We examine the internal structure of two-mode entanglement criteria for quadrature- and number-phase-squeezed states. For criteria obtained from the partial transpose of the Schrödinger--Robertson inequality, we show that the additional covariance term effectively performs an optimization over the intra-mode rotations entering the criterion. We demonstrate this both for quadrature variables and for number-phase-squeezed states. We further show that Simon's criterion carries out this optimization automatically, which motivates a Simon-like criterion for number-phase-squeezed states that performs the optimization directly in the number-phase plane. We also analyze entanglement in terms of the product of the noises of the two modes, which we call the noise area. Analytically and numerically, we explore whether widely used entanglement criteria can be interpreted as searches for a noise area below unity. In particular, for the product form of the Duan--Giedke--Cirac--Zoller criterion, we show numerically that the minimum noise area obtained after optimization over intra-mode rotations equals the input nonclassicality that a beam splitter needs to generate the same amount of entanglement as the state under consideration. Finally, for Gaussian states we introduce an alternative entanglement measure that can also be extended to multimode settings, and we outline several open questions, including a simpler definition of entanglement depth for number-phase-squeezed states.
We experimentally examine the nonclassical character of a class of non-Gaussian states known as phase-diffused squeezed states. These states may show no squeezing effect at all, and therefore provide an interesting example to test nonclassicality criteria. The characteristic function of the Glauber-Sudarshan representation (P function) proves to be a powerful tool to detect nonclassicality. Using this criterion we find that phase-diffused squeezed states are always nonclassical, even if the squeezing effect vanishes. Testing other criteria of nonclassicality based on higher-order squeezing and…
Single photon sources are necessary for optical quantum technology. The nonclassicality of emitted single photons manifests itself in diverse experiments and applications. The applications already require different nonclassical aspects of single photons, however, a suitable hierarchy of criteria is missing. We propose variable experimental tests, based on adjustable linear optical networks and single photon detectors, giving a hierarchy of such nonclassicality criteria. The hierarchy goes beyond Hanbury - Brown - Twiss test of photon antibunching and allows us to compare faithfully quality and…
A method is introduced for the verification of nonclassicality in terms of moments of nonclassicality quasiprobability distributions. The latter are easily obtained from experimental data and will be denoted as nonclassicality moments. Their relation to normally-ordered moments is derived, which enables us to verify nonclassicality by using well established criteria. Alternatively, nonclassicality criteria are directly formulated in terms of nonclassicality moments. The latter converge in proper limits to the usually used criteria, as is illustrated for squeezing and sub-Poissonian photon stat…
In this work we generalize the Bochner criterion addressing the characteristic function, i.e., the Fourier transform, of the Glauber-Sudarshan phase-space function. For this purpose we extend the Bochner theorem by including derivatives of the characteristic function. The resulting necessary and sufficient nonclassicality criteria unify previously known moment-based criteria with those based on the characteristic function. For applications of the generalized nonclassicality probes, we provide direct sampling formulas for balanced homodyne detection. A squeezed vacuum state is experimentally re…
We use the exact calculation of the quantum mechanical, temporal characteristic function $χ(η)$ and the degree of second-order coherence $g^{(2)}(τ)$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state, to study the different criteria for nonclassicality. In particular, we contrast criteria that involve only one-time functions of the dynamical system, for instance, the quasiprobability distribution $P(β)$ of the Glauber-Sudarshan coherent or P-representation of the density of state and the Mandel $Q_{M}(τ)$ parameter, …
We devise a new class of criteria to certify the nonclassicality of photon- and phonon-number statistics. Our criteria extend and strengthen the broadly used Klyshko's criteria, which require knowledge of only a finite set of Fock-state probabilities. This makes the criteria well-suited to experimental implementation in realistic conditions. Moreover, we prove the completeness of our method in some scenarios, showing that, when only two or three Fock-state probabilities are known, it detects all finite distributions incompatible with classical states. In particular, we show that our criteria d…
Nonclassicality criteria based on intensity moments and derived from the usual matrix approach are compared to those provided by the majorization theory. The majorization theory is shown to give a greater number of more suitable nonclassicality criteria. Fifteen experimentally useful criteria of the majorization theory containing the intensity moments up to the fifth order are identified. Their performance is experimentally demonstrated on the set of eleven potentially nonclassical states generated from a twin beam by postselection based on detecting a given number of photocounts in one arm by…
Numerous inequalities involving moments of integrated intensities and revealing nonclassicality and entanglement in bipartite optical fields are derived using the majorization theory, non-negative polynomials, the matrix approach, as well as the Cauchy-Schwarz inequality. Different approaches for deriving these inequalities are compared. Using the experimental photocount histogram generated by a weak noisy twin beam monitored by a photon-number-resolving iCCD camera the performance of the derived inequalities is compared. A basic set of ten inequalities suitable for monitoring the entanglement…
Driven by single photon detection requirements especially for quantum information sciences, the theory of arrays of off-on detectors has been well developed and applied. However for a comprehensive characterization of nonclassicality one also needs phase sensitive properties. This missing link is fulfilled by the theory of phase sensitive click counting measurements. This theory is presented. It unifies the balanced homodyne detection for high intensities with the click detection in the few photon regime. We formulate and apply a hierarchy of nonlinear squeezing conditions to probe quantum eff…
We consider a way to generate operational inequalities to test nonclassicality (or quantumness) of multimode (or multiparty) bosonic fields 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 P 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 phys…
Nonclassical properties of photon added and photon subtracted squeezed coherent states have been compared with specific focus on the higher order nonclassicalities, such as higher order squeezing, higher order sub-Poissonian photon statistics, higher order antibunching. It is observed that both photon added and photon subtracted squeezed coherent states are highly nonclassical as they satisfy criteria for all of the above mentioned nonclassicalities and a set of other criteria including negativity of Wigner function, Klyshko's criterion and Agarwal's $(A_{3})$ parameter. Further, the amount of…
Mode-entanglement based criteria and measures become insufficient for broadband emission, e.g. from spasers (plasmonic nano-lasers). We introduce criteria and measures for the (i) total entanglement of two wavepackets, (ii) entanglement of a wavepacket with an ensemble and (iii) total nonclassicality of a wavepacket~(WP). We discuss these criteria in the context of (i) entanglement of two WPs emitted from two initially entangled cavities (or two initially entangled atoms) and (ii) entanglement of an emitted WP with the ensemble/atom for the spontaneous emission and the single-photon superradia…
We study different representations of a many-particle system. We reveal the stunning beauty of the connections among the nonclassicalities of a many-particle, a single-mode and a two-mode systems. First, we demonstrate that (separable) atomic coherent states~(ACSs), of a many-particle system, mimic the coherent states of single-mode light. This enables the utilization of a many-particle entanglement~(MPE) criterion as a single-mode nonclassicality~(SMNc) condition via Holstein-Primakoff transformation. As an example, we show that spin-squeezing criterion, for MPE, can be transformed into the q…
We calculate exactly the quantum mechanical, temporal characteristic function $χ(η)$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. Knowledge of $χ(η)$ allows only the determination of the time development of arbitrary functions of equal-time products of creation and annihilation photon operators. We calculate, in particular, the fluctuations in photon number, quadrature operators, and quadrature variance. We contrast the very important difference between the nonclassicality criteria based on the one-time charact…
Interference lies at the heart of the behavior of classical and quantum light. It is thus crucial to understand the boundaries between which interference patterns can be explained by a classical electromagnetic description of light and which, on the other hand, can only be understood with a proper quantum mechanical approach. While the case of two-mode interference has received a lot of attention, the multimode case has not yet been fully explored. Here we study a general scenario of intensity interferometry: we derive a bound on the average correlations between pairs of output intensities for…
We introduce a kind of entangled state---photon-addition two-mode squeezed thermal state (TMSTS) by adding photons to each mode of the TMSTS. Using the P-representation of thermal state, the compact expression of the normalization factor is derived, a Jacobi polynomial. The nonclassicality is investigated by exploring especially the negativity of Wigner function. The entanglement is discussed by using Shchukin-Vogel criteria. It is shown that the photon-addtion to the TMSTS may be more effective for the entanglement enhancement than the photon-subtraction from the TMSTS. In addition, the quant…
The efficient certification of nonclassical effects of light forms the basis for applications in optical quantum technologies. We derive general correlation conditions for the verification of nonclassical light based on multiplexed detection. The obtained nonclassicality criteria are valid for imperfectly-balanced multiplexing scenarios with on-off detectors and do not require any knowledge about the detector system. In this sense they are fully independent of the detector system. In our experiment, we study light emitted by clusters of single-photon emitters, whose photon number may exceed th…
We devise a method to certify nonclassical features via correlations of phase-space distributions by unifying the notions of quasiprobabilities and matrices of correlation functions. Our approach complements and extends recent results that were based on Chebyshev's inequality [Phys. Rev. Lett. 124, 133601 (2020)]. The method developed here correlates arbitrary phase-space functions at arbitrary points in phase space, including multimode scenarios and higher-order correlations. Furthermore, our approach provides necessary and sufficient nonclassicality criteria, applies to phase-space functions…
Two twin beams with a shared signal beam and separated idler beams are used together with the photon-number-resolving postselection in the signal beam to arrive at two coupled beams with anticorrelations in photon-number fluctuations. Moreover, the beams exhibit the sub-Poissonian photon-number statistics in their marginal distributions under suitable conditions. The postselected fields with the increasing mean photon numbers are reconstructed from the experimental photocount histograms by the maximum likelihood approach. Also a suitable Gaussian fit of both original twin beams and simulation …
We present Bell-type tests of nonclassicality and non-Gaussianity for single-mode fields employing a generalized quasiprobability function. Our nonclassicality tests are based on the observation that two orthogonal quadratures in phase space (position and momentum) behave as independent realistic variables for a coherent state. Taking four (three) points at the vertices of a rectangle (right triangle) in phase space, our tests detect every pure nonclassical Gaussian state and a range of mixed Gaussian states. These tests also set an upper bound for all Gaussian states and their mixtures, which…
Various nonclassical and quantum phase properties of photon added then subtracted displaced Fock state have been examined systematically and rigorously. Higher-order moments of the relevant bosonic operators are computed to test the nonclassicality of the state of interest, which reduces to various quantum states (having applications in quantum optics, metrology and information processing) in different limits ranging from the coherent (classical) state to the Fock (most nonclassical) states. The nonclassical features are discussed using Klyshko's, Vogel's, and Agarwal-Tara's criteria as well a…
We study the evolution of higher-order nonclassicality and entanglement criteria in atmospheric fluctuating-loss channels. By formulating input-output relations for the matrix of moments, we investigate the influence of such channels on the corresponding quantumness criteria. This generalization of our previous work on Gaussian entanglement [M. Bohmann et al., Phys. Rev. A 94, 010302(R) (2016)] not only exploits second-order-based scenarios, but it also provides a detailed investigation of nonclassicality and entanglement in non-Gaussian and multimode radiation fields undergoing a fluctuating …
A beam splitter can generate entanglement if the input field is nonclassical, e.g. squeezed. Thus, beam splitter transformation can be utilized for obtaining single-mode nonclassicality conditions from two-mode entanglement criteria. Here, we demonstrate that (i) the generalized quadrature-squeezing condition $\langle\hat{a}^\dagger\hat{a}\rangle < |\langle \hat{a}^2\rangle|$ can be obtained from the two-mode entanglement criterion of Duan-Giedke-Cirac-Zoller~[PRL 84, 2722]. (ii) Single-mode nonclassicality condition, obtained from the two-mode entanglement criterion of Simon~[PRL, 84 2726], c…
We derive a family of inequalities involving different phase-space distributions of a quantum state which have to be fulfilled by any classical state. The violation of these inequalities is a clear signature of nonclassicality. Our approach combines the characterization of nonclassical effects via negativities in phase-space distributions with inequality conditions usually being formulated for moments of physical observables. Importantly, the obtained criteria certify nonclassicality even when the involved phase-space distributions are non-negative. Moreover, we show how these inequalities are…
Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.
Quantum features of correlated optical modes define a major aspect of the nonclassicality in quantized radiation fields. However, the phase-sensitive detection of a two-mode light field is restricted to interferometric setups and local intensity measurements. Even the full reconstruction of the quantum state of a single radiation mode relies on such detection layouts and the preparation of a well-defined reference light field. In this work, we establish the notion of the essential quantum correlations of two-mode light fields. It refers to those quantum correlations which are measurable by a g…
In the recent experimental and theoretical literature well-established nonclassicality criteria from the field of quantum optics have been directly applied to the case of excitations in matter-waves. Among these are violations of Cauchy-Schwarz inequalities, Glauber-Sudarshan P-nonclassicality, sub-Poissonian number-difference squeezing (also known as the two-mode variance) and the criterion of nonseparability. We review the strong connection of these criteria and their meaning in quantum optics, and point out differences in the interpretation between light and matter waves. We then calculate …
The interplay between the nonclassical features and the parity-time (PT) symmetry (or its breaking) is studied here by considering a PT symmetric system consisting of two cavities with gain and loss. The conditions for PT invariance is obtained for this system. The behavior of the average photon number corresponding to the gain and loss modes for different initial states (e.g., vacuum, NOON, coherent, and thermal states) has also been obtained. With the help of the number operators, quantum Zeno and anti-Zeno effects are studied, and the observed behavior is compared in PT symmetric (PTS) and …
In this paper, we follow our presented model in J. Opt. Soc. Am. B {\bf 30}, 1109--1117 (2013), in which the interaction between a $Λ$-type three-level atom and a quantized two-mode radiation field in a cavity in the presence of nonlinearities is studied. After giving a brief review on the procedure of obtaining the state vector of the atom-field system, some further interesting and important physical features (which are of particular interest in the quantum optics field of research) of the whole system state, i.e., the number-phase entropic uncertainty relation (based on the two-mode Pegg-Bar…
