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arXiv:quant-ph/0312114 · Physical Review A

Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry

Raphael C. Pooser, Olivier Pfister

Published Apr 22, 2004• 15 citations
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Abstract

We investigate the scaling of the phase sensitivity of a nonideal Heisenberg-limited interferometer with the particle number $N$, in the case of the Bayesian detection procedure proposed by Holland and Burnett [Phys. Rev. Lett. 71, 1355 (1993)] for twin boson input modes. Using Monte Carlo simulations for up to $10 000$ bosons, we show that the phase error of a nonideal interferometer scales with the Heisenberg limit if the losses are of the order of ${N}^{\ensuremath{-}1}$. Greater losses degrade the scaling which is then in ${N}^{\ensuremath{-}1∕2}$, such as the beam-splitter shot-noise limit, yet the sensitivity stays sub-shot-noise as long as photon correlations are present. These results give the actual limits of Bayesian detection for twin-mode interferometry and prove that it is an experimentally feasible scheme, contrary to what is implied by the coincidence-detection analysis of Kim et al. [Phys. Rev. A 60, 708 (1999)].

DOI
10.1103/physreva.69.043616

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We investigate the scaling of the phase sensitivity of a nonideal Heisenberg-limited interferometer with the particle number NNN, in the case of the Bayesian detection procedure proposed by Holland and Burnett [Phys. Rev. Lett. 71, 1355 (1993)] for twin boson input modes. Using Monte Carlo simulations for up to 1000010 00010000 bosons, we show that the phase error of a nonideal interferometer scales with the Heisenberg limit if the losses are of the order of N\ensuremath−1{N}^{\ensuremath{-}1}N\ensuremath−1. Greater losses degrade the scaling which is then in N\ensuremath−1∕2{N}^{\ensuremath{-}1∕2}N\ensuremath−1∕2, such as the beam-splitter shot-noise limit, yet the sensitivity stays sub-shot-noise as long as photon correlations are present. These results give the actual limits of Bayesian detection for twin-mode interferometry and prove that it is an experimentally feasible scheme, contrary to what is implied by the coincidence-detection analysis of Kim et al. [Phys. Rev. A 60, 708 (1999)].
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.
The feasibility of a Heisenberg-limited phase measurement using a Mach-Zehnder interferometer fed with twin photon correlated light is investigated theoretically. To take advantage of the Heisenberg limit, 1/N1/N1/N, for the phase sensitivity, one wants the number of correlated photons, NNN, to be high. This favors the use of parametric oscillation rather than the weaker but better correlated source given by parametric down-conversion. In real systems, decorrelation arising from photon absorption, mode mismatch, and nonideal detectors must be considered. In this paper we address the problem of detection when correlated photons are used as the input. We study the influence of photon statistics and of imperfect quantum correlation of the input light, and show that it is still possible to break the classical 1/N1/\sqrt{N}1/N​ phase sensitivity limit in nonideal experimental conditions. All the results are valid in the general case of quantum correlated bosons.
Bose-Einstein condensates (BEC), with their coherence properties, have attracted wide interest for their possible application to ultraprecise interferometry and ultraweak force sensors. Since condensates, unlike photons, are interacting, they may permit the realization of specific quantum states needed as input of an interferometer to approach the Heisenberg limit, the supposed lower bound to precision phase measurements. To this end, we study the sensitivity to external weak perturbations of a representative matter-wave Mach-Zehnder interferometer whose input are two Bose-Einstein condensates created by splitting a single condensate in two parts. The interferometric phase sensitivity depends on the specific quantum state created with the two condensates, and, therefore, on the time scale of the splitting process. We identify three different regimes, characterized by a phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ scaling with the total number of condensate particles NNN as (i) the standard quantum limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{1∕2}$, (ii) the sub shot-noise $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{3∕4}$, and the (iii) the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕N$. However, in a realistic dynamical BEC splitting, the 1∕N1∕N1∕N limit requires a long adiabaticity time scale, which is hardly reachable experimentally. On the other hand, the sub-shot-noise sensitivity $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{3∕4}$ can be reached in a realistic experimental setting. We also show that the 1∕N3∕41∕{N}^{3∕4}1∕N3∕4 scaling is a rigorous upper bound in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, while keeping constant all different parameters of the bosonic Mach-Zehnder interferometer.
Sensitivity bounds for a generic interferometric phase estimation problem are the shot noise and the Heisenberg limits. The shot noise is the highest sensitivity that can be reached with separable states, while the Heisenberg limit is the ultimate bound in sensitivity which can be saturated with entangled states. The scaling of these bounds with the number of particles NNN entering the interferometer depends on the specific Hamiltonian governing the same interferometer. In typical cases, the Hamiltonian is linear, and the shot-noise and Heisenberg limits scale with ${N}^{\ensuremath{-}1/2}$ and with ${N}^{\ensuremath{-}1}$, respectively. With interferometers described by generic, nonlinear Hamiltonian, the scalings with the number of particles can be rather different. Here we study the shot-noise and Heisenberg limits of Ising-like Hamiltonians in the presence of longitudinal and transverse fields, both in the nearest-neighbor and in the fully connected spin interaction cases. We provide the explicit forms of the states saturating the shot-noise and Heisenberg limits. These results can be relevant not only for precision measurement purposes but also to characterize quantum phase transitions and, more generally, the witnessing of multipartite entanglement.
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.
We propose a Heisenberg-limited quantum interferometer whose input is twin optical beams from which one or more photons have been indistinguishably subtracted. Such an interferometer can yield Heisenberg-limited performance while at the same time giving a direct fringe reading, unlike for the twin-beam input of the Holland-Burnett interferometer. We propose a feasible experimental realization using a photon-number correlated source, such as nondegenerate parametric down-conversion, and perform realistic analyses of performance in the presence of loss and detector inefficiency.