Strong quadrature squeezing and quantum amplification in a coupled Bose-Einstein condensate- optomechanical cavity via coherent modulation
Ali Motazedifard A. Dalafi M. H. Naderi R. Roknizadeh




 
motazedifard.ali@gmail.com
Quantum Optics Group, Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran





 
a_dalafi@sbu.ac.ir
Laser and Plasma Research Institute, Shahid Beheshti University, Tehran 19839-69411, Iran

 
mhnaderi@sci.ui.ac.ir

Quantum Optics Group, Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran



rokni@sci.ui.ac.ir

Quantum Optics Group, Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran








We consider an optomechanical cavity with a movable end-mirror as a quantum mechanical oscillator (MO) containing an interacting cigar-shaped Bose-Eisenstein condensate (BEC). It is assumed that both the MO and the BEC interact with the radiation pressure of the cavity field in the red-detuned and weak coupling regimes while the two-body atomic collisions frequency of the BEC and the mechanical spring coefficient of the MO are coherently modulated. By analyzing the scattering matrix we find that a frequency-dependent squeezing is induced to the three subsystems only due to the coherent modulations. In the largely different cooperativities regime together with strong modulations, the mechanical mode of the MO and the Bogoliubov mode of the BEC exhibit quadrature squeezing which can surpass the so-called 3dB limit (up to 75 dB) with high robustness to the thermal noises. Surprisingly, in this regime by controlling the system and modulation parameters, a very high degree of squeezing (up to 16 dB) together with high purity of quantum state for the output cavity field is achievable. Furthermore, one can attain simultaneous strong quantum amplification, added-noise suppression, and controllable gain-bandwidth for the complementary quadratures of squeezed ones in the subsystems.










Strong quadrature squeezing and quantum amplification in a coupled Bose-Einstein condensate- optomechanical cavity via coherent modulation

    R. Roknizadeh
    December 30, 2023
===========================================================================================================================================






§ INTRODUCTION


Over the past decade, there has been a significant theoretical and experimental progress in the field of quantum optomechanics which deals with linear or quadratic coupling of the electromagnetic radiation pressure to a mechanical oscillator (MO) <cit.>. Quantum optomechanical systems (OMSs) have had a remarkable contribution to the emergence of quantum behaviors on macroscopic scales and have made a significant contribution in probing fundamentals of physics. OMSs have many practical applications such as displacement and force sensing <cit.>, ground-state cooling of the MO <cit.>, generation of entanglement <cit.>, synchronization of MOs <cit.> and generation of nonclassical states of the mechanical and optical modes <cit.>. Very recently, OMSs have been proposed for the realization of the parametric-dynamical Casimir effect (P-DCE) <cit.> and the quantum simulation of the curved space-times <cit.>.




Among various applications of OMSs, quantum amplification <cit.> and generation of squeezing <cit.> have been very attractive fields of research in the realm of quantum optics over the recent years. In addition to the standard ponderomotive squeezing <cit.> and more recently proposed method of dissipative-squeezing <cit.> and also methods of Refs. <cit.>, there is a well-known method based on the modulation technique to achieve the quadrature squeezing in OMSs <cit.> which at best, can lead to 50% noise reduction below the zero-point level (the so-called 3 dB limit). Nevertheless, noise reduction much below the standard quantum limit (SQL) is also achievable by parametrically modulating the spring constant of
the MO in a bare optomechanical cavity <cit.>. As has been shown in Ref. <cit.> through the mentioned modulation method the system can also act as an optical quantum amplifier. Besides, there are some methods based on the feedback process <cit.>, quantum measurement <cit.>, and quadratic optomechanical coupling (QOC) <cit.> in order to surpass and beat the 3dB limit. Also, very recently, it has been shown <cit.> that the modulation of the amplitude of the driving laser in an optomechanical cavity containing an optical parametric amplifier can generate two-fold mechanical squeezing beyond the SQL.




On the other hand, hybrid OMSs assisted with trapped Bose-Einstein condensates (BECs) <cit.> or ultra cold atoms integrated in an atom chip <cit.>  in which the fluctuations in the collective excitations of the atoms, i.e., the so-called Bogoliubov mode, plays the role of an effective mechanical mode while the nonlinear atom-atom interaction plays the role of an atomic amplifier <cit.>, have attracted considerable attention to a number of studies such as normal mode splitting (NMS) <cit.>, noise reduction <cit.>, and controlling bistability, cooling, and bipartite entanglement <cit.>. Moreover, very recently, it has been shown <cit.> that through the effect of the QOC in the OMS assisted with the BEC, one can achieve a considerable degree of squeezing (up to 10 dB) in the mechanical mode and a strong stationary entanglement between the mechanical and atomic modes.
 

 

Motivated by the above-mentioned studies as well as inspired by the proposed scheme of Ref. <cit.>, in the present paper we are going to investigate a hybrid optomechanical system as a quantum linear amplifier/squeezer with an extra degree of freedom (the Bogoliubov mode of the BEC) in addition to the mechanical mode of the MO in which both the mechanical spring coefficient of the MO and the two-body atomic collisions frequency of the BEC are coherently modulated. The advantage of the present hybrid system in comparison to the bare one presented in Ref. <cit.> is that the presence of the extra Bogoliubov mode of the BEC which have much more controllability with respect to the mechanical mode of the MO, can lead to a considerable enhancement of the squeezing and amplification in both the mechanical and the optical modes.

On the other hand, we have shown in a very recent work <cit.> that the parametric modulations of the mechanical and atomic modes in such a hybrid system can lead to the realization of the parametric DCE of photons and phonons. However, in the present paper we investigate some other features of the system, including controllable realization of strong quadrature squeezing together with complementary-quadrature amplification, added-noise suppression, and controllable increasing of the gain-bandwidth. In other words, we study the system as a controllable optomechanical squeezer/amplifier.

Here, by determining the quadratures spectra with use of the scattering-matrix method we find that in the so-called regime of largely different cooperativities high degrees of quadrature squeezing for both the output cavity field (up to 16 dB) and phonon modes (up to 75 dB ) can be achieved through the modulation of the atomic collisions frequency and the mechanical spring coefficient. Such high degrees of quadratures squeezing are originated from a frequency-dependent squeezing effect induced to the cavity field, the MO, and the BEC. Besides, It is shown that the complementary quadratures of squeezed ones, can be strongly amplified in the large driving regime where the added noises can be strongly suppressed simultaneously. In this situation the system behaves as a noiseless phase-sensitive amplifier. Furthermore, their gain-bandwidth can be externally controlled by adjusting the amplitudes of modulations and atomic collisions frequency. 


The paper is organized as follows. In Sec. <ref>, we describe the physical system and its effective Hamiltonian.
In Sec. <ref>, by solving the quantum Langevin equations in frequency space we find the self-energies and induced-parametric squeezing coefficients. In Sec. <ref> we investigate the quadratures amplification spectra and the gain-bandwith using the scattering matrix method. In Sec. <ref> by calculating the quadratures squeezing spectra we discuss how one can achieve high degrees of quadrature squeezing for both photon and phonon modes. Finally, we summarize our conclusions in Sec. <ref> 




§ THE SYSTEM HAMILTONIAN 




As shown in Fig. (<ref>), we consider a red-detuned driven hybrid optomechanical cavity with length L whose end mirror with effective mass m and damping rate γ_m is free to oscillate at mechanical frequency  ω_m. The cavity contains a cigar-shaped BEC of N ultracold two-level atoms with mass m_a and transition frequency ω_a. The cavity is driven at rate E_L= √(κ P_L / ħω_L) through the fixed mirror by a laser with frequency ω_L and wavenumber k_0=ω_L/c (P_L is the laser power and κ is the cavity decay rate). 

We assume that the MO is parametrically driven by modulating its spring coefficient at twice its natural frequency <cit.>, i.e., k(t)= k+δ k cos(2ω_m t +ϕ_m) (ϕ_m being the phase of external modulation). Under these assumptions, the total Hamiltonian of the system in the frame rotating at laser frequency is given by

    Ĥ= Ĥ_0 + Ĥ_om + Ĥ_m(t) + i ħ E_L (â-â^† ) + Ĥ_BEC(t).

The first term in Eq. (<ref>) describes the free Hamiltonian of the cavity field and the MO, the second term describes the optomechanical interaction, and the third term corresponds to the mechanical parametric drive whose explicit forms are, respectively, given by

	
    Ĥ_0=ħΔ_c â^†â + ħω_m b̂^†b̂ , 
          Ĥ_om= -ħ g_0 â^†â (b̂ + b̂^†), 
           H_m(t)= i ħ/2 (λ_m b̂^† 2  e^-2iω_m t- λ_m^∗b̂^2 e^2iω_m t),

	
where â (b̂) is the annihilation operator of the cavity (mechanical) mode, Δ_c= ω_c-ω_L is the detuning of the cavity mode from the driving laser frequency, and g_0=x_zpω_c /L stands for the single-photon optomechanical coupling with x_zp= √(ħ / 2mω_m) being the mechanical zero-point position fluctuation. 
Besides, λ_m= |λ_m | e^iϕ_m characterizes the strength and phase of the mechanical parametric driving with |λ_m |= δ k x_zp^2 / 2ħ, which can be taken real by fixing the phase of modulation ϕ_m. Note that the Hamiltonian of Eq. (<ref>) describes the mechanical phonon analog of the degenerate parametric amplification (DPA) where the vibrational fluctuation of the MO plays the role of the signal mode in the DPA. In Ref. <cit.>, it has been shown that such a phonon analog of DPA leads to the amplification of quantum vacuum fluctuations of the MO, i.e., the DCE of mechanical phonons.

The last term in Eq. (<ref>) describes the many-body Hamiltonian of the BEC in the framework of second quantization <cit.>. In the dispersive regime of atom-field interaction where the laser pump is far-detuned from the atomic resonance (Δ_a=ω_a - ω_L ≫Γ_a where Γ_a is the atomic linewidth)  the Hamiltonian of the BEC can be written as <cit.>

    Ĥ_BEC=∫_-L/2^L/2dx ψ̂^†(x) [-ħ^2/2m_ad^2/dx^2,
           + ħ U_0 cos^2(k_0 x) â^†â+1/2U_sψ̂^†(x)  ψ̂^†(x)] ψ̂(x),

where ψ̂(x) is the annihilation operator of the atomic field, U_0=-g_a^2/Δ_a is the optical lattice barrier height per photon which represents the atomic backaction on the field in the dispersive regime, g_a is the vacuum Rabi frequency or atom-field coupling constant, U_s=4πħ^2 a_s/m_a, and a_s is the two-body s-wave scattering length <cit.>. In the weakly interacting regime <cit.>, and under the Bogoliubov approximation <cit.> the atomic field operator can be expanded as the following single-mode quantum field

    ψ̂(x)= √(N/L)+ √(2/L)cos(2k_0x) d̂ ,

where the so-called Bogoliubov mode d̂ corresponds to the quantum fluctuations of the atomic field around the classical condensate mode √(N/L).

On the other hand, as has been shown in Ref. <cit.>, the time modulation of the s-wave scattering frequency of atom-atom interaction, ω_sw, can be realized by the time modulation of the scattering length via the modulation of the electromagnetic trap, or the modulation of the density of the BEC by changing the trap stiffness via the intensity modulation of the pump laser. Now, considering the time modulation of the s-wave scattering frequency as ω_sw(t)=ω_sw(1+ϵcos(2ω_dt+ϕ_d))  where ϵ and ϕ_d refer, respectively, to the amplitude and the phase of modulation, ω_d=4ω_R+ω_sw with ω_R=ħ k_0^2/2m_a being the recoil frequency and ω_sw=8πħ N a_s/(m_a L w^2) (w is the waist radius of the optical mode), the Hamiltonian of the modulated BEC within the rotating wave approximation (RWA) can be written as <cit.>

	
    Ĥ_BEC=ħδ_0 â^†â  +  ħω_d d̂^†d̂ + ħ G_0â^†â (d̂  + d̂^†) +  Ĥ_sw(t) , 
          Ĥ_sw(t)=i ħ/2 (λ_d d̂^† 2  e^-2iω_d t- λ_d^∗d̂^2 e^2iω_d t),

	
where  δ_0=NU_0/2 and λ_d=-iϵω_sw e^-iϕ_d/4 can be taken real by fixing the phase ϕ_d. Furthermore, G_0=√(2N)U_0/4 is the strength of an optomechanical-like coupling between the Bogoliubov mode of the BEC and the intracavity field . The first term of the Hamiltonian of Eq. (<ref>) leads to a shift of the cavity detuning,i.e., Δ_c →Δ_0=Δ_c + NU_0/2 which can be interpreted as an effective Stark-shifted detuning.





§ DYNAMICS OF THE SYSTEM 

Here, we investigate the dynamics of the system under the conditions that the frequency of the Bogoliubov mode of the BEC, ω_d, can be matched to the mechanical frequency (ω_d ≈ω_m) via the atomic collisions frequency ω_sw or through the recoil frequency ω_R via the driving laser frequency. Therefore, we restrict ourselves to the red-detuned regime of cavity optomechanics where Δ_0 ≈ω_m ≈ω_d. In this regime and within the RWA where the BEC-cavity mode as well as the MO-cavity mode couplings are analogous to the beam-splitter interaction, the linearized QLEs describing the dynamics of the quantum fluctuations are given by  <cit.>
 
	
    δâ̇ = -κ/2δâ + i g  δb̂ - i G δd̂ + √(κ)â_in , 
          δḃ̂̇  = -γ_m/2δb̂ + i g δâ + λ_m δb̂^† +  √(γ_m)b̂_in, 
          δḋ̂̇  = - γ_d2δd̂  -i G  δâ + λ_d δd̂^†  +  √(γ_d)d̂_in,


where g=g_0 a̅ and G = G_0 a̅ are, respectively, the enhanced-optomechanical coupling strengths of the moving mirror and the Bogoliubov mode of the BEC to the intracavity field with a̅ being the optical mean-field value. Furthermore, γ_m and γ_d  denote the dissipation rates of the mechanical and the Bogoliubov modes, respectively.

According to Eqs.(<ref>a)-(<ref>c), three noise operators affect the system: the optical input vacuum noise, a_in, and the thermal Brownian noise operators b_in and d_in acting, respectively, on the MO and the Bogoliubov mode of the BEC. These noises which are assumed to be uncorrelated for the different modes of both the matter and light fields, satisfy the Markovian correlation functions <cit.> ⟨â_in(t)â_in^†(t^')⟩=(n̅_a^T+1)δ(t-t^'), ⟨â_in^†(t)â_in(t^')⟩=n̅_a^Tδ(t-t^'),  ⟨b̂_in(t)b̂_in^†(t^')⟩=(n̅_m^T+1)δ(t-t^'), ⟨b̂_in^†(t)b̂_in(t^')⟩=n̅_m^Tδ(t-t^'),
⟨d̂_in(t)d̂_in^†(t^')⟩=(n̅_d^T+1)δ(t-t^'), and ⟨d̂_in^†(t)d̂_in(t^')⟩=n̅_d^Tδ(t-t^') where n̅_j^T=[exp(ħω_j/k_B T_j)-1]^-1 with j=a,m,d are the thermal excitations of the optical, mechanical and Bogoliubov modes at temperature T_j.

Now, by defining the fluctuation quadratures δX̂_o=(δô+δô^†)/√(2) and δP̂_o=(δô-δô^†)/√(2)i (o=a,b,d) we can express the linearized Eqs. (<ref>a)-(<ref>c) in the compact matrix form as

    δû̇(t)= A  δû(t) + û_in(t),

where the vector of continuous-variable fluctuation operators and the corresponding vector of noises are given by δû=(δX̂_a,δP̂_a,δX̂_b, δP̂_b,δX̂_d,δP̂_d)^T and û_in=(√(κ)X̂_a^in,√(κ)P̂_a^in,√(γ_m)X̂_b^in,√(γ_m)P̂_b^in, √(γ_d)X̂_d^in,√(γ_d)P̂_d^in )^T, respectively. Here, X̂_o^in=(ô_in+ô_in^†)/√(2) and P̂_o^in=(ô_in-ô_in^†)/√(2)i (o=a,b,d).
Furthermore, the time-independent drift matrix A is given by 

    A=( -κ/2   0   0   -g   0   G
    0   -κ/2   g   0   -G   0
    0   -g   λ_m-γ_m/2   0   0   0
    g   0   0   -(λ_m+γ_m/2)   0   0
    0   G   0   0   λ_d -γ_d/2   0
    -G   0   0   0   0   -(λ_d+γ_d/2 )
    ).




In the following sections we will show how modulation of the atomic collisions frequency or the mechanical spring coefficient leads to the strong quadrature squeezing of the mechanical and Bogoliubov modes. As has been shown in Ref. <cit.>, based on the Routh-Hurwitz criterion for the optomechanical stability condition, the modulation parameters λ_m and λ_d should satisfy the condition

    λ_m(d)≤λ_m(d)^max=γ_m(d) + Γ_op^m(d)= γ_m(d)/2[1+ 𝒞_m(d)],

where the maximum optomechanically induced damping rate is Γ_op^m(d)=-2 ImΣ_b(d)(0) in which the self-energies Σ_b(d) have been given by Eqs. (26) of Ref. <cit.>. Furthermore, 𝒞_m(𝒞_d) is the collective optomechanical cooperativity which is given by 

    𝒞_m(d)=  𝒞_0(1)1+ 𝒞_1(0) - ξ_d(m)^2/(1+ 𝒞_1(0)-ξ_d(m)^2)^2- ξ_d(m)^2  𝒞_1(0)^2 ,

where 𝒞_0=4g^2/κγ_m and 𝒞_1= 4G^2/κγ_d are, respectively, the optomechanical and opto-atomic cooperativities, and ξ_d(m)=2λ_d(m)/γ_d(m) plays the role of an effective dimensionless amplitude of modulation. 

Before calculating the scattering matrix in the next section it is worthwhile, here, to consider the solutions of the linearized QLEs [Eqs. (<ref>a)-(<ref>c)] in frequency space which are given by <cit.>

	
    -iωδâ  = -[κ/2  + iΣ_a(ω)] δâ + λ̃_a δâ^† + √(κ)Â_in(ω) , 
           -iωδb̂= -[γ_m/2 -  iΣ_b(ω)] δb̂ + λ̃_b δb̂^† + √(γ_m)B̂_in(ω), 
           -iωδd̂=  -[γ_d/2  +  i Σ_d(ω)] δd̂ + λ̃_d δd̂^† + √(γ_d)D̂_in(ω).


Here, Σ_o(ω) (o=a,b,d) is the self-energy (given by Eq. (26) in Ref. <cit.>), Â_in, B̂_in, and D̂_in denote the generalized noise operators (given by Eq. (29) in Ref. <cit.>), and the induced frequency-dependent parametric amplification coefficients λ̃_o(ω) which are analogous to the gain factors in the conventional DPA are given by <cit.>

	
    λ̃_a(ω)= g^2  λ_m/(γ_m/2-i ω)^2-|λ_m |^2 + G^2 λ_d/(γ_d/2-i ω)^2-|λ_d |^2 , 
          λ̃_b(ω)= λ_m + g^2 Λ̅_s(ω)/[κ/2-i(ω-Σ_s(ω))]^2- Λ̅_s^2(ω) , 
          λ̃_d(ω)= λ_d + G^2 Λ_m(ω)/[κ/2 - i(ω-Σ_m(ω))]^2 - Λ_m^2(ω) ,


where the frequency-dependent functions Λ̅_s, Λ_m, Σ_m and Σ_s are given by Eq. (27) in Ref. <cit.>.
As is evident, by turning on the modulations (i.e., λ_m,d≠ 0), the mechanics and the atomic collisions do indeed mediate a parametric-amplifier-like effective squeezing interaction for the cavity mode which is also frequency-dependent (see Eq. (<ref>)), unlike the case of the conventional DPA. Furthermore, the time modulation of the atomic collisions (mechanical mode) induces indirectly the frequency-dependent squeezing through the second term in Eq. (<ref>) (Eq. (<ref>)) to the mechanical (Bogoliubov) mode. As we will show in the next sections, such an induced-frequency-dependent squeezing is responsible for the strong quadrature squeezing in the subsystems. 



§ QUADRATURES AMPLIFICATION SPECTRA 

Using the theory of linear quantum amplifiers <cit.>, in this and the following sections, we investigate the key role of the coherent modulation processes in the proposed hybrid optomechanical system as an amplifier/squeezer. For this purpose, it is more convenient to work in the frequency domain. Writing Eq. (<ref>) in the Fourier space gives

    δû(ω)=χ(ω) √(Γ)û_in(ω) ,
		
where √(Γ) =  Diag( √(κ),√(κ),√(γ_m),√(γ_m),√(γ_d),√(γ_d) ) and  the susceptibility matrix is given by

    χ(ω)  = ( χ_11(ω)   0   0   χ_14(ω)   0   χ_16(ω)
    0   χ_22(ω)   χ_23(ω)   0   χ_25(ω)   0
    0   χ_32(ω)   χ_33(ω)   0   χ_35(ω)   0
    χ_41(ω)   0   0   χ_44(ω)   0   χ_46(ω)
    0   χ_52(ω)   χ_53(ω)   0   χ_55(ω)   0
    χ_61(ω)   0   0   χ_64(ω)   0   χ_66(ω)
    )   ,
 
where the matrix elements χ_ij(ω) have been given in the Appendix. In the absence of the BEC (G_0=0, λ_d=0, γ_d=0, and ω_sw=0) χ(ω) reduces to the susceptibility matrix given in Ref. <cit.>.
Using input-output theory for the field operators, ô_out= ô_in + √(γ_o)ô (o=a,m,d with γ_a=κ), and going over to the quadrature basis, one can thus write

    û_out[ω]= s[ω] û_in[ω],

with the scattering matrix

    s[ω]= Î - √(Γ)χ(ω)  √(Γ),

which can be simplified as 

    s[ω]  = ( s_11   0   0   s_14   0    s_16
    0   s_22    s_23   0   s_25   0
    0   s_32   s_33   0   s_35   0
    s_41   0   0   s_44   0   s_46
    0    s_52    s_53   0    s_55   0
    
    	s_61   0   0    s_64   0    s_66
    ),
 
where s_11(ω)= 1 - κχ_11(ω), s_14(ω)=√(κγ_m)χ_14(ω), 
s_16(ω)= √(κγ_d)χ_16(ω), s_22(ω)=1 - κχ_22(ω), 
s_23(ω)=√(κγ_m)χ_23(ω), s_25(ω)=√(κγ_d)χ_25(ω), s_32(ω)= √(κγ_m)χ_32(ω), s_33(ω)= 1 -γ_m χ_33(ω), s_35(ω)= √(κγ_d)χ_35(ω), s_41(ω)=√(κγ_m)χ_41(ω), s_44(ω)= 1 - γ_m χ_44(ω), s_46(ω)=√(γ_m γ_d)χ_46(ω), s_52(ω)= √(κγ_d)χ_52 (ω)s_53(ω)= √(γ_m γ_d)χ_53(ω), s_55(ω)= 1 - γ_d χ_55(ω), s_61(ω)=√(κγ_d)χ_61(ω), s_64(ω) =√(γ_m γ_d)χ_64(ω), and s_66(ω)= 1 - γ_d χ_66(ω).

Note that the stability of the system can be determined from the poles of the susceptibility matrix χ (ω) <cit.>. In order to keep the system in the stable regime, and for  avoiding mode-splitting the poles should lie in the lower half-plane on the imaginary axis.





 §.§ gain, added noise and amplification 

Using the scattering matrix (<ref>) together with the symmetrized correlator which is defined as S̅_AB^out(ω)=1/2⟨Â_out(ω) B̂_out(-ω) + B̂_out(-ω) Â_out(ω) ⟩, one can calculate the amplified quadratures of the optical , mechanical, and Bogoliubov modes, respectively, as follows
 

    S̅_PP,a^out(ω)/𝒢_a(ω)= (n̅_a^T+1/2) + n̅_add,a^(amp)(ω) , 
          S̅_XX,m^out(ω)/𝒢_m(ω)= (n̅_m^T+1/2) + n̅_add,m^(amp)(ω), 
          S̅_XX,d^out(ω)/𝒢_d(ω)= (n̅_d^T+1/2) + n̅_add,d^(amp)(ω) ,


where 𝒢_a(ω)=| 1-κχ_22(ω) |^2 and  𝒢_m(d)(ω)=| 1-γ_m(d)χ_33(55)(ω) |^2 are the frequency-dependent gains. Also, the amplifier-added noises 
are given by
 

    n̅_add,a^(amp)(ω)=κ𝒢_a^-1(ω) [(n̅_m^T  + 1/2) γ_m |χ_23(ω) |^2   +  (n̅_d^T  + 1/2) γ_d |χ_25(ω) |^2 ] , 
          
          n̅_add,m^(amp)(ω) = γ_m 𝒢_m^-1(ω)[ (n̅_a^T  +1/2)  κ|χ_32(ω) |^2  +  (n̅_d^T  + 1/2) γ_d |χ_35(ω) |^2 ], 
          
          n̅_add,d^(amp)(ω)  = γ_d 𝒢_d^-1(ω) [(n̅_a^T  + 1/2) κ|χ_52(ω) |^2  +  (n̅_m^T  + 1/2) γ_m|χ_53(ω) |^2 ].



On-resonance (ω=0), the optical, mechanical and Bogoliubov gains, √(𝒢_j)≡𝒬_j/𝒫_j with j=a,m,d, can be simplified as
 

    √(𝒢_a)= 𝒞_0-(1-ξ_m)+𝒞_1 1-ξ_m/1-ξ_d/𝒞_0+(1-ξ_m)+𝒞_1 1-ξ_m/1-ξ_d , 
          √(𝒢_m)= 𝒞_0-(1+ξ_m)-𝒞_1 1+ξ_m/1-ξ_d𝒞_0+(1-ξ_m)+𝒞_1 1-ξ_m/1-ξ_d  , 
          √(𝒢_d)=𝒞_1-(1+ξ_d)-𝒞_0 1+ξ_d/1-ξ_m/𝒞_1+(1-ξ_d)+𝒞_0 1-ξ_d/1-ξ_m .


As is evident from Eqs. (<ref>b) and (<ref>c) under the replacements 𝒞_0 ↔𝒞_1 and ξ_m ↔ξ_d, one has 𝒢_m ↔𝒢_d.
Moreover, on-resonance amplifier-added noises are given by 
 
	
    n̅_add,a^(amp)(0)=( 1 - 1/√(𝒢_a))^2 [ (1/2 +n̅_m^T)𝒞_0/(1-ξ_m)^2 +(1/2+n̅_d^T)𝒞_1/(1-ξ_d)^2], 
          
          n̅_add,m^(amp)(0)=4𝒞_0/𝒬_m^2[ (n̅_a^T + 1/2 )+ (n̅_d^T + 1/2) 𝒞_1/(1-ξ_d)^2] , 
          n̅_add,d^(amp)(0)=4𝒞_1/𝒬_d^2[ (n̅_a^T + 1/2 )+ (n̅_m^T + 1/2) 𝒞_0/(1-ξ_m)^2] .


From Eqs.(<ref>a)-(<ref>c) one finds that for each of the three modes (optical, mechanical, and Bogoliubov modes) the corresponding added noise includes the contributions originating from the noises of the two other modes. Moreover, as is evident, under the replacements 𝒞_0 ↔𝒞_1, ξ_m ↔ξ_d, and n̅_m^T ↔n̅_d^T  not only the optical added noise n̅_add,a^(amp)(0) is invariant, but also n̅_add,m^(amp)(0) ↔n̅_add,d^(amp)(0). This indicates the symmetric roles of the mechanical and Bogoliubov modes in on-resonance optical added noise.

Interestingly, although our system is very complicated compared to a standard DPA but we will show that all added noises can be vanished (analogous to what happens in an ideal DPA) by controlling cooperativities and modulation parameters through the atomic collisions frequency and mechanical spring coefficient. On the other hand, it has been shown <cit.> that suppression of the mechanical thermal noise contribution to the optical added noise is achievable in the dissipative optomechanical amplification scheme. 

It should be noted that the added noise suppression in the present system is different from that of a simple optomechanical amplifier realized by driving an optomechanical cavity on its blue sideband regime which corresponds to the two-mode squeezing interaction. In fact, in a DPA the mechanical noise is not cooled, and can represent a significant source of added noise for the amplifier. Surprisingly, we will show that, one can achieve simultaneously large gains corresponding to the amplifier together with suppressed added noises by controlling the modulation parameters and cooperativities. In other words, we show that the scattering matrix can be controlled by the system parameters in order to behave as a noiseless quantum-limited phase-sensitive amplifier (n̅_add,j^(amp)→ 0 and 𝒢_j ≫ 1) <cit.> for each subsystem, in which the noiseless amplified subsystem is entirely decoupled from the other subsystems.






Figure (<ref>) shows the on-resonance logarithm of optical and mechanical gains versus normalized effective mechanical paramp, ξ_m/ξ_m^max, for different values of cooperativity, 𝒞_1, and effective atomic paramp, ξ_d. As is seen from Fig. <ref>(a), in the absence of the BEC (blue dashed line) or in its presence with no atomic parametric drive (red solid thick line), there is no optical gain except for ξ_m≈ξ_m^max. On the other hand, in the presence of the BEC and with turning on atomic paramp one can obtain a considerable gain for some specified values of the mechanical driving. Especially, in the largely different cooperativities regime, i.e., 𝒞_0 ≫𝒞_1 (black thin line), the optical gain shows a high peak at ξ_m≈ 0.9ξ_m^max. 

The mechanical gain 𝒢_m has been shown in Fig. <ref>(b). Just like what observed for the optical gain, in the absence of the BEC (blue dotted line) or in its presence with no atomic parametric drive (red dashed line), there is no mechanical gain except for the limit of ξ_m →ξ_m^max. However, by turning on the atomic paramp (ξ_d ≠ 0), for 𝒞_1 ≥𝒞_0 (black solid thick and green solid thin lines), one can have much larger mechanical gain in comparison to what obtained in Ref. <cit.> (blue dotted line). Therefore, one of the advantages of the present system in comparison to the one studied in Ref. <cit.>, where the cavity contains no BEC, is the capability of obtaining high optical and mechanical gains by controlling the atomic parameters. 

The appearance of the peaks in the optical and mechanical gains can be explained in terms of the set of Eqs. (<ref>)-(<ref>). Here, it should be noted that when the denominators go to zero, i.e., 𝒫_j → 0, one can have large gains (𝒢_j →∞) while if the numerators become zero, i.e., 𝒬_j → 0, the gains go to zero. For example for the optical gain, generally ξ_m →ξ_m^(p,a) and ξ_m →ξ_m^(Q,a) correspond, respectively, to the large and weak gains in which ξ_m^(p,a)=1+ 𝒞_0 (1-ξ_d)/ [𝒞_1+(1-ξ_d) ] and ξ_m^(Q,a)= 1+ 𝒞_0 (1-ξ_d)/ [𝒞_1-(1-ξ_d )] are, respectively, the zeros of the denominator and numerator of the optical gain. As is evident one always has ξ_m^(P,a)≤ξ_m^(Q,a)≤ξ_m^max= 1+𝒞_m, and that is why the optical gain in the presence of the atomic paramp is not always a monotonous increasing function of the mechanical paramp ξ_m. 







Figure (<ref>) shows the on-resonance added-noises and amplified quadratures spectra versus optomechanical cooperativity, 𝒞_0, for different values of 𝒞_1, r_m=ξ_m/ξ_m^max and r_d=ξ_d/ξ_d^max. In each panel the red line indicated by 1 corresponds to the case studied in Ref. <cit.> where there is no BEC in the cavity while the mechanical oscillator is modulated. As is seen, in such a system the optical added noise is very low [Fig. <ref>(a)] but due to the mechanical modulation the the P-quadrature of the optical field is effectively amplified [Fig. <ref>(b)]. On the other hand, the black line indicated by 2 shows the situation where the cavity contains a BEC but both the mechanical and atomic modulations are turned off (off-mods). Although in this case the optical and mechanical added noises are low [Fig. <ref>(a) and <ref>(c)] but there is no amplification in the P-quadrature of the optical field [Fig. <ref>(b)].

More interestingly, for the system equipped with a BEC with both the mechanical and the atomic modulations turned on [the green lines indicated by 4 in Fig. (<ref>)] the highest optical and mechanical amplification can be achieved [Figs. <ref>(b) and <ref>(d)] while the optical and mechanical added noises are still low enough [Figs. <ref>(a) and <ref>(c)]. This situation is achieved in the largely different cooperativities regime (|𝒞_1 - 𝒞_0 |≫ 1) with weak opto-atom cooperativity (𝒞_1 <1). In order to see the destructive effect of large values of opto-atomic cooperativity, one can compare the lines indicated by 3 and 5 which represent similar conditions with different opto-atomic cooperativities 𝒞_1 . As is seen, the increase of opto-atomic cooperativity leads to substantial increase in both the optical and the mechanical added noises while it does not much help to the amplifications.

Finally, it should be mentioned that here we have left out the investigation of the atomic amplification and its added noise since, if 𝒞_1 ↔𝒞_0 and ξ_d ↔ξ_m, the same results can be obtained for the atomic added noise and atomic amplification.




 §.§ controllable gain-bandwidth


In the previous subsection leaving out the frequency dependence of the added-noise and quadratures amplification, we studied the details of their on-resonance behaviors [Fig. (<ref>)]. In this subsection we are going to consider the gain-bandwidth limitation in our system, i.e., the full-width at half maximum (FWHM) of the amplified quadratures (the elements | s_jj[ω] |^2 in Eq. (<ref>) with j=2,3,5) specially the optical amplification bandwidth. 
To determine the amplifier bandwidth, we consider the s-matrix elements as s_jj[ω] ∝ 1/D_g[ω] where D_g[ω] is a cubic function of ω, i.e., D_g[ω]=iω^3 +a ω^2 +b ω +c with the coefficients given by 

    4b ≈κγ_m (𝒞_0+1-ξ_m) +κγ_d (𝒞_1+1-ξ_d) +γ̅^2 ξ_m ξ_d , 
           c ≈κγ̅^2/8[ 𝒞_0(1-ξ_d)+𝒞_1(1-ξ_m) + 1+ξ_m ξ_d -ξ_m - ξ_d   ] , 
           -2a ≈κ + γ_m(1-ξ_m ) +  γ_d(1-ξ_d ) ,

where γ̅=√(γ_m γ_d). In order to find the FWHM one should solve for the equation D_g[δω]=2D_g[0]. Under the conditions ξ_d=0 and ξ_m →ξ_m^max=1+𝒞_0/(1+𝒞_1) one can show straightforwardly that c → 0, therefore the gain-bandwidth can be easily calculated as follows

    Δω_g=--2b/a= κγ_m (𝒞_0+1-ξ_m) +κγ_d (𝒞_1+1) /κ + γ_m(1-ξ_m ) +  γ_d.

Here, one can ignore γ_d because in the normal dissipation regime γ_d≪κ. Moreover, in the limit of large optical gain (√(𝒢_a )≫ 1) which corresponds to ξ_m →ξ_m^max, the gain-bandwidth can be written in terms of the optical gain as follows

    Δω_g √(𝒢_a) =  2𝒞_0 γ_m 1 - √(𝒢_a)/2𝒞_0[ 𝒞_1 (1-ξ_m) -γ_dγ_m(1+𝒞_1) ]/1-γ_m /κ[ 𝒞_0+𝒞_1(1-ξ_m)  ] ,

where 2𝒞_0 γ_m=8g^2/κ. In the weak coupling regime ( g,G ≪κ) the gain-bandwidth takes the form

    Δω_g ≈8g^2/κ[1/√(𝒢_a)  + 𝒞_1 /2 𝒞_0(ξ_m-1) +γ_d/γ_m(1+𝒞_1)/2 𝒞_0].


As is seen from Eq. (<ref>), in the large gain limit and weak atomic modulation, i.e., ξ_d ≈ 0, the gain-bandwidth can be controlled by the ratios g/κ, γ_m/γ_d, the ratio of cooperativities 𝒞_1/𝒞_0 and also by the mechanical parametric drive ξ_m. In Eq. (<ref>), the second and third terms in the brackets which are due to the presence of the Bogoliubov mode of the BEC, have no counterparts in the bare optomechanical system studied in Ref. <cit.>. The presence of these terms can provide more controllability on the gain-bandwidth of the present hybrid system through the BEC parameters. Furthermore, the presence of the BEC
increases the gain-bandwidth which can be considered as another advantage of our optomechanical amplifier. Note that ξ_d can
be controlled by ω_sw and ϵ while ξ_m can be controlled via δ k.




§ QUADRATURES SQUEEZING SPECTRA 

Quadratures squeezing spectra can be investigated by complementary amplified quadratures which are given by
 
	
    S̅_XX,a^out(ω)/1/2= (2n̅_a^T+1) | 1-κχ_11(ω) |^2 
             + (2n̅_m^T+1)  κγ_m |χ_14(ω) |^2  +  (2n̅_d^T+1)  κγ_d |χ_16(ω) |^2 , 
          S̅_PP,m^out(ω)/1/2= (2n̅_m^T+1) | 1-γ_mχ_44(ω)|^2 
             +  (2n̅_a^T+1)  κγ_m |χ_41(ω) |^2  + (2n̅_d^T+1) γ_m γ_d |χ_46(ω) |^2 ,  
          S̅_PP,d^out(ω)/1/2=  (2n̅_d^T+1) | 1-γ_dχ_66(ω) |^2
              + (2n̅_a^T+1)  κγ_d |χ_61(ω) |^2  +  (2n̅_m^T+1) γ_m γ_d |χ_64(ω) |^2.


The on-resonance (ω= 0) output squeezing spectra are given by 
 
	
    S̅_XX,a^out(0)/1/2=(1+2n̅_a^T) ( 1-2/η_a)^2 
            + (1+2n̅_m^T) 4𝒞_0/( 1+ξ_m)^21/η_a^2+  (1+ 2n̅_d^T) 4𝒞_1/(1+ξ_d)^21/η_a^2, 
          S̅_PP,m^out(0)/1/2= 1/s_m^2[ (1+2n̅_m^T) η_m^2 + 4𝒞_0 (1+2n̅_a^T) . 
                          .  + 4𝒞_0 𝒞_1/(1+ξ_d)^2 (1+2n̅_d^T) ]  , 
          S̅_PP,d^out(0)/1/2=  1/s_d^2[ (1+2n̅_d^T) η_d^2  + 4𝒞_1 (1+2n̅_a^T)  . 
                          . + 4𝒞_1 𝒞_1/(1+ξ_m)^2 (1+2n̅_d^T) ],


with 
 
	
    η_a=1+𝒞_0/1+ξ_m +𝒞_1/1+ξ_d, 
          η_m(d)=𝒞_0(1)+ξ_m(d)-1+𝒞_1(0)ξ_m(d)-1/1+ξ_d(m), 
             s_m(d)= 𝒞_0(1)+1+ξ_m(d) +𝒞_1(0)1+ξ_m(d)/1+ξ_d(m) .


Squeezing occurs if the right-hand sides of Eqs. (<ref>a)-(<ref>c) (or Eqs. (<ref>a)-(<ref>c)) become smaller than unity. In addition to the degree of squeezing which is characterized by the quadrature spectrum, the purity or impurity of squeezing is another important feature. The impurity of output squeezing spectrum can be characterized by the effective thermal occupancy <cit.> which is given by (j=a,m,d)

    (n̅_ eff, j^ out [ω]  + 1/2)^2  =S̅_XX,j^out[ω] S̅_PP,j^out[ω]- S̅_XP,j^out[ω] S̅_PX,j^out[ω] ,

where the cross correlations are given by 
 
	
    S̅_XP,a^out[ω]=S̅_PX,a^out[ω]= -1/2 Im[ (1-κχ_11(ω)) (1-κχ_22^∗(ω))  . 
                    . +κγ_mχ_23(ω) χ_14^∗(ω)+ κγ_dχ_25(ω) χ_16^∗(ω) ], 
          S̅_XP,m^out[ω]=S̅_PX,m^out[ω]= -1/2 Im[ (1-γ_m χ_33(ω)) (1-γ_m χ_44^∗(ω))  . 
                    .+γ_mκχ_41(ω)χ_32^∗(ω)+ γ_m γ_dχ_35(ω) χ_46^∗(ω) ], 
          S̅_XP,d^out[ω]=S̅_PX,d^out[ω]= -1/2 Im[ (1-γ_d χ_55(ω)) (1-γ_d χ_66^∗(ω))  . 
                    .+γ_dκχ_61(ω)χ_52^∗(ω)+ γ_d γ_mχ_53(ω) χ_64^∗(ω) ].


It can be easily shown that in on-resonance (ω=0) the cross correlations become zero, i.e., S̅_xp,j^out[0]=0, which is because of the RWA. It should be noted that the impurity n̅_ eff,j^ out [ω] for any pure state of the output mode will be zero, and for a thermal state will be equal to the actual thermal occupancy of that mode <cit.>. In the following, we will show how our system has the potential to achieve simultaneously perfect squeezing with high purity.




 §.§ quadrature squeezing of the output cavity field



Let us show how we can overcome the decoherence effects in order to achieve perfect squeezing for the cavity output field. As can be seen from Eq. (<ref>), in the limit η_a→ 2  the contribution of the optical noise in the on-resonance output optical squeezing spectrum vanishes and thus  

    S̅_XX,a^out(0)/1/2→ (1+2n̅_m^T) 𝒞_0/( 1+ξ_m)^2+  (1+ 2n̅_d^T) 𝒞_1/(1+ξ_d)^2.

It is easy to verify that in the absence of atomic modulation (ξ_d=0) together with strong mechanical driving such that ξ_m →ξ_m,sq^a,max=[(1+𝒞_1)ξ_m^max-2]/(1-𝒞_1), the limit η_a→ 2  is achievable. Under such conditions, we get

    S̅_xx,a^out(0)/1/2→ (1-𝒞_1)^2 (1+2n̅_m^T) /𝒞_0+ 𝒞_1 (1+ 2n̅_d^T)  .


As is seen from Eq. (<ref>), in the regime of 𝒞_1 ≪ 1 and 𝒞_0 ≫ 1 (i.e., the regime of largely different cooperativities), and when n̅_d^T is negligibly small <cit.>, the squeezing of the on-resonance cavity output quadrature X̂_out(0) below zero-point is set by the ratio n̅_m^T/𝒞_0 . This means that significant degree of squeezing can be achieved by ground state cooling of the MO. However, in the limit n̅_m^T≪𝒞_0, which corresponds to the regime of strong optomechanical coupling between the MO and the intracavity field, i.e., when g^2≫κγ_m/4k_B T_m/ħω_m, one can achieve strong output field squeezing which is robust against thermal noises. It should be mentioned that interchanging the mechanical and the Bogoliubov modes (ξ_m ↔ξ_d, 𝒞_0 ↔𝒞_1, n̅_m^T↔n̅_d^T) leaves the results unchanged. 




In Figs. (<ref>a) and (<ref>b) we have plotted, respectively, the degree of squeezing of the output cavity field on-resonance (-10log_102S̅_XX,a^out(0) , in dB unit) and its purity (n̅^out_ eff,a(0)) versus the optomechanical cooperativity 𝒞_0 in the absence of atomic modulation (ξ_d=0) for different values of ξ_m and 𝒞_1 (𝒞_1≪ 1). As is evident, in the largely different cooperativities regime (|𝒞_0-𝒞_1 |≫ 1), one can beat the 3dB limit (50% squeezing) through the increase of mechanical modulation amplitude (ξ_m) without losing the purity of squeezing. Also, by increasing the difference between cooperativities both the degree of squeezing and its purity increase. However, in spite of the cavity squeezing enhancement, its purity decreases a little bit with increasing the mechanical modulation.

In fact, there is a competition between the maximum achievable squeezing and its purity which can be controlled by the difference between the cooperativities and the effective amplitude of the mechanical modulation ξ_m. 
This trade-off can be interpreted as follows: while heating the MO and BEC via the coherent modulation they are cooled by the intracavity light field through the radiation pressure in the red-detuned regime. These phononic fluctuations are transferred into the P-quadrature of the light field via the optomechanical coupling and therefore that trade-off originating from cooling and heating of the BEC and MO is induced into the cavity output-field squeezing and its purity. 

It should be noted that the maximum achievable squeezing can be even more than 16 dB which never occurs in the standard OMSs. On the other hand, as can be seen from Fig. (<ref>), the generated squeezing of the output cavity field and its purity are robust to the mechanical thermal noises. Moreover, the high degree of optical squeezing in our system can be attributed to the induced frequency-dependent squeezing [see Eq. (<ref>)] because of the cavity field interaction with BEC and MO in the presence of the external modulations of the MO and atomic collisions.



 §.§ quadrature squeezing of the mechanical phonons

Here, we show that in the present hybrid optomechanical system one can achieve perfect quadrature squeezing of the phononic mode of the MO or of the BEC with mediocre purity via controlling the atomic or mechanical modulation as well as cooperativities such that both conditions η_m→ 0 and s_m≫ 1 are satisfied.

To this end, we consider two special cases: (i) no atomic collisions modulation (ξ_d=0) and (ii) no mechanical frequency modulation (ξ_m=0). In the first case, if  ξ_m → 2-ξ_m^max then according to Eqs. (<ref>) and (<ref>), η_m→ 0 and s_m → 2(1+𝒞_1). Therefore in the limit of large opto-atomic cooperativity (𝒞_1≫ 1) one has s_m≫ 1 , and consequently the squeezing of the on-resonance P-quadrature of the MO below zero-point [Eq. (<ref>)] reads as

    S̅_PP,m^out(0)/1/2→𝒞_0/𝒞_1^2 (1+2n̅_a^T)+ 𝒞_0/𝒞_1 (1+2n̅_d^T)  ,

which shows that for 𝒞_0≪𝒞_1 together with n̅_a^T=n̅_d^T≈ 0, one can achieve high degree of mechanical P-quadrature squeezing much above the standard quantum limit of 3dB limit. On the other hand, in the second case, i.e., ξ_m=0, if ξ_d → [(1+𝒞_0)ξ_d^max-2𝒞_0]/(𝒞_0-1)=(𝒞_1+1-𝒞_0)/(𝒞_0-1) (with 𝒞_0 >1 and 𝒞_1>𝒞_0-1) then η_m→ 0 and s_m→ 2𝒞_0. In this case, we get

    S̅_PP,m^out(0)/1/2→(1+2n̅_a^T)/𝒞_0 + 𝒞_0-1/𝒞_1 (1+2n̅_d^T)  ,

which clearly shows that the squeezing of the P-quadrature of the MO can be significantly enhanced by requiring 𝒞_1≫𝒞_0≫ 1.
Note that in both cases the perfect mechanical squeezing can be achieved in the largely different cooperativities regime (|𝒞_0-𝒞_1 |≫ 1). It should be mentioned that because of the symmetry between the two phononic modes, i.e., mechanical and Bogoliubov modes, the same results can be obtained for the Bogoliubov-type phonon squeezing with the replacements 𝒞_0 ↔𝒞_1 and ξ_m  ↔ξ_d.



Figure (<ref>) shows the on-resonsnce mechanical P-quadrature squeezing and its impurity versus opto-atomic cooperativity 𝒞_1 in the red-detuned and weak coupling regimes. As is seen, there is no mechanical squeezing in the absence of modulations [brown line indicated by off-modulations] while the squeezing is generated as soon as the modulations are turned on and increases rapidly by increasing the opto-atomic cooperativity 𝒞_1 so that it surpasses the 3 dB limit in the largely different cooperativities regime. On the other hand, case (i), i.e., off-atomic modulation with large mechanical parametric drive leads to much more degree of squeezing than case (ii), i.e., off-mechanical modulation with large atomic parametric drive (compare lines 1 and 2 with lines 3 and 4 in Fig. <ref>(a).

Moreover, by increasing the difference between the cooprativities the impurity increases [Fig. <ref>(b)]. In the regime of large parametric drive, since n̅_ eff,m^ out [0] ∼n̅_m^T, one can conclude that the state of the MO is a thermal squeezed state which is in agreement with explanations given in Refs. <cit.> based on the collective mode in which two phononic subsystems evolve to a collective Bogoliubov mode as a thermal-squeezed and entangled state. In other words, the generated controllable strong mechanical squeezing reflects to overcome of the noise amplification in the X-quadratures of the MO and BEC to the noise suppression in the P-quadratures of the MO and BEC.

It should be pointed out that in spite of the standard ponderomotive squeezing which is maximized in the unresolved sideband regime or bad-cavity limit (κ≫ω_m), our system operates in the weak couplings, resolved-sideband (good-cavity limit) regimes  (g,G < κ≪ω_m ≈ω_d) where very high degree of squeezing (up to 75 dB) is achievable in the red detuned and largely different cooperativities regimes. Nevertheless, the standard ponderomotive squeezing mechanism can lead to higher purities than those obtained in our proposed scheme. Moreover, in the present scheme the squeezed quadrature angle can be controlled through the phases of mechanical and atomic modulations (ϕ_m and ϕ_d), while in the ponderomotive approach the squeezing angle is dependent on frequency. Furthermore, the squeezing bandwidth in our system can be controlled by cooperativities, g, G/κ, and γ_m/γ_d. In comparison to Ref. <cit.>, the additional coupling to the atomic BEC in the presence of atomic collisions frequency modulation lead to increase of the gain-bandwidth [see Eq. (<ref>)] which can be considered as an advantage of our proposed system.

Finally, we should emphasize that the present scheme is experimentally feasible because it is compatible with the experimental data applied in Refs.<cit.> as has been shown in our previous theoretical papers <cit.>. For this purpose, one can consider N=10^5 Rb atoms inside an optical cavity of length L=178 μm with bare frequency ω_c=2.41494× 10^15Hz corresponding to a wavelength of λ=780nm and damping rate κ=2π× 1.3MHz. The atomic D_2 transition corresponding to the atomic transition frequency ω_a=2.41419× 10^15Hz couples to the mentioned mode of the cavity. The atom-field coupling strength g_0=2π× 14.1MHz and the recoil frequency of the atoms is ω_R=23.7KHz. The movable end mirror can be assumed to have a mass of m=10^-9g and damping rate of γ_m=2π× 100Hz which oscillates with frequency ω_m=10^5Hz. Besides, the coherent modulation of the mechanical spring coefficient of the MO and also the time modulation of the s-wave scattering frequency of atom-atom interaction of the BEC can be realized experimentally as have been reported, respectively, in <cit.> and <cit.>.







§ CONCLUDING REMARKS  


In summary, we have theoretically proposed and investigated a feasible experimental scheme to generate controllable strong quadrature squeezing and noiseless phase-sensitive amplification in a hybrid optomechanical cavity containing an interacting cigar-shaped atomic BEC in the red-detuned and weak coupling regimes through coherent modulations of the atomic collisions frequency of the BEC and of the mechanical spring coefficient of the MO.

We have shown that by controlling the system parameters such as the amplitudes of modulations and cooperativities, the X-quadrature of the cavity mode and the P-quadrature of the MO/BEC can be strongly squeezed (up to 16 dB and 75 dB for the output cavity field and the MO, respectively) in the largely different cooperativities regime with large amplitude of modulations which never occurs in the red-detuned regime of standard cavity optomechanics. Moreover, the output cavity field is squeezed with considerable amount of purity while the squeezing of the MO/BEC is a thermal-like squeezed state. Furthermore, we have shown that the generated squeezings in the subsystems are robust against the thermal noises. 

On the other hand, by analyzing the complementary quadratures we have shown that the P-quadrature of the cavity mode and the X-quadrature of the MO/BEC can be strongly amplified with simultaneous suppression of added-noises, i.e., noiseless phase-sensitive quantum amplifications. What we should mention as another advantageous feature of the present system is the controllability of the gain-bandwidth and the possibility of increasing it through the modulation of the s-wave scattering frequency of the BEC which is experimentally controllable by the transverse trapping frequency.


















§  DEFINITION OF THE ELEMENTS OF THE SUSCEPTIBILITY MATRIX OF EQ. (<REF>) 

In Eq. (<ref>), the matrix elements χ_ij(ω) are given by 

	
    χ_11(ω)= [ χ_0^-1(ω)+g^2 χ_+m+G^2 χ_+d(ω)    ]^-1;  
    	              χ_14(ω)=-g [ χ_0^-1(ω) χ_+m^-1(ω)  +g^2  +G^2 χ_+d(ω)  χ_+m^-1(ω)   ]^-1 ,
          χ_16(ω)=G [ χ_0^-1(ω) χ_+d^-1(ω)  +G^2  +g^2 χ_+m(ω)  χ_+d^-1(ω)   ]^-1;    
    	      χ_22(ω)=  [ χ_0^-1(ω)+g^2 χ_-m+G^2 χ_-d(ω)    ]^-1 ,
          χ_23(ω)=g [ χ_0^-1(ω) χ_-m^-1(ω)  +g^2  +G^2 χ_-d(ω)  χ_-m^-1(ω)   ]^-1;   
    	      χ_25(ω)=-G [ χ_0^-1(ω) χ_-d^-1(ω)  +G^2  +g^2 χ_-m(ω)  χ_-d^-1(ω)   ]^-1, 
          χ_32(ω)=-g [  χ_0^-1(ω) χ_-m^-1(ω) + g^2 + G^2 χ_-d(ω) χ_-m^-1(ω)   ]^-1; 
    	    χ_33(ω)= 1+G^2 χ_0(ω) χ_-d(ω) /χ_-m^-1(ω) + g^2 χ_0(ω) + G^2 χ_0(ω) χ_-d(ω) χ_-m^-1(ω) , 
          χ_35(ω)=gG/χ_0^-1(ω) χ_-m^-1(ω)  χ_-d^-1(ω) + G^2 χ_-m^-1(ω) + g^2 χ_-d^-1(ω)   ;
    	    χ_41(ω)=g [  χ_0^-1(ω) χ_+m^-1(ω) + g^2 + G^2 χ_+d(ω) χ_+m^-1(ω)   ]^-1  , 
          χ_44(ω)= 1+G^2 χ_0(ω) χ_+d(ω) /χ_+m^-1(ω) + g^2 χ_0(ω) + G^2 χ_0(ω) χ_+d(ω) χ_+m^-1(ω) ;
    	    χ_46(ω)=gG/χ_0^-1(ω) χ_+m^-1(ω)  χ_+d^-1(ω) + G^2 χ_+m^-1(ω) + g^2 χ_+d^-1(ω)   , 
          χ_52(ω)=G [  χ_0^-1(ω) χ_-d^-1(ω) + G^2 + g^2 χ_-m(ω) χ_-d^-1(ω)   ]^-1   ;
    	    χ_53(ω)=gG/χ_0^-1(ω) χ_-m^-1(ω)  χ_-d^-1(ω) + G^2 χ_-m^-1(ω) + g^2 χ_-d^-1(ω)   , 
          χ_55(ω)= 1+g^2 χ_0(ω) χ_-m(ω) /χ_-d^-1(ω) + G^2 χ_0(ω) + g^2 χ_0(ω) χ_-m(ω) χ_-d^-1(ω) ;
    	    χ_61(ω)=-G [  χ_0^-1(ω) χ_+d^-1(ω) + G^2 + g^2 χ_+m(ω) χ_+d^-1(ω)   ]^-1  , 
          χ_64(ω)=gG/χ_0^-1(ω) χ_+m^-1(ω)  χ_+d^-1(ω) + G^2 χ_+m^-1(ω) + g^2 χ_+d^-1(ω)  ;
    	    χ_66(ω)= 1+g^2 χ_0(ω) χ_+m(ω) /χ_+d^-1(ω) + G^2 χ_0(ω) + g^2 χ_0(ω) χ_+m(ω) χ_+d^-1(ω) ,


with χ_0^-1(ω)= κ/2 - iω, χ_± m^-1(ω)= γ_m/2 ±λ_m - i ω and χ_± d^-1(ω)= γ_d/2 ±λ_d - i ω. It is clear that χ_0(-ω)= χ_0(ω)^∗ and χ_± m,± d(-ω)= χ_± m,± d(ω)^∗, so χ_ij(-ω)= χ_ij(ω)^∗.












 






















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Aspelmeyer M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, https://doi.org/10.1103/RevModPhys.86.1391Rev. Mod. Phys. 86, 1391 (2014).


Meystreoptomechanics P. Meystre, https://doi.org/10.1002/andp.201200226 Ann. Phys. 525, 215 (2013).


chenoptomechanics Y. Chen, 
https://doi.org/10.1088/0953-4075/46/10/104001J. Phys. B: At. Mol. Opt. Phys. 46 104001 (2013).


milburnoptomechanics G. J. Milburn and M. J. Woolley, 
https://doi.org/10.2478/v10155-011-0005-7Act. Phys. Slov. 61, 483 (2011).


xsensing1 T. J. Kippenberg and K. J. Vahala, 
https://doi.org/10.1364/OE.15.017172Opt. Exp. 15, 17172 (2007).


CQNCPRL M. Tsang and C.M. Caves, 
https://doi.org/10.1103/PhysRevLett.105.123601Phys. Rev. Lett. 105 123601 (2010).




CQNCmeystre F. Bariani, H. Seok, S. Singh, M. Vengalattore, and P. Meystre, 
https://doi.org/10.1103/PhysRevA.92.043817Phys. Rev. A 92, 043817 (2015).





aliNJP A. Motazedifard, F. Bemani, M. H. Naderi, R. Roknizadeh, and D. Vitali, 
http://stacks.iop.org/1367-2630/18/i=7/a=073040New J. Phys. 18, 073040 (2016) .


complexCQNC L. F. Buchmann, S. Schreppler, J. Kohler, N. Spethmann, and D. M. Stamper-Kurn, 
https://doi.org/10.1103/PhysRevLett.117.030801Phys. Rev. Lett. 117 030801 (2016) .


ground state cooling A. D. O' Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, 
https://doi.org/10.1038/nature08967Nature (London) 464, 697 (2010).


Sideband cooling J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, 
https://doi.org/10.1038/nature10261Nature (London) 475, 359 (2011).


Laser cooling J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, 
https://doi.org/10.1038/nature10461Nature (London) 478, 89 (2011).





Chan J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Grbölacher, M. Aspelmeyer, and O. Painter, https://doi.org/10.1038/nature10461Nature (London) 478, 89 (2011).


Palomaki2 T. A. Palomaki, J. D. Teufel, R.W. Simmonds, and K.W. Lehnert, https://doi.org/10.1126/science.1244563Science 342, 710 (2013).


Paternostro M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, https://doi.org/10.1103/PhysRevLett.99.250401Phys. Rev. Lett. 99, 250401 (2007).


genes-entangelment C. Genes, A. Mari, D. Vitali, and P. Tombesi, https://doi.org/10.1016/S1049-250X(09)57002-4Adv. At. Mol. Opt. Phys. 57, 33 (2009).


Mari1 A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, 
https://doi.org/10.1103/PhysRevLett.111.103605Phys. Rev. Lett. 111, 103605 (2013).


MianZhang M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, 
https://doi.org/10.1103/PhysRevLett.109.233906Phys. Rev. Lett. 109, 233906 (2012).


Bagheri M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, 
https://doi.org/10.1103/PhysRevLett.111.213902Phys. Rev. Lett. 111, 213902 (2013).




grebogi L. Ying, Y. Ch. Lai, and C. Grebogi, 
https://doi.org/10.1103/PhysRevA.90.053810Phys. Rev. A 90, 053810 (2014).


foroud F. Bemani, A. Motazedifard, R. Roknizadeh, M. H. Naderi, and D. Vitali, 
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.96.023805Phys. Rev. A 96, 023805 (2017).


BorkjeA. Nunnenkamp, K. Borkje, and S. M. Girvin, 
https://doi.org/10.1103/PhysRevLett.107.063602Phys. Rev. Lett. 107, 063602 (2011).


Hammerer K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, 
https://doi.org/10.1103/PhysRevA.79.063819Phys. Rev A. 79, 063819 (2009).


foroudnonlinearcoherent F. Bemani, R. Roknizadeh, and M. H. Naderi, 
https://doi.org/10.1364/JOSAB.32.001360J. Opt. Soc. Am. B 32, 2519 (2015).

unconditionalstate M. Brunelli, O. Houhou, D. W. Moore, A. Nunnenkamp, M. Paternostro, and A. Ferraro, 
https://arxiv.org/abs/1804.00014arXiv:1804.00014v1 [quant-ph] (2018).

adde-coherentstate J. Li, S. Groblacher, Sh. Zhu, and G. S. Agarwal, 
https://arxiv.org/abs/1803.06767arXiv:1803.06767v2 [quant-ph] (2018).



DCEQWell S. Mahajan, N. Aggarwal, T. Kumar, A. B. Bhattacherjee, and M. Mohan, 
https://doi.org/10.1139/cjp-2014-0255Can. J. Phys. 93,716 (2015).


aliDCE1 A. Motazedifard, M. H. Naderi, and R. Roknizadeh, 
https://doi.org/10.1364/JOSAB.32.001555J. Opt. Soc. Am. B 32, 1555 (2015).


aliDCE2 A. Motazedifard, M. H. Naderi, and R. Roknizadeh, 
https://doi.org/10.1364/JOSAB.34.000642J. Opt. Soc. Am. B 34, 642 (2017).


aliDCE3 A. Motazedifard, A. Dalafi, M. H. Naderi, and R. Roknizadeh, https://doi.org/10.1016/j.aop.2018.07.013Ann. Phys. 396, 202 (2018).




foroudCurvedspacetime F. Bemani, R. Roknizadeh, and M. H. Naderi, 
https://doi.org/10.1364/JOSAB.34.002519J. Opt. Soc. Am. B 34, 2519 (2017).




Refrevise2 F. Massel, T. T. Heikkilä J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen and M. A. Sillanpää, 
https://doi.org/10.1038/nature10628Nature 480, 351 (2011).

Refrevise3 C. F. Ockeloen-Korppi, T. T. Heikkilä, M. A. Sillanpää and F. Massel, 
https://orcid.org/0000-0002-6132-109XQuantum Sci. Technol. 2, 035002 (2017).

Refrevise4 C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, 
Phys. Rev. X 6, 041024 (2016).

Refrevise5 C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, 
https://doi.org/10.1103/PhysRevLett.118.103601Phys. Rev. Lett. 118, 103601 (2017).

Refrevise6 E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, K. C. Schwab, 
https://doi.org/10.1126/science.aac5138Science 349, 952 (2015).

Refrevise7 J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, 
https://doi.org/10.1103/PhysRevLett.115.243601Phys. Rev. Lett. 115, 243601 (2015).


Kippenbergsqueezing T. J. Kippenberg and K. J. Vahala, https://doi.org/10.1126/science.1156032Science 321, 1172 (2008).


ponderomotive V. B. Braginsky and A. B. Manukin, 
Sov. Phys. JETP 25 653 (1967). 


vitaliponderomotive D. Vitali and P. Tombesi, 
https://doi.org/10.1016/j.crhy.2010.12.007C. R. Phys. 12, 848 (2011). 


Clerkdissipativeoptomechanics A. Kronwald, F. Marquardt, and A. A. Clerk, 
https://doi.org/10.1088/1367-2630/16/6/063058New J. Phys. 16, 063058 (2014). 




gentlymodulating A. Mari and J. Eisert, 
https://doi.org/10.1103/PhysRevLett.103.213603Phys. Rev. Lett. 103, 213603 (2009).


giovanettimodulation A. Farace and V. Giovannetti, 
https://doi.org/10.1103/PhysRevA.86.013820Phys. Rev. A 86, 013820 (2012).


lawmodulation J.-Q. Liao and C. K. Law, 
https://doi.org/10.1103/PhysRevA.83.033820Phys. Rev. A 83, 033820 (2011).


schmidtmodulation M. Schmidt, M. Ludwig, and F. Marquardt, 
https://doi.org/10.1088/1367-2630/14/12/125005New J. Phys. 14, 125005 (2012).



milburnmodulation M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, 
https://doi.org/10.1103/PhysRevA.78.062303Phys. Rev. A 78, 062303 (2008).



pontinmodulation A. Pontin, M. Bonaldi, A. Borrielli, L. Marconi, F. Marino, G. Pandraud, G. A. Prodi,
P. M. Sarro, E. Serra, and F. Marin, 
https://doi.org/10.1103/PhysRevLett.116.103601Phys. Rev. Lett. 116, 103601 (2016).


optomechanicswithtwophonondriving B. A. Levitan, A. Metelmann, and A. A. Clerk, 
https://doi:10.1088/1367-2630/18/9/093014New J. Phys. 18, 093014 (2016).





clerkfeedback A. A. Clerk, F. Marquardt, and K. Jacobs, 
https://doi.org/10.1088/1367-2630/10/9/095010New J. Phys. 10, 095010 (2008).


Harris A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen, 
https://doi.org/10.1103/PhysRevLett.107.213603Phys. Rev. Lett. 107, 213603 (2011). 


Bowen A. Szorkovszky, G. A. Brawley, A. C. Doherty, and W. P. Bowen, 
https://doi.org/10.1103/PhysRevLett.110.184301Phys. Rev. Lett. 110, 184301 (2013).


Nunnenkamp A. Nunnenkamp, K. Borkje, J. G. E. Harris, and S. M. Girvin, 
https://doi.org/10.1103/PhysRevA.82.021806Phys. Rev. A 82, 021806 (2010).


kumar U. S. Sainadh and M. A. Kumar, 
https://doi.org/10.1103/PhysRevA.92.033824Phys. Rev. A 92, 033824 (2015).


asjad M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. Di Giuseppe, and D. Vitali, 
https://doi.org/10.1103/PhysRevA.89.023849Phys. Rev. A 89, 023849 (2014).


dalafiQOC A. Dalafi, M. H. Naderi, and A. Motazedifard, 
https://doi.org/10.1103/PhysRevA.97.043619Phys. Rev. A 97, 043619 (2018).


twofoldsqueezing Ch. Hu, Zh. Yang, H. Wu, Y. Li, and Sh. Zheng, 
https://arxiv.org/abs/1803.05147arXiv:1803.05147v1 [quant-ph] (2018).



Refrevise1 T. P. Purdy, D. W. C. Brooks, T. Botter, N. Brahms, Z.-Y. Ma, and D. M. Stamper-Kurn, 
https://doi.org/10.1103/PhysRevLett.105.133602Phys. Rev. Lett. 105, 133602 (2010).

MeystreBEC R. Kanamoto and P. Meystre, 
https://doi.org/10.1088/0031-8949/82/03/038111Phys. Scr. 82, 038111 (2010).


BrennBECexp F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, 
https://doi.org/10.1126/science.1163218Science 322, 235 (2008).

RitterBECexp S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, 
https://doi.org/10.1007/s00340-009-3436-9Appl. Phys. B 95, 213 (2009).


dalafi1 A. Dalafi, M. H. Naderi, M. Soltanolkotabi, and Sh. Barzanjeh, 
https://doi.org/10.1103/PhysRevA.87.013417Phys. Rev. A 87 013417 (2013) .

dalafi3 A. Dalafi, M. H. Naderi, and M. Soltanolkotabi, 
https://doi.org/10.1080/09500340.2014.935818J. Mod. Opt. 61, 1387 (2014).

dalafi4 A. Dalafi, M. H. Naderi, and M. Soltanolkotabi, 
https://doi.org/10.1088/0953-4075/48/11/115507J. Phys. B: At. Mol. Opt. Phys. 48, 115507 (2015).

BhattacherjeeNMS A. Bhattacherjee, 
https://doi.org/10.1103/PhysRevA.80.043607Phys. Rev. A 80, 043607 (2009).

Bhattacherjeenoisereduction A. Bhattacherjee, 
https://doi.org/10.1088/0953-4075/43/20/205301J. Phys. B: At. Mol. Opt. Phys. 43, 205301 (2010).

dalafi5 A. Dalafi and M. H. Naderi, 
https://doi.org/10.1088/0953-4075/49/14/145501J. Phys. B: At. Mol. Opt. Phys. 49, 145501 (2016).

dalafi2 A. Dalafi, M. H. Naderi, M. Soltanolkotabi, Sh. Barzanjeh, 
https://doi.org/10.1088/0953-4075/46/23/235502J. Phys. B: At. Mol. Opt. Phys. 46, 235502 (2013).

dalafi6 A. Dalafi and M. H. Naderi, 
https://doi.org/10.1103/PhysRevA.94.063636Phys. Rev. A 94, 063636 (2016).

dalafi7 A. Dalafi and M. H. Naderi, 
https://doi.org/10.1103/PhysRevA.95.043601Phys. Rev. A 95, 043601 (2017).

dalafi8 A. Dalafi and M. H. Naderi, 
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.96.033631Phys. Rev. A 96, 033631 (2017).






Ritsch C. Maschler and H. Ritsch, 
http://dx.doi.org/10.1016/j.optcom.2004.10.038Opt. Commun.  243, 145 (2004) .


Domokos2 G. Szirmai, D. Nagy, and P. Domokos, 
https://doi.org/10.1103/PhysRevA.81.043639 Phys. Rev. A 81, 043639 (2010).

Nagy D. Nagy, P. Domokos, A. Vukics, and H. Ritsch, 
https://doi.org/10.1140/epjd/e2009-00265-7Eur. Phys. J. D 55, 659 (2009).

JaskulaBECmodulationexp J. C. Jaskula, G. B. Partridge, M. Bonneau, R. Lopes, J. Ruaudel, D. Boiron, and C. I. Westbrook, 
https://doi.org/10.1103/PhysRevLett.109.220401Phys. Rev. Lett. 109, 220401 (2012).



Refrevise8 C. M. Caves, 
https://doi.org/10.1103/PhysRevD.26.1817Phys. Rev. D 26, 1817 (1982).

IntQN A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, 
https://doi.org/10.1103/RevModPhys.82.1155Rev. Mod. Phys. 82, 1155 (2010).



Quantum-Limited-Amplification A. Metelmann and A. A. Clerk, 
https://doi.org/10.1103/PhysRevLett.112.133904Phys. Rev. Lett. 112, 133904 (2014).



JLiNJP J. Li, I. M. Haghighi, N. Malossi, S. Zippilli, and D. Vitali, 
 https://doi:10.1088/1367-2630/17/10/103037New. J. Phys. 17, 103037 (2015).

JLiPRA J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, 
 https://doi.org/10.1103/PhysRevA.95.043819Phys. Rev. A 95, 043819 (2017).

gard C.W. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin, 2000).

giov V. Giovannetti and D. Vitali, https://doi.org/10.1103/PhysRevA.63.023812Phys. Rev. A 63, 023812 (2001).










