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arXiv:2105.08536

Frequency measurements beyond the Heisenberg time-energy limit with a single atom

Liam Paul McGuinness

Published May 18, 2021• 0 citations
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Abstract

The Heisenberg time-energy relation prevents determination of an atomic transition to better than the inverse of the measurement time. The relation generally applies to frequency estimation of a near-resonant field [1-3], since information on the field frequency can be used to infer the atomic transition [4, 5]. Here we demonstrate a frequency estimation technique that provides an uncertainty orders of magnitude below the Heisenberg limit with a single atom. With access to $N$ atoms, we propose a fundamental uncertainty limit improving as $\sqrt{N}$, regardless of whether entanglement is employed. We describe implementation of the quantum fourier transform to estimate an unknown frequency without using entanglement. A comparison to classical algorithms severely limits the benefit that quantum algorithms provide for frequency estimation and that entanglement provides to quantum sensing in general.

DOI
10.48550/arxiv.2105.08536

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The Heisenberg time-energy relation prevents determination of an atomic transition to better than the inverse of the measurement time. The relation generally applies to frequency estimation of a near-resonant field [1-3], since information on the field frequency can be used to infer the atomic transition [4, 5]. Here we demonstrate a frequency estimation technique that provides an uncertainty orders of magnitude below the Heisenberg limit with a single atom. With access to NNN atoms, we propose a fundamental uncertainty limit improving as N\sqrt{N}N​, regardless of whether entanglement is employed. We describe implementation of the quantum fourier transform to estimate an unknown frequency without using entanglement. A comparison to classical algorithms severely limits the benefit that quantum algorithms provide for frequency estimation and that entanglement provides to quantum sensing in general.
Resolving frequencies in a time-dependent field is classically limited by the measurement bandwidth. Using tools from quantum metrology and quantum control may overcome this limit, yet the full advantage afforded by entanglement so far remains elusive. Here we map the problem of frequency measurement to that of estimating a global dephasing quantum channel. In this way, we determine the ultimate quantum limits of frequency estimation in stochastic ac sensing. We find exact quantum Fisher information bounds for estimating frequency and frequency differences of stochastic fields. In particular, given two close signals with frequency separation ω_{r}, we find that the quantum Fisher information for the separation estimation is approximately 2/ω_{r}^{2}, i.e., inversely proportional to the separation parameter. The bounds are achievable in certain regimes by superpositions of Dicke states. GHZ states are suboptimal but improve precision over unentangled states, achieving Heisenberg scaling in the low-bandwidth limit. This Letter establishes a robust framework for stochastic ac signal sensing that can be extended to arbitrary time-dependent and stochastic fields.
Abstract According to the Heisenberg uncertainty principle, the energy or frequency uncertainty of a measurement can be at the best inversely proportional to the observation time ( TTT T ). The observation time in an experiment using a quantum mechanical probe is ultimately limited by the coherence time of the probe. Therefore the inverse proportionality of the statistical uncertainty of a frequency measurement to the observation time is also limited up to the coherence time of the probe, provided the systematic uncertainties are well below the statistical uncertainties. With a single laser-cooled barium ion as a quantum probe, we show that the uncertainty in the frequency measurement for a general time-dependent Hamiltonian scales as 1/T1.75±0.031/{T}^{1.75\pm 0.03}1/T1.75±0.03 1 ∕ T 1.75 ± 0.03 as opposed to 1/T1/T1/T 1 ∕ T , given by the Heisenberg limit for time-independent Hamiltonian. These measurements, based on controlled feedback Hamiltonian and implemented on a laser cooled single ion, allowed precise measurement of noise frequency in the kHz range. Moreover, based on the observed sensitivity of a single ion experiment presented here, we propose the use of a similar protocol with enhanced sensitivity as a tool to directly verify the existence of certain types of light mass axion-like dark matter particles where no direct measurement protocol exists.
We present an enhanced entangled-quantum-clock protocol that incorporates a quantum phase estimation algorithm to directly estimate proper-time differences as an unknown phase. By employing highly entangled multi-clock states, the achievable uncertainty scales inversely with the total number of quantum clocks, achieving the Heisenberg limit O(1/N)O(1/N)O(1/N) while remaining completely immune to dynamical phase noise. Furthermore, we discuss the feasibility of utilizing this ultra-high precision protocol for gravitational wave detection, highlighting its potential for quantum-enhanced relativistic sensing and fundamental physics tests.
In recent years, several estimation strategies have been formulated to determine the value of an unknown parameter in the most precise way, taking into account the presence of noise. These strategies typically rely on the use of quantum entanglement between the sensing probes and they have been shown to be optimal in the asymptotic limit in the number of probes, as long as one performs measurements on shorter and shorter time scales. Here, we present a different approach to frequency estimation, which exploits quantum coherence in the state of each sensing particle in the long time limit and is obtained by properly engineering the environment. By means of a commonly used master equation, we show that our strategy can overcome the precision achievable with entanglement-based strategies for a finite number of probes. We discuss a possible implementation of the scheme in a realistic setup that uses trapped ions as quantum sensors.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of NNN resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with NNN better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources NNN is strictly constrained, I show that the final measurement uncertainty is πππ times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.
We analyze an efficient frequency estimation scheme that is applied to measure the unknown frequency of an atomic state in Ramsey spectroscopy. The scheme is employing appropriate combinations of uncorrelated probe atoms and Greenberger–Horne–Zeilinger (GHZ) type correlated probe atoms to estimate its frequency. The estimation value of frequency is obtained through the Bayesian analysis of the final measurement outcomes. The proposed scheme allows us to obtain better precision than the scheme without quantum correlation and it also prevents us from ambiguity in the frequency estimation procedure with GHZ correlations only. We show that the scheme can beat the shot-noise limit and, in addition, it is found that there is a trade-off relation between the precision of the frequency estimation and the decoherence rate in the atomic states.