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arXiv:2309.08450

Lower bound on phase noise of classical states

Mark S. Hillery

Published Sep 15, 2023• 0 citations
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Abstract

An uncertainty relation for the number and phase of a single-mode field state is derived. It is then used to find a lower bound on the phase noise of a classical state. Any state that violates this condition is nonclassical. An example of such a nonclassical state is presented. Because a nonclassical state can have less phase noise than a classical state with the same average photon number, nonclassical states can play a role in the measurement of small phase shifts.

DOI
10.48550/arxiv.2309.08450

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An uncertainty relation for the number and phase of a single-mode field state is derived. It is then used to find a lower bound on the phase noise of a classical state. Any state that violates this condition is nonclassical. An example of such a nonclassical state is presented. Because a nonclassical state can have less phase noise than a classical state with the same average photon number, nonclassical states can play a role in the measurement of small phase shifts.
We derive upper bounds on the statistics of phase and phase difference that are satisfied by all classical states. They are obtained by finding the maximum projection of classical states on phase states. For a single-mode phase, meaningful bounds are obtained conditioned to a fixed mean number of photons. We also derive classical bounds for the projection on phase-coherent states, discussing their relation with phase-state bounds within the context of analytic representations. We find states with nonclassical phase properties disclosed by the violation of these classical bounds. These are quadrature and SU(222) squeezed states and phase-coherent states.
The total noise of a state is the sum of the fluctuations in the real part of the field amplitude and those in the imaginary part. It is a measure of the total fluctuations in the field amplitude and is a minimum for coherent states. Several examples of nonclassical fields are considered, and it is shown that as nonclassical behavior increases so does total noise. For example, total noise increases as a state becomes more squeezed or as the photon statistics become more sub-Poissonian. This result is then generalized by using the concept of nonclassical distance. Nonclassical distance δ is a measure of the extent to which the behavior of a field state differs from that of classical states. A lower bound for the total noise is derived which is an increasing function of δ. This means that the more nonclassical a state is the greater its total noise must be.
The total noise of a field state is a measure of the fluctuations of the field amplitude. It is a minimum for coherent states. As the behavior of a state becomes more nonclassical, its total noise increases. This is shown first for several specific types of nonclassical states, among them squeezed and sub-Poissonian states. These results are generalized by using nonclassical distance to measure how nonclassical a field state is. A lower bound for the total noise is derived that is an increasing function of nonclassical distance. From it one can conclude that highly nonclassical states have large amplitude fluctuations.
We study the effect of losses on the phase noise of single-mode field states. The losses are described by the standard loss master equation, and it is used to find an upper bound for the increase in the phase noise as a function of time. We compare the time dependence of the phase noise of an initial coherent state to that of a state that initially has very small phase noise. Both states have the same initial mean photon number. While the small-phase noise state is more susceptible to losses, the difference between its behaviour and that of the coherent state is not great.