1 Single Photon Generation in a Cavity Optomechanical System induced by a Nonlinear Photonic Crystal Bijita Sarma and Amarendra K. Sarma* This strong coupling condition is hard to achieve in Abstract— We propose a novel scheme for realizing experiments. single-photon blockade in a weakly driven hybrid cavity In this letter, we propose to use a photonic crystal optomechanical system consisting of a nonlinear photonic cavity showing third order nonlinearity in-between the mirrors crystal. Sub-Poissonian statistics is realized even when the of the cavity to circumvent the issue of weak nonlinearity. single-photon optomechanical coupling strength is smaller Optomechanical photonic crystals offer the privilege of added than the decay rate of the optical mode. The scheme material-induced nonlinearity such as third-order nonlinear relaxes the requirement of strong coupling for photon susceptibility, in addition to the inherent optomechanical Kerr- blockade in optomechanical systems. It is shown that type nonlinearity [15]. Photonic crystals offer the advantage of photon blockade could be generated at the availability of diffraction-limited mode volumes and ultrahigh telecommunication wavelength. quality factors. In this letter, we show that the stringent requirement to have 푔 > 휅, in order to achieve single photon Index Terms— Cavity optomechanics, photon blockade in an optomechanical system, could be relaxed using antibunching, single photon source. our proposal. Also, it is worthy to be noted that the prospective long distance communication of single-photon I. INTRODUCTION requires the workability around 1550 nm, i.e. the so-called telecommunication wavelength, as the photons generated Generation of single photons is a pivotal requirement in should also have a wavelength that would give low loss and quantum information and communication technologies [1,2]. attenuation while travelling through an optical fiber. This issue The fingerprint of a single photon source is the observation of has not been explored much in optomechanical systems. In sub-Poissonian radiation when driven by a classical light field. this letter, we have considered the wavelength of the cavity Photon blockade is a possible mechanism, in which a strong radiation to be 1550 nm and show that it is possible to obtain optical nonlinearity makes the presence of one photon in the strong photon antibunching at this wavelength. system enough to prohibit simultaneous presence of another photon. In recent years, tremendous progress has been made in II. SYSTEM AND METHOD demonstrating single photon sources in diverse platforms, such as diamond nanophotonic circuits [3], solid-state systems [4,5], optical fibers [6], Bose-Einstein condensate [7] and photonic crystals [8]. In most cases, the prototype system that has been widely explored is cavity quantum electrodynamics. 휒(3) Recently, engineered quantum devices based on Photonic crystal 휔퐿 optomechanical systems have garnered much attention owing to numerous possible applications towards ground state cooling of mesoscopic oscillators, entanglement of optical and Fig. 1. Schematic diagram of a nonlinear optomechanical photonic crystal- mechanical modes, non-classical state generation, quantum cavity system driven by a resonating coherent field. state transfer between different modes etc. [9,10]. A typical optomechanical system consists of an optical cavity in which The schematic of the proposed system is depicted in Fig. 1, one of the end-mirrors is movable by the force exerted by where a Kerr-type photonic crystal is inserted in an light. The system is intrinsically nonlinear due to the coupling optomechanical cavity with one mirror fixed while the other between optical field and mechanical motion and exhibits one is movable. The cavity system is weakly driven by a laser Kerr-type nonlinearity. This feature of optomechanical with frequency 휔퐿. The cavity field frequency is 휔푎 . The systems has been utilized to propose single photon source or electromagnetic field couples with the mechanical mirror emitter by using the phenomenon of photon blockade [10-14]. motion via radiation pressure and results in an optomechanical However, as the effect of one photon tends to be very tiny, coupling term. In a rotating frame at the laser frequency, the conventional photon blockade in optomechanical systems Hamiltonian of the driven system is given by [8,9]: † † † † † † requires the system to be in the strong coupling regime, i.e. 퐻 = ℏ∆푎푎 푎 + ℏ휔푏푏 푏 + 푈푎 푎 푎푎 + ℏ푔푎 푎(푏 + 푏 ) + 푔 > 휅, where 푔 is the single-photon optomechanical coupling ℏΩ(푎 + 푎†). (1) strength and 휅 is the decay rate of the optical mode [11,12]. Here, 푎 (푎†) and 푏 (푏†) are the annihilation (creation) operators of the cavity field and mechanical resonator The authors are with Department of Physics, Indian Institute of Technology respectively. ∆ = 휔 − 휔 is the detuning of the cavity field Guwahati, Guwahati-781039, Assam, India. * [email protected] 푎 푎 퐿 and 휔푏 is the mechanical oscillator resonance frequency. 푔 = 2 휔푎 −19 2 2 푥푍푃퐹 is the single-photon optomechanical coupling 10 푚 /푉 while for some nano-particle doped glasses its 퐿 of the order of 10−16 푚2/푉2 in near-infrared [18,19]. The strength, where 퐿 is the length of the cavity and 푥푍푃퐹 is the zero-point fluctuation of the mechanical oscillator. Ω = value of 푛̅푟 typically varies between 2 to 4 [20]. √휅푃푛⁄ℏ휔퐿 is the drive amplitude, 푃푛 and 휅 = 휔푎/푄푎 being the input power and optical mode decay rate respectively. 푄푎 2 is the quality factor of the optical cavity. 푈 = 3(ℏ휔푎) 퐷/ (4휀0푉푒푓푓) is the strength of nonlinear Kerr interaction, where (3) 2 (3) 퐷 = 휒̅ /휀푟̅ , where 휒̅ and 휀푟̅ are, respectively, the average nonlinear susceptibility and relative dielectric permittivity of 3 the photonic crystal. 푉푒푓푓~(휆푎⁄2푛̅푟) is the cavity mode volume, considering difraction-limited confinement volume for a Si-based system, where 휆푎 is the wavelength of the cavity radiation and 푛̅푟 = √휀푟̅ [16]. The first and second term in Eq. (1) gives the energy of the cavity mode and the mechanical mode respectively. The third term accounts for the interaction energy between the cavity mode and the photonic crystal. The optomechanical interaction between the cavity mode and the mechanical oscillator is described by the fourth Fig. 2. Variation of zero-time-delay second-order correlation function with (3) 2 term. The last term describes the laser driving. normalized cavity detuning for different values of 퐷 = 휒̅ /휀푟̅ . The statistical properties of the photon field could be 3 described by the normalized zero-time delay second order The cavity mode volume is taken to be 푉푒푓푓 = 0.01 μm , correlation function: while Ω = 0.01휅 . The figure exhibits strong photon 〈푎†(푡)푎†(푡)푎(푡)푎(푡)〉 antibunching near zero cavity detuning in the system, the 푔(2)(0) = (2) 〈푎†(푡)푎(푡)〉2 maximum occurring for highest value of 퐷. It should be In the next section, we find out the second-order photon noted that unlike bulk media, here even at low optical correlation function by numerically solving the quantum intensity, the nonlinear susceptibility is significant due to low master equation of the system: mode volume, thanks to recent advances in nanofabrication 푑휌 = [휌, 퐻] + 퐿푎(휌) + 퐿푏(휌), (3) techniques. The role of the cavity mode volume 푉푒푓푓 on 푑푡 ℏ (2) 휅 † † † 푔 (0) is demonstrated in Fig. 3. Here we have assumed 퐷 = where 퐿 (휌) = (푛 + 1)(2푎휌푎 − 푎 푎휌 − 휌푎 푎) + 푎 2 푡ℎ,푎 −17 2 2 (2) 휅 † † † 훾 10 m /V . For higher values of 푉 , 푔 (0) value tends ( ) ( ) 푒푓푓 2 푛푡ℎ,푎 2푎 휌푎 − 푎푎 휌 − 휌푎푎 and 퐿푏 휌 = 2 (푛푡ℎ,푏 + † † † 훾 † † to show Poissonian statistics. However, as 푉푒푓푓 is lowered, the 1)(2푏휌푏 − 푏 푏휌 − 휌푏 푏) + 푛 (2푏 휌푏 − 푏푏 휌 − 2 푡ℎ,푏 cavity-field tends to show sub-Poissonian characteristics †) 휌푏푏 are the Liouvillian operators for the optical and phonon indicating photon antibunching. These results indicates that, modes respectively [10]. 훾 is the phonon decay rate. 푛푡ℎ,푎 and with judicious engineering, it is possible to obtain photon 푛푡ℎ,푏 are the thermal photon and phonon numbers given by blockade in this system even if we have 푔 < 휅. −1 푛푡ℎ, = [푒푥푝{ℏ휔/푘퐵푇} − 1] , where 푘퐵 is the Boltzmann constant. Thermal photon number is considered to be zero due to high frequency of optical radiation. III. RESULTS AND DISCUSSIONS The steady-state value of 푔(2)(0) could be treated as a figure of merit in order to quantify single photon blockade behavior [11]. The condition 푔(2)(0) < 1 corresponds to sub-Poisson feature of the cavity field, which is a nonclassical effect, referred to as photon antibunching or photon blockade, while 푔(2)(0) > 1 indicates super-Poissonian statistics which is a classical effect. On the other hand, 푔(2)(0) → 0 refers to the case of full photon blockade. For our numerical calculations we choose, in the weak single-photon optomechanical coupling regime i.e. 푔 < 휅, the Fig. 3. Variation of 푔(2)(0)with normalized cavity detuning for different following experimentally realistic parameters [16,17]: values of effective confinement volume 푉푒푓푓. 휔푎⁄2휋 = 205.6 THz, 휆푎 = 1.55 μm , 휔푏⁄2휋 = 9.5 GHz, 푔⁄2휋 = 292 kHz, 푄 = 107 , 훾 = 0.001휅 . Fig. 2 depicts The role of Kerr-nonlinear interaction, 푈 , on photon 푎 antibunching is illustrated in Fig. 4. In the inset, the variation 푔(2)(0) as a function of the normalized cavity detuning for of 푔(2)(0), at ∆ = 0, with normalized 푈 is plotted. It can be different values of the ratio 퐷. It should be noted that, as 푎 seen that when 푈 = 0 , i.e. in absence of the nonlinear reported in literatures, the magnitude of 휒(3) tensor elements (2)( ) for typical semiconductors are of the order of 10−18 − photonic crystal, 푔 0 = 1, referring to Poissonian photon 3 statistics, a classical behavior.