Bell’s inequalities, multiphoton states and phase space distributions

The connection between quantum optical nonclassicality and the violation of Bell’s inequalities is explored. Bell type Ž . inequalities for the electromagnetic field are formulated for general states arbitrary number or photons, pure or mixed of quantised radiation and their violation is connected to other nonclassical properties of the field. Classical states are shown to obey these inequalities and for the family of centered Gaussian states the direct connection between violation of Bell-type inequalities and squeezing is established. PACS: 42.50.Wm; 03.65.Bz The violation of Bell’s inequalities is one of the w x most striking features of quantum theory 1 . The testing ground for these inequalities has mostly been w x the states of the electromagnetic field 2–4 . When a state does not obey Bell-type inequalities it definitely has essential quantum features which cannot be reconciled with the classical notions of reality and locality. In most treatments the Bell-type inequalities are formulated for specific quantum states. For the electromagnetic field there have been attempts to generalise the treatment and relate the violation of Bell-type inequalities with other general nonclassical w x features of the states 5–9 . 1 Email:[email protected] 2 Also at Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, 560 064, India. We develop the machinery for analysing the violation of Bell type inequalities for a general state of the 4-mode radiation field in a set-up of the type shown in Fig. 1. For the direction k, the two orthogonal polarisation modes are described by the annihilation operators a and a , with a and a being 1 2 3 4 similarly chosen for the direction k X. Without any loss of generality we choose k and k X to be in the plane of the paper. This allows a simple choice for the directions x, x X to be in the same plane while y and yX point out of this plane. The passive, total photon number conserving, canonical transformaŽ tions which will play an important role in our . analysis amount to replacing the a ’s by their comj plex linear combinations aX sU a , with U being a j jk k Ž . unitary matrix belonging to U 4 . P and P are 1 2 polarisers placed at angles u and u with respect to 1 2 the x and x X axes while D and D are photon 1 2 detectors. Fig. 1. Set-up to study the violation of Bell type inequalities for arbitrary states of the four mode radiation field. Usually states with strictly one photon in each direction are considered for violation of Bell-type inequalities; a general state however could have an arbitrary number of photons, and could even be a mixed state. To handle such states one needs to generalise the concept of coincidence counts, stipulate the polariser action on general quantum states and identify precisely the hermitian operators for which a hidden variable description is being assumed. As a result of this generalisation we will show that a classical state in the quantum optical sense always obeys these inequalities while a nonclassical state may violate them, possibly after a Ž . passive U 4 transformation. Starting with a general nonclassical state, we subject it to a general unitary evolution corresponding to passive canonical transŽ . formations U 4 before we look for the violation of Bell-type inequalities. A coincidence is defined to occur when both the detectors D and D click 1 2 simultaneously i.e., one or more photons are detected by each. The following coincidence count rates are considered: Ž . Ž . a P u ,u : P at u and P at u . 1 2 1 1 2 2 Ž . Ž . b P u , : P at u and P removed. 1 1 1 2 Ž . Ž . c P ,u : P removed and P at u . 2 1 2 2 Ž . Ž . d P , : Both P and P removed. 1 2 Before further analysis and calculation of these count rates we need to specify the precise action of the polarisers on a given quantum state. Classically, the action of a polariser is straightforward. The component of the electric field along the axis passes through unaffected while the orthogonal component is completely absorbed. The quantum action of the polariser is more complicated: for a given two-mode Ž density matrix r the two polarisation modes for a . fixed direction incident on a polariser placed at an Ž . Ž . angle u , the output single-mode state r u is obtained by taking the trace over the mode orthogonal to the linear polarisation defined by u . Explicitly in the number state basis: