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arXiv:1706.01700 · Physical Review A

Quantum interferometry in multimode systems

Jan Chwedeńczuk

Published Sep 12, 2017• 2 citations
arXivHTMLPDF

Abstract

We consider the situation when the signal propagating through each arm of an interferometer has a complicated multimode structure. We demonstrate that the shot-noise level for such a setup is the same as for the common two-mode case for as long as the interferometric transformation treats each arm as a whole and does not penetrate its inner structure. We find the relation between the particle entanglement and the possibility to surpass the shot-noise limit of the phase estimation. Our results are general---they apply to pure and mixed states of identical and distinguishable particles (or combinations of both) for a fixed and fluctuating number of particles. An important result is that the method for detecting the entanglement often used in a two-mode system---based on the measurement of the visibility of fringes and the population imbalance fluctuations---can give misleading results when applied to the multimode case. Our main result is that the additional modes do not boost the sensitivity for as long as they are not interacting. Therefore, the improvement above the shot-noise level established for the two-mode systems can safely be treated as an entanglement criterion also in the complex multimode case.

DOI
10.1103/physreva.96.032320

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We consider the situation when the signal propagating through each arm of an interferometer has a complicated multimode structure. We demonstrate that the shot-noise level for such a setup is the same as for the common two-mode case for as long as the interferometric transformation treats each arm as a whole and does not penetrate its inner structure. We find the relation between the particle entanglement and the possibility to surpass the shot-noise limit of the phase estimation. Our results are general---they apply to pure and mixed states of identical and distinguishable particles (or combinations of both) for a fixed and fluctuating number of particles. An important result is that the method for detecting the entanglement often used in a two-mode system---based on the measurement of the visibility of fringes and the population imbalance fluctuations---can give misleading results when applied to the multimode case. Our main result is that the additional modes do not boost the sensitivity for as long as they are not interacting. Therefore, the improvement above the shot-noise level established for the two-mode systems can safely be treated as an entanglement criterion also in the complex multimode case.
We provide general bounds of phase estimation sensitivity in linear two-mode interferometers. We consider probe states with a fluctuating total number of particles. With incoherent mixtures of states with different total number of particles, particle entanglement is necessary but not sufficient to overcome the shot noise limit. The highest possible phase estimation sensitivity, the Heisenberg limit, is established under general unbiased properties of the estimator. If coherences can be created, manipulated, and detected, the Heisenberg limit can only be set in the central limit, with a sufficiently large repetition of interferometric measurements.
We investigate the connection between the shot-noise limit in linear interferometers and particle entanglement. In particular, we ask whether sub-shot-noise sensitivity can be reached with all pure entangled input states of NNN particles if they can be optimized with local operations. Results on the optimal local transformations allow us to show that for N=2N=2N=2 all pure entangled states can be made useful for sub-shot-noise interferometry while for N>2N>2N>2 this is not the case. We completely classify the useful entangled states available in a bosonic two-mode interferometer. We apply our results to several states, in particular to multiparticle singlet states and to cluster states. The latter turn out to be practically useless for sub-shot-noise interferometry. Our results are based on the Cramer-Rao bound and the Fisher information.
We identify precision limits for the simultaneous estimation of multiple parameters in multimode interferometers. Quantum strategies to enhance the multiparameter sensitivity are based on entanglement among particles, modes, or combining both. The maximum attainable sensitivity of particle-separable states defines the multiparameter shot-noise limit, which can be surpassed without mode entanglement. Further enhancements up to the multiparameter Heisenberg limit are possible by adding mode entanglement. Optimal strategies that saturate the precision bounds are provided.
A major challenge of the phase estimation problem is the engineering of high-intensity entangled probe states. The goal is to significantly enhance above the shot-noise limit the sensitivity of two-mode interferometers. Here we show that this can be achieved by squeezing in input, and then measuring in output, the population fluctuations of a single mode. The second input mode can be left as an arbitrary nonvacuum (e.g., a bright coherent) state. This two-mode state belongs to a novel class of particle-entangled states which are not spin squeezed. Already a 2.4 db gain above shot noise can be obtained when just a single-particle Fock state is injected into the empty input port of a classical interferometer configuration. Higher gains, up to the Heisenberg limit, can be reached with squeezed states of a larger number of particles. We finally study the robustness of this protocol with respect to detection noise.
We critically analyze the commonly maintained statement that entanglement is necessary to beat the shot-noise limit or standard quantum limit (SQL) in the sensitivity with which certain parameters can be measured in interferometric experiments. We consider in detail three different physical realizations of a beam splitter and a Mach-Zehnder interferometer using photons, massive bosons, and distinguishable spins or qubits. We study several input states and the corresponding definitions of entanglement. We show that mode entanglement is not required for photons or massive bosons for beating the SQL. In particular, with a fluctuating number of two-mode bosons, the shot-noise limit can be beaten by nonentangled bosonic states with all bosons in one mode. As a consequence, the trivially present entanglement due to symmetrization that appears when writing bosonic states in terms of individual particles is a useful resource for precision measurements.
We study in detail a system of two interferometers aimed at detecting extremely faint phase fluctuations. This system can represent a breakthrough for detecting a faint correlated signal that would remain otherwise undetectable even using the most sensitive individual interferometric devices, as in the case of so-called holographic noise. The signature of this kind of noise emerges as a correlation between the output signals of the interferometers. On the other hand, when holographic noise is absent one expects uncorrelated signals since the time-averaged fluctuations due to shot noise and other independent contributions vanish (though limiting the overall sensitivity). We show how injecting quantum light in the free ports of the interferometers can reduce the photon noise of the system beyond the shot noise, enhancing the resolution in the phase-correlation estimation. We analyze the use of both the two-mode squeezed vacuum and two independent squeezed states. Our results confirm the benefit of using squeezed beams together with strong coherent beams in interferometry. We also investigate the possible use of the two-mode squeezed vacuum, discovering interesting and unexplored areas of application of bipartite entanglement, in particular the possibility of reaching in principle a surprising uncertainty reduction.