A photonic transistor device based on photons and phonons in a cavity electromechanical system
We present a scheme for photonic transistors based on photons and phonons in a cavity electromechanical system, which is consisted of a superconducting microwave cavity coupled to a nanomechanical resonator. Control of the propagation of photons is achieved through the interaction of microwave field (photons) and nanomechanical vibrations (phonons). By calculating the transmission spectrum of the signal field, we show that the signal field can be efficiently attenuated or amplified, depending on the power of a second ‘gating’ (pump) field. This scheme may be a promising candidate for single-photon transistors and pave the way for numerous applications in telecommunication and quantum information technologies.
pacs:42.50.Nn; 85.60.Dw; 85.85.+j; 85.25.-j
A photonic transistor is a device where the propagation of the signal photons is controlled by another ‘gate’ photons . In the past decade, such a device has received a lot of interest in view of its important applications ranging from optical communication and optical quantum computer  to quantum information processing . Schemes based on nanoscale surface plasmons , microtoroidal resonators , a single-molecule  and some others [7, 8, 9] have been proposed to realize photonic transistors. However, practical realization remains challenging because it necessitates large nonlinearities and low losses.
Recently, Weis et al.  reported a novel phenomenon based on the interaction between photons and phonons in a cavity optomechanical system, which is called optomechanically induced transparency (OMIT). The transition between the absorptive and transparent regime of the probe laser field was modulated by a second control laser field [10, 11]. In the presence of the control laser, a transparency window for the probe field appears due to the optomechanical interference effect when the beating of the two laser fields is resonant with the vibration frequency of the mechanical resonator. At the same time, electromagnetically induced transparency (EIT) and slow light with optomechanics have been also experimentally realized . As a conterpart of cavity optomechanical system, circuit cavity electromechanical system, usually consisted of a superconducting microwave cavity and a nanomechanical resonator, has been under extensive investigation in recent years [13, 14, 15, 16]. In particular, strong-coupling regime was reached recently , which paves the way for ground-state cooling of the nanomechanical resonator. Based on these achievements, and we note that they mainly consider the situation where the cavity is driven on the red sideband, in the present paper, we deal with a different case that the cavity is driven on its blue sideband in a coupled superconducting microwave cavity-nanomechanical resonator system [13, 14, 15, 16, 17]. Theoretical study shows that the transmitted signal (‘source’) field can be attenuated or amplified, depending on the power of the pump (‘gate’) field that controls the number of photons in the cavity. Therefore, such a system can be employed to serve as a photonic transistor and may be realized under the existing experimental conditions [16, 17]. Importantly, the photonic transistor proposed here does not require the large nonlinearities and is compatible with chip-scale processing.
2 Model and Theory
Our cavity electromechanical system, composed of a nanomechanical resonator capacitively coupled to a superconducting microwave cavity denoted by equivalent inductance and equivalent capacitance , is sketched in Fig. 1. A strong pump (‘gate’) field with frequency and a weak signal (‘source’) field with frequency are applied to the microwave cavity simultaneously. The beating of the two fields causes the nanomechanical resonator to vibrate which can change the capacitance of the microwave cavity and thus its resonance frequency. The coupling capacitance can be approximated by , where represents an equilibrium capacitance, d is the equilibrium nanoresonator-cavity separation, and is the displacement of the nanomechanical resonator from its equilibrium position. Therefore, the coupled cavity has an equivalent capacitance , such that resonance frequency of the microwave cavity is . In a rotating frame at the pump frequency , the system Hamiltonian reads as follows [13, 15]:
The first term is the energy of the microwave cavity, where is the creation (annihilation) operator of the microwave cavity and is the cavity-pump field detuning. The second term gives the energy of the nanomechanical resonator with creation (annihilation) operator , resonance frequency and effective mass . The third term corresponds to the capacitive coupling between the microwave cavity and the nanomechanical resonator, where is the coupling strength between the cavity and the resonator, is the effect of the displacement on the perturbed cavity resonance frequency, and is the zero-point motion of the nanomechanical resonator. The last two terms represent the interaction between the cavity field and the two rf fields with frequency and . is the signal-pump detuning. and are, respectively, amplitudes of the pump field and the signal field, and they are defined by and , where is the pump power, is the power of the signal field, and is the decay rate of the cavity.
where we have set be the resonator amplitude. The cavity mode decays at the rate and is affected by the input vacuum noise operator with zero mean value, which obeys the correlation function in the time domain,
The mechanical mode is affected by a vicious force with damping rate and by a Brownian stochastic force with zero mean value that has the following correlation function 
where is the Boltzmann constant and is the temperature of the reservoir of the mechanical resonator. Following standard methods from quantum optics, we derive the steady-state solution to Eqs. (2) and (3) by setting all the time derivatives to zero. They are given by
To go beyond weak coupling, we can always rewrite each Heisenberg operator as the sum of its steady-state mean value and a small fluctuation with zero mean value,
Such linearized Langevin equations have been used in investigating optomechanically induced transparency , parametric normal-mode splitting , and sideband cooling of mechanical motion  in optomechanical systems, where the strong-coupling is required. In the circuit cavity electromechanical system we study here, strong-coupling regime has been achieved recently, where the cooperativity , larger than those previously achieved in optomechanical systems [10, 22]. Here, ( is the effective coupling strength) is an equivalent opto- or electro-mechanical cooperativity parameter. In this situation, the eigenmodes of the driven system are hybrids of the original mechanical mode and the cavity mode. The coupled system shows the normal-mode splitting, a phenomenon well-known to both classical and quantum physics. Theoretical transmission spectrum obtained through the linearized Langevin equations is in good agreement with the experiment result . In the following, since the drives are weak, but classical coherent fields, we will identify all operators with their expectation values, and drop the quantum and thermal noise terms . Then the linearized Langevin equations can be written as:
where , , , , and . Here , approximately equal to the number of pump photons in the cavity, is determined by the following equation
The output field can be obtained by employing the standard input-output theory  , where is the output field operator, we have
The transmission of the signal field, defined by the ratio of the output and input field amplitudes at the signal frequency, is then given by
3 Numerical results and discussion
In what follows, we choose a realistic cavity electromechanical system to calculate the transmission spectrum of the signal field. The parameters used in the numerical simulation are : GHz, MHz, kHz, Hz, and where is the quality factor of the nanomechanical resonator, and the damping rate is given by We can see that , therefore the system operates in the resolved-sideband regime also termed good-cavity limit. When the cavity is driven on its red sideband, i.e. , the analogy of electromagnetically induced transparency (EIT) could appear, which has been extensively discussed [10, 11, 12, 17]. If we choose , we can also obtain the similar EIT effect, as shown in Fig. 2(a). When the pump power increases from zero, the transmission of the signal field at signal-cavity detuning increases to unity gradually, and then the transparency window is broadened by increasing the pump power further. Our result is good agreement with that in Ref. . Here, we mainly consider the situation where the cavity is driven on its blue sideband, i.e., . Under blue-detuned pumping, the effective interaction Hamiltonian for the cavity field and the mechanical phonon mode becomes one of the parametric amplification, , where is the effective coupling strength. Fig. 2(b) displays a series of transmission spectra of the signal field as a function of the signal-cavity detuning () for various pump powers. When the pump field is off, i.e., , the transmission spectrum of the signal field shows the usual Lorentzian line shape of the bare cavity. However, as the pump power is raised (pW and pW), we can see that the transmission is attenuated around the signal-cavity detuning compared to the situation where the pump field is off, a result of the increased feeding of photons into the cavity. If the pump power is increased further, the system switches from electromagnetically induced absorption (EIA)  to parametric amplification (PA) , leading to signal amplification (pW). When the pump power equals to 0.9pW, the transmitted signal field can be amplified greatly. Similar to Fig. 2(a), at the low pump power, the normal-mode splitting is not apparent even if the coupled system has entered the strong-coupling regime. When the pump power is large enough (nW), normal-mode splitting can be easily observed. In the strong-coupling regime, the eigenmodes of the coupled system are hybrids of the original cavity modes characterized by cavity photons and mechanical modes represented by mechanical quanta. Fig. 2(c) is the level diagram of the driven, coupled system. When the cavity is driven on its red sideband, corresponding to the transition between and , the system exhibits the mechanical analogue of EIT. However, the blue-detuned pump field induces a transition between and , which can efficiently amplify the signal field at the the cavity resonance. Fig. 2(b) demonstrates that such a circuit cavity electromechanical system can indeed act as a photonic transistor, where the pump (‘gate’) field regulates the flow of the signal (‘source’) field by controlling the number of photons in the cavity. We mainly use the narrow region around signal-cavity detuning to attenuate or amplify the signal field. The physical origin of this phenomenon comes from the radiation pressure force oscillating at the beat frequency between the pump field and the signal field, which induces the vibration of the nanomechanical resonator. When the beat frequency is resonant with the mechanical resonance frequency . The frequency of the pump field is downshifted to the Stokes frequency , which is degenerate with the signal field. Constructive interference between the Stokes field and the signal field amplifies the weak signal field. Similar amplification of a signal due to radiation pressure backaction in a detuned cavity optomechanical system was recently demonstrated by Verlot et al . Note that the phenomenon of parametric oscillation instability can occur at some pump power threshold when the the Stokes field coincides with the cavity resonance, which has been predicted by Braginsky  and demonstrated for the first time at Caltech . To better demonstrate the transistor action of the system, we plot the transmission of the signal field as a function of the signal-cavity detuning in Fig. 3(a) for pW, pW and pW, respectively. It is clearly seen that transmission is greatly enhanced around the signal-cavity detuning at the higher pump power. Fig. 3(b) summarizes the transistor characteristic curve by plotting the gain of the transmitted signal field as a function of the pump power when the two-photon detuning . The transmitted signal field can be amplified when the pump power increases above a critical value. Similar results have been obtained experimentally recently . Therefore, the transmitted signal field can be attenuated or amplified in this coupled system under the control of the strong pump field when the cavity-pump detuning .
In conclusion, we have demonstrated that the coupled nanomechanical resonator-microwave cavity system can serve as a photonic transistor when the cavity is pumped on its blue sideband. The transmitted signal field can be amplified greatly, depending on the power of the pump field. The photonic transistor is based on the interaction of photons and phonons, where the pump field (photons) can be converted into mechanical vibrations (phonons). Quantum interference effect between the generated Stokes field and the signal field is responsible for the transistor action of the coupled system. This scheme proposed here can be achievable immediately in current experiments .
-  H. M. Gibbs, Optical Bistability: Congrolling light with light (Academic, Orlando, 1985).
-  J. L. O¡¯Brien, Science 318, 1567 (2007).
-  D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, Berlin, 2000).
-  D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, Nat. Phys. 3, 807 (2007).
-  F. Y. Hong and S. J. Xiong, Phys. Rev. A 78, 013812 (2008).
-  J. Hwang, M. Pototschnig, R. Lettow, G. Zumofen, A. Renn, S. Götzinger, and V. Sandoghdar, Nature 460, 76 (2009).
-  J. Tominaga, C. Mihalcea, D. B¨¹chel, H. Fukuda, T. Nakano, N. Atoda, H. Fuji, and T. Kikukawa, Appl. Phys. Lett. 78, 2417 (2001).
-  S. Medhekar and R. K. Sarkar, Opt. Lett. 30, 887 (2005).
-  Y. Y. Huang and S. T. Ho, Opt. Express 16, 16806 (2008).
-  S. Weis, R. Rivière , S. Deléglise, E. Gavartin , O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
-  G. S. Agarwal and S. Huang, Phys. Rev. A 81, 041803 (2010).
-  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 472, 69 (2011).
-  C. A. Regal, J. D. Teufel, and K. W. Lehnert, Nat. Phys. 4, 555 (2008).
-  D. Vitali, P. Tombesi, M. J. Woolley, A. C. Doherty, and G. J. Milburn, Phys. Rev. A 76, 042336 (2007).
-  M. J. Woolley, A. C. Doherty, and G. J. Milburn and K. C. Schwab, Phys. Rev. A 78, 062303 (2008).
-  T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, Nature (London) 463, 72 (2010).
-  J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, Nature 471, 204 (2011).
-  J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, Phys. Rev. Lett. 101, 263602 (2008).
-  C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, Phys. Rev. A 77, 033804 (2008).
-  V. Giovannetti and D. Vitali, Phys. Rev. A 63, 023812 (2001).
-  F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys. Rev. Lett. 99, 093902 (2007).
-  S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, Nature 460, 724 (2009).
-  R. W. Boyd Nonlinear Optics (San Diego, CA: Academic) (2008).
-  T. J. Kippenberg and K. J. Vahala, Opt. Express. 15, 17172 (2007).
-  S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, Phys. Rev. Lett. 99, 213601 (2007).
-  R. Kanamoto and P. Meystre, Phys. Rev. Lett. 104, 063601 (2010).
-  C. W. Gardiner and P. Zoller Quantum Noise (Springer) (2004).
-  A. Lezama, S. Barreiro, and A. M. Akulshin, Phys. Rev. A 59, 4732 (1999).
-  B. R. Mollow, R. J. Glauber, Phys. Rev. 160, 1076 (1967).
-  P. Verlot, A. Tavernarakis, T. Briant, P.-F. Cohadon, and A. Heidmann, Phys. Rev. Lett. 104, 133602 (2010).
-  V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Phys. Lett. A 287, 331 (2001).
-  T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
-  F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, M. A. Sillanpää, Nature 480, 351-154 (2011).
Figure 1 Schematic of a nanomechanical resonator capacitively coupled to a microwave cavity denoted by equivalent inductance and equivalent capacitance in the presence of a strong pump (‘gate’) field and a weak signal (‘source’) field . The transmitted signal field can be probed using a low-noise high-electron-mobility-transistor (HEMT) microwave amplifier.
Figure 2 The normalized magnitude of the cavity transmission as a function of signal-cavity detuning for various pump powers with (a) and (b) , respectively. (c) Level diagram of the coupled system under red-detuned pumping () and blue-detuned pumping (), respectively. The signal field probes the transition in which the mechanical occupation is unchanged. Other parameters used are GHz, kHz, Hz, =40 Hz, and MHz.
Figure 3 (a) Amplification of the signal field around the region for three different pump powers with . (b) The photonic transistor characteristic curve by plotting the gain of the transmitted signal field with respect to the pump power when the signal field is resonant with the cavity resonance frequency.