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arXiv:0904.0456 · Physical Review A

Quantum phase estimation with lossy interferometers

Rafał Demkowicz-Dobrzański, U. Dorner, Brian J. Smith, Jeff S. Lundeen, Wojciech Wasilewski, Konrad Banaszek, Ian A. Walmsley

Published Jul 24, 2009• 314 citations
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Abstract

We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e., the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.

DOI
10.1103/physreva.80.013825

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We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e., the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.
By using a systematic optimization approach, we determine quantum states of light with definite photon number leading to the best possible precision in optical two-mode interferometry. Our treatment takes into account the experimentally relevant situation of photon losses. Our results thus reveal the benchmark for precision in optical interferometry. Although this boundary is generally worse than the Heisenberg limit, we show that the obtained precision beats the standard quantum limit, thus leading to a significant improvement compared to classical interferometers. We furthermore discuss alternative states and strategies to the optimized states which are easier to generate at the cost of only slightly lower precision.
We investigate an optimal distance of two components in an entangled coherent state for quantum phase estimation in lossy interferometry. The optimal distance is obtained by an economical point, representing the quantum Fisher information that we can extract per input energy. Maximizing the formula of the quantum Fisher information over an input mean photon number, we show that, as the losses of the interferometer increase, it can be more beneficial to prepare an initially entangled coherent state that is less entangled. This represents that the optimal distance of the two-mode components decreases with more loss in the interferometry. Under the constraint of the input mean photon number, we obtain that the optimal entangled coherent state is more robust than a separable coherent state, even in a high photon loss rate. The optimal entangled coherent state preserves quantum advantage over the standard interferometric limit of the separable coherent state. We also show that the corresponding optimal measurement requires correlation measurement bases.
We analyse the phase space representation of the optimal measurement of a phase shift in an interferometer with equal photon loss in both its arms. In the local phase estimation scenario with a fixed number of photons, we identify features of the spin Wigner function that warrant sub-shot-noise precision, and discuss their sensitivity to losses. We derive the asymptotic form of an integral kernel describing the process of photon loss in the phase space in the limit of large photon numbers. The analytic form of this kernel allows one to assess the ultimate precision limit for a lossy interferometer. We also provide a general lower bound on the quantum Fisher information in terms of spin Wigner functions.
We optimize two-mode entangled number states of light in the presence of loss in order to maximize the extraction of the available phase information in an interferometer. Our approach optimizes over the entire available input Hilbert space with no constraints, other than fixed total initial photon number. We optimize to maximize the Fisher information, which is equivalent to minimizing the phase uncertainty. We find that in the limit of zero loss, the optimal state is the maximally path-entangled (so-called N00N) state, for small loss, the optimal state gradually deviates from the N00N state, and in the limit of large loss, the optimal state converges to a generalized two-mode coherent state, with a finite total number of photons. The results provide a general protocol for optimizing the performance of a quantum optical interferometer in the presence of photon loss, with applications to quantum imaging, metrology, sensing, and information processing.
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.
Interferometry with quantum light is known to provide enhanced precision for estimating a single phase. However, depending on the parameters involved, the quantum limit for the simultaneous estimation of multiple parameters may not be attainable, leading to tradeoffs in the attainable precisions. Here we study the simultaneous estimation of two parameters related to optical interferometry: phase and loss, using a fixed number of photons. We derive a tradeoff in the estimation of these two parameters which shows that, in contrast to single-parameter estimation, it is impossible to design a strategy saturating the quantum Cram\'er-Rao bound for loss and phase estimation in a single setup simultaneously. We design optimal quantum states with a fixed number of photons achieving the best possible simultaneous precisions. Our results reveal general features about concurrently estimating Hamiltonian and dissipative parameters and have implications for sophisticated sensing scenarios such as quantum imaging.