Quantum Harmonic Oscillator Spectrum Analyzers

Jonas Keller, Pan-Yu Hou, Katherine C. McCormick, Daniel C. Cole, Stephen D. Erickson, Jenny J. Wu, Andrew C. Wilson, and Dietrich Leibfried

Phys. Rev. Lett. 126, 250507 – Published 25 June 2021

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.

Received 20 October 2020
Accepted 8 April 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.250507

Physics Subject Headings (PhySH)

Quantum control Quantum information with trapped ions Quantum sensing

Quantum harmonic oscillator Trapped ions

Atomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Jonas Keller ^1,2,*, Pan-Yu Hou ^1,2, Katherine C. McCormick^1,2,†, Daniel C. Cole ¹, Stephen D. Erickson ^1,2, Jenny J. Wu^1,2, Andrew C. Wilson¹, and Dietrich Leibfried¹

¹National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA
²Department of Physics, University of Colorado, Boulder, Colorado 80309, USA

^*Present Address: Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany., he/him/his
^†Present Address: Department of Physics, University of Washington, Seattle, Washington 98195, USA.

Quantum Oscillator Noise Spectroscopy via Displaced Cat States

Alistair R. Milne, Cornelius Hempel, Li Li, Claire L. Edmunds, Harry J. Slatyer, Harrison Ball, Michael R. Hush, and Michael J. Biercuk

Phys. Rev. Lett. 126, 250506 (2021)

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Vol. 126, Iss. 25 — 25 June 2021

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Images

Figure 1
Implementation of filter functions $s (t)$ using coherent displacements. (a) In the absence of noise [ $Δ (t) = 0$ ], the time-dependent displacement $α_{0} (t)$ along the real axis (solid line) is proportional to the integral over time of the coherent drive $Ω_{d} (t)$ (dashed line). (b) In the LO reference frame rotating at $ω_{0}$ , frequency fluctuations $Δ (t)$ cause azimuthal rotations. During an infinitesimal time step $d t$ , this produces a displacement perpendicular to $α (t)$ and equal to $- i α (t) Δ (t) d t$ . For small rotations, $Re [α (t)] \approx α_{0} (t)$ thus controls the sensitivity to $Δ (t)$ and implements a filter function $s (t)$ .
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Figure 2
Implementation of filter functions $s (t)$ using number state superpositions. (a) Sensitivity function approximating a Hann filter (dashed line) by a piecewise constant sensitivity $s (t) = n_{1} - n_{2}$ (solid line) in different superpositions of number states $| n_{1} ⟩$ and $| n_{2} ⟩$ , as indicated by the labels [cf. Eq. (7); phase and normalization omitted for clarity]. (b) The filter $| \tilde{s} (f) |^{2}$ corresponding to $s (t)$ shown in (a), calculated both analytically using a simple model and numerically [35].
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Figure 3
Experimental results of the coherent displacement method. (a) Response to a $f_{noise} = f_{0} = 8 kHz$ sinusoidal modulation of the HO frequency as a function of the filter amplification (symbols: experimental data, line: model with the modulation amplitude $Δ_{0}$ as a free parameter). (b) Measured response of various filter functions ( $t_{w} = 250 μ s$ , varied $k$ ) to sinusoidal modulations of varied frequency and random phase. The modulation amplitude $Δ_{0}$ and filter amplification are chosen to keep the signal within the first monotonic increase of the function shown in (a). Filter functions above 100 kHz are measured with increased modulation to compensate technical limitations of the filter amplitude in this regime. Data points are linked with solid lines to guide the eye. (c) Power spectral density estimate of the HO frequency noise, determined from the transmission through filter functions with $t_{w} = 1 ms$ and varied $k$ . The dashed line indicates a $1 / f$ slope to guide the eye. Experimental data in all subfigures are averages over 200 repetitions. Error bars: $1 σ$ (vertical), $δ_{rbw}$ [horizontal in (c)].
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Figure 4
Number state implementation results: Measured response of different filter functions to applied modulation of varied frequency $f_{noise}$ and random phase. The 5.0 kHz trace corresponds to the sensitivity function in Fig. 2. The solid lines show the expected response, adjusted by a global scaling factor to account for the dependence of the modulation depth on the applied voltage. Each data point represents 200 repetitions, error bars: $1 σ$ .
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