- Email: [email protected]
Two mode mechanical non-Gaussian squeezed number state in a two-membrane optomechanical system
Optics Communications 370 (2016) 55–61
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Two mode mechanical non-Gaussian squeezed number state in a two-membrane optomechanical system S. Shakeri a,n, Z. Mahmoudi a, M.H. Zandi a, A.R. Bahrampour b a b
Faculty of Physics, Shahid Bahonar University of Kerman, Kerman, Iran Sharif University of Technology, Department of Physics, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 27 November 2015 Received in revised form 26 February 2016 Accepted 27 February 2016
We consider an optomechanical system with two membranes when a bichromatic laser field with redsideband and blue-sideband frequencies is applied in the single photon strong coupling regime. It is shown that using the mode selecting method and under the Lamb–Dicke approximation, motion of membranes can evolve to single or two mode squeezed number states. By considering the environmental effect, a Wigner function is plotted for understanding the conditions that lead to the generation of nonGaussian states. The results show that, in this system, initial states of membranes are important to generation of non-Gaussian mechanical squeezed number states. & 2016 Elsevier B.V. All rights reserved.
Keywords: Two-membrane optomechanical system Single photon strong coupling regime Non-Gaussian state Motional squeezed state Wigner function
1. Introduction The implementation of strong optical nonlinearities on a single photon level is one of the main goals in quantum optics and is specifically interesting in cavity QED [1–3]. The other remarkable system which has been widely investigated in this context is the optomechanical system (OMS) where the radiation pressure coupling between light and mechanical motion is studied [4–6]. Nonclassical states are the characteristic elements of quantum theory. There has been an increasing interest in establishing the conditions under which nonclassical states among macroscopic objects can arise. The motional nonclassical states such as squeezed states, Schrödinger cats and Fock states [7–16] have been already investigated in various optomechanical systems with a strong or a weak coupling regime. Mechanical squeezed state is very important and several proposals have been studied to generate mechanical squeezing in an optomechanical oscillator, including the injection of nonclassical light, conditional quantum measurement, parametric amplification and quantum-reservoir engineering [17–20]. In many proposed procedures, created mechanical squeezed states are classified in Gaussian squeezed states. Non-Gaussian states allow to apply the quantum algorithm or to process on a classical computer [21]. Non-Gaussian states also find important applications in high-precision metrology, novel test for n
Corresponding author. E-mail address: [email protected] (S. Shakeri).
http://dx.doi.org/10.1016/j.optcom.2016.02.063 0030-4018/& 2016 Elsevier B.V. All rights reserved.
Bell-like inequalities and fundamental investigations [22,23]. Therefore, finding an effective way to prepare non-Gaussian states and interactions is crucial for the evolution of quantum based applications and also for the better understanding of physics [24]. Non-Gaussian squeezed states have been considered in quantum optics and OMS seriously. In Ref. [25], a protocol has been proposed for coherently transferring non-Gaussian quantum states from optical field to a mechanical oscillator. Squeezing of a strongly interacting opto-electromechanical system using a parametric device has been indicated in Ref. [26]. By employing realtime feedback [27,28] on the phase of the pump at twice the resonance frequencies the thermo-mechanical noise is squeezed beyond the 3 dB in stability limit. This method can also be used to generate highly nonlinear states. Also, in Ref. [7], the authors considered the generation of non-classical states of the mirror motion in the single photon strong coupling regime. They found that the motion of mirror can evolve into a squeezed coherent dark state and beyond the Lamb–Dicke limit, the state can become a squeezed non-Gaussian state. Two mode non-Gaussian squeezed number state has not been discussed in optomechanical systems up to now. In this paper, preparation of non-Gaussian single and two mode squeezed states is considered theoretically. Our system is an optomechanical system with two membranes and bichromatic laser is applied within the single photon strong coupling regime. Multiple membrane optomechanical systems are studied in the single photon strong coupling regime. In Ref. [12] the implied method is
56
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
converted to the method used in trapped ions. We have introduced and used this method to the generation of motional Schrödinger cat states in our previous paper [29]. The Hamiltonian of the squeezed operator is achieved clearly and the main idea is related to non-Gaussian single and two mode motional squeezed number state between two membranes. The two mode squeezed states are very important in quantum optics and can be employed to explain EPR paradox. More recently this kind of correlation has also played a leading role in the quantum teleportation of continuous variables. It seems that strong coupling regime leads to the strong nonlinearity in the system, and therefore the generation of non-Gaussian states is an undeniable fact [30–32]. Despite the strong nonlinearity condition, it has been shown that the initial state of membranes has an important role in creating the two mode non-Gaussian states. Also, creating the condition of single photon strong coupling is hard and complicated, in most experiments up to now. According to [33], multiple membrane hybrid electro-optomechanical systems are proposed to realise strong Kerr nonlinearities even in the weak-coupling regime and optimal conditions for achieving the non-Gaussian squeezed state are studied. Our paper is organized as follows. In Section 2, the theoretical model of single-photon multiple membrane optomechanical system is studied and Hamiltonians of motional single mode and two mode squeezed state are calculated, with Lamb–Dicke approximation. In Section 3, non-Gaussian state is proved by negativity of Wigner function (WF). Furthermore, the experimental feasibility is discussed in Section 4. The conclusions are finally given in Section 5.
Fig. 1. Schematic of an optomechanical system with two mechanical membranes inside the cavity which is driven by bichromatic laser. q0 and q3 denote the positions of the cavity mirrors, q2 and q3 denote the positions of two membranes. 2
2
∑ =ωmi bi† bi + =a†a ∑ gi (bi + bi† )
H = =ωc a†a +
i=1
i=1
2
+
∑ =Ω (a† exp ( − iωLi t ) + H·c·)
(4)
i=1
a†
b†
where (a) and (b) are the creation (annihilation) operators of the cavity field and the mechanical resonator, respectively. The coupling strength between the cavity field and the ith membrane is written as gi. Also, ωc and ωm are the optical resonance frequency and the frequency of mechanical oscillator, respectively. By employing the g 2 polaron transformation Up = exp (a†a ∑i = 1 [ ωi (bi† − bi )]), the Hai
miltonian in Eq. (1) can be written as: 2
Heff = = (ωc − Δ0 a†a) a†a +
2. Theoretical method
∑ =ωmi bi† bi i=1
In this paper, we focus on an optomechanical system with a cavity containing two non-absorptive membranes, each located in qi and vibrational frequencies are ωmi (i = 1, 2). The goal of this paper is the generation of non-Gaussian squeezed number state. In this condition the great nonlinearity has the key role. Therefore, multiple-membrane optomechanical array is the best because this model causes to increase the nonlinearity. The frequencies of the cavity modes can be given as [12]:
2 ⎛ ⎞ 2 [ ∑ ηi (bi† − bi )− iωLj t ] ⎜ † ⎟ + =Ω ⎜ a ∑ e i = 1 + H · c· ⎟ ⎜ j=1 ⎟ ⎝ ⎠
Here, the first two terms on the right-hand side display the energy structure of the photon Hamiltonian and become the anharmonic one due to the photon–photon interaction induced by the radiation pressure. The nonlinear photon–photon interaction term, 2
N
ω (qi ) = ω (qi0 ) +
∑ gi(1) (qi − qi0 ) + ∑ i =1
gi(,2j ) (qi − qi0 )(qj − q 0j ) + ⋯
i, j = 1
Under the condition
(qi − qi0 ) λ
⎡ ∂ω (q ) ⎤ i gi(1) = ⎢ ⎥ ⎣ ∂qi ⎦q = q 0
(1)
⪡1, we have:
(2)
qi = qi0, qj = q0j
(3)
1 ⎡⎢ ∂ 2ω (qi ) ⎤⎥ 2 ⎢⎣ ∂qi ∂qj ⎥⎦
gi2 ωmi
, with the strong coupling strength gi and low dis-
sipation of the cavity field, Δ0 > γc , guarantees the photon blockade in the optomechanical system. The nonlinearity enhances when the g number of membranes inside the cavity is increased. Here, ηi = ωi is i
i
i
gi(,2j ) =
Δ0 = ∑i = 1
N
(5)
where λ is the wavelength of the optical mode, and qi0 (i = 1, … , N ) is the position of the ith membrane when there is no radiation pressure. In our study below, we only consider that the frequency shift of the cavity mode is linearly dependent on the membrane displacement and we also assume that the cavity is driven by a dichromatic laser field with frequencies ωLi and amplitudes Ωi (i = 1, 2), Ω1 = Ω2 as shown in Fig. 1. The Hamiltonian of the system with two membranes is given by:
the Lamb–Dicke parameter in analogy to the trapped ions [34]. Therefore, the nonlinearity shift, Δ0, is large and the transition to higher excited cavity states can be neglected. In this case, the driving field couples only two lowest energy levels |g 〉 and |e〉 of the cavity field, and Eq. (5) by using the operators σz = |e〉〈e| − |g 〉〈g |, σ+ = |e〉〈g | and σ− = |g 〉〈e| can be reduced to:
Htwo
ω = = σz + 2
2
∑ i=1
=ω mi bi† bi
2 ⎛ ⎞ 2 [ ∑ ηi (bi† − bi )− iωLj t ] ⎜ ⎟ + =Ω ⎜ σ+ ∑ e i = 1 + H · c· ⎟ ⎜ j=1 ⎟ ⎝ ⎠ (6)
Here, ω = ωc − Δ0 . The Hamiltonian in Eq. (6) is similar to the interaction between a classical driving field and a single two-level trapped ion vibrating along one direction. This Hamiltonian can be further written as:
Htwo = H0 + Hint ,
(7)
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
57
Δbj1k = ωLk − ω − ω mj , Δbj2k = ωLk − ω − 2ω mj , Δrj1k = ωLk − ω + ω mj , Δrj2k = ωLk − ω + 2ω mj , Δbk = ωLk − ω − ω m1 − ω m2, Δrk = ωLk − ω + ω m1 + ω m2
(k = 1, 2).
(13)
Therefore, by the mode selecting method [9], each of the desired term of Hamiltonian can be obtained, the Hamiltonian of entangled states of photon–phonon or entangled motional state. Fig. 2. In this diagram transition processes are introduced by carrier ωL = ω , redsideband excitation ωL = ω − kωm and blue-sideband excitation ωL = ω + kωm .
H0 = =
ω σz + 2
2
∑ =ωmi bi† bi ,
(8)
i=1
⎡ 2 2 −ηi2 Hint = =Ωσ+⎢ ∑ e−iωLj t ∏ e 2 ⎢⎣ i=1 j=1
( − 1)ki ηi ji + ki
∑ ji , ki = 0
ji !ki !
⎤ bi† ji bik i ⎥ + H·c· ⎥⎦
(9)
From Eq. (9), we find that |k i − ji | phonons can be created (k i > ji ) or annihilated (k i < ji ) from the ith membrane when one photon is annihilated in the cavity with the assistance of the external field, as shown in Fig 2. The generation of non-classical states of mirror motion is studied under the Lamb–Dicke approximation condition as for the trapped ion case [12,13,34]:
ηi ni + 1 =
gi ni + 1 ⪡1 ω mi
(10)
with the average phonon number ni of the mechanical vibration. In this case, only a single phonon transition occurs with the help of the driving field. Thus the Hamiltonian in Eq. (6) is expanded up to the second order of η: 2 ⎡ Hint = =Ωσ+ ∑ e−iωLj t ⎢ 1 + ⎢⎣ j=1
2
∑ ηi (bi† − bi ) + i=1
⎤ 1 (ηi (bi† − bi ))2⎥ ⎥⎦ 2! (11)
+ H · c·
⎡⎛ V = =Ωσ+⎢ ⎜⎜ e−iΔc1t + ⎢ ⎣⎝ 2
2
j
Δri21 = 0,
2
+
∑
∑
⎞ + b1b2 e−iΔr1t ) ⎟⎟ ⎠
j ηj (b† e−iΔb12 t
j
−
(Vsm )i = =Ωσ+ηi2 (bi† 2 + bi2 ) + H·c· ,
+
(i = 1, 2)
(16)
Therefore, time evolution of Hamiltonian leads to a single mode squeezed number state in first or second membrane:
j bj e−iΔr12 t )
⎛ −i (Vsm )i t ⎞ ⎟ |ψ (0)〉, = S (ξi )|g 〉|m〉1|n〉2 , |ψ (t )〉 = exp ⎜ ⎝ ⎠ =
j=1
j ηj2 (b†j 2 e−iΔb22 t
(15)
ωL2 = ω + 2ω mi .
Accordingly, we have:
j=1
2
Δbi22 = 0,
ωL1 = ω − 2ω mi ,
j
∑ ηj2 (b†j 2 e−iΔb21t + b2j e−iΔr21t )
⎛ + ⎜⎜ e−iΔc 2 t + ⎝
(14)
ωL2 = ω − 2ω mi .
or:
j
j=1
+ 2η1η2 (b1† b2† e−iΔb1t
Δri22 = 0,
ωL1 = ω + 2ω mi ,
∑ ηj (b†j e−iΔb11t − bj e−iΔr11t ) j
The generation of entanglement phonon state has been studied in many papers [35]. As shown previously [36], intra-cavity photon–phonon entanglement is present within each optomechanical cavity. Here, the presence of entanglement between mechanical modes of an optomechanical system is considered. In the previous section, the Hamiltonian of squeezed operator of a membrane or between two membranes has been obtained beyond the Lamb–Dicke approximation. Here, a Wigner function (WF) is employed to show the effects of nonlinear coupling on the preparation of non-classical state of the oscillator motion [37] since the presence of negative regions in the Wigner distribution is a proof of non-Gaussian state [38]. In Ref. [39], the authors have presented a set of criteria to detect quantum non-Gaussian states including the Wigner distribution test. According to Hudson theory [37], the only pure states with positive WF are Gaussian states. Then all states that do not have a Gaussian WF are called nonGaussian states. Accordingly, in this paper WF test is implemented to display the non-Gaussian states. Here, the pumping field consists of two resonant frequencies: red-sideband and blue-sideband transitions. It is assumed that the initial state of system is |ψ (0)〉 = |g 〉|m〉1|n〉2, the optical cavity is in the ground state and both membranes are in Fock state. By adjusting the frequencies of laser ωL1 and ωL2, based on the mechanical frequencies ωm1 and ωm2, the single mode mechanical squeezed number state S (ξ1)|m〉1 or S (ξ2 )|n〉2 is prepared. Achieving this goal is possible when detuning conditions are:
Δbi21 = 0,
In the interaction picture with V = eiH0 t / = Hint e−iH0 t / = we have:
+
3. Non-Gaussian mechanical state beyond the Lamb–Dicke limit
j b2j e−iΔr22 t )
(i = 1 or 2)
(17)
j=1
⎞⎤ + 2η1η2 (b1† b2† e−iΔb2 t + b1b2 e−iΔr2 t ) ⎟⎟ ⎥ + H·c· ⎥ ⎠⎦ With detuning conditions:
S (ξi ) = exp (ξbi† 2 − ξ ⁎bi2 ), (12)
ξi = ri ei (π /2)
(18)
Here, ri is the squeezed parameter and ri = tΩηi2. Moreover, two mode squeezed state between two membranes is obtained under the conditions:
58
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
(b)
0.5
W(q,p)
W(q,p)
(a)
0 −0.5 0.5
0
q
−0.5 −0.4 −0.2
0
0.2
0.5 0 −0.5 0.5 q
p
0.5 0 −0.5 0.5
0
q
−0.5 −0.4 −0.2
0
0.2
p
(d)
W(q,p)
W(q,p)
(c)
0
−0.5 −0.4 −0.2
0
0.2
0.4 0.2 0 0.5
q
p
0
−0.5 −0.5
0
0.5
p
Fig. 3. Wigner function in the phase-space has been illustrated for squeezed single phonon state with nm = 0, m = 1, r = 0.3 and different values of γm t equal to 0.1 (a), 0.2 (b), 0.3 (c ) and 0.4 (d ) . The absolute value of the negative minimum of the WF decreases as γm t increases.
Δr1 = 0,
Δb2 = 0,
ωL1 = ω − ω m1 − ω m2,
ωL2 = ω + ω m1 + ω m2.
(19)
ωL2 = ω − ω m1 − ω m2.
(20)
or:
Δr2 = 0,
Δb1 = 0,
ωL1 = ω + ω m1 + ω m2,
The Hamiltonian of optomechanical system can be written as:
Vtm = =Ωσ+η1η2 (b1† b2† + b1b2 )
(21)
In this way, after the time evolution of Vtm, two mode mechanical squeezed state is calculated with squeezed parameter rtm = Ωtη1η2: ⁎ S (ξtm ) = exp (ξtm b1† b2† − ξtm b1b2 ),
ξtm = rtm ei (π /2)
(22)
If the initial state of mechanical mode consists of one phonon, time evolution of motional state becomes a squeezed single phonon. This assumption is reasonable because according to Eq. (10) under the Lamb–Dicke approximation the average phonon number is low. In quantum optics, a squeezed single photon, can be obtained by adding a photon to a squeezed vacuum as: ^ ^ a†S (r )|0〉 = coshr S (r )|1〉, where the right hand side is unnormalized due to the characteristics of the creation operator a† . It is also known that the squeezed single photon can also be obtained by subtracting a photon from a squeezed vacuum as: ^ ^ aS (r )|0〉 = − sinhr S (r )|1〉. Multiple studies derived the analytical expression of WF of the photon squeezed number state and discussed the nonclassicality in terms of the negativity of WF which implies the highly nonclassical properties of quantum states [40,41]. Consequently, they have proved that squeezed number state is a non-Gaussian state [40]. Similarly, in this paper, the single and two mode phonon squeezed number states and thus the mechanical non-Gaussian state have been created. Here, the important point is the damping of the membrane state due to its coupling to the thermal heat bath which destroys the mechanical states. For cavities with high-Q factor the rate of photon leakage will be very small. It is assumed that the non-classical state is generated by the effective Hamiltonian for times short enough that the photon leakage and thermal heating of the mechanical modes can be ignored. Then, the effective Hamiltonian is turned off and only dissipation evolution is considered. To gain more
details into the effect of decoherence, the reduced master equation for the density operator of the membranes ρm is transformed into the Fokker–Plank equation [40]. Then, time-evolution of the mechanical state and its WF are plotted and loss of mechanical nonGaussianity is discussed in reference of the negativity of WF due to decoherence. The reduced master equation and the Fokker–Plank equation for the damped mechanical oscillator in the Wigner representation become [42,40,43]:
dρm (t ) = dt
2
γmi (2bi ρbi† − bi† bi ρ − ρbi† bi ) 2
∑ i=1
2
+
∑ γmi nmi (bi ρbi† + bi† ρbi − bi† bi ρ − ρbi† bi ) i=1
(23)
γ ⎛ ∂ ∂W (q, p) ∂ ⎞ = m⎜ q + p⎟ W (q, p) ∂t ∂p ⎠ 2 ⎝ ∂q +
γm ⎛ ∂2 ⎞ 1 ⎞ ⎛ ∂2 ⎜ nm + ⎟ ⎜ 2 + 2 ⎟ W (q, p) ∂p ⎠ 4⎝ 2 ⎠ ⎝ ∂q
(24)
where γc and γm are the decay rates of the optical mode and mechanical oscillator, respectively. The two modes are assumed to have the same average energy and have the same average thermal phonon number, nm . This assumption is reasonable as the two mode squeezed state is in the same temperature of the environment [40,43]. We assume that the initial state of the first membrane is m ¼1 and the second membrane is in vacuum state, n ¼0. Generation of mechanical Fock state is very hard and challenging. Let us postpone the discussion to final section. Then, the external weak driving fields are employed to system with the frequencies ωL1 and ωL2 such that the effective Hamiltonian becomes Vsm = =Ωη12 (b1† 2 + b12 ). Hence, the squeezed number state is prepared in the first membrane as |ψ (t )〉 = S (ξ1)|1〉1|0〉2. The reduced density matrix of the first membrane is computed and the timeevolution of WF are plotted in Fig. 3(a), (b), (c) and (d) for different values of γm t = 0.1, 0.2, 0.3 and 0.4, respectively. Here, we assume that nm = 0 and the squeezed parameter r ¼ 0.3. There is some negative region of the WF in the phase-space which is evidence of non-Gaussianity of the state. Also, the absolute value of the
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
m
decreases as nm or r increases for a given time. Since squeezed parameter is related to time r = Ωtη2 (assuming η1 = η2), therefore the negativity of WF is dependent on both the parameters, γm t and r. The results reveal that if γm t ≈
1 2
ln
2nm + 2 , 2nm + 1
there exists a limit of
squeezed parameter which leads to obtain the minimum value of WF where r ≈ 0.7 − 1. Considering the upper bound, r ≈ 1 means Ω that (γm t )( γ η2) ≈ 1. By substituting the threshold value ofγmt we m
have:
2 Ω 2 η ≈ 2nm + 2 γm ln 2nm + 1
(25)
W(q1,q2,0,0)
0.1 0.05 0 −0.05 −0.1 −0.15
4 2 0
q2
−2 −4
−4
−3
−2
−1
0
1
2
3
4
q1
Fig. 4. The WFs for (m = 0, n = 1) are plotted in phase space (q1, q2, 0, 0) with n = 1, r ¼0.3 and γm t = 0.05. WF is negative and two mode squeezed state squeezed state is non-Gaussian.
0.4
W(q1,q2,0,0)
negative minimum of the WF decreases as γm t increases. The results show that the WF is always positive for γm t > 0.35 when nm = 0. Also, at long times when γm t → ∞, the non-Gaussian squeezed state of the first membrane turns into the Gaussian state 2 2 2 and the WF becomes π e−2 (q + p ) which corresponds to a vacuum state [40]. In the following, we consider the two mode squeezed number state. We assume that the initial conditions of two membranes are Fock state, |m〉1|n〉2. Two weak coherent laser modes are employed to system such that the effective Hamiltonian and the generated state become Vtm = =Ωσ+η1η2 (b1† b2† + b1b2 ) and S (ξtm )|m〉1|n〉2, respectively. The reduced density matrix of two membranes is applied according to Eq. (23). Also, according to Eq. (10), under the Lamb–Dicke approximation, the average of phonon number of membranes is low. Figs. 4 and 5 demonstrate WF for two mode squeezed state (entangled of two membranes) in the Lamb–Dicke approximation. When m ¼n ¼0 corresponds to the two mode squeezed vacuum state which is Gaussian state [41,40]. In Figs. 4 and 5, the phasespace Wigner distributions are illustrated for different parameter values m and n with a fixed values of p1 and p2. In Fig. 4, a negative region is shown in the WF plot and phase space (q1, q2, 0, 0) where (m = 0, n = 1) with nm = 1, γm t = 0.05 and r ¼0.3. Therefore, a nonGaussian entangled state is created by this special initial condition. In Fig. 5 the initial condition is considered where (m = 1, n = 1) with nm = 1, γm t = 0.05 and r¼ 0.3 in phase space (q1, q2, 0, 0). The results indicate that unexpectedly, WF is positive anyway and a non-Gaussian state is not created. Then, in the two mode squeezed number state in the optomechanical system, the initial states of membranes have an important role. On the other hand, if γm t → ∞ the two mode system reduces to two thermal state which is independent of initial conditions and depicts a Gaussian distribution. Actually there exists a threshold value for γmt in creating non-Gaussian two-mode squeezed number state. The WF is always positive if the γmt exceeds an upper 1 2n + 2 bound, γm t > 2 ln 2nm + 1 [40,41]. The partial negativity of the WF
59
0.3 0.2 0.1 0 −0.1 4 2 0
q2
−2 −4
−4
−3
−1
−2
0
1
2
3
4
q1
Fig. 5. The WFs for (m = 1, n = 1) are illustrated in phase space (q1, q2, 0, 0) with n = 1, r ¼0.3 and γm t = 0.05. In this case, WF is negative anyway and two mode motional squeezed state is Gaussian.
Notice that the Hamiltonian of mechanical single and two mode squeezed states have been presented with these limitations: under the Lamb–Dicke approximation η < 1, weak driving field γm < Ω⪡γc , photon blockade regime ωm, g > γc , γm and big photon–photon interaction,
g2 ωm
> γc . For example it is assumed that for a membrane
system, η = 0.1, γc = 0.3 MHz , γm = 0.1 Hz [44] and Ω = 1 KHz, consequently, only in the limit of nm < 25 a non-Gaussian state is created. Hence, the cooling methods are necessary conditions. The proposed method to generate non-Gaussian squeezed state is debatable as experimental feasibility. Let us discuss about it in the next section.
4. Discussion on experimental feasibility In this paper, the main goal is the generation of non-Gaussian motional squeezed number state by using sideband excitations and photon blockade effect. Experimental realization of photon blockade effect involves considerable details [45–47]: (1) The single photon strong coupling regime is g > γc , γm . (2) Resolved sideband regime that is ωm > γc . 2 (3) The nonlinear photon–photon interaction strength, g must ωm g2 be greater than decay rate of cavity, ω > γc . m
So far, in most experiments, g is much smaller than γc, only using ultracold atom in optical resonators or in optomechanical crystals [48,49], one can achieve the approach of single photon strong coupling regime. However one is typically far from the resolved sideband regime. Therefore, these systems are not suitable for the scheme presented here which is also based on the selection of resonant processes. Xuereb et al. displayed in Refs. [50,51], in optomechanical system with more massive and macroscopic systems like membranes, the large optomechanical coupling could be attained by using interference phenomenon and the collective interaction of many optical elements within an optical cavity. Then, the authors studied an optomechanical system of two membranes placed inside a cavity and provided the special conditions to create the large single photon optomechanical coupling. In the limit of highly reflective membranes very large single photon optomechanical coupling is achieved when the two membranes are placed very close to a resonance of the inner cavity formed by the two membranes [52]. In this paper, we require single photon strong coupling and apply these assumptions in two membrane optomechanical system. In Section 3, the main goal is the generation of non-Gaussian mechanical squeezed number state. Therefore, the Fock state has been used as the initial state of membranes. In Ref. [13], the generation of mechanical Fock state is investigated
60
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
by employing the photon blockade effect. The authors display a method to generate target states by using a series of classical pulses with desired frequencies, phases and durations. Our scheme is based on the photon blockade. Thus, the initial Fock state is prepared before preparation of conditions to Hamiltonian of mechanical squeezed state. Of course, with regard to the details which are mentioned in the previous paragraph about experimental conditions, it is difficult to achieve the mechanical Fock state experimentally. Generally, the generation of mechanical coherent state is easier than the Fock state. Thus, it is best to get the coherent state as initial states of membranes. Then, we should investigate the features of non-Gaussian squeezed coherent state. Recently, the squeezed coherent state has been discussed as a non-Gaussian regime [53]. In this scheme, in the limit of small squeezing, r < 1, the squeezed operator can be expanded to:
S (ξ ) = 1 +
ξ †2 ξ ⁎ 2 b − b 2 2
(26)
Therefore, in this approach, as a suggestion, the initial coherent state can be selected which is suitable experimentally. Eq. (26) can be implemented by using two ideas: 1. In the limit of small time, weak driving and Lamb–Dicke approximation we have, r = Ωtη2 < 1. Then we can use Eq. (26). 2. We try to extract S (ξ ) from Eq. (12). Here, one can use the quantum interference effects such as the optomechanically induced transparency (OMIT). In this case, the motion of the mirror evolves into a dark state. Compared with [7], where tuning the three frequencies of the driving field to be resonant to the carrier, red-sideband and bluesideband transitions, in our model, third laser frequency couples to carrier transition. We define, Δc 3 = ωL3 − ω and in the resonance condition becomes ωL3 = ω . For example, the dark state of first membrane when the state of second membrane is vacuum ∞ becomes |D〉 = ∑m = 0 N |g 〉|m〉1|0〉2 (N is a normalization constant) [7]. Then, we obtain Vsm = Ω (1 + ηb1† 2 + ηb12 ). Forasmuch as Ω is small so the approximation Ω ≈ 1 Hz is valid. In this case, when the average number of phonon is small the non-Gaussian squeezed coherent state of first membrane is created.
5. Conclusions In this paper, we focus on an optomechanical system with a cavity containing two non-absorptive membranes. In the strong single photon coupling, by using the similarity between the Hamiltonian of optomechanical system in photon blockade regime and Hamiltonian of ion trapping and also, by adjusting the input frequency, the motional non-Gaussian squeezed number state is generated. Non-Gaussian state is proved by negativity of Wigner function. It is found out that under the Lamb–Dicke approximation and low temperature, the single mode motional non-Gaussian squeezed state is prepared while the generation of two mode motional non-Gaussian squeezed state depends on the initial state of the membranes. The results show that in low temperature, if one of the membranes is in the ground state and the other in single phonon state, the two mode squeezed state is non-Gaussian. In the conditions that the membranes are in the single phonon state, even in the low temperature, two mode squeezed state is not non-Gaussian. Although it is believed that in the strong nonlinearity the non-Gaussian is generated, but in the two mode squeezed number state, the initial state of membranes has an important role.
References [1] T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A.V. Gorshkov, T. Pohl, M.D. Lukin, V. Vuletić, Quantum nonlinear optics with single photons enabled by strongly interacting atoms, Nature 488 (7409) (2012) 57–60. [2] N. Matsuda, R. Shimizu, Y. Mitsumori, H. Kosaka, K. Edamatsu, Observation of optical-fibre Kerr nonlinearity at the single-photon level, Nat. Photonics 3 (2) (2009) 95–98. [3] A. Imamolu, H. Schmidt, G. Woods, M. Deutsch, Strongly interacting photons in a nonlinear cavity, Phys. Rev. Lett. 79 (8) (1997) 1467. [4] G. Milburn, M. Woolley, An introduction to quantum optomechanics, Acta Phys. Slovaca 61 (5) (2011) 483–601. [5] D. Vitali, S. Mancini, L. Ribichini, P. Tombesi, Macroscopic mechanical oscillators at the quantum limit through optomechanical cooling, JOSA B 20 (5) (2003) 1054–1065. [6] J.-Q. Liao, C. Law, L.-M. Kuang, F. Nori, Enhancement of mechanical effects of single photons in modulated two-mode optomechanics, Phys. Rev. A 92 (1) (2015) 013822. [7] W.-j. Gu, G.-x. Li, S.-p. Wu, Y.-p. Yang, Generation of non-classical states of mirror motion in the single-photon strong-coupling regime, Opt. Express 22 (15) (2014) 18254–18267. [8] M. Woolley, A. Clerk, Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir, Phys. Rev. A 89 (6) (2014) 063805. [9] M. Schmidt, M. Ludwig, F. Marquardt, Optomechanical circuits for nanomechanical continuous variable quantum state processing, New J. Phys. 14 (12) (2012) 125005. [10] H. Tan, G. Li, P. Meystre, Dissipation-driven two-mode mechanical squeezed states in optomechanical systems, Phys. Rev. A 87 (2013) 033829. [11] Y.-D. Wang, A.A. Clerk, Using dark modes for high-fidelity optomechanical quantum state transfer, New J. Phys. 14 (10) (2012) 105010. [12] X.-W. Xu, Y.-J. Zhao, Y.-x. Liu, Entangled-state engineering of vibrational modes in a multimembrane optomechanical system, Phys. Rev. A 88 (2) (2013) 022325. [13] X.-W. Xu, H. Wang, J. Zhang, Y.-x. Liu, Engineering of nonclassical motional states in optomechanical systems, Phys. Rev. A 88 (6) (2013) 063819. [14] S. Bose, K. Jacobs, P. Knight, Preparation of nonclassical states in cavities with a moving mirror, Phys. Rev. A 56 (5) (1997) 4175. [15] U. Akram, W.P. Bowen, G.J. Milburn, Entangled mechanical cat states via conditional single photon optomechanics, New J. Phys. 15 (9) (2013) 093007. [16] S. Barzanjeh, M. Naderi, M. Soltanolkotabi, Generation of motional nonlinear coherent states and their superpositions via an intensity-dependent coupling of a cavity field to a micromechanical membrane, J. Phys. B: At. Mol. Opt. Phys.: At. 44 (10) (2011) 105504. [17] K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E.S. Polzik, P. Zoller, Cavityassisted squeezing of a mechanical oscillator, Phys. Rev. A 79 (6) (2009) 063819. [18] H. Tan, G. Li, P. Meystre, Dissipation-driven two-mode mechanical squeezed states in optomechanical systems, Phys. Rev. A 87 (3) (2013) 033829. [19] D.V. Muhammad Asjad, Reservoir engineering of a mechanical resonator: generating a macroscopic superposition state and monitoring its decoherence, J. Phys. B: At. Mol. Opt. Phys. 47 (4) (2014) 045502. [20] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. Büchler, P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4 (11) (2008) 878–883. [21] S.D. Bartlett, B.C. Sanders, S.L. Braunstein, K. Nemoto, Efficient classical simulation of continuous variable quantum information processes, Phys. Rev. Lett. 88 (2002) 097904. [22] S. Boixo, A. Datta, M.J. Davis, S.T. Flammia, A. Shaji, C.M. Caves, Quantum metrology: dynamics versus entanglement, Phys. Rev. Lett. 101 (4) (2008) 040403. [23] H. Nha, H. Carmichael, Proposed test of quantum nonlocality for continuous variables, Phys. Rev. Lett. 93 (2) (2004) 020401. [24] M. Stobińska, A.S. Villar, G. Leuchs, Generation of Kerr non-Gaussian motional states of trapped ions, EPL (Europhys. Lett.) 94 (5) (2011) 54002. [25] F. Khalili, S. Danilishin, H. Miao, H. Müller-Ebhardt, H. Yang, Y. Chen, Preparing a mechanical oscillator in non-Gaussian quantum states, Phys. Rev. Lett. 105 (7) (2010) 070403. [26] M. Poot, K.Y. Fong, H.X. Tang, Classical non-Gaussian state preparation through squeezing in an optoelectromechanical resonator, Phys. Rev. A 90 (6) (2014) 063809. [27] A. Pontin, M. Bonaldi, A. Borrielli, F. Cataliotti, F. Marino, G. Prodi, E. Serra, F. Marin, Squeezing a thermal mechanical oscillator by stabilized parametric effect on the optical spring, Phys. Rev. Lett. 112 (2) (2014) 023601. [28] A. Szorkovszky, G.A. Brawley, A.C. Doherty, W.P. Bowen, Strong thermomechanical squeezing via weak measurement, Phys. Rev. Lett. 110 (18) (2013) 184301. [29] Z. Mahmoudi, S. Shakeri, O. Hamidi, M. Zandi, A. Bahrampour, Generation of motional entangled coherent state in an optomechanical system in the single photon strong coupling regime 62 (19) (2015) 1685–169162. [30] T. Tyc, N. Korolkova, Highly non-Gaussian states created via cross-Kerr nonlinearity, New J. Phys. 10 (2) (2008) 023041. [31] M. Yanagisawa, Non-Gaussian state generation from linear elements via feedback, Phys. Rev. Lett. 103 (20) (2009) 203601. [32] M. Stobińska, G. Milburn, K. Wódkiewicz, Wigner function evolution of quantum states in the presence of self-Kerr interaction, Phys. Rev. A 78 (1)
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
(2013) 013810. [33] X.-Y. Lü, W.-M. Zhang, S. Ashhab, Y. Wu, F. Nori, Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems, Sci. Rep. 3 (2013). [34] y. Leibfried, R. Blatt, C. Monroe, D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75 (1) (2003) 281. [35] S.-C. Gou, J. Steinbach, P. Knight, Generation of mesoscopic superpositions of two squeezed states of motion for a trapped ion, Phys. Rev. A 55 (5) (1997) 3719. [36] D. Vitali, S. Gigan, A. Ferreira, H. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, M. Aspelmeyer, Optomechanical entanglement between a movable mirror and a cavity field, Phys. Rev. Lett. 98 (3) (2007) 030405. [37] R. Hudson, When is the Wigner quasi-probability density non-negative? Rep. Math. Phys. 6 (2) (1974) 249–252. [38] A. Kenfack, K. Źyczkowski, Negativity of the Wigner function as an indicator of non-classicality, J. Opt. B: Quantum Semiclass. Opt. 6 (10) (2004) 396. [39] M.G. Genoni, M.L. Palma, T. Tufarelli, S. Olivares, M. Kim, M.G. Paris, Detecting quantum non-Gaussianity via the Wigner function, Phys. Rev. A 87 (6) (2013) 062104. [40] H. Li-Yun, F. Hong-Yi, Two-variable hermite polynomial excitation of twomode squeezed vacuum state as squeezed two-mode number state, Commun. Theor. Phys. 50 (4) (2008) 965. [41] L.-j. Xu, G.-b. Tan, S.-j. Ma, Nonclassicality and decoherence of two-mode photon-added squeezed vacuum, Opt. Commun. 284 (13) (2011) 3335–3344. [42] H. Carmichael, An Open Systems Approach to Quantum Optics: Lectures Presented at the Université Libre de Bruxelles, October 28 to November 4, 1991, vol. 18, Springer Science & Business Media, 2009.
61
[43] A. Biswas, G.S. Agarwal, Nonclassicality and decoherence of photon-subtracted squeezed states, Phys. Rev. A 75 (3) (2007) 032104. [44] J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, J. Harris, Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane, Nature 452 (7183) (2008) 72–75. [45] P. Rabl, Photon blockade effect in optomechanical systems, Phys. Rev. Lett. 107 (6) (2011) 063601. [46] J.-Q. Liao, F. Nori, et al., Photon blockade in quadratically coupled optomechanical systems, Phys. Rev. A 88 (2) (2013) 023853. [47] A. Miranowicz, M. Paprzycka, Y.-x. Liu, J. Bajer, F. Nori, Two-photon and threephoton blockades in driven nonlinear systems, Phys. Rev. A 87 (2) (2013) 023809. [48] M. Asjad, P. Tombesi, D. Vitali, Quantum phase gate for optical qubits with cavity quantum optomechanics, Opt. Express 23 (6) (2015) 7786–7794. [49] D.W. Brooks, T. Botter, S. Schreppler, T.P. Purdy, N. Brahms, D.M. StamperKurn, Non-classical light generated by quantum-noise-driven cavity optomechanics, Nature 488 (7412) (2012) 476–480. [50] A. Xuereb, C. Genes, A. Dantan, Strong coupling and long-range collective interactions in optomechanical arrays, Phys. Rev. Lett. 109 (22) (2012) 223601. [51] A. Xuereb, C. Genes, A. Dantan, Collectively enhanced optomechanical coupling in periodic arrays of scatterers, Phys. Rev. A 88 (5) (2013) 053803. [52] J. Li, A. Xuereb, N. Malossi, D. Vitali, Cavity mode frequencies and large optomechanical coupling in two-membrane cavity optomechanics, arXiv preprint arxiv:1512.07536. [53] K. Jeong, J. Kim, S.-Y. Lee, Gaussian private quantum channel with squeezed coherent states, Sci. Rep. 5 (13974) (2015).
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Two mode mechanical non-Gaussian squeezed number state in a two-membrane optomechanical system S. Shakeri a,n, Z. Mahmoudi a, M.H. Zandi a, A.R. Bahrampour b a b
Faculty of Physics, Shahid Bahonar University of Kerman, Kerman, Iran Sharif University of Technology, Department of Physics, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 27 November 2015 Received in revised form 26 February 2016 Accepted 27 February 2016
We consider an optomechanical system with two membranes when a bichromatic laser field with redsideband and blue-sideband frequencies is applied in the single photon strong coupling regime. It is shown that using the mode selecting method and under the Lamb–Dicke approximation, motion of membranes can evolve to single or two mode squeezed number states. By considering the environmental effect, a Wigner function is plotted for understanding the conditions that lead to the generation of nonGaussian states. The results show that, in this system, initial states of membranes are important to generation of non-Gaussian mechanical squeezed number states. & 2016 Elsevier B.V. All rights reserved.
Keywords: Two-membrane optomechanical system Single photon strong coupling regime Non-Gaussian state Motional squeezed state Wigner function
1. Introduction The implementation of strong optical nonlinearities on a single photon level is one of the main goals in quantum optics and is specifically interesting in cavity QED [1–3]. The other remarkable system which has been widely investigated in this context is the optomechanical system (OMS) where the radiation pressure coupling between light and mechanical motion is studied [4–6]. Nonclassical states are the characteristic elements of quantum theory. There has been an increasing interest in establishing the conditions under which nonclassical states among macroscopic objects can arise. The motional nonclassical states such as squeezed states, Schrödinger cats and Fock states [7–16] have been already investigated in various optomechanical systems with a strong or a weak coupling regime. Mechanical squeezed state is very important and several proposals have been studied to generate mechanical squeezing in an optomechanical oscillator, including the injection of nonclassical light, conditional quantum measurement, parametric amplification and quantum-reservoir engineering [17–20]. In many proposed procedures, created mechanical squeezed states are classified in Gaussian squeezed states. Non-Gaussian states allow to apply the quantum algorithm or to process on a classical computer [21]. Non-Gaussian states also find important applications in high-precision metrology, novel test for n
Corresponding author. E-mail address: [email protected] (S. Shakeri).
http://dx.doi.org/10.1016/j.optcom.2016.02.063 0030-4018/& 2016 Elsevier B.V. All rights reserved.
Bell-like inequalities and fundamental investigations [22,23]. Therefore, finding an effective way to prepare non-Gaussian states and interactions is crucial for the evolution of quantum based applications and also for the better understanding of physics [24]. Non-Gaussian squeezed states have been considered in quantum optics and OMS seriously. In Ref. [25], a protocol has been proposed for coherently transferring non-Gaussian quantum states from optical field to a mechanical oscillator. Squeezing of a strongly interacting opto-electromechanical system using a parametric device has been indicated in Ref. [26]. By employing realtime feedback [27,28] on the phase of the pump at twice the resonance frequencies the thermo-mechanical noise is squeezed beyond the 3 dB in stability limit. This method can also be used to generate highly nonlinear states. Also, in Ref. [7], the authors considered the generation of non-classical states of the mirror motion in the single photon strong coupling regime. They found that the motion of mirror can evolve into a squeezed coherent dark state and beyond the Lamb–Dicke limit, the state can become a squeezed non-Gaussian state. Two mode non-Gaussian squeezed number state has not been discussed in optomechanical systems up to now. In this paper, preparation of non-Gaussian single and two mode squeezed states is considered theoretically. Our system is an optomechanical system with two membranes and bichromatic laser is applied within the single photon strong coupling regime. Multiple membrane optomechanical systems are studied in the single photon strong coupling regime. In Ref. [12] the implied method is
56
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
converted to the method used in trapped ions. We have introduced and used this method to the generation of motional Schrödinger cat states in our previous paper [29]. The Hamiltonian of the squeezed operator is achieved clearly and the main idea is related to non-Gaussian single and two mode motional squeezed number state between two membranes. The two mode squeezed states are very important in quantum optics and can be employed to explain EPR paradox. More recently this kind of correlation has also played a leading role in the quantum teleportation of continuous variables. It seems that strong coupling regime leads to the strong nonlinearity in the system, and therefore the generation of non-Gaussian states is an undeniable fact [30–32]. Despite the strong nonlinearity condition, it has been shown that the initial state of membranes has an important role in creating the two mode non-Gaussian states. Also, creating the condition of single photon strong coupling is hard and complicated, in most experiments up to now. According to [33], multiple membrane hybrid electro-optomechanical systems are proposed to realise strong Kerr nonlinearities even in the weak-coupling regime and optimal conditions for achieving the non-Gaussian squeezed state are studied. Our paper is organized as follows. In Section 2, the theoretical model of single-photon multiple membrane optomechanical system is studied and Hamiltonians of motional single mode and two mode squeezed state are calculated, with Lamb–Dicke approximation. In Section 3, non-Gaussian state is proved by negativity of Wigner function (WF). Furthermore, the experimental feasibility is discussed in Section 4. The conclusions are finally given in Section 5.
Fig. 1. Schematic of an optomechanical system with two mechanical membranes inside the cavity which is driven by bichromatic laser. q0 and q3 denote the positions of the cavity mirrors, q2 and q3 denote the positions of two membranes. 2
2
∑ =ωmi bi† bi + =a†a ∑ gi (bi + bi† )
H = =ωc a†a +
i=1
i=1
2
+
∑ =Ω (a† exp ( − iωLi t ) + H·c·)
(4)
i=1
a†
b†
where (a) and (b) are the creation (annihilation) operators of the cavity field and the mechanical resonator, respectively. The coupling strength between the cavity field and the ith membrane is written as gi. Also, ωc and ωm are the optical resonance frequency and the frequency of mechanical oscillator, respectively. By employing the g 2 polaron transformation Up = exp (a†a ∑i = 1 [ ωi (bi† − bi )]), the Hai
miltonian in Eq. (1) can be written as: 2
Heff = = (ωc − Δ0 a†a) a†a +
2. Theoretical method
∑ =ωmi bi† bi i=1
In this paper, we focus on an optomechanical system with a cavity containing two non-absorptive membranes, each located in qi and vibrational frequencies are ωmi (i = 1, 2). The goal of this paper is the generation of non-Gaussian squeezed number state. In this condition the great nonlinearity has the key role. Therefore, multiple-membrane optomechanical array is the best because this model causes to increase the nonlinearity. The frequencies of the cavity modes can be given as [12]:
2 ⎛ ⎞ 2 [ ∑ ηi (bi† − bi )− iωLj t ] ⎜ † ⎟ + =Ω ⎜ a ∑ e i = 1 + H · c· ⎟ ⎜ j=1 ⎟ ⎝ ⎠
Here, the first two terms on the right-hand side display the energy structure of the photon Hamiltonian and become the anharmonic one due to the photon–photon interaction induced by the radiation pressure. The nonlinear photon–photon interaction term, 2
N
ω (qi ) = ω (qi0 ) +
∑ gi(1) (qi − qi0 ) + ∑ i =1
gi(,2j ) (qi − qi0 )(qj − q 0j ) + ⋯
i, j = 1
Under the condition
(qi − qi0 ) λ
⎡ ∂ω (q ) ⎤ i gi(1) = ⎢ ⎥ ⎣ ∂qi ⎦q = q 0
(1)
⪡1, we have:
(2)
qi = qi0, qj = q0j
(3)
1 ⎡⎢ ∂ 2ω (qi ) ⎤⎥ 2 ⎢⎣ ∂qi ∂qj ⎥⎦
gi2 ωmi
, with the strong coupling strength gi and low dis-
sipation of the cavity field, Δ0 > γc , guarantees the photon blockade in the optomechanical system. The nonlinearity enhances when the g number of membranes inside the cavity is increased. Here, ηi = ωi is i
i
i
gi(,2j ) =
Δ0 = ∑i = 1
N
(5)
where λ is the wavelength of the optical mode, and qi0 (i = 1, … , N ) is the position of the ith membrane when there is no radiation pressure. In our study below, we only consider that the frequency shift of the cavity mode is linearly dependent on the membrane displacement and we also assume that the cavity is driven by a dichromatic laser field with frequencies ωLi and amplitudes Ωi (i = 1, 2), Ω1 = Ω2 as shown in Fig. 1. The Hamiltonian of the system with two membranes is given by:
the Lamb–Dicke parameter in analogy to the trapped ions [34]. Therefore, the nonlinearity shift, Δ0, is large and the transition to higher excited cavity states can be neglected. In this case, the driving field couples only two lowest energy levels |g 〉 and |e〉 of the cavity field, and Eq. (5) by using the operators σz = |e〉〈e| − |g 〉〈g |, σ+ = |e〉〈g | and σ− = |g 〉〈e| can be reduced to:
Htwo
ω = = σz + 2
2
∑ i=1
=ω mi bi† bi
2 ⎛ ⎞ 2 [ ∑ ηi (bi† − bi )− iωLj t ] ⎜ ⎟ + =Ω ⎜ σ+ ∑ e i = 1 + H · c· ⎟ ⎜ j=1 ⎟ ⎝ ⎠ (6)
Here, ω = ωc − Δ0 . The Hamiltonian in Eq. (6) is similar to the interaction between a classical driving field and a single two-level trapped ion vibrating along one direction. This Hamiltonian can be further written as:
Htwo = H0 + Hint ,
(7)
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
57
Δbj1k = ωLk − ω − ω mj , Δbj2k = ωLk − ω − 2ω mj , Δrj1k = ωLk − ω + ω mj , Δrj2k = ωLk − ω + 2ω mj , Δbk = ωLk − ω − ω m1 − ω m2, Δrk = ωLk − ω + ω m1 + ω m2
(k = 1, 2).
(13)
Therefore, by the mode selecting method [9], each of the desired term of Hamiltonian can be obtained, the Hamiltonian of entangled states of photon–phonon or entangled motional state. Fig. 2. In this diagram transition processes are introduced by carrier ωL = ω , redsideband excitation ωL = ω − kωm and blue-sideband excitation ωL = ω + kωm .
H0 = =
ω σz + 2
2
∑ =ωmi bi† bi ,
(8)
i=1
⎡ 2 2 −ηi2 Hint = =Ωσ+⎢ ∑ e−iωLj t ∏ e 2 ⎢⎣ i=1 j=1
( − 1)ki ηi ji + ki
∑ ji , ki = 0
ji !ki !
⎤ bi† ji bik i ⎥ + H·c· ⎥⎦
(9)
From Eq. (9), we find that |k i − ji | phonons can be created (k i > ji ) or annihilated (k i < ji ) from the ith membrane when one photon is annihilated in the cavity with the assistance of the external field, as shown in Fig 2. The generation of non-classical states of mirror motion is studied under the Lamb–Dicke approximation condition as for the trapped ion case [12,13,34]:
ηi ni + 1 =
gi ni + 1 ⪡1 ω mi
(10)
with the average phonon number ni of the mechanical vibration. In this case, only a single phonon transition occurs with the help of the driving field. Thus the Hamiltonian in Eq. (6) is expanded up to the second order of η: 2 ⎡ Hint = =Ωσ+ ∑ e−iωLj t ⎢ 1 + ⎢⎣ j=1
2
∑ ηi (bi† − bi ) + i=1
⎤ 1 (ηi (bi† − bi ))2⎥ ⎥⎦ 2! (11)
+ H · c·
⎡⎛ V = =Ωσ+⎢ ⎜⎜ e−iΔc1t + ⎢ ⎣⎝ 2
2
j
Δri21 = 0,
2
+
∑
∑
⎞ + b1b2 e−iΔr1t ) ⎟⎟ ⎠
j ηj (b† e−iΔb12 t
j
−
(Vsm )i = =Ωσ+ηi2 (bi† 2 + bi2 ) + H·c· ,
+
(i = 1, 2)
(16)
Therefore, time evolution of Hamiltonian leads to a single mode squeezed number state in first or second membrane:
j bj e−iΔr12 t )
⎛ −i (Vsm )i t ⎞ ⎟ |ψ (0)〉, = S (ξi )|g 〉|m〉1|n〉2 , |ψ (t )〉 = exp ⎜ ⎝ ⎠ =
j=1
j ηj2 (b†j 2 e−iΔb22 t
(15)
ωL2 = ω + 2ω mi .
Accordingly, we have:
j=1
2
Δbi22 = 0,
ωL1 = ω − 2ω mi ,
j
∑ ηj2 (b†j 2 e−iΔb21t + b2j e−iΔr21t )
⎛ + ⎜⎜ e−iΔc 2 t + ⎝
(14)
ωL2 = ω − 2ω mi .
or:
j
j=1
+ 2η1η2 (b1† b2† e−iΔb1t
Δri22 = 0,
ωL1 = ω + 2ω mi ,
∑ ηj (b†j e−iΔb11t − bj e−iΔr11t ) j
The generation of entanglement phonon state has been studied in many papers [35]. As shown previously [36], intra-cavity photon–phonon entanglement is present within each optomechanical cavity. Here, the presence of entanglement between mechanical modes of an optomechanical system is considered. In the previous section, the Hamiltonian of squeezed operator of a membrane or between two membranes has been obtained beyond the Lamb–Dicke approximation. Here, a Wigner function (WF) is employed to show the effects of nonlinear coupling on the preparation of non-classical state of the oscillator motion [37] since the presence of negative regions in the Wigner distribution is a proof of non-Gaussian state [38]. In Ref. [39], the authors have presented a set of criteria to detect quantum non-Gaussian states including the Wigner distribution test. According to Hudson theory [37], the only pure states with positive WF are Gaussian states. Then all states that do not have a Gaussian WF are called nonGaussian states. Accordingly, in this paper WF test is implemented to display the non-Gaussian states. Here, the pumping field consists of two resonant frequencies: red-sideband and blue-sideband transitions. It is assumed that the initial state of system is |ψ (0)〉 = |g 〉|m〉1|n〉2, the optical cavity is in the ground state and both membranes are in Fock state. By adjusting the frequencies of laser ωL1 and ωL2, based on the mechanical frequencies ωm1 and ωm2, the single mode mechanical squeezed number state S (ξ1)|m〉1 or S (ξ2 )|n〉2 is prepared. Achieving this goal is possible when detuning conditions are:
Δbi21 = 0,
In the interaction picture with V = eiH0 t / = Hint e−iH0 t / = we have:
+
3. Non-Gaussian mechanical state beyond the Lamb–Dicke limit
j b2j e−iΔr22 t )
(i = 1 or 2)
(17)
j=1
⎞⎤ + 2η1η2 (b1† b2† e−iΔb2 t + b1b2 e−iΔr2 t ) ⎟⎟ ⎥ + H·c· ⎥ ⎠⎦ With detuning conditions:
S (ξi ) = exp (ξbi† 2 − ξ ⁎bi2 ), (12)
ξi = ri ei (π /2)
(18)
Here, ri is the squeezed parameter and ri = tΩηi2. Moreover, two mode squeezed state between two membranes is obtained under the conditions:
58
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
(b)
0.5
W(q,p)
W(q,p)
(a)
0 −0.5 0.5
0
q
−0.5 −0.4 −0.2
0
0.2
0.5 0 −0.5 0.5 q
p
0.5 0 −0.5 0.5
0
q
−0.5 −0.4 −0.2
0
0.2
p
(d)
W(q,p)
W(q,p)
(c)
0
−0.5 −0.4 −0.2
0
0.2
0.4 0.2 0 0.5
q
p
0
−0.5 −0.5
0
0.5
p
Fig. 3. Wigner function in the phase-space has been illustrated for squeezed single phonon state with nm = 0, m = 1, r = 0.3 and different values of γm t equal to 0.1 (a), 0.2 (b), 0.3 (c ) and 0.4 (d ) . The absolute value of the negative minimum of the WF decreases as γm t increases.
Δr1 = 0,
Δb2 = 0,
ωL1 = ω − ω m1 − ω m2,
ωL2 = ω + ω m1 + ω m2.
(19)
ωL2 = ω − ω m1 − ω m2.
(20)
or:
Δr2 = 0,
Δb1 = 0,
ωL1 = ω + ω m1 + ω m2,
The Hamiltonian of optomechanical system can be written as:
Vtm = =Ωσ+η1η2 (b1† b2† + b1b2 )
(21)
In this way, after the time evolution of Vtm, two mode mechanical squeezed state is calculated with squeezed parameter rtm = Ωtη1η2: ⁎ S (ξtm ) = exp (ξtm b1† b2† − ξtm b1b2 ),
ξtm = rtm ei (π /2)
(22)
If the initial state of mechanical mode consists of one phonon, time evolution of motional state becomes a squeezed single phonon. This assumption is reasonable because according to Eq. (10) under the Lamb–Dicke approximation the average phonon number is low. In quantum optics, a squeezed single photon, can be obtained by adding a photon to a squeezed vacuum as: ^ ^ a†S (r )|0〉 = coshr S (r )|1〉, where the right hand side is unnormalized due to the characteristics of the creation operator a† . It is also known that the squeezed single photon can also be obtained by subtracting a photon from a squeezed vacuum as: ^ ^ aS (r )|0〉 = − sinhr S (r )|1〉. Multiple studies derived the analytical expression of WF of the photon squeezed number state and discussed the nonclassicality in terms of the negativity of WF which implies the highly nonclassical properties of quantum states [40,41]. Consequently, they have proved that squeezed number state is a non-Gaussian state [40]. Similarly, in this paper, the single and two mode phonon squeezed number states and thus the mechanical non-Gaussian state have been created. Here, the important point is the damping of the membrane state due to its coupling to the thermal heat bath which destroys the mechanical states. For cavities with high-Q factor the rate of photon leakage will be very small. It is assumed that the non-classical state is generated by the effective Hamiltonian for times short enough that the photon leakage and thermal heating of the mechanical modes can be ignored. Then, the effective Hamiltonian is turned off and only dissipation evolution is considered. To gain more
details into the effect of decoherence, the reduced master equation for the density operator of the membranes ρm is transformed into the Fokker–Plank equation [40]. Then, time-evolution of the mechanical state and its WF are plotted and loss of mechanical nonGaussianity is discussed in reference of the negativity of WF due to decoherence. The reduced master equation and the Fokker–Plank equation for the damped mechanical oscillator in the Wigner representation become [42,40,43]:
dρm (t ) = dt
2
γmi (2bi ρbi† − bi† bi ρ − ρbi† bi ) 2
∑ i=1
2
+
∑ γmi nmi (bi ρbi† + bi† ρbi − bi† bi ρ − ρbi† bi ) i=1
(23)
γ ⎛ ∂ ∂W (q, p) ∂ ⎞ = m⎜ q + p⎟ W (q, p) ∂t ∂p ⎠ 2 ⎝ ∂q +
γm ⎛ ∂2 ⎞ 1 ⎞ ⎛ ∂2 ⎜ nm + ⎟ ⎜ 2 + 2 ⎟ W (q, p) ∂p ⎠ 4⎝ 2 ⎠ ⎝ ∂q
(24)
where γc and γm are the decay rates of the optical mode and mechanical oscillator, respectively. The two modes are assumed to have the same average energy and have the same average thermal phonon number, nm . This assumption is reasonable as the two mode squeezed state is in the same temperature of the environment [40,43]. We assume that the initial state of the first membrane is m ¼1 and the second membrane is in vacuum state, n ¼0. Generation of mechanical Fock state is very hard and challenging. Let us postpone the discussion to final section. Then, the external weak driving fields are employed to system with the frequencies ωL1 and ωL2 such that the effective Hamiltonian becomes Vsm = =Ωη12 (b1† 2 + b12 ). Hence, the squeezed number state is prepared in the first membrane as |ψ (t )〉 = S (ξ1)|1〉1|0〉2. The reduced density matrix of the first membrane is computed and the timeevolution of WF are plotted in Fig. 3(a), (b), (c) and (d) for different values of γm t = 0.1, 0.2, 0.3 and 0.4, respectively. Here, we assume that nm = 0 and the squeezed parameter r ¼ 0.3. There is some negative region of the WF in the phase-space which is evidence of non-Gaussianity of the state. Also, the absolute value of the
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
m
decreases as nm or r increases for a given time. Since squeezed parameter is related to time r = Ωtη2 (assuming η1 = η2), therefore the negativity of WF is dependent on both the parameters, γm t and r. The results reveal that if γm t ≈
1 2
ln
2nm + 2 , 2nm + 1
there exists a limit of
squeezed parameter which leads to obtain the minimum value of WF where r ≈ 0.7 − 1. Considering the upper bound, r ≈ 1 means Ω that (γm t )( γ η2) ≈ 1. By substituting the threshold value ofγmt we m
have:
2 Ω 2 η ≈ 2nm + 2 γm ln 2nm + 1
(25)
W(q1,q2,0,0)
0.1 0.05 0 −0.05 −0.1 −0.15
4 2 0
q2
−2 −4
−4
−3
−2
−1
0
1
2
3
4
q1
Fig. 4. The WFs for (m = 0, n = 1) are plotted in phase space (q1, q2, 0, 0) with n = 1, r ¼0.3 and γm t = 0.05. WF is negative and two mode squeezed state squeezed state is non-Gaussian.
0.4
W(q1,q2,0,0)
negative minimum of the WF decreases as γm t increases. The results show that the WF is always positive for γm t > 0.35 when nm = 0. Also, at long times when γm t → ∞, the non-Gaussian squeezed state of the first membrane turns into the Gaussian state 2 2 2 and the WF becomes π e−2 (q + p ) which corresponds to a vacuum state [40]. In the following, we consider the two mode squeezed number state. We assume that the initial conditions of two membranes are Fock state, |m〉1|n〉2. Two weak coherent laser modes are employed to system such that the effective Hamiltonian and the generated state become Vtm = =Ωσ+η1η2 (b1† b2† + b1b2 ) and S (ξtm )|m〉1|n〉2, respectively. The reduced density matrix of two membranes is applied according to Eq. (23). Also, according to Eq. (10), under the Lamb–Dicke approximation, the average of phonon number of membranes is low. Figs. 4 and 5 demonstrate WF for two mode squeezed state (entangled of two membranes) in the Lamb–Dicke approximation. When m ¼n ¼0 corresponds to the two mode squeezed vacuum state which is Gaussian state [41,40]. In Figs. 4 and 5, the phasespace Wigner distributions are illustrated for different parameter values m and n with a fixed values of p1 and p2. In Fig. 4, a negative region is shown in the WF plot and phase space (q1, q2, 0, 0) where (m = 0, n = 1) with nm = 1, γm t = 0.05 and r ¼0.3. Therefore, a nonGaussian entangled state is created by this special initial condition. In Fig. 5 the initial condition is considered where (m = 1, n = 1) with nm = 1, γm t = 0.05 and r¼ 0.3 in phase space (q1, q2, 0, 0). The results indicate that unexpectedly, WF is positive anyway and a non-Gaussian state is not created. Then, in the two mode squeezed number state in the optomechanical system, the initial states of membranes have an important role. On the other hand, if γm t → ∞ the two mode system reduces to two thermal state which is independent of initial conditions and depicts a Gaussian distribution. Actually there exists a threshold value for γmt in creating non-Gaussian two-mode squeezed number state. The WF is always positive if the γmt exceeds an upper 1 2n + 2 bound, γm t > 2 ln 2nm + 1 [40,41]. The partial negativity of the WF
59
0.3 0.2 0.1 0 −0.1 4 2 0
q2
−2 −4
−4
−3
−1
−2
0
1
2
3
4
q1
Fig. 5. The WFs for (m = 1, n = 1) are illustrated in phase space (q1, q2, 0, 0) with n = 1, r ¼0.3 and γm t = 0.05. In this case, WF is negative anyway and two mode motional squeezed state is Gaussian.
Notice that the Hamiltonian of mechanical single and two mode squeezed states have been presented with these limitations: under the Lamb–Dicke approximation η < 1, weak driving field γm < Ω⪡γc , photon blockade regime ωm, g > γc , γm and big photon–photon interaction,
g2 ωm
> γc . For example it is assumed that for a membrane
system, η = 0.1, γc = 0.3 MHz , γm = 0.1 Hz [44] and Ω = 1 KHz, consequently, only in the limit of nm < 25 a non-Gaussian state is created. Hence, the cooling methods are necessary conditions. The proposed method to generate non-Gaussian squeezed state is debatable as experimental feasibility. Let us discuss about it in the next section.
4. Discussion on experimental feasibility In this paper, the main goal is the generation of non-Gaussian motional squeezed number state by using sideband excitations and photon blockade effect. Experimental realization of photon blockade effect involves considerable details [45–47]: (1) The single photon strong coupling regime is g > γc , γm . (2) Resolved sideband regime that is ωm > γc . 2 (3) The nonlinear photon–photon interaction strength, g must ωm g2 be greater than decay rate of cavity, ω > γc . m
So far, in most experiments, g is much smaller than γc, only using ultracold atom in optical resonators or in optomechanical crystals [48,49], one can achieve the approach of single photon strong coupling regime. However one is typically far from the resolved sideband regime. Therefore, these systems are not suitable for the scheme presented here which is also based on the selection of resonant processes. Xuereb et al. displayed in Refs. [50,51], in optomechanical system with more massive and macroscopic systems like membranes, the large optomechanical coupling could be attained by using interference phenomenon and the collective interaction of many optical elements within an optical cavity. Then, the authors studied an optomechanical system of two membranes placed inside a cavity and provided the special conditions to create the large single photon optomechanical coupling. In the limit of highly reflective membranes very large single photon optomechanical coupling is achieved when the two membranes are placed very close to a resonance of the inner cavity formed by the two membranes [52]. In this paper, we require single photon strong coupling and apply these assumptions in two membrane optomechanical system. In Section 3, the main goal is the generation of non-Gaussian mechanical squeezed number state. Therefore, the Fock state has been used as the initial state of membranes. In Ref. [13], the generation of mechanical Fock state is investigated
60
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
by employing the photon blockade effect. The authors display a method to generate target states by using a series of classical pulses with desired frequencies, phases and durations. Our scheme is based on the photon blockade. Thus, the initial Fock state is prepared before preparation of conditions to Hamiltonian of mechanical squeezed state. Of course, with regard to the details which are mentioned in the previous paragraph about experimental conditions, it is difficult to achieve the mechanical Fock state experimentally. Generally, the generation of mechanical coherent state is easier than the Fock state. Thus, it is best to get the coherent state as initial states of membranes. Then, we should investigate the features of non-Gaussian squeezed coherent state. Recently, the squeezed coherent state has been discussed as a non-Gaussian regime [53]. In this scheme, in the limit of small squeezing, r < 1, the squeezed operator can be expanded to:
S (ξ ) = 1 +
ξ †2 ξ ⁎ 2 b − b 2 2
(26)
Therefore, in this approach, as a suggestion, the initial coherent state can be selected which is suitable experimentally. Eq. (26) can be implemented by using two ideas: 1. In the limit of small time, weak driving and Lamb–Dicke approximation we have, r = Ωtη2 < 1. Then we can use Eq. (26). 2. We try to extract S (ξ ) from Eq. (12). Here, one can use the quantum interference effects such as the optomechanically induced transparency (OMIT). In this case, the motion of the mirror evolves into a dark state. Compared with [7], where tuning the three frequencies of the driving field to be resonant to the carrier, red-sideband and bluesideband transitions, in our model, third laser frequency couples to carrier transition. We define, Δc 3 = ωL3 − ω and in the resonance condition becomes ωL3 = ω . For example, the dark state of first membrane when the state of second membrane is vacuum ∞ becomes |D〉 = ∑m = 0 N |g 〉|m〉1|0〉2 (N is a normalization constant) [7]. Then, we obtain Vsm = Ω (1 + ηb1† 2 + ηb12 ). Forasmuch as Ω is small so the approximation Ω ≈ 1 Hz is valid. In this case, when the average number of phonon is small the non-Gaussian squeezed coherent state of first membrane is created.
5. Conclusions In this paper, we focus on an optomechanical system with a cavity containing two non-absorptive membranes. In the strong single photon coupling, by using the similarity between the Hamiltonian of optomechanical system in photon blockade regime and Hamiltonian of ion trapping and also, by adjusting the input frequency, the motional non-Gaussian squeezed number state is generated. Non-Gaussian state is proved by negativity of Wigner function. It is found out that under the Lamb–Dicke approximation and low temperature, the single mode motional non-Gaussian squeezed state is prepared while the generation of two mode motional non-Gaussian squeezed state depends on the initial state of the membranes. The results show that in low temperature, if one of the membranes is in the ground state and the other in single phonon state, the two mode squeezed state is non-Gaussian. In the conditions that the membranes are in the single phonon state, even in the low temperature, two mode squeezed state is not non-Gaussian. Although it is believed that in the strong nonlinearity the non-Gaussian is generated, but in the two mode squeezed number state, the initial state of membranes has an important role.
References [1] T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A.V. Gorshkov, T. Pohl, M.D. Lukin, V. Vuletić, Quantum nonlinear optics with single photons enabled by strongly interacting atoms, Nature 488 (7409) (2012) 57–60. [2] N. Matsuda, R. Shimizu, Y. Mitsumori, H. Kosaka, K. Edamatsu, Observation of optical-fibre Kerr nonlinearity at the single-photon level, Nat. Photonics 3 (2) (2009) 95–98. [3] A. Imamolu, H. Schmidt, G. Woods, M. Deutsch, Strongly interacting photons in a nonlinear cavity, Phys. Rev. Lett. 79 (8) (1997) 1467. [4] G. Milburn, M. Woolley, An introduction to quantum optomechanics, Acta Phys. Slovaca 61 (5) (2011) 483–601. [5] D. Vitali, S. Mancini, L. Ribichini, P. Tombesi, Macroscopic mechanical oscillators at the quantum limit through optomechanical cooling, JOSA B 20 (5) (2003) 1054–1065. [6] J.-Q. Liao, C. Law, L.-M. Kuang, F. Nori, Enhancement of mechanical effects of single photons in modulated two-mode optomechanics, Phys. Rev. A 92 (1) (2015) 013822. [7] W.-j. Gu, G.-x. Li, S.-p. Wu, Y.-p. Yang, Generation of non-classical states of mirror motion in the single-photon strong-coupling regime, Opt. Express 22 (15) (2014) 18254–18267. [8] M. Woolley, A. Clerk, Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir, Phys. Rev. A 89 (6) (2014) 063805. [9] M. Schmidt, M. Ludwig, F. Marquardt, Optomechanical circuits for nanomechanical continuous variable quantum state processing, New J. Phys. 14 (12) (2012) 125005. [10] H. Tan, G. Li, P. Meystre, Dissipation-driven two-mode mechanical squeezed states in optomechanical systems, Phys. Rev. A 87 (2013) 033829. [11] Y.-D. Wang, A.A. Clerk, Using dark modes for high-fidelity optomechanical quantum state transfer, New J. Phys. 14 (10) (2012) 105010. [12] X.-W. Xu, Y.-J. Zhao, Y.-x. Liu, Entangled-state engineering of vibrational modes in a multimembrane optomechanical system, Phys. Rev. A 88 (2) (2013) 022325. [13] X.-W. Xu, H. Wang, J. Zhang, Y.-x. Liu, Engineering of nonclassical motional states in optomechanical systems, Phys. Rev. A 88 (6) (2013) 063819. [14] S. Bose, K. Jacobs, P. Knight, Preparation of nonclassical states in cavities with a moving mirror, Phys. Rev. A 56 (5) (1997) 4175. [15] U. Akram, W.P. Bowen, G.J. Milburn, Entangled mechanical cat states via conditional single photon optomechanics, New J. Phys. 15 (9) (2013) 093007. [16] S. Barzanjeh, M. Naderi, M. Soltanolkotabi, Generation of motional nonlinear coherent states and their superpositions via an intensity-dependent coupling of a cavity field to a micromechanical membrane, J. Phys. B: At. Mol. Opt. Phys.: At. 44 (10) (2011) 105504. [17] K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E.S. Polzik, P. Zoller, Cavityassisted squeezing of a mechanical oscillator, Phys. Rev. A 79 (6) (2009) 063819. [18] H. Tan, G. Li, P. Meystre, Dissipation-driven two-mode mechanical squeezed states in optomechanical systems, Phys. Rev. A 87 (3) (2013) 033829. [19] D.V. Muhammad Asjad, Reservoir engineering of a mechanical resonator: generating a macroscopic superposition state and monitoring its decoherence, J. Phys. B: At. Mol. Opt. Phys. 47 (4) (2014) 045502. [20] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. Büchler, P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4 (11) (2008) 878–883. [21] S.D. Bartlett, B.C. Sanders, S.L. Braunstein, K. Nemoto, Efficient classical simulation of continuous variable quantum information processes, Phys. Rev. Lett. 88 (2002) 097904. [22] S. Boixo, A. Datta, M.J. Davis, S.T. Flammia, A. Shaji, C.M. Caves, Quantum metrology: dynamics versus entanglement, Phys. Rev. Lett. 101 (4) (2008) 040403. [23] H. Nha, H. Carmichael, Proposed test of quantum nonlocality for continuous variables, Phys. Rev. Lett. 93 (2) (2004) 020401. [24] M. Stobińska, A.S. Villar, G. Leuchs, Generation of Kerr non-Gaussian motional states of trapped ions, EPL (Europhys. Lett.) 94 (5) (2011) 54002. [25] F. Khalili, S. Danilishin, H. Miao, H. Müller-Ebhardt, H. Yang, Y. Chen, Preparing a mechanical oscillator in non-Gaussian quantum states, Phys. Rev. Lett. 105 (7) (2010) 070403. [26] M. Poot, K.Y. Fong, H.X. Tang, Classical non-Gaussian state preparation through squeezing in an optoelectromechanical resonator, Phys. Rev. A 90 (6) (2014) 063809. [27] A. Pontin, M. Bonaldi, A. Borrielli, F. Cataliotti, F. Marino, G. Prodi, E. Serra, F. Marin, Squeezing a thermal mechanical oscillator by stabilized parametric effect on the optical spring, Phys. Rev. Lett. 112 (2) (2014) 023601. [28] A. Szorkovszky, G.A. Brawley, A.C. Doherty, W.P. Bowen, Strong thermomechanical squeezing via weak measurement, Phys. Rev. Lett. 110 (18) (2013) 184301. [29] Z. Mahmoudi, S. Shakeri, O. Hamidi, M. Zandi, A. Bahrampour, Generation of motional entangled coherent state in an optomechanical system in the single photon strong coupling regime 62 (19) (2015) 1685–169162. [30] T. Tyc, N. Korolkova, Highly non-Gaussian states created via cross-Kerr nonlinearity, New J. Phys. 10 (2) (2008) 023041. [31] M. Yanagisawa, Non-Gaussian state generation from linear elements via feedback, Phys. Rev. Lett. 103 (20) (2009) 203601. [32] M. Stobińska, G. Milburn, K. Wódkiewicz, Wigner function evolution of quantum states in the presence of self-Kerr interaction, Phys. Rev. A 78 (1)
S. Shakeri et al. / Optics Communications 370 (2016) 55–61
(2013) 013810. [33] X.-Y. Lü, W.-M. Zhang, S. Ashhab, Y. Wu, F. Nori, Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems, Sci. Rep. 3 (2013). [34] y. Leibfried, R. Blatt, C. Monroe, D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75 (1) (2003) 281. [35] S.-C. Gou, J. Steinbach, P. Knight, Generation of mesoscopic superpositions of two squeezed states of motion for a trapped ion, Phys. Rev. A 55 (5) (1997) 3719. [36] D. Vitali, S. Gigan, A. Ferreira, H. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, M. Aspelmeyer, Optomechanical entanglement between a movable mirror and a cavity field, Phys. Rev. Lett. 98 (3) (2007) 030405. [37] R. Hudson, When is the Wigner quasi-probability density non-negative? Rep. Math. Phys. 6 (2) (1974) 249–252. [38] A. Kenfack, K. Źyczkowski, Negativity of the Wigner function as an indicator of non-classicality, J. Opt. B: Quantum Semiclass. Opt. 6 (10) (2004) 396. [39] M.G. Genoni, M.L. Palma, T. Tufarelli, S. Olivares, M. Kim, M.G. Paris, Detecting quantum non-Gaussianity via the Wigner function, Phys. Rev. A 87 (6) (2013) 062104. [40] H. Li-Yun, F. Hong-Yi, Two-variable hermite polynomial excitation of twomode squeezed vacuum state as squeezed two-mode number state, Commun. Theor. Phys. 50 (4) (2008) 965. [41] L.-j. Xu, G.-b. Tan, S.-j. Ma, Nonclassicality and decoherence of two-mode photon-added squeezed vacuum, Opt. Commun. 284 (13) (2011) 3335–3344. [42] H. Carmichael, An Open Systems Approach to Quantum Optics: Lectures Presented at the Université Libre de Bruxelles, October 28 to November 4, 1991, vol. 18, Springer Science & Business Media, 2009.
61
[43] A. Biswas, G.S. Agarwal, Nonclassicality and decoherence of photon-subtracted squeezed states, Phys. Rev. A 75 (3) (2007) 032104. [44] J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, J. Harris, Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane, Nature 452 (7183) (2008) 72–75. [45] P. Rabl, Photon blockade effect in optomechanical systems, Phys. Rev. Lett. 107 (6) (2011) 063601. [46] J.-Q. Liao, F. Nori, et al., Photon blockade in quadratically coupled optomechanical systems, Phys. Rev. A 88 (2) (2013) 023853. [47] A. Miranowicz, M. Paprzycka, Y.-x. Liu, J. Bajer, F. Nori, Two-photon and threephoton blockades in driven nonlinear systems, Phys. Rev. A 87 (2) (2013) 023809. [48] M. Asjad, P. Tombesi, D. Vitali, Quantum phase gate for optical qubits with cavity quantum optomechanics, Opt. Express 23 (6) (2015) 7786–7794. [49] D.W. Brooks, T. Botter, S. Schreppler, T.P. Purdy, N. Brahms, D.M. StamperKurn, Non-classical light generated by quantum-noise-driven cavity optomechanics, Nature 488 (7412) (2012) 476–480. [50] A. Xuereb, C. Genes, A. Dantan, Strong coupling and long-range collective interactions in optomechanical arrays, Phys. Rev. Lett. 109 (22) (2012) 223601. [51] A. Xuereb, C. Genes, A. Dantan, Collectively enhanced optomechanical coupling in periodic arrays of scatterers, Phys. Rev. A 88 (5) (2013) 053803. [52] J. Li, A. Xuereb, N. Malossi, D. Vitali, Cavity mode frequencies and large optomechanical coupling in two-membrane cavity optomechanics, arXiv preprint arxiv:1512.07536. [53] K. Jeong, J. Kim, S.-Y. Lee, Gaussian private quantum channel with squeezed coherent states, Sci. Rep. 5 (13974) (2015).