Generation of macroscopic superposition states with small nonlinearity

We suggest a scheme to generate a macroscopic superposition state (“Schrödinger cat state”) of a free-propagating optical field using a beam splitter, homodyne measurement, and a very small Kerr nonlinear effect. Our scheme makes it possible to reduce considerably the required nonlinear effect to generate an optical cat state using simple and efficient optical elements.

Figure 1

Schematic of generating a cat state using small Kerr nonlinearity, a beam splitter, and a homodyne detection.

Figure 2

(a) The probability distribution of the quadrature variable $P$ for the state after passing the small nonlinear medium but before the beam splitter in Fig. 1. The interaction time $\lambda \tau =\pi ∕20$ and the initial amplitude ${\alpha }_i=20$. (b) The probability distribution for the state after the beam splitter and homodyne measurement of result $X=0$. It is obvious from the figures that a cat state with well separated peaks is obtained after the process.

Figure 3

Fidelity $F$ of the generated cat state against the measurement outcome $X$ where $N=20$ (solid line), $N=40$ (dashed line), $N=60$ (dotted line). The initial amplitude is ${\alpha }_i=20$.

Figure 4

The probability distribution of $P$ for the state conditioned on the homodyne measurement of result $X=0$ after the beam splitter when the interaction time is $\lambda =\pi ∕200$ and the initial amplitude is ${\alpha }_i=20$.

Figure 5

The average fidelity between the ideal cat state and the phase-fluctuated state with standard deviation $\sigma$ during the process in Fig. 1. The cases for $N=20$ (solid line), $N=40$ (dashed line), and $N=60$ (dotted line) have been plotted with the initial amplitude ${\alpha }_i=20$. The result of the homodyne measurement is considered $X=0$.