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arXiv:2010.10438 · Physical Review Letters

Quantum Harmonic Oscillator Spectrum Analyzers

Jonas Keller, Pan‐Yu Hou, Katherine C. McCormick, Daniel C. Cole, Stephen D. Erickson, Jenny J. Wu, A. C. Wilson, Dietrich G. Leibfried

Published Jun 25, 2021• 21 citations
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Abstract

Characterization and suppression of noise are essential for the control of harmonic oscillators in the quantum regime. We measure the noise spectrum of a quantum harmonic oscillator from low frequency to near the oscillator resonance by sensing its response to amplitude modulated periodic drives with a qubit. Using the motion of a trapped ion, we experimentally demonstrate two different implementations with combined sensitivity to noise from 500 Hz to 600 kHz. We apply our method to measure the intrinsic noise spectrum of an ion trap potential in a previously unaccessed frequency range.

DOI
10.1103/physrevlett.126.250507

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Characterization and suppression of noise are essential for the control of harmonic oscillators in the quantum regime. We measure the noise spectrum of a quantum harmonic oscillator from low frequency to near the oscillator resonance by sensing its response to amplitude modulated periodic drives with a qubit. Using the motion of a trapped ion, we experimentally demonstrate two different implementations with combined sensitivity to noise from 500 Hz to 600 kHz. We apply our method to measure the intrinsic noise spectrum of an ion trap potential in a previously unaccessed frequency range.
Quantum harmonic oscillators are central to many modern quantum technologies. We introduce a method to determine the frequency noise spectrum of oscillator modes through coupling them to a qubit with continuously driven qubit-state-dependent displacements. We reconstruct the noise spectrum using a series of different drive phase and amplitude modulation patterns in conjunction with a data-fusion routine based on convex optimization. We apply the technique to the identification of intrinsic noise in the motional frequency of a single trapped ion with sensitivity to fluctuations at the sub-Hz level in a spectral range from quasi-dc up to 50 kHz.
Characterization of noise of a quantum harmonic oscillator is important for many experimental platforms. We experimentally demonstrate motional spin-locking spectroscopy, a method that allows us to directly measure the motional noise spectrum of a quantum harmonic oscillator. We measure motional noise of a single trapped ion in a frequency range from 200 Hz to 5 kHz with a power spectral density that resolves noise over two orders of magnitude. Coherent modulations in the oscillation frequency of the oscillator can be probed with a relative frequency sensitivity at the 10−610^{-6}10−6 level.
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.
Native noise processes in quantum processors are often difficult to isolate because multiple error mechanisms contribute to the same measured loss of coherence. Here we use dephasing-robust quantum noise spectroscopy to identify and characterize control noise induced by axial motion in an individually-addressed trapped-ion processor. When the ion motion is transverse to the addressing beam, thermal axial motion couples to the beam profile and produces effective amplitude control noise. We show that this noise is governed primarily by the local beam curvature and appears as a low-frequency contribution to the reconstructed control-noise spectrum. By varying the ion position within the beam profile, we separate curvature-dependent axial-motion noise from curvature-independent native control noise and extract motional parameters that are otherwise difficult to access on this platform. We also apply the protocol in parallel to a four-ion register, demonstrating a spectroscopic method for simultaneous characterization of position-dependent control noise across multiple qubits. The results of this study identify beam inflection points as operating regions that suppress axial-motion-induced noise at the cost of reduced Rabi rate, as found in PRX Quantum 3, 010334 (2022).
Coherently displaced number states of a harmonically bound ion can be coupled to two internal states of the ion by a laser-induced motional sideband interaction. The internal states can subsequently be read out in a projective measurement via state-dependent fluorescence, with near-unit fidelity. This leads to a rich set of line shapes when recording the internal-state excitation probability after a sideband excitation, as a function of the frequency detuning of the displacement drive with respect to the ion's motional frequency. We precisely characterize the coherent displacement based on the resulting line shapes, which exhibit sharp features that are useful for oscillator frequency determination from the single quantum regime up to very large coherent states with average occupation numbers of several hundred. We also introduce a technique based on multiple coherent displacements and free precession for characterizing noise on the trapping potential in the frequency range of 500 Hz–400 kHz. Signals from the ion are directly used to find and eliminate sources of technical noise in this typically unaccessed part of the spectrum.
We extend quantum noise spectroscopy (QNS) of amplitude control noise to settings where dephasing noise or detuning errors make significant contributions to qubit dynamics. Previous approaches to characterize amplitude noise are limited by their vulnerability to low-frequency dephasing noise and static detuning errors, which can overwhelm the target control noise signal and introduce bias into estimates of the amplitude noise spectrum. To overcome this problem, we leverage optimal control to identify a family of amplitude control waveforms that optimally suppress low-frequency dephasing noise and detuning errors, while maintaining the spectral concentration in the amplitude filter essential for spectral estimation. The waveforms found via numerical optimization have surprisingly simple analytic forms, consisting of oscillating sine waves obeying particular amplitude and frequency constraints. In numerically simulated QNS experiments, these waveforms demonstrate superior robustness, enabling accurate estimation of the amplitude noise spectrum in regimes where existing approaches are biased by low-frequency dephasing noise and detuning errors.