Phil. Trans. R. Soc. A (2012) 370, 5408–5421 doi:10.1098/rsta.2011.0524 Single photon quantum filtering using non-Markovian embeddings BY JOHN E. GOUGH 1 ,MATTHEW R. JAMES 2, * AND HENDRA I. NURDIN 3 1 Institute of Mathematics and Physics, Aberystwyth University, Aberystwyth SY23 3BZ, UK 2 ARC Centre for Quantum Computation and Communication Technology, Research School of Engineering, Australian National University, Canberra, Australian Capital Territory 0200, Australia 3 School of Electrical Engineering and Telecommunications, The University of New South Wales at Sydney, Sydney, New South Wales 2052, Australia We determine quantum master and filter equations for continuous measurement of systems coupled to input fields in certain non-classical continuous-mode states, specifically single photon states. The quantum filters are shown to be derivable from an embedding into a larger non-Markovian system, and are given by a system of coupled stochastic differential equations. 1. Introduction In recent years, single photon states of light and superpositions of coherent states have become increasingly important owing to applications in quantum technology, in particular quantum computing and quantum information systems [1–5]. For instance, the light may interact with a system, say an atom, quantum dot or cavity, and this system may be used as a quantum memory [3], or to control the pulse shape of the single photon state [4]. When light interacts with a quantum system, information about the system is contained in the scattered light (output) and this may be used to monitor or control the system. The problem of extracting information from continuous measurement of the scattered light is a problem of quantum filtering [6–13]; however, this has tended to consider inputs only in a vacuum or other Gaussian state, with quadrature or counting measurements. The purpose of this paper is to solve a quantum filtering problem for systems driven by fields in single photon states. When the input field is in a single photon state, the master equation describing unconditional dynamics was shown to be a system of coupled equations in Gheri et al.[14], an apparently non-Markovian feature. Markovian embeddings were used in Breuer [15] to derive quantum trajectory equations (quantum filtering equations) for a class of non-Markovian master equations. In recent work, the authors have shown how to construct ancilla systems to combine with the *Author for correspondence (matthew.james@anu.edu.au). One contribution of 15 to a Theo Murphy Meeting Issue ‘Principles and applications of quantum control engineering’. This journal is © 2012 The Royal Society 5408 on March 24, 2016 http://rsta.royalsocietypublishing.org/ Downloaded from