Generation of optical Schrödinger cat states by generalized photon subtraction

We propose a high-rate generation method of optical Schrödinger cat states. Thus far, photon subtraction from squeezed vacuum states has been a standard method in cat-state generation, but its constraints on experimental parameters limit the generation rate. In this paper, we consider the state generation by photon number measurement in one mode of two-mode Gaussian states, which is a generalization of conventional photon subtraction, and derive the conditions to generate high-fidelity and large-amplitude cat states. Our method relaxes the constraints on experimental parameters, allowing us to optimize them and attain a high generation rate. Supposing realistic experimental conditions, the generation rate of cat states with large amplitudes ($\left|\alpha \right|\ge 2)$ can exceed megacounts per second, about ${10}^3$ to ${10}^6$ times better than typical rates of conventional photon subtraction. This rate would be improved further by the progress of related technologies. The ability to generate non-Gaussian states at a high rate is important in quantum computing using optical continuous variables, where scalability has been demonstrated but preparation of non-Gaussian states of light remains as a challenging task for universality and fault tolerance. Our proposal reduces the difficulty of the state preparation and opens a way for practical applications in quantum optics.

Figure 1

(a) Photon subtraction method. A weak beam is tapped by a beam splitter (BS) from a squeezed vacuum. When a photon number resolving detector (PNRD) detects photons in the tapping channel, cat-like states are heralded. (b) Generalized photon subtraction (GPS). Two squeezed vacuum states squeezed in orthogonal directions interfere at a beam splitter. Photon number measurement heralds cat-like states.

Figure 2

Schematic of GPS. First, a two-mode Gaussian state $|G\rangle$ is prepared. Detection of $n$ photons in one mode of $|G\rangle$ heralds $|{\psi }_n\rangle$ in the other mode. By preparing proper $|G\rangle$, the outcome state $|{\psi }_n\rangle$ approximates cat states well (Sec. 2b). Such $|G\rangle$ can be generated from two squeezed vacuum states and a beam splitter (Sec. 2c).

Figure 3

(a) The plots of the functions ${\varphi }_n\left(x_1\right), {\varphi }_n\left(x_1\right)G(x_1,x_2)$, and ${\psi }_n\left(x_2\right)$ in the case of $r_1=-r_2=0.576$ (5 dB squeezing) and $n=10$. From the left, ${\sigma }_{11}=0.6(R=0.10),{\sigma }_{11}=1(R=0.24),$ and ${\sigma }_{11}=1.4(R=0.38)$. The black broken lines show the wave function of $\widehat{S}\left(r_c\right)\left|{\mathrm{Cat}}_{\sqrt{10},10}\right.〉$ in the case of ${\sigma }_{11}=1$ [see Eq. (18)]. When ${\sigma }_{11}=1$, the fidelity of $|{\psi }_n\rangle$ and the target state is $F_{10}\approx 0.997$. (b) Similar plots about $p$.

Figure 4

(a), (b) Comparison of the success probability to generate a Schrödinger cat state with $\alpha =\sqrt{n}$. The compared methods are GPS, the homodyne conditioning method, and conventional photon subtraction. In each method, we suppose the squeezing parameters of inputs are $\left|r\right|=0.576$ (5 dB squeezing) in (a) and $\left|r\right|=1.15$ (10 dB squeezing) in (b).

Figure 5

The probability to detect 5, 10, and 20 photons in GPS in the case of $r_1=-r_2$ and ${\sigma }_{11}=1$.