PHYSICAL REVIEW A 88, 013822 (2013) Mode reconstruction of a light field by multiphoton statistics Elizabeth A. Goldschmidt, 1,2,* Fabrizio Piacentini, 3 Ivano Ruo Berchera, 3 Sergey V. Polyakov, 2 Silke Peters, 4 Stefan K¨ uck, 4 Giorgio Brida, 3 Ivo P. Degiovanni, 3 Alan Migdall, 1,2 and Marco Genovese 3 1 Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA 2 National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899, USA 3 Istituto Nazionale di Ricerca Metrologica INRIM, Strada delle Cacce 91, 10135 Torino, Italy 4 Physikalisch-Technische Bundesanstalt Braunschweig, Bundesallee 100, 38116 Braunschweig, Germany (Received 31 January 2013; published 15 July 2013) We present a simple method to reconstruct the mode distribution of multimode classical and nonclassical optical fields using a single measurement of higher-order photon number correlation functions. Knowing the underlying number and structure of occupied modes of a light field plays a crucial role in minimizing loss and decoherence of quantum information. Typically, full characterization of the mode structure involves a series of several separate measurements in spatial, temporal, frequency, and polarization domains. We experimentally demonstrate reconstruction of up to three modes with excellent agreement and study the robustness of our method in experimentally realizable regimes. DOI: 10.1103/PhysRevA.88.013822 PACS number(s): 42.50.Ar, 03.65.Wj, 42.50.Dv I. INTRODUCTION Characterizing the underlying processes contributing to a light field has wide ranging applications throughout physics. For instance, knowledge of the mode structure is vital for engineering sources of nonclassical light that minimize loss and decoherence of quantum information due to coupling to unwanted modes. Such applications include mode-matching biphoton collection [1], producing factorizable states of photon pairs [2], minimizing classical background emission from single-emitter sources [3], and characterizing the number and degree of squeezing in multimode continuous variable entangled states [1,4–8]. Photon-number statistics are used to characterize a variety of optical systems including single-photon sources [9–11], photon pair sources [12–14], cavity QED [15,16], and lasers [17,18]. In most cases, these measurements have been lim- ited to single- and twofold photodetection, or first- and second-order optical coherence. In terms of understanding the underlying processes contributing to the light field, this can provide only limited information, such as a measure of the purity of the system. Recent developments in photon-number resolving (PNR) detectors [19–21] allow simpler measurement of higher-order correlations, and such measurements should continue to become more routine [22–24]. We show that this additional information can allow a full characterization of the various quantum and classical modes present in a light field. We present and implement experimentally a method to reconstruct the underlying mode structure of an optical field using high-order photon-number statistics. Typically, full characterization of the mode structure in- volves a series of separate measurements in spatial, temporal, frequency, and polarization domains, requiring a range of instrumentation. However, our method can be easily inte- grated into existing optical systems as it uses only a single measurement of the photon-number distribution of a field. * egolds@nist.gov Also, full mode reconstruction allows a more subtle distinction between classical and nonclassical fields. We show how a full reconstruction of the underlying mode structure of a field can provide information about nonclassical components of a nominally classical field. We consider multimode light from a single source, such as multimode thermal light from spontaneous parametric down conversion (SPDC), as well as from multiple sources, each producing light in one or more modes, such as attenuated single photons from a single emitter and coherent light from a laser. We note, however, that this method is extremely general and can be used for any combination of sources, though only the total fraction of the power with underlying Poissonian statistics can be determined. We perform a proof of principle experiment using PNR detection and mixed states with contributions from one or more modes with thermal statistics, up to one mode with attenuated single-photon statistics, and up to one mode with Poissonian statistics. We successfully identify the distribution of contribu- tions from up to three total modes of classical and nonclassical light. We also theoretically study the robustness and prospects of our method in experimentally accessible regimes. II. METHOD It is straightforward to write down the full photon-number probability distribution for a given mode structure with mean photon numbers μ i in the modes. For thermal and Poissonian statistics μ =〈n〉 is the mean photon number and for attenuated single-photon statistics μ is the probability of finding a single photon. The photon-number probability distribution is uniquely described by a probability generating function G(s ), which is the product of the generating functions for all the underlying modes where G thermal (s ) = [1 + μ(1 − s )] −1 , G single photon (s ) = [1 − μ(1 − s )], and G Poissonian (s ) = e −μ(1−s ) [25]. It is convenient to translate this into a set of relations between the μ i and the intensity autocorrelation functions (at zero time difference), g (k) (0) = g (k) =〈:ˆ n k :〉/〈 ˆ n〉 k , where 〈::〉 denotes normal ordering of the operators. We find that g (k) = G (k) (s = 1)/(μ total ) k , where 013822-1 1050-2947/2013/88(1)/013822(5) ©2013 American Physical Society