Efficient noiseless linear amplification for light fields with larger amplitudes

We suggest and investigate a scheme for non-deterministic noiseless linear amplification of coherent states using successive photon addition, $(\hat a^{\dagger})^2$, where $\hat a^\dagger$ is the photon creation operator. We compare it with a previous proposal using the photon addition-then-subtraction, $\hat a \hat a^\dagger$, where $\hat a$ is the photon annihilation operator, that works as an appropriate amplifier only for weak light fields. We show that when the amplitude of a coherent state is $|\alpha| \gtrsim 0.91$, the $(\hat a^{\dagger})^2$ operation serves as a more efficient amplifier compared to the $\hat a \hat a^\dagger$ operation in terms of equivalent input noise. Using $\hat a \hat a^\dagger$ and $(\hat a^{\dagger})^2$ as basic building blocks, we compare combinatorial amplifications of coherent states using $(\hat a \hat a^\dagger)^2$, $\hat a^{\dagger 4}$, $\hat a \hat a^\dagger\hat a^{\dagger 2}$, and $\hat a^{\dagger 2}\hat a \hat a^\dagger$, and show that $(\hat a \hat a^\dagger)^2$, $\hat a^{\dagger 2}\hat a \hat a^\dagger$, and $\hat a^{\dagger 4}$ exhibit strongest noiseless properties for $|\alpha| \lesssim 0.51$, $0.51 \lesssim |\alpha| \lesssim 1.05 $, and $|\alpha|\gtrsim 1.05 $, respectively. We further show that the $(\hat a^{\dagger})^2$ operation can be used for amplifying superpositions of the coherent states. In contrast to previous studies, our work provides efficient schemes to implement a noiseless amplifier for light fields with medium and large amplitudes.