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arXiv:1710.04673 · New Journal of Physics

Fundamental limits to frequency estimation: a comprehensive microscopic perspective

Jan F. Haase, Andrea Smirne, Jan Kołodyński, Rafał Demkowicz-Dobrzański, Susana F. Huelga

Published Mar 14, 2018• 74 citations
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Abstract

We consider a metrology scenario in which qubit-like probes are used to sense an external field that affects their energy splitting in a linear fashion.Following the frequency estimation approach in which one optimizes the state and sensing time of the probes to maximize the sensitivity, we provide a systematic study of the attainable precision under the impact of noise originating from independent bosonic baths.Specifically, we invoke an explicit microscopic derivation of the probe dynamics using the spin-boson model with weak coupling of arbitrary geometry.We clarify how the secular approximation leads to a phase-covariant (PC) dynamics, where the noise terms commute with the field Hamiltonian, while the inclusion of non-secular contributions breaks the PC.Moreover, unless one restricts to a particular (i.e., Ohmic) spectral density of the bath modes, the noise terms may contain relevant information about the frequency to be estimated.Thus, by considering general evolutions of a single probe, we study regimes in which these two effects have a non-negligible impact on the achievable precision.We then consider baths of Ohmic spectral density yet fully accounting for the lack of PC, in order to characterize the ultimate attainable scaling of precision when N probes are used in parallel.Crucially, we show that beyond the semigroup (Lindbladian) regime the Zeno limit imposing the 1/N 3/2 scaling of the mean squared error, recently derived assuming PC, generalises to any dynamics of the probes, unless the latter are coupled to the baths in the direction perfectly transversal to the frequency encoding-when a novel scaling of 1/N 7/4 arises.As our microscopic approach covers all classes of dissipative dynamics, from semigroup to non-Markovian ones (each of them potentially non-phase-covariant), it provides an exhaustive picture, in which all the different asymptotic scalings of precision naturally emerge.

DOI
10.1088/1367-2630/aab67f

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We consider a metrology scenario in which qubit-like probes are used to sense an external field that affects their energy splitting in a linear fashion.Following the frequency estimation approach in which one optimizes the state and sensing time of the probes to maximize the sensitivity, we provide a systematic study of the attainable precision under the impact of noise originating from independent bosonic baths.Specifically, we invoke an explicit microscopic derivation of the probe dynamics using the spin-boson model with weak coupling of arbitrary geometry.We clarify how the secular approximation leads to a phase-covariant (PC) dynamics, where the noise terms commute with the field Hamiltonian, while the inclusion of non-secular contributions breaks the PC.Moreover, unless one restricts to a particular (i.e., Ohmic) spectral density of the bath modes, the noise terms may contain relevant information about the frequency to be estimated.Thus, by considering general evolutions of a single probe, we study regimes in which these two effects have a non-negligible impact on the achievable precision.We then consider baths of Ohmic spectral density yet fully accounting for the lack of PC, in order to characterize the ultimate attainable scaling of precision when N probes are used in parallel.Crucially, we show that beyond the semigroup (Lindbladian) regime the Zeno limit imposing the 1/N 3/2 scaling of the mean squared error, recently derived assuming PC, generalises to any dynamics of the probes, unless the latter are coupled to the baths in the direction perfectly transversal to the frequency encoding-when a novel scaling of 1/N 7/4 arises.As our microscopic approach covers all classes of dissipative dynamics, from semigroup to non-Markovian ones (each of them potentially non-phase-covariant), it provides an exhaustive picture, in which all the different asymptotic scalings of precision naturally emerge.
Quantum metrology protocols allow us to surpass precision limits typical to classical statistics. However, in recent years, no-go theorems have been formulated, which state that typical forms of uncorrelated noise can constrain the quantum enhancement to a constant factor and, thus, bound the error to the standard asymptotic scaling. In particular, that is the case of time-homogeneous (Lindbladian) dephasing and, more generally, all semigroup dynamics that include phase covariant terms, which commute with the system Hamiltonian. We show that the standard scaling can be surpassed when the dynamics is no longer ruled by a semigroup and becomes time inhomogeneous. In this case, the ultimate precision is determined by the system short-time behavior, which when exhibiting the natural Zeno regime leads to a nonstandard asymptotic resolution. In particular, we demonstrate that the relevant noise feature dictating the precision is the violation of the semigroup property at short time scales, while non-Markovianity does not play any specific role.
Abstract We study the estimation precision attainable by entanglement-enhanced Ramsey interferometry in the presence of spatiotemporally correlated non-classical noise. Our analysis relies on an exact expression of the reduced density matrix of the qubit probes under general zero-mean Gaussian stationary dephasing, which is established through cumulant-expansion techniques and may be of independent interest in the context of non-Markovian open dynamics. By continuing and expanding our previous work (Beaudoin et al 2018 Phys. Rev. A 98 020102(R)), we analyze the effects of a non-collective coupling regime between the qubit probes and their environment, focusing on two limiting scenarios where the couplings may take only two or a continuum of possible values. In the paradigmatic case of spin–boson dephasing noise from a thermal environment, we find that it is in principle possible to suppress, on average, the effect of spatial correlations by randomizing the location of the probes, as long as enough configurations are sampled where noise correlations are negative. As a result, superclassical precision scaling is asymptotically restored for initial entangled states, including experimentally accessible one-axis spin-squeezed states.
The problem of estimating the frequency of a two-level atom in a noisy environment is studied. Our interest is to minimise both the energetic cost of the protocol and the statistical uncertainty of the estimate. In particular, we prepare a probe in a 'GHZ-diagonal' state by means of a sequence of qubit gates applied on an ensemble of n atoms in thermal equilibrium. Noise is introduced via a phenomenological time-non-local quantum master equation, which gives rise to a phase-covariant dissipative dynamics. After an interval of free evolution, the n -atom probe is globally measured at an interrogation time chosen to minimise the error bars of the final estimate. We model explicitly a measurement scheme which becomes optimal in a suitable parameter range, and are thus able to calculate the total energetic expenditure of the protocol. Interestingly, we observe that scaling up our multipartite entangled probes offers no precision enhancement when the total available energy is limited. This is at stark contrast with standard frequency estimation, where larger probes—more sensitive but also more 'expensive' to prepare—are always preferred. Replacing by the resource that places the most stringent limitation on each specific experimental setup, would thus help to formulate more realistic metrological prescriptions.
We consider the estimation of an unknown parameter θ via a many-body probe. The probe is initially prepared in a product state and many-body time-independent interactions enhance its θ sensitivity during the dynamics and/or in the steady state. We present bounds on the quantum Fisher information, and corresponding optimal interacting Hamiltonians, for two paradigmatic scenarios for encoding θ : (i) via unitary Hamiltonian dynamics (dynamical metrology), and (ii) in the Gibbs and diagonal ensembles (time-averaged dephased state), two ubiquitous steady states of many-body open dynamics. We then move to the specific problem of estimating the strength of a magnetic field via interacting spins and derive two-body interacting Hamiltonians that can approach the fundamental precision bounds. In this case, we additionally analyze the transient regime leading to the steady states and characterize trade-offs between equilibration times and measurement precision. Overall, our results provide a comprehensive picture of the potential of many-body control in quantum sensing.
Measuring stochastic signals ("noise metrology") constitutes a central task in quantum sensing and the characterization of open quantum systems. Here we establish ultimate precision bounds for multiparameter estimation of stochastic signals encoded through Markovian Lindblad dynamics, allowing for arbitrary quantum control and noiseless ancillae. Although Markovianity enforces standard-quantum-limit scaling with sensing time TTT, our bounds reveal Heisenberg-type scaling in the number of dissipative channels, RRR: when the stochastic signal exhibits high-rank correlations across the RRR channels and the probe is entangled, the average variance (per parameter) scales no better than Ω(1/(TR2))Ω(1/(TR^2))Ω(1/(TR2)). For collective kkk-body dissipation, R=Θ(Nk)R=Θ(N^k)R=Θ(Nk), signifying super-Heisenberg scaling with the system size NNN. We further show that, when the unknown parameters enter through the dissipative eigenrates, a Rapid Prepare-and-Measure (RPM) protocol that tracks many distinct quantum jumps in parallel attains these limits. In this regime, the estimation problem reduces to a multi-Poisson counting model, providing a conceptually clean route to optimal quantum noise metrology. We illustrate the breadth of the framework with applications to networked noise metrology, collective many-body dissipation, learning Pauli noise, and subdiffraction quantum imaging.
In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as 1/N21/N^21/N2 with the number of particles involved---substantially surpassing the 1/N1/N1/N-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic NNN limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large NNN the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.