Abstract
Two-mode interferometers lay the foundations for quantum metrology. Instead of exploring quantum entanglement in the two-mode interferometers, a single bosonic mode also promises a measurement precision beyond the shot-noise limit (SNL) by taking advantage of the infinite-dimensional Hilbert space of Fock states. Here, we demonstrate a single-mode phase estimation that approaches the Heisenberg limit (HL) unconditionally. Due to the strong dispersive nonlinearity and long coherence time of a microwave cavity, quantum states of the form left( {left| 0 rightrangle + left| N rightrangle } right)/sqrt 2 can be generated, manipulated and detected with high fidelities, leading to an experimental phase estimation precision scaling as ∼N−0.94. A 9.1 dB enhancement of the precision over the SNL at N = 12 is achieved, which is only 1.7 dB away from the HL. Our experimental architecture is hardware efficient and can be combined with quantum error correction techniques to fight against decoherence, and thus promises quantum-enhanced sensing in practical applications.
Highlights
Two-mode interferometers lay the foundations for quantum metrology
By preppaffirffi ing the superpositions of Fock states as jΨðNÞi 1⁄4 ðj0i þ jNiÞ= 2 up to N = 12, we demonstrate a phase estimation precision that scales as δ~θ $ NÀ0:94 and approaches the Heisenberg limit (HL)
According to the quantum Cramér–Rao bound[29], the estimation precision of parameter θ encoded in the state |ψ(θ)〉 = e−iθH|ψ〉 is lower bounded as δ~θ ! 2Δ1H, where δ~θ is the standard deviation of an unbiased estimator ~θ, and (ΔH)2 = 〈ψ|H2|ψ〉 − 〈ψ|H|ψ〉2 is the variance of the Hamiltonian H with the initial probe |ψ〉
Summary
Instead of exploring quantum entanglement in the two-mode interferometers, a single bosonic mode promises a measurement precision beyond the shot-noise limit (SNL) by taking advantage of the infinite-dimensional Hilbert space of Fock states. Based on a single bosonic mode, quantum metrology schemes have been proposed[21,22] by taking advantage of the infinite-dimensional Hilbert space of Fock states. Such single-mode quantum sensors hold the advantages of hardware efficiency, compactness, and robustness against non-local perturbations. By utilizing coherence and implementing phase estimation algorithms, a quantum-enhanced magnetometry was recently demonstrated with a single artificial atom[25,26,27], with a precision approaching the Heisenberg limit (HL).
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