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arXiv:quant-ph/0110118

Limits to phase information in the parametric oscillator

Paul Kinsler

Published Oct 19, 2001• 0 citations
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Abstract

Light in which the quantum fluctuations have been squeezed is often proposed as a means of obtaining an improved phase reference compared to that available from coherent light. Such a phase reference contains information about the phase of the squeezed light, so it is important to calculate the limits to the amount of ``phase information'' available. I define a phase resolution and show how this scales as we increase the number of photons available when using squeezed light generated by a parametric oscillator. Simple schemes for creating squeezed coherent light using a beam-splitter and an interferometer are analyzed, and it is shown that the results agree with earlier claims [Kinsler] that when using squeezed coherent light as a phase reference, the gain is not one of improved accuracy, but of lower power.

DOI
10.48550/arxiv.quant-ph/0110118

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Light in which the quantum fluctuations have been squeezed is often proposed as a means of obtaining an improved phase reference compared to that available from coherent light. Such a phase reference contains information about the phase of the squeezed light, so it is important to calculate the limits to the amount of ``phase information'' available. I define a phase resolution and show how this scales as we increase the number of photons available when using squeezed light generated by a parametric oscillator. Simple schemes for creating squeezed coherent light using a beam-splitter and an interferometer are analyzed, and it is shown that the results agree with earlier claims [Kinsler] that when using squeezed coherent light as a phase reference, the gain is not one of improved accuracy, but of lower power.
We investigate the time evolution of squeezing in the signal mode of the undamped parametric amplifier. Both the degenerate and nondegenerate cases are considered. These are the simplest models that give limits to squeezing caused by both the intrinsic pump noise and pump depletion. Conflicting analytic expressions have been reported. We resolve this by the use of exact numerical methods, employing both number state and stochastic techniques. Our central result is that in both cases we find that the minimum in the squeezed quadrature variance scales as N${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}1/2}$, where N\ifmmode\bar\else\textasciimacron\fi{} is the initial coherent pump photon number. This bound on the amount of squeezing obtainable, coupled with the loss in energy due to the conversion efficiency, serves to limit the phase information in the squeezed channel to no more than that of the coherent pump. We believe this to be a fundamental limit for using the squeezed radiation generated in coherently driven parametric amplifiers.
Squeezed-state interferometry plays an important role in quantum-enhanced optical phase estimation, as it allows the estimation precision to be improved up to the Heisenberg limit by using ideal photon-number-resolving detectors at the output ports. Here we show that for each individual NNN-photon component of the phase-matched coherent $\ensuremath{\bigotimes}$ squeezed vacuum input state, the classical Fisher information always saturates the quantum Fisher information. Moreover, the total Fisher information is the sum of the contributions from each individual NNN-photon component, where the largest NNN is limited by the finite number resolution of available photon counters. Based on this observation, we provide an approximate analytical formula that quantifies the amount of lost information due to the finite photon number resolution; e.g., given the mean photon number n‾\overline{n}n in the input state, over 96% of the Heisenberg limit can be achieved with the number resolution larger than 5n‾5\overline{n}5n.
A squeezed state of light is usually taken to mean a state in which the fluctuations of one of the quadratures are below the level associated with a coherent or vacuum state. These squeezed fluctuations are phase sensitive in the sense that they vary periodically as the field evolves, but the relationship between squeezing and phase fluctuations is not trivial. While squeezed states can exhibit an arbitrarily low level of phase fluctuations, the phase states, which are eigenstates of the Hermitian optical phase operator are not themselves quadrature squeezed. A phasesqueezed state is one with a lower level of phase fluctuations than a coherent state with the same mean photon number. Phase squeezing is fundamentally different from quadrature squeezing in much the same way that sub-Poissonian photon statistics is different. For this reason optimally squeezed states and states with minimum phase noise will be distinct. We derive the form of the states which minimum phase noise for a given intensity. These have an Airy function behavior for the photon number probability distribution. These phase squeezed states have a far better phase resolution than coherent states of similar intensity, but do not seem to offer a significant reduction of phase fluctuations compared with conventional squeezed states.
Squeezed states of the electromagnetic field are characterized by a reduction of fluctuations for one of two quadrature phase amplitudes below the level of fluctuations for the vacuum state. This reduction is a manifestly quantum or nonclassical feature which is of great interest in its own right but also potentially important for precision measurement with sensitivity beyond the shot-noise or vacuum-state limit. We describe three different experiments in which squeezed light has been generated in our laboratory involving parametric downconversion, intracavity frequency doubling, and optical bistability with two-state atoms. By employing the squeezed light produced by a subthreshold optical parametric oscillator, we have achieved improvements in sensitivity beyond the vacuum-state limit for the detection of both phase and amplitude changes of the electromagnetic field. An alternate route to sensitivity beyond the vacuum-state limit is through quantum nondemolition measurement, where we have suggested and are implementing a scheme involving parametric conversion and polarization mixing.
We have previously [Phys. Rev. A 65, 043803 (2002)] analyzed adaptive measurements for estimating the continuously varying phase of a coherent beam, and a broadband squeezed beam. A real squeezed beam must have finite photon flux N\mathcal{N}N and hence can be significantly squeezed only over a limited frequency range. In this paper we analyze adaptive phase measurements of this type for a realistic model of a squeezed beam. We show that, provided it is possible to suitably choose the parameters of the beam, a mean-square phase uncertainty scaling as ${(\mathcal{N}∕\ensuremath{\kappa})}^{\ensuremath{-}5∕8}$ is possible, where $\ensuremath{\kappa}$ is the linewidth of the beam resulting from the fluctuating phase. This is an improvement over the ${(\mathcal{N}∕\ensuremath{\kappa})}^{\ensuremath{-}1∕2}$ scaling found previously for coherent beams. In the experimentally realistic case where there is a limit on the maximum squeezing possible, the variance will be reduced below that for coherent beams, though the scaling is unchanged.