Mach-Zehnder Interferometry at the Heisenberg Limit with Coherent and Squeezed-Vacuum Light

We show that the phase sensitivity $\Delta \theta$ of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit $\Delta \theta \sim 1/N_T$, where $N_T$ is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\Delta \theta \sim 1/\sqrt{|\alpha {|}^2{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter $r$ and the amplitude of the coherent mode $\alpha$ by using a Bayesian phase inference protocol.

Figure 1

Schematic representation of the Mach-Zehnder interferometer. The input modes $a$, $b$ are a coherent and a squeezed-vacuum field, respectively.

Figure 2

Comparison between Eq. (1) (dashed line) and Eq. (3) (solid line). Circles are the results of a Bayesian analysis. (a) Phase sensitivity $\Delta \theta \sqrt{\overline{n}p}$ as a function of the squeezing parameter $r$, for $\theta =\pi /2$ and $|\alpha {|}^2=10$. Notice that Eq. (1) diverges at ${sinh}^{2}r=|\alpha {|}^2$ (dotted vertical line). For $r\gg 1$, Eq. (3) gives $\Delta \theta \sqrt{\overline{n}p}\to 1/\sqrt{4|\alpha {|}^2+1}$ (dotted horizontal line). (b) $\Delta \theta \sqrt{\overline{n}p}$ in the limit $p\to \infty$ as a function of the true value of the phase shift. Here $|\alpha {|}^2=10$ and $r=1$.

Figure 3

Demonstration of the Heisenberg limit $\Delta \theta \sim 1/N_T$. (a) Circles are the phase sensitivity obtained with the Bayesian analysis as a function of the number of measurements $p$ with fixed total number of particles $N_T=p\overline{n}$. The optimal value, $p_{\mathrm{opt}}=30$, corresponds to the minimum of $\Delta \theta$ and does not depend on $\overline{n}$. Dashed lines are guides to the eye. (b) Phase sensitivity as a function of $N_T$ calculated with the optimal number of measurements $p_{\mathrm{opt}}=30$. The dashed line is the asymptotic limit $\Delta \theta =7.12/N_T$, the solid line is $\Delta \theta \sim 1/(\sqrt{p_{\mathrm{opt}}}\sqrt{|\alpha {|}^2{e}^{2r}+{sinh}^{2}r})$. Shot noise has been included for comparison (dot-dashed line).

Figure 4

Relative number of particles distribution $P(\mu )$ for (a) the input state $|{\psi }_N⟩$ with a postselected number of particles $N=\overline{n}$ and $|\alpha {|}^2={sinh}^{2}r$ and (b) the state created by the first beam splitter, $|{\psi }_N^{\mathrm{BS}}⟩$. (c) Phase distribution $P(\varphi )$ obtained by projecting $|{\psi }_N^{\mathrm{BS}}⟩$ over phase states. (d) Probability to obtain a NOON state after the first beam splitter, $P_{\mathrm{NOON}}$, as a function of $|\alpha {|}^2/\overline{n}$. The maximum is reached at $|\alpha {|}^2\simeq \overline{n}/2$. Here $\overline{n}=20$.