Quantum metrology for nonlinear phase shifts with entangled coherent states

We investigate the phase enhancement of quantum states subject to nonlinear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states with the same average particle number $\langle n\rangle$ and nonlinearity exponent $k$. We investigate the creation of an approximate entangled coherent state (AECS) from a photon-subtracted squeezed vacuum with current optical technology methods and show that a pure AECS is even better than an even ECS for large $\langle n\rangle$. Finally, we examine the simple, but physically relevant, cases of loss in the nonlinear interferometer for a fixed average photon number $\langle n\rangle$.

Figure 1

The inequality $\delta {\phi }_{N^k}\ge \delta {\phi }_{E_{-}^k}\ge \delta {\phi }_{E_{+}^k}$ with respect to $N=2\langle n_N^1\rangle =\langle n_{E_{\pm }}^1\rangle$ ($k=1,2,3$).

Figure 2

Difference of the optimal phase estimation between $|{\psi }_N\rangle$ and $|{\text{ECS}}_{-}^k\left({\alpha }_{-}\right)\rangle$, such as $\delta {\phi }_{N^k}-\delta {\phi }_{E_{-}^k}$ for $k=1,2,3$.

Figure 3

Schematics of generating $\left|\text{AECS}\right(r,{\alpha }_A)\rangle$ from a squeezed vacuum $S\left(r\right)|0\rangle$ and a coherent state $|\alpha \rangle$. After single-photon subtraction at stage 1, state $\left|\text{ACSS}\right(r)\rangle$ is very similar to $|{\text{CSS}}_{-}\left(\alpha \right)\rangle$. The coherent state in stage 2 has the same amplitude of the old CSS (${\text{BS}}^{\eta }$ and $D$ are an unbalanced BS with a transmission rate $\eta$ and a single-photon detector).

Figure 4

The coefficient $H(m,m^{\prime })$ of $\left|\text{AECS}\right(r_0,{\alpha }_A)\rangle$ (${\alpha }_A\approx 1.9807$). The major contribution of the NOON state and the minor of $m$ and $m^{\prime }$ states is clear. Blue (dark gray) indicates a positive value, and red (light gray) indicates a negative value.

Figure 5

The amplitude of $H_{m,0}\left({\alpha }_A\right)$ (Amp) for ${\alpha }_A=2.0$ and that of coherent state $|2.0\rangle$ with respect to photon number $m$. Both states contain the same average photon number. This implies that the AECS reaches the peak amplitude in smaller $m$ and has a longer tail than the coherent state.

Figure 6

The bound of the optimal phase sensitivity ($1/\sqrt{C_k^Q}$) with small lossy conditions for $k=1,2$ ($T$ is the transmission rate of the BS mimicking photon losses). The thick solid line is for NOON states, and the thin solid line is for AECS, which is very similar to the dashed lines for even (short-dashed line) and odd (long-dashed line) ECSs. The classical precision limit (uncorrelated states) does not depend on the nonlinearity $k$, and its value starts above 0.5 for $\langle n^1\rangle =2.0$.

Figure 7

Phase uncertainties of even ECSs and NOON states using the maximum-likelihood method (ML) and quantum Fisher information with $N=4$ in Eq. (21). The phase sensitivities are indicated as red crosses for ECSs with ML, blue triangles for NOON states with ML, black circles for ECSs given by $1/\sqrt{\mu F_{E_{+}}^Q}$, and green squares for NOON states with $1/\sqrt{\mu F_N^Q}$. We observe that ECSs always have better phase sensitivity than NOON states in cases of both ML and $F^Q$ for the same number of measurements $\mu$.