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arXiv:quant-ph/0701158 · Physical Review Letters

Phase Detection at the Quantum Limit with Multiphoton Mach-Zehnder Interferometry

Luca Pezzè, Augusto Smerzi, G. Khoury, Juan F. Hodelin, Dirk Bouwmeester

Published Nov 27, 2007• 135 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.99.223602

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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.
We show that the phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1/{N}_{T}$, where NT{N}_{T}NT​ is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter rrr and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.
A recently proposed phase-estimation protocol that is based on measuring the parity of a two-mode squeezed-vacuum state at the output of a Mach-Zehnder interferometer shows that Cramer-Rao bound sensitivity can be obtained [P. M. Anisimov, et al., Phys. Rev. Lett. {\bf104}, 103602 (2010)]. This sensitivity, however, is expected in the case of an infinite number of parity measurements made on an infinite number of photons. Here we consider the case of a finite number of parity measurements and a finite number of photons, implemented with photon-number-resolving detectors. We use Bayesian analysis to characterize the sensitivity of the phase estimation in this scheme. We have found that our phase estimation becomes biased near 0 or π/2\pi/2π/2 phase values. Yet there is an in-between region where the bias becomes negligible. In this region, our phase estimation scheme saturates the Cramer-Rao bound and beats the shot-noise limit.
Quantum phase measurement with multiphoton twin-Fock states has been shown to be optimal for detecting equal numbers of photons at the output ports of a Mach–Zehnder interferometer (i.e., the so-called single-fringe detection), since the phase sensitivity can saturate the quantum Cramér–Rao lower bound at certain values of phase shift. Here we report a further step to achieve a global phase estimation at the Heisenberg limit by detecting the particle-number difference (i.e., the J ^ z measurement). We show the role of experimental imperfections on the ultimate estimation precision with the six-photon twin-Fock state of light. Our results show that both the precision and the sensing region of the J ^ z measurement are better than those of the single-fringe detection, due to combined contributions of the measurement outcomes. We numerically simulate the phase estimation protocol using an asymptotically unbiased maximum likelihood estimator.
We derive a general expression of the expectation value of the parity operator in a Mach–Zehnder interferometer with a Fock state in one input port and an arbitrary state in the other input. We obtain the quantum Crámer–Rao bound via the error propagation method from parity detection for the phase shift around zero. In addition, for an arbitrary state with the same mean photon number, our results show that the same phase resolution and the same phase uncertainty can be obtained for the phase shift around zero. That is to say, when a Mach–Zehnder interferometer is powered by a Fock state into one input port, all quantum states with same mean photon number at the other input may have the same performance in the phase estimation via parity detection. As some specific applications, we consider the arbitrary state being a Fock state, a coherent state, and a squeezed-vacuum state.
Mach-Zehnder interferometry based on mixing the coherent and the squeezed vacuum states of light has Heisenberg limited capabilities for phase estimation. This is also, because the quantum Cramer-Rao bound on sensitivity of phase estimation with the above interferometric scheme reaches the Heisenberg limit when the inputs are mixed in hear equal proportions. We show that a detection strategy based on the measurement of parity of photon number in one of the output modes of the interferometer saturates the quantum Cramer-Rao bound of the interferometric scheme, and therefore- as a consequence- hits the Heisenberg limit when the inputs are mixed in equal intensities.
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.