Large and robust mechanical squeezing of optomechanical systems in a highly unresolved sideband regime via Duffing nonlinearity and intracavity squeezed light
aa r X i v : . [ qu a n t - ph ] J u l Large and robust mechanical squeezing ofoptomechanical systems in a highly unresolvedsideband regime via Duffing nonlinearity andintracavity squeezed light J IAN -S ONG Z HANG AND A I -X I C HEN Department of Applied Physics, East China Jiaotong University, Nanchang 330013, People’s Republicof China Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, People’s Republic of China Institute for Quantum Computing, University of Waterloo, Ontario N2L3G1, Canada * [email protected] Abstract:
We propose a scheme to generate strong and robust mechanical squeezing in anoptomechanical system in the highly unresolved sideband (HURSB) regime with the help ofthe Duffing nonlinearity and intracavity squeezed light. The system is formed by a standardoptomechanical system with the Duffing nonlinearity (mechanical nonlinearity) and a second-order nonlinear medium (optical nonlinearity). In the resolved sideband regime, the second-order nonlinear medium may play a destructive role in the generation of mechanical squeezing.However, it can significantly increase the mechanical squeezing (larger than 3dB) in the HURSBregime. Finally, we show the mechanical squeezing is robust against thermal fluctuations of themechanical resonator. The generation of large and robust mechanical squeezing in the HURSBregime is a combined effect of the mechanical and optical nonlinearities. © 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optomechanical systems has received a lot of attentions due to the wide range of applications suchas highly sensitive measurement of tiny displacement and quantum information processing [1–7].In the highly sensitive measurement of tiny displacement, quantum squeezing of mechanicalmode is indispensible. In principle, quantum squeezing can be accomplished by the parametricinteraction of a quantum system [8]. However, quantum squeezing in this scheme can not belarger than 3dB since a quantum system becomes unstable if the quantum squeezing is largerthan 3dB as pointed out by Milburn and Walls [9].In recent years, many schemes have been proposed to generate strong mechanical squeezingbeyond the 3dB limit including continuous weak measurement and feedback [10–13], squeezedlight [14, 15], quantum-reservoir engineering [16–24], strong intrinsic nonlinearity [25, 26],and frequency modulation [27]. For instance, large steady-state mechanical squeezing can beachieved by applying two driving lasers to a cavity in an optomechanical system [22]. Inthis scheme, the power of the red-detuned driving field should be larger than that of the blue-detuned driving field. This scheme was realized experimentally in 2015 [23]. Very recently,the authors of [24] have shown that larger mechanical squeezing can also be achieved withonly one periodically amplitude-modulated external driving field. The Duffing nonlinearity ofthe mechanical mode can be used to generate strong mechanical squeezing beyond the 3dBlimit [26]. In addition, the mechanical squeezing is robust against the thermal fluctuations ofthe mechanical resonator with the help of the Duffing nonlinearity.It is worth noting that the above schemes [10–26] are not valid to realize larger mechanicalsqueezing beyond the 3dB limit in the HURSB regime with κ ≫ ω m . Here, κ is decay rateof the cavity and ω m is the frequency of the mechanical resonator. In order to generate largeechanical squeezing beyond the resolved sideband regime, the authors of [27] suggested touse frequency modulation acting on both the cavity field and mechanical resonator. Theyhave shown that mechanical squeezing beyond 3dB can be achieved in the presence of frequencymodulation beyond the resolved sideband and weak-coupling limits. It was shown that the strongmechanical squeezing beyond 3dB in the unresolved sideband regime ( κ ≈ ω m ) can also beachieved by adding two auxiliary cavities since the unwanted counter-rotating terms could besuppressed significantly with the help of quantum interference from the auxiliary cavities [28].However, the decay rates of the auxiliary cavities must be much smaller than the frequencyof the mechanical resonator in the above scheme. Later, we proposed a scheme to generatelarge mechanical squeezing beyond 3dB in the HURSB regime by adding two two-level atomicensembles and two driving lasers with different amplitudes [29]. Very recently, it was shownthat the quantum ground-state cooling of mechanical resonator in an optomechanical system canbe accomplished using intracavity squeezed light produced by a second-order nonlinear mediumin the optomechanical system [30–32].In the present work, we propose a scheme to generate large and robust mechanical squeezingin the HURSB regime via the Duffing nonlinearity of the mechanical mode and a second-ordernonlinear medium in the cavity. The mechanical squeezing of the mechanical resonator can belarger than 3dB and is robust against the thermal fluctuations of the mechanical resonator. Thisis a combined effect of nonlinearity-induced parametric amplification (Duffing nonlinearity) andquantum ground-state cooling of the optomechanical system (intracavity squeezed light). On theone hand, in the resolved sideband regime, the second-order nonlinear medium may decrease themechanical squeezing. On the other hand, the second-order nonlinear medium can significantlyincrease the mechanical squeezing in the HURSB regime for realistic parameters.
2. Model and Hamiltonian
In the present work, we consider an optomechanical system formed by two mirrors. One mirror isfixed and partially transmitting. The other mirror is movable and perfectly reflecting. In addition,a second-order nonlinear medium χ ( ) is put into the Fabry-Perot cavity. The fundamental modeand second-order optical mode are represented by a and a with frequencies ω c and 2 ω c . Thedecay rates of the two optical modes are κ and κ . The movable mirror (mechanical oscillator)is denoted by b with frequency ω m and decay rate γ m . In addition, two driving fields withamplitudes ε and ε are applied to fundamental and second-order modes. The Hamiltonian ofthe present model is (we set ~ = H = H + H dr + H I + H D + H N , (1) H = ω c a † a + ω c a † a + ω m b † b , (2) H dr = i ( ε e − i ω L t a † + ε e − i ω L t a † − H . c . ) , (3) H I = − g a † a ( b † + b ) − g a † a ( b † + b ) , (4) H D = η ( b † + b ) , (5) H N = i χ ( a † a − a a † ) , (6)where H is the free Hamiltonian of the whole system. H dr is the Hamiltonian for drivingfields applied to the fundamental and second-order modes with frequencies ω L and 2 ω L . H I is the interaction between the optical and mechanical modes. The coupling strength betweenthe mechanical mode and fundamental mode (second-order mode) is denoted by g ( g ). TheDuffing nonlinearity of the mechanical mode is represented by H D . It was pointed out that anonlinear amplitude of η = − ω m can be achieved by coupling the mechanical mode to an ig. 1. Schematic representation of our model. The movable mirror is perfectly re-flecting. However, the fixed mirror is partially transmitting. A second-order nonlinearmedium denoted by χ ( ) is put into the cavity. The fundamental and second-orderoptical modes with frequencies ω c and 2 ω c are denoted by a and a . Here, κ and κ are the decay rates of the fundamental and second-order optical modes, respectively.The mechanical resonator (movable mirror) with frequency ω m and decay rate γ m isdenoted by b . ancilla system [26]. The Hamiltonian of a second-order nonlinear medium is denoted by H N with χ being the interaction between the fundamental and second-order optical modes [30–32].In a rotating frame defined by the unitary transformation U ( t ) = exp {− i ω L t ( a † a + a † a )} ,we obtain the Hamiltonian as follows H = U † HU − iU † Û U = ∆ c a † a + ∆ c a † a + ω m b † b + i ( ε a † + ε a † − ε a − ε a )−( g a † a + g a † a )( b † + b ) + η ( b † + b ) + i χ ( a † a − a a † ) , (7)with ∆ c = ω c − ω L .
3. Quantum Langevin equations
First, we linearize the above Hamiltonian by employing the following displacement transforma-tions a → α + δ a , a → α + δ a , and b → β + δ b . The quantum Langevin equationsre Û α = −( i ∆ c + κ ) α + χ α ∗ α + ε , Û α = −( i ∆ ′ c + κ ) α − χ α + ε , Û β = −( i ω m + γ m ) β − i η ( β + β ) + i g | α | + i g | α | ,δ Û a = −( i ∆ c + κ ) δ a + iG ( δ b † + δ b ) + χ α δ a † + χ α ∗ δ a + √ κ a , in , (8) δ Û a = −( i ∆ ′ c + κ ) δ a + iG ( δ b † + δ b ) − χ α δ a + √ κ a , in ,δ Û b = −( i ω m + γ m ) δ b − i Λ ( δ b † + δ b ) + i ( G δ a † + G ∗ δ a ) + i ( G δ a † + G ∗ δ a ) + √ γ m b in , where ∆ c = ∆ c − g ( β ∗ + β ) , ∆ ′ c = ∆ c − g ( β ∗ + β ) , Λ = η ( β + ) , and G , = g , α , .In the limit of large κ , the fluctuations of mode a can be neglected and the adiabaticapproximation is valid [31]. Thus, the quantum Langevin equations can be reduced to δ Û a = −( i ∆ c + κ ) δ a + iG ( δ b † + δ b ) + χδ a † + √ κ a , in δ Û b = −( i ω m + γ m ) δ b − i Λ ( δ b † + δ b ) + iG ( δ a † + δ a ) + √ γ m b in , (9)where χ = χ α = | χ | e i φ . Without loss of generality, G has been assumed to be real.Now, we define the following quadrature operators X O = a , b = ( δ O † + δ O )/√ Y O = a , b = i ( δ O † − δ O )/√
2. The noise quadrature operators are defined as X inO = a , b = ( O † in + O in )/√ Y inO = a , b = i ( O † in − O in )/√
2. From the above quantum Langevin equations, we obtain Û® f = A ® f + ® n , (10)where ® f = ( X a , Y a , X b , Y b ) T and ® n = (√ κ X ina , √ κ Y ina , √ γ m X inb , √ γ m Y inb ) T , (11) A = © « | χ | cos 2 φ − κ | χ | sin 2 φ + ∆ c | χ | sin 2 φ − ∆ c −| χ | cos 2 φ − κ G
00 0 − γ m ω m G − ω m − Λ − γ m ª®®®®®®®¬ . (12)Note that the dynamics of the present system described by Eq. (10) can be completelydescribed by a 4 × V with V jk = h f j f k + f k f j i/
2. Using the definitions of V , ® f , and the above equations, we obtain the evolution of the covariance matrix V as follows Û V = AV + V A T + D , (13)where D is the noise correlation defined by D = dia g [ κ , κ , γ m ( n th + ) , γ m ( n th + )] . Here, n th is the mean phonon number of the mechanical resonator. / ω m (a) G = 0.1 ω m Λ / ω m (b) G = 0.8 ω m φ / π Λ / ω m (c) G = 1.6 ω m Fig. 2. Stability of the present model. Here, χ = χ α = | χ | e i φ and Λ = η ( β + ) are directly related to the second-order nonlinear medium and the Duffing nonlinearityof the mechanical mode. We consider the HURSB case with κ ≫ ω m . The stableand unstable regimes are represented by the red and blue regions, respectively. Otherparameter values are γ m / ω m = − , ∆ c / ω m = κ / ω m = n th = | χ | = p ( κ / ) + ( ∆ c − ω m ) [31].
4. Large and robust mechanical squeezing beyond resolved sideband regime
We first investigate influence of the Duffing nonlinearity and optomechanical coupling strengthon the the stability of the present system. The Duffing nonlinearity and optomechanical couplingconstant are related to parameters Λ = η ( β + ) and G = g α , respectively. It is well knownthat the system described by Eq. (13) is stable only if all the real parts of the eigenvalues of thematrix A are negative according to the Routh-Hurwitz criterion [33].From Fig.2, we find the system is unstable for 0 . < φ < .
97. Comparing three panels ofthis figure, one can see that the areas of the unstable regions could increase with the increaseof the coupling constant G . For example, the system is stable with φ = . Λ = ω m and G = . ω m as one can find in the upper panel of Fig.2. However, if we increase theeffective optomechanical coupling strength G from 0 . ω m to 1 . ω m the system is not stablewith Λ = ω m and φ = .
2. Fortunately, the system can be stable if the Duffing nonlinearity Λ is increased to about 7 . ω m (see the lower panel of this figure). Thus, the present systemcould be stable even for the strong and deep-strong coupling cases in the presence of the Duffing / ω m (a) κ = 0.1 ω m with | χ |=0 Λ / ω m (b) κ = 10 ω m with | χ |=0 log ( ∆ c / ω m ) Λ / ω m (c) κ = 100 ω m with | χ |=0 κ = 0.1 ω m with |x| >0 κ = 10 ω m with | χ | > 0 log ( ∆ c / ω m ) (f) κ = 100 ω m with | χ | > 0 Fig. 3. Mechanical squeezing of the mechanical resonator (in units of dB) versus Λ and ∆ c for different values of κ and | χ | . The white solid lines correspond to mechanicalsqueezing at 3dB. Other parameter values are γ m / ω m = − , n th = G / ω m = . φ = . π , and | χ | = p ( κ / ) + ( ∆ c − ω m ) for Figs.2(d)-2(f). nonlinearity which is consistent with the results of [26]. In Fig.3, we plot the mechanical squeezing of the mechanical resonator (in units of dB) versus Λ and ∆ c for different values of κ and | χ | . In Figs.3(a)-3(c), the second-order nonlinear mediumis not put into the cavity. From Fig.3(a), one can see the mechanical squeezing can be largerthan 3dB if the Duffing nonlinearity is strong enough in the resolved sideband regime in theabsence of second-order nonlinear medium χ ( ) . This is consistent with the results of [26]. Themechanical squeezing decreases with the increase of the decay rate of cavity κ . For instance,if the decay rate of the cavity is much larger than the frequency of the mechanical resonator( κ = ω m ), then the mechanical squeezing of the mechanical resonator could not be largerthan 3dB (see Fig.3(c)). If the second-order nonlinear medium is put into the cavity, then themechanical squeezing overcomes the 3dB limit even in the HURSB regime as one can see fromFigs.3(d)-3(f).On the one hand, in the resolved sideband regime, the second-order nonlinear medium mayplay a destructive role in the generation of mechanical squeezing of the mechanical resonator(see Figs.3(a) and 3(d)). On the other hand, the situation is very different for the sideband χ |/ ω m Λ / ω m Fig. 4. Mechanical squeezing of the mechanical resonator (in units of dB) versus Λ and | χ | . The white solid line corresponds to mechanical squeezing at 3dB. Otherparameter values are γ m / ω m = − , n th = G / ω m = . ∆ c / ω m = κ / ω m = φ = . π . unresolved regime. In the absence of the second-order nonlinear medium with | χ | =
0, themechanical squeezing depends heavily on the decay rate κ , i.e., it decreases with the increase ofthe decay rate κ significantly (Figs.3(b)-3(c)). However, if we put the second-order nonlinearmedium into the cavity, the mechanical squeezing is insensitive to the decay rate κ as one canfind in Figs.3(e)-3(f). In Fig.4, we plot the mechanical squeezing of the mechanical resonator versus Λ and | χ | with γ m / ω m = − , n th = G = . ω m , ∆ c = ω m , κ = ω m , and φ = . π . If thereis no Duffing nonlinearity or second-order nonlinear medium, large mechanical squeezing cannot be achieved. However, if the second-order nonlinear medium and Duffing nonlinearity arechosen appropriately the mechanical squeezing can overcome the 3dB limit even in the HURSBregime and in the presence of thermal fluctuation of the mechanical mode with n th = χ ( ) in thecavity.In order to show the influence of the thermal fluctuations on the mechanical squeezing more S d B (a) G = 0.5 ω m , | χ | = 0 (d) G = 0.5 ω m , | χ | > 0 S d B (b) G = ω m , | χ | = 0 (e) G = ω m , | χ | > 0 log ( ∆ c / ω m ) S d B (c) G = 2.5 ω m , | χ | = 0 (f) G = 2.5 ω m , | χ | > 0 log ( ∆ c / ω m ) Fig. 5. Mechanical squeezing of the mechanical resonator (in units of dB) versus ∆ c for different values of G and | χ | . The black solid lines correspond to mechanicalsqueezing at 3dB. The red, green, blue, and cyan lines correspond to n th = n th = n th = n th = γ m / ω m = − , Λ / ω m = φ = . π , and | χ | = p ( κ / ) + ( ∆ c − ω m ) for Figs.3(d)-3(f). clearly, we plot the mechanical squeezing (in units of dB) as functions of ∆ c for different valuesof G and | χ | with n th = n th =
500 (green lines), n th = n th = | χ | = G = . ω m as one can find from Figs.5(a)-5(c). However, if the second-ordernonlinear medium χ ( ) is considered, the situation is very different. In the case of G = . ω m and | χ | > n th = n th =
500 while the 3dB limit can not been overcome for n th = n th = G , then the 3dB limitcan be surpassed even in the high temperature n th =
5. Conclusion
In the present work, we have proposed an efficient scheme to generate large and robust mechanicalsqueezing beyond the 3dB limit in the HURSB regime for realistic parameters. The system wasformed by a standard optomechanical system with a second-order nonlinear medium χ ( ) in acavity and the Duffing nonlinearity of the mechanical mode. In fact, a strong Duffing nonlinearityould be achieved by coupling the mechanical mode to an ancilla system as point out in [26].There are two modes in the cavity. One is the fundamental mode. The other is the second-order mode. We assumed the decay rate of the second-order mode is very large. In the adiabaticapproximation, we derived effective quantum Langevin equations of the model. The influence ofthe second-order nonlinear medium χ ( ) and Duffing nonlinearity on the mechanical squeezingwas discussed carefully.In the absence of the second-order nonlinear medium χ ( ) , the mechanical squeezing S dB decreases with the increase of the decay rate of the cavity significantly and it could be negativefor HURSB regime. However, if we put the second-order nonlinear medium into the cavity, themechanical squeezing is insensitive with the decay rate of the cavity and S dB can be larger than3dB even when the decay rate of the cavity is much larger than the frequency of the mechanicalresonator.Then, we discussed the influence of the thermal fluctuations of the mechanical mode on themechanical squeezing in the HURSB regime. On the one hand, the mechanical squeezing cannot be larger than 3dB without the second-order nonlinear medium if the thermal fluctuationsof the mechanical mode is considered. On the other hand, the mechanical squeezing could belarger than 3dB even for high temperature when the second-order nonlinear medium is put intothe cavity. Thus, we have shown that large and robust mechanical squeezing beyond the 3dBlimit can be generated in the HURSB regime. Funding
This research was supported by Zhejiang Provincial Natural Science Foundation of Chinaunder Grant No. LZ20A040002; National Natural Science Foundation of China (11047115,11365009 and 11065007); Scientific Research Foundation of Jiangxi (20122BAB212008 and20151BAB202020.)
Disclosures
The authors declare no conflicts of interest.
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