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arXiv:1903.11282 · Physical Review A

Quantum enhancement of sensitivity achieved by photon-number-resolving detection in the dark port of a two-path interferometer operating at high intensities

Jun-Yi Wu, Toda Norifumi, Holger Friedrich Hofmann

Published Jul 9, 2019• 18 citations
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Abstract

It is shown that the maximal phase sensitivity of a two-path interferometer with high-intensity coherent light and squeezed vacuum in the input ports can be achieved by photon-number-resolving detection of only a small number of photons in a dark output port. It is then possible to achieve the quantum Cram\'er-Rao bound of the two-path interferometer using only the field displacement dependence of the photon number statistics in the single mode output of the dark port represented by a field-displaced squeezed vacuum state. We find that, at small field displacements, it is not sufficient to use the average photon number as the estimator, indicating that an optimal phase estimation depends critically on measurements of the precise photon number. We therefore analyze the effect of detection efficiency on the Fisher information and show that there is a transition from low robustness against photon losses associated with quantum interference effects at low field displacements to high robustness against photon losses at high field displacements. The transition between the two regimes occurs at field shifts proportional to the third power of the squeezing factor, indicating that squeezing greatly enhances the phase interval in which quantum effects are relevant in optimal phase estimations using photon resolving detectors. The case under study could thus be understood as a ``missing link'' between genuine multiphoton interference and the straightforward suppression of noise usually associated with squeezed light.

DOI
10.1103/physreva.100.013814

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It is shown that the maximal phase sensitivity of a two-path interferometer with high-intensity coherent light and squeezed vacuum in the input ports can be achieved by photon-number-resolving detection of only a small number of photons in a dark output port. It is then possible to achieve the quantum Cram'er-Rao bound of the two-path interferometer using only the field displacement dependence of the photon number statistics in the single mode output of the dark port represented by a field-displaced squeezed vacuum state. We find that, at small field displacements, it is not sufficient to use the average photon number as the estimator, indicating that an optimal phase estimation depends critically on measurements of the precise photon number. We therefore analyze the effect of detection efficiency on the Fisher information and show that there is a transition from low robustness against photon losses associated with quantum interference effects at low field displacements to high robustness against photon losses at high field displacements. The transition between the two regimes occurs at field shifts proportional to the third power of the squeezing factor, indicating that squeezing greatly enhances the phase interval in which quantum effects are relevant in optimal phase estimations using photon resolving detectors. The case under study could thus be understood as a ``missing link'' between genuine multiphoton interference and the straightforward suppression of noise usually associated with squeezed light.
Two-path interferometry with coherent states and a squeezed vacuum can achieve phase sensitivities close to the Heisenberg limit when the average photon number of the squeezed vacuum is close to the average photon number of the coherent light. Here, we investigate the phase sensitivity of such states in the presence of photon losses. It is shown that the Cramer-Rao bound of phase sensitivity can be achieved experimentally by using a weak local oscillator and photon counting in the output. The phase sensitivity is then given by the Fisher information FFF of the state. In the limit of high squeezing, the ratio (F−−N)/N2(F--N)/{N}^{2}(F−−N)/N2 of Fisher information above shot noise to the square of the average photon number NNN depends only on the average number of photons lost, nloss{n}_{\mathrm{loss}}nloss​, and the fraction of squeezed vacuum photons $\ensuremath{\mu}$. For $\ensuremath{\mu}=1/2$, the effect of losses is given by (F−−N)/N2=1/(1+2nloss)(F--N)/{N}^{2}=1/(1+2{n}_{\mathrm{loss}})(F−−N)/N2=1/(1+2nloss​). The possibility of increasing the robustness against losses by lowering the squeezing fraction $\ensuremath{\mu}$ is considered, and an optimized result is derived. However, the improvements are rather small, with a maximal improvement by a factor of 2 at high losses.
In the field of quantum precision measurement, enhancing phase sensitivity is crucial for various applications, including quantum metrology and quantum sensing technologies. We theoretically investigate the improvement in phase sensitivity and quantum Fisher information achieved through multiphoton subtraction operations at the output port of an SU(1,1) interferometer under conditions of photon loss. We use vacuum and coherent states as the inputs and detect the outputs by intensity detection. The results indicate that internal photon losses within the SU(1,1) interferometer have a more significant impact on the phase sensitivity compared to external photon losses. Moreover, increasing the number of photon subtractions m effectively enhances both the phase sensitivity and the quantum Fisher information. Notably, even under conditions of severe photon loss, the multiphoton subtraction operations can enable the phase sensitivity to surpass the standard quantum limit, approaching both the Heisenberg limit and the quantum Cramér-Rao bound. This study provides a new theoretical basis for enhancing the phase sensitivity in the SU(1,1) interferometer.
We consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating the quantum Fisher information of the mixed quantum state in one of the interferometer's outputs about the difference phase. We show that, in the vicinity of the black fringe, this information is asymptotically close to the quantum Fisher information about this phase for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and allows for the Heisenberg scaling of precision. We also show that the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.
In the field of quantum precision measurement, enhancing phase sensitivity is crucial for various applications, including quantum metrology and quantum sensing technologies. We theoretically investigate the improvement in phase sensitivity and quantum Fisher information achieved through multiphoton subtraction operations at the output port of an SU(1,1) interferometer under conditions of photon loss. We use vacuum and coherent states as the inputs and detect the outputs by intensity detection. The results indicate that internal photon losses within the SU(1,1) interferometer have a more significant impact on the phase sensitivity compared to external photon losses. Moreover, increasing the number of photon subtractions m effectively enhances both the phase sensitivity and the quantum Fisher information. Notably, even under conditions of severe photon loss, the multiphoton subtraction operations can enable the phase sensitivity to surpass the standard quantum limit, approaching both the Heisenberg limit and the quantum Cramér-Rao bound. This study provides a new theoretical basis for enhancing the phase sensitivity in the SU(1,1) interferometer.
We theoretically study the phase estimation based on a Mach-Zehnder interferometer (MZI) with a two-mode squeezed coherent state. By maximizing the quantum Fisher information, we find that the quantum Cramér-Rao bounds (QCRB) can reach sub-Heisenberg limit under the phase-matched condition. The optimal phase sensitivity can reach the sub-shot noise limit (SNL) and approach the QCRB by employing the intensity difference detection. Meanwhile, compared with the MZI fed with a coherent plus a single-mode squeezed vacuum state, this scheme can have better performance by adjusting the squeezing parameter and the mean photon number. With the same parameter, our scheme shows more sensitive phase measurement than the SU(1,1) interferometer with a coherent plus a vacuum state. We also show that the phase sensitivity of our proposal can still reach the SNL when the loss of the photon is 36%. This scheme can provide potential applications in optical sensors.
We study a Mach-Zehnder interferometer fed by a coherent state in one input port and vacuum in the other. We explore a Bayesian phase estimation strategy to demonstrate that it is possible to achieve the standard quantum limit independently from the true value of the phase shift and specific assumptions on the noise of the interferometer. We have been able to implement the protocol by using parallel operation of two photon-number-resolving detectors and multiphoton coincidence logic electronics at the output ports of a weakly illuminated Mach-Zehnder interferometer. This protocol is unbiased, saturates the Cramer-Rao phase uncertainty bound, and, therefore, is an optimal phase estimation strategy.