Multiphoton quantum interference at a beam splitter and the approach to Heisenberg-limited interferometry

We study multiphoton quantum-interference effects at a beam splitter and its connection to the prospect of attaining interferometric phase-shift measurements with noise levels below the standard quantum limit. Specifically, we consider the mixing of the most classical states of light coherent states with the most nonclassical states of light number states at a 50 : 50 beam splitter. Multiphoton quantum-interference effects from mixing photon-number states of small photon numbers with coherent states of arbitrary amplitudes are dramatic even at the level of a single photon. For input vacuum and coherent states, the joint photon-number distribution after the beam splitter is unimodal, a product of Poisson distributions for each of the output modes but with the input of a single photon, the original distribution is symmetrically bifurcated into a bimodal distribution. With a two-photon-number state mixed with a coherent state, a trimodal distribution is obtained, etc. The obtained distributions are shown to be structured so as to be conducive for approaching Heisenberg-limited sensitivities in photon-number parity-based interferometry. We show that mixing a coherent state with even a single photon results in a significant reduction in noise over that of the shot-noise limit. Finally, based on the results of mixing coherent light with single photons, we consider the mixing coherent light with the squeezed vacuum and the squeezed one-photon states and find the latter yields higher sensitivity in phase-shift measurements for the same squeeze parameter owing to the absence of the vacuum state.

Figure 1

A Mach-Zehnder interferometer with input coherent and number states where $\phi$ is the relative phase shift between the two arms where (a) represents the usual detection scheme wherein the output beam photocurrents are subtracted and (b) represents the detection scheme where photon-number parity measurements are performed on just one of the output beams.

Figure 2

For $\alpha =3$, (a) the joint photon number distribution $P(m_a,m_b|0)$ plotted against $m_a$ and $m_b,$ (b) for $P(m_a,m_b|1),$ (c) for $P(m_a,m_b|2),$ and (d) for $P(m_a,m_b|3)$.

Figure 3

Expectation value of the parity operator for the output $b$ beam plotted against the phase shift $\phi$ and $\left|\alpha \right|$ for (a) $N=0,$ (b) $N=1,$ (c) $N=2,$ and (d) $N=3$.

Figure 4

The phase uncertainty $\Delta \phi$ versus the total average photon number $\overline{n}={\left|\alpha \right|}^2+N$ with the choice $\phi ={10}^{-4}\mathrm{rad}$ for (a) $N=0,$ (b) $N=1,$ (c) $N=2,$ and (d) $N=3.$ The upper dashed line represents the corresponding standard quantum limit, and the lower dashed line represents the Heisenberg limit. The dots represent the results obtained for the Cramér-Rao bound given by Eq. (28). We include (a) the case $N=0$ as a reminder that, when performing parity measurements for input coherent light only, one reaches the standard quantum limit for low values of the phase shift in contrast to what happens using subtraction of photocurrents leading to Eq. (16).

Figure 5

Photon-number probability distributions for (a) the single-mode squeezed vacuum state and (b) the single-mode squeezed one-photon states, both for squeeze parameter $r=1.2$.

Figure 6

The joint photon-number distribution after beam splitting for equal intensity coherent and squeezed vacuum states corresponding to $\alpha =\sqrt{1.2}$ and $r=0.947$.

Figure 7

Phase uncertainties against the total average photon number for coherent light mixed with (a) the squeezed vacuum and (b) with squeezed one-photon states for the choice of $r=0.3$ and for $\phi ={10}^{-4}$.

Figure 8

Same as Fig. 7 but for $r=0.9$.

Figure 9

(a) Plot of the joint photon-number probability distribution after the mixing of a coherent state with a squeezed one-photon state for the case of $\alpha =2$ and $r=0.9$, and the average total photon number for this state is $\overline{n}=8.133$. (b) Plot of the distribution for the mixed coherent squeezed vacuum states with the same parameters. The average total photon number for this state is $\overline{n}=5.026$.

Figure 10

Plot of the expectation value of the output $b$-mode photon-number parity operator versus $\phi$ for mixed coherent, squeezed vacuum, and squeezed one-photon states for $\alpha =2$ and $r=0.9$. The total average photon number in the case of the squeezed vacuum state is $\overline{n}=5.054$ where, for the case of the squeezed one-photon state, it is $\overline{n}=8.161$. The curve for the latter case is narrower than that for the former indicating an increase in resolution.