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arXiv:1803.01724 · npj Quantum Information

A single atom noise probe operating beyond the Heisenberg limit

Tarun Dutta, M. Mukherjee

Published Jan 8, 2020• 6 citations
arXivHTMLPDF

Abstract

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 ( $$T$$ 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 $$1/{T}^{1.75\pm 0.03}$$ 1 ∕ T 1.75 ± 0.03 as opposed to $$1/T$$ 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.

DOI
10.1038/s41534-019-0234-z

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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.
Quantum metrology for frequency measurement has been used to surpass the Heisenberg scaling of the measurement uncertainty with observation time. In a single ion experiment, we show that the uncertainty in the frequency measurement scales as 1/T21/T^21/T2 for over 100 μ\muμs time scale limited by the de-phasing time of the probe state. The proof-of-principle experiment also sets a limit on the coupling strength of axion like dark matter particle wind to interact with an electron spin. Even though the limit is a weak limit, we provide feasible experiment by which the limit can be strengthened to complement the other dark matter search experiments. The scheme is based on the proposal by Pang et al.~\cite{Pang2017} implemented on a laser cooled single ion probe which has been subjected to an external time dependent field coupling.
Abstract The Heisenberg limit is acknowledged as the ultimate precision limit for parameter estimation in quantum metrology. Recent studies have demonstrated super-Heisenberg-scaling limits, where the root mean square errors of estimation decrease faster-than-linearly with the number of probes or evolution processes. This casts doubt on what the ultimate precision limit of quantum metrology really is and, in particular, whether there is a deeper connection with Heisenberg's uncertainty principle. By linking the dynamics of the metrological process with properties in parameter space, we construct a Heisenberg's uncertainty relation between the unknown parameter and its canonical momentum. We propose a novel quantum metrological scheme that utilizes a generating process with indefinite time direction to increase the canonical momentum nonlinearly. This approach achieves a nonlinear-scaling precision limit that improves quadratically with the increase in evolution time length or the number of processes, in accordance with Heisenberg's uncertainty relation. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed amount of energy in the probes. Consequently, our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on improving the precision of practical metrological and sensing tasks in realistic quantum systems.
The Heisenberg scaling is an ultimate precision limit of parameter estimation allowed by the principles of quantum mechanics, with no counterpart in the classical realm, and has been a long-pursued goal in quantum metrology. It has been known that interactions between the probes can help reach the Heisenberg scaling without entanglement. In this paper, we show that interactions between the probes and the additional dimensions of an ancillary system may also increase the precision of parameter estimation to surpass the standard quantum limit and attain the Heisenberg scaling without entanglement, if the measurement scheme is properly designed. The quantum Fisher information exhibits periodic patterns over the evolution time, implying the existence of optimal time points for measurements that can maximize the quantum Fisher information. By implementing optimizations over the Hamiltonian, the initial states of the probes and the ancillary system, the interaction strength, and the time points for measurements, our protocol achieves the Heisenberg scaling for the parameter of the probe Hamiltonian, in terms of both evolution time and probe number. Our protocol features two aspects: (i) the Heisenberg scaling can be achieved by a product state of the probes and (ii) mere local measurement on the ancilla is sufficient, both of which reduce the quantum resources and the implementation complexity to achieve the Heisenberg scaling. The paper is concluded by the investigation of the effects of noise on this protocol.
A central feature of quantum metrology is the possibility of Heisenberg scaling, a quadratic improvement over the limits of classical statistics. This scaling, however, is notoriously fragile to noise. While for some noise types it can be restored through error correction, for other important types, such as dephasing, the Heisenberg scaling appears to be irremediably lost. Here we show that this limitation can sometimes be lifted if the experimenter has the ability to probe physical processes in a coherent superposition of alternative configurations. As a concrete example, we consider the problem of phase estimation in the presence of a random phase kick, which in normal conditions is known to prevent the Heisenberg scaling. We provide a parallel protocol that achieves Heisenberg scaling with respect to the probes' energy, as well as a sequential protocol that achieves Heisenberg scaling with respect to the total probing time. In addition, we show that Heisenberg scaling can also be achieved for frequency estimation in the presence of continuous-time dephasing noise, by combining the superposition of paths with fast control operations.
Learning the Hamiltonian governing a quantum system is a central task in quantum metrology, sensing, and device characterization. Existing Heisenberg-limited Hamiltonian learning protocols either require multi-qubit operations that are prone to noise, or single-qubit operations whose frequency or strength increases with the desired precision. These two requirements limit the applicability of Hamiltonian learning on near-term quantum platforms. We present a protocol that learns a quantum Hamiltonian with the optimal Heisenberg-limited scaling using only single-qubit control in the form of static fields with strengths that are independent of the target precision. Our protocol is robust against the state preparation and measurement (SPAM) error. By overcoming these limitations, our protocol provides new tools for device characterization and quantum sensing. We demonstrate that our method achieves the Heisenberg-limited scaling through rigorous mathematical proof and numerical experiments. We also prove an information-theoretic lower bound showing that a non-vanishing static field strength is necessary for achieving the Heisenberg limit unless one employs an extensive number of discrete control operations.
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.