of coherent state 0.5, and thus the mechanical motion is squeezed. Moreover, to quantitatively describe how close of the analytical state and the numerical result, we resort to fidelity F = |〈ϕnum|ϕsqcoh〉|2, where |ϕnum〉 denotes the numerical result and |ϕsqcoh〉 denotes squeezed coherent state. Numerical calculation of fidelity between the two states is F = 99.9% with the parameters in Fig. 3, which can display the consistency between two results.
Thus in the Lamb-Dicke limit, the dark state of the mirror motion is well described by a squeezed coherent state. As compared with the case of two-tone laser applied on the cavity , with use of a third frequency component to drive the blue-sideband transition, the dark state becomes squeezed. In physics, it is analogous to the technique of reservoir engineering in the linearized optomechanical system in , where an additional field is applied to couple the blue-sideband transition based on the red-sideband cooling model, and the motion of the oscillator evolves into a dark state which is a squeezed vacuum state. Moreover, if we remove the frequency element ω1, i.e. Ω1 = 0, the model in Fig. 2 is similar to the model in , and the recursive relation (23) becomes the recursive equation of squeezed vacuum state . The motion of the mirror is in a squeezed vacuum state. And when a trichromatic field is applied, the motion of the mirror is in a squeezed coherent state.
In order to gain insight of quantum distribution of the dark state in phase space, we use the Wigner function to present the differences between the classical and quantum behaviors in phase space. The Wigner function is defined as42]. Wigner function of the dark state is plotted in phase space in Fig. 4, where X and Y are position and momentum. The distribution is suppressed along the X̂ quadrature component and presents crescent shape, which can provide a clear insight: the state is close to a squeezed coherent state . Thus in the Lamb-Dicke limit, the dark state of mechanical motion is Gaussian. According to Hudson’s theorem  for pure states, the border between Gaussian and non-Gaussian states coincides exactly with the one between states with positive and negative Wigner functions. Obviously there is no negative region in Fig. 4. However, we know that negative Wigner functions of non-Gaussian quantum states indicate strong nonclassicality and genuine quantum behavior, and non-Gaussian states have advantages over Gaussian ones. The use of mechanical nonlinearity will help generate the non-Gaussian state, and thus we will investigate the mechanical behavior in the single-photon strong-coupling regime.
3.2. Non-Gaussian mechanical state beyond Lamb-Dicke limit
Further, when beyond the Lamb-Dicke limit, the nonlinear optomechanical effects become significant and may play a vital role in preparation of non-Gaussian steady state of the mechanical oscillator , such as a strongly negative Wigner density and photon-photon correlation function if the optomechanical coupling is comparable to or larger than the optical decay rate and the mechanical frequency [45–47]. It is also possible to extend the squeezing beyond squeezed coherent state and consider the squeezed non-Gaussian state, which can provide a basis for several proposals of quantum information protocols [48,49]. The squeezed non-Gaussian state has been investigated using the cross-Kerr nonlinearity and the corresponding Wigner function has a specific crescent shape as well as negative values .
In the paper we also resort to Wigner function to show the effects of nonlinear coupling on preparation of nonclassical state of the oscillator motion, since for pure states, according to Hudson’s theorem  the presence of negative regions in the Wigner distribution is an authentic sign of quantum character. Beyond the Lamb-Dicke limit, such as that when the motion state is confined within the number state N = 13, the single-photon coupling strength is gN = 0.5124ωm, and Wigner function of the dark state in Fig. 5 presents the crescent shape of the squeezed state as well as negative values. Here we choose Ω2 = 0.0225ωm, Ω3 = −0.015ωm that govern the degree of squeezing in the Lamb-Dicke regime, and then numerically determine Ω1 = 0.0029ωm from Eq. (20) with the requirement of rank(M) = N. The variance of the quadrature operator X̂ is numerically obtained, which is equal to 0.386(1.12dB) and below the fluctuation of the coherent state. Thus the mechanical state is squeezed even in the single-photon strong-coupling limit, and the multi-tone driving approaches to generate squeezing in  can be also useful beyond the linearized optomechanical system. Meanwhile, the minimum of the Wigner function is −0.1. Therefore, in the strong-coupling optomechanical system driven by a trichromatic field, the mirror motion can be in a squeezed non-Gaussian state.
In physics, because of nonlinear coupling in the single-photon strong-coupling limit, the effective coupling strength between different phonon states varies, which means that the coupling strength is dependent on phonon number state. For example, we can write the coupling coefficient in the formEq. (15) becomes nonlinear. The nonlinear coupling between the mechanical states can modify the coherence between different mechanical Fock states and has pronounced quantum interference effects, which is of benefit to the generation of non-Gaussian state [50, 52].
In order to gain a comprehensive insight into the changes of dark states of mechanical motion with the coupling strengths, we choose another group of parameters: gN = 0.3678ωm(N = 26), Ω2 = 0.0225ωm, Ω3 = −0.015ωm, and Ω1 = 0.00963ωm fulfilling the requirement of rank(M) = N, where the coupling strength is between the values of well within and beyond the Lamb-Dicke regimes. The Wigner function and phonon number distribution are shown in Fig. 6, where the minimum of Wigner function is −0.05 and the variance of the quadrature operator X̂ is 0.142(5.5dB). The dark state is also a squeezed non-Gaussian state. However the negativity of Wigner function is smaller than that of strong single-photon regime while the degree of squeezing is smaller than the squeezed coherent state in Lamb-Dicke regime. With the increasing single-photon coupling strength, the degree of squeezing decreases and negativity of Wigner function increases. It seems that although the squeezing and negativity of Wigner function are the signatures for the non-classical mechanical motional state, they do not possess the correspondence. This is because Wigner function is connected to all orders of correlations, while squeezing is just connected to the second-order correlation, and they do not have a one-to-one match .
Moreover, if we change the field amplitudes Ω2 and Ω3, in the Lamb-Dicke limit, the dark state of mechanical motion is still a squeezed coherent state except that the degree of squeezing varies followed the ratio of Ω2 and Ω3. We now discuss the changes of dark states in the single-photon strong-coupling regime. We choose a group of parameters: gN = 0.5124ωm(N = 13), Ω2 = 0.016ωm, Ω3 = −0.01ωm, and Ω1 = 0.002ωm fulfilling the requirement of rank(M) = N. The Wigner function and phonon number distribution are shown in Fig. 7, where the minimum of Wigner function is −0.069 and the variance of the quadrature operator X̂ is 0.239(3.21dB). The dark state is a squeezed non-Gaussian state with the smaller negativity of Wigner function and larger degree of squeezing compared with those in Fig. 5.
In addition, one should test the validity of the dark state as well as the effective Hamiltonian (11). We will solve the evolution of the system state |Ψ(t)〉 which obeys the Schrödinger equation with the Hamiltonian (2) without the rotating-wave approximation. Here we consider the time interval t ≫ 1/κ, where 1/κ represents the readout timescale of the cavity field. For example, a cavity-Cooper pair transistor (cCPT) mechanical resonator, ωm/κ ∼ 10 is achievable. The time-dependent dark state governed by effective Hamiltonian (11) is given byFig. 8 we plot the fidelity as a function of time. Fidelity is F ≈ 1 for a long time, i.e. t ≫ κ−1, and thus the system state |Ψ(t)〉 can be well described by the dark state |D(t)〉 and the effective Hamiltonian (11) is valid. The high frequency oscillation of the fidelity is caused by the off-resonant transitions in Eq. (12), and the phenomena of collapse and revival can be physically understood from the summation of different Rabi oscillations which are related to different values of detunings and the coupling strengths that vary for different phonon number states. However, due to the too many oscillations in the system the revival is hard to reach for the unity.
3.3. Effects of cavity and mechanical dissipations on the mechanical dark state
In the following we will discuss the generation of the dark state for the mirror motion under the help of the cavity dissipation together with the influence of mechanical damping. In the frame of Hamiltonian given in Eq. (2), the evolution of the system with the cavity and phonon damping is governed by the master equationEq. (14) is also the eigenvector of Liouvillian operator of cavity damping with zero eigenvalue, which can help to generate the dark state. The mechanical quality factor can reach Q ∼ 105, and the damping rate γ ∼ 10−5ωm. Around ground-state cooling of mechanical oscillator and in the timescale κ−1 ≪ t ≪ γ−1, we can neglect the affect of the mechanical damping on the generation of dark state.
We numerically solve the master equation (29) with the initial ground state of system, and to measure to probability of the system in the dark state we also resort to the fidelityFig. 9 we plot the fidelity that increases with time and at time T = 2500ω−1 the fidelity can reach the value 0.91. It should be noted here the line in Fig. 9 also has small oscillations as there in Fig. 8, but much smaller compared to the degree scale in Y-axis. Thus the dark state can be generated from the ground state of the cavity-mirror system with the help of the cavity damping, and the Wigner function also presents negative values and the minimal negativity is about −0.055.
Finally, there are a number of techniques capable of detecting tiny mechanical vibrations , and also various proposals to perform quantum state reconstruction (QSR) of mechanical motion. For example, mechanical quadrature tomography using back-action-evading interactions can provide the position or momentum distributions of mechanical state and Wigner function for the typical optomechanical system . Thus the squeezed non-Gaussian state can be visualized with this approaching techniques.
In summary, we have investigated the dark state of mirror motion in a single-photon optomechanical system when a trichromatic field is applied to drive a cavity mode. In the displaced representation, three frequency components couple to the carrier, red-sideband and blue-sideband transitions respectively, the mirror motion can evolve into a dark state which arises from the quantum interference effects. Moreover, via carefully tuning single-photon coupling strength, the phonon state can be confined within a low number. In the Lamb-Dicke limit, the mirror motion can evolve into a squeezed coherent state, and beyond the Lamb-Dicke limit, the state evolves into a squeezed non-Gaussian state, where Wigner function of mirror motion state presents the crescent shape of the squeezed state as well as negative values. Finally, we apply the fidelity to exhibit the validity of the non-classical dark state of mirror motion in the system and discuss the effects of cavity damping on the generation of dark state.
This work is supported by the National Natural Science Foundation of China (Grant No. 61275123), the National Basic Research Program of China (Grant No. 2012CB921602), and the Key Laboratory of Advanced Micro-Structure of Tongji University.
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