Optical Squeezing for an Optomechanical System Without Quantizing the Mechanical Motion
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PHYSICAL REVIEW RESEARCH 2, 023208 (2020) Optical squeezing for an optomechanical system without quantizing the mechanical motion Yue Ma, 1,* Federico Armata,1,† Kiran E. Khosla,1,‡ and M. S. Kim1,2,§ 1QOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom 2Korea Institute of Advanced Study, Seoul 02455, South Korea (Received 16 October 2019; revised manuscript received 24 March 2020; accepted 30 March 2020; published 21 May 2020) Witnessing quantumness in mesoscopic objects is an important milestone for both quantum technologies and foundational reasons. Cavity optomechanics offers the ideal system to achieve this by combing high-precision optical measurements with mechanical oscillators. However, mechanical quantumness can only be established if the behavior is incompatible with any classical description of an oscillator. After explicitly considering classical and hybrid quantum-classical descriptions of an optomechanical system, we rule out squeezing of the optical field as such a witness by showing it is also predicted without quantizing the mechanical oscillator. DOI: 10.1103/PhysRevResearch.2.023208 I. INTRODUCTION [24–29]. Characterizing which aspects of the optical field can demonstrate the quantumness of the mechanical oscillator is Witnessing the quantum nature of a physical system is a of foundational interest. This has been briefly investigated central and recurrent goal in physics. Quantumness can only in optomechanics in the context of tracing the origin of be unambiguously demonstrated when predictions based on experimentally observed sideband asymmetry [30–32]. A di- all possible classical theories are violated [1], for example by rect theoretical comparison between quantum and classical the violation of a Bell inequality [2–4], or detecting Wigner descriptions of the optomechanically generated phase has negativity via tomographic reconstruction [5–8]. However, for been considered in light of interferometric experiments [33]. experiments without such unambiguous witnesses, a classical Here, we focus on optomechanically generated squeezing description may predict the observed result. In this case, even of a single-mode optical field [34,35], previously observed in a nonclassical state may not necessarily have its nonclassical- several experiments [36–38], and investigate whether optical ity revealed. This has already been pointed out in a variety of squeezing is a signature for mechanical quantumness. The systems, including Josephson oscillator [9], Rabi oscillation experiments operate in the linearized regime and are there- in light-atom interaction [10], and optical Berry phase [11,12]. fore expected to reveal less quantumness than the nonlinear Here and in the following, we use the word “quantumness” to optomechanical interaction [39,40]. In this work, we consider mean the necessity of quantization given certain experimental the latter, thereby maximizing the possibility for detecting me- setups, irrespective of any nonclassical features in the state as chanical quantumness in optomechanics. We examine classi- described by quantum mechanics. cal, quantum, and hybrid quantum-classical theories, focusing Several general, operational criteria for characterizing on the temporal evolution of the field quadrature variance. We quantumness have been proposed [1,13–15], most of which show that optical squeezing is predicted without quantizing are directly applicable to photonics where one has direct the mechanical oscillator and is therefore eliminated as a access to the relevant mode. However, one would also like mechanical quantumness witness, regardless of the oscillator to study the quantumness criterion of a mode without direct temperature. We emphasize the aim of the work as ruling access, with optomechanics being a typical example [6]. out optical quadrature squeezing as evidence for mechanical Optomechanics uses high-precision control of optical or mi- quantumness, instead of proposing a new experimental wit- crowave fields to manipulate and read out the micromechan- ness for it. ical motion of massive oscillators, and has been widely stud- ied for both foundational [16–23] and practical applications II. QUANTUM DESCRIPTION *[email protected] We examine a closed optomechanical system with the in- †[email protected] tracavity field initialized to a coherent state. The joint system ‡[email protected] undergoes unitary evolution and we compute the time evo- §[email protected] lution of the intracavity field variance. Note that this setup is different from most experiments [6], where the cavity is driven Published by the American Physical Society under the terms of the by an external laser, the output field from the cavity is mon- Creative Commons Attribution 4.0 International license. Further itored, and the system is subject to dissipation. Laser driving distribution of this work must maintain attribution to the author(s) and dissipation introduce open system dynamics, smearing and the published article’s title, journal citation, and DOI. out any mechanical quantumness signature. We neglect these 2643-1564/2020/2(2)/023208(9) 023208-1 Published by the American Physical Society MA, ARMATA, KHOSLA, AND KIM PHYSICAL REVIEW RESEARCH 2, 023208 (2020) FIG. 1. (a) A Fabry-Pérot cavity with one movable mirror described as a harmonic oscillator. (b) Field variance as a function of time for the quantum (blue, “Q”) and classical (red, “C”) description, mean-field hybrid descriptions with constant I (gray, “SC1”), Poisson random I (orange, “SC2”), and Gaussian random I (green, “SC3”). All plots use α = 20, k = 0.01. A line containing two alternating colors means the two corresponding descriptions predict indistinguishable results. (c) Field variance as a function of time for the hybrid measurement model with k = 0 (blue line), k = 0.1, x(0) = 0, p(0) = 0 (red line), k = 0.1, x(0), and p(0) as classical random variables simulating the zero-point fluctuation (pink line). All curves use α = 2, = 0.01ω. For comparison, the effect of cavity dissipation is shown for the hybrid model (green dashed line) with photon dissipation rate κ = ω, and the quantum model (brown dotted line) with κ = 0.3ω. The linewidth for numerical solutions represents the mean ± standard deviation. elements to maximize the mechanical quantumness effect in is the mechanical period, there is a quantum revival (half optical field variance. revival in the form of an optical cat state when t is an integer We consider a Fabry-Pérot cavity with a movable mirror, multiple of τ) where the variance rapidly returns to 1for shown in Fig. 1(a), the equilibrium frequency of the oscillator several mechanical periods before again stabilizing at 2α2 + (field) given by ω (). The Hamiltonian in the frame rotating 1. There is another qualitatively different revival after another with the optical frequency is [41,42] interval of t ≈ τ/4k2 (where |α→|−α). The quantum † g0 † † variance again rapidly decreases, this time below 1, indicating Hˆ /h¯ = ωbˆ bˆ − √ aˆ aˆ(bˆ + bˆ), (1) optical squeezing. The two quantum revivals are repeated 2 periodically. The behaviors are similar to the time evolution of wherea ˆ √(bˆ) is the optical (mechanical) annihilation operator phase-space distribution of an optical field in a Kerr medium and g0/ 2 is the single-photon optomechanical coupling [45], as the Hamiltonian given by Eq. (1) can be understood strength. The time evolution of the field in the Heisenberg as an (oscillator-mediated) intensity-dependent optical phase picture is [43] shift [46]. Note that at the end of each mechanical period, −iωt iωt † † the field and oscillator decouple [33]. The field variance is aˆ(t ) = eiA(t )/2ek[(1−e )bˆ−(1−e )bˆ ]eiA(t )ˆa aˆ aˆ, (2) √ independent of the oscillator state, and thus the appearance 2 where A(t ) = 2k (ωt − sin ωt ) and k = g0/( 2ω). In the of squeezing and revival are independent of the mechanical following, we will be interested in the θ-dependent quadrature temperature (see the Appendix). −iθ † iθ operator Xˆ θ (t ) = aˆ(t )e + aˆ (t )e and the corresponding The quantum description will serve as the reference with 2 2 variance Varθ (t ) =Xˆ θ (t )−Xˆ θ (t ) . Squeezing is defined which alternative classical or hybrid descriptions will be based on the minimum of the variance taken over all quadra- compared. Irreproducible field variance behaviors by other de- ture angles, Var(t ) ≡ minθ {Var θ (t )}.IfVar(t ) < 1, the field scriptions are candidates of mechanical quantumness witness. is squeezed at time t. Note that the effect of the mechanical commutation relation only comes in as the phase eiA(t )/2, and III. CLASSICAL DESCRIPTION thus is not able to change Var(t ). The intracavity field is initialized to a coherent state |αL The initial states for both the field and the oscillator have with α real (without loss of generality), and the mechanical well-defined positive phase-space distributions. Classical en- oscillator to a vacuum state (for generalization to the ini- semble dynamics can thus be defined by averaging over dif- tial thermal states, see the Appendix). The time evolution ferent time evolutions where the initial conditions are sampled of Var(t ) is shown in blue in Fig. 1(b) (see the analytical from probability distributions matching the initial quantum expressions in the Appendix). For simplicity, we only plot the states. nontrivial parts where different descriptions to be considered The Hamiltonian (1) is mapped into a classical Hamil- later predict distinguishable variances.