Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics

Experimental photon addition and subtraction in multi-mode and entangled optical fields

Multiple photon addition and subtraction applied to multi-mode thermal and sub-Poissonian fields as well as twin beams are mutually compared using one experimental setup. Twin beams (TWBs) with tight spatial correlations detected by an intensified CCD camera with high spatial resolution are used to prepare the initial fields. Up to three photons are added or subtracted to arrive at the nonclassical and non-Gaussian states. Only the photon-subtracted thermal states (TSs) remain classical. In general, the experimental photon-added states exhibit greater nonclassicality and non-Gaussianity than the comparable photon-subtracted states. Once photons are added or subtracted in twin beams, both processes result in comparable properties of the obtained states owing to twin-beam photon pairing.

Coherence-Based Operational Nonclassicality Criteria

The nonclassicality of quantum states is a fundamental resource for quantum technologies and quantum information tasks, in general. In particular, a pivotal aspect of quantum states lies in their coherence properties, encoded in the nondiagonal terms of their density matrix in the Fock-state bosonic basis. We present operational criteria to detect the nonclassicality of individual quantum coherences that use only data obtainable in experimentally realistic scenarios. We analyze and compare the robustness of the nonclassical coherence aspects when the states pass through lossy and noisy channels. The criteria can be immediately applied to experiments with light, atoms, solid-state system, and mechanical oscillators, thus providing a toolbox allowing practical experiments to more easily detect the nonclassicality of generated states.

Direct Measurement of Higher-Order Nonlinear Polarization Squeezing

We report on nonlinear squeezing effects of polarization states of light by harnessing the intrinsic correlations from a polarization-entangled light source and click-counting measurements. Nonlinear Stokes operators are obtained from harnessing the click-counting theory in combination with angular-momentum-type algebras. To quantify quantum effects, theoretical bounds are derived for second- and higher-order moments of nonlinear Stokes operators. The experimental validation of our concept is rendered possible by developing an efficient source, using a spectrally decorrelated type-II phase-matched waveguide inside a Sagnac interferometer. Correlated click statistics and moments are directly obtained from an eight-time-bin quasi-photon-number-resolving detection system. Macroscopic Bell states that are readily available with our source show the distinct nature of nonlinear polarization squeezing in up to eighth-order correlations, matching our theoretical predictions. Furthermore, our data certify nonclassical correlations with high statistical significance, without the need to correct for experimental imperfections and limitations. Also, our nonlinear squeezing can identify nonclassicality of noisy quantum states which is undetectable with the known linear polarization-squeezing criterion.

Sub-shot-noise-limited phase estimation via single-mode inputs

In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer with single-mode inputs is bounded by the shot-noise limit. The quantum Fisher information analysis shows that this statement holds true for the scenario where two antisymmetric phase shifts occur in two arms, but it is invalid for the scenario where an unknown phase is embedded in one of two arms. In this paper, we focus on the phase sensitivity directed against the latter scenario. The optimal single-mode input is discussed by analyzing common states, including displaced squeezed states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. We find that the best choice is a squeezed vacuum state and show the specific measurement scheme which is capable of saturating the corresponding phase sensitivity limit. In addition, we study the effects of several realistic factors-anti-squeezing noise, photon loss and dark counts-on the phase sensitivity. Our results suggest that sub-shot-noise-limited phase sensitivity is attainable with low noise or loss, which paves the way for practical metrology.

Room-Temperature Mechanical Resonator with a Single Added or Subtracted Phonon

A room-temperature mechanical oscillator undergoes thermal Brownian motion with an amplitude much larger than the amplitude associated with a single phonon of excitation. This motion can be read out and manipulated using laser light using a cavity-optomechanical approach. By performing a strong quantum measurement (i.e., counting single photons in the sidebands imparted on a laser), we herald the addition and subtraction of single phonons on the 300 K thermal motional state of a 4 GHz mechanical oscillator. To understand the resulting mechanical state, we implement a tomography scheme and observe highly non-Gaussian phase-space distributions. Using a maximum likelihood method, we infer the density matrix of the oscillator, and we confirm the counterintuitive doubling of the mean phonon number resulting from phonon addition and subtraction.

Two-beam light with 'checkered-pattern' photon-number distributions

Photon-number-resolved post-selection on one beam out of a correlated system of three beams with bi-partite photon-number correlations gives rise to joint photon-number distributions with the probabilities forming checkered patterns. These patterns originate in the convolution of two constituting photon-number distributions, one endowed with correlations in photon numbers, the other exhibiting anti-correlations in photon-number fluctuations. Using three twin beams of comparable intensity whose constituting beams suitably overlap on the photocathode of a photon-number-resolving iCCD camera, we experimentally as well as theoretically analyze the properties of such states as they change with the varying ratio of the correlated and anti-correlated contributions. The experimental photocount 2D histograms of the fields post-selected by the iCCD camera that are reconstructed by the maximum-likelihood approach confirm their non-classicality though the limited detection efficiency in post-selection conceals the checkered patterns. Contrary to this, the maximum-likelihood reconstruction of the experimental 3D photocount histogram similarly as a suitable 3D Gaussian fit, that reveal the states as they would be obtained by ideal post-selection, provide the photon-number distributions with the checkered patterns. The corresponding quasi-distributions of integrated intensities are determined. Nonclassical properties of the generated states are investigated using suitable non-classicality criteria and the corresponding non-classicality depths. These states with their correlations of varying intensity are prospective for two-photon excitations of atoms and molecules as well as two-photon spectroscopy.

Single-Phonon Addition and Subtraction to a Mechanical Thermal State

Adding or subtracting a single quantum of excitation to a thermal state of a bosonic system has the counter-intuitive effect of approximately doubling its mean occupation. We perform the first experimental demonstration of this effect outside optics by implementing single-phonon addition and subtraction to a thermal state of a mechanical oscillator via Brillouin optomechanics in an optical whispering-gallery microresonator. Using a detection scheme that combines single-photon counting and optical heterodyne detection, we observe this doubling of the mechanical thermal fluctuations to a high precision. The capabilities of this joint click-dyne detection scheme adds a significant new dimension for optomechanical quantum science and applications.

Statistical Benchmarking of Scalable Photonic Quantum Systems

Targeting at the realization of scalable photonic quantum technologies, the generation of many photons, their propagation in large optical networks, and a subsequent detection and analysis of sophisticated quantum correlations are essential for the understanding of macroscopic quantum systems. In this experimental contribution, we explore the joint operation of all mentioned ingredients. We benchmark our time-multiplexing framework that includes a high-performance source of multiphoton states and a large multiplexing network, together with unique detectors with high photon-number resolution, readily available for distributing quantum light and measuring complex quantum correlations. Using an adaptive approach that employs flexible time bins, rather than static ones, we successfully verify high-order nonclassical correlations of many photons distributed over many modes. By exploiting the symmetry of our system and using powerful analysis tools, we can analyze correlations that would be inaccessible by classical means otherwise. In particular, we produce on the order of ten photons and distribute them over 64 modes. Nonclassicality is verified with correlation functions up to the 128th order and statistical significances of up to 20 standard deviations.

Experimental Certification of Nonclassicality via Phase-Space Inequalities

In spite of its fundamental importance in quantum science and technology, the experimental certification of nonclassicality is still a challenging task, especially in realistic scenarios where losses and noise imbue the system. Here, we present the first experimental implementation of the recently introduced phase-space inequalities for nonclassicality certification, which conceptually unite phase-space representations with correlation conditions. We demonstrate the practicality and sensitivity of this approach by studying nonclassicality of a family of noisy and lossy quantum states of light. To this end, we experimentally generate single-photon-added thermal states with various thermal mean photon numbers and detect them at different loss levels. Based on the reconstructed Wigner and Husimi Q functions, the inequality conditions detect nonclassicality despite the fact that the involved distributions are nonnegative, which includes cases of high losses (93%) and cases where other established methods do not reveal nonclassicality. We show the advantages of the implemented approach and discuss possible extensions that assure a wide applicability for quantum science and technologies.

Non-classicality of optical fields as observed in photocount and photon-number distributions

Non-classicality criteria for optical fields based on the probabilities of photocount and photon-number distributions are derived. Relations among the criteria obtained by the applied methods are revealed. Redundant criteria are identified. The performance of the fundamental criteria is tested on a set of potentially sub-Poissonian fields generated by photon-number-resolved post-selection from a mesoscopic twin beam. The corresponding non-classicality depths are determined to quantitatively compare the used criteria.

Phase-Space Inequalities Beyond Negativities

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 related to correlation measurements. The strength of the derived conditions is demonstrated by different examples, including squeezed states, lossy single-photon states, and even coherent states.

Quadrature Coherence Scale Driven Fast Decoherence of Bosonic Quantum Field States

We introduce, for each state of a bosonic quantum field, its quadrature coherence scale (QCS), a measure of the range of its quadrature coherences. Under coupling to a thermal bath, the purity and QCS are shown to decrease on a timescale inversely proportional to the QCS squared. The states most fragile to decoherence are therefore those with quadrature coherences far from the diagonal. We further show a large QCS is difficult to measure since it induces small scale variations in the state's Wigner function. These two observations imply a large QCS constitutes a mark of "macroscopic coherence." Finally, we link the QCS to optical classicality: optical classical states have a small QCS and a large QCS implies strong optical nonclassicality.

Simultaneous observation of higher-order non-classicalities based on experimental photocount moments and probabilities

Using a sub-Poissonian optical field generated from a weak twin beam by photon-number resolving post-selection we have simultaneously observed higher-order non-classicalities in photocount moments (sub-Poissonian statistics) and probabilities (witnessed by the Klyshko inequalities). Up to the seventh-order non-classicalities in photocount moments simultaneously with up to the eleventh-order non-classicalities in photocount probabilities have been experimentally observed. Non-classicality counting parameters of different orders as experimental counterparts of the theoretical Lee non-classicality depth have been suggested to quantify and also mutually compare the robustness of these non-classicalities against the noise.

Measuring Nonclassicality of Bosonic Field Quantum States via Operator Ordering Sensitivity

We introduce a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways. We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasiprobabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce a distance-based measure of nonclassicality. We analyze the link between this nonclassicality measure and a recently introduced quantum macroscopicity measure, showing how the two notions are distinct.

Thermal-difference states of light: true states of heralded photons

We introduce a three-parameter family of single-mode optical states whose density operator is a weighted difference of two thermal states, the thermal-difference states. We identify the parameter values for which these states have a negative non-singular P-function, implying they are nonclassical. We show that the states of the “heralded photons” generated via spontaneous parametric downconversion belong to this family, with the three parameters corresponding to the nonlinear gain and the losses in the signal and the idler channels. The thermal-difference states yield new benchmark states for the analysis of nonclassicality and quantum macroscopicity criteria.

Identification of nonclassical properties of light with multiplexing layouts

In Sperling et al. [Phys. Rev. Lett. 118, 163602 (2017)], we introduced and applied a detector-independent method to uncover nonclassicality. Here, we extend those techniques and give more details on the performed analysis. We derive a general theory of the positive-operator-valued measure that describes multiplexing layouts with arbitrary detectors. From the resulting quantum version of a multinomial statistics, we infer nonclassicality probes based on a matrix of normally ordered moments. We discuss these criteria and apply the theory to our data which are measured with superconducting transition-edge sensors. Our experiment produces heralded multiphoton states from a parametric down-conversion light source. We show that the known notions of sub-Poisson and sub-binomial light can be deduced from our general approach, and we establish the concept of sub-multinomial light, which is shown to outperform the former two concepts of nonclassicality for our data.

Detector-Independent Verification of Quantum Light

We introduce a method for the verification of nonclassical light which is independent of the complex interaction between the generated light and the material of the detectors. This is accomplished by means of a multiplexing arrangement. Its theoretical description yields that the coincidence statistics of this measurement layout is a mixture of multinomial distributions for any classical light field and any type of detector. This allows us to formulate bounds on the statistical properties of classical states. We apply our directly accessible method to heralded multiphoton states which are detected with a single multiplexing step only and two detectors, which are in our work superconducting transition-edge sensors. The nonclassicality of the generated light is verified and characterized through the violation of the classical bounds without the need for characterizing the used detectors.

Experimental detection of nonclassicality of single-mode fields via intensity moments

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 using an iCCD camera.

Quantum Correlations from the Conditional Statistics of Incomplete Data

We study, in theory and experiment, the quantum properties of correlated light fields measured with click-counting detectors providing incomplete information on the photon statistics. We establish a correlation parameter for the conditional statistics, and we derive the corresponding nonclassicality criteria for detecting conditional quantum correlations. Classical bounds for Pearson's correlation parameter are formulated that allow us, once they are violated, to determine nonclassical correlations via the joint statistics. On the one hand, we demonstrate nonclassical correlations in terms of the joint click statistics of light produced by a parametric down-conversion source. On the other hand, we verify quantum correlations of a heralded, split single-photon state via the conditional click statistics together with a generalization to higher-order moments. We discuss the performance of the presented nonclassicality criteria to successfully discern joint and conditional quantum correlations. Remarkably, our results are obtained without making any assumptions on the response function, quantum efficiency, and dark-count rate of photodetectors.

Unbalanced Homodyne Correlation Measurements

A method is introduced that allows one to measure normal-ordered moments of the displaced photon-number operator up to higher orders, without the need of photon-number resolving detectors. It is based on unbalanced homodyne correlation measurements, with the local oscillator being replaced by a displaced dephased laser. The measured moments yield a simple approximation of quasiprobabilities, representing the full quantum state. Quantum properties of light are efficiently certified through normal-ordered observables directly accessible by our method, which is illustrated for a weakly squeezed vacuum and a single-photon-added thermal state.

Deterministic nonclassicality from thermal states

Coupling an oscillator to a single two-level system is one of the most fundamental interactions in quantum physics. We report on a dynamical effect during which a thermal state of an oscillator is unconditionally transformed to a highly nonclassical state with negative Wigner function values by mere absorbtion by a single uncontrolled two-level system. This complements the traditional test of Rabi oscillations and it serves as a simply measurable witness that the process in question is highly nonclassical. The process is experimentally feasible with possible experimental implementation in a number of experimental platforms with intrinsic Jaynes-Cummings interaction and it has the potential of enabling deterministic generation of nonclassical quantum states.

Sub-binomial light

The click statistics from on-off detector systems is quite different from the counting statistics of the more traditional detectors. This necessitates the introduction of new parameters to characterize the nonclassicality of fields from measurements using on-off detectors. To properly replace the Mandel Q(M) parameter, we introduce a parameter Q(B). A negative value represents a sub-binomial statistics. This is possible only for quantum fields, even for super-Poisson light. It eliminates the problems encountered in discerning nonclassicality using Mandel's Q(M) for on-off data.

Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields

We present the experimental realization of a scheme, based on single-photon interference, for implementing superpositions of distinct quantum operations. Its application to a thermal light field (a well-categorized classical entity) illustrates quantum superposition from a new standpoint and provides a direct and quantitative verification of the bosonic commutation relation between creation and annihilation operators. By shifting the focus towards operator superpositions, this result opens interesting alternative perspectives for manipulating quantum states.

Nonclassicality of quantum states: a hierarchy of observable conditions

A necessary and sufficient hierarchy of conditions is derived that is completely equivalent to the failure of the Glauber-Sudarshan P function to be a probability density. The conditions are formulated in terms of experimentally accessible characteristic functions of quadratures.