Deterministic Quantum Phase Estimation beyond N00N States

The modern scientific method is critically dependent on precision measurements of physical parameters. A classic example is the measurement of the optical phase enabled by optical interferometry, where the error on the measured phase is conventionally bounded by the so-called Heisenberg limit. To achieve phase estimation at the Heisenberg limit, it has been common to consider protocols based on highly complex N00N states of light. However, despite decades of research and several experimental explorations, there has been no demonstration of deterministic phase estimation with N00N states reaching the Heisenberg limit or even surpassing the shot noise limit. Here we use a deterministic phase estimation scheme based on a source of Gaussian squeezed vacuum states and high-efficiency homodyne detection to obtain phase estimates with an extreme sensitivity that significantly surpasses the shot noise limit and even beats the conventional Heisenberg limit as well as the performance of a pure N00N state protocol. Using a high-efficiency setup with a total loss of about 11%, we achieve a Fisher information of $15.8(6){\mathrm{rad}}^{-2}$ per photon—a significant increase in performance compared to state of the art and beyond an ideal six photon N00N state scheme. This work represents an important achievement in quantum metrology, and it opens the door to future quantum sensing technologies for the interrogation of light-sensitive biological systems.

Figure 1

Principles and limits of quantum phase estimation. (a) Left: Schematics of three different phase estimation schemes. A quantum state of light undergoes a phase shift that is measured with either a homodyne detector (HD) or a N00N state detector (involving photon counters) from which estimators are used to estimate the phase shift. Note that the N00N state scheme is based on a two-beam interferometer in which only half of the photons traverse the sample. We therefore use the conservative sensitivity bound of $1/2⟨\widehat{\mathbf{n}}⟩$ (where $N=2⟨\widehat{\mathbf{n}}⟩$) for the comparison to our approach. Right: The optimal sensitivities $\sigma$ for the three schemes. (b) Phase space pictures of a displaced squeezed state and a vacuum squeezed state, and the resulting quadrature measurements as a function of the phase. The phase is estimated using the estimators $⟨X⟩$ or $⟨X^2⟩$ for the displaced squeezed state and vacuum squeezed state, respectively.

Figure 2

Experimental scheme and measurement method. (a) Schematic of the experimental setup comprising an optical parametric oscillator (OPO) for squeezed light generation and a high-efficiency homodyne detector with a controllable local oscillator (LO). As the estimated phase shift is relative between the squeezed vacuum and the local oscillator, in the experimental realization, we imposed the phase shift onto the local oscillator. (b) Squeezed and antisqueezed variances relative to the shot noise variance as a function of pump power at a sideband frequency of 5 MHz (left) and frequency at a pump power of 3.5 mW (right). (c) Quadrature measurement outcomes. The data are acquired while slowly varying the phase of the local oscillator and down-mixed to a 5 MHz sideband frequency with a bandwidth of 1 MHz. (d) An example of how the estimated phase converges as a function of homodyne samples used in the Bayesian estimation process. The gray area marks the standard deviation.

Figure 3

Quantum phase estimation results. (a) The variance of the phase estimate ${\sigma }_{\mathrm{est}}^2$ based on 1000 quadrature samples of a squeezed vacuum state with 1.8 photons on average. This is compared to the SNL and the limit for a lossless N00N state with $2⟨\widehat{\mathbf{n}}⟩=N=4$. (b) The variance of the phase estimated for different average photon numbers represented in a polar diagram and compared to the SNLs of the respective realizations (the curves are color coded). It is clear from these plots that the variance is minimized for certain phases. (c) The optimal estimated phases ${\varphi }_{\mathrm{est},\mathrm{opt}}$ with corresponding estimation variance ${\sigma }_{\mathrm{est},\mathrm{opt}}^2$ are presented for different photon numbers and compared with theory. (d) The experimental sensitivities $\sigma$ versus photon numbers are presented and related to the theoretical predictions for the SNL, squeezed vacuum limit, and the N00N state limit. We include theoretical predictions for the ideal limits and the practical limits with 11% loss as measured in our system.

Figure 4

Quantum-enhanced tracking of a phase signal. (a) Power spectral density (PSD) of a measurement of an estimated dynamically varying phase signal using squeezed vacuum (with six photons) and Bayesian inference. A 3 kHz induced signal as well as some low-frequency noise is apparent. (b) Time trace of the same signal, but bandpass filtered at 3 kHz with a 2 kHz bandwidth. The enlargement of the time trace as well as the frequency spectrum clearly shows the 3 kHz modulation. The $y$ axis $\mathrm{\Delta }\varphi$ is the relative phase shifts compared to the preset measurement phase.