Nonlinear Phonon Interferometry at the Heisenberg Limit H. F. H. Cheung, Y. S. Patil, L. Chang, S. Chakram and M. Vengalattore Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853∗ Interferometers operating at or close to quantum limits of precision have found wide applica- tion in tabletop searches for physics beyond the standard model, the study of fundamental forces and symmetries of nature and foundational tests of quantum mechanics. The limits imposedp by quantum fluctuations and measurement backaction on conventional interferometers (δφ ∼ 1= N) have spurred the development of schemes to circumvent these limits through quantum interference, multiparticle interactions and entanglement. A prominent example of such schemes, the so-called SU(1; 1) interferometer, has been shown to be particularly robust against particle loss and inefficient detection, and has been demonstrated with photons and ultracold atoms. Here, we realize a SU(1; 1) interferometer in a fundamentally new platform in which the interfering arms are distinct flexural modes of a millimeter-scale mechanical resonator. We realize up to 15.4(3) dB of noise squeezing and demonstrate the Heisenberg scaling of interferometric sensitivity (δφ ∼ 1=N), corresponding to a 6-fold improvement in measurement precision over a conventional interferometer. Our work ex- tends the optomechanical toolbox for the quantum manipulation of macroscopic mechanical motion and presents new avenues for studies of optomechanical sensing and the nonequilibrium dynamics of multimode optomechanical systems. Interferometers are an indispensable metrological tool interactions via radiation pressure [21], geometric design in the study of fundamental forces [1,2], the search for [22] or reservoir engineering [23]. This raises prospects of physics beyond the standard model [3] and the measure- manipulating macroscopic mechanical states in the quan- ment of fundamental constants [4]. The realization that tum regime with techniques similar to those in quantum conventional interferometry is limited by quantum fluc- or atom optics. tuations and measurement backaction has led to the con- In this work, we realize a nonlinear phonon interfer- cept of the standard quantum limit (SQL) [5]. This has ometer in a millimeter-scale mechanical resonator and spurred efforts to observe quantum effects in macroscopic demonstrate Heisenberg scaling of phase sensitivity with interferometers [6] and to circumvent the SQL via entan- phonon number. By using quantum-compatible two- glement [7,8], multiparticle interactions [9] and quantum mode nonlinearities to create strong correlations between interference [10, 11]. In a broader sense, these efforts have the two modes of the interferometer, we demonstrate up led to theoretical studies aimed at elucidating the metro- to 15.4(3) dB of noise squeezing and a 6-fold enhance- logical precision of a macroscopic quantum many-body ment in measurement sensitivity over a conventional in- system and the interplay between entanglement, many- terferometer. The features of our nonlinear interferom- body interactions, topology and nonlinearities [12{15], as eter are accurately captured by a model applicable to a well as experimental efforts to investigate these questions generic pair of parametrically coupled oscillators which in atomic, solid state and hybrid quantum systems. also shows that the achievable noise reduction is, in prin- Optomechanical systems have emerged as a promis- ciple, unbounded, enabling an unrestricted improvement arXiv:1601.02324v1 [quant-ph] 11 Jan 2016 ing arena for the investigation of these foundational as- in signal-to-noise ratio. pects of quantum metrology and the innovation of novel A schematic of the nonlinear phonon interferometer is precision measurement technologies [16]. The enormous shown in Fig.1. While this schematic highlights the nom- range of size and mass, spanning the nanoscale to the inal similarities to a Mach-Zehnder or a Ramsey interfer- macroscale, the long coherence times that compare fa- ometer, we note two key differences. First, the arms of the vorably with those realized in atomic or solid state spin interferometer consist of two distinct mechanical modes systems [17{20], and the ability to cool, probe and control of a silicon nitride (SiN) membrane resonator. The mo- mechanical motion with radiation pressure have aided tion of these modes can be spectroscopically resolved and these efforts. While optomechanical interactions have independently measured via an optical interferometer as thus far been mainly in the weak coupling regime, recent described in previous work [23]. Unlike in the optical do- work has demonstrated the possibility of realizing strong, main, the phonons in this interferometer are necessarily quantum-compatible nonlinear or multimode mechanical confined within a cavity, i.e. the mechanical resonator, (a) (b) squeezed state preparation PA t ϕ PA coherent mixing signal mode BS tBS PA BS weak measurement idler mode 0 tf Time FIG. 1. A SU(1; 1) phonon interferometer. (a) The two arms of the interferometer are distinct mechanical modes at frequences !s and !i. A parametric amplifier interaction (PA) between the two modes generates strong correlations between these modes. A phase shift of interest ' is then imparted to the signal mode. A timed and pulsed beam splitter interaction (BS) between the modes coherently mixes the two correlated arms resulting in reduced quadrature noise at the outputs. (b) The timing sequence : The input to the interferometer is the coherent state jαs; 0i prepared at t < 0. The signal and idler get correlated during the parametric amplifier pulse for time tPA. After a variable interaction period, the two modes are coherently mixed by the beam splitter pulse for time tBS , followed by a weak measurement of the output modes. and do not freely propagate. In this sense, the mechanical terferometric modes, hereafter referred to as the `signal' modes are more analogous to intracavity optical fields. and `idler' modes, with resonance frequencies !s and !i While these modes are coupled to a thermal reservoir, respectively,g ~S; g~D are coupling strengths between the their finite response time allows us to transiently over- two modes at the sum and difference frequencies, and come the deleterious effects of the environmental coupling XS(t);XD(t) are the amplitudes of the supporting sub- and generate strong two-mode correlations. Second, in strate modes (`pump') at the sum and difference frequen- contrast to a conventional interferometer, a nondegen- cies. The first term represents the nondegenerate para- erate parametric amplifier takes the place of the input metric oscillator that causes the correlated production of beamsplitter. As proposed in [24], such a configuration, down-converted phonons in the signal and idler modes. also referred to as a SU(1; 1) interferometer, exhibits in- The second term signifies the beamsplitter interaction terferometric sensitivity surpassing the SQL due to the that results in the coherent exchange of phonons between two-mode correlations created by the parametric interac- the signal and idler modes. tion [25, 26]. Importantly, in contrast to interferometry The parametric amplifier and beamsplitter interactions with squeezed or entangled input states, the Heisenberg in our system are independently ascertained. For the ex- scaling of sensitivity in the SU(1; 1) interferometer has periments described below, the resonance frequencies and been shown to be robust to particle loss and inefficient damping rates of the signal and idler modes are !s=2π = detection [27]. 1:233 MHz, !i=2π = 1:466 MHz and γs=2π = 0:083(2) Hz, γi=2π = 0:108(3) Hz. As is well known in quantum The nonlinear phonon interferometer is described by optics, the parametric amplifier shows an instability when the interaction Hamiltonian (see SI) driven past a critical pump amplitude, XS;th, where the system is characterized by a divergent mechanical sus- g~ X (t) H (t) = i S S (ayay − a a ) ceptibility and critical dynamics. This instability can be int ~ 2 s i s i regarded in terms of a nonequilibrium continuous phase g~DXD(t) y y + i~ (asai − asai ) (1) transition [28, 29]. When the substrate is driven beyond 2 this threshold, the signal and idler modes self-oscillate, where as; ai are the annihilation operators of the two in- achieving a steady state when their decay rate matches 2 q (a) 5 2 2 rs;i = xs;i + ys;i [30]. The cross-quadrature and ampli- 4 tude sum, difference squeezing phase diagram is shown 3 in Fig.2(a), and shows excellent agreement with a no- free-parameter calculation based on the model and inde- . v 2 pendently measured damping and frequency parameters. The growth of the steady state amplitude of the modes Std. De above threshold shown in Fig.2(b) is measured to have 1 9 8 a power-law growth with exponent 0:53 ± 0:03, in close 7 agreement with the theoretical prediction (1=2). Also, 6 5 the exponential growth rate of the modes' amplitudes in- creases linearly in the drive strength µ, as predicted by 0 1 2 3 μ our parametric amplifier model. (b) The mechanical beamsplitter interaction [31, 32] real- ] 3 izes a coherent transfer of quanta between the signal and Amplitude 3.0 1.5 Inverse Rise Signal idler modes. By making weak optical measurements of Idler the signal and idler modes, this interaction can also be 2.0 1.0 realized in post hoc analysis. In this work, we perform weak, independent measurements on the signal and idler T ime [/s] Amplitude [/10 output modes with minimal backaction, and coherently 1.0 0.5 combine the measured quadratures to effect the coherent beamsplitter. Back-action evading measurements [33] al- Inverse Rise Time Signal low for such a realization of the beamsplitter to be ex- Normalized Idler 0 0.0 tended into the quantum regime. 1.0 1.5 2.0 2.5 3.0 As shown in Fig.1(b), the nonlinear interferometer is μ realized in the time domain.