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arXiv:quant-ph/0103080 · Journal of Optics B Quantum and Semiclassical Optics

Optimal quantum estimation of the coupling between two bosonic modes

G Mauro D'Ariano, Matteo G A Paris, Paolo Perinotti

Published Sep 26, 2001• 13 citations
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Abstract

We address the problem of the optimal quantum estimation of the coupling parameter of a bilinear interaction, such as the transmittivity of a beamsplitter or the internal phase-shift of an interferometer. The optimal measurement scheme confirms Heisenberg scaling of precision versus the total energy as an unsurpassable bound, but with a largely reduced multiplicative constant.

DOI
10.1088/1464-4266/3/5/309

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We address the problem of the optimal quantum estimation of the coupling parameter of a bilinear interaction, such as the transmittivity of a beamsplitter or the internal phase-shift of an interferometer. The optimal measurement scheme confirms Heisenberg scaling of precision versus the total energy as an unsurpassable bound, but with a largely reduced multiplicative constant.
Quantum information theory is applied to practical interferometer-based phase measurements to deduce the optimal phase measurement scheme with two optical modes. Optimal phase measurements, given ideal input states, reveal an asymptotic 1/n1/n1/n decrease in phase uncertainty $\ensuremath{\Delta}\ensuremath{\theta}$ for nnn the mean photon number of the input state. In contradistinction to previous schemes for realizing the number-phase uncertainty limit, the 1/n1/n1/n limit achieved here is independent of the interferometer phase shift; prior information about the expected phase shift is not necessary to attain this limit. These results apply more generally to su(2) and so(3) phase parameter estimation.
We discuss an adaptive interferometric approach to quantum estimation of phase shifts. In a first step we find optimized two-mode input states for a generalized Mach-Zehnder interferometer. These two states show a small phase variance at the so-called Heisenberg limit. It turns out that they can be generated by entangling a coherent state and a squeezed vacuum at a beam splitter. Moreover, the interferometric setup allows us to describe an adaptive estimation scheme for arbitrary phase shifts if we assume a fixed mean photon number as a quantum resource.
We study the best attainable measurement precision when a double-well trap with bosons inside acts as an interferometer to measure the energy difference of the atoms on the two sides of the trap. We introduce time-independent perturbation theory as the main tool in both analytical arguments and numerical computations. Nonlinearity from atom-atom interactions will not indirectly allow the interferometer to beat the Heisenberg limit, but in many regimes of the operation the Heisenberg limit scaling of measurement precision is preserved in spite of added tunneling of the atoms and atom-atom interactions, often even with the optimal prefactor.
We address high-precision measurements by active and passive interferometric schemes based on Gaussian states and operations. In particular, we look for the best states to be injected into their ports according to the quantum Cramer–Rao bound, i.e., maximizing the quantum Fisher information over all the involved parameters, given a constraint on the overall mean number of photons entering into the interferometer. We found that for passive interferometers involving only beam splitters, the optimal input leading to Heisenberg scaling is a pair of identical squeezed-coherent states with at most one-third of the total energy employed in squeezing. For active interferometers involving optical amplifiers, input coherent signals are enough to achieve Heisenberg scaling, given an optimal value of the amplification gain. For passive schemes our results clarify the role of squeezing in improving both the reference phase and the signal phase of an interferometer.
We address the issue of precisely estimating small parameters encoded in a general linear transformation of the modes of a bosonic quantum field. Such Bogoliubov transformations frequently appear in the context of quantum optics. We provide a set of instructions for computing the quantum Fisher information for arbitrary pure initial states. We show that the maximally achievable precision of estimation is inversely proportional to the squared average particle number and that such Heisenberg scaling requires nonclassical but not necessarily entangled states. Our method further allows us to quantify losses in precision arising from being able to monitor only finitely many modes, for which we identify a lower bound.