Entanglement and non-classicality of generalized quantum optical vortex states Anindya Banerji 1 * Ravindra Pratap Singh 2 † Abir Bandyopadhyay 1 ‡ 1 Hooghly Engineering and Technology College, Vivekananda Road, Pipulpati, Hooghly 712103 2 Physical Research Laboratory, Navrangpura, Ahmedabad 380009 Abstract. The negativity of the Wigner function is discussed as a measure of the non classicality and the quantum interference pattern obtained therein as a possible measure of the entanglement between the two modes of the vortex states formed by photon subtraction from two mode squeezed vacuum. This measure of entanglement is compared with the results obtained from concurrence and log negativity. Keywords: Entanglement, Wigner function, concurrence, quantum elliptical vortex 1 Introduction Entanglement is a fundamental characteristic of quan- tum mechanics. It is this feature of quantum mechan- ics which is being increasingly studied and analyzed to achieve a variety of quantum phenomena like quantum teleportation [1] and quantum information processing [2]. In the not so distant future these emerging technologies will define how we communicate and compute. Photonic states of light are speculated to play an ever increasing role in this particular field [3]. It is therefore important to study the entanglement features of various such states of light which can be generated and put to use for a prac- tical model of quantum information processing. 2 Formulation and results In our present work we study the entanglement present in a class of generalized quantum optical vortex states [4]. The states that we consider here are engineered by sub- traction of photons from one of the modes of the output of a type II spontaneous parametric down conversion. These states have the form |ξ 〉 (s) k = A k b k exp ( ξa † b † − ξ * ab ) |0, 0〉 (1) We call these two mode squeezed vortex (TMSV) states. The order of the vortex is determined by the difference in the number of photons between the two modes. If k pho- tons are subtracted, the state, after some simplification, can be written as, |ξ 〉 (s) k = e ikθ cosh 2 r a †k |ξ 〉 (2) An interesting property of the TMSV states is the ap- pearance of the vortex structure in the quadrature space. The order of the vortex can be determined from the num- ber of singular points in the phase plot [5]. It is interest- ing to note that the vortex structure in the quadrature distribution becomes less prominent with increasing or- der. * abanerji09@gmail.com † rpsingh@prl.res.in ‡ abir@hetc.ac.in Figure 1: A schematic of the setup for producing TMSV state of order 2. Two 99% beam splitters(BS 1 and BS 2) are used for subtracting photons. NLC is a nonlin- ear crystal. SPDC generates the signal (mode a) and the idler (mode b). APD 1 and 2 are Avalanche Photo Diodes. We use the Wigner function [6] to study the non classi- cality of this state. For the current state under study it can be written as W (˜ α, ˜ β)= 4 π 2 (−1) k L k  4| ˜ α| 2  exp  −2  | ˜ α| 2 + | ˜ β| 2  , (3) where k is the number of photons subtracted. The pres- ence of the negative regions is a signature of non classi- cality [7]. The cross correlaton between different quadra- tures of two different modes show interesting quantum inteference patterns. This can also be interpreted as a signature of entanglement. In support of this we study the entanglement present in this state with the help of logarithmic negativity [2, 8] and entanglement of forma- tion [9, 10] or concurrence and compare the observations from the two approaches. The concurrence is a measure of the amount of entanglement needed to create the en- tangled state and defined as [11], C (ρ) = min pi,ψi  i p i E (|ψ i 〉〈ψ i |) (4) The minimization is taken over all possible decomposi- tions of pure states ψ i with probabilities p i , which taken