Fourier Transform Noise Spectroscopy

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Arian Vezvaee,1, ∗Nanako Shitara,1, 2, ∗Shuo Sun,2, 3 and Andr´es Montoya-Castillo1, †

1Department of Chemistry, University of Colorado Boulder, Colorado 80309, USA

2Department of Physics, University of Colorado Boulder, Colorado 80309, USA

3JILA, University of Colorado Boulder, Colorado 80309, USA

Abstract

Spectral characterization of noise environments that lead to the decoherence of qubits is crit-

ical to developing robust quantum technologies. While dynamical decoupling oﬀers one of the

most successful approaches to characterize noise spectra, it necessitates applying large sequences

of πpulses that increase the complexity and cost of the method. Here, we introduce a noise

spectroscopy method that utilizes only the Fourier transform of free induction decay or spin

echo measurements, thus removing the need for the application many πpulses. We show that

our method faithfully recovers the correct noise spectra for a variety of diﬀerent environments

(including 1/f-type noise) and outperforms previous dynamical decoupling schemes while signi-

ﬁcantly reducing their experimental overhead. We also discuss the experimental feasibility of our

proposal and demonstrate its robustness in the presence of statistical measurement error. Our

method is applicable to a wide range of quantum platforms and provides a simpler path toward a

more accurate spectral characterization of quantum devices, thus oﬀering possibilities for tailored

decoherence mitigation.

Keywords: Quantum noise spectroscopy, dynamical decoupling, Open quantum systems,

Decoherence

∗These authors have contributed equally to this work.

†Andres.MontoyaCastillo@colorado.edu

arXiv:2210.00386v4 [quant-ph] 10 Apr 2024

INTRODUCTION

Nearly all current quantum technology applications rely on a two-level quantum system (qubit)

that is subject to environmental noise. In the pure dephasing limit this environmental noise

causes ﬂuctuations in the frequency of the qubit that lead to decoherence. Spectral char-

acterization of such environments is the most crucial step in successfully controlling and sup-

pressing decoherence. Indeed, characterizing the noise spectrum allows for a ﬁlter-design ap-

proach that suppresses the noise and improves the coherence of the qubit [1–4]. Therefore,

developing methods that can recover the noise spectrum of qubit environments has been one

of the most active ﬁelds of research over the past two decades [5–8]. Among these eﬀorts,

dynamical decoupling noise spectroscopy (DDNS) [9–12] has been one of the most success-

ful approaches. In this method, applying a sequence of π-pulses turns the qubit into a noise

probe (approximated as a frequency comb) that isolates contributions from particular fre-

quencies of the noise spectrum. The dynamical decoupling framework has been studied ex-

tensively theoretically and implemented experimentally in various platforms such as super-

conducting circuits [13, 14], ultracold atoms [15], quantum dots [16–18], and nitrogen-vacancy

(NV) centers in diamonds [19, 20]. A DDNS protocol based on the Carr-Purcell-Meiboom-

Gill (CPMG) sequence [21, 22] was proposed by ´

Alvarez and Suter [9] which would ideally

yield a system of equations and unknowns from the measured values of the qubit coherence

C(t) = |⟨ρ01(t)⟩|/|⟨ρ01(0)⟩|, and speciﬁc frequencies of the spectrum. However, this method

oﬀers reasonable performance only when the number of π-pulses in each sequence is large.

Beyond a pulse economy standpoint, other diﬃculties, such as deviations from the ideal fre-

quency comb approximation [23], have recently inspired utilizing neural networks as ‘uni-

versal function approximators’ to reconstruct the noise spectrum from the coherence func-

tion of the qubit [24]. The success of this deep learning method suggests the existence of a

one-to-one mapping between the two quantities.

Here, we present a simple and inexpensive method that uniquely maps the measured coher-

ence function of a qubit to its noise power spectrum, removing the need for long sequences

of π-pulses at the heart of DDNS or turning to neural networks. In fact, we show that the

map obtained using neural networks in Ref. [24] can be found explicitly and analytically and

then translated to a simple and eﬀective noise spectroscopy method. This approach only re-

quires free induction decay or spin-echo measurements of the qubit and employs a simple

Fourier transform to accurately reconstruct the noise spectrum of the system. While Four-

ier spectroscopy has been implemented in Nuclear Magnetic Resonance and on diﬀerent types

of quantum processors [7, 25, 26], it has not been utilized in the context of pure dephasing

with the ﬁlter function formalism. Here, we combine the Fourier transform technique with

the ﬁlter function formalism to introduce an approach we call Fourier transform noise spec-

troscopy (FTNS) that signiﬁcantly enhances one’s ability to reconstruct the power spectrum

while dramatically reducing the required experimental overhead. We show that FTNS en-

ables the reconstruction of the noise spectrum over a frequency range that is otherwise in-

accessible through DDNS — information that is critical for eﬀective noise mitigation. We

then extend the FTNS method to directly extract the noise spectrum from a spin-echo sig-

nal, which becomes necessary when the system of interest is dominated by strong low-frequency

noise. While this FTNS method requires taking two time derivatives of the signal and is there-

fore sensitive to measurement noise in the time domain, we show that simple signal processing

steps can mitigate the eﬀect of such noise and yield accurate results.

RESULTS AND DISCUSSIONS

Theoretical description

We begin by laying out the theoretical basis for the ﬁlter function formalism in a pure de-

phasing setup [1, 6, 10, 27]. In this case, the qubit relaxation process (quantiﬁed by T1) takes

much longer than phase randomization (quantiﬁed by T⋆

2), implying that the decoherence

time T−1

2= (2T1)−1+T⋆−1

2≈T⋆−1

2becomes a measure of how fast the phase informa-

tion is lost due to environmental ﬂuctuations. Frequency ﬂuctuations of a qubit subject to

a stationary, Gaussian noise, ˆ

β(t), can be described by the Hamiltonian ˆ

H=1

2[Ω+ ˆ

β(t)]ˆσz,

where Ω is the natural frequency of the qubit. Here, the coherence function is C(t) = e−χ(t),

where the attenuation function χ(t) is given by the overlap of the noise spectrum and a ﬁl-

ter function that incorporates the eﬀect of the pulses on the system:

χ(t) = −ln[C(t)] = 1

4πZ∞

−∞

dω S(ω)F(ωt).(1)

The noise spectrum, S(ω) = R∞

−∞ dt eiωtS(t), is the Fourier transform of the equilibrium

time correlation function of the environmental noise, S(t) = ⟨{ˆ

β(t),ˆ

β(0)}⟩/2, where {A, B}=

AB+BA is the anticommutator. The ﬁlter function, F(ωt), encodes the sign switching (±1)

of the environmental ﬂuctuations upon application of each πpulse in the sequence [1].

The use of the absolute value in the deﬁnition of C(t)∝ |⟨ρ01(t)⟩| merits further comment.

Without the absolute value, ˜

C(t) = ⟨ρ01(t)⟩contains both a real and an imaginary com-

ponent, which is the output of the full coherence measurement, i.e., ⟨σx(t)⟩+i⟨σy(t)⟩. Here,

⟨σx(t)⟩refers to the Ramsey measurement of the real part that involves the sequence

RY(π/2) −t−RY(−π/2), giving access to Re[ρ01(t)], whereas ⟨σy(t)⟩refers to the Ram-

sey measurement of the imaginary part that involves the sequence RY(π/2) −t−RX(π/2),

giving access to Im[ρ01(t)] [14]. For quantum noise sources that obey Gaussian statistics, this

measurement can be written as ˜

C(t)∼e−χ(t)+iΦ(t)[28–31]. We consider the absolute value

of this measurement, which leads to C(t) = |˜

C(t)| ∼ e−χ(t). While removing the depend-

ence on Φ(t) may appear to cause information loss, it is not so as Φ(t) contains the same

information about the noise spectrum as χ(t). Indeed, Φ(t) is related to χ(t) via detailed

balance, with:

Φ(t) = Z∞

−∞

dω S(ω) coth ω

2kBTG(ωt),(2)

where G(ωt) encodes the eﬀect of the DD sequence on the imaginary-part Ramsey proced-

ure, Tdenotes temperature, and kBthe Boltzmann constant. Hence, knowledge of either χ(t)

or Φ(t) implies knowledge of the other. Other noise spectroscopy works have distinguished

between classical and quantum noise sources, with classical noise leading to a signal where

C(t)∼e−χ(t). However, such a measurement would indicate the breakdown of detailed bal-

ance. Instead, we articulate the problem in terms of C(t) = |˜

C(t)|and emphasize that such

a measurement does not imply that the source of noise is classical. We also note that pre-

vious work has shown that Φ(t) appears in the case of biased coupling [29] or in the M2 model [28],

when the interaction of the qubit with the bath has the form 1

2λ(ˆσz+ηˆ

I)⊗ˆ

V, where ˆ

Vis

a bath operator and η̸= 0. This case is particularly relevant for qubits based on the m=

0,±1 levels of the NV center in diamond.

To demonstrate the advantages of our proposed FTNS, we ﬁrst consider what is arguably

the state-of-the-art approach to noise spectroscopy: the ´

Alvarez-Suter protocol. The main

insight of the ´

Alvarez-Suter method is that when the number of pulses is suﬃciently large,

the ﬁlter function reaches the spectroscopic limit. In this limit, one can approximate the ﬁl-

ter function by a δ-function (frequency comb) with various harmonics: χ(t)≈tPkc

k=1 |Akω0|2S(kω0),

where Akω0are the Fourier coeﬃcients for a given pulse sequence, truncated at kc(for the

CPMG sequence, Akω0= 0 for even k). Applying many π-pulses is necessary for each peak

to better resemble a δ-function. The extreme case of kc= 1 approximates the ﬁlter func-

tion as a single δfunction, discarding many details of the noise spectrum. This is referred

to as the single δ-function approximation or the ﬁrst harmonic approximation. Often, one

can still account for a limited number of harmonics (set by the cut-oﬀ kc), which attenuates

the loss of spectral information [6, 24]. In the latter case, by appropriately varying the delay

time between pulses and the total time of the sequence, one can form a linear system of equa-

tions consisting of coherence values at selected times and a matrix of contributing Fourier

coeﬃcients. Inverting this system of equations yields the noise spectrum at the probed fre-

quencies, which are bounded by π/τmax ≤ |ωDDNS| ≤ π/τmin. Here, τmax(min) is the max-

imum (minimum) delay between consecutive π-pulses required to minimize the overlap between

subsequent pulses and validate the instantaneous pulse assumption. Furthermore, since A(k=0) =

0 for balanced pulse sequences like CPMG, the zero-frequency part of the spectrum cannot

be accessed directly. Thus, going beyond the π/τmax ≤ |ωDDNS| ≤ π/τmin limit and extract-

ing S(ω= 0) requires imbalanced sequences such as concatenated dynamical decoupling

(CDD) [11]. Hence, the experimental overhead, frequency restrictions, and accuracy depend-

ence on harmonic inclusions of ´

Alvarez-Suter [23] motivate the development of a more ac-

cessible scheme.

FTNS directly maps FID coherence to the noise spectrum

We introduce a radically more straightforward approach by inverting Eq. (1) directly to ob-

tain the noise power spectrum. We ﬁrst demonstrate this in the context of free induction

decay, noting that FFID(ωt) = (4/ω2) sin2(ωt/2) [1]. Substituting FFID(ωt) in Eq. (1), and

diﬀerentiating twice with respect to time, we obtain

¨χFID(t) = 1

2πZ∞

−∞

dω S(ω) cos(ωt).(3)

We Fourier transform both sides to ﬁnd

S(ω) = √2πF¨χFID(t),(4)

noting that S(−ω) = S(ω). This straightforward derivation demonstrates that there is a

simple and invertible one-to-one map between the noise power spectrum S(ω) and the second

time derivative of the logarithm of the experimentally measured coherence function.

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FourierTransformNoiseSpectroscopyArianVezvaee,1,∗NanakoShitara,1,2,∗ShuoSun,2,3andAndr´esMontoya-Castillo1,†1DepartmentofChemistry,UniversityofColoradoBoulder,Colorado80309,USA2DepartmentofPhysics,UniversityofColoradoBoulder,Colorado80309,USA3JILA,UniversityofColoradoBoulder,Colorado80309,USAAbstrac...

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Fourier Transform Noise Spectroscopy

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