Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields

The parity of photonic number states is known to be an important observable for quantized electromagnetic fields with applications to quantum information processing and to Heisenberg-limited measurement of phase shifts in quantum interferometry performed with maximally entangled states and with twin number states. In this paper we describe an approach to the quantum nondemolition measurement of parity for quantized optical fields. The method proposed involves the use of a cross-Kerr interaction where we assume a large Kerr nonlinearity is available through the techniques of electromagnetically induced transparency. Our proposed method does not require the measurement of photon number but rather measures parity directly. The method not only allows for the quantum nondemolition measurement of parity but also allows for the von Neumann projection of parity eigenstates from an arbitrary field state. The generation and detection of higher-order parity eigenstates is also discussed. Losses from dissipation and the effects of detector efficiency are considered.

Figure 1

Schematic of the proposed quantum nondemolition measurement of parity. A coherent state of amplitude $\sqrt{2}\alpha$ is split into two coherent states each of amplitude $\alpha$. The cross-Kerr interaction couples modes a mode internal to the interferometer to an external mode $c$. The state of whose parity is to be measured, or to be reduced to a parity eigenstate, in the case of a single mode traveling-wave field is injected into the Kerr medium (mode $c$). In the case of a two-mode field state (modes $c$ and $d$) where there are correlations between the modes, only one of the modes, again $c$ needs to pass though the Kerr medium. The angle $\theta$ represents and adjustable phase shift.

Figure 2

Same as Fig. 1 but where both $c$ and $d$ modes pass through the Kerr medium. This allows for the generation of entangled states in those modes with uncorrelated (unentangled) input states.