Amplifying Single-Photon Nonlinearity Using Weak Measurements

We show that weak measurement can be used to “amplify” optical nonlinearities at the single-photon level, such that the effect of one properly postselected photon on a classical beam may be as large as that of many unpostselected photons. We find that “weak-value amplification” offers a marked improvement in the signal-to-noise ratio in the presence of technical noise with long correlation times. Unlike previous weak-measurement experiments, our proposed scheme has no classical equivalent.

Figure 1

The single-photon system is prepared in an equal superposition of arms $a$ and $b$ by the first beam-splitter (BS1). After a weak cross-phase-modulation (XPM) interaction with the probe, prepared in a coherent state $|\alpha {⟩}_p$, the system is postselected on a nearly orthogonal state by detecting the single photon in the nearly dark port, $D1$. The success probability of postselection depends on the imbalance $\delta$ in the reflection and transmission coefficients of the beam splitter BS2 and the backaction of the probe on the system. Using the lower interferometer to read out the phase shift of the probe amounts to a measurement of the system observable $n_b$, the photon number in arm $b$. The phase shifter $\theta$ is used to maximize the sensitivity of the measurement.

Figure 2

The enhancement factor versus $|\alpha {|}^2{\varphi }_0$. The parameters used are ${\varphi }_0=2\pi \times {10}^{-5}$ and $\delta =0.01$. The enhancement factor is calculated by using the state of Eq. (3) without any approximations. The dashed line shows the enhancement factor if the average phase written by the probe on the system, $|\alpha {|}^2{\varphi }_0$, is compensated; otherwise, enhancement occurs whenever $|\alpha {|}^2{\varphi }_0$ is close to an integer multiple of $2\pi$ (solid curve). The inset shows the enhancement factor as a function of postselection parameter, $\delta$, in two different regimes: (i) $|\alpha {|}^2={10}^5$, in which case the imparted phase on the system by the probe, $ϵ$, is 0 (solid blue line); (ii) $|\alpha {|}^2={10}^2$, where $ϵ$ is a small nonzero phase (dashed green line). For large values of $\delta$ the weak-measurement prediction is valid; however, as $\delta$ decreases, the backaction from the probe plays a more dominant role. The dashed line shows the prediction of the weak-measurement formalism.

Figure 3

The SNR as a function of the single-photon rate $\Gamma$. The technical noise is modeled by an exponential correlation function with an amplitude, $\overline{\eta }$, 10 times larger than the quantum noise. The dashed line shows the nonpostselected SNR for the phase shift due to one photon in mode $b$. The postselected SNR for ${\delta }_1=0.1$ (weak-measurement regime—dash-dotted red line) and ${\delta }_2=0.01$ (the optimum value of measured phase shift—solid green line) are also shown; the dotted line shows the quantum-limited SNR for comparison. The nonpostselected SNR approaches a maximum value $S_0$ due to low-frequency noise. However, for the postselected SNR, we see enhancement by a factor of $\delta /2P$, compared to the nonpostselected SNR $S_0$ for measurements with a high enough rate. For low rates the enhancement is given by $\delta /2\sqrt{P}$, and therefore the weak-measurement results in the best possible postselected SNR. We have taken $T/{\tau }_c={10}^3$, $\varphi =2\pi \times {10}^{-5}$, $|\alpha {|}^2={10}^5$, and therefore $P_1=0.01$ and $P_2=3\times {10}^{-4}$.