PHYSICAL REVIEW A 90, 043839 (2014)
Formation and manipulation of optomechanical chaos via a bichromatic driving Jinyong Ma,1,* Cai You,1 Liu-Gang Si,1 Hao Xiong,1 Jiahua Li,1,2 Xiaoxue Yang,1 and Ying Wu1,† 1
Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China 2 MOE Key Laboratory of Fundamental Quantities Measurement, Wuhan 430074, People’s Republic of China (Received 7 April 2014; published 20 October 2014) We propose a scheme to efficiently manipulate optomechanical systems into and out of chaotic regimes. Here the optical system is coherently driven by a continuous-wave bichromatic laser field consisting of a pump field and a probe field, where the beat frequency of the bichromatic components plays an important role in controlling the appearance of chaotic motion and the corresponding chaotic dynamics. With state-of-the-art experimental parameters, we find that a broad chaos-absent window with sharp edges can be formed by properly adjusting the powers of the bichromatic input field. Moreover, the lifetime of the transient chaos and the chaotic degree of the optomechanical system can be well tuned simply by changing the initial phases of the bichromatic input field. This investigation may be useful for harnessing the optomechanical nonlinearity to manipulate rich chaotic dynamics and find applications in chaos-based communication. DOI: 10.1103/PhysRevA.90.043839
PACS number(s): 42.65.Sf, 42.50.Wk, 05.45.Gg
I. INTRODUCTION
The coupling of a movable mirror (mechanical resonator) and a cavity via radiation pressure is achieved in cavity optomechanical systems, leading to numerous momentous topics [1–4]. As this field develops rapidly, some nonlinear [5–8] and nonperturbative effects [9,10], which are the intrinsic property of the optomechanical system, have been observed clearly and studied in-depth. It is demonstrated experimentally [9] that the oscillation of a light field evolves from periodic into chaotic as the optical power of a single input laser increases. Recently, a few works [11–13] have mentioned some physical processes associated with chaotic motion as well. To our knowledge, however, the research about formation and manipulation of optomechanical chaos is still almost in blank (for recent reviews, see, e.g., [14–18]), and further insight into the chaotic regime may open up a new and broad prospective for the properties of the optomechanical nonlinear interactions. Clearly seeking for a way to control the optomechanical chaos is an undoubted challenge from both scientific and technological viewpoints. The purpose of this paper is to explore chaotic dynamics of the optomechanical system in a controlled way by introducing a continuous-wave (cw) bichromatic laser field (a pump field and a probe field) to coherently drive a single-mode cavity [see Fig. 1(a)]. Using state-of-the-art experimental parameters, the proposed setup enables us to extremely lower optical powers for the achievement of the optomechanical chaos, as well as to effectively manipulate the optomechanical chaos simply by modulating the bichromatic laser field. Some nonlinear processes associated with chaos, such as period doubling [19] and aperiodic oscillations, are clearly shown under appropriate optical powers of the bichromatic laser field. More interestingly, we find that the beat frequency of the bichromatic input field can be used to well control the emergence of chaotic motion and the corresponding chaotic
dynamics. At the same time, we open a broad chaos-absent window (CAW) with sharp edges, whose physical process shares some internal similarities with optomechanically induced transparency (OMIT) [20–23]. We emphasize, however, that OMIT is just a linear and perturbative effect while the appearance of CAW belongs to a nonlinear and nonperturbative phenomenon. Therefore, the features of CAW can be tuned by adjusting the power of the pump field and probe field. Since OMIT is an analog of electromagnetically induced transparency (EIT) [24,25], the effect consistent with CAW is likely to be found in an atomic system or the other EIT-like platforms. On account of the importance of OMIT [26,27] and EIT [28,29], the presence of CAW may provide a novel and valuable method for the operation of light. In particular, we uncover two phase-dependent effects on account of the strong dependence of chaotic motion upon the system parameters as follows. (i) For a kind of beat frequency, the oscillation of optical field is chaotic initially, but this chaotic motion cannot live permanently and will evolve into periodical oscillation finally. With regard to this physical process, we find that the lifetime of the chaos can be modulated by changing the initial phases of the bichromatic input laser field. (ii) For another kind of beat frequency, it is shown that the optomechanical chaos can exist permanently, and furthermore the chaotic degree of the optomechanical system, characterized by positive Lyapunov exponent, can also be controlled by changing the phases. In addition, it should be pointed out that chaos-based communication has made remarkable progress in recent years [30,31] and the related work [32] provides a strong argument for wireless communication with chaos. Hence the formation and manipulation of chaos based upon optomechanics in our work, including CAW and phase-dependent chaos, may imply great potential application on chaotic communication. II. MODEL
*
[email protected] †
[email protected] 1050-2947/2014/90(4)/043839(6)
We consider an optomechanical system [Fig. 1(a)] driven by a cw bichromatic input field, which consists of a pump field (initial phase ϕl , central frequency ωl , pump power Pl , 043839-1
©2014 American Physical Society
MA, YOU, SI, XIONG, LI, YANG, AND WU 20 (b)
(a) Out
PHYSICAL REVIEW A 90, 043839 (2014)
Pp =3.60mW
15
Cavity field
period
10 In
Pp = 3.61mW
15 Movable mirror
Fixed mirror
(c)
Pp = 3.64mW
0.05
15
period doubling
10 0
lnS(ω)
p(kg pm/s)
period doubling
10
the investigation of chaos due to the sensitivity for small changes of parameter in this chaotic optomechanics. With optical and mechanical damping processes included, the dynamics of the optomechanical system excited by a cw bichromatic laser field can be described by the HeisenbergLangevin equations of motion [9]. Here we define the mean value of the bosonic annihilation as c = cr + ici (cr and ci are real numbers) for facilitating the discussion related to chaotic property of the system. We therefore give the equations of motion in the absence of imaginary number as
−0.05
−10
−5
0
q(pm)
5
10
18
Pp = 4.50mW
chaos
14 0
2
ω/ω
4
6
m
FIG. 1. (Color online) (a) Schematic diagram of an optomechanical system driven by a probe field with frequency ωp and a pump field whose frequency is detuned by from the cavity resonance frequency, in which one mirror is fixed and the other is movable. (b) The spectrum of the intracavity field for four different values of the probe-field power Pp at Pl = 60 mW, = 0.6ωm , and ϕl = ϕp = 0. (c) The trajectory of mechanical motion in phase space at Pl = 60 mW, Pp = 4.5 mW, and = 0.6ωm . For the simulations, we employ the following experiment parameters from Ref. [21] throughout this work: m = 20 ng, ωm = 2π × 51.8 MHz, γm = 2π × 41.0 kHz, G = −2π × 12 GHz/nm, κ = 2π × 15.0 MHz, and = −ωm , respectively. We define the initial value of σ = (q,p,cr ,ci ) as σ0 = (q0 ,p0 ,cr0 ,ci0 ), where q0 , p0 , cr0 , and ci0 correspond to the initial values of q, p, cr , and ci , respectively. We set σ0 = (0,0,0,0) for this figure.
√ and amplitude sl = Pl /ωl ) and a probe field (initial phase ϕ p , central frequency ωp , power Pp , and amplitude sp = Pp /ωp ), in which the movable mirror (eigenfrequency ωm ) is coupled to optical cavity (eigenfrequency ωc ) via radiation pressure produced by these two lasers (coupling constant G). The power of the probe laser field may exceed the power of the pump field in the present work, but we still call it the “probe” field like the previous work [21] for simplifying the description. The Hamiltonian of the whole system in a frame rotating at the frequency ωl of the pump laser reads 1 pˆ 2 2 2 + mωm Hˆ = qˆ − cˆ† cˆ + Gqˆ cˆ† cˆ 2m 2 √ + i ηc κ{[sl e−iϕl + sp e−i(t+ϕp ) ]cˆ† − H.c.}, (1) where qˆ and pˆ are, respectively, the position and momentum operators of the mechanical resonator with effective mass m. Here = ωp − ωl and = ωc − ωl are the detunings of the probe laser frequency ωp and the cavity resonance frequency ωc from the pump laser frequency ωl . cˆ (cˆ† ) is the bosonic annihilation (creation) operator of the cavity mode. The cavity linewidth κ, including an external loss rate (waveguide coupling) κex and an intrinsic loss rate κ0 , is for the total loss rate of the cavity (i.e., κ = κex + κ0 ). Moreover, ηc = κex /κ characterizes the cavity coupling parameter and we choose ηc = 1/2 according to Ref. [21]. It is worth noting that the initial phases (ϕl and ϕp ) of the two input components, which are neglected in many previous works, become important for
p dq = , dt m
(2)
dp 2 q − γm p − G cr2 + ci2 , = −mωm dt
(3)
dcr 1 = (− + Gq)ci − κcr dt 2 √ + ηc κ[sl cosϕl + sp cos(t + ϕp )],
(4)
dci 1 = ( − Gq)cr − κci dt 2 √ − ηc κ[sl sinϕl + sp sin(t + ϕp )].
(5)
Equations (2) and (3) describe the dynamics of the mechanical resonator. γm is the damping rate of the mechanical resonator. Equations (4) and (5) describe the dynamics of the cavity mode. The quadratic term [−G(cr2 + ci2 )] and the mixed terms [(− + Gq)ci and ( − Gq)cr ] in Eqs. (3)–(5) present the nonlinearity of this system and the coupling of the movable mirror with the cavity field. Equation (3) suggests that the mechanical resonator can be driven readily by the radiation pressure when the beat frequency of the cavity field is close to the corresponding eigenfrequency ωc . Before proceeding, we note that Carmon et al. [9] proposed a scheme for chaotic quivering of a micron-scaled on-chip resonator that involves a monochromatic driving field, but is fundamentally different from the one which is discussed here. With respect to the optomechanical system driven by a monochromatic light field (i.e., sp = 0) [9], the effective beat frequency can be achieved via the interaction among the laser and the excited cavity field, but the extremely high optical power is required for the excitation. Hence one cannot observe these obvious nonlinear and nonperturbative effects (such as the high-order sidebands [6] and chaos [9]) of the coupled system unless the input field is strong enough. Numerical calculation confirms that, only when the optical power of the single laser is beyond about 5 W, the permanent chaos can emerge for the experimental parameters from Ref. [21], which is very difficult to achieve experimentally. Moreover, in the dielectric high-Q resonator, when the power exceeds 150 mW level, a third-order optical nonlinear effect would inevitably appear [33], such as four-wave mixing or stimulated Raman scattering. These effects would make the system too complicated for optomechanics study. Fortunately, in our work, as a bichromatic driving field is applied (i.e., a second probe laser field joins), the radiation pressure enables the resonator to oscillate effortlessly since the beat frequency of the pump field and probe field (which is
043839-2
FORMATION AND MANIPULATION OF OPTOMECHANICAL . . .
PHYSICAL REVIEW A 90, 043839 (2014)
identical with the probe frequency in a frame rotating at ωl ) is able to access to the eigenfrequency of the resonator, lowering the optical power required for inducing optomechanical chaos greatly. Since the evolution of the adjacent trajectories in phase space characterizes the dynamical behaviors of such a system, we consider how the displacement of two nearby points [defined as ε = (εq ,εp ,εcr ,εci )] in phase space moves forward over time. The equations characterizing the evolution of ε are obtained by linearizing Eqs. (2)–(5) [9] as follows: dεq εp = , dt m
(6)
dεp 2 = −mωm εq − γm εp − 2G cr εcr + ci εci , dt
(7)
dεcr 1 = Gci εq − κεcr + (− + Gq)εci , dt 2
(8)
dεci 1 = −Gcr εq − κεci + ( − Gq)εcr . (9) dt 2 The numerical solution of Eqs. (2)–(9) solved by using the Runge-Kutta method uncovers the intrinsic cavity property. Then we define Ic = cr2 + ci2 as the intensity of the cavity field, whose spectrum S(ω) can be attained by using fast Fourier transform of Ic , to describe the dynamical features of this optomechanical system. As shown in Fig. 1(b), in the case of a strong pump field (Pl = 60 mW), the cavity behaves in distinguishable features when the corresponding optical power of probe field is applied, which are similar to the processes in the previous work [9]. For a weak probe field (e.g., Pp = 3.60 mW), S(ω) is periodically tuned by the beat frequency of the probe field and pump field. The occurrence of the higherorder sidebands presents the nonlinear property of the coupled system. As the optical power of the probe field increases gradually (Pp = 3.61 mW and Pp = 3.64 mW), the period doubling showing the route to chaos is observed obviously [19]. At a higher power of the probe field (Pp = 4.5 mW), a set of discrete lines in the spectrum S(ω) transforms into a continuum [34], which shows the chaotic signatures of the cavity field. Under the identical condition, the trajectory of mechanical motion in phase space is shown in Fig. 1(c). It is easy to see that the mechanical resonator also oscillates chaotically and its motion evolves within a physical regime, which verifies that the numerical results of evolution of the cavity field are reliable. III. CHAOS-ABSENT WINDOW
For the sake of investigating the correlation between the beat frequency of the probe and pump fields and the dynamical features of the cavity, we take account of the evolution of a nearby point (Ic + εIc ) of Ic to characterize the property of the cavity field. The calculated exponential variation of εIc manifests how the initial states of the cavity field evolve in temporal domain and phase space. Under the fixed optical powers of the two input laser fields, the evolution of Ic is remarkably disparate for different beat frequencies (see Fig. 2). For the case of = ωm in Fig. 2(a), the decrease of ln(εIc ) over time indicates that all the nearby points of Ic in
FIG. 2. (Color online) Evolution of ln(εIc ) and cavity fields in temporal domain and phase space for four different values of the probe laser frequency under the fixed optical powers of the two input lasers (Pl = 60 mW and Pp = 20 mW). The phase space here is attained from the plots of the first derivative of Ic in time vs Ic . We use ϕl = ϕp = 0 and σ0 = (0,0,0,0), and set the initial value of ε as ε = (10−20 ,10−20 ,10−13 ,10−13 ) throughout our work.
phase space will finally oscillate in the identical trajectory at frequencies n (n is an integer) on account of the nonlinearity of the coupled system. They cannot converge to a point since the time appears in Eqs. (2)–(5). The flat evolution of ln(εIc ) for the case of = 1.5ωm in Fig. 2(b) implies the period-doubling bifurcation, which is observed in temporal domain and phase space. At = 0.54ωm in Fig. 2(c), the aperiodic oscillation and the exponential divergence of εIc emerge, which reveals a different regime that is extremely sensitive to initial conditions. This feature, however, just lasts a few microseconds before the oscillation of the cavity field becomes periodical ultimately, which is referred to as the transient chaos. Under the condition of = 1.2ωm in Fig. 2(d), εIc increases exponentially with the time; meanwhile the trajectory in phase space becomes greatly complicated and unpredictable, and we call such an oscillation the “permanent chaos.” Now, we turn to introduce the Lyapunov exponent (LE) [35], defined by the logarithmic slope of the curve of εIc versus the time t, to describe the dynamical behaviors of the cavity field. Then the following correspondence is obtained from the previous analysis: (i) the cavity field oscillates regularly when
043839-3
MA, YOU, SI, XIONG, LI, YANG, AND WU 10
PHYSICAL REVIEW A 90, 043839 (2014)
Pl = 10mW , Pp = 10mW
Pl = 60mW , Pp = 20mW
(a)
(e)
150 100
0
50
−10
0 −20
Lyapunov exponent
Pl = 60mW , Pp = 40mW
(b)
(f) 80
50
40
0
0 Pl = 60mW , Pp = 10mW
100
Pl = 60mW , Pp = 60mW
(c)
(g) 80
50
40
0
0 Pl = 100mW , Pp = 10mW
100
Pl = 60mW , Pp = 80mW
(d)
(h) 80
50
40
0
0 0.6
0.8
1
Ω/ωm
1.2
1.4
0.6
0.8
1
Ω/ωm
1.2
Lyapunov exponent
Pl = 40mW , Pp = 10mW
100
1.4
FIG. 3. (Color online) (a)–(d) Lyapunov exponent varies with for four different values of Pl under the condition of a fixed Pp = 10 mW. (e)–(h) Lyapunov exponent varies with for four different values of Pp at a fixed Pl = 60 mW. Here, ϕl = 0 and ϕp = 0 are used. Lyapunov exponent is taken from the slope of the curve of ln(εIc ) versus time t (from 0.84 μs to 1.2 μs) by using linear fitting. We set σ0 = (0,0,0,0) for all panels.
LE is less than zero, (ii) the cavity field exhibits the period doubling when LE is equal to zero, and (iii) the cavity field becomes chaotic when LE is greater than zero. In order to provide a clear vision of the frequency-dependent effect, we further look into LE for different beat frequencies as shown in Fig. 3. At low optical powers of the two laser fields, the oscillation remains periodical for different even though the period doubling emerges near = ωm [see Fig. 3(a)]. As the pump-field power increases, LE become positive for some beat frequencies but negative at ≈ ωm [see Figs. 3(b)–3(h)]; that is, chaotic oscillation turns up in some frequency regions but chaos is absent around = ωm . The process similar to OMIT provides some physical insight for the emergence of this chaos-absent window. The oscillation of the mechanical resonator driven by radiation pressure gives rise to strong anti-Stokes scattering of light from the pump field at ≈ −. Then the buildup of an intracavity probe field at ≈ − is suppressed as a result of the destructive interference of the near-resonant probe field and anti-Stokes field. The suppressed intracavity probe field fails to effectively influence the behaviors of the cavity field even if the input probe field is strong, leading to the absence of chaos at ≈ −. Strong intracavity probe field, however, can be built for some other beat frequencies since the anti-Stokes field is quite weak in these cases, which gives rise to the chaotic oscillation of cavity field. Therefore, we obtain a chaos-absent window. It is significant to note that OMIT is based upon the perturbation method but CAW belongs to a nonperturbative effect. Therefore, the window of OMIT is modulated only by adjusting the pump power and cannot be affected by the
probe power, but CAW is tuned by adjusting the power of both pump field [Figs. 3(b)–3(d)] and probe field [Figs. 3(e)–3(h)] simultaneously. Moreover, the enhancement of the pump power just makes CAW more obvious but cannot control the width of CAW remarkably. Nevertheless, the increase of the power of the probe field is able to narrow CAW effectively. Furthermore, both the numerous discrete lines in Fig. 3 and the sharp edges of CAW reflect the strong dependence of this chaotic system upon probe frequency. IV. PHASE-DEPENDENT CHAOS
Previous analysis suggests that the change of the system parameters can affect the dynamical signatures of the optomechanical system remarkably, which inspires one to consider some phase-dependent effects. We therefore recall the importance of the initial phases of the two input lasers, which is abandoned in most of the previous works. In order to discuss the phase-dependent effects in depth, we focus on the Hamiltonian [see Eq. (1)] at first. We are able to redefine the creation (or annihilation) operator cˆ† by absorbing the phase of one of the input fields. We consider the † ˆ iϕp ) here. Therefore, the transformation cˆt = cˆ† e−iϕp (cˆt = ce Hamiltonian can be transformed to the following form for any given initial phases of probe field:
043839-4
1 pˆ 2 † † 2 2 + mωm qˆ − cˆt cˆt + Gqˆ cˆt cˆt Hˆ = 2m 2 √ † + i ηc κ{[sl e−iϕ + sp e−it ]cˆt − H.c.},
(10)
FORMATION AND MANIPULATION OF OPTOMECHANICAL . . .
where ϕ is the relative phase of the two input lasers, i.e., ϕ = ϕl − ϕp . Thereupon, we are capable of obtaining identical equations of motion for any given ϕp . But at the same time, we should take account of the differences in initial conditions since the initial values of both c0∗ and c0 (which are the mean † values of cˆ0 and cˆ0 , respectively) are transformed as well during the transformation of Hamiltonian, i.e., ct∗0 = c0∗ e−iϕp and ct0 = c0 eiϕp (ct∗0 and ct0 are respectively the initial values of ct∗ and ct ). In the case that there is no photon in the cavity when t = 0, both initial conditions and equations of motion remain unchanged even if the initial phase of the probe laser varies, which suggests that the evolution of this optomechanical system only depends on the relative phase of the bichromatic driving. In view of this, we just find out the dependence of optomechanical chaos on relative phase when c0 = 0 (Fig. 4). On the one hand, the significance of relative phase is exhibited in the manipulation of transient chaos. That is, the lifetime of chaos is modulated by changing ϕ [Figs. 4(a)–4(d)]. Such an approach to controlling the lifetime of chaos by tuning the relative phases may guide potential applications on the 20
φ = 3 π 16
30
(a)
φ = 7 π 16
(b)
20
ln
ln
10
10
0 0 −10 0
1
2
3
4
0
1
Time(μs) 30
φ = 8 π 16
30
(c)
3
4
φ = 9 π 16
(d)
20
ln
20
ln
2
Time(μs)
10
PHYSICAL REVIEW A 90, 043839 (2014)
operation of light. On the other hand, the phases are capable of influencing the chaotic degree of the system with regard to the permanent chaos induced by this optomechanical system. The chaotic degree is quantified by the value of LE; hence the evolution of ln(εIc ) is considered for various ϕ [Fig. 4(e)]. Figure 4(e) shows that the nearly flat evolution of ln(εIc ) emerges at ϕ = π/2, but ln(εIc ) increases quickly for ϕ = 7π/8. Such an effect illustrates that the chaotic motion in this system intensely relies on the relative phase of the input fields. However, Fig. 4 just tells part of the story about phasedependent chaos. Supposing that there are some photons existing in the optical cavity, we would see a different picture of phase-dependent chaos. It’s because we acquire different initial conditions for various ϕp in this case after † the transformation cˆt = cˆ† e−iϕp . Therefore, we could obtain completely disparate processes of evolution of the system for different ϕp due to the remarkable sensitivity to initial condition in the chaotic system, which indicates that chaotic oscillation of cavity field depends on the phases of two input lasers, respectively, rather than the relative phase between them (when c0 = 0). The evolutions of system in two different cases (i.e., c0 = 0 and c0 = 0) are compared by using Fig. 5, in which, for identical relative phase (ϕ = π/2) and different phases of two input drives, LE is invariant when c0 = 0 [Fig. 5(a)], but varies when c0 = 0 [Fig. 5(b)]. We provide a reasonable explanation to the phase-dependent effect when c0 = 0: the selection of initial phases of the two drives leads to the difference of initial condition of the system, which definitely affects the process of period-doubling bifurcation. Then we obtain different results for different initial phases. We therefore conclude that the changes of the phases can affect the chaotic degree of the system prominently.
10
80
50 0 0
1
2
3
4
(b)
(a)
0
45
0
1
Time(μs)
2
3
70
4
Time(μs)
40 60
40
35
(e) φ = 7π 8
ln εIc
30
25 20
φ=π 4
25
50
30
ln εIc
35
40 30
ln
15 20
20
10
15
5
φ= π 2
10
0 0
5 0
0.5
1
1.5
Time(μs)
FIG. 4. (Color online) (a)–(d) With respect to the transient chaos ( = 0.6ωm , Pp = 20 mW), the lifetime of chaos is modulated by different relative phases ϕ. (e) For the chaotic oscillation of cavity field ( = 1.1ωm , Pp = 10 mW), Lyapunov exponent can be tuned by different ϕ. We use Pl = 60 mW and σ0 = (0,0,0,0) for all panels.
10
0.5
1
Time(µs)
1.5
2
0 0
0.5
1
Time(µs)
1.5
FIG. 5. (Color online) (a) For the chaotic oscillation of cavity field, Lyapunov exponent does not vary with initial phases of the two input drives for identical ϕ when cr0 = ci0 = 0; that is, Lyapunov exponent does not change under the following three different conditions: (i) ϕl = π/2,ϕp = 0; (ii) ϕl = π,ϕp = π/2; (iii) ϕl = 3π/2,ϕp = π . (b) We set cr0 = ci0 = 100; then Lyapunov exponent can be modulated by different phases of two input lasers even if ϕ remains unchanged. We use Pl = 60 mW, Pp = 20 mW, = 1.1ωm , p0 = 0, and q0 = 0 for these two panels.
043839-5
MA, YOU, SI, XIONG, LI, YANG, AND WU
PHYSICAL REVIEW A 90, 043839 (2014)
Properly speaking, we just illustrate the dependence of the chaotic degree of the system on initial phases and the accurate correlation of them relies on experiment. The dependence of LE on phase may provide an effective method for chaosencrypted communication [36] based upon optomechanics, because it is highly difficult to detect the initial phases from a transmitter time series alone unless the correlation between LE and initial phases in corresponding system parameters is given. In addition, recent work [32] demonstrates that the chaosbased information is not modified by the channel since LE cannot be altered during communicating wirelessly, suggesting that the manipulation of optomechanical chaos in our work may be applied to chaos-based wireless communication.
oscillation upon the system parameters, we have indicated that a series of nonperturbative effects can appear. A broad chaos-absent window with sharp edges can be observed in this optomechanical system and might be found in other OMIT-like systems. The phase-dependent effects, including the lifetime of the transient chaos and the chaotic degree of the system, are also investigated in detail. Since the restriction of the input power is released in the proposed setup with the two-tone laser, we believe that the observation and manipulation of the optomechanical chaos will be highly accessible in experiments. ACKNOWLEDGMENTS
We have presented an effective approach for achieving optomechanical chaos at low optical powers of the two cw optical inputs. Due to the strong dependence of the chaotic
The work is supported in part by the National Fundamental Research Program of China (Grant No. 2012CB922103), the National Science Foundation (NSF) of China (Grants No. 11375067, No. 11275074, No. 11374116, No. 11204096, and No. 11405061), and the Fundamental Research Funds for the Central Universities, HUST (Grant No. 2014QN193).
[1] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, Nature (London) 475, 359 (2011). [2] J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Grblacher, M. Aspelmeyer, and O. Painter, Nature (London) 478, 89 (2011). [3] G. Amalino-Camelia, Nature (London) 398, 216 (1999). [4] J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, Nature (London) 471, 204 (2011). [5] H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, Phys. Rev. A 86, 013815 (2012). [6] S. Huang and G. S. Agarwal, Phys. Rev. A 81, 033830 (2010). [7] T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, Phys. Rev. Lett. 94, 223902 (2005). [8] M. A. Lemonde, N. Didier, and A. A. Clerk, Phys. Rev. Lett. 111, 053602 (2013). [9] T. Carmon, M. C. Cross, and K. J. Vahala, Phys. Rev. Lett. 98, 167203 (2007). [10] H. Xiong, L. G. Si, X. Y. L¨u, X. Yang, and Y. Wu, Opt. Lett. 38, 353 (2013). [11] K. Zhang, W. Chen, M. Bhattacharya, and P. Meystre, Phys. Rev. A 81, 013802 (2010). [12] J. Larson and M. Horsdal, Phys. Rev. A 84, 021804(R) (2011). [13] G. Wang, L. Huang, Y. C. Lai, and C. Grebogi, Phys. Rev. Lett. 112, 110406 (2014). [14] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, arXiv:1303.0733. [15] P. Meystre, Ann. Phys. (N.Y.) 525, 215 (2013). [16] F. Marquardt and S. M. Girvin, Physics 2, 40 (2009). [17] T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008). [18] M. Aspelmeyer, P. Meystre, and K. Schwab, Physics Today 65, 29 (2012). [19] M. J. Feigenbaum, J. Stat. Phys. 21, 669 (1979).
[20] G. S. Agarwal and S. Huang, Phys. Rev. A 81, 041803 (2010). [21] S. Weis, R. Rivi`ere, S. Del´eglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010). [22] A. Kronwald and F. Marquardt, Phys. Rev. Lett. 111, 133601 (2013). [23] K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, Phys. Rev. Lett. 111, 053603 (2013). [24] Y. Wu and X. Yang, Phys. Rev. A 76, 013832 (2007). [25] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005). [26] F. Massel, T. T. Heikkil¨a, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanp¨aa¨ , Nature (London) 480, 351 (2011). [27] A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, Nature (London) 472, 69 (2011). [28] A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 74, 2447 (1995). [29] C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature (London) 409, 490 (2001). [30] A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garc´ıa-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature (London) 438, 343 (2005). [31] M. Eisencraft, R. Attux, and R. Suyama, Chaotic Signals in Digital Communications (CRC Press, Boca Raton, FL, 2013). [32] H. P. Ren, M. S. Baptista, and C. Grebogi, Phys. Rev. Lett. 110, 184101 (2013). [33] M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, Nat. Phys. 5, 276 (2009). [34] B. A. Huberman and A. B. Zisook, Phys. Rev. Lett. 46, 626 (1981). [35] V. I. Oseledec, Trans. Mosc. Math. Soc. 19, 179 (1968). [36] T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Els¨aßer, IEEE J. Quant. Electron. 38, 1162 (2002).
V. CONCLUSION
043839-6
