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Journal of the Optical Society of Korea

The Phase Sensitivity of the Coincidence Detection in one Output Port of a Mach-Zehnder Interferometer

Harim Shin, Henoh Kim, Goo-Dong Park, Taesoo Kim

Published Dec 1, 2005• 1 citation
Publisher

Abstract

The phase sensitivity of the coincidence detection in one output port of a Mach-Zehnder interferometer is analysed for twin Fock state inputs. Firstly, the ideal detectors with quantum efficiency of unity are assumed for the detection of the output photons. The sensitivity is found out to be independent of the photon number of input light, which means that the Heisenberg limit cannot be reached in the coincidence detection even with ideal detectors. Secondly, the practical detectors with quantum efficiencies less than unity are discussed.

DOI
10.3807/josk.2005.9.4.169

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The phase sensitivity of the coincidence detection in one output port of a Mach-Zehnder interferometer is analysed for twin Fock state inputs. Firstly, the ideal detectors with quantum efficiency of unity are assumed for the detection of the output photons. The sensitivity is found out to be independent of the photon number of input light, which means that the Heisenberg limit cannot be reached in the coincidence detection even with ideal detectors. Secondly, the practical detectors with quantum efficiencies less than unity are discussed.
The effect of quantum efficiencies of detectors on the phase sensitivity of a Mach-Zehnder is analyzed for the Fock state input. The phase uncertainty of two different interference experiments, the second order and the fourth order, are compared with two practical detectors with less than unit quantum efficiencies. It is found out that in the second-order interference experiment we cannot beat the classical limit in the phase sensitivity, even with ideal detectors. We show that almost ideal detectors are required to get the Heisenberg limit in the phase sensitivity of the fourth-order interference for two Fock state light inputs. There exists an optimum angle given by the efficiency of the detectors.
Holland and Burnett [Phys. Rev. Lett. 71, 1355 (1993)] have argued that twin Fock states of equal photon number NNN injected at both input ports of a Mach-Zehnder interferometer lead to phase measurements with accuracies approaching the Heisenberg limit $\ensuremath{\Delta}{\ensuremath{\varphi}}_{\mathrm{HL}}=1/(2N).$ However, the method of phase detection suggested by those authors, obtaining the difference of the photocurrents at the output ports of the interferometer, is not sensitive to the phase difference between the two interferometer paths; in fact, the photocurrent vanishes. In this paper we show that the use of parity measurements on just one of the output modes not only is sensitive to the phase difference but that the sensitivity approaches the Heisenberg limit for large N.N.N.
We have studied the sensitivity of four different phase shift measurement schemes with a Mach-Zehnder interferometer. The input light is considered to be in a coherent state and the detectors are assumed to be ideal with the quantum efficiency of unity. It is shown by direct calculation of the operators corresponding to the measurement schemes that the uncertainty of the phase-shift measurement is limited to the classical one 1m\frac{1}{\sqrt{m}}m​1​ (m is the average number of the photons in the input state) regardless of the phase-shift measurement schemes.
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.
A two-photon interference experiment is presented in which an entangled pair of photons generated from a parametric down-conversion was incident on two input ports of a Mach-Zehnder interferometer. The experiment was carried out using two photon coincidence detection with two detectors at the two output ports of the interferometer. The measured coincidence counts exhibit an interference effect with visibility of 0.75 at simultaneous inputs and 0.38 at inputs with different arrival times according to the degree of photon number entanglement.
We quantitatively investigate phase measurement in a Mach-Zehnder interferometer (MZI), which is injected with a weak coherent and a squeezed vacuum generated from a spontaneous parametric down-conversion. The measured three-photon coincidence in this system is quantified as a function of a ratio between the amplitude of the coherent state and the squeezed parameter of squeezed vacuum when the photon number is detected at one output of MZI. It shows that the phase sensitivity reaches the Heisenberg limit when an optimal ratio is chosen. This may introduce one approach to quantify phase sensitivity in precision phase measurement.