C44 Vol. 34, No. 6 / June 2017 / Journal of the Optical Society of America B Research Article Effects of photon scattering torque in off-axis levitated torsional cavity optomechanics 1, 1 1 1 2 M. BHATTACHARYA, *B.RODENBURG, W. WETZEL, B. EK, AND A. K. JHA 1School of Physics and Astronomy, Rochester Institute of Technology, Rochester, New York 14623, USA 2Department of Physics, Indian Institute of Technology, Kanpur 208016, India *Corresponding author: [email protected] Received 23 January 2017; revised 28 April 2017; accepted 28 April 2017; posted 3 May 2017 (Doc. ID 285304); published 25 May 2017 We consider theoretically a dielectric nanoparticle levitated in an optical ring trap inside a cavity and probed by an angular lattice, with all electromagnetic fields carrying orbital angular momentum. Analyzing the torsional mo- tion of the particle about the cavity axis, we find that photon scattering from the trap beam plays an important role in the optomechanical system. First we show that the presence of the torque introduces an instability. Subsequently, we demonstrate that for bound motion near a stable equilibrium, varying the optical torque strength allows for tuning the linear optomechanical coupling. Finally, we indicate that the relative strengths of the linear and quadratic couplings can be detected directly by homodyning the cavity output. Our studies should be of interest to researchers exploring quantum mechanics using torsional optomechanics. © 2017 Optical Society of America OCIS codes: (080.4865) Optical vortices; (140.4780) Optical resonators; (260.6042) Singular optics. https://doi.org/10.1364/JOSAB.34.000C44 1. INTRODUCTION optomechanical coupling in the system. To support our analy- Levitated systems have lately emerged as exciting platforms for sis, we first present a quantum master equation for the physical exploring optomechanical effects such as cooling [1], sensing configuration of interest. We then consider the ensuing [2,3], and matter–wave interferometry [4]. A unique advantage Langevin equations in the classical limit, which provide quan- presented by such systems is their lack of mechanical clamping titative details regarding stability and optomechanical coupling. to any substrate or support. One consequence of this feature is To conclude, we demonstrate that homodyning the output of excellent isolation from environmental, particularly thermal, the cavity allows one to measure the amount of linear optome- disturbances. Studies of linear mechanical oscillations of opti- chanical coupling present in the system. cally levitated particles have been carried out both with [5] and without [6] cavities. Torsional optomechanics of levitated particles has also been theoretically proposed [7,8] and exper- 2. MODEL OF THE PHYSICAL CONFIGURATION imentally realized [9]. The configuration of interest is shown in Fig. 1. In this section In this paper we consider the torsional optomechanics of a we develop the mathematical model corresponding to the physi- levitated subwavelength-size dielectric nanoparticle confined in cal system, culminating in the presentation of the quantum a ring trap in a cavity and probed by an azimuthal lattice, as master equation and the ensuing classical Langevin equations. shown in Fig. 1. This configuration was recently proposed as a rotational analog of linear cavity optomechanics, in the case A. Electric Fields where the particle is free to rotate fully about the cavity The net electric field has four contributions: axis [10]. In contrast to previous work on torsional optomechanics E E E E E ; (1) [7–9], the oscillations in the proposed configuration are off- t p c b beam axis. All the optical beams in the configuration carry orbital angular momentum (OAM). As has been shown earlier, where E t is the electric field of the trap beam, Ep is the azimu- the photons scattered from such beams exert an optical torque thal lattice probe used to excite the cavity mode E c, and E b is on particles in vacuum [11]. In the present work we show that the field of all the background modes into which photons depending on the angular location of the particle, this scatter from the nanoparticle. We now describe each of these torque can lead to an instability or enable tunable linear fields individually. 0740-3224/17/060C44-08 Journal © 2017 Optical Society of America Research Article Vol. 34, No. 6 / June 2017 / Journal of the Optical Society of America B C45 and assumed to be mode-matched to the cavity mode at one of M1 the mirrors. The energy of this mode is Z ∞ z f ℏω † ω ω ω H p pap ap d ; (6) 0 LG 2 LG 2 trap lattice and its coupling with the cavity field is given by Z y ∞ R i ℏ γ ω † ω − ω H cp i a ap h:c:d ; (7) x m pffiffiffiffiffiffiffiffi 0 where γ ω ≅ κ∕π is the coupling constant in terms of the cavity decay rate κ [12]. M 2 3. Background Fields The background field is the continuum of modes that the Fig. 1. Torsional optomechanical setup considered in this paper. A nanoparticle scatters light into. In the plane wave basis the dielectric nanosphere of mass m is trapped on an optical ring of radius electric field is given by [13] R created by a beam carrying OAM l t 2 and by an azimuthal lattice Z X 1 arising from the interference of two degenerate OAM l 2 beams. ℏω 2 3 k ik:r E k eμ k μ k (8) The resulting motion of the particle consists of harmonic oscillations b i d π3ε a e h:c:; μ 16 0 about the beam axis. Mirrors M 1 and M 2 constitute the cavity. and the free field energy follows as X Z † 1. Trap Field b ℏω μ k k 3k H f ka aμ d ; (9) The trap field is a Laguerre–Gaussian (LG) beam carrying μ OAM l t , where the index μ indicates the polarization of the photon. R2−ρ2 ρ jl t j ϕ− ω ω ω2 ilt i t t z E E 2 t 4. Total Field Hamiltonian t 0 e 0 cos c:c:; (2) R c The total field Hamiltonian is where E is the fieldpffiffiffiffi amplitude, ρ is the cylindrical radial co- 0 f f f i b ω H H c H p H H : (10) ordinate, R 0 l t is the radius at which the field intensity is field cp f ω ϕ a maximum at z 0, 0 is the trap beam waist, is the ω cylindrical angular coordinate, t is the trap beam frequency, B. Mechanical Degrees of Freedom and c is the speed of light. Here we have ignored the curvature The Hamiltonian of mechanical motion is of the beam wavefront as well as the contribution due to the 2 p2 2 2 f jpj ρ pz Lz Gouy phase. H m ; (11) 2m 2m 2m 2I 2. Probe and Cavity Fields where m is the mass; pρ and pz are the radial and axial linear ρ2 ≈ 2 The angular lattice in Fig. 1 arises from a superposition of two momenta, respectively; and Lz and I m mR are the ω copropagating LG beams with OAM l, beam waist 0, and angular momentum and moment of inertia about the cavity ω ≠ ω degenerate frequencies c t . The polarizations of the trap (i.e., z) axis of the nanoparticle, respectively. and probe beams will be assumed to be orthogonal, although we will not assign them explicit polarizations below. For the C. Optomechanical Interaction moment we consider thes probeffiffiffiffiffiffiffiffi beam quantum mechanically, The dielectric nanoparticle interacts with the total electric field through the induced dipole interaction given by ℏω Z E cψ ra h:c:; (3) ε ε c ϵ 1 0 r 0 H i − P r · E rd3r ≅ − V jEj2; (12) inf 2 2 where a and a† are the canonical creation and annihilation V π 3∕ ε operators, respectively, of the cavity mode obeying the commu- where V 4 r 3 is the volume, r is the radius, and r is the tation rule a; a†1, with the mode function relative dielectric permittivity of the nanoparticle. A number of terms arise when Eq. (1) is used in Eq. (12). We will describe ρ jlj − ρ2 ω jE j2 jE j2 E E 1 2ω2 cz below the terms containing t , c , and t · b only, ψ rpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi e 0 cos lϕ cos ; L π2ω2jlj! ω l c which yield the optical trap, the optomechanical coupling, c 0 0 and the scattering of photons from the trap beam, respectively. (4) The remaining terms either vanish identically (due to the where Lc is the length of the cavity. The free-field Hamiltonian polarization orthogonality of the trap and probe beams) or are of this mode is small shifts in position and frequency which we have absorbed into the appropriate definitions. f ℏω † (5) H c ca a; For the parameters considered in this paper, photon scatter- where we have dropped a factor of 1∕2 which has no dynamical ing from the probe is negligible in comparison to the scattering significance. The probe is a field defined outside of the cavity from the trap. C46 Vol. 34, No. 6 / June 2017 / Journal of the Optical Society of America B Research Article Z 1. Optical Trap ε ε ε ε i − 0 r E r E r 3r ≅ − 0 r E r E r H inf t · b d V t · b : Using Eq. (2) in Eq. (12), expanding around the points 2 V 2 ρ − qρ R 0 and qz z 0, up to second order in the (21) coordinates, the optical trapping potential is given by ε ε 0 r 2 m 2 2 2 2 2.