Single-photon-added coherent states are the result of the most elementary amplification process of classical light fields by a single quantum of excitation. Being intermediate between a single-photon Fock state (fully quantum-mechanical) and a coherent (classical) one, these states offer the opportunity to closely follow the smooth transition between the particle-like and the wavelike behavior of light. We report the experimental generation of single-photon-added coherent states and their complete characterization by quantum tomography. Besides visualizing the evolution of the quantum-to-classical transition, these states allow one to witness the gradual change from the spontaneous to the stimulated regimes of light emission.

verse order, as in this case, is the real essence of the Heisenberg uncertainty principle. Besides its fundamental importance, the experimental implementation of such a sequence of basic quantum operations is an essential tool for the full-scale engineering of a quantum light state optimized for a multitude of different tasks (15), including robust quantum communication. As any quantum operation, including non-Gaussian operations, is composed of photon additions and subtractions (i.e, it can be expressed as f ð% a; % a † Þ), our experimental results constitute a step toward the full quantum control of a field and the generation of highly entangled states (16). 16. J. Eisert, S. Scheel, M. B. Plenio, Phys. Rev. Lett. 89, 137903 (2002). 17. We thank F. T. Arecchi, A. Montina, and E. Park for helpful comments and for a critical reading of the manuscript, and P. Poggi for the improvement of detection electronics.

Noise is the price to pay when trying to clone or amplify arbitrary quantum states. The quantum noise associated to linear phase-insensitive amplifiers can only be avoided by relaxing the requirement of a deterministic operation. Here we present the experimental realization of a probabilistic noiseless linear amplifier that is able to amplify coherent states at the highest level of effective gain and final state fidelity ever reached. Based on a sequence of photon addition and subtraction, and characterized by a significant amplification and low distortions, this high-fidelity amplification scheme may become an essential tool for quantum communications and metrology, by enhancing the discrimination between partially overlapping quantum states or by recovering the information transmitted over lossy channels.Comment: 5 pages, 4 figure

Entanglement between quantum and classical objects is of special interest in the context of fundamental studies of quantum mechanics and potential applications to quantum information processing. In quantum optics, single photons are treated as light quanta while coherent states are considered the most classical among all pure states. Recently, entanglement between a single photon and a coherent state in a free-traveling field was identified to be a useful resource for optical quantum information processing. However, it was pointed out to be extremely difficult to generate such states since it requires a clean cross-Kerr nonlinear interaction. Here, we devise and experimentally demonstrate a scheme to generate such hybrid entanglement by implementing a coherent superposition of two distinct quantum operations. The generated states clearly show entanglement between the two different types of states. Our work opens a way to generate hybrid entanglement of a larger size and to develop efficient quantum information processing using such a new type of qubits.Quantum entanglement is of crucial importance for fundamental tests of quantum mechanics and implementations of quantum information processing. The idea of entangling "classical" and "quantum" states is found in Schrödinger's famous cat paradox [1], where a microscopic atom -as a quantum particle-and a cat -as a classical object-were assumed to be entangled to each other. In quantum optics, coherent states are considered the most classical among all pure states [2]. In many situations they can be treated semi-classically, i.e. as a classical light field with the addition of stochastic noise, and are typically most robust against decoherence [3]. On the other hand, single photons are normally treated as discrete light quanta containing the minimum quantized amount of energy available at a given frequency and any attempt to describe them with an effective noise theory leads to negative probabilities.Recently, entanglement between a single photon and a coherent state was identified to be a very useful resource for optical quantum information processing, enabling one to perform nearly deterministic quantum teleportation and universal gate operations for quantum computation using linear optics [4]. This type of hybrid entanglement, however, is difficult to generate in spite of its conceptual interest and potential usefulness. It is well known that a clean cross-Kerr type interaction between a single photon and a coherent state may generate such a state [5,6]. However, many fundamental problems lie in the way of realizing a suitable interaction of this kind [7][8][9]. Therefore, an experimentally accessible scheme to replace the cross-Kerr nonlinearity and create entanglement between a single photon and a coherent state would be highly desirable. In this article, we introduce such a scheme and use it to experimentally generate small-scale hybrid entanglement. We also outline extensions whereby our methods can be generalized to produce larger scale hybrid entanglement.Th...

We report the experimental realization and tomographic analysis of novel quantum light states obtained by exciting a classical thermal field by a single photon. Such states, although completely incoherent, possess a tunable degree of quantumness which is here exploited to put to a stringent experimental test some of the criteria proposed for the proof and the measurement of state non-classicality. The quantum character of the states is also given in quantum information terms by evaluating the amount of entanglement that they can produce.Comment: 7 pages, 6 figures, accepted for publication in Phys. Rev.

When a single quantum of electromagnetic field excitation is added to the same spatio-temporal mode of a coherent state, a new field state is generated that exhibits intermediate properties between those of the two parents. Such a single-photon-added coherent state is obtained by the action of the photon creation operator on a coherent state and can thus be regarded as the result of the most elementary excitation process of a classical light field. Here we present and describe in depth the experimental realization of such states and their complete analysis by means of a novel ultrafast, time-domain, quantum homodyne tomography technique clearly revealing their nonclassical character.

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.PACS numbers: PACS number(s); 03.65. Ta, 42.50.Ar, 42.50.Xa The superposition principle is one of the pillars upon which the entire structure of quantum mechanics is built [1]. A quantum system in a pure state can always be described as a superposition of linearly independent states; thus once one has a quantum system represented by a pure state, the superposition is naturally there. An inter-body superposition state, the so-called entangled state, is somewhat trickier to generate than a single-body superposition state. However, it has been demonstrated that entanglement can be achieved by various methods, including a series of unitary operations [2,3,4] or by post-selection of events after unitary operations [5]. On the other hand, the discussion about superpositions of classical mixed states is not as clear as for a pure state [6].Quantum operators, besides quantum states, play a crucial role in describing physical operations including unitary transformations and measurements in quantum theory. If one can implement a superposition of operators, one can also construct state superpositions by applying the superposed operators to a given state, unless it is a simultaneous eigenstate of the component operations. In fact, also the Schrödinger's cat paradox [7] can be understood as the quantum-mechanical impact of the superposition of macroscopically distinct operations (to kill or not to kill ) on a classical object (the cat ).Several groups have recently succeeded in applying simple quantum operators to different quantum states. For example, in the optical domain, basic operations, like single-photon addition and subtraction, have been demonstrated to produce highly nonclassical [8,9,10,11] and non-Gaussian states [12] even when applied to classical states of light [13,14]. Both photon addition and subtraction are performed in a conditional way upon the detection of a single photon in an ancillary (herald) light mode. Sequences of photon additions and subtractions have also been implemented to show that the two sequencesââ † andâ †â , whereâ † andâ are the bosonic creation and annihilation operators, give different results when applied to the same input light state [15]. This is an important corner stone for the proof of the bosonic commutation relationwhich is at the heart of many important consequences of quantum mechanics. However, the complete demonstration of the commutation relation was out of reach because of the lack of an important element in the qu...

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...