Autocorrelative weak-value amplification and simulating the protocol under strong Gaussian noise

By choosing more orthogonality between preselection and postselection states, one can significantly improve the sensitivity in general optical quantum metrology based on the weak-value amplification (WVA) approach. However, increasing the orthogonality decreases the probability of detecting photons and makes the weak measurement difficult, especially when the weak measurement is disturbed by strong noise and the pointer is drowned in noise with a negative-decibel signal-to-noise ratio (SNR). In this article we introduce and numerically evaluate a modified weak-measurement protocol with a temporal pointer, namely, the autocorrelative weak-value amplification (AWVA) approach. Specifically, a small longitudinal time delay (tiny phase shift) $\tau$ of a Gaussian pulse is measured by implementing two simultaneous autocorrelative weak measurements under Gaussian white noise with different SNRs. The small quantities $\tau$ are obtained by measuring the autocorrelation coefficient of the pulses instead of fitting the shift of the mean value of the probe in the standard WVA technique. Simulation results show that the AWVA approach outperforms the standard WVA technique in the time domain with smaller statistical errors, remarkably increasing the precision of weak measurement under a strong noise background.

Figure 1

Scheme of the standard WVA technique. The Gaussian beam is produced by the laser and modulator, Then photons are preselected by polarizer 1 (P1) with the optical axis set at ${45}^{\circ }$. A time delay $\tau$ (corresponding to a phase shift between the horizontal polarized state $|H\rangle$ and vertical polarized state $|V\rangle$) is induced by a birefringent crystal. Finally, the photons are postselected by the second polarizer (P2) with an optical axis set at $\alpha -{45}^{\circ }$ and the arrival time of single photons is measured with an avalanche photodiode (APD).

Figure 2

Scheme of the AWVA technique. The light path is similar to that in the WVA scheme (Fig. 1), except for a 50:50 beam splitter being inserted between polarizer P1 and the birefringent crystal to add a light path for an autocorrelative measurement. The optical axis of polarizer 3 (P3) is also set at $\alpha -{45}^{\circ }$.

Figure 3

Scheme of the signal processing module with the AWVA technique

Figure 4

Simulation results with WVA and AWVA schemes in the absence of noise: (a) signals $I_1^{\mathrm{out}}(t;\tau )$ with $\tau =0\times {10}^{-9}$ s and $\tau =3\times {10}^{-9}$ s in the WVA scheme, (b) signals $I_{21}^{\mathrm{out}}(t;\tau )$ as well as $I_{22}^{\mathrm{out}}(t;\tau )$ with $\tau =3\times {10}^{-9}$ s in the AWVA scheme, and (c) signals $I_A^{AC}$ with $\tau =0\times {10}^{-9}$ s and $\tau =3\times {10}^{-9}$ s in the AWVA scheme. The quantities $I_{21}^{\mathrm{out}}(t;\tau )$, $I_{21}^{\mathrm{out}}(t;\tau )$, and $I_{22}^{\mathrm{out}}(t;\tau )$ are in units of $I_0$ and the quantity $I_A^{AC}$ is in units of voltage.

Figure 5

(a) Gaussian noise signals $\mathbf{N}(t,{\sigma }^2=1.0\times {10}^{-5},\xi =0)$ in the time domain, (b) its power spectral densities $\mathbf{PSD}(f,{\sigma }^2=1.0\times {10}^{-5},\xi =0)$, and (c) its FFT result $\mathbf{FFT}(f,{\sigma }^2=1.0\times {10}^{-5},\xi =0)$.

Figure 6

Simulation results in two schemes under the Gaussian white noise with different SNR: (a), (d), (g), and (j) signals $I_1^{\mathrm{out}}(t;\tau )$ in the WVA scheme under noise $\mathbf{N}(t,{\sigma }^2,\xi =0)$; (b), (e), (h), and (k) signals $I_{21}^{\mathrm{out}}(t;\tau )$ and $I_{22}^{\mathrm{out}}(t;\tau )$ in the AWVA scheme; and (c), (f), (i), and (l) autocorrelative intensity $\mathrm{\Theta }$ in the AWVA scheme under noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1=0)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2=700)$ with sampling frequency $1/T=1$ MHz. The quantities $I_{21}^{\mathrm{out}}(t;\tau )$, $I_{21}^{\mathrm{out}}(t;\tau )$, and $I_{22}^{\mathrm{out}}(t;\tau )$ are in units of $I_0$ and the quantity $I_A^{AC}$ is in units of voltage.

Figure 7

Autocorrelative intensity ${\mathrm{\Theta }}_{\mathrm{NN}}(t;\tau )$ of various noises with different sampling frequencies $1/T$ of the APD: (a) autocorrelative results of noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2)$ with different seeds ${\xi }_1$ and ${\xi }_2$ at ${\mathrm{SNR}}^{*}=-18.6$ dB and $1/T=1$ MHz, (b) autocorrelative results of noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1=0)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2=700)$ at sampling frequency $1/T=1$ MHz with different ${\mathrm{SNR}}^{*}$, and (c) autocorrelative results of noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1=0)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2=700)$ at ${\mathrm{SNR}}^{*}=-18.6$ dB with different sampling frequencies $1/T$.

Figure 8

Simulation results of the autocorrelative intensity $\mathrm{\Theta }$ in the AWVA scheme noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1=0)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2=700)$ at ${\mathrm{SNR}}^{*}=-18.6$ dB with different sampling frequencies: (a) $1/T=1$ MHz, (b) $1/T=10$ MHz, and (c) $1/T=100$ MHz.

Figure 9

Examples of estimating the SNR and the sensitivity the maximum sensitivity $K_2^M$: (a) example of estimating the SNR in the WVA scheme under Gaussian white noise $\mathbf{N}(t,{\sigma }^2={10}^{-11},\xi =0)$ with different sampling frequencies, (b) dependence of the sensitivity with different ${\mathrm{SNR}}^{*}$ on the integral time $t$ under the noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1=0)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2=700)$ at sampling frequency $1/T=1$ MHz, and (c) dependence of the sensitivity with different sampling frequencies $1/T$ on the integral time $t$ under the noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1=0)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2=700)$ at ${\mathrm{SNR}}^{*}=-18.6$ dB. The gray band represents the measurements failing to effectively detect the final signal.

Figure 10

Normalized sensitivity in the WVA scheme and in the AWVA scheme under Gaussian noises with different SNRs and sampling frequencies: (a) normalized sensitivity in the WVA scheme under noises $\mathbf{N}(t,{\sigma }^2,\xi )$ with different SNRs, (b) normalized sensitivity in the AWVA scheme under noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2)$ at sampling frequency $1/T=1$ MHz with different SNR, and (c) normalized sensitivity in the AWVA scheme under noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2)$ at $\text{SNR}=-18.6$ dB with different sampling frequencies $1/T$. The gray band represents the measurements failing to effectively detect the final signal. The red data points with the error bar represent the final results of seven statistical averages in the AWVA scheme.

Figure 11

Shifts $\mathrm{\Delta }\left(\delta t\right)$ as well as (a)–(c) shifts $\mathrm{\Delta }\left(\mathrm{\Theta }\right)$ and (d)–(f) corresponding sensitivity with respect to the coupling strength $\tau$ under the Gaussian noise with $\text{SNR}=-13.2$ dB. Results are shown (a) and (d) under noises $\mathbf{N}(t,{\sigma }^2,\xi )$ with different seeds $\xi$ in the WVA scheme, (b) and (e) at sampling frequency $1/T=10$ MHz under noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2)$ with different seeds ${\xi }_1$ and ${\xi }_2$ in the AWVA scheme, and (c) and (f) at sampling frequency $1/T=100$ MHz under noises $\mathbf{N}(t,{\sigma }^2,{\xi }_1)$ and $\mathbf{N}(t,{\sigma }^2,{\xi }_2)$ with different seeds ${\xi }_1$ and ${\xi }_2$ in the AWVA scheme. The red data points with the error bar represent the final results of seven statistical averages in the AWVA scheme.

Figure 12

Scheme of the WVA measurement in simulink. The signal01.mat block and signal02.mat block represent the array of the signals $I_1^{\mathrm{out}}(t;\tau )$ with different $\tau$ in the matlab workplace. The Random Number 1 block represents the random number generator. The Add 1 and Add 2 blocks represent adding inputs. The Scope 1 block represents the oscilloscope. The Bus Creator block combines a set of signals into a bus.

Figure 13

Block parameters of Random Number 1.

Figure 14

Example of the display of Scope 1. The data are taken from the simulation of the measurement $\tau =0$ ns and the measurement $\tau =3$ under Gaussian noise with seed equal to 600 at SNR equal to 11.5 dB.

Figure 15

Example of Gaussian fitting. The data are taken from the simulation of (a) the measurement $\tau =0$ ns and (b) the measurement $\tau =3$ under Gaussian noise with seed equal to 600 at SNR equal to 11.5 dB.

Figure 16

Scheme of the AWVA measurement in simulink. The signal1.mat block and signal2.mat block represent the array of signals $I_{21}^{\mathrm{out}}(t;\tau )$ and $I_{22}^{\mathrm{out}}(t;\tau )$ with different $\tau$ in the matlab workplace. The Random Number 2 and Random Number 3 blocks represent the random number generator. The Add 3, Add 4, Add 4, and Add 6 blocks represent adding inputs. The Product 1 and Product 2 blocks represent multiplying inputs. The Integrator 1 and Integrator 2 blocks represent multiplying inputs. The Scope 2 and Scope 3 blocks output the value of the integral of its input signal with respect to time. The Bus Creator block combines a set of signals into a bus.

Figure 17

Example of the parameter set of (a) the Random Number 2 block and (b) the Random Number 3 for the measurement $\tau =0$ ns and the measurement $\tau =3$ ns under Gaussian noise, respectively, with (a) seed equal to 000 in the Random Number 2 block and (b) seed equal to 700 in the Random Number 3 block at SNR equal to 11.5 dB.

Figure 18

Example of the display of Scope 3. The data are taken from the simulation of the measurement $\tau =0$ ns and the measurement $\tau =3$ ns under Gaussian noise with seed equal to 000 in the Random Number 2 block and seed equal to 700 in the Random Number 3 block at SNR equal to 11.5 dB.

Figure 19

Example of the display of Scope 2. The data are taken from the simulation of the measurement $\tau =0$ ns and the measurement $\tau =3$ ns under Gaussian noise with seed equal to 000 in the Random Number 2 block and seed equal to 700 in the Random Number 3 block at SNR equal to 11.5 dB.

Figure 20

Example of the display of Scope 2. The data are taken from the simulation of the measurement $\tau =0$ ns and the measurement $\tau =3$ ns under Gaussian noise with seed equal to 000 in the Random Number 2 block and seed equal to 700 in the Random Number 3 block at SNR equal to $-13.2$ dB.