# Experimental demonstration of robustness under scaling errors for
superadiabatic population transfer in a superconducting circuit

Shruti Dogra<EMAIL_ADDRESS>QTF Centre of Excellence, Department of
Applied Physics, Aalto University School of Science, P.O. Box 15100, FI-00076
AALTO, Finland  Antti Vepsäläinen  QTF Centre of Excellence, Department of
Applied Physics, Aalto University School of Science, P.O. Box 15100, FI-00076
AALTO, Finland  Gheorghe Sorin Paraoanu<EMAIL_ADDRESS>QTF Centre of
Excellence, Department of Applied Physics, Aalto University School of Science,
P.O. Box 15100, FI-00076 AALTO, Finland

###### Abstract

We study experimentally and theoretically the transfer of population between
the ground state and the second excited state in a transmon circuit by the use
of superadiabatic stimulated Raman adiabatic passage (saSTIRAP). We show that
the transfer is remarkably resilient against variations in the amplitudes of
the pulses (scaling errors), thus demostrating that the superadiabatic process
inherits certain robustness features from the adiabatic one. In particular, we
put in evidence a new plateau that appears at high values of the
counterdiabatic pulse strength, which goes beyond the usual framework of
saSTIRAP.

##  I Introduction

In recent years, shortcuts to adiabaticity have emerged as a powerful
technique of quantum control. While in the past both adiabatic and Rabi types
of control have been of tremendous utility, modern quantum control based on
shortcuts to adiabaticity aims at combining the advantageous features of both
of these techniques. Specifically, adiabatic processes are known to be
insensitive to small errors in the shape of the pulses, but their time of
operation is slow, limited by the adiabatic theorem. On the other hand,
control methods based on Rabi oscillations, as typically used for quantum
gates, are fast but needs precise timings and pulse-shape control. Thus,
processes that are both fast and resilient against variations of the pulse
parameters are of significant interest for the field of quantum information
processing. Here we discuss one such process, the superadiabatic correction of
STIRAP (stimulated Raman adibatic passage) and observe its robustness features
in an experiment of population transfer in a transmon qubit.

Adiabatic control methods based on the stimulated Raman adiabatic passage
(STIRAP) have been widely studied and integrated into various experimental
schemes [1, 7]. Recently, a lot of interest has been raised by the concept of
superadiabatic or transitionless process, introduced by Demirplak and Rice [2,
3] and by Berry [4]. Superadiabatic processes belong to a more general class
of protocols generically referred to as shortcuts to adiabaticity. When
applied to STIRAP, this results in the so-called saSTIRAP protocol, where a
counteradiabatic pulse is applied in addition to the usual pump and Stokes
STIRAP pulses [5, 6].

Due to the fact that experiments are subjected to various sources of errors
and imperfections, a key task is to characterize the robustness of these
protocols. For any practical implementation, we would need to understand if
the generic features of a process typically designed for ideal operation
remain valid in realistic conditions. For STIRAP, the effects of noise have
been studied already theoretically as well as on some experimental platforms.
For example, in superconducting circuits the reduction of population transfer
due to broadband colored noise was discussed in Ref. [8, 9]. In Ref. [10], it
was shown that saSTIRAP works well in the noisy environment consisting of
dissipation and Ornstein-Uhlenbeck dephasing. It was found that the
performance of shortcuts to adiabaticity decreases with the correlation time
of the Ornstein-Uhlenbeck noise [14], see also [15]. In the context of
shortcuts to adiabaticity using the Lewis-Riesenfeld method, various noises
(white noise, Ornstein-Uhlembeck noise, flicker noise, constant error) in the
spring constant of the trap have been studied in Ref. [11, 12]. Other errors,
such as the accordion noise in the wave vector of the trap, and errors in the
amplitude and phase of the trap have been addressed systematically in Ref.
[13].

For superadiabatic processes, which is our main interest in this work, several
theoretical studies have been performed. Already in 2005, Demirplak and Rice
have addressed the sensitivity of the counterdiabatic technique to errors such
as pulse width, location of the peak, and intensity, demonstrating that a
window of errors exist where high-fidelity transfer can be maintained [16]. In
the case of two-level systems, the robustness of several driving schemes
against off-resonance effects, as well as the tradeoff with respect to the
speed of the process, has been analyzed recently in Ref. [17]. The effect of
dephasing for superadiabatic STIRAP in three-level system has been considered
in [18]. In Ref. [19] transfer protocols in a system of coupled 1/2 spins have
been analyzed, where the systematic errors due to shifts in the magnetic field
and due to the Dzyalozynskii-Moryia interaction between the two spins are
treated perturbatively.

In the present work we study experimentally and theoretically the robustness
of sa-STIRAP in a three-level system with respect to scaling errors. For a
three-level system, the superadiabatic STIRAP (saSTIRAP) is characterized by
the simultaneous application of three pulses, the pump and Stokes pulses of
STIRAP (coupling the levels $|0\rangle-|1\rangle$ and $|1\rangle-|2\rangle$
respectively) and the counterdiabatic pulse coupling into the
$|0\rangle-|2\rangle$ transition. These three pulse drive the three-level
transmon concurrently in a loop (also called $\Delta$-driving). Scaling errors
are errors which come into play due to technical limitations, causing
miscalibrations during the experimental implementation. For a given pulse
shape (say a Gaussian), the errors can originate from the miscalibration of
the pulse-amplitude, pulse-width, phase and relative placement of the pulse in
the pulse sequence. Robustness under scaling has also been discussed
theoretically in Ref. [20], where a cost functional has been introduced while
observing random scaling with STIRAP pulses, and in Ref. [21], where optimal
STIRAP population transfer was observed as a function of the relative delay
between the pump and Stokes pulses.

Here we analyse the robustness of saSTIRAP in wide ranges of experimental
parameters, providing a systematic characterization of robustness under
scaling errors. We also analyse the imperfections related to the
counterdiabatic term in detail. We first present a method of characterizing
this term based on the Wigner-Ville spectrum. Then we focus on the phase of
the drive, and the saSTIRAP performance resulting from the use of two-photon
resonant drive as compared to the direct coupling between the initial and the
target states. We characterize the experimentally-achieved quantum speed limit
during the state evolution in STIRAP and saSTIRAP based on the Bures distance.
Next, we observe the formation of plateaus of nearly-constant populations in
the final state as a function of STIRAP and counterdiabatic pulse areas. This
demonstrates that saSTIRAP is largely insensitive to scaling errors.
Furthermore, we extend our study to counterdiabatic pulsea areas significantly
larger than the superadiabatic value of $\pi$. Suprisingly, we observe the
existence of regions where population transfer still occurs with high
efficiency.

##  II Experimental setup

Figure 1: a) The three-level system is driven by two microwave fields applied
resonantly to the 0-1 and 1-2 transitions and an additional field applied off-
resonantly with respect to state $|1\rangle$. b) Simplified schematic of
experiment, showing the transmon device coupled to a readout resonator and
placed at the mixing chamber in a dilution refrigerator, as well as the
electronic blocks used to control the system.

The experiments are performed on the three lowest energy levels ($|0\rangle$,
$|1\rangle$, $|2\rangle$) of an artificial atom (see Fig. 1) constituted by a
pair of Josephson junctions with transition frequencies
$\omega_{01}=2\pi\times 7.395$ GHz and $\omega_{12}=2\pi\times 7.099$ GHz,
with anharmonicity $2\Delta=(\omega_{01}-\omega_{12})=2\pi\times 296$ MHz.
Here, we consider Gaussian shaped pulsed fields:
$\Omega_{01}(t)=\Omega_{01}e^{-t^{2}/2\sigma^{2}}$ and
$\Omega_{12}(t)=\Omega_{12}e^{-(t-t_{s})^{2}/2\sigma^{2}}$, which couple with
transitions $|0\rangle-|1\rangle$ and $|1\rangle-|2\rangle$ respectively.
These Gaussians have the same values of standard deviation $\sigma$ and are
being truncated at $\pm n\sigma$. The counter-intuitive STIRAP pulse sequence
requires the drive $\Omega_{12}(t)$ to set in before the drive
$\Omega_{01}(t)$ with an overlap for time, $t_{s}=-\kappa\sigma$. The
counterdiabatic drive $\Omega_{02}(t)=2\dot{\Theta}$, where
$\Theta=\tan^{-1}(\Omega_{01}(t)/\Omega_{12}(t))$ [24]. In two-photon
resonance excitation, amplitudes of the effective couplings
$\tilde{\Omega}_{12}=\sqrt{2}\tilde{\Omega}_{01}$ and phases of the drives,
$\Phi=\phi_{01}+\phi_{12}+\phi_{20}=-\pi/2$.

In the rotating wave approximation applied to the transmon circuit,

$\displaystyle H_{01}(t)$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\Omega_{01}(t)e^{i\phi_{01}}|0\rangle\langle
1|+h.c.,$ (1) $\displaystyle H_{12}(t)$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\Omega_{12}(t)e^{i\phi_{12}}|1\rangle\langle
2|+h.c.,$ (2)

The 0-2 coupling is

$\displaystyle\tilde{H}_{\rm 2ph}(t)$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\left[\tilde{\Omega}_{01}(t)e^{-i\Delta
t+i\tilde{\varphi}}|0\rangle\langle 1|+\tilde{\Omega}_{12}(t)e^{+i\Delta
t+i\tilde{\varphi}}|1\rangle\langle 2|\right]+h.c.$ (3)

resulting in a 0-2 Rabi frequency
$\Omega_{02}=\tilde{\Omega}_{01}\tilde{\Omega}_{12}/(2\Delta)$ and phase
$\phi_{02}=2\tilde{\varphi}+\pi$,

$\displaystyle H_{02}(t)$ $\displaystyle=$
$\displaystyle-\frac{\hbar\tilde{\Omega}_{01}\tilde{\Omega}_{12}}{4\Delta}e^{2i\tilde{\varphi}}|0\rangle\langle
2|+h.c.$ (4)

The evolution of the system can be formulated as a dynamical map with trace-
class generator $\mathbb{L}$,

$\dot{\rho}(t)=\mathbb{L}[\rho(t)],$ (5)

where $\mathbb{L}[\rho(t)]=(i/\hbar)[\rho(t),H(t)]+\mathcal{L}[\rho(t)]$, with
$\cal{L}$ the Lindblad superoperator,

${\cal L}[\rho(t)]=\sum_{{\rm c}}\Gamma_{{\rm c}}\left(\Lambda_{{\rm
c}}\rho(t)\Lambda_{{\rm c}}^{{\dagger}}-\frac{1}{2}\\{\Lambda_{{\rm
c}}^{{\dagger}}\Lambda_{{\rm c}},\rho(t)\\}\right),$ (6)

where ${\rm c}$ is an index denoting the dissipation channel with Lindblad
operator $\Lambda_{c}$. For this sample, the dissipation results predominantly
from relaxation processes, namely two decay channels are present: the decay
from level 2 to level 1 with relaxation rate $\Gamma_{21}$ and
$\Lambda_{21}=|2\rangle\langle 1|$ and the decay rate from level 1 to the
ground state, with $\Lambda_{10}=|1\rangle\langle 0|$. We get $\Gamma_{10}=$
5.0 MHz and $\Gamma_{21}=$ 7.0 MHz, with dephasing times being dominated by
the energy relaxation for this sample. For a three-level system [25, 26], the
Lindbladian is ${\cal L}[\rho]=-\Gamma_{21}\rho_{22}|2\rangle\langle
2|-(\Gamma_{10}\rho_{11}-\Gamma_{21}\rho_{22})|1\rangle\langle
1|+\Gamma_{10}\rho_{11}|0\rangle\langle 0|$.

##  III Numerical analysis and optimization of drive parameters

Superadiabatic STIRAP is driven jointly by the drives $H_{01}(t)$,
$H_{12}(t)$, and $\tilde{H}_{\rm 2ph}(t)$. Thus, a precise saSTIRAP
implementation involves a balance between the choice of several parameters. In
the following subsections, we analyse the robustness of saSTIRAP with respect
to each of these parameters with wide ranges of experimentally feasible
values. It is noteworthy that saSTIRAP is found to display remarkable
resilience against variation in these parameter values. We perform a
quantitative analysis of the saSTIRAP protocol using the population
transferred to the second excited state $p_{2}$ as the figure of merit.

###  III.1 Wigner-Ville analysis of the counterdiabatic pulse

For such non-stationary noises the Wiener-Kinchin theorem, which connects the
correlation function with the power spectrum, is not valid. In this case, an
appropriate charaterization of the spectral content of the correlations is
provided by the Wigner-Ville spectrum. For a time-dependent variable $x(t)$
with aging correlation function $\langle x(t+\tau/2)x(t-\tau/2)\rangle$ we
define the Wigner-Ville spectrum

$S_{WV}(\omega,t)=\int_{-\infty}^{\infty}e^{-i\omega\tau}\langle
x(t+\tau/2)x(t-\tau/2)\rangle d\tau$ (7)

For stationary systems, one recovers the usual definition of power spectral
density as $S(\omega)=S_{WV}=(\omega,0)$. The instantaneous power at any
moment $t$ can be obtained by summing the Wigner-Ville spectrum over all
infinitesimal bandwidths

$\langle x^{2}(t)\rangle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega
S_{WV}(\omega,t)$ (8)

Figure 2: (a) Contour plot for the Wigner-Ville distribution Eqs. (10) as a
function of frequency and time is plotted with continuous lines along with the
approximated dotted lines for $\kappa=-1.5$, $\sigma=30$ ns. (b) Surface maps
of the Wigner-Ville distribution depicting self-correlations of the two-photon
resonant drive with $\phi_{20}=-\pi/2$ (c) Zero frequency spectra as a
function of time (d) Normalized full width at half maxima (FWHM) of
$S_{WV}^{(02)}(0,t)$ and $\omega_{02}(t)$.

In the case of saSTIRAP, all ac-Stark energy shifts are related to the same
function, the two-photon coupling. Thus, all cross-correlations bewteen noises
become self-correlations of the effective 0 - 2 coupling

$\Omega_{02}(t)=-\frac{t_{s}}{\sigma^{2}}\frac{1}{\cosh\left[-\frac{t_{s}}{\sigma^{2}}\left(t-\frac{t_{s}}{2}\right)\right]}$
(9)

Thus we can obtain the Wigner-Ville spectrum

$\displaystyle S_{WV}^{(02)}(\omega,t)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}e^{-i\omega\tau}\langle\Omega_{02}(t+\tau/2)\Omega_{02}(t-\tau/2)\rangle
d\tau,$ (10) $\displaystyle=$
$\displaystyle\frac{2|ts|}{\sigma^{2}}\int_{-\infty}^{\infty}\frac{e^{-i\frac{\sigma^{2}}{|t_{s}|}\omega\tau}}{\cosh[\frac{2t_{s}}{\sigma^{2}}(t-\frac{t_{s}}{2})]+\cosh[\frac{t_{s}}{\sigma^{2}}\tau]}d\tau.$

This function is quantified numerically and plotted verses frequency
($\omega$) and time ($t$) as shown by continuous lines in Fig. 2(a). To get an
understanding of the behaviour of this spectrum, we noticed that it has
maximal values at $\omega=0$ and $t=t_{s}/2$, otherwise
$S_{WV}^{(02)}(\omega,t)$ drops fast to zero. Wigner-Ville spectrum in the
time-domian for $\omega=0$ is given by

$\displaystyle S_{WV}^{(02)}(0,t)$ $\displaystyle=$
$\displaystyle\frac{8t_{s}^{2}}{\sigma^{4}}\left(t-\frac{t_{s}}{2}\right)\,\textrm{cosech}\left[\frac{2|t_{s}|}{\sigma^{2}}\left(t-\frac{t_{s}}{2}\right)\right],$
(11)

and in the frequency domain corresponding to $t=t_{s}/2$ is given by

$S_{WV}^{(02)}(\omega,t_{s}/2)=4\pi\omega\,\textrm{cosech}[\sigma^{2}\pi\omega/|t_{s}|].$
(12)

Each of these attain a maximum value of $4|t_{s}|/\sigma^{2}$, which is four
times the maximum of $\Omega_{02}$ drive. Fig. 2(c) presents the zero
frequnecy Wigner-Ville distribution ($S_{WV}^{(02)}(0,t)$) and
$4\times\Omega_{02}$ with continuous and dashed lines respectively.
$S_{WV}^{(02)}(0,t)$ has a steeper rise and fall than $\Omega_{02}$, which can
be quantified in terms of full width at half maxima (FWHM). We obtain the FWHM
at diffferent values of $\sigma\in[15,45]$ ns and $\kappa\in[1.5,3]$ and
analyze these considering the normalized time durations of the pulse
($T=t_{f}-t_{i}=2n\sigma-t_{s}$) with $n=0.5$. As shown in Fig. 2(d) , FWHM of
$S_{WV}^{(02)}(0,t)$ , following Eq.(10), decreases with increasing $\kappa$,
following a trend, which is similar to that of FWHM of $\Omega_{02}$. Thus,
with increasing $\kappa$, we observe the squeezing of
$S_{WV}^{(02)}(\omega,t)$ along the time-axis. As expected, a squeezed
distribution for a given $\sigma$ and $\kappa$ has higher peak value of
$S_{WV}^{(02)}(\omega,t)$ at $t=t_{s}/2$ and $\omega=0$. On the other hand,
FWHM of $S_{WV}^{(02)}(\omega,t_{s}/2)$ in frequency domain is larger than the
FWHM of the Fourier transform of the $\Omega_{02}(t)$ drive.

An approximation of the Wigner-Ville distribution function for small $t$ and
$\tau~{}t$ i.e. with vanishing self-correlations at larger $\tau$, is
$S_{WV}^{(02)}(\omega,t)\approx
2\sqrt{\pi}|t_{s}|/\sigma^{2}e^{-\frac{t_{s}^{2}}{\sigma^{4}}(t-\frac{t_{s}}{2})^{2}}e^{-\frac{\sigma^{4}}{t_{s}^{2}}\omega^{2}}$.
This is plotted as a function of frequency and time as shown in Fig. 2(a),
with dotted lines.

The experimental implementation of the $\Omega_{02}(t)$ drive is achieved here
via a two-photon resonance pulse,
$\Omega_{2\rm{ph}}(t)=\sqrt{\sqrt{2}\Delta\Omega_{02}(t)}e^{-i\phi_{2ph}}e^{\pm
i\Delta t}$, which is off-resonant from $|0\rangle-|1\rangle$ and
$|1\rangle-|2\rangle$ transitions by $-\Delta$ and $\Delta$ respectively.
These are represented by $\tilde{\Omega}_{01}(t)$ and $\tilde{\Omega}_{12}(t)$
transitions in Fig. 1(a). The Wigner-Ville distribution of two-photon
resonance is thus linearly shifted in frequency, with maximum at
$(t=t_{s}/2,\omega=\pm\Delta)$ as shown in Fig. 2(b) for
$\tilde{\Omega}_{12}(t)$.

###  III.2 Optimal phases of the drives

Figure 3: Maximum attained population $p_{2}$ is simulated for saSTIRAP as a
function of (a) initial phase of the counter diabatic drive ($\phi_{02}$) and
standard deviation of the Gaussian drives ($\sigma$) with $\kappa=2.7$,
$\phi_{01}=\phi_{12}=0$. (b) initial phase of the counter diabatic drive
($\phi_{02}$) and total time taken by the saSTIRAP drives (T01), where
$\phi_{01}=\phi_{12}=0$, $\sigma\in[13.5,45]$ ns and $\kappa\in[1.5,3]$; the
pulse sequence is truncated in the middle of the Gaussian drive
$\Omega_{01}(t)$ such that T01=$3\sigma+\kappa\sigma$. Results from the
simulations are obtained (a1, b1) without decoherence, (a2, b2) with
decoherence, and the phase $\phi_{02}$ is constant in time. Vertical dashed
line in each plot corresponds to $\phi_{02}=\pi/2$.

We simulate saSTIRAP protocols for different phases of the counter diabatic
drive ($\phi_{02}$), with $\phi_{01}=\phi_{12}=0$. For best transfer in a
saSTIRAP, $\phi_{01}+\phi_{12}+\phi_{20}=-\pi/2$ [6] which is reflected in
Fig. 3(a1,a2) presenting the maximum population attained in the second excited
state ($p_{2}$); given different values of the initial phases $\phi_{02}$ and
standard deviations $\sigma$ of the Gaussian drives in the range
$\sigma\in[13.5,57.5]$ ns, without and with decoherence respectively. The
value of $\kappa$ is fixed at $\kappa=2.7$ such that $t_{s}=-2.7\sigma$.

In Fig. 3(b1,b2) the vertical axis represents the time interval $T_{01}$ which
extends from the starting of the saSTIRAP pulse sequence ($t=t_{i}$) to the
middle of the $0-1$ drive. Here $\sigma\in[13.5,45]$ns, $\kappa\in[3,1.5]$,
such that $T_{01}=n\sigma+\kappa\sigma$, where $n=3$ (the Gaussians are
truncated at $\pm n\sigma$). The range of $T_{01}$ is taken along the diagonal
connecting ($\sigma=13.5$ ns, $\kappa=3.0$) to ($\sigma=45$ ns, $\kappa=1.5$)
in Fig. 4. Fig. 3(b1,b2) show the maximum population in the second excited
state without and with decoherence respectively, with constant phases of the
drives. This surface plot has a flattened profile for maximum population
transfer around $\phi_{02}=\pi/2$. The distribution is more sensitive to
$\phi_{02}$ around an optimal transfer time, $T_{01}=170$ ns, which also
corresponds to maximal population transfer in STIRAP.

##  IV Experimental results

###  IV.1 STIRAP vs saSTIRAP

An ideal STIRAP may be designed for a perfect transfer of the population from
the ground state to the second excited state in a three-level system. On the
other hand, sa-STIRAP is much more robust against the parameters such as
amplitudes, standard-deviations, and the relative separation between the pair
of counter-intuitive Gaussian profiles, wherein the imperfections of these
parameters is accounted for by an additional two-photon resonance.

Figure 4: Population $p_{2}$ in the state $|2\rangle$ (a) ideally expected,
(b) simulated with decoherence and (c) experimental data for the STIRAP
process and saSTIRAP with parameters $\Omega_{01}/(2\pi)=44$ MHz,
$\Omega_{12}/(2\pi)=37$ MHz as a function of the pulse width $\sigma$ and the
normalized pulse separation $\kappa=|t_{\rm s}|/\sigma$. Optimal populations
resulting from STIRAP are shown with black mesh and for saSTIRAP without mesh.

Next, we study experimentally the effect of changing the STIRAP pulse width
$\sigma$ and the normalized STIRAP pulse separation $k=|t_{s}|/\sigma$, see
Fig. 4. For the data presented we took $\Omega_{01}=2\pi\times 44$ MHz and
$\Omega_{12}=2\pi\times 37$ MHz, which are both much smaller than the qubit
anharmonicity $2\Delta=2\pi\times 296$ MHz. Gaussian drives are truncated at
$n=\pm 3$. It is clear from Fig. 4 that saSTIRAP is insensitive to variations
in $\sigma$ and $\kappa$ for a significantly wide range of values, whereas
STIRAP typically works well when the pulses are relatively close to each
other, corresponding to $|t_{\rm s}|/\sigma=1.5$. Figs. 4 (b,c) show that
simulation and experimental results are in very good agreement with each
other. These plots are obtained under strong decoherence, whose effect is
partially mitigated by optimally truncating the STIRAP and saSTIRAP sequences
in the middle of the drive $\Omega_{01}(t)$. By this time, the maximum
transfer has already taken place and beyond this, the decay of $p_{2}$
dominates over the slow transfer of $p_{0}$ to $p_{2}$. Fig. 4(a) presents the
ideal $p_{2}$ resulting from ideal STIRAP and saSTIRAP drives in the absence
of decoherence, wherein there is a similar decline in STIRAP population
transfer.

The plots which include the counterdiabatic correction demonstrate that the
protocol is effective in counteracting the diabatic excitations, and we reach
experimental values for $p_{2}$ in the range 0.8 - 0.9. These results can be
also analyzed from the point of view of the quantum speed limit, by looking at
the transfer time $t_{\rm tr}$ over which population transfer occurs. We
define the transfer time as

$t_{\rm tr}=t_{f}-t_{i}=(2n+\kappa)\sigma$ (13)

between an initial dark state $|D(t_{i})\rangle$ and a final state
$|D(t_{f})\rangle$. A convenient choice for a dissipative system is to take an
initial state with 99 % population in $|0\rangle$ (mixing angles
$\Theta(t_{i})=0.03\pi$) and a final state with 80 % population in $|2\rangle$
(mixing angle $\Theta(t_{f})=0.35\pi$). For dissipationless systems the latter
is usually taken 90%, see [27].

To further confirm these results, we realized another set of experiments where
we observed the transfer efficiency of saSTIRAP against STIRAP area
($\mathcal{A}$) by varying the width ($\sigma$) of the STIRAP drive at
constant amplitudes $\Omega_{01}/(2\pi)=25$ MHz and $\Omega_{12}/(2\pi)=16$
MHz. We perform five different experiments where the aim was to reach a target
value $p_{2}=0.55$ by adjusting the parameters ($\sigma$, $\mathcal{A}_{02}$).
We found experimentally ($10$ ns, $\pi/2$), ($17$ ns, $3\pi/2$), ($17$ ns,
$\pi/3$), ($25$ ns, $5\pi/4$), and ($25$ ns, $0$). The experimental points
correspond well to the simulation.

Several bounds on the speed of state transfer have been derived in open
systems, for example based on Fisher information [28], on relative purity
[29], and on the Bures distance [22, 23]. To find the quantum speed for
evolution from a pure initial state $\rho_{0}=|\psi(0)\rangle\langle\psi(0)|$
at $t=0$ to a mixed state $\rho_{\tau}$ at $t=\tau$, we use the bounds derived
in Ref. [22],

$T_{\rm QSL}=\frac{\sin^{2}(d_{B}(\rho_{0},\rho_{\tau}))}{\lambda_{\tau}^{\rm
op}}$ (14)

where $d_{B}(\rho_{0},\rho_{\tau})=\arccos(\sqrt{F(\rho_{\tau},\rho_{0})})$ is
the Bures distance and $F(\rho_{\tau},\rho_{0})=\left({\rm
tr}\sqrt{\sqrt{\rho_{0}}\rho_{\tau}\sqrt{\rho_{0}}}\right)^{2}$ is the
fidelity ($F(\rho_{\tau},\rho_{0})=\langle\psi(0)|\rho_{\tau}|\psi(0)\rangle$
for a pure $\rho_{0}$). This yields

$T_{\rm QSL}=\frac{1-F(\rho_{\tau},\rho_{0})}{\lambda_{\tau}^{\rm op}}$ (15)

and

$\lambda_{\tau}^{\rm
op}=\frac{1}{\tau}\int_{0}^{\tau}||\mathbb{L}[\rho(t)]||_{\rm op}dt$ (16)

and $||\mathbb{L}[\rho(t)]||_{\rm op}={\rm max}_{i}\\{s_{i}(t)\\}$ is the
operator norm ($s_{i}$ are the singular values of $L[\rho(t)]$ defined as
eigenvalues of $\sqrt{\mathbb{L}^{{\dagger}}[\rho(t)]\mathbb{L}[\rho(t)]}$,
where $\mathbb{L}^{{\dagger}}$ is the adjoint of $\mathbb{L}$). In the case
when the dissipation is mainly due to relaxation processes, see Eqns. (5) and
(6), operator $\mathbb{L}[\rho(t)]$ is Hermitian. We obtain the Bures distance
and corresponding quantum speed for the mixed state evolution under STIRAP and
saSTIRAP drives (see Eqns. 1, 2, 3), with $\sigma=30$ ns, $\kappa=1.5$,
$\Omega_{01}/2\pi=44$ MHz, $\Omega_{12}/2\pi=37$ MHz, and $n=\pm 3$. In the
ideal situation, where the initial state is $|0\rangle$ and the final state is
$-|2\rangle$, the Bures distance between the initial and the final state is
$0.5\pi$. However, with given STIRAP and saSTIRAP parameters, the maximum
Bures distance is $0.48\pi$ without considering decoherence and $0.33\pi$ in
the presence of decoherence. Correspondingly, the instantaneous rate of change
of quantum state also varies with and without decoherence as shown in Fig.
5(a), where continuous curves are obtained by considering the pure state
evolution (no decoherence) and dotted curves are obtained from the mixed state
evolution (with decoherence). The rate of change $||\dot{\rho}(t)||$ of a
quantum state at an arbitrary time ($t$) is defined as the operator norm of
$\dot{\rho}(t)$, which is in fact the same as $||\mathbb{L}[\rho(t)]||_{\rm
op}$, see Eq. (5). It is noteworthy that the rate of change of quantum state
in saSTIRAP (in red) is much higher than that of STIRAP (in blue), while both
of these peak around the middle of the sequence. In Fig. 5(b), the quantum
speed limit ($T_{QSL}$) under saSTIRAP drive is shown as a function of time
for pure and mixed states. In case of pure state evolution, $T_{QSL}$ reaches
a maximum of $13.9$ ns while the mixed state evolution has a lower bound at
$T_{QSL}=12.6$ ns. This faster mixed state evolution is deceptive, as it
results from the smaller Bures distance and infact has lower value of
$||\dot{\rho}(t)||$.

Figure 5: (a) Instantaneous rate of change of quantum state resulting from
STIRAP and saSTIRAP evolutions without (continuous) and with (dotted)
decoherence. Here $\sigma=30$ ns, $\kappa=1.5$, $\Omega_{01}/2\pi=44$ MHz,
$\Omega_{12}/2\pi=37$ MHz, and $n=\pm 3$. (b) Corresponding quantum speed
limit without (continuous) and with (dotted) decoherence as a function of
time, and (c) $T_{QSL}$ resulting from saSTIRAP with decoherence for different
values of $\sigma$ and $\kappa$.

Considering Fig. 5(a) again, it is evident that $T_{QSL}$ for STIRAP is an
order of magnitude larger than that of saSTIRAP, which is attributed to almost
the same Bures distance while an order of magnitude difference exists in the
rate of evolution of the quantum states. Finally, we compare the quantum speed
limits of the mixed state evolutions for different STIRAP parameters:
($\sigma=15$ ns, $\kappa=3$), where saSTIRAP evolution is predominantly
achieved by the counterdiabatic drive, as can be seen from Fig. 4 and for
($\sigma=15$ ns, $\kappa=1.5$), which corresponds to much shorter pulse
duration. Results from the comparisons are shown in Fig. 5(c), where red curve
presents the fastest evolution with $T_{QSL}=9.6$ ns. Another interesting
point would be to compare the quantum state evolution under STIRAP and
saSTIRAP for ($\sigma=15$ ns, $\kappa=3$), where ill performance of STIRAP is
readily seen in Fig. 4(a) as well as from
$d_{B}(\rho_{t_{i}},\rho_{t_{f}})=0.19\pi$ (without decoherence). In this
case, the maximum value of the rate of change of quantum state under STIRAP
($||\dot{\rho}(t)||\approx 215$ MHz) is of the same order of magnitude as that
of saSTIRAP ($||\dot{\rho}(t)||\approx 250$ MHz). However, in the same time
$T=t_{f}-t_{i}$, STIRAP leads to nowhere close to the expected final state,
while saSTIRAP driven $\rho_{t_{i}}$ is close to the final state with
$d_{B}(\rho_{t_{i}},\rho_{t_{f}})=0.43\pi$.

For pure states the Bures distance becomes the Fubini-Study distance

$d_{\mathrm{FS}}(|\psi_{1}\rangle,|\psi_{2}\rangle)=\cos^{-1}\sqrt{F(|\psi_{1}\rangle,|\psi_{2}\rangle)}$
(17)

where the fidelity is defined as
$F(|\psi_{1}\rangle,|\psi_{2}\rangle)=|\langle\psi_{1}|\psi_{2}\rangle|$. In
the case of dark states $|D\rangle=\cos\theta|0\rangle-\sin\Theta|1\rangle$ we
have

$d_{\mathbf{FS}}(\Theta_{1},\Theta_{2})=|\Theta_{1}-\Theta_{2}|^{2}$ (18)

and the Bures distance is
$d_{\mathbf{Bures}}=\sqrt{2(1-\cos(\Theta_{1}-\Theta_{2}))}$.

###  IV.2 Pulse area

Figure 6:  Transferred population $p_{2}$ from a) simulation, b) experiment-
as a function of the STIRAP pulse area $\mathcal{A}$ and the two-photon pulse
area $\mathcal{A}_{02}$. The STIRAP Rabi couplings are varied linearly from
zero to a maximum value of $\Omega_{01}/(2\pi)=80$ MHz and
$\Omega_{12}/(2\pi)=58$ MHz while keeping the asymmetry constant at
$\eta=0.16$. Similarly, the two-photon pulse was varied from zero to maximum
values of $\tilde{\Omega}_{12}/(2\pi)=\sqrt{2}\tilde{\Omega}_{01}/(2\pi)=118$
MHz. The separation time was $t_{\rm s}=-45$ ns and the width was $\sigma=30$
ns.

STIRAP is known to be insensitive to the amplitudes of the drives once it
satisfies the adiabaticity criteria, $\Omega\sigma>>\sqrt{\pi}/4$ (assuming
$\Omega_{01}=\Omega_{12}=\Omega$). Thus for a fixed $\sigma$, there is a
threshold $\Omega$ beyond which the value of $\Omega$ ceases to matter anymore
in a STIRAP. This robust feature of STIRAP is passed onto saSTIRAP, for which
the inessentiality of a lower bound to $\Omega$ is already seen. These robust
features are clearly reflected in Fig. 6, where the population transferred
from the ground state to the second excited state $p_{2}$ is plotted as a
function of the STIRAP area,
$\mathcal{A}=\int_{-\infty}^{\infty}\sqrt{\Omega_{01}(t)^{2}+\Omega_{12}(t)^{2}}dt$
and two-photon pulse area, $\mathcal{A}_{02}=2\Theta(t_{f})$. The STIRAP area
is spanned by linearly varying the drive-amplitudes in the ranges:
$\Omega_{01}\in 2\pi\times[0,80]$ MHz and $\Omega_{12}\in 2\pi\times[0,58]$
MHz with asymmetry
$\eta=(\Omega_{01}-\Omega_{12})/(\Omega_{12}+\Omega_{01})=0.16$, while keeping
$\sigma=30$ ns, $t_{s}=-1.5\sigma$, $t_{i}=-3\sigma+t_{s}$, and
$t_{f}=3\sigma$ fixed. The area of the two-photon drive is spanned by simply
varying the target mixing angle ($\Theta(t_{f})$), such that
$\mathcal{A}_{02}\in[0,4\pi]$. In Fig. 6, $\mathcal{A}_{02}=0$ corresponds to
purely STIRAP implementation. In that case, population transfer efficiency
becomes significant for $\mathcal{A}>4\pi$. From the adiabaticity criteria, we
have $\Omega>>14.8$ MHz and to an approximation $\mathcal{A}>>3.17\pi$. Fixing
the STIRAP area $\mathcal{A}=4\pi$, and amplitude ratio,
$a=(1-\eta)/(1+\eta)=\Omega_{12}/\Omega_{01}=0.725$, we obtain
$\Omega_{01}/2\pi=18.6$ MHz and $\Omega_{12}/2\pi=13.48$ MHz. The efficiency
of STIRAP and its robustness against the drive-amplitudes can be observed
along the vertical stretch for $\mathcal{A}_{02}=0$. The area of the counter-
diabatic saSTIRAP drive,
$\mathcal{A}_{02}=\int_{-\infty}^{\infty}\Omega_{02}(t)dt=\int_{t_{i}}^{t_{f}}2\dot{\Theta}(t)dt=2\theta(t_{f})$,
is equal to $\pi$ for saSTIRAP. The drive $\Omega_{02}(t)$ has a maximum
amplitude of $-t_{s}/\sigma^{2}$ at $t=t_{s}/2$, see Eq. (9) and the
corresponding amplitudes of two-photon drives can be obtained by using:
$\tilde{\Omega}_{12}=\sqrt{2}\tilde{\Omega}_{01}=\sqrt{\sqrt{2}\Delta\Omega_{02}}$.
In the experiments and simulations a linear variation of
$\tilde{\Omega}_{12}\in 2\pi[0,118]$ MHz corresponds to
$\mathcal{A}_{02}\in[0,4\pi]$. As $\mathcal{A}_{02}$ is increased from zero,
the population transfer begins to improve and for $\mathcal{A}_{02}=\pi$, we
have saSTIRAP with a near perfect population transfer. This narrow region
around $\mathcal{A}_{02}=\pi$ is clearly insensitive to the STIRAP area
$\mathcal{A}$. As STIRAP begins to work properly, this narrow region gets
wider and we obtain a highly efficient population transfer. With $A_{02}$
approaching $2\pi$, the counter-diabatic two-photon drive of saSTIRAP
effectively corresponds to a $2\pi$ rotation in the $|0\rangle-|2\rangle$
subspace. For $\mathcal{A}_{02}=3\pi$, a saSTIRAP like transfer is observed
again. Here the two-photon drive effectively implements a $3\pi$ rotation such
that initial state $|0\rangle$ ends up close to $i|2\rangle$, which interferes
constructively with the output state ($-|2\rangle$) from STIRAP. The phase
difference between these two output states prevents them to interfere
destructively. The periodicity of the pattern for relatively small values of
$\mathcal{A}$ in Fig. 6 is infact a consequence of Rabi oscillations in the
$|0\rangle-|2\rangle$ subspace due to two-photon drives:
$\tilde{\Omega}_{01}(t)$, detuned by $-\Delta$ from $\omega_{01}$ and
$\tilde{\Omega}_{12}(t)$, detuned by $\Delta$ from $\omega_{12}$.

##  V Discussion: comparison with saSTIRAP by direct-coupling counterdibatic
drive

Figure 7: Results from simulations of saSTIRAP with a direct $0-2$
counterdibatic drive and in the absence of decoherence. (a) Final populations
- $p_{0}$, $p_{1}$, and $p_{2}$ resulting from saSTIRAP, as a function of
amplitudes of STIRAP drives ($\Omega_{01}$, $\Omega_{12}$) and area of the
superadiabatic drive ($\mathcal{A}_{02}$). Here $\sigma=30$ ns, $t_{s}=-45$
ns, $t_{i}=-3\sigma+t_{s}$, $t_{f}=3\sigma$, $\phi_{01}=\phi_{12}=0$, and
$\phi_{20}=-\pi/2$. Corresponding population-transfer profiles over time are
plotted for $\Omega_{01}=\Omega_{12}=2\pi\times 8$ MHz with (b)
$\mathcal{A}_{02}=0$, (c) $\mathcal{A}_{02}=\pi$, and (d)
$\mathcal{A}_{02}=5\pi$. These points are also marked with red cross marks in
the surface map of $p_{2}$.

In order to understand better the previous results, we analyze here the case
of saSTIRAP using a counterdiabatic drive that couples directly with
transition $|0\rangle-|2\rangle$. This is the standard way to obtain a highly
efficient desired population transfer. We study here what happens when the
strength of this counterdibatic drive area exceeds the value of $\pi$ obtained
in the standard superadiabatic theory.

The superadiabatic(sa)-STIRAP closely follows the dynamics of adiabatic
population transfer via STIRAP while correcting for any non-adiabatic
excitations. The counterdiabatic ($0-2$) drive can be understood as working in
parallel with the STIRAP sequence; they interfere constructively to achieve
the final dark state
$|\psi_{f}\rangle=|D\rangle=\cos(\Theta(t_{f}))|0\rangle-\sin(\Theta(t_{f}))|2\rangle$
with mixing angle $\Theta(t_{f})=2n\pi+\pi/2$, where $n\in\mathbb{Z}$.
saSTIRAP is strictly defined for $\mathcal{A}_{02}=\pi$ and $\phi_{20}=-\pi/2$
where the phases $\phi_{01}$ and $\phi_{12}$ of the STIRAP pulses are fixed to
$0$ in Eq. (1) and Eq. (2). However, now the 0-2 drive, which in Eq. (4) was
obtained from the effective two-photon drive, is now realized directly as
$H_{02}=(i\hbar\Omega_{02}/2)\left(|0\rangle\langle 2|-|2\rangle\langle
0|\right)$. Importantly, this means that energy-level shifts, which inevitably
appear in the two-photon drive, do not appear. We simulate the dynamics of a
three-level system initialized in the dark state
$|\psi_{i}\rangle=|D\rangle=\cos(\Theta(t_{i}))|0\rangle-\sin(\Theta(t_{i}))|2\rangle$
with mixing angle $\Theta(t_{i})=0$. The decoherence is neglected altogether.
We extend the simulation to a much broader range of $\mathcal{A}_{02}$ beyond
the standard area of $\mathcal{A}_{02}=\pi$ and observe a periodically
repeating pattern with features similar to saSTIRAP, see Fig. 7.

The counterdiabatic pulse area
$\mathcal{A}_{02}=\int_{-\infty}^{\infty}\Omega_{02}(t)dt=2\int_{-t_{i}}^{t_{f}}\dot{\Theta}(t)dt=2\Theta(t_{f})$
since $\Theta(t_{i})=0$. Therefore, the final angle is
$\Theta(t_{f})=\mathcal{A}_{02}/2$ and the corresponding final state is

$|\psi_{f}(\mathcal{A}_{02})\rangle=\cos(\mathcal{A}_{02}/2)|0\rangle+e^{i\phi}\sin(\mathcal{A}_{02}/2)|2\rangle$
(19)

which is a dark state
($|D(\theta)\rangle=\cos\theta|0\rangle-\sin\theta|2\rangle$) for
$\mathcal{A}_{02}=2n\pi+\theta$ and $\phi=(2m\pi+\phi_{20}-\pi/2)$, where
$n,m\in\mathbb{Z}$. Populations of the ground state – $p_{0}$, first excited
state – $p_{1}$, and second excited state – $p_{2}$ obtained in a saSTIRAP are
plotted with resect to maximum amplitudes in the STIRAP drive
($\Omega_{01}=\Omega_{12}$) and area of the counterdiabatic pulse-
$\mathcal{A}_{02}$, as shown in Fig. 7. Simulations in Fig. 7 have STIRAP
drives with fixed width $\sigma=30$ ns and overlap $t_{s}=-1.5\times\sigma$
with each Gaussian being truncated at $\pm 3\sigma$. In Fig. 7(a),
$\Omega_{01}=0$ corresponds to Rabi oscillations in the $0-2$ subspace, such
that $p_{0}+p_{2}=1$, while $p_{1}=0$. $\mathcal{A}_{02}=0$ corresponds to the
STIRAP implementation alone, which begins to work well beyond
$\Omega_{01}=2\pi\times 15$ MHz. STIRAP population dynamics with time, in the
region of inefficient implementation with $\Omega_{01}=\Omega_{12}=2\pi\times
8$ MHz is shown in Fig. 7(b). As the counterdiabatic pulse of saSTIRAP comes
into picture, there is a dramatic improvement with a perfect population
transfer as shown in Fig. 7(c). Interestingly, highly efficient population
transfer appears also at other values of $\mathcal{A}_{02}$, most notable near
values $\mathcal{A}_{02}=4n\pi+\pi$ ($n\in\mathbb{Z}$, $n\neq 0$). An example
of time-domain population transfer is shown in Fig. 7(d) for
$\mathcal{A}_{02}=5\pi$. However, it is important to notice that in this case
has additional oscillations in the middle of the sequence, therefore the non-
adiabatic terms are not suppressed. Still, such protocols are also interesting
and can be used to realize holonomic gates, see for example Ref. [30] for an
experimental realization.

Another interesting situation arises for $\mathcal{A}_{02}=3\pi$, where there
is destructive interference between states resulting from STIRAP and counter-
diabatic drive for $\Omega_{01}=\Omega_{12}=2\pi\times 8$ MHz. The destructive
interference is due to an overall phase of $e^{i\pi}$ acquired by the target
dark state in the case of direct 0 – 2 drive (with $\mathcal{A}_{02}=3\pi$),
_i.e._ under $H_{02}(t)$ drive the corresponding final states, see Eq. (19)
for $\mathcal{A}_{02}=\pi$ and $3\pi$ are related as
$|\psi_{f}(\pi)\rangle=e^{i\pi}|\psi_{f}(3\pi)\rangle$.

This interference is absent in the results discussed in previous section with
reference to Fig. 6. In fact under two-photon drive the state does not develop
an overall phase of $e^{i\pi}$ for the case of $\mathcal{A}_{02}=\pi$ vs
$\mathcal{A}_{02}=3\pi$.

The final state from the two-photon drive contains both real and imaginary
parts, which do not get nullified by the final state resulting from STIRAP,
which has only real coefficients. The difference between the actions of direct
0 – 2 drive and the two-photon drive can be explained by considering the
details of the effective two-photon drive in the $0-2$ subspace. Let us
consider the evolution of an arbitrary state
$\alpha|0\rangle+\beta|1\rangle+\gamma|2\rangle$ driven by two-photon drive
Hamiltonian given in Eq. (3). Here we use the method of adiabatic elimination
by assuming $\Omega_{02}<<\Delta$ and $\dot{\beta}=0$ as described in Ref.
[31]. This leads to an effective Hamiltonian in the $0-2$ subspace,

$\displaystyle H_{\rm 2ph}^{\rm eff}(t)$ $\displaystyle=$
$\displaystyle-\frac{\hbar\tilde{\Omega}_{01}\tilde{\Omega}_{12}}{4\Delta}\left(\frac{1}{2\lambda}|0\rangle\langle
0|+\frac{\lambda}{2}|2\rangle\langle 2|+e^{2i\tilde{\varphi}}|0\rangle\langle
2|\right)+h.c..$ (20)

This Hamiltonian is clearly different from the directly coupled drive
$H_{02}(t)$ from Eq. (4 in the $0-2$ subspace. These additional diagonal terms
in Eq. (20) are responsible for generating additional relative phases during
the evolution, which lead to different phases of the complex coefficients,
even though their absolute values remain close to the ones expected from a
direct 0–2 drive.

In addition, from Fig. 6(a) and Fig. 7(a), imperfections in the $p_{2}$
surface maps in the form of tilted arcs with tails for increasing values on
$\mathcal{A}_{02}$ occur. This happens because $\Omega_{02}$ becomes
comparable with the anharmonicity $\Delta$, and also because of hiher
truncation errors due to increasing $0-2$ pulse amplitude. Despite these
imperfections, a much wider plateau is formed, depicting efficient population
transfer and therefore, insensitivity to pulse parameters. The structure of
this plateau is preserved even in the presence of decoherence. An added
advantage is seen for $A_{02}=\pi$, $5\pi$ in Fig. 7(a), and $A_{02}=\pi$,
$3\pi$ in Fig. 6, where population transfer is resilient to the STIRAP drive
amplitudes. The large flexilibility in the choice of parameters without
compromising the efficiency of the protocol makes the protocol remarkably
robust.

##  VI Conclusion

In the loop conﬁguration for a transmon device, we have implemented a
superadiabatic protocol where two couplings produce the standard stimulated
Raman adiabatic passage, while the third is a counterdiabatic ﬁeld that
suppresses the nonadiabatic excitations. This technique enables fast operation
and it is remarkably robust against errors in the shape of the pulses. The
superadiabatic method enables a continuos interpolation between speed and
insensitivity to errors, allowing one to select the optimal values under
realistic experimental constraints. We also observe the existence of highly
efficient transfer of population for large values of the counterdiabatic pulse
area.

##  Appendix A Appendix: Dynamic phase corrections

Figure 8: (a) Population $p_{2}$ resulting from saSTIRAP with dynamically
varying phases of the drives is plotted as a function of initial phase of the
$0-2$ drive. (a1, a3), (a2, a4) show the transferred population ($p_{2}$)
without and with decoherence respectively in case of dynamically varying phase
($\phi_{02}(t)$). Vertical dashed line in each plot corresponds to
$\phi_{02}=\pi/2$. $p_{2}$ as a function of $\kappa$ and $\sigma$ in the case
of saSTIRAP with constant phases of the drives and with dynamic phases of the
drives. (b) $p_{2}$ from saSTIRAP driven dynamics as a function of $\kappa$
and $\sigma$.

Following Ref. [31], dynamic phase corrections implemented to drives
$\Omega_{01}(t)$, $\Omega_{12}(t)$, and $\Omega_{02}(t)$ (with
$\mathcal{A}_{02}=\pi$) are $\phi_{01}(t)=\phi_{01}+2\sqrt{2}\hbar\Theta(t)$,
$\phi_{12}(t)=\phi_{12}-(5\hbar/\sqrt{2}\Theta(t)$, and
$\phi_{02}(t)=\phi_{02}-(\hbar/\sqrt{2}\Theta(t)$ respectively. We simulated
saSTIRAP with dynamically varying phases by incorporating these ac-Stark shift
corrections. Fig. 8(a) presents the resulting surface maps of $p_{2}$ as a
function of initial phase of the $0-2$ drive plotted with $\sigma$ in parts
(a1,a2) and as a function of the transfer time in parts (a3, a4), similar to
that of Fig. 3. Interestingly, by attributing time-dependent dynamical phases
to the drives in order to compensate the ac-Stark shifts resulting from the
two-photon resonanant drive, intial choice of $\phi_{02}(t=0)=\pi/2$ is not
necessarily optimal. A clear shift in the initial $\phi_{02}(t)$ is observed
as shown in Fig. 3(a3,a4) without and with decoherence.

In Fig. 8(b), results from saSTIRAP driven dynamics is shown as 3D map of
$p_{2}$ as a function of $\kappa$ and $\sigma$. The map in which $p_{2}$
decreases for smaller values of $\sigma$ and relatively larger
$t_{s}/\sigma^{2}$ is acquired with constant phases
$\phi_{01}=\phi_{12}=0,\phi_{20}=-\pi/2$. The map acquired with dynamic phase
corrections as described above has higher values for the shown range. Thus,
ac-Stark shift compensations using dynamic phases of the drives results into
highly efficient population transfer. Constant phases of the drives are at
their worst performance for small $\sigma$ values, which is the region where
STIRAP drives are also not effective. In this region, the saSTIRAP efficacy
highly depends upon the $0-2$ drive. Therefore, this region faces maximum
errors due to the detuned two-photon drives and hence there is a lot of room
for the improvement by compensating the corresponding ac-Stark shifts.

##  Appendix B Appendix: Majorana representation

Figure 9: Majorana trajectories of a qutrit driven by (a) STIRAP, (b) saSTIRAP
are shown, where the red and blue trajectories follow two Majorana stars
representing a qutrit and the black trajectory is the path traversed by the
averaged magnetization vector. (c) saSTIRAP drives and corresponding variation
of populations in levels $|0\rangle$ (black), $|1\rangle$ ( red), and
$|2\rangle$ (blue) with optimal transfer time and total transfer time are
shown with solid lines (thick for saSTIRAP and thin for STIRAP) and dashed
lines (for STIRAP or dotted for saSTIRAP) respectively.

To compare further the efficacies of STIRAP and saSTIRAP, we present a
geometrical visualization of the results from STIRAP and saSTIRAP on the
Majorana sphere [32]. The transmon, initialized in its ground state
($|0\rangle$), is driven by the STIRAP and saSTIRAP Hamiltonians with
$\sigma=35$ ns, $\kappa=2.7$ and $n=\pm 3$ in a decohering environment.
Subject to the strong decoherence (as mentioned in the main text), the
mixedness keeps on increasing and the purity of the single qutrit state drops
to as low as 0.45. However, by considering a optimal truncation of the pulse
sequence, which falls close to the middle of the $0-1$ Gaussian drive, the
purity saturates close to $0.8$. Majorana trajectories in Fig. 9(a), (b)
present the mixed state dynamics for the maximum population transfer, for
STIRAP and saSTIRAP drives respectively. Here we show the trajectories
followed by the dominant eigenstates alone, as other two eigenvalues are order
of magnitude lower. The corresponding variation of populations in levels
$|0\rangle$ (black lines), $|1\rangle$ (red lines), and $|2\rangle$ (blue
lines) are shown in Fig. 9(c). The optimally truncated dynamics is shown with
continuous lines (thick for saSTIRAP and thin for STIRAP) until a maximum
transfer of population from $|0\rangle-|2\rangle$ is achieved, while the
respective curves continue as dashed (STIRAP) and dotted (saSTIRAP) lines. The
levels of decoherence and other experimental parameters of STIRAP and saSTIRAP
are the same, but the results from saSTIRAP are clearly better.

## data access

Data related to the experiments and simulatons can be obtained from the
authors under reasonable request.

## Author contribution

AV carried out the measurements under the supervision of GSP, and did the
preliminary data analysis. SD did an in-depth analysis of the data and
contributed significantly to the theoretical analysis. SD and GSP wrote the
manuscript. All authors read and approved the manuscript.

## Acknowledgement

We thank Dr. Sergey Danilin for fabricating the sample used in these
experiments and for assistance with the measurements. We acknowledge financial
support from the Finnish Center of Excellence in Quantum Technology QTF
(projects 312296, 336810) of the Academy of Finland. We also are grateful for
financial support from the RADDESS programme (project 328193) of the Academy
of Finland and from Grant No. FQXi-IAF19-06 (“Exploring the fundamental limits
set by thermodynamics in the quantum regime”) of the Foundational Questions
Institute Fund (FQXi), a donor advised fund of the Silicon Valley Community
Foundation. This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under grant agreement no.
824109 (European Microkelvin Platform project, EMP). This work used the
experimental facilities of the Low Temperature Laboratory of OtaNano.

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