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Physical Review A

Some limits to precision phase measurement

Samuel L. Braunstein

Published Jan 1, 1994• 16 citations
Publisher

Abstract

States near the vacuum having 〈n^〉\ensuremath{\ll}1 are used for the precision determination of a phase shift. When this phase shift is sampled by enough of these near-vacuum states, its size may be extracted via data analysis. We calculate the achievable sensitivity of such schemes when we are limited in our resources to a mean total number of quanta ${\mathit{N}}_{\mathrm{tot}}$. We study in detail some examples of these near-vacuum schemes and show that in the absence of loss or noise their sensitivity approaches that of squeezed-state interferometry.

DOI
10.1103/physreva.49.69

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States near the vacuum having 〈n^〉\ensuremath{\ll}1 are used for the precision determination of a phase shift. When this phase shift is sampled by enough of these near-vacuum states, its size may be extracted via data analysis. We calculate the achievable sensitivity of such schemes when we are limited in our resources to a mean total number of quanta Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​. We study in detail some examples of these near-vacuum schemes and show that in the absence of loss or noise their sensitivity approaches that of squeezed-state interferometry.
We investigate the quantum mechanical bound to how precisely we can determine a phase shift given only a constraint on the mean total number of photons available. By considering how quickly one can gain information from data analysis, we derive the sensitivity achievable (in principle) for measurements involving even highly non-Gaussian noise. Using these results we calculate the sensitivity of several recent proposals for precision phase measurement, and show that no proposal to date beats the sensitivity believed achievable by squeezed state interferometry.
We study the feasibility of sub-shot-noise interferometry with imperfect detectors, starting from twin Fock states and two-mode squeezed vacuum states. We derive analytical expressions for the corresponding phase uncertainty. We find that one can achieve phase shift measurements below the standard quantum limit, as long as the losses are smaller than a given threshold, and that the measured phase is close enough to an optimal value. We provide our analytical formulas in a python package, accessible online.
In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer with single-mode inputs is bounded by the shot-noise limit. The quantum Fisher information analysis shows that this statement holds true for the scenario where two antisymmetric phase shifts occur in two arms, but it is invalid for the scenario where an unknown phase is embedded in one of two arms. In this paper, we focus on the phase sensitivity directed against the latter scenario. The optimal single-mode input is discussed by analyzing common states, including displaced squeezed states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. We find that the best choice is a squeezed vacuum state and show the specific measurement scheme which is capable of saturating the corresponding phase sensitivity limit. In addition, we study the effects of several realistic factors-anti-squeezing noise, photon loss and dark counts-on the phase sensitivity. Our results suggest that sub-shot-noise-limited phase sensitivity is attainable with low noise or loss, which paves the way for practical metrology.
We consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating the quantum Fisher information of the mixed quantum state in one of the interferometer's outputs about the difference phase. We show that, in the vicinity of the black fringe, this information is asymptotically close to the quantum Fisher information about this phase for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and allows for the Heisenberg scaling of precision. We also show that the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.
Many works have stated that nonlinear interactions can improve phase sensitivity beyond the Heisenberg limit scaling of 1/N1/N1/N with NNN being the mean photon number. This raises some open questions---among them the conclusive sensitivity limits with respect to single-mode inputs. Namely, when one of two inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase shifts, we make an attempt to provide an answer to this question. Based upon phase-averaging approach, this puzzle is partially resolved with careful calculations of the quantum Fisher information regarding three kinds of common inputs: Gaussian states, squeezed number states, and Schrödinger cat states. The results suggest that shot-noise-style sensitivity limit is no longer available, and the ideal candidate is squeezed vacuum.
Improvements in precision beyond the limit set by the vacuum state or zero-point fluctuations of the field are reported for two optical experiments. Both experiments employ the squeezed light from an optical parametric oscillator to reduce the level of fluctuations in one of two quadrature phase amplitudes below the shot-noise level (SNL). The first experiment follows the work of Caves1 on precision interferometry with squeezed states. An increase in signal-to-noise of 2.5 dB relative to the SNL is demonstrated for the detection of phase modulation in a Mach-Zehnder interferometer. In the second experiment amplitude modulation is encoded on a weak signal beam with an acoustooptic modulator. For the case of a squeezed signal beam compared with a beam in a coherent state, an improvement in sensitivity of 2.5 dB is achieved for the detection of amplitude changes of the field. The observed increases in sensitivity in both experiments are currently limited by simple linear losses in propagation and detection and not by the degree of available squeezing.