Abstract Non-orthogonal quantum states cannot be distinguished with complete certainty, making their underlying physical parameters difficult to resolve. In frequency spectroscopy of two closely spaced electromagnetic fields, this challenge has traditionally been associated with the Fourier-transform-limited linewidth, implying that the measurement time required to resolve a frequency difference Δ ω diverges as Δ ω → 0. Here, we show that this linewidth does not fundamentally limit the resolution of two closely spaced frequencies when each signal has sufficient amplitude for efficient detection. We propose and experimentally demonstrate a broadband quantum harmonic oscillator protocol for super-resolution spectroscopy in the radio-frequency and microwave domains. A Quantum Fluctuation Suppression (QFS) sequence encodes Δ ω into the oscillator state while suppressing parameter-independent shot noise. We resolve two randomly chosen electric fields near 100 MHz separated by 5 Hz, measuring 5.0(1.6) Hz with a 1ms probe time, 200 × beyond the conventional spectral-resolution limit. The method reduces acquisition time by more than five orders of magnitude and, through the motional Raman framework, is applicable across a broad frequency range.
Abstract Non-orthogonal quantum states cannot be distinguished with complete certainty, making their underlying physical parameters difficult to resolve. In frequency spectroscopy of two closely spaced electromagnetic fields, this challenge has traditionally been associated with the Fourier-transform-limited linewidth, implying that the measurement time required to resolve a frequency difference Δ ω diverges as Δ ω → 0. Here, we show that this linewidth does not fundamentally limit the resolution of two closely spaced frequencies when each signal has sufficient amplitude for efficient detection. We propose and experimentally demonstrate a broadband quantum harmonic oscillator protocol for super-resolution spectroscopy in the radio-frequency and microwave domains. A Quantum Fluctuation Suppression (QFS) sequence encodes Δ ω into the oscillator state while suppressing parameter-independent shot noise. We resolve two randomly chosen electric fields near 100 MHz separated by 5 Hz, measuring 5.0(1.6) Hz with a 1ms probe time, 200 × beyond the conventional spectral-resolution limit. The method reduces acquisition time by more than five orders of magnitude and, through the motional Raman framework, is applicable across a broad frequency range.
Due to quantum fluctuations, non-orthogonal quantum states cannot be distinguished with complete certainty, making their underlying physical parameters difficult to resolve. Traditionally, it has been believed that the linewidth of a system behaves like these quantum fluctuations to set the ultimate limit on frequency resolution as two oscillating electromagnetic fields are applied. Consequently, the measurement time required to resolve a frequency difference Δω was assumed to diverge as Δω→0. Here, we show that linewidth does not play a defining role in resolving two closely spaced frequencies. Instead, the ultimate limit is set by parameter-independent quantum fluctuations, such as shot noise in our case. We propose and experimentally demonstrate the first general broadband protocol for super-resolution spectroscopy. Specifically, our protocol uses a Walsh-modulated dynamical decoupling (WMDD) sequence to encode Δω between two unknown tones into a quantum state. This leverages phase information to suppress parameter-independent shot noise, thereby enhancing the signal-to-noise ratio and enabling super-resolution spectroscopy. With this approach, we resolve two randomly chosen oscillating electric fields of order 100 MHz separated by 5 Hz, with a measured frequency difference of 5.0(1.6) Hz using a measurement time per run of just 1 ms, representing an improvement of 200 beyond the traditional resolution limit. As such, our technique accelerates data acquisition by more than 105 magnitude compared to conventional methods. Crucially, as our protocol is rooted in the motional Raman (quantum vector signal analyzer) framework, it is effective across an arbitrary frequency range and thus promises to enhance broadband sensing of electromagnetic fields and improve spectral efficiency of next-generation communication systems.
A key advantage of quantum metrology is the ability to surpass the standard quantum limit (SQL) for measurement precision through the use of nonclassical states. However, there is typically little to no improvement in precision with the use of nonclassical states for measurements whose duration exceeds the decoherence time of the underlying quantum states. Measurements aimed at the ultimate possible precision are thus performed almost exclusively with classical states and, therefore, are constrained by the SQL. Here, we demonstrate that by using the phenomenon of subharmonic excitation, in combination with a recently demonstrated technique of Raman excitation of a harmonic oscillator, the frequency of an electric field can be measured at a resolution below the SQL of the corresponding linear generator. With this method we measure a radio-frequency electrical signal with a fractional frequency uncertainty of 0.56 Hz/80 MHz=7×10^{-9}, which to our knowledge is the most precise frequency measurement of a radio-frequency electrical signal using a quantum harmonic oscillator. Because the input states can be classical, the coherence time is not degraded by the enhanced decoherence typically associated with nonclassical states, thereby improving the ultimate achievable precision. While we demonstrate this technique using motional Raman subharmonic excitation of a single ^{40}Ca^{+} ion through engineered Floquet states, this technique is expected to be extendable to other platforms, such as NV centers, solid-state qubits, and neutral atoms, where it can provide metrological gain for sensing across the radio frequency, microwave, and optical domains.
The ability to determine precisely the separation of two frequencies is fundamental to spectroscopy, yet the resolution limit poses a critical challenge: distinguishing two incoherent signals becomes impossible when their frequencies are sufficiently close. Here, we demonstrate a simple and powerful approach, dubbed superresolution quantum sensing, which experimentally resolves two nearly identical incoherent signals using a solid-state spin quantum sensor. By carefully choosing interrogation times that satisfy the superresolution condition, we eliminate quantum projection noise, overcoming the vanishing distinguishability of signals with near-identical frequencies. This leads to improved resolution, which scales as t^{-2} in comparison to the standard t^{-1} scaling. Together with a greatly reduced classical readout noise assisted by a nuclear spin, we are able to achieve sub-kHz resolution with a signal detection time of 80 μs. Our results highlight the potential of quantum sensing to overcome conventional frequency resolution limitations, with broad implications for precision measurements.
Existing super-resolution methods of optical imaging hold a solid place as an application in natural sciences, but many new developments allow for beating the diffraction limit in a more subtle way. One of the recently explored strategies to fully exploit information already present in the field is to perform a quantum-inspired tailored measurements. Here we exploit the full spectral information of the optical field in order to beat the Rayleigh limit in spectroscopy. We employ an optical quantum memory with spin-wave storage and an embedded processing capability to implement a time-inversion interferometer for input light, projecting the optical field in the symmetric-antisymmetric mode basis. Our tailored measurement achieves a resolution of 15 kHz and requires 20 times less photons than a corresponding Rayleigh-limited conventional method. We demonstrate the advantage of our technique over both conventional spectroscopy and heterodyne measurements, showing potential for application in distinguishing ultra-narrowband emitters, optical communication channels, or signals transduced from lower-frequency domains.
Abstract The field of quantum sensing explores the use of quantum phenomena to measure a broad range of physical quantities, of both static and time-dependent types. While for static signals the main figure of merit is sensitivity, for time dependent signals it is spectral resolution, i.e. the ability to resolve two different frequencies. Here we study this problem, and develop new superresolution methods that rely on quantum features. We first formulate a general criterion for superresolution in quantum problems. Inspired by this, we show that quantum detectors can resolve two frequencies from incoherent segments of the signal, irrespective of their separation, in contrast to what is known about classical detection schemes. The main idea behind these methods is to overcome the vanishing distinguishability in resolution problems by nullifying the projection noise.
Existing super-resolution methods of optical imaging hold a solid place as an application in multiple sciences, but many new developments allow beating of diffraction limit in better and more subtle ways. An avenue was opened by suggesting to fully exploit information already present in the field by performing quantum-inspired tailored measurements. Here we exploit the full spectral information of the optical field in order to beat the Rayleigh limit in spectroscopy. We employ optical quantum memory with spin-wave storage and an embedded processing capability to implement a timeinversion interferometer for input light, projecting the optical field in the symmetricanti-symmetric mode basis. Our tailored measurement achieves a resolution of 15 kHz and requires 20.1 times less photons than a corresponding Rayleigh-limited conventional method. We demonstrate the advantage of our technique over both conventional spectroscopy and heterodyne measurements, showing potential for application in distinguishing ultra-narrowband emitters, optical communication channels, or signals transduced from lower-frequency domains.