Generating “squeezed” superpositions of coherent states using photon addition and subtraction

We study how photon addition and subtraction can be used to generate squeezed superpositions of coherent states (SSCSs) in free-traveling fields with high fidelities and large amplitudes. It is shown that an arbitrary $N$-photon subtraction results in the generation of a SSCS with nearly the perfect fidelity $(F>0.999)$ regardless of the number of photons subtracted. In this case, the amplitude of the SSCS increases as the number of the subtracted photons gets larger. For example, two-photon subtraction from a squeezed vacuum state of $6.1\mathrm{dB}$ can generate a SSCS of $\alpha =1.26$, while in the case of the four-photon subtraction a SSCS of a larger amplitude $\alpha =1.65$ is obtained under the same condition. When a photon is subtracted from a squeezed vacuum state and another photon is added subsequently, a SSCS with a lower fidelity $(F\approx 0.96)$ yet higher amplitude $(\alpha \approx 2)$ can be generated. We analyze some experimental imperfections including inefficiency of the detector used for the photon subtraction.

Figure 1

(Color online) (a) Optimal fidelity $F_{\mathit{opt}}$ between the ideal TPSS $\left(a^2\right)$ and the squeezed SCS of amplitude $\alpha$ (solid curve) and the optimal fidelity between the ideal TPSS and the corresponding regular SCS (dashed curve). The squeezing parameter $r_{\mathit{opt}}^{\prime }$ of the initial squeezed state is $r=-0.7$. (b) The squeezing parameter $r^{\prime }$ of the target SSCS for which the fidelity is optimized.

Figure 2

(Color online) (a) Optimal fidelity $F_{\mathit{opt}}$ between the ideal PSAS $\left(a^{†}a\right)$ and the SSCS of amplitude $\alpha$ (solid curve) and the optimal fidelity between the ideal PSAS and the corresponding regular SCS (dashed curve). The squeezing parameter $r_{\mathit{opt}}^{\prime }$ of the initial squeezed state is $r=-0.7$. (b) The squeezing parameter of the target SSCS for which the fidelity is optimized.

Figure 3

(Color online) (a) The Wigner function of the TPSS with $r=-0.7$ $(\approx 6.1\mathrm{dB})$ and (b) the Wigner function of the ideal SSCS with $\alpha =1.26$ and $r^{\prime }=-0.425$. The fidelity between the two states is nearly perfect as $F>0.999$.

Figure 4

(Color online) (a) The Wigner function of the PSAS (left) with $r=-0.7$ $(\approx 6.1\mathrm{dB})$ and (b) the Wigner function of the ideal SSCS with $\alpha =2$ and $r^{\prime }=-0.14$ (right). The fidelity between the two states is $F\approx 0.955$.

Figure 5

(Color online) (a) Optimal fidelity $F_{\mathit{opt}}$ between the TPSS $\left(a^2\right)$ generated using ideal avalanche photon detectors and the SSCS of amplitude $\alpha$ (solid curve) and the optimal fidelity with the corresponding regular SCS (dashed curve). The squeezing parameter of the initial squeezed state is $r=-0.7$. The parameter of real transformation is $T=0.99$. (b) The squeezing parameter of the target SSCS for which the fidelity is optimized.

Figure 6

(Color online) Optimal fidelity $F_{\mathit{opt}}$ between the PSAS $\left(a^{†}a\right)$ generated using ideal avalanche photon detectors and the SSCS of amplitude $\alpha$ (solid curve) and the optimal fidelity with the corresponding regular SCS (dashed curve). The squeezing parameter of the initial squeezed state is $r=-0.7$. (b) The squeezing parameter of the target SSCS for which the fidelity is optimized. The parameters of real transformations are $T=0.99$, $G=1.01$.

Figure 7

(Color online) (a) Optimal fidelity $F_{\mathit{opt}}$ between the TPSS $\left(a^2\right)$ generated using realistic avalanche photon detectors and the SSCS of amplitude $\alpha$ (solid curve) and the optimal fidelity with the corresponding regular SCS (dashed curve). The squeezing parameter of the initial squeezed state is $r=-0.7$. The parameters of realistic transformations are $T=0.99$ and $\eta =0.6$. (b) The squeezing parameter of the target SSCS for which the fidelity is optimized.

Figure 8

(Color online) (a) Optimal fidelity $F_{\mathit{opt}}$ between the PSAS $\left(a^{†}a\right)$ generated using realistic avalanche photon detectors and the SSCS of amplitude $\alpha$ (solid curve) and the optimal fidelity with the corresponding regular SCS (dashed curve). The squeezing parameter of the initial squeezed state is $r=-0.7$. (b) The squeezing parameter of the target SSCS for which the fidelity is optimized. The parameters of realistic transformations are $T=0.99$, $G=1.01$, and $\eta =0.6$.

Figure 9

The optimal fidelity between the NPSSs with $\mathcal{R}(=T\tanh r)=0.6$ and the SSCSs of $r^{\prime }=0.44$ from $N=3$ (left) to $N=8$ (right). The $x$ axis represents the amplitude $\alpha$ of the target SSCS. The optimal fidelity is $F>0.999$ regardless of the number of $N$.