Production of Superposition of Coherent States Using Kerr Non Linearity

Abstract

The coherent states are often called as Glauber coherent states and were named after the American Scientist Glauber who was first to realize the extraordinary usefulness of these coherent states for explanation and analysis of many optical phenomena. These states were first introduced by Sudarshan also and are now been extensively studied and applied to quantum-optical problems. The most explicit form of these states are expressed as, $\left| {\,\alpha \,} \right\rangle = \,\sum\nolimits_{n = 0}^{\infty } {e^{{ - \frac{1}{2}\,\left| {\,\alpha \,} \right|^{2} }} \frac{{\alpha^{n} }}{\sqrt n !}\,\left| {\,n\,} \right\rangle }$ where, the Fock states $\left| {\,n\,} \right\rangle$ is the eigen state of the number operator $N = \,a^{{{\dag }}} a$, i.e., $N\,\left| {\,n\,} \right\rangle = \,n\,\left| {\,n\,} \right\rangle$ and $\alpha \, = \,\alpha_{r} + \,i\alpha_{i}$ is a complex number. These Glauber coherent states are the eigen states of annihilation operator and are well known. They play a very important role in many applications of quantum information processing including quantum teleportation. But it has been a long dream for physicists to generate these superposed coherent states in the most general desired form $\left| {\,\psi \,} \right\rangle \, = \,N\,\left( {\cos \,\frac{\theta }{2}\,\left| {\,\alpha \,} \right\rangle \, \pm \sin \,\frac{\theta }{2}\,e^{i\,\varphi } \,\left| { - \alpha \,} \right\rangle \,} \right)\,$ where, N is the normalization factor. In this paper, we propose a scheme to generate any such general superposition of coherent states $\left| {\,\alpha \,} \right\rangle$ and $\left| { - \alpha \,} \right\rangle$ using Kerr effect, two beams in coherent states, a single photon beam and optical devices like polarization beam splitter and mirrors. In the output, if a single photon is detected in a polarization state defined by angle $\theta$ and $\varphi$, the desired superposition of coherent states $\left| {\,\alpha \,} \right\rangle$ and $\left| { - \alpha \,} \right\rangle$ results. If the photon is detected in an orthogonal polarization state (the state in which the electric field strength at a given point in space is normal to the direction of propagation), a superposition state different from the desired one results.