Experimental determination of a nonclassical Glauber-SudarshanPfunction

Abstract

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...

Non-Gaussianity of quantum states: An experimental test on single-photon-added coherent states

“…We quantify experimentally the amount of non-Gaussianity obtained by adding a photon to a coherent state [19,[35][36][37]. Differently from previous investigations [35,[38][39][40][41], we can explicitly address the two aspects of non-Gaussianity and nonclassicality at once. For the former, we adopt the non-Gaussianity measure δ[ ] proposed in [42,43], which is defined as the quantum relative entropy between the quantum state itself and a reference Gaussian state τ having the same covariance matrix as .…”

Sub-Binomial Light

“…For this purpose let us study a singlephoton-added thermal state (SPATS) [19]. This state has been experimentally realized [20], and its nonclassicality has been verified by reconstructing its P function [21]. The P function of the SPATS is given by…”

A measure of non-Gaussianity for quantum states

“…These nonclassical states have been generated experimentally [28,29,30,31]. Their special importance to the present work arises from the fact that the Q functions of these states are scaled versions of those of the Fock states, and therefore one will expect any good measure of non-Gaussianity to return the same values for both classes of states.…”