Necessary and sufficient conditions for the nonclassicality of bosonic quantum states are formulated by introducing nonclassicality filters and nonclassicality quasiprobability distributions. Regular quasiprobabilities are constructed from characteristic functions, which can be directly sampled by balanced homodyne detection. Their negativities uncover the nonclassical effects of general quantum states. The method is illustrated by visualizing the nonclassical nature of a squeezed state.
A quantum state is nonclassical if its Glauber-Sudarshan P function fails to be interpreted as a probability density. This quantity is often highly singular, so that its reconstruction is a demanding task. Here we present the experimental determination of a well-behaved P function showing negativities for a single-photon-added thermal state. This is a direct visualization of the original definition of nonclassicality. The method can be useful under conditions for which many other signatures of nonclassicality would not persist. PACS numbers: 42.50.Dv, 42.50.Xa, 03.65.Ta, 03.65.Wj Einstein's hypothetical introduction of light quanta, the photons, was the first step toward the consideration of nonclassical properties of radiation [1]. But what does nonclassicality mean in a general sense? A radiation field is called nonclassical when its properties cannot be understood within the framework of the classical stochastic theory of electromagnetism. For other systems, nonclassicality can be defined accordingly. Here we will focus our attention on harmonic quantum systems, such as radiation fields or quantum-mechanical oscillators, for example trapped atoms.In this context the coherent states, first considered by Schrödinger in the form of wave packets [2], play an important role. They represent those quantum states that are most closely related to the classical behavior of an oscillator or an electromagnetic wave. For a single radiation mode, the coherent states |α are defined as the right-hand eigenstates of the non-Hermitian photon annihilation operatorâ,â|α = α|α ; cf. e.g. [3]. A general mixed quantum stateρ,can be characterized by the Glauber-Sudarshan P function [3,4]. In this form the quantum statistical averages of normally ordered operator functions can be written aswhere the normal ordering prescription :f (â,â † ) : means that all creation operatorsâ † are to be ordered to the left of all annihilation operatorsâ. Formally, the resulting expressions (2) for expectation values are equivalent to classical statistical mean values. However, in general, the P function does not exhibit all the properties of a classical probability density. It can become negative or even highly singular. Within the chosen representation of the theory, the failure of the Glauber-Sudarshan P function to show the properties of a probability density is taken as the key signature of quantumness [5,6].In this Rapid Communication we demonstrate the experimental determination of a nonclassical P function. Within the experimental precision it clearly attains negative values. This is a direct demonstration of nonclassicality: the negativity of the P function prevents its interpretation as a classical probability density.Why is it so difficult to demonstrate the nonclassicality directly on the basis of this original definition? Let us go back to a single photon as postulated by Einstein. Its P function iscf. e.g. [7]. Already in this case we get a highly singular distribution in terms of derivatives of the δ distribution, which cannot be in...
Although squeezed states are nonclassical states, so far, their nonclassicality could not be demonstrated by negative quasiprobabilities. In this work we derive pattern functions for the direct experimental determination of so-called nonclassicality quasiprobabilities. The negativities of these quantities turn out to be necessary and sufficient for the nonclassicality of an arbitrary quantum state and are therefore suitable for a direct and general test of nonclassicality. We apply the method to a squeezed vacuum state of light that was generated by parametric down-conversion in a second-order nonlinear crystal.
We report the experimental reconstruction of a nonclassicality quasiprobability for a single-photon-added thermal state. This quantity has significant negativities, which is necessary and sufficient for the nonclassicality of the quantum state. Our method exhibits several advantages compared to the reconstruction of the P function, since the nonclassicality filters used in this case can regularize the quasiprobabilities as well as their statistical uncertainties. A priori assumptions about the quantum state are therefore not necessary. We also demonstrate that, in principle, our method is not limited by small quantum efficiencies.
We propose a definition of nonclassicality for a single-mode quantum-optical process based on its action on coherent states. If a quantum process transforms a coherent state to a nonclassical state, it is verified to be nonclassical. To identify nonclassical processes, we introduce a representation for quantum processes, called the process-nonclassicality quasiprobability distribution, whose negativities indicate nonclassicality of the process. Using this distribution, we derive a relation for predicting nonclassicality of the output states for a given input state. We experimentally demonstrate our method by considering the single-photon addition as a nonclassical process and predicting nonclassicality of the output state for an input thermal state.
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