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.
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.
Quantum sensing uses nonclassical states to improve measurement sensitivity, but the same states that provide metrological gain also decohere more rapidly. This limits the usable interrogation time and, in practice, often precludes improvement in ultimate sensitivity - realized useful quantum advantage has consequently remained rare. Here, we restore persistent quantum advantage by embedding Fock-state enhancement within a quantum heterodyne (Qdyne) protocol, decoupling sensitivity from the decoherence-limited interrogation time τ. Measurements are acquired at short τ where the quantum-enhanced gain is optimal, while precision accumulates with the total measurement time. Demonstrated on the motional mode of a trapped 40Ca+ ion, we observe quantum-enhanced precision that persists to measurement times seven orders of magnitude beyond the dephasing limit, scaling as T^{-3/2} with no indication of saturation. Using the n=3 Fock state, we reach a frequency precision of 0.5uHz relative to an 86MHz carrier, achieving a fractional precision ~6x10^{-15}. This represents a quantum-enhanced gain of 7.1(10) dB over the n=0 state, in agreement with Fisher information predictions. This is the first demonstration of Qdyne beyond solid-state spin-defects. Further, by using the quantum harmonic oscillator to perform frequency mixing, we extend operation beyond 1GHz, two orders of magnitude above the ceiling of pulsed dynamical-decoupling implementations. These results recover quantum advantage at the long timescales required to improve ultimate sensitivity, with direct implications for nanoscale NMR and quantum logic spectroscopy.
Trapped atomic ions are among the most capable platforms for quantum computation, yet leading architectures using laser-based gates suffer photon-scattering errors, demanding cooling requirements, and optical delivery challenges that grow more burdensome at scale. Electric-field-gradient gates (EGGS) bypass these obstacles by driving qubit transitions with radio-frequency fields applied directly to easily scalable trap electrodes - the transitions addressed render photon scattering negligible and tolerate thermal motion. The low-lying, electric-dipole transitions necessary for EGGS are found only in molecules, and this dissertation details the pursuit of EGGS-based quantum-logic spectroscopy of the lambda-doublet qubit in H35Cl+ with a co-trapped 40Ca+ ion in a cryogenic system built to that end. This pursuit required wideband calibration of the EGGS field gradients across the 10 MHz to 1000 MHz band of the molecular transitions - for which no suitably wideband interaction existed. We resolve this with our discovery of motional Raman transitions, which downconvert an arbitrary-frequency drive into a near-resonant motional displacement, and apply them toward quantum vector signal analysis, a protocol for complete waveform sensing of an electric field, which we demonstrate from 100 kHz to 1 GHz, a range 800× wider than prior work. The protocol is compatible with quantum amplification, which we use to reach an amplitude sensitivity 3.4(20) dB below the standard quantum limit. We then extend this family of interactions toward sensing beyond conventional limits. We use subharmonic motional Raman transitions to achieve nonlinearity-enhanced frequency sensitivity, surpassing the Fourier-transform limit with classical states alone to reach a frequency uncertainty of 0.56(32) Hz at 80 MHz - to our knowledge, the most precise frequency measurement of a radio-frequency signal on a quantum harmonic oscillator. Separately, we demonstrate broadband frequency super-resolution: a phase-reversal sequence encodes the separation between two conventionally unresolvable tones into a displacement, resolving a 5 Hz splitting near 80 MHz to 1.6 Hz, 200x below the spectral limit. Finally, combining Fock-state enhancement with synchronized readout recovers a quantum advantage that persists to long timescales, improving the ultimate frequency precision to 2 uHz.
Abstract A technique that allows a harmonic oscillator to be used as a wideband vector signal analyzer is described and demonstrated using a single trapped 40ca+ ion cooled near its motional ground state. Further, the analyzer is shown to be compatible with both quantum amplification via squeezing and measurement in the Fock basis, allowing performance beyond the standard quantum limit. In addition to providing an attractive platform for quantum sensing of small fields, the technique allows in situ calibration of qubit control lines in systems using quantum harmonic oscillators and transduction of external, non-resonant drives into oscillator motion. This technique is extendable to other quantum harmonic oscillator systems.
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.
A trapped-ion harmonic oscillator represents a rich and well-controlled platform for tests of quantum physics, with applications to quantum simulations, quantum sensing, and quantum information processing. Owing to its net charge, the ion's motion is affected by changes in the electric potential. Consequently, electric field and potential curvature noise will limit the coherence of the ion's harmonic oscillation. While in some respects, this serves as a limitation in the performance of the trapped ion system (for example, in the fidelity of two-qubit gates or quantum simulations with motional quanta), it also opens an opportunity to use the ion as a precise field sensor and to use engineered quantum states of motion to enhance the sensitivity to these fields beyond the standard quantum limit (SQL). This thesis describes the implementation of such applications on a single beryllium ion confined in a Paul trap. In the thesis, I present the theory of an ideal harmonic oscillator and the implementation of this system using the motion of a single trapped ion, including producing special quantum states of motion; these states include the energy eigenstates of the harmonic oscillator, called number states, up to n=100, superpositions of number states of the form (1/√2)(|0> + |n>), with n up to 18, and coherently displaced number states with an average occupation up to approximately 300. I describe how we use these quantum states of motion for precise sensing of fluctuations of the harmonic oscillator frequency. The sensitivity of the (1/√2)(|0> + |n>) state ideally follows the 1/n Heisenberg scaling for frequency sensitivity. Additionally, I describe investigations of the spectrum of motional frequency noise using a series of coherent displacements of the motion of the ion, with features similar to Ramsey and dynamical decoupling sequences for two-level systems. While these displaced states don't improve sensitivity over the SQL, they can be simply and rapidly implemented with trapped ions. I discuss extensions to all of these experiments, including to multiple ions in a new trap designed to confine 3 or 4 ions in individually tunable potential wells.
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.