Abstract

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

1. Introduction

A great surge of interest in macroscopic mechanical systems covers manipulation of the center-of-mass motion [14], preparation of nonclassical states of mechanical oscillators [5] and light fields [6, 7], etc. In special, preparations of the nonclassical states of mechanical oscillators can demonstrate the quantum control at large scales [8], which possess potential applications in such as precision measurement [9], long-distance quantum communication and networking [1012]. Nonclassical states such as squeezed state [13], Schrödinger ‘cats’ [14] and Fock states [15] 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 [24]. For example, Ren et al. [25] proposed a heralded probabilistic scheme to generate mechanical NOON state via manipulating photon transport in the single-photon strong-coupling optomechanical systems, Rabl [26] 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. [27] 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 [28], which are much useful in the optomechanical system such as manipulation of electromagnetically induced transparency (EIT) to produce desired transmission [2932], 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 [28] 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 [3335]. Moreover, in contrast to earlier generation of non-classical mechanical states relying on conditional measurement [36] and non-Gaussian input light [5], 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)

H^=ωca^a^+ωmb^b^gaa^(b^+b^)+[(Ω1eiω1t+Ω2eiω2t+Ω3eiω3t)a^+H.c.],
where â and are the annihilation operators of cavity and mechanical modes and their resonant frequencies are ωc and ωm respectively. Single-photon coupling strength is g = ωcx0/L, where x0=/(2Mωm) is the zero-point fluctuation of the mirror with its mass M, and cavity’s rest length is L. In a frame rotating at the cavity’s frequency ωc, the transformed Hamiltonian reads
H^=ωmb^b^ga^a^(b^+b^)+[(Ω1eiΔ1t+Ω2eiΔ2t+Ω3eiΔ3t)a^+H.c.],
where Δi = ωiωc are the detunings of the laser components from the cavity frequency.

 

Fig. 1 Sketch of the optomechanical system consisting of a fixed end mirror and a movable end mirror. The cavity field couples to the movable mirror via radiation pressure, which is driven by a trichromatic laser field.

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The first two terms of Ĥ′ correspond to a Hamiltonian Ĥ0 without driving, which conserves the photon number, i.e., [ââ, Ĥ0] = 0, and it can be diagonalized by introducing a displaced oscillator basis as

H^0=ωmb^b^ga^a^(b^+b^)=n,pεn,p|ψn,pψn,p|.
The eigenvector of Ĥ0 is
|ψn,p=|ncD(ng/ωm)|pm=|nc|p˜(n)m,
in which |nc and |pm are the number states of the cavity and mirror, and D(ng/ωm)=exp[ngωm(b^b^)] is the displacement operator. The state |(n)〉m is an n-photon displaced number state [37]. The corresponding eigenvalue is εn,p = mn2g2/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 form

a^=n,pn,p|ψn,pψn,p|a^|ψn,pψn,p|=n,p,pAp,p(n)|ψn1,pψn,p|,
with the coefficients Ap,p(n)=np|D(g/ωm)|p. The general form of the coefficients is given by
Ap,p(n)={np!p!eξ22(ξ)ppLppp(ξ2),pp,np!p!eξ22(ξ)ppLppp(ξ2),p>p,
in which ξ = g/ωm indicates the oscillator displacement in units of 2x0 that is related to effects of the radiation pressure of a single photon [38], and Lrs(x) are associated Laguerre polynomials. Then, from Eq. (5) the Hamiltonian Ĥ′ can be reexpressed as
H^=n,pεn,p|ψn,pψn,p|+n,p,p[Ap,p(n)(Ω1eiΔ1t+Ω2eiΔ2t+Ω3eiΔ3t)|ψn,pψn1,p|+H.c.].
Further, we go into the interaction picture in which the Hamiltonian becomes
H^int=n,p,p[Ap,p(n)(Ω1eiΔ1t+Ω2eiΔ2t+Ω3eiΔ3t)×ei(εn,pεn1,p)t|ψn,pψn1,p|+H.c.].
By assuming that the cavity decay rate κc is much smaller than the oscillating frequency ωm of mirror, the energy levels are clearly separated. Due to the strong single-photon radiation pressure interaction, the eigenstates of the system are anharmonic and then through appropriately choosing the detunings Δi the different non-classical states of the movable mirror may be generated.

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:

Δ1=ε1,pε0,p=g2/ωm,Δ2=ε1,pε0,p+1=ωmg2/ωm,Δ3=ε1,pε0,p1=ωmg2/ωm,
which correspond to the resonance to the carrier, red-sideband and blue-sideband transitions in the displaced representation respectively. Then the Hamiltonian (8) can be decomposed into two parts
H^int=H^eff+V^,
in which Ĥeff describes the resonant interactions and is given in the form
H^eff=p[(Ap,p(1)Ω1|ψ1,pψ0,p|+Ap+1,p(1)Ω2|ψ1,pψ0,p+1|+Ap,p+1(1)Ω3|ψ1,p+1ψ0,p|)+H.c.],
while describes the off-resonance transitions
V^=Σn,p,p[Ap,p(n)(Ω1eiδ1(n,p,p)t+Ω2eiδ2(n,p,p)t+Ω3eiδ3(n,p,p)t)|ψn,pψn1,p|+H.c.].
Here the primed summation in excludes those terms contained in Ĥeff, and the off-resonance detunings are
δ1(n,p,p)=(pp)ωm2(n1)g2/ωm,δ2(n,p,p)=(pp+1)ωm2(n1)g2/ωm,δ3(n,p,p)=(pp1)ωm2(n1)g2/ωm.
If the detunings δi(n, p, p′) are sufficiently large compared with the effective coupling strength Ap,p(n)Ωi, the transitions described by can be neglected in the rotating-wave approximation. Consequently, if the cavity is initially in the vacuum state, the system will be confined within the zero- and single-photon states. This result can be also interpreted as the photon blockade effect [26], which is based on the fact that the excitation to two-photon space from one-photon states is far off-resonance, i.e., δi(2,p,p)Ap,p(2)Ωi. 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 [28]. The explicit expression for the dark state is given by

|D=Cp=0βp|0c|pm,
where β0 = 1 and C is a normalization constant. It should be noted here the cavity mode is in vacuum and in the following explicit discussions we can numerically prove that the cavity mode is empty. Moreover, when we consider the damping rate of the cavity field, which is represented by the Liouvillian operator 𝒦̂ρ̂ = κ/2(2âρ̂âââρ̂ρ̂ââ), the dark state is also the eigenvector of operator 𝒦̂ with zero eigenvalue. Thus the cavity damping does not influence the dark state and may even help to generate the dark state. By substituting the expression into the equation, we obtain a chained set of recursive relation
βp1Ω3Ap1,p(1)+βpΩ1Ap,p(1)+βp+1Ω2Ap+1,p(1)=0.
In particular, the effective Hamiltonian in Eq. (11) becomes the form discussed in [28] 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.

 

Fig. 2 The transitions between the energy levels of the optomechanical system with cavity field confined within the zero- and single-photon states. Each component of trichromatic field is applied to establish a set of resonant transitions, and we denote |Pm = |(0)〉m and |m = |(1)〉m for simplicity. By choosing an appropriate g = gN, there will be no transitions to N + 1 phonons states shown by dashed lines.

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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 Ω2AN+1,N(1) and Ω3AN,N+1(1) 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 AN+1,N(1) and AN,N+1(1) read

AN+1,N(1)=1/(N+1)eξ2/2ξLN(1)(ξ2),AN,N+1(1)=1/(N+1)eξ2/2(ξ)LN(1)(ξ2).
According to the behavior of the associated Laguerre polynomial LN1(ξ2), it becomes zero at some special values g = gN. Here we take gN the smallest positive value, and the phonon state can be limited within N phonon number space.

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 equation

βN1Ω3AN1,N(1)+βNΩ1AN,N(1)=0
should be also fulfilled. This can be achieved by reducing the degrees of freedom of the laser driving strengths, i.e., determining Ω1 from the values of Ω2 and Ω3. In order to facilitate the mathematical treatment, we write the chained set of finite linear recursive equations in Eq. (15) into a matrix form
M|β]=0,
where
|β]=(β0,β1,,βN1,βN)T
is a (N + 1) × 1 column vector and T denotes the transpose of a vector, and the coefficient matrix M of linear equations is a (N + 1) × (N + 1) tridiagonal matrix and written as
M=[Ω1A0,0(1)Ω2A1,0(1)Ω3A0,1(1)Ω1A1,1(1)Ω2A2,1(1)Ω3A1,2(1)Ω2AN,N1(1)Ω3AN,N1,N(1)Ω1AN,N(1)].

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 [39], 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 gωm, i.e., ξ ≪ 1. From the expressions of the associated Laguerre polynomials, Ap1,p(1), Ap,p(1) and Ap+1,p(1) in the Lamb-Dicke limit become

Ap1,p(1)p(ξ)eξ2/2,Ap,p(1)eξ2/2,Ap+1,p(1)p+1ξeξ2/2.
The recursive relation in Eq. (15) becomes
βp1p(Ω3ξ)+βpΩ1+βp+1p+1(Ω2ξ)=0.
If we redefine the parameters: μ=Ω3Ω22Ω33, ν=Ω2Ω22Ω32 and β=Ω1Ω22Ω32ξ, the recursive relation (22) is simplified as
μβp1p+νβp+1p+1=ββp,
which is the same recursive relation as squeezed coherent state in the Fock space. The amplitude coefficients are
βp=1p!ν(μ2ν)p2exp(|β|22+μ*2νβ2)Hp(β2μν),
where Hp(z) are the Hermite polynomials [40]. To fulfill the relation gωm, 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 X^=(b^+b^)/2 and the variance of is determined by the relation
(ΔX^)2=X^2X^2.
The variance calculated from the analytical coefficient in Eq. (24) is (Δ)2 = 0.5(νμ)2 = 0.1(7dB), and from the numerical results we can obtain the variance (Δ)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.

 

Fig. 3 Numerical results of phonon number distribution Pm of the dark state compared with that of squeezed coherent state in Eq. (24). The coupling strength is gN = 0.1916ωm with phonon number N = 99, laser driving strength is Ω2 = 0.0225ωm, Ω3 = −0.015ωm, and Ω1 = −0.0038ωm is determined from Eq. (20) with the requirement of rank(M) = N.

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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 [28], 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 [41], 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 [41], and the recursive relation (23) becomes the recursive equation of squeezed vacuum state [40]. 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 as

W(α)=2πTr[D^(α)ρ^D^(α)(1)b^b^],
in which ρ̂ is the density operator and (α) = exp(αb̂α*) is the displacement operator [42]. 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 quadrature component and presents crescent shape, which can provide a clear insight: the state is close to a squeezed coherent state [43]. Thus in the Lamb-Dicke limit, the dark state of mechanical motion is Gaussian. According to Hudson’s theorem [44] 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.

 

Fig. 4 Wigner function of the dark state with gN = 0.1916ωm corresponding to the phonon number N = 99, laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm, and Ω1 = −0.0038ωm determined from Eq. (20) with the requirement of rank(M) = N. The crescent shape is clearly visible.

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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 [38], 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 [4547]. 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 [50].

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 [44] 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 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 [51] 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.

 

Fig. 5 Wigner function and the phonon number distribution Pm of the dark state with the coupling strength gN = 0.5124ωm (N = 13), laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.0029ωm determined from Eq. (20) with the requirement of rank(M) = N. The crescent shape and negative values of the Wigner function are clearly visible.

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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 form

Ap+1,p(1)=1p+1eξ2/2ξLp(1)(ξ2)=p+1eξ2/2ξ[1p2!ξ2+(p1)p3!ξ4+]=p+1eξ2/2ξf(p),
where f (p) is nonlinear function of phonon number p. Then the recursive relation in Eq. (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 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 [53].

 

Fig. 6 Wigner function and the phonon number distribution Pm of the dark state with the coupling strength gN = 0.3678ωm (N = 26), laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.00963ωm determined from Eq. (20) with the requirement of rank(M) = N. Here the phonon number population is mainly distributed within N = 15.

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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 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.

 

Fig. 7 Wigner function and the phonon number distribution Pm of the dark state with the coupling strength gN = 0.5124ωm (N = 13), laser driving strength Ω2 = 0.016ωm, Ω3 = −0.01ωm and Ω1 = 0.002ωm determined from Eq. (20) with the requirement of rank(M) = N.

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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 iddt|Ψ(t)=H^(t)|Ψ(t) 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 ∼ 10 is achievable. The time-dependent dark state governed by effective Hamiltonian (11) is given by

|D(t)=Cp=0Nβpeipωmt|0c|pm,
due to eigenenergies ε0,p = m. In particular, we choose a dark state as the initial system state |Ψ(0)〉 = |D(0)〉 and calculate the fidelity F = |〈D(t)|Ψ(t)〉|2 between |Ψ(t)〉 and |D(t)〉 to verify the rotating-wave approximation, where system state |Ψ(t)=|Ψ(0)i0tdtH^(t)|Ψ(t). In Fig. 8 we plot the fidelity as a function of time. Fidelity is F ≈ 1 for a long time, i.e. tκ−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.

 

Fig. 8 Fidelity between the system state and the dark state as a function of time. The initial system state is a dark state with the coupling strength gN = 0.5124ωm (N = 13), laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.0029ωm determined from Eq. (20) with the requirement of rank(M) = N. Laser detunings are given in Eq. (9).

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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 equation

ddtρ^=i[H^,ρ^]+κ2(2a^ρ^a^a^a^ρ^ρ^a^a^)+γ(n¯+1)2(2b^ρ^b^b^b^ρ^ρ^b^b^)+γn¯2(2b^ρ^b^b^b^ρ^ρ^b^b^),
where ρ̂ is the density matrix of the photon-mirror system, γ is the mechanical damping rate and is the phonon number in the thermal environments. It should be noted here that the dark state in Eq. (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−1tγ−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 fidelity

F=Tr[|D(t)D(t)|ρ^(t)].
In the Fig. 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.

 

Fig. 9 Time evolution of Fidelity between the system state and the dark state. The initial system is in its ground state with the coupling strength gN = 0.5124ωm (N = 13), cavity damping rate κ/ωm = 0.08, laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.0029ωm determined from Eq. (20) with the requirement of rank(M) = N. Laser detunings are given in Eq. (9).

Download Full Size | PPT Slide | PDF

Finally, there are a number of techniques capable of detecting tiny mechanical vibrations [54], 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 [55]. Thus the squeezed non-Gaussian state can be visualized with this approaching techniques.

4. Conclusion

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.

Acknowledgments

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.

References and links

1. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011). [CrossRef]   [PubMed]  

2. Z. Q. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009). [CrossRef]  

3. J. Zhang, Y. X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79, 052102 (2009). [CrossRef]  

4. Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011). [CrossRef]  

5. F. Khalili, S. Danilishin, H. X. Miao, H. Müller-Ebhardt, H. Yang, and Y. B. Chen, “Preparing a mechanical oscillator in non-gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010). [CrossRef]   [PubMed]  

6. D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature 488, 476–480 (2012). [CrossRef]   [PubMed]  

7. Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013). [CrossRef]   [PubMed]  

8. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007). [CrossRef]   [PubMed]  

9. C. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981). [CrossRef]  

10. H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008). [CrossRef]   [PubMed]  

11. B. Kraus and J. I. Cirac, “Discrete entanglement distribution with squeezed light,” Phys. Rev. Lett. 92, 013602 (2004). [CrossRef]   [PubMed]  

12. M. Paternostro, W. Son, and M. S. Kim, “Complete conditions for entanglement transfer,” Phys. Rev. Lett. 92, 197901 (2004). [CrossRef]   [PubMed]  

13. W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013). [CrossRef]  

14. S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997). [CrossRef]  

15. S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, “Steady-state negative Wigner functions of nonlinear nanomechanical oscillators,” New J. Phys. 14, 023042 (2012). [CrossRef]  

16. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007). [CrossRef]   [PubMed]  

17. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007). [CrossRef]   [PubMed]  

18. K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4, 561–564 (2008). [CrossRef]  

19. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein Condensate,” Science 322, 235–238 (2008). [CrossRef]   [PubMed]  

20. A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012). [CrossRef]   [PubMed]  

21. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010). [CrossRef]   [PubMed]  

22. T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the Josephson effect,” arXiv:1311.3802.

23. A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D. Nation, “A cavity-Cooper pair transistor scheme for investigating quantum optomechanics in the ultra-strong coupling regime,” New J. Phys. 16, 055008 (2014). [CrossRef]  

24. J. Eisert, S. Scheel, and M. B. Plenio, “Distilling gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002). [CrossRef]   [PubMed]  

25. X. X. Ren, H. K. Li, M. Y. Yan, Y. C. Liu, Y. F. Xiao, and Q. H. Gong, “Single-photon transport and mechanical NOON-state generation in microcavity optomechanics,” Phys. Rev. A 87, 033807 (2013). [CrossRef]  

26. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011). [CrossRef]   [PubMed]  

27. U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013). [CrossRef]  

28. G. F. Xu and C. K. Law, “Dark states of a moving mirror in the single-photon strong-coupling regime,” Phys. Rev. A 87, 053849 (2013). [CrossRef]  

29. G. S. Agarwal and Sumei Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010). [CrossRef]  

30. H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012). [CrossRef]  

31. S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011). [CrossRef]  

32. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011). [CrossRef]   [PubMed]  

33. A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009). [CrossRef]  

34. A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012). [CrossRef]  

35. W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21, 020423 (2013). [CrossRef]  

36. M. R. Vanner, “Selective linear or quadratic optomechanical coupling via measurement,” Phys. Rev. X 1, 021011 (2011).

37. F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990). [CrossRef]   [PubMed]  

38. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011). [CrossRef]   [PubMed]  

39. M. E. A. El-Mikkawy, “On the inverse of a general tridiagonal matrix,” Appl. Math. Comput. 150, 669–679 (2004). [CrossRef]  

40. J. S. Peng and G. X. Li, Introduction to Modern Quantum Optics (World Scientific, 1998).

41. P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004). [CrossRef]  

42. S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 1997).

43. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).

44. R. L. Hudson, “When is the wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974). [CrossRef]  

45. J. Qian, A. A. Clerk, K. Hammerer, and F. Marquardt, “Quantum signatures of the optomechanical instability,” Phys. Rev. Lett. 109, 253601 (2012). [CrossRef]  

46. P. D. Nation, “Nonclassical mechanical states in an optomechanical micromaser analog,” Phys. Rev. A 88, 053828 (2013). [CrossRef]  

47. N. Lörch, J. Qian, A. Clerk, F. Marquardt, and K. Hammerer, “Laser theory for optomechanics: limit cycles in the quantum regime,” Phys. Rev. X 4, 011015 (2014).

48. W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005). [CrossRef]  

49. J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A 67, 022304 (2003). [CrossRef]  

50. T. Tyc and N. Korolkova, “Highly non-Gaussian states created via cross-Kerr nonlinearity,” New. J. Phys. 10, 023041 (2008). [CrossRef]  

51. M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014). [CrossRef]  

52. R. L. de, M. Filho, and W. Vogel, “Nonliear coherent states,” Phys. Rev. A 54, 4560–4563 (1996). [CrossRef]  

53. E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005). [CrossRef]  

54. Y. Greenberg, Y. A. Pashkin, and E. II’ichev, “Nanomechanical resonators,” Phys.-Usp. 55, 382 (2012). [CrossRef]  

55. M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” arXiv:1406.1013.

References

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  • |

  1. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
    [Crossref] [PubMed]
  2. Z. Q. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009).
    [Crossref]
  3. J. Zhang, Y. X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79, 052102 (2009).
    [Crossref]
  4. Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
    [Crossref]
  5. F. Khalili, S. Danilishin, H. X. Miao, H. Müller-Ebhardt, H. Yang, and Y. B. Chen, “Preparing a mechanical oscillator in non-gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010).
    [Crossref] [PubMed]
  6. D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature 488, 476–480 (2012).
    [Crossref] [PubMed]
  7. Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
    [Crossref] [PubMed]
  8. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
    [Crossref] [PubMed]
  9. C. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [Crossref]
  10. H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
    [Crossref] [PubMed]
  11. B. Kraus and J. I. Cirac, “Discrete entanglement distribution with squeezed light,” Phys. Rev. Lett. 92, 013602 (2004).
    [Crossref] [PubMed]
  12. M. Paternostro, W. Son, and M. S. Kim, “Complete conditions for entanglement transfer,” Phys. Rev. Lett. 92, 197901 (2004).
    [Crossref] [PubMed]
  13. W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013).
    [Crossref]
  14. S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
    [Crossref]
  15. S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, “Steady-state negative Wigner functions of nonlinear nanomechanical oscillators,” New J. Phys. 14, 023042 (2012).
    [Crossref]
  16. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
    [Crossref] [PubMed]
  17. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
    [Crossref] [PubMed]
  18. K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4, 561–564 (2008).
    [Crossref]
  19. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein Condensate,” Science 322, 235–238 (2008).
    [Crossref] [PubMed]
  20. A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
    [Crossref] [PubMed]
  21. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
    [Crossref] [PubMed]
  22. T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the Josephson effect,” arXiv:1311.3802.
  23. A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D. Nation, “A cavity-Cooper pair transistor scheme for investigating quantum optomechanics in the ultra-strong coupling regime,” New J. Phys. 16, 055008 (2014).
    [Crossref]
  24. J. Eisert, S. Scheel, and M. B. Plenio, “Distilling gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
    [Crossref] [PubMed]
  25. X. X. Ren, H. K. Li, M. Y. Yan, Y. C. Liu, Y. F. Xiao, and Q. H. Gong, “Single-photon transport and mechanical NOON-state generation in microcavity optomechanics,” Phys. Rev. A 87, 033807 (2013).
    [Crossref]
  26. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
    [Crossref] [PubMed]
  27. U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
    [Crossref]
  28. G. F. Xu and C. K. Law, “Dark states of a moving mirror in the single-photon strong-coupling regime,” Phys. Rev. A 87, 053849 (2013).
    [Crossref]
  29. G. S. Agarwal and Sumei Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
    [Crossref]
  30. H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
    [Crossref]
  31. S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
    [Crossref]
  32. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
    [Crossref] [PubMed]
  33. A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009).
    [Crossref]
  34. A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
    [Crossref]
  35. W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21, 020423 (2013).
    [Crossref]
  36. M. R. Vanner, “Selective linear or quadratic optomechanical coupling via measurement,” Phys. Rev. X 1, 021011 (2011).
  37. F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
    [Crossref] [PubMed]
  38. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
    [Crossref] [PubMed]
  39. M. E. A. El-Mikkawy, “On the inverse of a general tridiagonal matrix,” Appl. Math. Comput. 150, 669–679 (2004).
    [Crossref]
  40. J. S. Peng and G. X. Li, Introduction to Modern Quantum Optics (World Scientific, 1998).
  41. P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004).
    [Crossref]
  42. S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 1997).
  43. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).
  44. R. L. Hudson, “When is the wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974).
    [Crossref]
  45. J. Qian, A. A. Clerk, K. Hammerer, and F. Marquardt, “Quantum signatures of the optomechanical instability,” Phys. Rev. Lett. 109, 253601 (2012).
    [Crossref]
  46. P. D. Nation, “Nonclassical mechanical states in an optomechanical micromaser analog,” Phys. Rev. A 88, 053828 (2013).
    [Crossref]
  47. N. Lörch, J. Qian, A. Clerk, F. Marquardt, and K. Hammerer, “Laser theory for optomechanics: limit cycles in the quantum regime,” Phys. Rev. X 4, 011015 (2014).
  48. W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
    [Crossref]
  49. J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A 67, 022304 (2003).
    [Crossref]
  50. T. Tyc and N. Korolkova, “Highly non-Gaussian states created via cross-Kerr nonlinearity,” New. J. Phys. 10, 023041 (2008).
    [Crossref]
  51. M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
    [Crossref]
  52. R. L. de, M. Filho, and W. Vogel, “Nonliear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
    [Crossref]
  53. E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
    [Crossref]
  54. Y. Greenberg, Y. A. Pashkin, and E. II’ichev, “Nanomechanical resonators,” Phys.-Usp. 55, 382 (2012).
    [Crossref]
  55. M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” arXiv:1406.1013.

2014 (3)

A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D. Nation, “A cavity-Cooper pair transistor scheme for investigating quantum optomechanics in the ultra-strong coupling regime,” New J. Phys. 16, 055008 (2014).
[Crossref]

N. Lörch, J. Qian, A. Clerk, F. Marquardt, and K. Hammerer, “Laser theory for optomechanics: limit cycles in the quantum regime,” Phys. Rev. X 4, 011015 (2014).

M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
[Crossref]

2013 (7)

W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21, 020423 (2013).
[Crossref]

P. D. Nation, “Nonclassical mechanical states in an optomechanical micromaser analog,” Phys. Rev. A 88, 053828 (2013).
[Crossref]

X. X. Ren, H. K. Li, M. Y. Yan, Y. C. Liu, Y. F. Xiao, and Q. H. Gong, “Single-photon transport and mechanical NOON-state generation in microcavity optomechanics,” Phys. Rev. A 87, 033807 (2013).
[Crossref]

U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
[Crossref]

G. F. Xu and C. K. Law, “Dark states of a moving mirror in the single-photon strong-coupling regime,” Phys. Rev. A 87, 053849 (2013).
[Crossref]

Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[Crossref] [PubMed]

W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013).
[Crossref]

2012 (7)

S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, “Steady-state negative Wigner functions of nonlinear nanomechanical oscillators,” New J. Phys. 14, 023042 (2012).
[Crossref]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature 488, 476–480 (2012).
[Crossref] [PubMed]

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
[Crossref]

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

J. Qian, A. A. Clerk, K. Hammerer, and F. Marquardt, “Quantum signatures of the optomechanical instability,” Phys. Rev. Lett. 109, 253601 (2012).
[Crossref]

Y. Greenberg, Y. A. Pashkin, and E. II’ichev, “Nanomechanical resonators,” Phys.-Usp. 55, 382 (2012).
[Crossref]

2011 (7)

M. R. Vanner, “Selective linear or quadratic optomechanical coupling via measurement,” Phys. Rev. X 1, 021011 (2011).

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
[Crossref]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
[Crossref]

2010 (3)

F. Khalili, S. Danilishin, H. X. Miao, H. Müller-Ebhardt, H. Yang, and Y. B. Chen, “Preparing a mechanical oscillator in non-gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010).
[Crossref] [PubMed]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

G. S. Agarwal and Sumei Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[Crossref]

2009 (3)

A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009).
[Crossref]

Z. Q. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009).
[Crossref]

J. Zhang, Y. X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79, 052102 (2009).
[Crossref]

2008 (4)

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref] [PubMed]

K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4, 561–564 (2008).
[Crossref]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein Condensate,” Science 322, 235–238 (2008).
[Crossref] [PubMed]

T. Tyc and N. Korolkova, “Highly non-Gaussian states created via cross-Kerr nonlinearity,” New. J. Phys. 10, 023041 (2008).
[Crossref]

2007 (3)

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref] [PubMed]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
[Crossref] [PubMed]

2005 (2)

E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
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W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
[Crossref]

2004 (4)

B. Kraus and J. I. Cirac, “Discrete entanglement distribution with squeezed light,” Phys. Rev. Lett. 92, 013602 (2004).
[Crossref] [PubMed]

M. Paternostro, W. Son, and M. S. Kim, “Complete conditions for entanglement transfer,” Phys. Rev. Lett. 92, 197901 (2004).
[Crossref] [PubMed]

M. E. A. El-Mikkawy, “On the inverse of a general tridiagonal matrix,” Appl. Math. Comput. 150, 669–679 (2004).
[Crossref]

P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004).
[Crossref]

2003 (1)

J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A 67, 022304 (2003).
[Crossref]

2002 (1)

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[Crossref] [PubMed]

1997 (1)

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
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1996 (1)

R. L. de, M. Filho, and W. Vogel, “Nonliear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
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1990 (1)

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
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1981 (1)

C. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
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1974 (1)

R. L. Hudson, “When is the wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974).
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Agarwal, G. S.

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
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G. S. Agarwal and Sumei Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
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U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
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Armour, A. D.

A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D. Nation, “A cavity-Cooper pair transistor scheme for investigating quantum optomechanics in the ultra-strong coupling regime,” New J. Phys. 16, 055008 (2014).
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Aspelmeyer, M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 1997).

Beausoleil, R. G.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
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Blencowe, M. P.

A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D. Nation, “A cavity-Cooper pair transistor scheme for investigating quantum optomechanics in the ultra-strong coupling regime,” New J. Phys. 16, 055008 (2014).
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Børkje, K.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
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Bose, S.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
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Botter, T.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature 488, 476–480 (2012).
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Bowen, W. P.

U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
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Brahms, N.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature 488, 476–480 (2012).
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Brennecke, F.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein Condensate,” Science 322, 235–238 (2008).
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Brooks, D. W. C.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature 488, 476–480 (2012).
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Buzek, V.

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
[Crossref] [PubMed]

Caves, C.

C. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
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Chan, J.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

Chang, D. E.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Chen, Y. B.

F. Khalili, S. Danilishin, H. X. Miao, H. Müller-Ebhardt, H. Yang, and Y. B. Chen, “Preparing a mechanical oscillator in non-gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010).
[Crossref] [PubMed]

Cirac, J. I.

B. Kraus and J. I. Cirac, “Discrete entanglement distribution with squeezed light,” Phys. Rev. Lett. 92, 013602 (2004).
[Crossref] [PubMed]

Clerk, A.

N. Lörch, J. Qian, A. Clerk, F. Marquardt, and K. Hammerer, “Laser theory for optomechanics: limit cycles in the quantum regime,” Phys. Rev. X 4, 011015 (2014).

Clerk, A. A.

M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
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Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
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J. Qian, A. A. Clerk, K. Hammerer, and F. Marquardt, “Quantum signatures of the optomechanical instability,” Phys. Rev. Lett. 109, 253601 (2012).
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F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Danilishin, S.

F. Khalili, S. Danilishin, H. X. Miao, H. Müller-Ebhardt, H. Yang, and Y. B. Chen, “Preparing a mechanical oscillator in non-gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010).
[Crossref] [PubMed]

de, R. L.

R. L. de, M. Filho, and W. Vogel, “Nonliear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
[Crossref]

de Oliveira, F. A. M.

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
[Crossref] [PubMed]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Donner, T.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein Condensate,” Science 322, 235–238 (2008).
[Crossref] [PubMed]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Eisert, J.

A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009).
[Crossref]

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[Crossref] [PubMed]

El-Mikkawy, M. E. A.

M. E. A. El-Mikkawy, “On the inverse of a general tridiagonal matrix,” Appl. Math. Comput. 150, 669–679 (2004).
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Esslinger, T.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein Condensate,” Science 322, 235–238 (2008).
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A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
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Filho, M.

R. L. de, M. Filho, and W. Vogel, “Nonliear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
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Filip, R.

J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A 67, 022304 (2003).
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Fiurášek, J.

J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A 67, 022304 (2003).
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Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
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Giovannetti, V.

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
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Girvin, S. M.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Gong, Q. H.

X. X. Ren, H. K. Li, M. Y. Yan, Y. C. Liu, Y. F. Xiao, and Q. H. Gong, “Single-photon transport and mechanical NOON-state generation in microcavity optomechanics,” Phys. Rev. A 87, 033807 (2013).
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Y. Greenberg, Y. A. Pashkin, and E. II’ichev, “Nanomechanical resonators,” Phys.-Usp. 55, 382 (2012).
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Gröblacher, S.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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Gu, W. J.

W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21, 020423 (2013).
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W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013).
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Gupta, S.

K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4, 561–564 (2008).
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Hammerer, K.

N. Lörch, J. Qian, A. Clerk, F. Marquardt, and K. Hammerer, “Laser theory for optomechanics: limit cycles in the quantum regime,” Phys. Rev. X 4, 011015 (2014).

J. Qian, A. A. Clerk, K. Hammerer, and F. Marquardt, “Quantum signatures of the optomechanical instability,” Phys. Rev. Lett. 109, 253601 (2012).
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S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, “Steady-state negative Wigner functions of nonlinear nanomechanical oscillators,” New J. Phys. 14, 023042 (2012).
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T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the Josephson effect,” arXiv:1311.3802.

Hill, J. T.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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Huang, S.

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
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Huang, Sumei

G. S. Agarwal and Sumei Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
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R. L. Hudson, “When is the wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974).
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Y. Greenberg, Y. A. Pashkin, and E. II’ichev, “Nanomechanical resonators,” Phys.-Usp. 55, 382 (2012).
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S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
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F. Khalili, S. Danilishin, H. X. Miao, H. Müller-Ebhardt, H. Yang, and Y. B. Chen, “Preparing a mechanical oscillator in non-gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010).
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Khan, R.

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the Josephson effect,” arXiv:1311.3802.

Kiffner, M.

S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, “Steady-state negative Wigner functions of nonlinear nanomechanical oscillators,” New J. Phys. 14, 023042 (2012).
[Crossref]

Kim, M. S.

M. Paternostro, W. Son, and M. S. Kim, “Complete conditions for entanglement transfer,” Phys. Rev. Lett. 92, 197901 (2004).
[Crossref] [PubMed]

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
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M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” arXiv:1406.1013.

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H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
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S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
[Crossref] [PubMed]

Knight, P. L.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
[Crossref]

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
[Crossref] [PubMed]

Korolkova, N.

T. Tyc and N. Korolkova, “Highly non-Gaussian states created via cross-Kerr nonlinearity,” New. J. Phys. 10, 023041 (2008).
[Crossref]

Kraus, B.

B. Kraus and J. I. Cirac, “Discrete entanglement distribution with squeezed light,” Phys. Rev. Lett. 92, 013602 (2004).
[Crossref] [PubMed]

Krause, A.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).

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W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21, 020423 (2013).
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W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013).
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Li, H. K.

X. X. Ren, H. K. Li, M. Y. Yan, Y. C. Liu, Y. F. Xiao, and Q. H. Gong, “Single-photon transport and mechanical NOON-state generation in microcavity optomechanics,” Phys. Rev. A 87, 033807 (2013).
[Crossref]

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[Crossref]

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N. Lörch, J. Qian, A. Clerk, F. Marquardt, and K. Hammerer, “Laser theory for optomechanics: limit cycles in the quantum regime,” Phys. Rev. X 4, 011015 (2014).

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[Crossref]

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Figures (9)

Fig. 1
Fig. 1

Sketch of the optomechanical system consisting of a fixed end mirror and a movable end mirror. The cavity field couples to the movable mirror via radiation pressure, which is driven by a trichromatic laser field.

Fig. 2
Fig. 2

The transitions between the energy levels of the optomechanical system with cavity field confined within the zero- and single-photon states. Each component of trichromatic field is applied to establish a set of resonant transitions, and we denote |Pm = |(0)〉m and |m = |(1)〉m for simplicity. By choosing an appropriate g = gN, there will be no transitions to N + 1 phonons states shown by dashed lines.

Fig. 3
Fig. 3

Numerical results of phonon number distribution Pm of the dark state compared with that of squeezed coherent state in Eq. (24). The coupling strength is gN = 0.1916ωm with phonon number N = 99, laser driving strength is Ω2 = 0.0225ωm, Ω3 = −0.015ωm, and Ω1 = −0.0038ωm is determined from Eq. (20) with the requirement of rank(M) = N.

Fig. 4
Fig. 4

Wigner function of the dark state with gN = 0.1916ωm corresponding to the phonon number N = 99, laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm, and Ω1 = −0.0038ωm determined from Eq. (20) with the requirement of rank(M) = N. The crescent shape is clearly visible.

Fig. 5
Fig. 5

Wigner function and the phonon number distribution Pm of the dark state with the coupling strength gN = 0.5124ωm (N = 13), laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.0029ωm determined from Eq. (20) with the requirement of rank(M) = N. The crescent shape and negative values of the Wigner function are clearly visible.

Fig. 6
Fig. 6

Wigner function and the phonon number distribution Pm of the dark state with the coupling strength gN = 0.3678ωm (N = 26), laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.00963ωm determined from Eq. (20) with the requirement of rank(M) = N. Here the phonon number population is mainly distributed within N = 15.

Fig. 7
Fig. 7

Wigner function and the phonon number distribution Pm of the dark state with the coupling strength gN = 0.5124ωm (N = 13), laser driving strength Ω2 = 0.016ωm, Ω3 = −0.01ωm and Ω1 = 0.002ωm determined from Eq. (20) with the requirement of rank(M) = N.

Fig. 8
Fig. 8

Fidelity between the system state and the dark state as a function of time. The initial system state is a dark state with the coupling strength gN = 0.5124ωm (N = 13), laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.0029ωm determined from Eq. (20) with the requirement of rank(M) = N. Laser detunings are given in Eq. (9).

Fig. 9
Fig. 9

Time evolution of Fidelity between the system state and the dark state. The initial system is in its ground state with the coupling strength gN = 0.5124ωm (N = 13), cavity damping rate κ/ωm = 0.08, laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.0029ωm determined from Eq. (20) with the requirement of rank(M) = N. Laser detunings are given in Eq. (9).

Equations (30)

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H ^ = ω c a ^ a ^ + ω m b ^ b ^ g a a ^ ( b ^ + b ^ ) + [ ( Ω 1 e i ω 1 t + Ω 2 e i ω 2 t + Ω 3 e i ω 3 t ) a ^ + H . c . ] ,
H ^ = ω m b ^ b ^ g a ^ a ^ ( b ^ + b ^ ) + [ ( Ω 1 e i Δ 1 t + Ω 2 e i Δ 2 t + Ω 3 e i Δ 3 t ) a ^ + H . c . ] ,
H ^ 0 = ω m b ^ b ^ g a ^ a ^ ( b ^ + b ^ ) = n , p ε n , p | ψ n , p ψ n , p | .
| ψ n , p = | n c D ( n g / ω m ) | p m = | n c | p ˜ ( n ) m ,
a ^ = n , p n , p | ψ n , p ψ n , p | a ^ | ψ n , p ψ n , p | = n , p , p A p , p ( n ) | ψ n 1 , p ψ n , p | ,
A p , p ( n ) = { n p ! p ! e ξ 2 2 ( ξ ) p p L p p p ( ξ 2 ) , p p , n p ! p ! e ξ 2 2 ( ξ ) p p L p p p ( ξ 2 ) , p > p ,
H ^ = n , p ε n , p | ψ n , p ψ n , p | + n , p , p [ A p , p ( n ) ( Ω 1 e i Δ 1 t + Ω 2 e i Δ 2 t + Ω 3 e i Δ 3 t ) | ψ n , p ψ n 1 , p | + H . c . ] .
H ^ int = n , p , p [ A p , p ( n ) ( Ω 1 e i Δ 1 t + Ω 2 e i Δ 2 t + Ω 3 e i Δ 3 t ) × e i ( ε n , p ε n 1 , p ) t | ψ n , p ψ n 1 , p | + H . c . ] .
Δ 1 = ε 1 , p ε 0 , p = g 2 / ω m , Δ 2 = ε 1 , p ε 0 , p + 1 = ω m g 2 / ω m , Δ 3 = ε 1 , p ε 0 , p 1 = ω m g 2 / ω m ,
H ^ int = H ^ eff + V ^ ,
H ^ eff = p [ ( A p , p ( 1 ) Ω 1 | ψ 1 , p ψ 0 , p | + A p + 1 , p ( 1 ) Ω 2 | ψ 1 , p ψ 0 , p + 1 | + A p , p + 1 ( 1 ) Ω 3 | ψ 1 , p + 1 ψ 0 , p | ) + H . c . ] ,
V ^ = Σ n , p , p [ A p , p ( n ) ( Ω 1 e i δ 1 ( n , p , p ) t + Ω 2 e i δ 2 ( n , p , p ) t + Ω 3 e i δ 3 ( n , p , p ) t ) | ψ n , p ψ n 1 , p | + H . c . ] .
δ 1 ( n , p , p ) = ( p p ) ω m 2 ( n 1 ) g 2 / ω m , δ 2 ( n , p , p ) = ( p p + 1 ) ω m 2 ( n 1 ) g 2 / ω m , δ 3 ( n , p , p ) = ( p p 1 ) ω m 2 ( n 1 ) g 2 / ω m .
| D = C p = 0 β p | 0 c | p m ,
β p 1 Ω 3 A p 1 , p ( 1 ) + β p Ω 1 A p , p ( 1 ) + β p + 1 Ω 2 A p + 1 , p ( 1 ) = 0 .
A N + 1 , N ( 1 ) = 1 / ( N + 1 ) e ξ 2 / 2 ξ L N ( 1 ) ( ξ 2 ) , A N , N + 1 ( 1 ) = 1 / ( N + 1 ) e ξ 2 / 2 ( ξ ) L N ( 1 ) ( ξ 2 ) .
β N 1 Ω 3 A N 1 , N ( 1 ) + β N Ω 1 A N , N ( 1 ) = 0
M | β ] = 0 ,
| β ] = ( β 0 , β 1 , , β N 1 , β N ) T
M = [ Ω 1 A 0 , 0 ( 1 ) Ω 2 A 1 , 0 ( 1 ) Ω 3 A 0 , 1 ( 1 ) Ω 1 A 1 , 1 ( 1 ) Ω 2 A 2 , 1 ( 1 ) Ω 3 A 1 , 2 ( 1 ) Ω 2 A N , N 1 ( 1 ) Ω 3 A N , N 1 , N ( 1 ) Ω 1 A N , N ( 1 ) ] .
A p 1 , p ( 1 ) p ( ξ ) e ξ 2 / 2 , A p , p ( 1 ) e ξ 2 / 2 , A p + 1 , p ( 1 ) p + 1 ξ e ξ 2 / 2 .
β p 1 p ( Ω 3 ξ ) + β p Ω 1 + β p + 1 p + 1 ( Ω 2 ξ ) = 0 .
μ β p 1 p + ν β p + 1 p + 1 = β β p ,
β p = 1 p ! ν ( μ 2 ν ) p 2 exp ( | β | 2 2 + μ * 2 ν β 2 ) H p ( β 2 μ ν ) ,
( Δ X ^ ) 2 = X ^ 2 X ^ 2 .
W ( α ) = 2 π Tr [ D ^ ( α ) ρ ^ D ^ ( α ) ( 1 ) b ^ b ^ ] ,
A p + 1 , p ( 1 ) = 1 p + 1 e ξ 2 / 2 ξ L p ( 1 ) ( ξ 2 ) = p + 1 e ξ 2 / 2 ξ [ 1 p 2 ! ξ 2 + ( p 1 ) p 3 ! ξ 4 + ] = p + 1 e ξ 2 / 2 ξ f ( p ) ,
| D ( t ) = C p = 0 N β p e i p ω m t | 0 c | p m ,
d d t ρ ^ = i [ H ^ , ρ ^ ] + κ 2 ( 2 a ^ ρ ^ a ^ a ^ a ^ ρ ^ ρ ^ a ^ a ^ ) + γ ( n ¯ + 1 ) 2 ( 2 b ^ ρ ^ b ^ b ^ b ^ ρ ^ ρ ^ b ^ b ^ ) + γ n ¯ 2 ( 2 b ^ ρ ^ b ^ b ^ b ^ ρ ^ ρ ^ b ^ b ^ ) ,
F = Tr [ | D ( t ) D ( t ) | ρ ^ ( t ) ] .

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