Phase sensitivity of a Mach-Zehnder interferometer

The best performance of a Mach-Zehnder interferometer is achieved with the input state $\mid N_T∕2+1⟩\mid N_T∕2-1⟩+\mid N_T∕2-1⟩\mid N_T∕2+1⟩$, $N_T$ being the total number of atoms or photons. This gives (i) a phase-shift error confidence $C_{68\%}=2.67∕N_T$ with (ii) a single interferometric measurement. Different input quantum states can achieve the Heisenberg scaling $\sim 1∕N_T$ but with higher prefactors and at the price of a statistical analysis of two or more independent measurements.

Figure 1

(Color online) Schematic representation of the Mach-Zehnder interferometer. Atoms or photons enter the $a$ and $b$ input ports, mix, and recombine in the beam splitters, and are finally detected in $D1$ and $D2$. The phase shift is inferred from the number of atoms or photons measured in each output port.

Figure 2

(Color online) (a) and (b) Probability distribution of the $m=0$ twin-Fock state with (a) $N=40$ particles and $p=1$ measurement; (b) $N=20$ particles and $p=2$ measurements. By combining the independent measurements it is possible to strongly reduce the weight of the tail of the distribution with respect to the central peak. (c) and (d) Amplitude $d_{1,1}^j\left(\varphi \right)$ (red line) and $d_{1,-1}^j\left(\varphi \right)$ (blue line), for $N=40$ particles $(j=20)$ [23], (c); the tails of the amplitudes oscillate out of phase and interfere destructively, so to enhance the central peak of the probability distribution $P(\varphi \mid j,0)\sim {\mid d_{1,1}^j\left(\varphi \right)+d_{1,-1}^j\left(\varphi \right)\mid }^2$, (d).

Figure 3

(Color online) Confidence interval as a function of the number of independent measurements $p$ for (a) the twin-Fock $m=0$ and (b) the twin $m=1$ states. Here $N_T=2000$ particles. In both plots, from the bottom line to the top one, the black line is the confidence at 38.29%, the blue at 68.27%; the green at 95.45%, the red at 99.73%, the yellow at 99.994%, and the brown at 99.999 94%. These correspond to $\sigma ∕2$, $\sigma$, $2\sigma$, $3\sigma$, $4\sigma$, and $5\sigma$ when the distribution is Gaussian.

Figure 4

(Color online) Phase uncertainty $\Delta \theta$ as a function of $N_T$. (a) Twin-Fock state: (from the top line to the bottom one) black line, $\sigma$ for $p=1(\sigma \approx 1∕\ln N_T)$; brown line, 68% confidence for $p=1(\Delta \theta =0.8∕N_T^{1∕3})$; yellow line, $\sigma$ for $p=2(\sigma \approx 1∕N_T^{1∕2})$; violet line, $\sigma$ for $p=4(\sigma =4∕N_T)$; green line, 68% confidence for $p=2(\Delta \theta =2.91∕N_T)$. (b) Phase sensitivity of the NOON state (blue line), $\sigma =\sqrt{2∕N_T}$, independent of $p$, and of the twin $m=1$ state (red line), $\Delta \theta =2.67∕N_T$ at $p=1$. Points indicate numerical data.