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arXiv:1903.11845

Generic Contractive States and Quantum Monitoring of Free Masses and Oscillators

Priyanshi Bhasin, Ujan Chakraborty, Shasanka Mohan Roy

Published Mar 28, 2019• 0 citations
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Abstract

Monitoring photon quadratures and free masses are useful tools to detect small disturbances such as gravitational waves.Here we report a large class of states for photon quadratures and free masses potentially useful for this purpose: (1)'generic coherent states' (GCS) of photons, whose width is independent of time and uncertainty product $σ(x) σ(p) $ is arbitrarily large (a generalization of the minimum uncertainty Schrödinger coherent states and (2) `squeezed generic contractive states' (SGCS) for photons and free masses (a generalization of the Yuen states \cite{Yuen}) whose width decreases with time ,uncertainty product is arbitrarily large, and the covariance squared $ ^2 $ has an arbitrary value within the allowed range $(0,4 σ^2 (x) σ^2 (p) -1\>)$. Dedicated to the 125th birth anniversary of S. N. Bose.

DOI
10.48550/arxiv.1903.11845

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Monitoring photon quadratures and free masses are useful tools to detect small disturbances such as gravitational waves.Here we report a large class of states for photon quadratures and free masses potentially useful for this purpose: (1)'generic coherent states' (GCS) of photons, whose width is independent of time and uncertainty product σ(x)σ(p)σ(x) σ(p) σ(x)σ(p) is arbitrarily large (a generalization of the minimum uncertainty Schrödinger coherent states and (2) `squeezed generic contractive states' (SGCS) for photons and free masses (a generalization of the Yuen states \cite{Yuen}) whose width decreases with time ,uncertainty product is arbitrarily large, and the covariance squared 2^22 has an arbitrary value within the allowed range (0,4σ2(x)σ2(p)−1 )(0,4 σ^2 (x) σ^2 (p) -1\>)(0,4σ2(x)σ2(p)−1). Dedicated to the 125th birth anniversary of S. N. Bose.
There are heuristic arguments proposing that the accuracy of monitoring position of a free mass mmm is limited by the standard quantum limit (SQL): ${\ensuremath{\sigma}}^{2}(X(t))\ensuremath{\ge}{\ensuremath{\sigma}}^{2}(X(0))+({t}^{2}/{m}^{2}){\ensuremath{\sigma}}^{2}(P(0))\ensuremath{\ge}\ensuremath{\hbar}t/m$, where ${\ensuremath{\sigma}}^{2}(X(t))$ and ${\ensuremath{\sigma}}^{2}(P(t))$ denote variances of the Heisenberg representation position and momentum operators. Yuen [Phys. Rev. Lett. 51, 719 (1983)] discovered that there are contractive states for which this result is incorrect. Here I prove universally valid rigorous quantum limits (RQL), viz. rigorous upper and lower bounds on ${\ensuremath{\sigma}}^{2}(X(t))$ in terms of ${\ensuremath{\sigma}}^{2}(X(0))$ and ${\ensuremath{\sigma}}^{2}(P(0))$, given by Eq. (12) for a free mass and by Eq. (36) for an oscillator. I also obtain the maximally contractive and maximally expanding states which saturate the RQL, and use the contractive states to set up an Ozawa-type measurement theory with accuracies respecting the RQL but beating the standard quantum limit. The contractive states for oscillators improve on the Schr\"odinger coherent states of constant variance and may be useful for gravitational wave detection and optical communication.
The familiar minimum-uncertainty wave packets for masses are generalized in analogy with the two-photon coherent states of the radiation field. The free evolution of a subclass of these states, the contractive states, leads to a narrowing of the position uncertainty in contrast with the usual minimum-uncertainty wave packets. As a consequence the standard quantum limit for monitoring the positions of a free mass can be breached. Further implications on quantum nondemolition measurements are discussed.
We discuss quantum measurements of position and repeated-measurement schemes for monitoring free-mass position to an arbitrary accuracy. Using these schemes, weak classical forces can be monitored to an arbitrary accuracy. To illustrate various measurement uncertainties and intrinsic uncertainties, we consider the case of a Michelson interferometer. In the low-intensity limit, two measurement uncertainties, photon counting error and radiation pressure error, are independent of the intrinsic uncertainty. The uses of squeezed states and number eigenstates are discussed.
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Contractive states for a free quantum particle were introduced by Yuen (Yuen H P 1983 Phys. Rev. Lett. 51 719) in an attempt to evade the standard quantum limit for repeated position measurements. We show how appropriate families of two- and three-component `Schrödinger cat states' are able to support non-trivial correlations between the position and momentum observables leading to contractive behaviour. The existence of contractive Schrödinger cat states is suggestive of potential novel roles of non-classical states for precision measurement schemes.
Using only elementary quantum-mechanical concepts, the statistical properties of various harmonic oscillator states that are linear superpositions of its energy eigenfunctions are described. These superpositions include coherent states and squeezed (or two-photon coherent) states. The resulting Gaussian, minimum-uncertainty wave packets are shown to oscillate back and forth for both coherent and squeezed states, but with an oscillating ‘‘width’’ for the squeezed states. Also examined are the principles underlying the production of squeezed electromagnetic waves via parametric amplification or four-wave mixing, their measurement by homodyne detection, and the connection between squeezing and non-Poissonian counting statistics.