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 $$\left( {\left| 0 \right\rangle + \left| N \right\rangle } \right)/\sqrt 2$$ 0 + N ∕ 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.
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 (∣0⟩+∣N⟩)/2 0 + N ∕ 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.
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 (∣0⟩+∣N⟩)/2 0 + N ∕ 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.
The use of quantum resources can provide measurement precision beyond the shot noise limit (SNL). The task of ab initio optical phase measurement-the estimation of a completely unknown phase-has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able-even in principle-to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techniques: entanglement, multiple samplings of the phase shift, and adaptive measurement. Our experimental demonstration of the scheme, with photonic qubits and N = 3 photon-passes, achieves a precision that is within 4% of the HL, surpassing the best precision theoretically achievable with simpler techniques. This work represents a fundamental achievement of the ultimate limits of metrology, and the scheme can be extended to higher N and other physical systems.
Quantum metrology offers the potential to surpass its classical counterpart, pushing the boundaries of measurement precision toward the ultimate Heisenberg limit. This enhanced precision is normally attained by utilizing large squeezed states or multiparticle entangled quantum states, both of which are often challenging to implement and prone to decoherence in real quantum devices. In this work, we present a versatile and on-demand protocol for deterministic parameter estimation that leverages two efficient state-transfer operations on a single bosonic mode. Specifically, we demonstrate this protocol in the context of phase estimation using the superposition of coherent states in the bosonic circuit quantum electrodynamics (cQED) platform. With low average photon numbers of only up to 1.76, we achieve quantum enhanced precision approaching the Heisenberg scaling, reaching a metrological gain of 7.5 ( 6 ) dB. Importantly, we show that the gain or sensitivity range can be further enhanced on the fly by tailoring the input states, with different superposition weights, based on specific system constraints. The realization of this versatile and efficient scheme affords a promising path toward practical quantum enhanced sensing, not only for bosonic cQED hardware but also readily extensible to other continuous-variable platforms.
Abstract The use of quantum resources can provide measurement precision beyond the shot-noise limit (SNL). The task of ab initio optical phase measurement—the estimation of a completely unknown phase—has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able—even in principle—to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techniques: entanglement, multiple samplings of the phase shift, and adaptive measurement. Our experimental demonstration of the scheme uses two photonic qubits, one double passed, so that, for a successful coincidence detection, the number of photon-passes is N = 3. We achieve a precision that is within 4% of the HL. This scheme can be extended to higher N and other physical systems.
Quantum metrology promises measurement precision beyond classical limits by exploiting large-scale quantum states, yet realizing this advantage faces two fundamental challenges: the deterministic preparation of non-trivial quantum probes and the efficient extraction of metrological information in high-dimensional Hilbert spaces. Here, we introduce quantum confocal microscopy in Fock space that simultaneously resolves both challenges. Drawing a direct analogy between classical wave optics and quantum state evolution in a bosonic mode, we construct a confocal system with two Fock-space lenses. The first lens deterministically focuses a coherent state into a quantum probe with a tightly concentrated photon-number distribution, while the second lens maps the metrological information back to the vacuum state for efficient readout. Using a superconducting circuit QED platform, we prepare focused probe states with mean photon numbers up to N=500, achieving a 21.5$\pm$1.1 dB compression of the photon-number uncertainty relative to a coherent state, with a scalable quantum circuit of O(1) operational depth. We demonstrate a displacement sensitivity scaling as N−0.416, approaching the Heisenberg scaling (N−0.5), and achieve a record metrological gain of 19.06$\pm$0.13 dB beyond the standard quantum limit. This work establishes quantum confocal microscopy as a scalable and practical framework for quantum-enhanced precision measurement, readily extendable to other bosonic platforms and high-dimensional quantum many-body systems.
The use of quantum resources can provide measurement precision beyond the shot-noise limit (SNL). The task of ab initio optical phase measurement-the estimation of a completely unknown phase-has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able-even in principle-to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techniques: entanglement, multiple samplings of the phase shift, and adaptive measurement. Our experimental demonstration of the scheme uses two photonic qubits, one double passed, so that, for a successful coincidence detection, the number of photon-passes is N = 3. We achieve a precision that is within 4% of the HL. This scheme can be extended to higher N and other physical systems.