Sensitivity of quantum gate delity to laser phase and intensity noise X. Jiang1J. Scott1Mark Friesen1and M. Saman1 2 1Department of Physics University of Wisconsin-Madison 1150 University Avenue Madison WI 53706

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Sensitivity of quantum gate fidelity to laser phase and intensity noise
X. Jiang,1J. Scott,1Mark Friesen,1and M. Saffman1, 2
1Department of Physics, University of Wisconsin-Madison, 1150 University Avenue, Madison, WI 53706
2Infleqtion, Inc., Madison, WI 53703
(Dated: April 3, 2023)
The fidelity of gate operations on neutral atom qubits is often limited by fluctuations of the
laser drive. Here, we quantify the sensitivity of quantum gate fidelities to laser phase and intensity
noise. We first develop models to identify features observed in laser self-heterodyne noise spectra,
focusing on the effects of white noise and servo bumps. In the weak-noise regime, characteristic
of well-stabilized lasers, we show that an analytical theory based on a perturbative solution of a
master equation agrees very well with numerical simulations that incorporate phase noise. We
compute quantum gate fidelities for one- and two-photon Rabi oscillations and show that they can
be enhanced by an appropriate choice of Rabi frequency relative to spectral noise peaks. We also
analyze the influence of intensity noise with spectral support smaller than the Rabi frequency. Our
results establish requirements on laser noise levels needed to achieve desired gate fidelities.
I. INTRODUCTION
Logical gate operations on matter qubits rely on coher-
ent driving with electromagnetic fields. For solid state
qubits these are generally at microwave frequencies of 1-
10 GHz. For atomic qubits microwave as well as optical
fields with carrier frequencies of several hundred THz are
used for gates. High fidelity gate operations require well
controlled fields with very low phase and amplitude noise.
In this paper we quantify the influence of control field
noise on the fidelity of gate operations on qubits. While
we mainly focus on the case of optical control with lasers,
our results are also applicable to high fidelity control of
solid state qubits with microwave frequency fields.
Since the limits imposed on qubit coherence and gate
fidelity by control field noise are of central importance
in the quest for improving performance, the topic has
been treated in a number of earlier works. Relaxation
of qubits in the presence of noise, with and without a
driving field was analyzed in [1–8]. Using a filter func-
tion methodology the influence of control field noise on
gate fidelity was analyzed in a series of papers by Bier-
cuk and collaborators [9–12]. In Ref. [11] experimental
measurements based on adding noise to microwave con-
trol signals were compared with theoretical results. Sub-
sequent work [13–15] has concentrated on qubit control
with optical frequency fields, including the application to
Rydberg gates for neutral atom qubits [13]. It was shown
convincingly in [16] that filtering of the laser phase noise
spectrum improves the fidelity of coherent Rydberg atom
excitation and in the work of Day et. al. [15], an aver-
age gate fidelity based on the filter function formalism
was calculated numerically which provided a prediction
of achievable performance based on measured laser noise
power spectra. Here we take a complementary approach
to [15] and use models for servo bump noise with Gaus-
sian distributed amplitude as well as underlying white
noise to provide compact analytical expressions that can
be used to predict gate fidelity based on fits to measured
laser noise spectra.
In this paper we develop a detailed theory of the depen-
dence of gate fidelity on the noise spectrum of the driv-
ing field based on a perturbative solution of the master
equation. Results for the cases of one- and two-photon
driving are presented as well as average control fideli-
ties together with the fidelity achieved when the qubit
starts in a computational basis state, which is of partic-
ular relevance to Rydberg excitation experiments. We
show analytically using a Gaussian model for the spec-
tral shape of servo bump noise that the spectral distri-
bution of phase and amplitude noise relative to the Rabi
frequency of the qubit drive is an important parameter
that determines the extent to which noise impacts gate
fidelity. Related numerical results for the impact of servo
bumps on gate fidelity were presented in Ref. [15]. When
the noise spectrum is peaked near the Rabi frequency the
deleterious effects are most prominent. Our one-photon,
state-averaged results for the influence of the noise spec-
trum on gate error are similar to, yet quantitatively dif-
ferent from the predictions of filter function theory [10].
As is shown in Appendix A the differences can be traced
to the use of different gate fidelity measures here, and
in [10].
We proceed in Sec. II with a summary of the theory
of the laser lineshape and its relation to self-heterodyne
spectral measurements. In Sec. III we show how the the-
ory can be used to extract parameters describing the laser
phase noise spectrum from experimental self-heterodyne
measurements. In Sec. IV we present a master equation
description for the coherence of Rabi oscillations with a
noisy drive field. A Schr¨odinger equation-based numer-
ical simulation is given in Sec. V, followed by a quasi-
static approximation in Sec. VI. The effect of intensity
noise on gate fidelity is presented in Sec. VII. The results
obtained, as well as a comparison with filter function the-
ory, are summarized in Sec. VIII and the appendices.
arXiv:2210.11007v3 [quant-ph] 31 Mar 2023
2
II. LASER NOISE ANALYSIS
The self-heterodyne interferometer is a powerful tool
for characterizing laser noise [17]. In a typical arrange-
ment, the heterodyne circuit outputs a current I(t) (or
normalized current i(t), as defined below) containing the
noise signal. The resulting power spectral density, Si(f),
provides a convenient proxy for laser field and frequency
fluctuations, SE(f) and Sδν (f), although the correspon-
dence is not one-to-one. In this section, we derive these
three functions and show how they are related, focusing
on the regime of weak frequency noise. While many of
the results in this section have been obtained previously,
we re-derive them here to establish a common framework
and notation. We then apply our results to two types of
noise affecting atomic qubit experiments: white noise and
servo bumps. This analysis forms the starting point for
the master equation calculations that follow.
The Rabi oscillations of a qubit are driven by a classical
laser field, which we define as
E(t) = ˆ
eE0(t)
2e−ı[2πνt+φ(t)] + c.c.,(1)
where c.c. stands for complex conjugate. Here we assume
the polarization vector ˆ
eand E0may be complex. Fluc-
tuations of the laser field are a significant source of deco-
herence in current atomic qubit experiments, and are the
focus of the present work. The fluctuations may occur in
any of the field parameters, ˆ
e,E0, or φ, where the latter
is the phase of the drive. For lasers of interest, the fluctu-
ations predominantly occur in the phase and amplitude
variables. In this work, we therefore ignore noise in the
polarization vector and focus on the fluctuations of φ(t).
The effect of relative intensity noise (RIN), where the in-
tensity is proportional to |E0(t)|2is considered briefly in
Sec. VII.
Phase fluctuations may alternatively be analyzed in
terms of fluctuations of the frequency, ν(t) = ν0+δν(t),
which are related to phase fluctuations through the rela-
tion
φ(t) = Zt0+t
t0
2πδν(t0)dt0,(2)
where t0is a reference time. The fluctuations of δν(t) [or
φ(t)] have a direct influence on the Rabi oscillations, and
must therefore be carefully characterized.
A compact description of a general fluctuating variable
X(t) is given by its autocorrelation function. Making
use of the ergodic theorem, we can equate ensemble and
time averages to obtain the following definition for the
autocorrelation function:
RX(τ) = hX(t)X∗(t+τ)i
≡lim
T→∞
1
2TZT
−T
X(t)X∗(t+τ)dt. (3)
Throughout this work, we will only consider random vari-
ables, X(t), that are wide-sense stationary.
According to the Wiener-Khintchine theorem, the au-
tocorrelation function of X(t) is related to its noise power
spectrum by the Fourier transform,
SX(f) = Z∞
−∞
RX(τ)e−i2πf τ dτ, (4)
and its inverse transform,
RX(τ) = Z∞
−∞
SX(f)ei2πf τ df, (5)
where in this work, we only consider two-sided power
spectra.
The main goal of this work is to characterize the noise
spectrum of E(t). However, SE(f) (also called the laser
lineshape) cannot be measured directly, due to the high
frequency of the carrier. We must therefore transduce the
power spectrum to lower frequencies. Here, we consider
the self-heterodyne transduction technique, in which the
laser field is split, delayed, and recombined to perform in-
terferometric measurements. The resulting signal is read
out as a photocurrent containing a direct imprint of the
underlying noise spectrum. For the dimensionless pho-
tocurrent i(t), which we define below, the self-heterodyne
power spectrum is denoted Si(f).
In this section, we derive the interrelated power spec-
tra of SE(f) and Si(f), which are in turn functions of
the underlying noise spectrum Sδν (f) [or Sφ(f)]. To per-
form noisy gate simulations, as discussed in later sections,
one would like to use actual self-heterodyne experimen-
tal data to characterize the underlying noise spectra. In
principle, such a deconvolution cannot be implemented
exactly [18]. However, we will show that reliable results
for the noise power may indeed be obtained, particularly
for lasers with very low noise levels, such as the locked
and filtered lasers used in recent qubit experiments.
A. Laser Lineshape
The autocorrelation function for the laser field is de-
fined as [18, 19]
RE(τ) = hE(t)E(t+τ)i,(6)
where we note that E(t) is the real, scalar amplitude
of E(t). This function contains information about both
the carrier signal, centered at frequency ν0, and the fluc-
tuations, which are typically observed as a fundamental
broadening of the carrier peak. Additional features of im-
portance for qubit experiments include structures away
from the peak that may be caused by the laser locking
and filtering circuitry, such as the “servo bump”, dis-
cussed in detail below.
The time average in Eq. (6) has been evaluated by
a number of authors. For completeness, we summarize
these derivations here, following the approach of Ref. [19].
Let us begin by assuming the noise process is strongly
3
stationary, so that Eq. (6) does not depend on t; for sim-
plicity, we set t= 0. Using Eqs. (1) and (6) and trigono-
metric identities, defining E0=|E0|e−iα, and neglecting
fluctuations of E0, we then have
RE(τ) = |E0|2
2ncos(2πν0τ)hcos[φ(τ)−φ(0)]i
+ cos(2πν0τ)hcos[φ(τ) + φ(0) + 2α]i
−sin(2πν0τ)hsin[φ(τ)−φ(0)]i
−sin(2πν0τ)hsin[φ(τ) + φ(0) + 2α]io.(7)
We then assume the phase difference Φ(τ)≡φ(τ)−φ(0)
to be a Gaussian random variable centered at Φ(τ) = 0,
with probability distribution
p(Φ) = 1
σΦ√2πe−Φ2/2σ2
Φ,
and variance Φ2=σ2
Φ. Here, the bar denotes an ensemble
average. According to the ergodic theorem, ensemble and
time averages should give the same result, so that
σ2
Φ(τ) = h[φ(τ)−φ(0)]2i= 2Rφ(0) −2Rφ(τ),(8)
where we note that
hφ2(0)i=hφ2(τ)i=Rφ(0).
Again, making use of the ergodic theorem, we have
hcos(Φ)i=e−σ2
Φ/2and hsin(Φ)i= 0,(9)
which is also known as the moment theorem for Gaus-
sian random variables. Finally we note that only biased
variables like φ(τ)−φ(0) are Gaussian. An unbiased vari-
able like φ(τ) + φ(0) is simply a random phase, for which
hcos[φ(τ) + φ(0)]i=hsin[φ(τ) + φ(0)]i= 0. Combining
these facts, we obtain the important relation [20]
RE(τ) = |E0|2
2cos(2πν0τ)e[Rφ(τ)−Rφ(0)].(10)
Note that since φcan take any value, Rφ(0) does not
have physical significance on its own; only the difference
Rφ(τ)−Rφ(0) is meaningful.
Another useful form for Eq. (10) can be obtained from
the relation 2πδν(t) = ∂φ/∂t, together with Eq. (4) and
the stationarity of Rφ(t), yielding
Sδν (f) = f2Sφ(f).(11)
Applying trigonometric identities, we then obtain the fol-
lowing, well-known results for the laser lineshape [18]:
RE(τ) = |E0|2
2cos(2πν0τ) exp −2Z∞
−∞
Sδν (f)sin2(πf τ)
f2df,(12)
and
SE(f) = |E0|2
2Z∞
−∞
cos(2πfτ) cos(2πν0τ) exp −2Z∞
−∞
Sδν (f0)sin2(πf 0τ)
(f0)2df0dτ. (13)
It is common to adopt a lineshape that is centered at
zero frequency; henceforth, we therefore set ν0= 0.
We note that SE(f) is properly normalized here, with
R∞
−∞ SE(f)df =|E0|2/2. Thus, fluctuations that broaden
the lineshape also reduce the peak height.
Some additional interesting results follow from
Eq. (13). First, in the absence of noise [Sδν = 0], we
see that the laser lineshape immediately reduces to an
unbroadened carrier signal: SE(f)=(|E0|2/2)δ(f). Sec-
ond, when Sδν is nonzero but small, as is typical for a
locked and filtered laser, the exponential term in Eq. (13)
may be expanded to first order, yielding [21]
2SE(f)/|E0|2≈[1 −Rφ(0)]δ(f) + Sφ(f).(14)
This approximation is generally very good, but breaks
down in the asymptotic limit τ→ ∞ of the τintegral,
and therefore in the limit f→0. To see this, we note
that sin2(πfτ) may be replaced by its average value of
1/2 in the integral; for nonvanishing values of Sδν (0), the
argument of the exponential then diverges. To estimate
the frequency fx, below which Eq. (14) breaks down, we
set the argument of the exponential in Eq. (14) to 1/2:
2Z∞
fx
Sδν (f)
f2df ≈1
2.(15)
This criterion clearly depends on the noise spectrum.
To conclude, we note that for some analytical calcu-
lations (such as the servo-bump analysis, described be-
low), it may be convenient or pedagogical to separate
the noise spectrum into distinct components: Sδν (f) =
Sδν,1(f) + Sδν,2(f), corresponding to different physical
noise mechanisms. From Eq. (12), the resulting line-
4
Laser
AOMSMF
delay, td
shift, 𝝂s
PD SA
FIG. 1. Self-heterodyne setup. The laser signal is split
equally between two paths. One path passes through a single-
mode fiber (SMF), where it is delayed by time td. It then
passes through an acousto-optic modulator (AOM), where its
frequency is shifted by νs. The interfering signals are com-
bined and measured by a photodiode (PD), and finally pro-
cessed through a spectrum analyzer (SA).
shapes can then be written as
RE(τ) = 2
|E0|2RE,1(τ)RE,2(τ),(16)
where RE,1(τ) and RE,2(τ) are the autocorrelation func-
tions corresponding to Sδν,1(f) and Sδν,2(f). Applying
the Fourier convolution theorem, we obtain
SE(f) = 2|E0|−2Z∞
−∞
SE,1(f−f0)SE,2(f0)df0.(17)
B. Self-Heterodyne Spectrum
We consider the self-heterodyne optical circuit shown
in Fig. 1. As depicted in the diagram, one of the paths is
delayed by time td, through a long optical fiber, and then
shifted in frequency by νs, by means of an acousto-optic
modulator. Here, the delay loop allows us to interfere
phases at different times, while the frequency shift pro-
vides a beat tone that is readily accessible to electronic
measurements, since it occurs at submicrowave frequen-
cies, νs≈100 MHz. The two beams are then recombined
and the total intensity is measured by a photodiode, us-
ing conventional measurement techniques.
For simplicity, we assume the laser signal is split
equally between the two paths, although unequal split-
tings may also be of interest [22]. The recombined field
amplitude is defined as
E(t) = |E0|
2nexp[−i2πν0t−iφ(t)−iα]
+ exp[−i2π(ν0+νs)(t−td)−iφ(t−td)−iα]o+ c.c.
(18)
The output current of the photodiode is then propor-
tional to |E(t)|2. For convenience, we consider instead a
dimensionless photocurrent i(t), defined as
i(t) = 1
2ncos [2πν0t+φ(t) + α]
+ cos [2π(ν0+νs)(t−td) + φ(t−td) + α]o2.(19)
The corresponding autocorrelation function is defined as
Ri(τ) = hi(t)i(t+τ)i.(20)
The evaluation of Ri(τ) is greatly simplified by noting
that cosine terms with ν0in their argument average to
zero in a physically realistic measurement. Again making
use of the fact that unbiased variables like φ(τ) + φ(0)
are random phases (i.e., nongaussian), we find that
Ri(τ)=4
+2hcos[2πνsτ+φ(t)−φ(t−td)−φ(t+τ)+φ(t+τ−td)]i.
(21)
Taking the same approach as in the derivation of RE(τ),
we take Φ0=φ(t)−φ(t−td)−φ(t+τ) + φ(t+τ−td)
to be a Gaussian random variable centered at zero, and
apply the Gaussian moment relations,
hcos(Φ0)i=e−σ2
Φ0/2and hsin(Φ0)i= 0,(22)
where
σ2
Φ0=h[φ(t)−φ(t−td)−φ(t+τ) + φ(t+τ−td)]2i.(23)
In this way we obtain
Ri(τ) = 4 + 2 cos(2πνsτ)
×exp[2Rφ(τ)+2Rφ(td)−2Rφ(0)
−Rφ(τ−td)−Rφ(τ+td)].(24)
Here, the cosine function represents the beat tone, and
the noise information is reflected in its amplitude. It can
be seen that the corresponding power spectrum, Si(f),
includes a central peak, δ(f), which contains no infor-
mation about the laser noise, and two broadened but
identical satellite peaks, centered at f=±νs. We now
recenter Ri(τ) at one of the satellite peaks, as consistent
with typical self-heterodyne measurements, such that
Ri(τ)→Ri(τ) = exp[2Rφ(τ)+2Rφ(td)−2Rφ(0)
−Rφ(τ−td)−Rφ(τ+td)].(25)
Applying trigonometric identities, we then obtain
Ri(τ) = exp −8Z∞
−∞
Sδν (f)sin2(πf τ) sin2(πftd)
f2df.
(26)
Taking Sδν (f) to be an even function, we can write
Si(f) = Z∞
−∞
cos(2πfτ)Ri(τ)dτ. (27)
We note that the self-heterodyne peak defined in this way
is normalized such that R∞
−∞ Si(f)df =Ri(0) = 1.
In the absence of noise [Sδν = 0], we see from Eqs. (26)
and (27) that the self-heterodyne power spectrum re-
duces to the bare carrier: Si(f) = δ(f). For nonzero but
5
small Sδν , we can expand the exponential in Eq. (26), as
was done in Eq. (14), to obtain
Si(f)≈1+2Rφ(td)−2Rφ(0)δ(f)
+ 4 sin2(πftd)Sφ(f).(28)
The second term in this expression is closely related to
the envelope-ratio power spectral density described in
Ref. [23], following on the earlier work of Ref. [24], and
provides a theoretical basis for the former.
As in the derivation of Eq. (14), the expansion leading
to Eq. (28) breaks down at low frequencies. However,
the well-known “scallop” features in the power spectrum
are seen to arise from the factor sin2(πftd). This result
clarifies the relation between the self-heterodyne signal,
the underlying laser noise, and the laser lineshape. The
latter relation is given by
|E0|2Si(f)≈2 sin2(πftd)SE(f),(29)
where we have omitted the central carrier peak.
To conclude, we again consider the possibility that the
noise spectrum may be separated into distinct compo-
nents, Sδν (f) = Sδν,1(f) + Sδν,2(f). As for the laser line-
shape, the self-heterdyne autocorrelation function may
then be written as
Ri(τ) = Ri,1(τ)Ri,2(τ),(30)
yielding the combined power spectrum
Si(f) = Z∞
−∞
Si,1(f−f0)Si,2(f0)df0.(31)
C. White Noise
The self-heterodyne laser noise measurements, re-
ported below, are well described by a combination of
white noise and a Gaussian servo bump. We now obtain
analytical results for the SE(f) and Si(f) power spectra,
for these two noise models. The results for white noise
are well-known [25]. However we reproduce them here
for completeness.
The underlying noise spectrum for white noise is given
by
Sδν =h0or Sφ(f) = h0
f2(32)
where h0is the amplitude of the power spectral density
of the frequency noise and has units of Hz2/Hz.
Note that it is common to use a one-sided noise spec-
trum for such calculations; however we use a two-sided
spectrum here. Our results may therefore differ by a fac-
tor of 2 from others reported in the literature. The most
straightforward calculation of Rφ(τ), from Eq. (5), im-
mediately encounters a singularity. We therefore proceed
Frequency, f (MHz)
0 0.2 0.4 0.6 0.8 1
Freq. (MHz)
-100
-80
-60
-40
-20
Power spectra (dB)
102104106
Freq. (Hz)
-100
-80
-60
-40
-20
Power spectra (dB)
Freq., f (Hz)
(dB)
(dB)
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2SE/E2
0,.
FIG. 2. White-noise power spectral densities, Sφ(f)
(blue), 2SE(f)/E2
0(gold), and Si(f)/4 (red), defined in
Eqs. (32), (35), and (38), respectively, for noise strength
h0= 100 Hz2/Hz. Here we have omitted the δ-function peak
in Si(f). The inset shows the same quantities plotted on a
logarithmic frequency scale. An approximate form for Si/4
(red, dotted) is obtained from Eq. (29). The cutoff fx(verti-
cal black), defined in Eq. (15), indicates the frequency where
Eq. (14) begins to fail.
by calculating RE(τ) from Eq. (12). Setting ν0= 0, to
center the power spectrum, then gives
RE(τ) = |E0|2
2e−2π2h0|τ|.(33)
From Eq. (10), we can identify
Rφ(τ)−Rφ(0) = −2π2h0|τ|,(34)
where the singularity has now been absorbed into Rφ(0).
Solving for the laser lineshape yields
SE(f) = |E0|2
2
h0
f2+ (πh0)2,(35)
for which the full-width-at-half-maximum (FWHM)
linewidth is 2πh0. Away from the carrier peak, which
is very narrow for a locked and well-filtered laser, we find
2SE(f)/E2
0≈h0
f2=Sφ(f),(36)
which is consistent with Eq. (14).
We can also evaluate the self-heterodyne autocorrela-
tion function, Eq. (25), obtaining
Ri(τ) = exp −2π2h0(2td+ 2|τ|−|τ−td|−|τ+td|,
(37)
and the corresponding power spectrum,
Si(f) = 2h0
f2+ (2πh0)2+e−4π2h0tdδ(f) (38)
−2h0
f2+ (2πh0)2cos(2πftd) + 2πh0
fsin(2πftd).
摘要：

Sensitivityofquantumgate delitytolaserphaseandintensitynoiseX.Jiang,1J.Scott,1MarkFriesen,1andM.Sa man1,21DepartmentofPhysics,UniversityofWisconsin-Madison,1150UniversityAvenue,Madison,WI537062Ineqtion,Inc.,Madison,WI53703(Dated:April3,2023)The delityofgateoperationsonneutralatomqubitsisoftenlimited...

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Sensitivity of quantum gate delity to laser phase and intensity noise X. Jiang1J. Scott1Mark Friesen1and M. Saman1 2 1Department of Physics University of Wisconsin-Madison 1150 University Avenue Madison WI 53706.pdf

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