Theory of qubit noise characterization using the long-time cavity transmission Philipp M. Mutterand Guido Burkard Department of Physics University of Konstanz D-78457 Konstanz Germany

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Theory of qubit noise characterization using the long-time cavity transmission

Philipp M. Mutter∗and Guido Burkard†

Department of Physics, University of Konstanz, D-78457 Konstanz, Germany

Noise induced decoherence is one of the main threats to large-scale quantum computation. In an

attempt to assess the noise aﬀecting a qubit we go beyond the standard steady-state solution of the

transmission through a qubit-coupled cavity in input-output theory by including dynamical noise

in the description of the system. We solve the quantum Langevin equations exactly for a noise-free

system and treat the noise as a perturbation. In the long-time limit the corrections may be written as

a sum of convolutions of the noise power spectral density with an integration kernel that depends on

external control parameters. Using the convolution theorem, we invert the corrections and obtain

relations for the noise spectral density as an integral over measurable quantities. Additionally,

we treat the noise exactly in the dispersive regime, and again ﬁnd that noise characteristics are

imprinted in the long-time transmission in convolutions containing the power spectral density.

I. INTRODUCTION

In the absence of quantum error correction [1], ﬂuctu-

ations from the environment severely impair the perfor-

mance of quantum hardware, as such external noise leads

to decoherence, i.e., the permanent loss of quantum co-

herence [2]. Even with quantum error correction at hand,

the presence of noise can force the system to exceed the

critical error threshold of fault tolerant quantum compu-

tation [3–7], thereby rendering large-scale computations

impossible [8]. To mitigate the eﬀects of the environment,

it can be advantageous to have a precise knowledge of

the form of the temporal ﬂuctuations. As a consequence,

the determination of noise characteristics is an impor-

tant task in present-day quantum information processing

units.

All qubit realizations suﬀer from noise in one way or

the other, either directly or indirectly. This holds es-

pecially for solid-state qubits which are surrounded by

a macroscopic number of atoms that make up the host

crystal [9–12]. Semiconductor charge qubits in quantum

dots suﬀer from charge noise stemming from ﬂuctuat-

ing gate voltages [13–15], spin qubits suﬀer from eﬀec-

tive random magnetic ﬁelds due to the interaction with

the host atomic nuclei [16–19], and superconducting cir-

cuits can be aﬀected by magnetic ﬂux noise, quasiparti-

cles and two-level ﬂuctuators [20–23]. Via the spin-orbit

interaction, charge noise can aﬀect the spin degree of

freedom as well [24–26]. Hybrid systems such as ﬂop-

ping mode qubits or other spin qubits that rely on the

spin-orbit interaction for gate operations, e.g., hole sys-

tems in germanium [27–32], may even suﬀer from both

magnetic and charge noise [33]. In general, enhanced

qubit performance from an extended control parameter

space typically comes at the price of additional decoher-

ence channels and increased susceptibility of the logical

two-level system to noise.

Here, we address the problem of characterizing ﬂuc-

∗philipp.mutter@uni-konstanz.de

†guido.burkard@uni-konstanz.de

port 1 port 2

κ1

⟨bin⟩ ⟨bout⟩

κ2

ωc

δωq(t)=λδX(t)

ωq

|1⟩

|0⟩

A(t)

average

extract convolution

theorem

FIG. 1. The system under consideration consists of a cavity-

coupled qubit (green dot) with ﬂuctuating energy separation

ωqdue to noise δX(t)aﬀecting its control parameters. The

cavity mirrors (grey) allow for an input ﬁeld bin at the left

port and an output ﬁeld bout at the right port, thereby creat-

ing a measurable transmission signal A(t). We ﬁnd that the

averaged transmission probability ⟪∣A∣2⟫receives noise cor-

rections in the form of sums over convolutions C=S⋆Kof

the power spectral density S(ω)of δX with an integration

kernel K. As a result, the spectral density can be extracted

from the averaged transmission probability by using the con-

volution theorem.

tuations in a quantum bit (qubit), which represents the

fundamental building block of a quantum computer. To

obtain information about the noise power spectral den-

sity, we propose a scheme based on the long-time trans-

mission through an electromagnetic cavity interacting

with the qubit (Fig. 1). The steady-state transmission

through a resonator-qubit system has become an impor-

tant tool in extracting system parameters such as val-

ley splittings [34,35] and in qubit readout [36–41]. The

cavity transmission signal has played an important role

in providing evidence for the achievement of the strong

coupling regime of cavity quantum electrodynamics for

superconducting coplanar microwave cavities coupled to

arXiv:2210.11891v1 [cond-mat.mes-hall] 21 Oct 2022

either superconducting qubits [42] or semiconducting

qubits [43–46] and has been used to experimentally char-

acterize charge noise in a semiconductor double quantum

dot [47]. Recent studies have theoretically investigated

the eﬀect of general dynamical noise on the transient cav-

ity transmission in the dispersive regime [48], as well as

the eﬀect of quantum noise [49]. Here, we study the ef-

fect of dynamical noise on the long-time transmission.

We complement the analysis in Ref. [48] in the dispersive

regime, and in addition obtain results that are valid for

arbitrary qubit-photon detunings given that the decay

rates in the system are suﬃciently large to allow for a

quickly converging perturbation series in the dynamical

noise parameter. In both regimes we ﬁnd that the aver-

aged transmission probability contains measurable infor-

mation on spectral features of the noise in the system.

The remainder of the paper is structured as follows:

Sec. II introduces the theoretical model and displays the

quantum Langevin equations that we aim to solve in dif-

ferent regimes in the subsequent sections. We treat the

noise as a perturbation in Sec. III and distinguish be-

tween two cases: In Sec. III A we work in the regime

where the matrix Lappearing in the system of diﬀerential

equations is diagonalizable, while we focus on the special

parameter regime in which Lbecomes non-diagonalizable

in Sec. III B. In both cases we ﬁnd that the long-time

transmission receives noise corrections which can be ex-

pressed as convolutions of the spectral density with in-

tegration kernels depending on system parameters. We

discuss and compare our ﬁndings in Sec. III C, and show

how the noise power spectral density can be extracted

in real measurements by using the convolution theorem

in Sec. III D. Additionally, we investigate the long-time

limit in the dispersive regime in Sec. IV and thereby

complement the results on the transient transmission in

Ref. [48]. Again, we ﬁnd that the cavity transmission con-

tains information on the ﬂuctuations aﬀecting the qubit

in terms of convolutions containing the noise power spec-

tral density. Finally, Sec. Vprovides a conclusion and an

outlook on possible future research directions.

II. SYSTEM AND TRANSMISSION

We study a generic noisy two-level system with ﬂuc-

tuating qubit energy separation ωq+δωq(t)and to-

tal noise-independent intrinsic decoherence rate γ=

coth(ωq2T)γ12+γϕ, where Tis the temperature, γ1is

the relaxation rate at zero Kelvin and γϕis the dephas-

ing rate [50]. The ﬂuctuations may be written in leading

order as δωq=λδX(t), where λ=∂XωqδX=0is the noise

sensitivity of the qubit, Xis the noise-aﬀected qubit con-

trol parameter, and δX(t)describes the time-dependent

classical random ﬂuctuations of this parameter. We re-

mark that ﬂuctuations of the coupling constant gmay

also be present in the system but can be controlled exter-

nally (e.g. via the detuning in a charge qubit, see below)

and we focus on the regime where they may be neglected.

The qubit is allowed to interact with a single cavity mode

of frequency ωc, and the qubit-cavity coupling is assumed

to be linear with coupling strength g(Fig. 1). The cavity

decay rate is given by κ=κ1+κ2+κint, where κjis the

decay rate at port j∈{1,2}and κint the intrinsic photon

loss rate, and in the following we choose a classical input

ﬁeld of amplitude binand frequency ωpto be present at

port 1, while no input ﬁeld is assumed to be present at

port 2. In a frame corotating with the probe frequency ωp

and within the rotating wave approximation, the quan-

tum Langevin equations for the expectation values of the

spin ladder operator σ−and the photon annihilation op-

erator atake the form,

dt 



σ−

a



+[L+δL(t)]



σ−

a



=





√κ1bin



,

L=





i∆q+γ2−iσzg

ig i∆c+κ2



, δL(t)=





iλδX(t)0

0 0



,

(1)

where we introduce the probe-qubit (probe-cavity) de-

tuning ∆q=ωq−ωp(∆c=ωc−ωp). The dynamics of

the operators under consideration are governed by the

non-Hermitian system matrix Land the noise corrections

appearing in the dynamical matrix δL(t). We remark

that σz≡σ(0)

zis taken to be the zeroth-order term

in the photon-qubit coupling gwhile higher orders are

neglected, a standard assumption in input-output the-

ory [34,41,51], which is justiﬁed as follows: The only

equation featuring σzis the one for σ−in the term

gσzawhich to leading order in greads gσ(0)

zawith

exact afor separable initial qubit-cavity states. As the

solution of σ−is only required to ﬁrst order in gin the

expansion of σzto calculate the transmission via the

exact Langevin equation for a, the procedure is con-

sistent. Physically this means that the level populations

are assumed to be unaﬀected by the interaction with the

photons. In the remainder of this paper we assume a

constant thermal distribution of the qubit energy levels,

σz=−tanh(ωq2T)<0. This is justiﬁed at the level

of the diﬀerential equation (1) by the fact that the time-

dependence of σzwill decay due to relaxation processes

in the long-time limit even in the presence of noise. A

proof is given in Appendix A.

While the model is general and describes any two-level

system with ﬂuctuating energy separation, we discuss the

potential tunability of the noise couplings by considering

the example of a charge qubit, i.e., a qubit that encodes

the logical states into the bonding and anti-bonding or-

bital states of a charge in a double quantum dot. The

qubit splitting is given by ωq=2+4t2

c, where is the

detuning between the two dots and tcis the tunnel matrix

element [52,53]. In the presence of charge noise due to

imperfect gate voltages, the detuning ﬂuctuates in time,

→+δ(t), and we ﬁnd λ=∂ωqδ=0=ωq. On the

other hand, the qubit-photon coupling strength has the

parameter dependence g∝tc2+4t2

cand hence may

also be aﬀected by noise. To leading order the noise cou-

ples to gwith strength given by λ′=∂gδ=0=−gω2

and the ratio λ′λ=−gωq∝12for ≫tccan be con-

trolled via the detuning. Therefore, if the system is oper-

ated at suﬃciently large detunings, the assumption that

only the energy separation is aﬀected by noise is valid.

Additionally, the prefactor of higher order noise terms

δXn⩾2in the expansion of the qubit energy ﬂuctuations

δωq(t)=λδX +∑∞

n=2λ(n)δXnn!is suppressed in this

regime. Speciﬁcally, one has λ(n)=∂n

ωqδ=0∼1n+1,

while the ﬁrst order term λis asymptotically constant at

large detunings.

Finally, we turn to the general framework for the de-

scription of the resonator transmission. The input ﬁeld

binappears in the Langevin equations (1), and stan-

dard input-output theory states the simple relation for

the output ﬁeld bout(t)=−√κ2a(t) [54–56]. Conse-

quently, the time-dependent transmission amplitude A

through the cavity is found to be

A(t)=bout(t)

bin=−√κ2a(t)

bin.(2)

In order to compute the transmission amplitude A, it is

therefore necessary to solve the system of coupled dif-

ferential equations (1) for the expectation value of the

photon annihilation operator a(t). Since the dimen-

sion of the matrix Lis larger than one, a general solu-

tion for arbitrary time-dependent noise can only be given

as a time-ordered series. To obtain tractable expressions

that can lead to physical insights and allow us to extract

noise characteristics, we apply time-dependent perturba-

tion theory. In Secs. III A and III B, the perturbation

parameter is the ratio of the maximum value of the noise

taken until measurement and the smaller of the two decay

rates κand γ. Hence, this approach is expected to be ap-

plicable in open cavities or in cavities containing quickly

relaxing qubits. In Sec. IV, on the other hand, we expand

the exact solution in the parameter gmax{δ0,κ−γ},

where δ0=ωc−ωqis the noise-free qubit-cavity detuning.

The results obtained in this fashion are expected to be

valid in the dispersive regime δ0≫gor when there is

a large discrepancy in the qubit and cavity decay rates.

III. PERTURBATION THEORY IN THE NOISE

The system of diﬀerential equations in (1) may not be

solved exactly for generic noise. In this section we ob-

tain an approximate analytical solution by treating the

noise as a perturbation, deriving the conditions of valid-

ity of the approach as we go along. Since the zeroth-

order transmission amplitude may then be obtained by

exponentiating the matrix L, we distinguish between the

cases where Lis diagonalizable (Sec. III A) and where it

is not (Sec. III B). We discuss and compare our ﬁndings

in Sec. III C and ﬁnally propose a scheme for extracting

noise characteristics from the measured average trans-

mission probability in Sec. III D.

A. Diagonalizable system matrix

For δ0≠0or δ0=0while (κ−γ)2≠−16σzg2, the non-

Hermitian matrix Lis diagonalizable with eigenvalues

l±=Γ+i∆±Λ,Λ=1

4(κ−γ+2iδ0)2+16σzg2,(3)

where Γ=(κ+γ)4and ∆=(∆c+∆q)2. In the absence

of noise the diﬀerential equations (1) can be decoupled

by expressing the matrix Lin its eigenbasis. Treating the

matrix δL and therefore λδX(t)as a perturbation then

allows us to solve for a(t)order by order in the noise

(Appendix A). Going up to quadratic order and using

the limiting cases discussed in Appendix B, we ﬁnd for

the transmission amplitude as given by (2) at long times

Re(l±)t≫1,

A(t)

A0=1+σzg2

Λ(i∆q+γ2)i

2(f+(t)−f−(t))

+

γ−κ−2iδ0±4Λ

16Λ (f±±(t)−f∓±(t)),

(4)

where

A0=−√κ1κ2

i∆c+κ2−σzg2(i∆q+γ2),

fp(t)=λe−lptt

dt1δX(t1)elpt1,

fpq(t)=λe−lptt

dt1δX(t1)elpt1fq(t1).

(5)

Here, A0is the zeroth-order transmission which would be

the steady state in an noise-free system, and the functions

fpand fpq with p, q ∈{±}are the ﬁrst- and second-order

noise corrections, respectively. They depend on time,

and hence a true steady state is not reached for a single

noise realization. While it may appear that the limit

Λ→0could lead to divergences, a closer analysis reveals

that the expressions f+−f−and ∑±±(f±± −f∓±)only

have terms Λnwith n≥1, i.e., no constant term, and

the expression ∑±(f±± −f∓±)only has terms Λnwith

n≥2. These are precisely the powers needed to cancel

the negative powers of Λin (4), and hence the limit Λ→0

exists and is well behaved (Appendix C). Note, however,

that at Λ=0, the matrix Lis non-diagonalizable, and

must be treated in a diﬀerent approach (Sec. III B).

As can be seen from Eq. (4) the noise integrals serve

as the perturbation parameters, and at long times,

Re(l±)t≫1, they are bounded by

fp⩽fmax

p=λδXmax

Γ+pRe(Λ)δ0=0

fpq⩽fmax

pq =fmax

pfmax

q,

(6)

where δXmax=max{δX(τ)∶τ∈[0, t]}is the maximal

value taken by the noise until measurement at time t.

The regime in which the perturbation expansion is valid

is therefore characterized by fmax

p≪1for p∈{±}. Note

that this condition is independent of the probe frequency

ωpand the qubit-cavity detuning δ0which are the con-

trol parameters of choice in transmission experiments.

Physically, the condition fmax

p≪1can be understood

as follows: in time-dependent perturbation theory the

quality of the expansion is a function of time as the dy-

namical noise will move the solution away from a steady

state. At long times the accuracy is determined solely by

the interplay of noise and decay rates and independent of

any oscillatory processes. The decay rates must then be

large enough to tame the deviations from the (by then

time-independent) zeroth-order solution induced by the

noise.

Relevant quantities in input-output measurements in-

clude the transmission Aand the transmission prob-

ability A2. In the literature both Aand A2are

used, and there is no universal convention as to which

of the two quantities is used to characterize cavity trans-

mission. Clearly, this is justiﬁed by the very simple

relation between the two quantities in noise-free sys-

tems. However, when a noise average ⋅is involved,

one must be cautious and include the variance of A,

A2=A2+Var(A). In Ref. [48] and Sec. IV of

this paper the variance vanishes to the order of interest in

perturbation theory, and the transmission and transmis-

sion probability can be used interchangeably. In contrast,

the variance does not vanish in general when performing

a perturbation expansion in the noise parameter as we

do in this section. Hence, we must decide on using one

ﬁgure of merit, and we choose the averaged transmission

probability A2. This is because we aim to extract

noise features from measurable convolutions of the noise

power spectral density with an integration kernel, and we

ﬁnd that the mean A and variance Var(A) of the

transmission both contain a term that cannot be written

as a sum over such convolutions (Appendix D), while this

problematic term is not present in the averaged transmis-

sion probability.

Squaring the transmission and averaging over many

noise conﬁgurations is a cumbersome task, and it con-

tains the additional complication that the radicand in Λ

has both real and imaginary parts. If the random process

governing the noise dynamics is assumed to be station-

ary, the mean of the noise is time independent and any

unconditional joint probability distribution is invariant

under time translations. In this case the general solution

has the form

A2

A02=1+σzg2

4∆2

q+γ2



λ%δX+λ2



j=1

ψjCj(∆)



,(7)

where A0is the absolute value of the well-known steady-

state transmission amplitude (5) in the absence of noise.

The corrections to the noise-free steady-state solution

are twofold: The term linear in the qubit-noise coupling

strength λis only present for biased noise, i.e., ﬂuctua-

tions with non-zero mean δX. The eﬀect of a non-zero

noise mean can be thought of as a shift in the qubit en-

ergy splitting, ωq→ωq+λδX, as can be shown by

expanding A02to leading order in λδX, and hence

we set δX=0in the remainder of this paper. On the

other hand, the term quadratic in λis present even for

zero-mean noise and describes the eﬀect of the noise cor-

relations. The functions %and ψjdepend on the system

parameters in a rather complicated fashion and are dis-

played in Appendix E. Finally, the quantities Cj(∆)are

convolutions,

Cj(∆)=(S⋆Kj)(∆)=1

2π∞

−∞ S(ω)Kj(∆−ω)dω. (8)

Here, S(ω)is the noise power spectral density deﬁned

as the Fourier transform of the noise autocorrelator, and

the integration kernels read

K1(∆)=

[Γ±Re(Λ)]2+[∆±Im(Λ)]2,

K2/3(∆)=1

[Γ±Re(Λ)]2+[∆±Im(Λ)]2,

K4/5(∆)=∆±Im(Λ)

[Γ±Re(Λ)]2+[∆±Im(Λ)]2,

(9)

where the positive (negative) sign belongs to the even

(odd) index in the latter two equations. A notable special

case is δ0=0,(κ−γ)2>−16g2σzfor which Im(Λ)=0.

The kernel K1can then be written as a linear combi-

nation of the kernels K2and K3by means of a partial

fraction decomposition. Since the coeﬃcients are inde-

pendent of ∆, we ﬁnd for the sum of convolutions,



j=1

ψjCj(∆)=5



j=2

ψjCj(∆),(10)

where the functions ˜

ψjrelate to the original functions ψj

via

ψ2/3=ψ2/3∓σzg2

Re(Λ)Γ,˜

ψ4/5=ψ4/5.(11)

Reducing the number of convolutions in this way can ease

their experimental extraction as we detail in Sec. III D.

B. Non-diagonalizable system matrix

For δ0=0and (κ−γ)2=−16g2σzone has Λ=0,

and the eigenvalues of Lbecome degenerate, l+=l−≡l.

Since in this case the algebraic multiplicity exceeds the

geometric multiplicity, the matrix is no longer diagonal-

izable and may only be brought into Jordan normal form,

LJ=





lg1

0lg



, l =Γ+i∆,(12)

where now ∆=∆c=∆q. In the generalized eigenbasis,

the system of diﬀerential equations is then only partially

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Theoryofqubitnoisecharacterizationusingthelong-timecavitytransmissionPhilippM.MutterandGuidoBurkardDepartmentofPhysics,UniversityofKonstanz,D-78457Konstanz,GermanyNoiseinduceddecoherenceisoneofthemainthreatstolarge-scalequantumcomputation.Inanattempttoassessthenoiseaectingaqubitwegobeyondthestand...

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Theory of qubit noise characterization using the long-time cavity transmission Philipp M. Mutterand Guido Burkard Department of Physics University of Konstanz D-78457 Konstanz Germany

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