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arXiv:1708.03294 · Physical Review Letters

Imaging Correlations in Heterodyne Spectra for Quantum Displacement Sensing

A. Pontin, J. E. Lang, Avishek Chowdhury, Paolo Vezio, Francesco Marino, Bruno Morana, Enrico Serra, Francesco Marin, T. S. Monteiro

Published Jan 10, 2018• 16 citations
arXivHTMLPDF

Abstract

The extraordinary sensitivity of the output field of an optical cavity to small quantum-scale displacements has led to breakthroughs such as the first detection of gravitational waves and of the motions of quantum ground-state cooled mechanical oscillators. While heterodyne detection of the output optical field of an optomechanical system exhibits asymmetries which provide a key signature that the mechanical oscillator has attained the quantum regime, important quantum correlations are lost. In turn, homodyning can detect quantum squeezing in an optical quadrature but loses the important sideband asymmetries. Here we introduce and experimentally demonstrate a new technique, subjecting the autocorrelators of the output current to filter functions, which restores the lost heterodyne correlations (whether classical or quantum), drastically augmenting the useful information accessible. The filtering even adjusts for moderate errors in the locking phase of the local oscillator. Hence we demonstrate the single-shot measurement of hundreds of different field quadratures allowing the rapid imaging of detailed features from a simple heterodyne trace. We also obtain a spectrum of hybrid homodyne-heterodyne character, with motional sidebands of combined amplitudes comparable to homodyne. Although investigated here in a thermal regime, the method's robustness and generality represents a promising new approach to sensing of quantum-scale displacements.

DOI
10.1103/physrevlett.120.020503

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The extraordinary sensitivity of the output field of an optical cavity to small quantum-scale displacements has led to breakthroughs such as the first detection of gravitational waves and of the motions of quantum ground-state cooled mechanical oscillators. While heterodyne detection of the output optical field of an optomechanical system exhibits asymmetries which provide a key signature that the mechanical oscillator has attained the quantum regime, important quantum correlations are lost. In turn, homodyning can detect quantum squeezing in an optical quadrature but loses the important sideband asymmetries. Here we introduce and experimentally demonstrate a new technique, subjecting the autocorrelators of the output current to filter functions, which restores the lost heterodyne correlations (whether classical or quantum), drastically augmenting the useful information accessible. The filtering even adjusts for moderate errors in the locking phase of the local oscillator. Hence we demonstrate the single-shot measurement of hundreds of different field quadratures allowing the rapid imaging of detailed features from a simple heterodyne trace. We also obtain a spectrum of hybrid homodyne-heterodyne character, with motional sidebands of combined amplitudes comparable to homodyne. Although investigated here in a thermal regime, the method's robustness and generality represents a promising new approach to sensing of quantum-scale displacements.
Homodyne and heterodyne detection represent "twin-pillars" of quantum displacement sensing using optical cavities, having permitted major breakthroughs including detection of gravitational waves and of the motion of quantum ground-state cooled mechanical oscillators. Both can suffer disadvantages as diagnostics in quantum optomechanics, either through symmetrisation (homodyne), or loss of correlations (heterodyne). We show that, for modest heterodyne beat frequencies (Ω∼ωM/10≫ΓΩ\sim ω_M/10 \gg ΓΩ∼ωM​/10≫Γ), judicious construction of the autocorrelation of the measured current can either recover (i) a spectrum with strong sidebands but without an imprecision noise floor (ii) a spectrum which is a hybrid, combining both homodyne and heterodyne sideband features. We simulate an experimental realisation with stochastic numerics and find excellent agreement with analytical quantum noise spectra. We term such retrospective recovery of lost heterodyne correlations "r-heterodyning": as the method simply involves post-processing of a normal heterodyne time signal, there is no additional experimental constraint other than on the magnitude of ΩΩΩ.
Abstract Cavity optomechanics has achieved the major breakthrough of the preparation and observation of macroscopic mechanical oscillators in non-classical states. The development of reliable indicators of the oscillator properties in these conditions is important also for applications to quantum technologies. We compare two procedures to infer the oscillator occupation number, minimizing the necessity of system calibrations. The former starts from homodyne spectra, the latter is based on the measurement of the motional sideband asymmetry in heterodyne spectra. Moreover, we describe and discuss a method to control the cavity detuning, that is a crucial parameter for the accuracy of the latter, intrinsically superior procedure.
We consider a system where an optical cavity mode is parametrically coupled to a mechanical oscillator. A laser beam driving the cavity at its resonance frequency will acquire red- and blue-shifted sidebands due to noise in the position of the mechanical oscillator. In a classical theory without noise in the electromagnetic field, the powers of these sidebands are of equal magnitude. In a quantum theory, however, an asymmetry between the sidebands can be resolved when the oscillator's average number of vibrational excitations (phonons) becomes small, i.e., comparable to 1. We discuss the interpretation of this sideband asymmetry in a heterodyne photodetection measurement scheme and show that it depends on the choice of detector model. In the optical regime, standard photodetection theory leads to a photocurrent noise spectrum given by normal- and time-ordered expectation values. The sideband asymmetry is in that case a direct reflection of the quantum asymmetry of the position noise spectrum of the mechanical oscillator. Conversely, for a detector that measures symmetric, nonordered expectation values, we show that the sideband asymmetry can be traced back to quantum optomechanical interference terms. This ambiguity in interpretation applies not only to mechanical oscillators, but to any degree of freedom that couples linearly to noise in the electromagnetic field. Finally, we also compare the quantum theory to a fully classical model, where sideband asymmetry can arise from classical optomechanical interference terms. We show that, due to the oscillator's lack of zero-point motion in a classical theory, the sidebands in the photocurrent spectrum differ qualitatively from those of a quantum theory at sufficiently low temperatures. We discuss the observable consequences of this deviation between classical and quantum theories.
As part of the effort to make use of squeezed states of light for detection of sub-shot-noise optical signals, we study the balanced heterodyne scheme, for which the corresponding spectral density of the photocurrent fluctuations produced at the output of the detector is calculated as the Fourier transform of their autocorrelation function. Our analysis shows that, for maximal signal-to-noise ratio enhancement by use of squeezed states of light, an optical signal to be measured must be carried in the squeezed quadrature of the carrier field. We discuss how "the additional heterodyne noise" can be eliminated in this scheme and its potential application to gravitational-wave searching. To demonstrate the practical feasibility, we propose and study a phase-locking technique for this scheme.
High-frequency squeezed states are important for quantum metrology and information processing, but quadrature measurements at gigahertz frequencies remain challenging. Balanced homodyne detection (BHD), the standard approach, requires closely matched complex transfer functions in both photodetection channels to suppress local-oscillator (LO) noise and maintain a well-defined readout quadrature. This matching becomes increasingly difficult at high bandwidths. Strongly asymmetric single-photodiode homodyne detection avoids this requirement, but achieving high signal efficiency and shot-noise clearance typically requires hundreds of milliwatts of LO power and does not suppress technical LO sidebands. I propose cavity-assisted homodyne detection, in which the quantum field and LO enter separate ports of an impedance-matched traveling-wave cavity. The resonant LO carrier is transmitted to the photodetector, where it beats with off-resonant signal sidebands reflected from the cavity. At the same time, technical LO sidebands outside the cavity linewidth are suppressed. I derive the quantum input-output relations and linearized photocurrent including intracavity loss, and show that near-unity signal-transfer efficiency can be achieved at moderate incident LO power. The cavity linewidth and free spectral range determine the usable detection band.
We propose an ultrasensitive displacement measurement scheme to overcome the standard quantum limit (SQL) in the unresolved sideband cavity optomechanical system with nonlinear optomechanical coupling and squeezed light injection. By introducing the optimized quantum correlation, which is enabled by suitable choices of the squeezing angle, squeezing level, power of the probe light, and measurement angle of homodyne detection, the off-resonant displacement sensitivity reaches 6 dB below the SQL in linear optomechanical coupling. In contrast, displacement sensitivities with a coherent probe plus variational readout and squeezed probe plus fixed measurement angle (phase quadrature) are 2.6 dB and 4.6 dB below the SQL, respectively. By combining linear and quadratic optomechanical coupling, we show that the displacement sensitivity can be further improved to 9.6 dB below the SQL. Our results have potential applications in gravitational-wave detectors, quantum metrology, and the search for dark matter.