Two-dimensional optomechanics formed by the graphene sheet and photonic crystal cavity
We theoretically study photon transmission and mechanical ground state cooling in a two-dimensional optomechanical system that is formed by suspending a graphene sheet on a one-dimensional optomechanical crystal. When the frequencies of graphene resonator and nanobeam resonator(phononic mode of optomechanical crystal) are approximately the same, the -type degenerate four-level structure of two-dimensional optomechanical system shows the two-color optomechanically-induced transparency, and the transparency window could be switched among probe signal’s absorption, transparency, and amplification. According to our calculations, the graphene resonator could also assist the ground state cooling of nanobeam resonator in the proposed two-dimensional optomechanics.
pacs:42.79. Gn, 42.50.Wk, 42.50.Lc
Cavity optomechanics supports a platform to explore nonlinear and nonclassical effects Vahala (); Painter2 (); Thompson (); Aspelmeyer1 (); Poot (). Significant progresses have been made on the strong optomechanical coupling and mechanical ground state cooling in optomechancis Marquardt2 (); Wilson (); Schliesser (); Vanner (); Chan (); Hertzberg (); Mathew (); Lecocq (); Cohadon (); Park (). The photonic nonlinear and nonclassical effects induced by optomechanical interaction have been widely studied, such as the OMIT (optomechanically-induced transparency) Agarwal (); Agarwal2 (); Agarwal3 (); Kippenberg (); Painter (); Luki (), photon blockade Rabl (); xu1 (); Li (); Liao1 (); Liao2 (); Kurn (); Liew (); Savona (), nonclassical mechanical motions Teufel (); Alegre (); Cohadon (); Park (); xu3 (); xu4 (), optical nonreciprocity Dong (); Peterson (); Fang (); Lipson (), and so on. The hybrid system formed by optomechanics coupling to TLS (two-level system) or mechanical resonator also attracts a lot of interest recently Bienert (); Ma (); Hui1 (); Hui2 (); Hui3 (); Akram ().
Cavity optomechanics has been realized in different systems, such as Whispering-gallery cavity, photonic crystal cavity, microwave circuit, dielectric membrane placed between two high-finesse mirrors, and so on Vahala (); Painter2 (); Harris (). Graphene sheet has been used to build optomechanics with microwave resonator through radiation pressure interaction Singh (); Bachtold (); Weber (); Song (). Moreover, the interaction between photonic crystal cavity and graphene sheet are more interesting, because the photonic cavity might be able to detect mechanical motions and nonlinear mechanical properties of graphene sheet with extra high precision Eichler (). In addition recent experiments have shown that the graphene sheet could be used to tune the optical, electrical, and heat transport of photonic crystal cavity Gan (); Huan (); Xu (); Majumdar (); Barton (); Shih ().
However, single layer graphene is almost transparent for visible and infrared lasers Falkovsky (); Geim (), so it is difficult to build optomechanics with single layer graphene sheet through radiation pressure type interaction. As verified by the perturbation calculations and recent experiments Gan (); Meade (); Huan (), the graphene sheet leads to resonant frequency shift and an additional damping for photonic cavity mode through optical gradient or absorption force. Thus, the optical gradient or absorption force could be used to build 2D (two-dimensional) optomechanical system consisting of the graphene sheet and 1D (one-dimensional) optomechanical crystal; as for radiation pressure type optomechanical interaction, the multi-layer graphene sheets might be a choice.
With a suspended single-layer graphene sheet above a 1D optomechanical crystal, we study photon transmission and mechanical ground state cooling in 2D optomechanics (see Fig.1). The optomechanical type interactions exist not only between photonic and mechanical modes of nanobeam, but also between the photonic cavity field and graphene resonator. Compared with other type multi-mode optomechanics, the steady position of graphene resonator could be easily tuned via a control voltage between graphene sheet and silicon substrate Bunch (); Chen (); thus, the damping rate of photonic cavity and graphene-cavity optomechanical interaction strength could be controlled. Here we mainly focus on the effects of a graphene sheet on photon transmission and ground state cooling of the nanobeam resonator in proposed 2D optomechanical system.
The paper is organized as follows: In Section II, we introduce the proposed 2D optomechanics model. In Section III, we analyze the effects of a graphene resonator on photon transmission. In Section IV, we study the graphene resonator assisted ground state cooling of nanobeam resonator. In Section V, we conclude this work.
Ii 2D optomechanical model
We study a 2D optomechanical system as shown in Fig. 1, the photonic and phononic modes of nanobeam in an 1D optomechanical crystal interact with each other in a plane parallel to silicon substrate; the single-layer graphene sheet interacts with photonic mode of 1D optomechanical crystal in the direction perpendicular to silicon substrate. Thus, the Hamiltonian of free 2D optomechanical system:
Here (), (), and () are annihilation (creation) operators of photon, nanobeam phonon, and graphene phonon, respectively. is the resonant frequency of photonic cavity, and () corresponds to vibration frequency of nanobeam (graphene) resonator. And, and describe the optomechanical interaction strengths of the photonic mode with the nanobeam and graphene resonators, respectively. The graphene-cavity optomechanical coupling strength and cavity damping rate could be changed by adjusting graphene sheet’s steady position which could be easily adjusted by a control voltage between graphene and silicon substrate Bunch (); Chen ().
In a rotating frame defined by an unitary transformation , the effective Hamiltonian becomes . The direct photon-phonon interactions disappear in and are replaced by a Kerr nonlinearity term with the coefficient , where and are the photon nonlinear coefficients induced by nanobeam and graphene resonators, respectively. The energy level structure of 2D optomechanical system could be obtained from as
Here is the cavity photon number, while and are phonon numbers on nanobeam and graphene resonators, respectively. Compared with the energy level structure of the standard optomechanicsRabl (); Aspelmeyer1 (); Girvin (), the energy levels in 2D optomechanics have an additional mechanical freedom. For degenerate mechanical modes , the degree of energy level degeneracy could be calculated from Eq. (2) as . The low excitation state energy levels are shown in Fig. 2, the degree of degeneracy for zero phonon state is , and it is (or ) for single (or double) phonon excited states, respectively. From the expression of , the eigenstates of 2D optomechanical system can be obtained as , where () are the Fock states of photon and phonons, respectively, while and are the phonon displaced Fock states in the case of photon number .
Iii Tunable Photon transmission
iii.1 Controllable Photons blockade
The photon blockade in optomechanical system has been widely studied Rabl (); xu1 (); Li (); Liao1 (); Liao2 (); Kurn (). If the photonic cavity is driven by a weak laser field with frequency and amplitude , that is . Thus, the Hamiltonian in a rotating frame at driving frequency , defined by an unitary transformation , becomes
here is the frequencies detuning of the driving laser and photonic mode. Define , , and as the damping rates of photonic cavity, nanobeam resonator, and graphene resonator, respectively. The photon blockade requires a weak driving field () and it is usually described by the second-order correlation function which could be defined with density operator as . The master equation of density operator:
The Lindblad terms in Eq. (4) are: , with corresponding to optical and mechanical variables, respectively. Under weak pumping, the density matrix can be numerically calculated by truncating to limited photon and phonon numbers in Eq. (4) Liew (); Tan (); Johansson1 (); Johansson2 (). The thermal photon and phonon numbers are defined as (), with and are environmental temperature and Boltzmann constants, respectively. Because of the extra high frequency, in this article the thermal photon numbers can be set as zero, that is .
The numerical calculation results of the photon second-order correlation function in degenerate 2D optomechanical system are shown in Fig. 3. The blue-solid curve describes photon blockade of standard optomechanics, which is equivalent to a large gap between graphene and photonic crystal cavity so that their interaction can be neglected. The peaks and dips in blue-solid curve can be explained analytically as follows. In weak pumping limits and , the Hamiltonian in Eq. (3), experiencing an unitary transformation , becomes . There is no direct interaction between photon and phonons in , if we neglect the excitations of phonons Li (); Bamba (); Savona (); xu1 (); thus, the low excited states wave-function of 2D optomechanics can be assumed as , then the second-order correlation function could be obtained as:
Equation (5) could be used to explain single-photon blockade dip and two-photon resonant peak in Fig. 3. The dip at in blue-solid curve satisfies and describes single photon blockade of standard optomechanics(), and the peak at satisfies and corresponds to two-photon resonant transition. The small peak at corresponds to photon resonant transitions between states and xu1 ().
The green-dashed and red-dotted curves in Fig. 3 describe photon blockade in 2D optomechanics (), which is equivalent to a limited gap between the graphene sheet and photonic crystal cavity. In the case of and , the photon nonlinearity coefficient in green-dashed curve is doubled compared with standard optomechanics (blue-solid curve), that is . So the position of single photon blockade dip (two-photon resonant peak) shifts to () in green-dashed curve.
The coupling strength in the red-dotted curve, the positions of photon blockade dip or multi-photon resonant transition peaks in -axis deviate from those of green-dashed curve (). From above discussions, it is shown that the graphene resonator can be used to tune photon nonlinearity, photon blockade, and photon tunneling in 2D optomechanical system.
iii.2 Optomechanically induced transparency
The OMIT in optomechanical system has been theoretically and experimentally studied Agarwal (); Kippenberg (); Painter (); Luki (). The mechanical resonator, TLS, and condensed states coupling to optical or mechanical mode of optomechanical system can affect photon transmission in optomechanical system Luki (); Ma (); Bienert (); Hui1 (); Hui2 (); Hui3 (). If the frequencies of nanobeam and graphene resonators are approximately the same (), the ground state and single excited states in 2D optomechanical system form a degenerate four-level structure as shown in Fig. 4. Thus, some additional photon transition channels are created by graphene resonator, this should affect photon transition of 1D optomechanical crystal. The energy level structure in Fig.4 is similar to that of the hybrid TLS-optomechanics system, where the TLS splits phonon energy levels and leads to double-transparency windows Hui1 ().
If the photonic crystal cavity is driven by a strong driving field and a weak probe field, that is , where (or ) and (or ) are the amplitudes (or frequencies) of the driving and probe fields, respectively. In a rotating frame at the driving field frequency , with the mean field approximation, the steady value equations for photonic and mechanical modes are obtained as
where is the frequency detuning between probe and driving fields. When the amplitude of pumping field is much larger than that of the probe field (), up to the first order small quantity of , the solutions of Eqs. (III.2) can be approximately expanded as , with and . Here and are respectively the steady values and first-order responses to weak probe field, and . Then the steady value of cavity field can be calculated as , and mechanical modes’ steady values are and , respectively.
The first-order small quantity describes probe field’s absorption and dispersion in 2D optomechanical system, and it could be calculated from Eqs. (III.2) as
with (), and . With the input-output relation, the output optical field can be written as Agarwal (); Zoller (). Define the quadratures of field as , here describes probe field’s absorption in 2D optomechanical system, while the imaginary part corresponds to dispersion Agarwal ().
The effects of mechanical resonators’ detuning on transparency window are shown in Figs. 5(a) and 5(b). There is no graphene-cavity interaction in blue-solid curve which describes the OMIT of standard optomechanics. When the frequencies of two mechanical resonators are the same (), the local minimal value within the transparency window moves down to negative regime in red-dotted curve, this means that the graphene resonator leads to the amplification of weak probe field which should be transparent in the 1D optomechanical crystal (blue-solid curve). This phenomenon originates from the additional photon transition channels () induced by graphene resonator(see Fig. 4), which changes the photon numbers of states and also the destructive coherent transitions in the type three-level structure of 1D optomechanical crystal. For a small mechanical detuning (), two transparency windows appear in the green-dashed (black-dashed-dotted) curve, the transparency window at is the same as that of standard optomechanics (blue-solid curve) and a new narrow transparency window appears at . This is similar to the double-color transparency window in the TLS-optomechanics coupled systems Hui1 ().
For the same mechanical frequencies (), the effects of graphene-cavity optomechanical coupling strength on transparency window are shown in Figs. 5(c) and 5(d). If the motions of the nanobeam and graphene resonators are inphase (), the probe signal is amplified in the green-dashed curve. If motions are out of phase , the black-dashed-dotted curve shows that a absorption peak appears within the transparency window of standard optomechanics (blue-solid curve). When the graphene and nanobeam resonators have the same frequencies, damping rates, and optomechanical coupling strengths, but oppositive motion phases (), the transparency window vanishes in red-dot curve in Fig. 5(c). The destructive coherent transitions induced by two mechanical resonators offset with each other. Through above discussions, it is shown that the graphene resonator can control photon transmission in one-dimensional optomechanics crystal.
Iv Ground state cooling
iv.1 Mechanical Spectrum
In the vacuum weak coupling regime , the photon operator can be split into , with an average coherent amplitude and a small fluctuation term Marquardt2 (); Painter3 (). From Eq. (3), we can obtain the linearized Heisenberg equations for , , and as follows:
with and . Under strong pumping field, the classical and nonlinear terms , , and have been neglected. Thus, the linearized Hamiltonian of 2D optomechanics could be written as . Here () are input noise operators of optical and mechanical modes, respectively; their environmental average values are , and the nonzero noise input correlation functions are and .
With the Fourier transformation, the motion equations in frequency domain are as follows:
Eliminating the variables , , , , and in Eqs. (IV.1), with (), then can be expressed with input noise operators , , and Mancini (). The mechanical spectrum of nanobeam resonator: Marquardt2 (); Painter3 ():
with , with and are assumed as real numbers. The optical and mechanical responsive functions are defined as , , and , respectively. The self-energy term of optomechanical crystal is defined as , which decides frequency shift and extra damping rate of nanobeam resonator in standard optomechanics.
Here describes the contribution of the optomechanical interaction between the nanobeam resonator and cavity field, originates from the nanobeam resonator’s thermal noises, and comes from graphene resonator. Compared with standard optomechanics, an effective amplification factor , which is induced by the graphene resonator, appears in nanobeam resonator’s responsive function (see Eq. (IV.1)). The equations and have been adopted during the above calculations.
The blue-solid curve in Fig. 6(a) describes mechanical spectrum of standard optomechanics( Hz), which has been widely studied in both theories and experiments Vahala (); Aspelmeyer1 (); Marquardt2 (). If , under blue-detuning driving laser (), the narrow peaks appear at in green-dashed and red-dotted curves representing the local amplification for mechanical spectrum. The FWHM (Full width at half maximum width) of narrow peak is close to (), which indicates that the narrow peak (or dip) originates from the graphene resonator. For the red-detuning driving laser () in Fig. 6(b), a narrow dip appears at in green-dashed and red-dotted curves, which corresponds to local suppression for mechanical spectrum. The dip in red-dotted curve is deeper than that of green-dashed curve, which means that a larger suppression for mechanical spectrum of nanobeam resonator could be realized with a larger graphene-cavity interaction strength. The numerical calculations show that the amplification factor is responsible for the narrow peak or dip in the mechanical spectrum of the graphene resonator. For a weak graphene-cavity coupling strength, , the narrow peak or dip will disappear.
The effects of mechanical detuning on mechanical spectrum of nanobeam resonator are shown in Fig. 6(c) (blue-detuning) and 6(d) (red-detuning). For nonzero mechanical detuning (), under blue detuning driving laser, the -axis positions of narrow peaks in Fig. 6(c) shift to . For red detuning driving laser in Fig. 6(d), the narrow dips shift to . Thus, we can control the positions of amplification peaks and suppression dips in mechanical spectrum of nanobeam resonator by tuning the frequency of the graphene resonator.
iv.2 Dynamical Backaction
The optomechanical interaction changes mechanical resonator’s frequencies and damping rates Vahala (); Painter2 (); Thompson (); Aspelmeyer1 (); Marquardt2 (), which can be used for mechanical self-oscillation, phonon laser, mechanical ground state cooling, and so on Barton (); Cohadon (); Zhang (); Capuj (); Park (); Durand (); Grudinin (); Yang (). In this section we discuss the shifts of nanobeam resonator’s frequency and damping rate in 2D optomechanical system.
If the frequencies of graphene and nanobeam resonators are approximately the same (), we can obtain from Eqs. (IV.1) that , with . The effective optomechanical self-energy in 2D optomechanics can be defined as . In weak coupling limit , the frequency shift and extra damping rate of nanobeam resonator can be defined as and , respectively. The graphene resonator’s frequency shift and extra damping rate can be calculated similarly. Because of the symmetry of two resonators, the and can be simply obtained from and by interchanging subscript and in corresponding expressions, respectively.
The nanobeam resonator’s effective frequency and damping rate in resolved-sideband regime () are shown in Figs. 7(a) and 7(b), respectively. The blue-solid curves in Figs. 7(a) and 7(b) correspond to effective frequency and damping rate of standard optomechanics (). With the increase of graphene-cavity optomechanical coupling strength, the changes of effective frequency and damping rate become larger in green-dashed and red-dotted curves, which means that graphene resonator can assist mechanical amplification and ground state cooling in 2D optomechanical system.
According to recent experiments, the coupled mechanical system shows coherent mixing of mechanical modes in the bad-cavity limitRosenberg (); Zhou (). The changes of effective frequencies and damping rates for two mechanical resonators in unresolved-sideband regime () are shown in Fig. 8, where and . If , the changes of two resonators’ effective frequencies (Fig. 8(a)) and damping rates (Fig. 8(b)) are always in step, the effects of one mechanical resonator on another resonator’s optomechanical dynamical backaction are very weak. For a large graphene-cavity coupling strength(), the shifts of effective frequencies (Fig. 8(c)) and damping rates (Fig. 8(d)) in blue-solid and red-dashed curves are the opposite of paces. The damping rates of nanobeam and graphene resonators always have oppositive signs in Fig. 8(c), this means when one mechanical resonator is amplified, the other one should be simultaneously cooled, moreover similar results can also be obtained in resolved-sideband regime. The blue-solid and red-dashed curves in Figs. 8(c) and 8(d) cross with each other, which means that coherent mixing of two mechanical resonators can be realized in 2D optomechanical system with optical field serving as an intermediacy.
iv.3 Graphene assisted Ground State Cooling
As shown in Fig. 7, the graphene resonator could enhance damping rate of the nanobeam resonator, which should make positive contributions to the ground state cooling of the nanobeam resonator. However, such a graphene resonator also introduces a noise term in Eq. (10) which makes negative contributions to ground state cooling. The relative value of above two terms decide the net contribution of the graphene sheet to ground state cooling of the nanobeam resonator. With the mechanical spectrum in Eq. (10), the phonon number on nanobeam resonator can be obtained as Marquardt2 (); Painter3 ()
The variation of phonon number on nanobeam resonator as a function of graphene-cavity coupling strength is shown in Fig. 9, the value corresponds to phonon number of standard optomechanics. Figure 9(a) shows the effects of different cavity damping rates on the ground state cooling of nanobeam resonator. In blue-solid (or green-dashed) curve of Fig. 9(a), the phonon number of 1D optomechanical crystal is larger than () for the case of no graphene-cavity interaction; however, the phonon number is quickly suppressed and reaches to its ground state () as the increase of graphene-cavity coupling strength. The green-dashed and red-dotted curves show that the suppression ability of the graphene sheet on nanobeam resonator’s phonon number becomes weaker in the case of a larger cavity damping rate (compared with blue-solid curve).
The ground state cooling of nanobeam resonators of different damping rates in 2D optomechanical system are shown in Fig. 9(b). Compared with small damping rates of blue-solid and green-dashed curves, the phonon number on nanobeam resonator with larger damping rate can be more effectively suppressed by the graphene resonator in red-dotted curve. The effects of graphene resonator’s damping rate and optomechanical coupling strength in 1D optomechanical crystal on the ground state cooling of nanobeam resonator are shown in Figs. 9(c) and 9(d), respectively. From above results, it is clear that the graphene resonator can enhance ground state cooling of nanobeam resonator in some parameter regimes, especially for the high damping nanobeam resonator.
V Discussion and Conclusion
We proposed a new type of multi-mode optomechanics formed by a suspended graphene resonator coupled to the photonic mode of a 1D optomechanical crystal. This kind of 2D optomechanical system could be experimentally realized by transferring graphene sheets above a released 1D photonic crystal cavity on silicon substrate, and the gap between graphene sheet and photonic cavity is the main difficulty for experimental success. By tuning the control voltage between graphene and silicon substrate, the steady-state position of the graphene sheet can be easily adjusted by a control voltage, and this can affect the damping rate of photonic cavity and graphene-cavity optomechanical interaction strength.
We have theoretically studied photon transmission and mechanical ground state cooling in the proposed 2D optomechanical system. Additional transitions channels induced by the graphene resonator could enhance or weaken the destructive coherent transitions of photons, and the tranmission of probe signal in 2D optomechanics can be switched among absorption, transparency, and amplification by tuning graphene sheet’s vibration frequency and graphene-cavity optomechanical interaction strength.
The graphene resonator could induce narrow amplification peak and suppression dip in the noise spectrum of nanobeam resonator, and affects effective damping rate of the nanobeam resonator. According to our numerical calculations, the graphene resonator can assist the ground state cooling of nanobeam resonator in certain parameter regime.
Additionally, the graphene sheet has some special properties, such as nonlinear mechanical motion and nonlinear mechanical damping, which could be interesting research topics in the future.
Guangya Zhou is supported by MOE Academic Research Fund of Singapore under Grant No. R-265-000-557-112. X.W.X. is supported by the National Natural Science Foundation of China (NSFC) under Grants No.11604096 and the Startup Foundation for Doctors of East China Jiaotong University under Grant No. 26541059. Y.J.Z. is supported by the China Postdoctoral Science Foundation under grant No. 2017M620945.
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