Physical Review A

Phase uncertainties of a squeezed state

Matthias Freyberger, W. P. Schleich

Published Jun 1, 1994• 27 citations

Abstract

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}${\mathit{cphi}}_{\mathit{g}}$${)}^{2}$, (ii) the dispersion (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{d}}$${)}^{2}$, and (iii) the rotational width (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{r}}$${)}^{2}$. Whereas (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{g}}$${)}^{2}$ and (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{r}}$${)}^{2}$ decay quadratically with the average number of photons 〈n〉, the dependence of (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{d}}$${)}^{2}$ is only of the form ln〈n〉/〈n${\mathrm{〉}}^{2}$. We give physical pictures for these scaling laws.

DOI: 10.1103/physreva.49.5056

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Physical Review A

Phase uncertainties of a squeezed state

Matthias Freyberger, W. P. Schleich

Published Jun 1, 1994• 27 citations

Publisher

Abstract

{\mathit{cphi}}_{\mathit{g}}$${)}^{2}

, (ii) the dispersion (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{d}}$${)}^{2}

, and (iii) the rotational width (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{r}}$${)}^{2}

. Whereas (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{g}}$${)}^{2}

and (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{r}}$${)}^{2}

decay quadratically with the average number of photons 〈n〉, the dependence of (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{d}}$${)}^{2}

is only of the form ln〈n〉/〈n

{\mathrm{〉}}^{2}

. We give physical pictures for these scaling laws.

DOI: 10.1103/physreva.49.5056

Related papers

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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.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.

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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.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.

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{\mathit{cphi}}_{\mathit{g}}$${)}^{2}

, (ii) the dispersion (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{d}}$${)}^{2}

, and (iii) the rotational width (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{r}}$${)}^{2}

. Whereas (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{g}}$${)}^{2}

and (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{r}}$${)}^{2}

decay quadratically with the average number of photons 〈n〉, the dependence of (\ensuremath{\delta}

{\mathit{cphi}}_{\mathit{d}}$${)}^{2}

is only of the form ln〈n〉/〈n

{\mathrm{〉}}^{2}

. We give physical pictures for these scaling laws.

We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number

N

of uncorrelated copies. We find that it scales with the mean photon number

n

, as dictated by the Heisenberg limit, i.e., as ${n}^{\ensuremath{-}2}$, only for

N>4

. For $N\ensuremath{\leqslant}4$ this fundamental scaling breaks down and it becomes ${n}^{\ensuremath{-}N∕2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling ${n}^{\ensuremath{-}2}$ is recovered for large samples.