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arXiv:1312.5707 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum metrology for relativistic quantum fields

Mehdi Ahmadi, David Edward Bruschi, Ivette Fuentes

Published Mar 20, 2014• 108 citations
arXivHTMLPDF

Abstract

In quantum metrology quantum properties such as squeezing and entanglement are exploited in the design of a new generation of clocks, sensors and other measurement devices that can outperform their classical counterparts. Applications of great technological relevance lie in the precise measurement of parameters which play a central role in relativity, such as proper accelerations, relative distances, time and gravitational field strengths. In this paper we generalize recently introduced techniques to estimate physical quantities within quantum field theory in flat and curved space-time. We consider a bosonic quantum field that undergoes a generic transformation, which encodes the parameter to be estimated. We present analytical formulas for optimal precision bounds on the estimation of small parameters in terms of Bogoliubov coefficients for single-mode and two-mode Gaussian channels.

DOI
10.1103/physrevd.89.065028

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In quantum metrology quantum properties such as squeezing and entanglement are exploited in the design of a new generation of clocks, sensors and other measurement devices that can outperform their classical counterparts. Applications of great technological relevance lie in the precise measurement of parameters which play a central role in relativity, such as proper accelerations, relative distances, time and gravitational field strengths. In this paper we generalize recently introduced techniques to estimate physical quantities within quantum field theory in flat and curved space-time. We consider a bosonic quantum field that undergoes a generic transformation, which encodes the parameter to be estimated. We present analytical formulas for optimal precision bounds on the estimation of small parameters in terms of Bogoliubov coefficients for single-mode and two-mode Gaussian channels.
Relativistic quantum metrology provides an optimal strategy for the estimation of parameters encoded in quantum fields in flat and curved spacetime. These parameters usually correspond to physical quantities of interest such as proper times, accelerations, gravitational field strengths, among other spacetime parameters. The precise estimation of these parameters can lead to novel applications in gravimeters, spacetime probes and gravitational wave detectors. Previous work in this direction only considered pure probe states. In realistic situations, however, probe states are mixed. In this paper, we provide a framework for the computation of optimal precision bounds for mixed single- and two-mode Gaussian states within quantum field theory. This enables the estimation of spacetime parameters in case the field states are initially at finite temperature.
We present a framework for relativistic quantum metrology that is useful for both Earth-based and space-based technologies. Quantum metrology has been so far successfully applied to design precision instruments such as clocks and sensors which outperform classical devices by exploiting quantum properties. There are advanced plans to implement these and other quantum technologies in space, for instance Space-QUEST and Space Optical Clock projects intend to implement quantum communications and quantum clocks at regimes where relativity starts to kick in. However, typical setups do not take into account the effects of relativity on quantum properties. To include and exploit these effects, we introduce techniques for the application of metrology to quantum field theory. Quantum field theory properly incorporates quantum theory and relativity, in particular, at regimes where space-based experiments take place. This framework allows for high precision estimation of parameters that appear in quantum field theory including proper times and accelerations. Indeed, the techniques can be applied to develop a novel generation of relativistic quantum technologies for gravimeters, clocks and sensors. As an example, we present a high precision device which in principle improves the state-of-the-art in quantum accelerometers by exploiting relativistic effects.
We consider quantum communication schemes where quantum optical signals are exchanged between a source on Earth and a satellite. The background curved spacetime affects the quantum state of the propagating photons. We employ quantum-metrology techniques to obtain optimal bounds for the precision of quantum measurements of relevant physical parameters encoded in the final state. We focus on satellites in low Earth orbits and we find that our scheme improves the precision of the measurement of the Schwarzschild radius obtained within previous studies. Therefore, our techniques can provide the theoretical tools for novel developments that can potentially outperform the state-of-the-art obtained through classical means. We also review the impact of the relativistic effects on a simple quantum key distribution protocol within satellite schemes and find that such effects can be greatly damaging if they are not properly accounted for.
n n n The precision of physical parameters is fundamentally limited by irreducible levelsof noise set by quantum mechanics. Quantum metrology is the study of reaching theselimits of noise by employing optimal schemes for parameter estimation. Techniquesin quantum metrology can assist in developing devices to measure the fundamentalinterplay between quantum mechanics and general relativity at state-of-the-artprecision. An example of this is the recent detection of gravitational waves bythe LIGO interferometer. In this thesis, we focus on using quantum metrologyfor estimating space-time parameters. We show the optimal quantum resourcesthat are needed for estimating the gravitational redshift of light propagating in theSchwarzschild space-time of Earth including the inevitable losses due to atmosphericdistortion. We also propose a quantum interferometer using higher order Kerrnon-linearities to improve the sensitivity of estimating gravitational time dilation.In principle, we would be able to downsize interferometers and probe gravity over asmall scale potentially making it practical for measuring gravitational gradients. Wethen study the interesting features of the metric around a rotating massive bodyknown as the Kerr metric and propose implementing a stationary interferometer tomeasure the effect of frame dragging. Finally, we consider loss in the visibility ofquantum interference of single photons in rotating reference frames, and analogouslyin the Kerr metric. In essence, the quantum interference of photons will be affectedby the relativistic effect of rotation. We find experimentally feasible parametersrequiring long optical fibre for long coherence lengths of photons. Our results willhopefully contribute to the efforts of building future quantum technologies that willenter a new regime where general relativistic effects can be measured.
In this thesis we focus on Gaussian quantum metrology in the phase-space formalism and its applications in quantum sensing and the estimation of space-time parameters. We derive new formulae for the optimal estimation of multiple parameters encoded into Gaussian states. We discuss the discontinuous behavior of the figure of merit - the quantum Fisher information. Using derived expressions we devise a practical method of finding optimal probe states for the estimation of Gaussian channels and we illustrate this method on several examples. We show that the temperature of a probe state affects the estimation generically and always appears in the form of four multiplicative factors. We also discuss how well squeezed thermal states perform in the estimation of space-time parameters. Finally we study how the estimation precision changes when two parties exchanging a quantum state with the encoded parameter do not share a reference frame. We show that using a quantum reference frame could counter this effect.
Optical and optomechanical systems are exploited for quantum technologies to perform highly precise measurements for fundamental physics as well as biological and engineering tasks. Theoretical studies using tools from statistics and quantum information can greatly aid studies of the sensing capabilities of quantum systems and experimental designs. These allow us to quantify the amount of information about a parameter encoded in a quantum state itself and examine how that can be extracted through measurements. In this thesis we show how mechanical squeezing and measurements beyond position can be utilised to improve the precision of wavepacket expansion measurements which can test collapse models of quantum mechanics. Particularly, that squeezing can compensate for the free-fall time which is often the most significant limiting factor in such experiments, and measurements of other quadratures can offer increased precision. We demonstrate that the use of additional optical fields to measure the displacement of a free mass in a radiation-pressure limited interferometer cannot surpass the ultimate precision of the single-mode interferometer. This work applies to the likes of laser-interferometric gravitational wave detectors. Finally, we venture into multi-parameter estimation: we calculate bounds for multi-mode Gaussian states used to estimate many phase shifts and optimise over the input states. In contrast to the—far less experimentally feasible—multi-mode generalisations of N00N states we see no significant improvement when using multi-mode Gaussian states for the task. Given their comparable performance in single-phase estimation this is a new limitation of Gaussian states appearing only at the multi-parameter level.