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Physical Review A

Exponential decrease in phase uncertainty

Wolfgang P. Schleich, Jonathan P. Dowling, R. J. Horowicz

Published Sep 1, 1991• 31 citations
Publisher

Abstract

The phase probability curve of a recently proposed photon state consists of a broad background with a sharp central peak [J. H. Shapiro, S. R. Shepard, and N. Wong, Phys. Rev. Lett. 62, 2377 (1989)]. These authors argue that the inverse peak-height phase uncertainty \ensuremath{\delta}cphi of this distribution decreases inversely as the square of the mean photon number 〈m〉---an improvement over either coherent or highly squeezed states. We show that the width \ensuremath{\Delta}cphi of the best-fitting Gaussian to the central peak--a measure of phase uncertainty tailored to this narrow feature--decreases exponentially with increasing 〈m〉. The importance of this result may be offset by the observation that the area under this peak also vanishes very rapidly.

DOI
10.1103/physreva.44.3365

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The phase probability curve of a recently proposed photon state consists of a broad background with a sharp central peak [J. H. Shapiro, S. R. Shepard, and N. Wong, Phys. Rev. Lett. 62, 2377 (1989)]. These authors argue that the inverse peak-height phase uncertainty \ensuremath{\delta}cphi of this distribution decreases inversely as the square of the mean photon number 〈m〉---an improvement over either coherent or highly squeezed states. We show that the width \ensuremath{\Delta}cphi of the best-fitting Gaussian to the central peak--a measure of phase uncertainty tailored to this narrow feature--decreases exponentially with increasing 〈m〉. The importance of this result may be offset by the observation that the area under this peak also vanishes very rapidly.
We study the width of the semiclassical phase distribution of a quantum state in its dependence on the average number of photons (m) in this state. As a measure of phase noise, we choose the width, delta phi, of the best Gaussian approximation to the dominant peak of this probability curve. For a coherent state, this width decreases with the square root of (m), whereas for a truncated phase state it decreases linearly with increasing (m). For an optimal phase state, delta phi decreases exponentially but so does the area caught underneath the peak: all the probability is stored in the broad wings of the distribution.
Recent theoretical studies have considered various possibilities for experimental measurements of phase using quantum states of a finite mean number of photons. Currently, the most experimentally accessible of these possibilities is an interferometric phase measurement using a squeezed state that has minimum variance for the phase quadrature; this possibility can yield, in principle, a phase measurement with uncertainty of order l/(mean number of photons). Shapiro, Shepard, and Wong1 (SSW) have proposed another possibility. They maximize the peak likelihood in an abstract measurement of the Susskind-Glogower phase operator, and they claim a phase sensitivity that goes as l/(mean number of photons).2 The SSW phase probability distribution consists of an extremely narrow peak sitting on a broad base that covers the entire 2 radians of phase; as a result, a single SSW phase measurement has poor sensitivity. Shapiro, Shepard, and Wong argue that their claimed sensitivity can be achieved in a sequence of measurements. We study the statistics of the maximum-likelihood estimator in a sequence of SSW phase measurements, and we report how the uncertainty in the estimator scales with the mean number of photons used in the entire sequence.
Shapiro, Shepard, and Wong [Phys. Rev. Lett. 62, 2377 (1989)] suggested that a scheme of multiple phase measurements, using quantum states with minimum ``reciprocal peak likelihood,'' could achieve a phase sensitivity scaling as 1/Ntot2{\mathit{N}}_{\mathrm{tot}}^{2}Ntot2​, where Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​ is the mean number of photons available for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3\ensuremath{\le}Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​\ensuremath{\le}120; a power-law fit to the simulated data yields a phase sensitivity that scales as 1/${\mathit{N}}_{\mathrm{tot}}^{0.82\ifmmode\pm\else\textpm\fi{}0.01}$. By using a combination of numerical and analytical techniques, we extend our results to higher values of Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​ than are accessible to our simulations; we find no evidence for phase sensitivities better than the benchmark 1/Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​ sensitivity of squeezed-state interferometry. We conclude that reciprocal peak likelihood is not a good measure of phase sensitivity. We discuss other factors that are important to phase sensitivity.
Shapiro, Shepard, and Wong [Phys. Rev. Lett. 62, 2377 (1989)] suggested that a scheme of multiple phase measurements, using quantum states with minimum ``reciprocal peak likelihood,'' could achieve a phase sensitivity scaling as 1/Ntot2{\mathit{N}}_{\mathrm{tot}}^{2}Ntot2​, where Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​ is the mean number of photons availavle for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3\ensuremath{\le}Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​\ensuremath{\le}120. A power-law fit to the simulated data yields a phase sensitivity that scales as 1/${\mathit{N}}_{\mathrm{tot}}^{0.85\ifmmode\pm\else\textpm\fi{}0.01}$. We conclude that reciprocal peak likelihood is not a good measure of sensitivity.
We investigate various measures of phase uncertainty in their dependence on the average number of photons for the case of a phase squeezed state. We use the following three measures: (i) the geometrical uncertainty (\ensuremath{\delta}cphig{\mathit{cphi}}_{\mathit{g}}cphig​)2{)}^{2})2, (ii) the dispersion (\ensuremath{\delta}cphid{\mathit{cphi}}_{\mathit{d}}cphid​)2{)}^{2})2, and (iii) the rotational width (\ensuremath{\delta}cphir{\mathit{cphi}}_{\mathit{r}}cphir​)2{)}^{2})2. Whereas (\ensuremath{\delta}cphig{\mathit{cphi}}_{\mathit{g}}cphig​)2{)}^{2})2 and (\ensuremath{\delta}cphir{\mathit{cphi}}_{\mathit{r}}cphir​)2{)}^{2})2 decay quadratically with the average number of photons 〈n〉, the dependence of (\ensuremath{\delta}cphid{\mathit{cphi}}_{\mathit{d}}cphid​)2{)}^{2})2 is only of the form ln〈n〉/〈n〉2{\mathrm{〉}}^{2}〉2. We give physical pictures for these scaling laws.