Dimension of quantum phase space measured by photon correlations *We dedicate this article to Margarita and Vladimir Man'ko on the occasion of their birthdays. The topic of our contribution is deeply rooted in and motivated by their love for non-classical light, quantum mechanical phase space distribution functions and orthogonal polynomials. Indeed, through their articles, talks and most importantly by many stimulating discussions and intensive collaborations with us they have contributed much to our understanding of physics. Happy birthday to you both!. (June 2015)

• Record Type: Journal Article
• Authors: Leuchs, Gerd; Glauber, Roy J; Schleich, Wolfgang P
• Is Part Of: Physica scripta. Volume 90:Number 7(2015:Jul.)
• Page Start: 149
• Publication Date: 2015-06
• Subjects: Physics -- Periodicals; 530.05
• DOI: 10.1088/0031-8949/90/7/074066
• Languages: English
• ISSNs: 0031-8949

Abstract

<abstract> <title>Abstract</title> <p>We show that the different values 1, 2 and 3 of the normalized second-order correlation function <inline-formula><tex-math><?CDATA ${g}^{(2)}(0)$?></tex-math><?MML <mml:math> <mml:msup> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2c9cqxbr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> corresponding to a coherent state, a thermal state and a highly squeezed vacuum originate from the different dimensionality of these states in phase space. In particular, we derive an exact expression for <inline-formula><tex-math><?CDATA ${g}^{(2)}(0)$?></tex-math><?MML <mml:math> <mml:msup> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2c9cr18p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> in terms of the ratio of the moments of the classical energy evaluated with the Wigner function of the quantum state of interest and corrections proportional to the reciprocal of powers of the average number of photons. In this way we establish a direct link between <inline-formula><tex-math><?CDATA ${g}^{(2)}(0)$?></tex-math><?MML <mml:math> <mml:msup> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2c9cr03d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> and the shape of the state in phase space. Moreover, we illuminate this connection by demonstrating that in the semi-classical limit the familiar photon statistics of a thermal state arise from an area in phase space weighted by a two-dimensional Gaussian, whereas those of a highly squeezed state are governed by a line-integral of a one-dimensional Gaussian.</p> </abstract>