# Qubit dynamics driven by smooth pulses of finite duration

Ivo S. Mihov and Nikolay V. Vitanov  Center for Quantum Technologies,
Department of Physics, St Kliment Ohridski University of Sofia, 5 James
Bourchier blvd, 1164 Sofia, Bulgaria

###### Abstract

We present a study of the dynamics of a qubit driven by a pulsed field of
finite duration. The pulse shape starts and ends linearly in time. The most
typical example of such a shape is the sine function between two of its nodes,
but several other pulse shapes are also studied. All of them present smooth
alternatives to the commonly used rectangular pulse shape, resulting in much
weaker power broadening, much faster vanishing wings in the excitation line
profile and hence much reduced sidebands. In the same time, such shapes with a
well-defined finite duration do not suffer from the spurious effects arising
when truncating a pulse of infinite duration, e.g. Gaussian. We derive two
approximate analytic solutions which describe the ensuing quantum dynamics.
Both approximations assume that the field changes linearly at the beginning
and the end of the driving pulse, and adiabatically in between. The first
approximation matches the linear and adiabatic parts at an appropriate instant
of time and is expressed in terms of Weber’s parabolic cylinder functions. The
second, much simpler, approximation uses the asymptotics of the Weber function
in order to replace it by simpler functions, and some additional
transformations. Both approximations prove highly accurate when compared to
experimental data obtained with two of the IBM Quantum processors. Both the
greatly reduced power broadening and the greatly suppressed sidebands are
observed for all pulse shapes, in a nearly complete agreement between theory
and experiment.

##  I Introduction

The profound understanding and control of qubit dynamics is increasingly
important in the quest for a scalable fault-tolerant quantum computer, where
one of the key steps is achieving high-fidelity gate operations. Quantum gates
are implemented by electromagnetic pulses, the shape of which has a
significant impact on the gate properties. The pulse shape has been recognized
as an important control parameter in numerous applications of quantum control,
including STIRAP [1, 2, 3, 4], quantum memories [5, 6], quantum optics [7, 8,
9], spectroscopy and sensing [10, 11], ultrafast effects [12], optimization of
quantum gates [13, 14, 15, 16, 17, 18, 19, 20, 21], including suppression of
leakage [22, 23, 24], to name just a few.

The most popular pulse shapes used for qubit control are the rectangular and
Gaussian pulses and in superconducting qubits, the DRAG pulse [22]. The
rectangular pulse packs most pulse area in the shortest time duration but it
suffers from excitation sidebands and power broadening which may lead to
detrimental cross talk to other states. The Gaussian pulse assumes an infinite
duration and has to be truncated in practical use; the truncation can cause
some artefacts in its excitation profile and in order to avoid these one has
to move the truncation times further away from its center thereby making it
too long and the gate too slow. Similarly, truncation is needed in the case of
the commonly used DRAG pulses [22, 25]. They are the state-of-the-art when
working with superconducting qubits, where the anharmonicity (the detuning
from the transition to the third energy level) is only 5-10% of the pulse’s
Rabi frequency.

To this end, the analytic description of qubit dynamics is an invaluable tool
because, contrary to numerical optimizations, it provides qualitative
understanding of the role and significance of various experimental parameters.
In this paper, we consider a very important class of pulse shapes, which
involves smooth pulses of well-defined finite duration, which start and end at
zero with a linear slope. A typical representative of this class is the sine
pulse. Other examples include a variety of bell-shaped smooth pulses which are
truncated and their ensuing background is subtracted: Gaussian, hyperbolic-
secant and its powers, Lorentzian and its powers, etc.

This class of pulses have some clear advantages. Compared to rectangular
pulses, they are smooth, with no sudden amplitude jumps in the beginning and
the end, meaning no large sidebands in the excitation profile and less power
broadening. Compared to bell-shaped pulses (e.g. Gaussian, hyperbolic-secant,
Lorentzian, etc.), they are finite and there is no need to truncate their
wings (and compensate for the ensuing adverse effects). Compared to these
latter pulses, the present type of pulses offers significant “filling ratio”,
i.e. large pulse area confined in a short duration. For instance, for a given
pulse area, the sine pulse requires a pulse duration which is a factor of only
$\pi/2$ longer than that for a rectangular pulse, while mitigating the
drawbacks of the latter pulse. Therefore these smooth pulses of finite
duration can be considered as candidates for the “sweet spot” of qubit
control.

In order to describe and understand the qubit dynamics when driven by such
shapes, we need some analytic description. Unfortunately, none of the existing
exact or approximate methods are applicable to such pulse shapes. There are
several exactly soluble analytic models for the coherent interaction of a two-
state quantum system with an external field, e.g. the models of Rabi [26],
Rosen-Zener [27], Allen-Eberly-Hioe [28, 29], Demkov-Kunike [30], Landau-
Majorana-Stückelberg-Zener (LMSZ) [31, 32, 33, 34], Demkov [35], Carroll-Hioe
[36], etc. All these models, except for the ones with a rectangular pulse
shape (the Rabi model and the finite LMSZ model), assume an infinite pulse
duration, e.g. hyperbolic secant or exponential. In addition, there are
several approximate methods, such as the perturbation theory, the Magnus
approximation [37], the Davis-Pechukas approach [38], the Rosen-Zener
conjecture [27, 39, 40], etc., which assume different limitations and deliver
different accuracy. The first two of these work in the weak-field regime and
are not suitable for qubit operations. The beautiful and powerful Davis-
Pechukas approach demands analytic pulse shapes (implying infinite pulse
duration). The Rosen-Zener conjecture is an interesting but rarely used
proposition, which has never been rigourously proved.

Pulse shaping techniques have lately been found to provide new approaches to
old issues in quantum control. Lorentzian pulses have been demonstrated as a
treatment for power broadening, and other pulse shapes have also been found to
provide reduced power broadening, in comparison with the rectangular pulse
[42, 41]. Trying to balance between these extreme cases, the sinusoidal pulse
shape provides a valuable option for uses where the duration of the pulse is
important. They have been the object of many scientific works in the recent
years [43, 10, 44, 45, 46]. Unfortunately, these applications could not make
use of an analytic solution for the sinusoidal pulse shape because the sine
model is not exactly soluble.

To this end, in order to describe the qubit dynamics we derive here two highly
accurate analytic approximations which are especially designed (and
applicable) for such pulse shapes. They are based on the assumption that the
qubit dynamics in the beginning and the end of the pulse is described by the
rotated half-crossing LMSZ model, which is applicable for linearly changing
pulse shapes. In the vicinity of the pulse maximum the evolution is assumed to
be adiabatic. We derive two approximate formulae, one using the exact solution
for the rotated LMSZ model, which involves Weber’s parabolic cylinder
function, and another using approximations to it in terms of simpler
functions. The latter approximation turns out to be very simple and, to our
surprise, very accurate. We test these two approximations on a variety of
pulse shapes by comparing them to data acquired on one of the quantum
processors of IBM Quantum.

We note that some features of excitation produced by the sine pulse have been
considered in Ref. [46], namely the line width $\Delta_{\frac{1}{2}}$ of the
excitation profile. The approach to estimate the line width used the
nonanalyticities at the beginning and the end of the pulse. It predicted power
broadening with the square root of the peak Rabi frequency,
$\Delta_{\frac{1}{2}}\propto\Omega_{0}^{\frac{1}{2}}$, and an excitation
profile vanishing as $\Delta^{-4}$ for large detuning $\Delta$. The approach
is only valid for large detunings and large Rabi frequencies and fails to
correctly predict the entire line profile, e.g. at resonance and small
detunings. Here we provide a complete description of the excitation by such
pulses, which is valid for any values of the experimental parameters, as long
as the pulse shape obeys some restrictions.

The paper is organized as follows. Section II introduces the problem and the
main assumptions made, and Secs. III and IV present the two approaches for
derivation of the analytic approximations. Section V shows experimental data
and comparisons to the analytic approximations, Sec. VI discusses the
conditions of validity of the approximations, and Sec. VII summarizes the
conclusions.

##  II General

In this paper, we consider a class of driving pulses with constant detuning
and smooth temporal shapes, which (i) have a finite duration $T$, (ii) are
symmetric in time, (iii) start from zero and end to zero with a nonzero
derivative,

$\displaystyle\Omega(t)$
$\displaystyle=\left\\{\begin{array}[]{cc}\Omega_{0}f(t),&0\leqq t\leqq T,\\\
0,&\text{otherwise},\end{array}\right.$ (1c) $\displaystyle\Delta(t)$
$\displaystyle=\text{const},$ (1d)

where

$\displaystyle f(T-t)$ $\displaystyle=f(t),$ (2a) $\displaystyle f(0)$
$\displaystyle=f(T)=0,$ (2b) $\displaystyle f^{\prime}(0)$
$\displaystyle=-f^{\prime}(T)>0.$ (2c)

and (iv) their initial and final segments are approximately linear in time,
i.e.

$f^{\prime}(t)\sim\text{const for }t\ll T/2\text{ and }t-T\ll T/2.$ (3)

Figure 1:  Illustration of the assumptions used in the derivation of the split
LMSZ-adiabatic approximation which assumes rotated-LMSZ evolution at
$t\in[0,\lambda]\cup[1-\lambda,1]$ of the driving pulse and adiabatic
evolution for $t\in[\lambda,1-\lambda]$.

We shall derive two analytic approximations to the solution of the Schrödinger
equation for such pulse shapes in the most general case when only the above
properties are imposed and no other restriction on the specific shape are
assumed. In both approximations, we first separate the evolution into two
intervals: $[0,\frac{1}{2}T]$ and $[\frac{1}{2}T,T]$. Because the pulse shape
is assumed to be symmetric in time, it can be shown that if the propagator for
the first half-interval is $\mathbf{U}$, then the propagator for the second
half-interval is $\mathbf{U}^{T}$. Therefore, the overall propagator in the
entire interval is $\mathbf{U}^{T}\mathbf{U}$ and it suffices to consider the
evolution in the first half interval $[0,\frac{1}{2}T]$ only.

In the second step of both methods, we first rotate the basis at an angle
$\pi/4$, described by the matrix (which amounts to the Hadamard gate)

$\mathbf{G}=\tfrac{1}{\sqrt{2}}\left[\begin{array}[]{cc}-1&1\\\
1&1\end{array}\right],$ (4)

so that the Rabi frequency and the detuning in the initial Hamiltonian,

$\mathbf{H}(t)=\tfrac{1}{2}\left[\begin{array}[]{cc}-\Delta(t)&\Omega(t)\\\
\Omega(t)&\Delta(t)\end{array}\right],$ (5)

exchange their places in the rotated Hamiltonian,

$\widetilde{\mathbf{H}}(t)=\mathbf{G}\mathbf{H}(t)\mathbf{G}=\tfrac{1}{2}\left[\begin{array}[]{cc}-\Omega(t)&\Delta(t)\\\
\Delta(t)&\Omega(t)\end{array}\right].$ (6)

Note that $\mathbf{G}^{-1}=\mathbf{G}$ and hence $\mathbf{G}^{2}=\mathbf{1}$.
If the propagator for the original Hamiltonian is
$\mathbf{U}(t_{\rm{f}},t_{\rm{i}})$ then the propagator for the rotated
Hamiltonian reads

$\widetilde{\mathbf{U}}(t_{\rm{f}},t_{\rm{i}})=\mathbf{G}\mathbf{U}(t_{\rm{f}},t_{\rm{i}})\mathbf{G}.$
(7)

In this rotated picture, the problem becomes one with a constant coupling
$\Delta$ and a pulse-shaped detuning $\Omega(t)$. This allows us to model the
evolution as two LMSZ half-crossings at the beginning and at the end of the
pulse, and adiabatic evolution in between. Both assumptions — the linear
behavior in the two ends and the adiabatic evolution in between — impose some
restrictions on the experimental parameters which will be discussed in due
course.

In the first approximation, it is assumed that the rot-LMSZ part lasts in the
interval $[0,\lambda]$, and the adiabatic part is in the interval
$[\lambda,\frac{1}{2}T]$ (see Fig. 1), where $\lambda$ is a free fitting
parameter, which depends on the pulse shape. The rot-LMSZ propagator is
expressed in terms of Weber’s parabolic cylinder function, whereas the
adiabatic solution is given in terms of trigonometric functions and an
adiabatic phase.

The second approximation is more sophisticated as it intermixes the rot-LMSZ
and adiabatic solutions in the adiabatic interaction representation, uses the
so-called strong-coupling asymptotics of the Weber function and replaces the
rot-LMSZ dynamic phase with the adiabatic phase. The result is a much simpler
analytic formula, which is expressed in terms of elementary functions, the
argument of the Euler’s Gamma function, and the adiabatic phase. Moreover,
this much simpler analytic formula, which does not include any fitting
parameters, fits the exact solution equally well as the first approximation.

##  III First method: Split LMSZ-adiabatic approximation

In the first method, we separate the first half interval $[0,\frac{1}{2}T]$
into two further sub-intervals: $[0,\lambda]$ and $[\lambda,\frac{1}{2}T]$,
see Fig. 1, where $\lambda$ is a free fitting parameter. In the first sub-
interval $[0,\lambda]$, we approximate the solution by the rot-LMSZ propagator
$U_{\text{LMSZ}}$ due to the near linearity of the pulse shape near $t=0$.
Obviously, the smaller the parameter $\lambda$, the better the linear
assumption and the better the rot-LMSZ approximation. In the second sub-
interval $[\lambda,\frac{1}{2}T]$, we approximate the solution by the
adiabatic solution. This choice is justified if in this interval $\Omega(t)$
is sufficiently large (meaning that its maximum $\Omega_{0}$ is large enough).

The rot-LMSZ propagator in the first sub-interval $[0,\lambda]$ is already
available in the literature [47]; it reads

$\mathbf{U}_{\text{LMSZ}}=\left[\begin{array}[]{cc}\Re a-i\Im b&\Re b+i\Im
a\\\ -\Re b+i\Im a&\Re a+i\Im b\end{array}\right],$ (8)

with

$\displaystyle a$
$\displaystyle=\frac{2^{\frac{1}{2}i\delta^{2}}(1+e^{-\pi\delta^{2}})\Gamma\left(\frac{1}{2}+\frac{1}{2}{i\delta^{2}}\right)}{2\sqrt{\pi}}D_{-i\delta^{2}}(\alpha
e^{\frac{i\pi}{4}})$
$\displaystyle+\frac{(1-e^{-\pi\delta^{2}})\Gamma\left(1-\frac{1}{2}{i\delta^{2}}\right)}{2^{\frac{1}{2}i\delta^{2}}\sqrt{2\pi}}D_{i\delta^{2}-1}(\alpha
e^{-\frac{i\pi}{4}}),$ (9a) $\displaystyle b$
$\displaystyle=\frac{e^{-i\pi/4}2^{\frac{1}{2}i\delta^{2}}(1-e^{-\pi\delta^{2}})\Gamma(1+\frac{1}{2}{i\delta^{2}})}{\delta\sqrt{2\pi}}D_{-i\delta^{2}}(\alpha
e^{\frac{i\pi}{4}})$
$\displaystyle-\frac{e^{-i\pi/4}(1+e^{-\pi\delta^{2}})\delta\Gamma(\frac{1}{2}-\frac{1}{2}{i\delta^{2}})}{2^{1+\frac{1}{2}i\delta^{2}}\sqrt{\pi}}D_{i\delta^{2}-1}(\alpha
e^{-\frac{i\pi}{4}}),$ (9b)

where $D_{\nu}(z)$ is Weber’s parabolic cylinder function [48], $\Gamma(z)$ is
Euler’s gamma function [49], and

$\alpha=\beta\lambda,\quad\delta=\frac{\Delta}{2\beta},\quad\beta=\sqrt{\Omega_{0}f(\lambda)}.$
(10)

The adiabatic solution for the second sub-interval $[\lambda,\frac{1}{2}T]$
reads

$\mathbf{U}_{\text{adb}}=\left[\begin{array}[]{cc}c&d\\\
-d^{*}&c^{*}\end{array}\right],$ (11a) where $\displaystyle c$
$\displaystyle=e^{-\frac{1}{2}i\eta_{\lambda}}\sin\theta_{\rm{i}}\sin\theta_{\rm{f}}+e^{\frac{1}{2}i\eta_{\lambda}}\cos\theta_{\rm{i}}\cos\theta_{\rm{f}},$
(11b) $\displaystyle d$
$\displaystyle=e^{-\frac{1}{2}i\eta_{\lambda}}\cos\theta_{\rm{i}}\sin\theta_{\rm{f}}-e^{\frac{1}{2}i\eta_{\lambda}}\sin\theta_{\rm{i}}\cos\theta_{\rm{f}}.$
(11c)

Here the mixing angle $\theta(t)$ is defined by

$\cos 2\theta(t)=\frac{\Delta}{\Lambda(t)},\quad\sin
2\theta(t)=\frac{\Omega(t)}{\Lambda(t)},$ (12)

and $\theta_{\rm{i}}=\theta(t_{\rm{i}})=\theta(\lambda)$ and
$\theta_{\rm{f}}=\theta(t_{\rm{f}})=\theta(\frac{1}{2}T)$. The eigenfrequency
splitting $\Lambda(t)$ and the adiabatic phase $\eta_{\lambda}$ read

$\displaystyle\Lambda(t)$ $\displaystyle=\sqrt{\Delta^{2}+\Omega(t)^{2}},$
(13a) $\displaystyle\eta_{\lambda}$
$\displaystyle=\int_{\lambda}^{\frac{1}{2}T}\Lambda(t)\,dt.$ (13b)

Most quantities above — the LMSZ parameters $\alpha$, $\beta$, $\delta$, the
mixing angle $\theta_{\rm{i}}$, and the adiabatic phase $\eta$ — depend on the
value of the point $\lambda$, which separates the LMSZ and adiabatic
intervals. It looks plausible to choose $\lambda=\frac{1}{4}T$, but the
accuracy of the formula can be improved if it is left as a free fitting
parameter, which has a different value for each pulse shape.

The propagator for the half interval $[0,\frac{1}{2}T]$ is
$\mathbf{U}_{\text{adb}}\mathbf{U}_{\text{LMSZ}}$, and the propagator for the
full interval $[0,T]$ is

$\mathbf{U}=\mathbf{U}_{\text{LMSZ}}^{T}\mathbf{U}_{\text{adb}}^{T}\mathbf{U}_{\text{adb}}\mathbf{U}_{\text{LMSZ}}.$
(14)

The two adjacent adiabatic propagators can be combined into a single adiabatic
propagator, which describes the evolution in the time interval
$[\lambda,T-\lambda]$, which can be obtained from Eq. (11) by setting
$\theta_{\rm{f}}=\theta_{\rm{i}}$ and $\eta_{\lambda}\rightarrow
2\eta_{\lambda}$. The transition probability for the full interval $[0,T]$ can
be obtained from the overall propagator (14) as $P=|U_{12}|^{2}$; explicitly,

$\displaystyle P=$ $\displaystyle\left|\Re[(a^{2}-b^{2})\sin
2\theta_{\rm{i}}+2ab^{*}\cos 2\theta_{\rm{i}}]\sin\eta_{\lambda}\right.$
$\displaystyle+\left.\Im(a^{2}-b^{2})\cos\eta_{\lambda}\right|^{2}.$ (15)

A much simpler expression for the transition probability is obtained by the
second method in the next section.

##  IV Second method: Integrated LMSZ-adiabatic approximation

For the derivation of the second analytic formula we follow an approach
similar to the one in Appendix D of Ref. [50]. As noted above, it suffices to
consider the evolution in the first half interval $[0,\frac{1}{2}T]$. Now we
do not split this half interval further, as in the first approach, but treat
it as a whole. We parameterize the propagator for the rotated Hamiltonian (6)
in this interval as

$\widetilde{\mathbf{U}}(\tfrac{1}{2}T,0)=\left[\begin{array}[]{cc}u&v\\\
-v^{*}&u^{*}\end{array}\right].$ (16)

Then the overall propagator in the original (non-rotated) basis in the entire
interval $[0,T]$ is expressed as

$\mathbf{U}(T,0)=\mathbf{G}\widetilde{\mathbf{U}}(\tfrac{1}{2}T,0)^{T}\widetilde{\mathbf{U}}(\tfrac{1}{2}T,0)\mathbf{G}=\left[\begin{array}[]{cc}x&y\\\
-y^{*}&x^{*}\end{array}\right],$ (17)

with

$\displaystyle x$
$\displaystyle=\frac{(u^{*}+v^{*})^{2}+(u-v)^{2}}{2}=(u_{\rm{r}}-iv_{\rm{i}})^{2}+(v_{\rm{r}}-iu_{\rm{i}})^{2},$
(18a) $\displaystyle y$
$\displaystyle=\frac{u^{*2}-u^{2}+v^{2}-v^{*2}}{2}=-2i(u_{\rm{r}}u_{\rm{i}}-v_{\rm{r}}v_{\rm{i}}).$
(18b)

Here, in order to avoid confusion, we explicitly keep the arguments of the
propagator.

Next we transform the rotated basis to the adiabatic interaction picture. The
reason to use this picture is that, contrary to other bases, the propagator
elements (and especially their phases) in the original LMSZ model (i.e. with
infinite interaction duration) are convergent. We do this in two steps: first,
we transform the rotated basis to the adiabatic rotated basis, and second, we
transform the latter to the adiabatic interaction representation.

We begin with the rotated Hamiltonian (6) and transform it to the adiabatic
representation — the basis of its eigenvectors,

$|-\rangle=\left[\begin{array}[]{c}\cos\vartheta(t)\\\
-\sin\vartheta(t)\end{array}\right],\quad|+\rangle=\left[\begin{array}[]{c}\sin\vartheta(t)\\\
\cos\vartheta(t)\end{array}\right],$ (19)

where

$\sin 2\vartheta(t)=\frac{\Delta}{\Lambda(t)},\quad\cos
2\vartheta(t)=\frac{\Omega(t)}{\Lambda(t)},$ (20)

and the eigenvalue splitting $\Lambda(t)$ is defined in Eq. (13a). Note that
the mixing angles in the original basis $\theta$ and in the rotated basis
$\vartheta$ are related as $2\vartheta=\frac{1}{2}\pi-2\theta$, see Eqs. (12)
and (20). We form the transformation matrix composed of the eigenvectors (19),

$\widetilde{\mathbf{R}}(\vartheta(t))=\left[\begin{array}[]{cc}|-\rangle,&|+\rangle\end{array}\right].$
(21)

Note that
$\widetilde{\mathbf{R}}(\vartheta(t))^{-1}=\widetilde{\mathbf{R}}(\vartheta(t))^{T}=\widetilde{\mathbf{R}}(-\vartheta(t))$.
In the adiabatic rotated representation, the Hamiltonian reads
$\widetilde{\mathbf{H}}_{a}(t)=\widetilde{\mathbf{R}}(-\vartheta(t))\widetilde{\mathbf{H}}(t)\widetilde{\mathbf{R}}(\vartheta(t))$;
explicitly,

$\widetilde{\mathbf{H}}_{a}(t)=\tfrac{1}{2}\left[\begin{array}[]{cc}-\Lambda(t)&-2i\dot{\vartheta}(t)\\\
2i\dot{\vartheta}(t)&\Lambda(t)\end{array}\right],$ (22)

where the overdot means $d/dt$.

Next, we perform the transformation to the adiabatic interaction
representation via the transformation matrix

$\mathbf{F}(\eta(t))=\left[\begin{array}[]{cc}e^{i\eta(t)/2}&0\\\
0&e^{-i\eta(t)/2}\end{array}\right],$ (23)

with the adiabatic phase

$\eta(t)=\int_{t_{\rm{i}}}^{t}\Lambda(t^{\prime})dt^{\prime},$ (24)

which casts the Hamiltonian into the form $\widetilde{\mathbf{H}}_{\rm
ai}(t)=\mathbf{F}(-\eta(t))\widetilde{\mathbf{H}}_{a}(t)\mathbf{F}(\eta(t))$;
explicitly,

$\widetilde{\mathbf{H}}_{\rm
ai}(t)=\left[\begin{array}[]{cc}0&-i\dot{\vartheta}(t)e^{-i\eta(t)}\\\
i\dot{\vartheta}(t)e^{i\eta(t)}&0\end{array}\right].$ (25)

The propagator (16), corresponding to the Hamiltonian (6), is transformed in a
similar fashion and in the adiabatic interaction representation it reads

$\widetilde{\mathbf{U}}_{\rm
ai}(\tfrac{1}{2}T,0)=\mathbf{F}(-\eta_{\rm{f})})\widetilde{\mathbf{R}}(-\vartheta_{\rm{f}})\widetilde{\mathbf{U}}(\tfrac{1}{2}T,0)\widetilde{\mathbf{R}}(\vartheta_{\rm{i}})\mathbf{F}(\eta_{\rm{i}}),$
(26)

where $\eta_{\rm{i}}=\eta(0)=0$,
$\eta_{\rm{f}}=\eta(\tfrac{1}{2}T)\equiv\eta$,
$\vartheta_{\rm{i}}=\vartheta(0)=\frac{1}{4}\pi$,
$\vartheta_{\rm{f}}=\vartheta(\tfrac{1}{2}T)=\arctan(\Delta/\Omega(\frac{1}{2}T))/2$.
From here, we have

$\widetilde{\mathbf{U}}(\tfrac{1}{2}T,0)=\widetilde{\mathbf{R}}(\vartheta)\mathbf{F}(\eta)\widetilde{\mathbf{U}}_{\rm
ai}(\tfrac{1}{2}T,0)\widetilde{\mathbf{R}}(-\pi/4).$ (27)

Next, we make the central assumption of this method: the evolution in the
vicinity of the initial time $t=0$ follows the half-crossing LMSZ model. The
rationale behind this assumption is that, due to the nonzero derivative of the
pulse at the initial (and final) time, its departure from (and arrival to)
zero can be considered linear in time. Moreover, we assume that the essential
part of the dynamics occurs in the vicinity of the half-crossing (as
$t\rightarrow 0$) and very little changes away from it (as
$t\rightarrow\frac{1}{2}T$). The rationale behind this assumption is that the
nonadiabatic coupling $\dot{\vartheta}(t)$ is maximal as $t\rightarrow 0$,
because $\dot{\Omega}(t)$ is maximal there, whereas in the other end of the
half-interval ($t\rightarrow\frac{1}{2}T$) we have $\dot{\Omega}(t)\rightarrow
0$ and hence $\dot{\vartheta}(t)\rightarrow 0$. Moreover, it is well known
that the transition time in the adiabatic basis is much shorter than the
transition time in the original basis [51], which means that the LMSZ
transition can be assumed to be well localized in the vicinity of the point
$t=0$.

The half-LMSZ evolution is described by the propagator (Appendix A)

$\widetilde{\mathbf{U}}_{\text{LMSZ}}=\tfrac{1}{\sqrt{2}}\left[\begin{array}[]{cc}\sqrt{1+e^{-\pi\delta^{2}}}e^{i\chi_{1}}&\sqrt{1-e^{-\pi\delta^{2}}}e^{i\chi_{2}}\\\
-\sqrt{1-e^{-\pi\delta^{2}}}e^{-i\chi_{2}}&\sqrt{1+e^{-\pi\delta^{2}}}e^{-i\chi_{1}}\end{array}\right],$
(28)

with

$\displaystyle\chi_{1}$
$\displaystyle=\frac{\delta^{2}}{2}-\frac{\delta^{2}}{2}\ln\left(\frac{\delta^{2}}{2}\right)+\arg\Gamma\left(\frac{1+i\delta^{2}}{2}\right),$
(29a) $\displaystyle\chi_{2}$
$\displaystyle=\frac{\delta^{2}}{2}-\frac{\delta^{2}}{2}\ln\left(\frac{\delta^{2}}{2}\right)+\arg\Gamma\left(\frac{i\delta^{2}}{2}\right)+\frac{\pi}{4},$
(29b)

and

$\delta=\frac{\Delta}{2\sqrt{\pi\,\Omega\left(\frac{1}{4}T\right)}}.$ (30)

The approximation consists of replacing the actual propagator
$\widetilde{\mathbf{U}}_{\rm ai}(\frac{1}{2}T,0)$ in Eq. (27) by
$\widetilde{\mathbf{U}}_{\text{LMSZ}}$ of Eq. (28), while keeping the other
matrices in Eq. (27) intact, as they are related to the actual (nonlinearized)
pulse shape. Thereby we find for the full interval $[0,T]$

$\displaystyle\mathbf{U}(T,0)$
$\displaystyle=\mathbf{G}\widetilde{\mathbf{R}}(\vartheta_{\rm{i}})\widetilde{\mathbf{U}}_{\text{LMSZ}}^{T}\mathbf{F}(\eta)\widetilde{\mathbf{R}}(-\vartheta_{\rm{f}})$
$\displaystyle\times\widetilde{\mathbf{R}}(\vartheta_{\rm{f}})\mathbf{F}(\eta)\widetilde{\mathbf{U}}_{\text{LMSZ}}\widetilde{\mathbf{R}}(-\vartheta_{\rm{i}})\mathbf{G}$
$\displaystyle=\mathbf{G}\widetilde{\mathbf{R}}(\vartheta_{\rm{i}})\widetilde{\mathbf{U}}_{\text{LMSZ}}^{T}\mathbf{F}(2\eta)\widetilde{\mathbf{U}}_{\text{LMSZ}}\widetilde{\mathbf{R}}(-\vartheta_{\rm{i}})\mathbf{G},$
(31)

since $\mathbf{F}(\eta)\mathbf{F}(\eta)=\mathbf{F}(2\eta)$. The transition
probability is $P=|U_{12}(T,0)|^{2}$; explicitly,

$\displaystyle P$
$\displaystyle=\left[\frac{1+e^{-\pi\delta^{2}}}{2}\sin\left(\eta-2\chi_{1}\right)-\frac{1-e^{-\pi\delta^{2}}}{2}\sin\left(\eta-2\chi_{2}\right)\right]^{2}$
(32a)
$\displaystyle=\left[\sin\left(\eta-2\chi_{1}\right)-(1-e^{-\pi\delta^{2}})\cos\chi_{-}\sin\left(\eta-\chi_{+}\right)\right]^{2}$
(32b)
$\displaystyle=\left[\sin\chi_{-}\cos\left(\eta-\chi_{+}\right)-e^{-\pi\delta^{2}}\cos\chi_{-}\sin\left(\eta-\chi_{+}\right)\right]^{2}$
(32c)
$\displaystyle=\left(\sin^{2}\chi_{-}+e^{-2\pi\delta^{2}}\cos^{2}\chi_{-}\right)\sin^{2}\left(\eta-\chi_{+}-\nu\right),$
(32d)

where $\chi_{\pm}=\chi_{1}\pm\chi_{2}$ and

$\nu=\arctan(e^{\pi\delta^{2}}\tan\chi_{-}).$ (33)

These expressions can be useful in different contexts. Expression (32b) is
useful for small $\delta$ when the second term vanishes. Expression (32c) is
useful for large $\delta$ when its second term vanishes. In particular, on
resonance ($\delta=0$) we have $\chi_{1}=0$,
$\eta=A/2=\frac{1}{2}\int_{0}^{T}\Omega(t)dt$ and from Eq. (32b) we recover
the resonant solution, $P=\sin^{2}(A/2)$. The last expression (32d) gives
explicitly the amplitude (the first factor) and the phase (in the sin2) of the
Rabi oscillations.

The expressions for the phases $\chi_{1}$ and $\chi_{2}$ can be simplified in
the limits of small and large $\delta$. For small $|\delta|\ll 1$, we have

$\displaystyle\chi_{1}$
$\displaystyle\sim\frac{\delta^{2}}{2}\left[1-\gamma-\ln(2\delta^{2})\right],$
(34a) $\displaystyle\chi_{2}$
$\displaystyle\sim\frac{\delta^{2}}{2}\left[1-\gamma-\ln(\delta^{2}/2)\right]-\frac{\pi}{4},$
(34b) $\displaystyle\chi_{+}$
$\displaystyle\sim\delta^{2}\left[1-\gamma-\ln(\delta^{2})\right]-\frac{\pi}{4},$
(34c) $\displaystyle\chi_{-}$ $\displaystyle\sim\frac{\pi}{4}-\delta^{2}\ln
2,$ (34d)

where $\gamma\approx 0.577216\ldots$ is Euler’s constant.

For large $|\delta|\gg 1$, the asymptotic behavior is

$\displaystyle\chi_{1}$ $\displaystyle\sim\frac{1}{12\delta^{2}},$ (35a)
$\displaystyle\chi_{2}$ $\displaystyle\sim-\frac{1}{6\delta^{2}},$ (35b)
$\displaystyle\chi_{+}$ $\displaystyle\sim-\frac{1}{12\delta^{2}},$ (35c)
$\displaystyle\chi_{-}$ $\displaystyle\sim\frac{1}{4\delta^{2}}.$ (35d)

These expressions make it possible to estimate the large-$\delta$ behavior of
the transition probability. Because the second term in Eq. (32c) vanishes for
large $\delta$, the asymptotics is determined by the first term, which behaves
as

$P\sim\sin^{2}\chi_{-}\cos^{2}\left(\eta-\chi_{+}\right)\sim\frac{\cos^{2}\eta}{16\delta^{4}}.$
(36)

Hence the wings of the transition probability vanish as $1/\delta^{4}$ — much
faster than the wings for the rectangular pulse which vanish as
$1/\delta^{2}$.

Equation (32d) allows us to readily estimate the width of the excitation
profile, i.e. the value $\Delta_{\frac{1}{2}}$ at which
$P(\Delta)=\frac{1}{2}$. Equation (32d) describes Rabi oscillations with the
first factor being the amplitude and the second factor containing the
frequency of the oscillation through the adiabatic phase $\eta$. The first
factor in Eq. (32d) depends on $\delta$ only, whereas the second factor
depends also on the adiabatic phase $\eta$.

We can estimate the linewidth of the excitation profile — the transition
probability versus the detuning $\Delta$ — by setting the first term in Eq.
(32d) to $\frac{1}{2}$. The solution is $\delta_{\frac{1}{2}}\approx\pm 3.78$.
By recalling the definition of $\delta$ of Eq. (30), we find

$\Delta_{\frac{1}{2}}T\approx\pm 13.4\sqrt{\Omega_{0}T}.$ (37)

We conclude that the line width scales as
$\Delta_{\frac{1}{2}}\propto\sqrt{\Omega_{0}}$.

Furthermore, the availability of a relatively simple analytic formula allows
us to estimate the magnitude of the satellites — another drawback of the
rectangular pulse, which is greatly reduced here. For a rectangular pulse with
a temporal area of $\pi$ and width $T$, the first satellite occurs at
detunings $\Delta\approx\pm 8.42/T$ and has a magnitude of 11.64%. For
comparison, for a sine pulse with a temporal area of $\pi$, the first
satellite occurs at detunings $\Delta\approx\pm 11.1/T$ and has a magnitude of
just 1.48%, a reduction by a factor of 8. This is a consequence of the fact
that for large detunings, the transition probability decreases as
$1/\Delta^{2}$ for the rectangular pulse and as $1/\Delta^{4}$ for the sine
pulse [46], as well as for the other pulses of this class considered here, as
noted above. Indeed, this behavior can readily be deduced from Eq. (32d) and
the asymptotic behavior of $\chi_{-}$ in Eq. (35).

It is important to note that because the oscillation amplitude in Eq. (32d)
depends on the parameter $\delta$ only, so do the line width and the
magnitudes of the sideband peaks. The parameter $\delta$, as defined in Eq.
(30) depends on the detuning $\Delta$ and the maximum Rabi frequency
$\Omega_{0}$ but it does not depend on the particular pulse shape. The
implication is that the conclusions about the line width and the magnitude of
the first sideband apply to all pulse shapes, which start and end linearly in
time and change smoothly in between.

##  V Demonstration on Superconducting Qubits

###  V.1 Qubit Specs

We have conducted series of experiments on the qubit 0 of ibmq_quito, one of
the IBM Quantum Falcon Processors [52]. The latter consisted of 5 transmon
qubits, and was an open-access quantum processor. The exact date of the
demonstration was 27 April 2023. The parameters of qubit 0 of the ibmq_quito
system at the time of the demonstration were as follows: the qubit frequency
was $5.3007$ GHz, with anharmonicity $-0.3315$ GHz. The $T_{1}$ coherence time
was $62.24$ µs, the $T_{2}$ coherence time was $82.71$ µs and the readout
assignment error was $7.14\%$.

The comparison between the excitation landscapes of the sinusoidal and the
rectangular pulses was performed on a larger quantum system – ibm_kyoto. As of
taking the measurements on 8 December 2023, the $T_{1}$ and $T_{2}$ coherence
times were 305.22 and $58.05~{}\mu$s, while the readout assignment error was
$0.29$%.

The demonstration aims to examine the accuracy of the two derived methods in
the context of finite pulses with (nearly) linear side slopes in real context.
To this end, pulses with multiple envelopes (described in Table 1) were
applied to the qubit and the post-excitation transition probability was
measured and compared to the theoretical predictions.

####  V.1.1 Measurement Analytics: Fit Functions and Metrics

The fits are constructed using the expressions for the transition probability
found in Eq. (15) by using $P=\left|U_{12}\right|^{2}$ for the split LMSZ-
adiabatic model and Eq. (32a) for the integrated LMSZ-adiabatic model. In
addition, a linear correction
$P_{\text{final}}=\epsilon_{1}+\epsilon_{2}(2P-1)$ is applied to model the
leakage, dephasing, SPAM and readout errors. The observations are then fitted
with the expressions that follow from the two models by adjusting the free
parameters $\lambda$, $\epsilon_{1}$ and $\epsilon_{2}$, where
$0\leq\lambda\leq T/2$, and $\pm\lambda$ are the border points between the
rot-LMSZ evolution and the adiabatic evolution. The parameter $\lambda$ is
only relevant for the split LMSZ-adiabatic model.

The accuracy of the models is analyzed using two metrics: the mean absolute
error (MAE) and the standard deviation of the resonant frequency (SDRF). The
MAE measures the agreement of the data with the fit, while the SDRF shows the
error in determining the resonant frequency.

| Type of Error | MAE(E-3) | SDRF(kHz)
---|---|---|---
Pulse Shape | Rabi Freq. | Model | Split | Int. | Split | Int.
Sine | $f(t)=\sin{\left(\frac{\pi t}{T}\right)}$ | 5.5 | 6.0 | 33.6 | 41.7
Lorentzian | $f_{0}(t)=\left(1+\left(\frac{t}{\tau}\right)^{2}\right)^{-1}$ | 5.5 | 6.4 | 37.7 | 48.1
Lorentzian2 | $f_{0}(t)=\left(1+\left(\frac{t}{\tau}\right)^{2}\right)^{-2}$ | 5.9 | 7.6 | 34.7 | 55.1
Sech | $f_{0}(t)=\operatorname{sech}{\left(\frac{t}{\tau}\right)}$ | 6.2 | 6.1 | 40.9 | 45.1
Sech2 | $f_{0}(t)=\operatorname{sech}^{2}{\left(\frac{t}{\tau}\right)}$ | 5.7 | 6.0 | 37.5 | 42.0
Gaussian | $f_{0}(t)=\exp{\left(-\left(\frac{t}{\tau}\right)^{2}\right)}$ | 6.2 | 6.5 | 34.8 | 40.4
Table 1: This table presents the raw Rabi frequency envelope $f_{0}$ for each
pulse type before the background subtraction and the amplitude rescaling. For
the sine pulse the pulse envelope is presented in its final form. The mean
absolute errors (MAE) and standard deviations of the resonant frequency (SDRF)
are also shown for the 6 shapes, obtained with pulse duration equal to the
pulse width $\tau=42.67~{}$ns.

####  V.1.2 Pulses

In order to test the relevance of the two approaches presented in Secs. III
and IV, a set of 6 pulse shapes was selected: sine, Lorentzian, Lorentzian2,
sech, sech2 and Gaussian (see Table 1). The sine function has natural endings
and the pulse could be limited to the interval from $0$ to the duration $T$,
so the pulse has its endpoints at zero. In comparison, the model demands that
any infinite pulses are truncated at carefully chosen locations. We truncate
the shapes at times $0$ and $T$ so that the pulse is centered at
$t=\frac{1}{2}T$, and then we subtract the background and rescale the
amplitude.

All pulses that were used in this work, both to test the accuracy of the two
models and to demonstrate scenarios outside the scope of the models, were of
the same pulse width $\tau$. They included the pulse shapes in Table 1, as
well as the exponential shape
$f_{0}(t)=\exp{\left(-\left|t/\tau\right|\right)}$. Because of the equal pulse
widths, their transition line profiles owed the differences in their
characteristics mainly to the shape of the pulses. The two approximations
assume linearity in the initial and final segments of the pulse, as stated in
Sec. IV. This condition is met closely for the sine function, since the first-
order term in its Taylor series at $t=0$ and $t=T$ is linear in $t$. The other
5 pulses were also truncated at the same duration $T=\tau$ to analyze the
applicability of the models.

The tests used a pulse width of $\tau=42.67$ ns for all continuous pulses in
the study. For the sine pulse this value corresponds to half a period, where
the pulse is halted. The other models’ duration values were equal to their
pulse width $\tau$, so as to truncate close to the points of steepest
gradient. Note that the points of steepest gradient stand at the zeroes of the
second derivative of the pulse shape; hence at this point of time the pulse
shape is as linear as it gets. The area of the pulse envelopes is calibrated
prior to the tests to be equal to $\pi$. The Qiskit framework only reports the
value of the Rabi frequency in arbitrary units. Thus, the value in SI units
was calculated using the calibrated pulse area and the envelope.

Furthermore, a study of the limitations of the two models was made using two
pulse shapes: Lorentzian2, to demonstrate what happens when slopes are non-
linear and the linearity assumption is not fulfilled, and exponential, to
analyze what happens when the adiabatic assumption is broken in the central
part of the pulse. A duration value of $1.67\tau$ was used for the Lorentzian2
pulse such that the effects from increasing non-linearity could be compared
with the Lorentzian2 pulse with duration $\tau$. The exponential pulse profile
was demonstrated with a single duration $T=\tau$.

The amplitude parameter in Qiskit is used to control the Rabi frequency of the
driving field and allows the application of pulsed excitation on the qubit.
However, its relation to the Rabi frequency is not linear, and lags behind
with greater amplitude values. Therefore, a mapping is used to match the
amplitude values to the actual Rabi frequency of the field, using a fit with a
custom function.

###  V.2 Results and Discussion

Figure 2: (Color online) Sech (red), exponential (blue) and Lorentzian2
(brown) pulse envelopes with the same pulse width $\tau$. The dotted vertical
lines show the truncation line scaled for the pulses with equal pulse width
and duration, which is the case for most pulses used in this demonstration.

####  V.2.1 Sine pulse

Figure 3: (Color online) The observed (left) and numerical (right) excitation
landscapes (transition probability against detuning and peak Rabi frequency)
of rectangular (top) and sinusoidal (bottom) pulsed excitation enable us to
compare the transition linewidth power expansion of the two models.

The tests of the model applicability were preceded by a detailed demonstration
of its relevance. We begin with the sine pulse in its role of a prime example
of the type of pulses studied here. The Taylor decomposition of the sine
function around the $t=0$ point shows linear evolution in the first order,
with zero second order contribution, making it the perfect case studies.
Furthermore, the natural ending points at sine arguments $0$ and $\pi$ suggest
no need for post-processing (truncation, background subtraction and amplitude
rescaling), which is required for the other pulse shapes.

Figure 4: (Color online) Measurements of off-resonant Rabi oscillations of a
sine pulse, with the same width $\tau=96.0$ ns as in Fig. 3, at detuning
$\Delta=-46.2$ MHz (blue crosses) fitted with the analytical expression for
the sine model found using the integrated model (red dashed line), and with
the numerical solution (green dotted line).

The excitation generated by a sine pulse was measured and compared against
that of a rectangular pulse. Demonstrations of the excitation landscapes of
rectangular and sinusoidal pulses with widths of $96.0$ ns were performed on
IBM Quantum’s ibm_kyoto processor. The results are displayed in Fig. 3. We
identify the first 5 maxima of the transition probability (corresponding to
temporal pulse areas of $\pi$, $3\pi$, $\ldots,$ $9\pi$) in the excitation
landscapes. The figure shows that sinuisoidal pulsed excitation delivers a
more compact, less power-broadened transition than the rectangular pulse while
sustaining its finite span. The transition line width is proportional to the
square root of the driving amplitude for the sine pulse [see Eq. (37)],
compared to the linear dependence in the rectangular pulse. This signals a
quadratic decrease in power broadening effects.

The off-resonant Rabi oscillations of a sine pulse are shown in Fig. 4.
Generally, the plot indicates good agreement between the measurements on the
quantum hardware and the integrated model predictions. Nevertheless, minor
inconsistencies between them are present, especially around the first maximum.
It is evident that the experimental points consistently are delayed on the
left-hand side of this figure and then begin to overtake the theoretical
prediction toward the right-hand side. We attribute this minor discrepancy to
the non-linearity of the Qiskit amplitude parameter with respect to the Rabi
frequency (mentioned at the end of Sec. V.1.2), which introduces a non-
negligible error on the Rabi oscillations.

It is evident in Fig. 3 that the measured excitation landscapes (left column)
are slightly skewed to the right – in the responses to both the rectangular
and the sine pulse. We note that such asymmetry might be an indication of
leakage. Indeed, while the closed qubit dynamics is symmetric to the detuning,
adding more states to the qubit makes excitation asymmetric versus the
detuning, which can also be explained in terms of an induced ac Stark shift
due to the presence of higher levels.

Figure 5: (Color online) The measured transition line profile of the
sinusoidal pulse envelope ($T=42.67$ ns) is plotted (blue hexagons) along with
fits based on the split approximation (in red), and the integrated
approximation (in green). The residuals of the two models are shown in the
thin axes below the plot.  Figure 6: (Color online) The measured transition
line profiles (blue hexagons) is plotted along with fits based on the
integrated approximation (in green) for 6 pulse shapes found in Table 1 with
$\tau=T=42.67$ ns. Top row: (left to right) Sine, Lorentzian, Lorentzian2;
Bottom row: Sech, Sech2 and Gaussian. The residuals of the two models are
shown in the thin axes below each plot.

The two approximations derived in this work were tested against the transition
line profile of a sinusoidal pulse with Rabi frequency $\Omega_{0}=115.7$ MHz
and pulse width $42.67$ ns. The transition line profile acquired on the
ibmq_quito system is presented in Fig. 5, along with fits based on the two
models. A brief analysis of the residuals at the bottom of Fig. 5 reveals
general agreement between the two approximate models. Furthermore, they
reflect the behavior of the data, following its pattern closely even in the
neighborhood of the sensitive satellite peaks at $\Delta\approx\pm 40$ MHz.
The MAE values of the two fits set a standard against which the compatibility
of the models with other candidate pulses can be evaluated. The two models
deliver similar accuracy when applied to the sine model. The integrated model
results in a 9% increase of the MAE and 24% increase of the SDRF when compared
to the split method. Because the two approximations deliver comparable
results, despite their very different appearances, we shall use hereafter the
simpler, integrated approximation only.

####  V.2.2 Other pulse shapes of finite duration

Similar demonstrations on a total of six pulse shapes (incl. the sine pulse)
are presented in Fig. 6, and their performance indicators are listed in Table
1. In order to satisfy the assumptions of the analytic models, the continuous
waveform shapes were symmetrically truncated on the steep segments of their
slopes, as shown in Fig. 2. The resulting additive background was subtracted,
so that the pulses begin and end at zero Rabi frequency. The measured
transition probability for the six pulses was then fit using the integrated
model. The data for each of the pulses can be examined in Fig. 6, where the
corresponding residuals of the fits are shown underneath each of the plots.
The examination of the transition line profiles of the 6 pulses reveals that
the accuracy of the approximation is sufficient when the two principal
assumptions of the models are satisfied by the pulse shape. The central
assumptions of the models are outlined in Secs. II, III and IV: linearity of
the slopes at the beginning and end of the pulse, and adiabaticity of the
pulse around its peak. It is quite remarkable that the plots look very
similar, including the line widths and the small sidebands, which confirms the
assertion that these are governed by the parameter $\delta$ of Eq. (30).

##  VI Validity of the model

The range of pulse shapes where the two models are applicable was analyzed
with the help of the Lorentzian2 and the exponential shapes. They both violate
the initial assumptions of the models, albeit in a different manner. The
Lorentzian2 pulse’s initial and final slopes can be made less linear by moving
the truncation point away from the steepest decent of the shape, thereby
introducing nonlinearities to it. In so doing both $\dot{\vartheta}(t)$
decreases and $\ddot{\vartheta}(t)$ increases, thereby rendering the linearity
assumption break down. The exponential shape violates the other condition:
that the pulse is nearly adiabatic in the vicinity of the central pulse
summit.

Figure 7: (Color online) The magnitudes of the first and second derivatives of
a Lorentzian2 pulse are shown and the interval where the pulse is approx.
linear is highlighted in red.

###  VI.1 Breaking the linearity condition

The suitability of the models for a given pulse shape can be estimated before
experiment, see Fig. 7. The linearity requirement can be estimated using
different metrics and evaluated against a threshold. Likewise, the second
derivative of its Rabi frequency $\Omega(t)$ in the vicinity of the truncation
point can be compared against an upper limit on its magnitude to check the
appropriateness of the model beforehand. In general, the most optimal
truncation point is in the vicinity of the inflection point where both the
second derivative is zero and the first derivative is maximal in magnitude.

Figure 8: (Color online) The measured transition line profiles of the
Lorentzian2 (top) and exponential (bottom) pulse envelopes with durations –
$71.11$ ns and $42.67$ ns respectively – are plotted (blue hexagons) along
with fits based on the split (in black), and the integrated method (in red).

However, displacing the truncation points would immediately be reflected in
the error of the fit, because of the non-linearity that would be present close
to the ends of the pulse. The Lorentzian2 pulse has a steep slope, which makes
it highly susceptible to movement of the truncation points. This is evident
from Fig. 8 where the Lorentzian2 pulse from Fig. 6 was truncated further from
its inflection point, and its duration is $71.11$ ns, or 1.67$\tau$, compared
to $42.67$ ns at the inflection point (see Fig. 6). Figure 8 exhibits some
discrepancies between theory and experiment both in the central part of the
profile and in the predicted but unobserved sidebands. These issues are
reflected in the MAE and SDRF of the fit, which increase in comparison with
the shorter-duration Lorentzian2 pulse on Fig. 6. For the longer Lorentzian2
pulse (Fig. 8), the split and integrated models have MAE’s of $15.7\times
10^{-3}$ and $17.8\times 10^{-3}$ and SDRF’s of $87.3$ and $99.4$ kHz
respectively, which signal a three-fold increase of the error, compared to the
$T=42.67$ ns Lorentzian2 pulse in Fig. 6.

These values are consistent with the truncation location analysis, as shown in
Fig. 7. It is evident that the second derivative of the Lorentzian2 pulse is
smaller than its first derivative for durations $T\in(0.7\tau,\tau)$, where
the ends lie in the vicinity of the inflection points. Notably, the integrated
model is comparable in predicting the behaviour of the Lorentzian2 pulse to
the split model. It is likely that this is because the adiabatic phase (see
Eq. (24)) is taken exactly, where the split model approximates it as that of a
linear coupling in the time interval $[0,\lambda]$. Nevertheless, the
assumption of slope linearity is still applied in the integrated model to find
the propagator in Eq. (28).

###  VI.2 Breaking the adiabatic condition

The adiabatic condition can similarly be calculated for the corresponding
pulse shape. However, one should take into account that the location where the
adiabatic condition is most unstable depends on the exact shape of the pulse.
For example, the exponential pulse, shown in Fig. 2, has a discontinuity in
the derivative exactly at its tip. This suggest that the point of highest non-
adiabaticity is its central point. In contrast, pulses with smooth peaks are
usually least adiabatic on their slopes, close to the inflection points, where
their first derivatives have high magnitude values. In the limiting case of
the exponential model, presented in the bottom plot of Fig. 8, the apparent
non-adiabatic behaviour at the central point significantly disrupts the
applicability of the models. It is the outlier in this demonstration.
Regardless of the highly linear slopes at the ends of the exponential pulse,
the non-adiabatic behavior at its summit ruins the fits. The MAE and SDRF
errors exceed these of the other pulses by factors between 5 and 8. To this
end, we note that as far as the exponential model is concerned, one can derive
an analytic approximation by adapting the derivation of the Demkov model [35]
for a finite-duration pulse.

##  VII Conclusions

In this paper we have presented comprehensive theoretical description and
experimental demonstration of the dynamics of a qubit driven by a pulse of
finite duration with linearly ending wings. Such pulse shapes, the most
prominent member of which is the sine pulse between two of its nodes, hold
some advantages over both the rectangular pulse and the truncated pulses of
otherwise infinite duration, such as Gaussian. Compared to the former, the
present pulse shape exhibit much less power broadening and an order of
magnitude smaller sidebands. Compared to the latter, the present pulse shape
has a well defined finite duration and avoids truncation artefacts, e.g.
truncation-induced power broadening.

We have derived two analytic approximations in order to describe the
excitation dynamics. Both of them make use of a rotated picture wherein the
problem resembles two LMSZ half-crossings separated by smooth adiabatic
evolution. One of these, the split approximation, splits the evolution into a
half crossing, adiabatic evolution and another half crossing and gives the
solution in terms of Weber’s parabolic cylinder function. The other,
integrated approximation, uses the asymptotics of the Weber’s function and
integrates the LMSZ and adiabatic evolution within the adiabatic interaction
representation, where the LMSZ dynamics is localized in the vicinity of the
half crossings. The integrated approximation delivers a much simple analytic
formula which allows one to draw important conclusions about the excitation
profile features. Among these, we explicitly find square-root power
broadening, $\Delta_{\frac{1}{2}}\propto\Omega_{0}^{\frac{1}{2}}$, the
$\Delta^{-4}$ law for the amplitude of the excitation profile, and an order of
magnitude smaller sidebands compared to the rectangular pulse. Moreover, the
integrated approximation shows that the transition probability depends
primarily on the slope of the pulse shape in the turn-on and turn-off times
and the adiabatic phase over its course, but not so much on the specific pulse
shape.

By comparing the two approximations to the experimental data acquired on two
of IBM Quantum processors we have found an excellent agreement between theory
and experiment in all cases when the linear and adiabatic conditions have been
fulfilled. We have exploited 6 different pulse shapes which show very similar
excitation profiles, thereby verifying the analytic findings that, as noted
above, the specific pulse shape is of no great significance as long as the
pulse shape turns on and off linearly and lacks sharp features in between. We
have also studied pulse shapes which violate either the linearity or the
adiabaticity conditions in order to find out the limitations of the analytic
approximations.

###### Acknowledgements.

We gratefully acknowledge the Karoll Knowledge Foundation for providing
financial support to Ivo S. Mihov during the preparation of this manuscript.
Their contribution was invaluable in encouraging the author’s ongoing
scientific endeavours. This research is supported by the Bulgarian national
plan for recovery and resilience, Contract No. BG-RRP-2.004-0008-C01 (SUMMIT),
Project No. 3.1.4 and by the European Union’s Horizon Europe research and
innovation program under Grant Agreement No. 101046968 (BRISQ). We acknowledge
the use of IBM Quantum services for this work. The views expressed are those
of the authors, and do not reflect the official policy or position of IBM or
the IBM Quantum team.

##  Appendix A Half LMSZ propagator in the adiabatic interaction
representation

For the LMSZ model, the propagator reads

$\left[\begin{array}[]{cc}a&b\\\ -b^{*}&a^{*}\end{array}\right]$ (38)

where the Cayley-Klein parameters are given by Eq. (9). We will use the so-
called strong-coupling asymptotics of Weber’s parabolic cylinder functions
($\alpha^{2}+\delta^{2}\gg 1$),

$\displaystyle D_{-i\delta^{2}}(\alpha e^{\frac{i\pi}{4}})$
$\displaystyle\sim\cos\zeta\exp\left(\frac{\pi\delta^{2}}{4}-i\eta^{\prime}\right),$
(39a) $\displaystyle D_{i\delta^{2}-1}(\alpha e^{-\frac{i\pi}{4}})$
$\displaystyle\sim\frac{\sin\zeta}{\delta}\exp\left(\frac{\pi\delta^{2}}{4}+\frac{i\pi}{4}+i\eta^{\prime}\right),$
(39b)

where

$\zeta=\frac{1}{2}\arctan\frac{2\delta}{\alpha}.$ (40)

and

$\eta^{\prime}=\frac{\alpha}{2}\sqrt{\alpha^{2}+4\delta^{2}}+2\delta^{2}\ln\left(\frac{\sqrt{\alpha^{2}+4\delta^{2}}+\alpha}{2}\right)-\delta^{2}\ln\delta^{2}.$
(41)

We note that $\eta^{\prime}$ is exactly the adiabatic phase for this model,

$\eta^{\prime}=\int_{0}^{\frac{1}{2}T}\sqrt{\alpha^{2}t^{2}+4\delta^{2}}\,dt.$
(42)

By using these asymptotics we find

$\displaystyle a$
$\displaystyle=e^{i\xi_{1}}\sqrt{\frac{1+e^{-\pi\delta^{2}}}{2}}\cos\zeta+e^{-i\xi_{2}}\sqrt{\frac{1-e^{-\pi\delta^{2}}}{2}}\sin\zeta\,b$
$\displaystyle=e^{i\xi_{2}}\sqrt{\frac{1-e^{-\pi\delta^{2}}}{2}}\cos\zeta\,-e^{-i\xi_{1}}\sqrt{\frac{1+e^{-\pi\delta^{2}}}{2}}\sin\zeta,$

where

$\xi_{1}=\chi_{1}-\frac{\eta^{\prime}}{2},\quad\xi_{2}=\chi_{2}-\frac{\eta^{\prime}}{2},$
(44)

where $\chi_{1}$ and $\chi_{2}$ are defined by Eqs. (29). We can write the
propagator of Eq. (38) with $a$ and $b$ from Eqs. (43) as (accounting for
$t_{\rm{i}}=0$ and $t_{\rm{f}}=T/2$)

$\mathbf{U}=\mathbf{R}(-\zeta)\mathbf{U}_{\text{adb}}\mathbf{U}_{\text{LMSZ}},$
(45)

where

$\mathbf{R}(-\zeta)=\left[\begin{array}[]{cc}\cos\zeta&-\sin\zeta\\\
\sin\zeta&\cos\zeta\end{array}\right],$ (46)
$\mathbf{U}_{\text{adb}}=\left[\begin{array}[]{cc}e^{-\frac{1}{2}i\eta^{\prime}}&0\\\
0&e^{\frac{1}{2}i\eta^{\prime}}\end{array}\right],$ (47)
$\mathbf{U}_{\text{LMSZ}}=\left[\begin{array}[]{cc}q&s\\\
-s^{*}&q^{*}\end{array}\right],$ (48)

with

$q=\sqrt{\frac{1+e^{-\pi\delta^{2}}}{2}}e^{i\chi_{1}},\quad
s=\sqrt{\frac{1-e^{-\pi\delta^{2}}}{2}}e^{i\chi_{2}}.$ (49)

Equation (48) represents the half-LMSZ propagator in the adiabatic interaction
representation.

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