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arXiv:1503.00250 · Journal of Physics A Mathematical and Theoretical

Sub-Poissonian and anti-bunching criteria via majorization of statistics

Daniel G Medranda, Alfredo Luis

Published Jun 5, 2015• 3 citations
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Abstract

We use majorization and confidence intervals as a convenient tool to compare the uncertainty in photon number for different quantum light states. To this end, majorization is formulated in terms of confidence intervals. As a suitable case study we apply this tool to sub- and super-Poissonian behavior and bunching and anti-bunching effects. We focus on the most significant classical and nonclassical states, such as Glauber coherent, thermal, photon number, and squeezed states. We show that majorization provides a more complete analysis that in some relevant situations contradicts the predictions of variance.

DOI
10.1088/1751-8113/48/25/255302

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We use majorization and confidence intervals as a convenient tool to compare the uncertainty in photon number for different quantum light states. To this end, majorization is formulated in terms of confidence intervals. As a suitable case study we apply this tool to sub- and super-Poissonian behavior and bunching and anti-bunching effects. We focus on the most significant classical and nonclassical states, such as Glauber coherent, thermal, photon number, and squeezed states. We show that majorization provides a more complete analysis that in some relevant situations contradicts the predictions of variance.
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A great deal of effort has been devoted during the last decade to the theory of nonclassical states of the radiation field. From a practical point of view some of the most interesting predictions of these theories concern the generation of light with sub-Poissonian photon counting statistics for which Such light is evidently less noisy than laser radiation with obvious implications for the improved performance of optical systems. When the temporal autocorrelation function of the photon counts increases as the delay time increases from zero the light is also said to be antibunched.
After developing the concept of displaced squeezed vacuum states in the nonunitary approach and establishing the connection to the unitary approach we calculate their quasiprobabilities and expectation values † k a a in general form.Then we consider the displacement of the squeezed vacuum states and calculate their photon statistics and their quasiprobabilities.The expectation values of the displaced states are related to the expectation values of the undisplaced states and are calculated for some simplest cases which are sufficient to discuss their categorization as sub-Poissonian and super-Poissonian statistics.A large set of these states do not belong to sub-or to super-Poissonian states but are also not Poissonian states.We illustrate in examples their photon distributions.This shows that the notions of sub-and of super-Poissonian statistics and their use for the definition of nonclassicality of states are problematic.In Appendix A we present the most important relations for ( ) 1,1 SU treatment of squeezing and the disentanglement of their operators.Some initial members of sequences of expectation values for squeezed vacuum states are collected in Appendix E.
Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.