Heisenberg Scaling in a Continuous-Wave Interferometer

TL;DR

This work addresses the challenge of achieving Heisenberg-scale precision in continuous-wave interferometry. By co-designing a Mach-Zehnder interferometer fed with a pair of CW squeezed vacua and a nonlinear maximum-likelihood estimator applied to joint homodyne records, the authors realize a CW sensor whose phase-imprecision PSD scales faster than classical limits and approaches the Heisenberg bound, subject to realistic losses. The experimental results show sub-SQL performance across a wide spectral band and demonstrate that, with reduced losses, the scaling can approach arbitrarily close to the Heisenberg limit, highlighting the importance of integrating quantum input states with tailored measurement and estimation strategies. The findings have implications for low-power, high-precision sensing applications and establish a practical CW paradigm for quantum-enhanced metrology.

Abstract

Continuous-wave (CW) interferometry has stood at the frontier of precision measurement science since its inception, where it was used to search for the luminiferous ether, to the present day, where it forms the basis of interferometric gravitational-wave detection. Quantum theory predicts that this frontier can be expanded more rapidly by employing certain quantum resources, compared with the case of using only classical resources. In the quantum case, we can achieve ``Heisenberg scaling'', which manifests as a quadratic improvement over the best possible classical precision scaling. Although Heisenberg scaling has been demonstrated in pulsed operation, it has not been demonstrated for continuous operation. The challenge in doing so is two-fold: continuous measurements capable of Heisenberg scaling were previously unknown, and the requisite CW quantum states are fragile. Here we overcome these challenges and demonstrate the first CW interferometer exhibiting resource efficiency approaching Heisenberg scaling. Our scheme comprises a Mach-Zehnder interferometer illuminated with a pair of squeezed light sources, followed by a nonlinear estimator of the output homodyne record to estimate a differential phase modulation signal that drives the interferometer. We observe that this signal can be extracted with a precision that scales faster than what is allowed classically, and approaches the Heisenberg scaling limit.

Figures (4)

• Figure 1: Schematic of the squeezed light interferometer and phase estimator.(a) A pair of CW squeezed states, with relative phase $\theta_s$, are used to sense a phase signal, $\delta \phi(t)$, injected into an arm of a Mach-Zehnder interferometer. A pair of homodyne detectors transduce the quadratures of the output field corresponding to the angles $\theta_{1,2}$. The homodyne emits a record from which the phase signal is estimated by squaring and subtracting them. (b) The sinusoidal phase signal injected into the interferometer. (c) The resulting homodyne records are zero-mean with variances that oscillate like the phase signal, but out of phase. (d,e) illustrate the principle of our nonlinear estimator. (d) The rectified homodyne records, whose means oscillate out of phase. (e) Subtracting the rectified records recovers the injected phase signal. (f) The PSD of the estimator shows the phase signal as a spike at its frequency, riding atop white noise. This white noise is below shot noise (gray dashed line), and it exhibits Heisenberg scaling with photon flux, and saturates the spectral QCRB.
• Figure 2: Quantum noise characterization.(a) Variance of the quantum noise of the two squeezed states as a function of their quadrature angle, referenced to that of the vacuum. Quadrature angle is tuned via the lockpoint of the demodulation phase of the homodyne detectors used for the measurement (see text and Methods for details). Variance is estimated by integrating the homodyne photocurrent spectrum from 200 kHz to 700 kHz, which corresponds to the highest dark noise clearance of our homodyne detector (see Supplementary Information). (b, c) Simultaneous measurement of the quantum noise variances at the outputs of the interferometer using a pair of homodyne detectors. The homodyne demodulation phases $\theta_{1,2}^\prime$ are simultaneously swept for each relative squeezed field demodulation phase $\theta_\text{s}^\prime$. The black dots indicate the optimal operating point. (d) Model of the squeezing level on one homodyne versus the physical phases; model parameters are inferred from the fits in panel (a). (e) Numeric fit of the data in panels (b, c) to the model (see Supplementary Information). (f) SNR of the reconstructed phase signal $\delta \phi(t)$ as the relative squeezing angle ($\theta_\mathrm{s}'$) and one of the homodyne demodulation phase ($\theta_1'$) is swept. The other demodulation phase ($\theta_2^\prime$) is set to maximize anti-squeezing in panel (c). SNR of unity indicates that the estimated signal lies below the measurement noise.
• Figure 3: Phase precision vs photon flux.(a) PSD of the phase estimator as photon flux through the interferometer is increased. The injected phase signal at $2$ kHz is visible above the quantum noise-limited noise floor. (b) Phase precision (in units of phase PSD) inferred from the data in panel (a) as a function of photon flux, extracted from the maps in in \ref{['main:fig:calibration']}(b,c). Top axis shows the equivalent photon occupancy per mode. The error bars in the photon flux are estimated from measurements taken before and after each PSD measurement. The error bars in the phase precision are 1-sigma standards of deviation from a series of five PSD measurements and arise from system drifts, rather than from PSD estimation uncertainty. Solid black line shows the theoretical phase precision with the mean level of measured loss.
• Figure 4: Extended figure: Control scheme for the MZI experiment. All electronic generators for the same frequencies (95.5 MHz, 104.5 MHz), as well as the FPGA that controls the CLF-LO angles on the homodyne detection benches are clocked together to limit phase drift.