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Optics Letters

General criteria for nonclassical photon statistics in multimode radiations

C. T. Lee

Published Dec 1, 1990• 14 citations
Publisher

Abstract

Using Muirhead's inequality based on the concept of majorization, general criteria for nonclassical photon statistics of arbitrary orders and involving any number of modes of radiations are derived. For the case of second-order correlations, it implies that both intramode antibunching and intermode bunching can contribute toward making multimode radiations nonclassical.

DOI
10.1364/ol.15.001386

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Using Muirhead's inequality based on the concept of majorization, general criteria for nonclassical photon statistics of arbitrary orders and involving any number of modes of radiations are derived. For the case of second-order correlations, it implies that both intramode antibunching and intermode bunching can contribute toward making multimode radiations nonclassical.
The concept of photon antibunching as a manifestation of nonclassical character in single-mode radiation is extended to two-mode radiation. A simple criterion for the existence of nonclassical effects in two-mode radiation is established. It implies that the intermode photon bunching, rather than antibunching, can play a key role in rendering two-mode radiation nonclassical. The two-mode squeezed vacuum states are used as simple examples to illustrate the essential point of the criterion. The criterion is then used to study the nonclassical properties of two types of two-mode squeezed states. A state generated by applying the squeeze operator to the two-mode vacuum first, followed by the displacement operator, is called a two-mode coherent squeezed state. If the order of the two operators is reversed, a two-mode squeezed coherent state is obtained. It is found that the latter has a much stronger tendency to become a nonclassical state than the former, assuming that the parameters involved are all the same. A measure of nonclassical depth of radiation is adopted. It is found that, according to this measure, the effect of squeezing in making two-mode coherent squeezed states nonclassical is not monotonic. Such phenomena do not occur in squeezed coherent states.
Based on a classical inequality, which is a generalization by Muirhead [Proc. Edinburg Math. Soc. 21, 144 (1903)] of the arithmetic-geometric mean inequality, and the expression for factorial moments of photon numbers in the P representation, the concept of two-photon antibunching is extended to involve many photons from the same mode, or from two different modes, of the radiation field. A systematic study is carried out to examine the conditions for the existence of many-photon antibunching in pair coherent states introduced recently by Agarwal [J. Opt. Soc. Am. B 5, 1940 (1988)]. It is proved analytically that the intramode antibunching always exists and that intermode antibunching occurs whenever a certain condition is satisfied by the relevant parameters; otherwise, numerical study shows that it occurs only when the eigenvalue of the pair-annihilation operator is large enough. The pair coherent states are then generalized by using a more realistic initial condition for their generation through the competition of different nonlinear processes in a two-photon medium. It is again established analytically that intramode antibunching always exists in these generalized pair coherent states; and numerical study reveals the general features of the intermode antibunching.
We consider a way to generate operational inequalities to test nonclassicality (or quantumness) of multimode bosonic fields (or multiparty bosonic systems) that unifies the derivation of many known inequalities and allows to propose new ones. The nonclassicality criteria are based on Vogel's criterion corresponding to analyzing the positivity of multimode PPP functions or, equivalently, the positivity of matrices of expectation values of, e.g., creation and annihilation operators. We analyze not only monomials but also polynomial functions of such moments, which can sometimes enable simpler derivations of physically relevant inequalities. As an example, we derive various classical inequalities which can be violated only by nonclassical fields. In particular, we show how the criteria introduced here easily reduce to the well-known inequalities describing (a) multimode quadrature squeezing and its generalizations, including sum, difference, and principal squeezing; (b) two-mode one-time photon-number correlations, including sub-Poisson photon-number correlations and effects corresponding to violations of the Cauchy-Schwarz and Muirhead inequalities; (c) two-time single-mode photon-number correlations, including photon antibunching and hyperbunching; and (d) two- and three-mode quantum entanglement. Other simple inequalities for testing nonclassicality are also proposed. We have found some general relations between the nonclassicality and entanglement criteria, in particular those resulting from the Cauchy-Schwarz inequality. It is shown that some known entanglement inequalities can be derived as nonclassicality inequalities within our formalism, while some other known entanglement inequalities can be seen as sums of more than one inequality derived from the nonclassicality criterion. This approach enables a deeper analysis of the entanglement for a given nonclassicality.
We address the problem of constructing witnesses for nonclassical light that are applicable in state-of-the-art photon-counting devices. The key ingredient for the criteria we derive are generalized and directly measurable counting statistics and matrices of counting moments. Beyond common criteria, we find classes of witnesses that are based on half-integer powers of click moments and counts. Remarkably, this leads to an exponential increase of the number of nonclassicality criteria one can construct and apply. With this finding, special attention is payed to probing even and odd parity states, requiring such distinct witnesses. Our method is applicable to spatial and time-bin multiplexing in optical systems, where each spatial and temporal mode can be measured with both on-off detectors and detectors with partial internal quasi-photon-number resolution. Generalizations to multimode scenarios are provided, allowing for the direct measurement of nonclassical correlations and coincidence counts between an arbitrary number of modes.