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

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 − |1 and |1 − |2 respectively) and the counterdiabatic pulse coupling into the |0 − |2 transition. These three pulse drive the three-level transmon concurrently in a loop (also called ∆-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 π. Suprisingly, we observe the existence of regions where population transfer still occurs with high efficiency. FIG. 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 . 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.

III. NUMERICAL ANALYSIS AND OPTIMIZATION OF DRIVE PARAMETERS
Superadiabatic STIRAP is driven jointly by the drives H 01 (t), H 12 (t), andH 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.

A. 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 x(t+τ /2)x(t−τ /2) we define the Wigner-Ville spectrum For stationary systems, one recovers the usual definition of power spectral density as S(ω) = S W V = (ω, 0). The instantaneous power at any moment t can be obtained by summing the Wigner-Ville spectrum over all infinitesimal bandwidths 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 Thus we can obtain the Wigner-Ville spectrum This function is quantified numerically and plotted verses frequency (ω) 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 ω = 0 and t = t s /2, otherwise S W V (ω, t) drops fast to zero. Wigner-Ville spectrum in the time-domian for ω = 0 is given by and in the frequency domain corresponding to t = t s /2 is given by Each of these attain a maximum value of 4|t s |/σ 2 , which is four times the maximum of Ω 02 drive. Fig. 2(c) presents the zero frequnecy Wigner-Ville distribution (S W V (0, t)) and 4 × Ω 02 with continuous and dashed lines respectively. S (02) W V (0, t) has a steeper rise and fall than Ω 02 , which can be quantified in terms of full width at half maxima (FWHM). We obtain the FWHM at diffferent values of σ ∈ [15,45] ns and κ ∈ [1.5, 3] and analyze these considering the normalized time durations of the pulse (T = t f − t i = 2nσ − t s ) with n = 0.5. As shown in Fig. 2 W V (0, t) , following Eq.(10), decreases with increasing κ, following a trend, which is similar to that of FWHM of Ω 02 . Thus, with increasing κ, we observe the squeezing of S (02) W V (ω, t) along the time-axis. As expected, a squeezed distribution for a given σ and κ has higher peak value of S (02) W V (ω, t) at t = t s /2 and ω = 0. On the other hand, FWHM of S (02) W V (ω, t s /2) in frequency domain is larger than the FWHM of the Fourier transform of the Ω 02 (t) drive. An approximation of the Wigner-Ville distribution function for small t and τ t i.e. with vanishing self-correlations This is plotted as a function of frequency and time as shown in Fig. 2(a), with dotted lines.
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 σ ∈ [13.5, 45]ns, κ ∈ [3, 1.5], such that T 01 = nσ + κσ, where n = 3 (the Gaussians are truncated at ±nσ). The range of T 01 is taken along the diagonal connecting (σ = 13.5 ns, κ = 3.0) to (σ = 45 ns, κ = 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 φ 02 = π/2. The distribution is more sensitive to φ 02 around an optimal transfer time, T 01 = 170 ns, which also corresponds to maximal population transfer in STIRAP.

A. 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.
Next, we study experimentally the effect of changing the STIRAP pulse width σ and the normalized STIRAP pulse separation k = |t s |/σ, see Fig. 4. For the data presented we took Ω 01 = 2π × 44 MHz and Ω 12 = 2π × 37 MHz, which are both much smaller than the qubit anharmonicity 2∆ = 2π × 296 MHz. Gaussian drives are truncated at n = ±3. It is clear from Fig. 4 that saSTIRAP is insensitive to variations in σ and κ for a significantly wide range of values, whereas STIRAP typically works well when the pulses are relatively close to each other, corresponding to |t s |/σ = 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 Ω 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 tr over which population transfer occurs. We define the transfer time as between an initial dark state |D(t i ) and a final state |D(t f ) . A convenient choice for a dissipative system is to take an initial state with 99 % population in |0 (mixing angles Θ(t i ) = 0.03π) and a final state with 80 % population in |2 (mixing angle Θ(t f ) = 0.35π). For dissipationless systems the latter is usually taken 90%, see [27].
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 ρ 0 = |ψ(0) ψ(0)| at t = 0 to a mixed state ρ τ at t = τ , we use the bounds derived in Ref. [22], where d B (ρ 0 , ρ τ ) = arccos( F (ρ τ , ρ 0 )) is the Bures distance and F (ρ τ , ρ 0 ) = tr √ ρ 0 ρ τ √ ρ 0 2 is the fidelity (F (ρ τ , ρ 0 ) = ψ(0)|ρ τ |ψ(0) for a pure ρ 0 ). This yields and and ||L[ρ(t)]|| op = max i {s i (t)} is the operator norm (s i are the singular values of L[ρ(t)] defined as eigenvalues of , where L † is the adjoint of L). In the case when the dissipation is mainly due to relaxation processes, see Eqns. (5) and (6), operator L[ρ(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 σ = 30 ns, κ = 1.5, Ω 01 /2π = 44 MHz, Ω 12 /2π = 37 MHz, and n = ±3. In the ideal situation, where the initial state is |0 and the final state is −|2 , the Bures distance between the initial and the final state is 0.5π. However, with given STIRAP and saSTIRAP parameters, the maximum Bures distance is 0.48π without considering decoherence and 0.33π 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 ||ρ(t)|| of a quantum state at an arbitrary time (t) is defined as the operator norm ofρ(t), which is in fact the same as ||L[ρ(t)]|| 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 ||ρ(t)||. 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: (σ = 15 ns, κ = 3), where saSTIRAP evolution is predominantly achieved by the counterdiabatic drive, as can be seen from Fig. 4 and for (σ = 15 ns, κ = 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 (σ = 15 ns, κ = 3), where ill performance of STIRAP is readily seen in Fig. 4(a) as well as from d B (ρ ti , ρ t f ) = 0.19π (without decoherence). In this case, the maximum value of the rate of change of quantum state under STIRAP (||ρ(t)|| ≈ 215 MHz) is of the same order of magnitude as that of saSTIRAP (||ρ(t)|| ≈ 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 ρ ti is close to the final state with d B (ρ ti , ρ t f ) = 0.43π.

B. Pulse area
STIRAP is known to be insensitive to the amplitudes of the drives once it satisfies the adiabaticity criteria, Ωσ >> √ π/4 (assuming Ω 01 = Ω 12 = Ω). Thus for a fixed σ, there is a threshold Ω beyond which the value of Ω 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 Ω 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, A = ∞ −∞ Ω 01 (t) 2 + Ω 12 (t) 2 dt and two-photon pulse area, A 02 = 2Θ(t f ). The STIRAP area is spanned by linearly varying the drive-amplitudes in the ranges: Ω 01 ∈ 2π × [0, 80] MHz and Ω 12 ∈ 2π × [0, 58] MHz with asymmetry η = (Ω 01 − Ω 12 )/(Ω 12 + Ω 01 ) = 0.16, while keeping σ = 30 ns, t s = −1.5σ, t i = −3σ + t s , and t f = 3σ fixed. The area of the two-photon drive is spanned by simply varying the target mixing angle (Θ(t f )), such that A 02 ∈ [0, 4π]. In Fig. 6, A 02 = 0 corresponds to purely STIRAP implementation. In that case, population transfer efficiency becomes significant for A > 4π. From the adiabaticity criteria, we have Ω >> 14.8 MHz and to an approximation A >> 3.17π. Fixing the STIRAP area A = 4π, and amplitude ratio, a = (1 − η)/(1 + η) = Ω 12 /Ω 01 = 0.725, we obtain Ω 01 /2π = 18.6 MHz and Ω 12 /2π = 13.48 MHz. The efficiency of STIRAP and its robustness against the drive-amplitudes can be observed along the vertical stretch for A 02 = 0. The area of the counter-diabatic saSTIRAP drive, A 02 = ∞ −∞ Ω 02 (t)dt = t f ti 2Θ(t)dt = 2θ(t f ), is equal to π for saSTIRAP. The drive Ω 02 (t) has a maximum amplitude of −t s /σ 2 at t = t s /2, see Eq. (9) and the corresponding amplitudes of two-photon drives can be obtained by using:Ω 12 = √ 2Ω 01 = √ 2∆Ω 02 . In the experiments and simulations a linear variation ofΩ 12 ∈ 2π[0, 118] MHz corresponds to A 02 ∈ [0, 4π]. As A 02 is increased from zero, the population transfer begins to improve and for A 02 = π, we have saSTIRAP with a near perfect population transfer. This narrow region around A 02 = π is clearly insensitive to the STIRAP area A. As STIRAP begins to work properly, this narrow region gets wider and we obtain a highly efficient population transfer. With A 02 approaching 2π, the counter-diabatic two-photon drive of saSTIRAP effectively corresponds to a 2π rotation in the |0 − |2 subspace. For A 02 = 3π, a saSTIRAP like transfer is observed again. Here the two-photon drive effectively implements a 3π rotation such that initial state |0 ends up close to i|2 , which interferes constructively with the output state (−|2 ) 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 A in Fig. 6 is infact a consequence of Rabi oscillations in the |0 − |2 subspace due to two-photon drives:Ω 01 (t), detuned by −∆ from ω 01 andΩ 12 (t), detuned by ∆ from ω 12 .

V. DISCUSSION: COMPARISON WITH SASTIRAP BY DIRECT-COUPLING COUNTERDIBATIC DRIVE
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 − |2 . 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 π obtained in the standard superadiabatic theory.
(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 Ω 02 /2) (|0 2| − |2 0|). 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 |ψ i = |D = cos(Θ(t i ))|0 − sin(Θ(t i ))|2 with mixing angle Θ(t i ) = 0. The decoherence is neglected altogether. We extend the simulation to a much broader range of A 02 beyond the standard area of A 02 = π and observe a periodically repeating pattern with features similar to saSTIRAP, see Fig. 7.
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 (Ω 01 = Ω 12 ) and area of the counterdiabatic pulse-A 02 , as shown in Fig. 7. Simulations in Fig. 7 have STIRAP drives with fixed width σ = 30 ns and overlap t s = −1.5 × σ with each Gaussian being truncated at ±3σ. In Fig. 7(a), Ω 01 = 0 corresponds to Rabi oscillations in the 0 − 2 subspace, such that p 0 + p 2 = 1, while p 1 = 0. A 02 = 0 corresponds to the STIRAP implementation alone, which begins to work well beyond Ω 01 = 2π × 15 MHz. STIRAP population dynamics with time, in the region of inefficient implementation with Ω 01 = Ω 12 = 2π × 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 A 02 , most notable near values A 02 = 4nπ + π (n ∈ Z, n = 0). An example of time-domain population transfer is shown in Fig. 7(d) for A 02 = 5π. 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 A 02 = 3π, where there is destructive interference between states resulting from STIRAP and counter-diabatic drive for Ω 01 = Ω 12 = 2π × 8 MHz. The destructive interference is due to an overall phase of e iπ acquired by the target dark state in the case of direct 0 -2 drive (with A 02 = 3π), i.e. under H 02 (t) drive the corresponding final states, see Eq. (19) for A 02 = π and 3π are related as |ψ f (π) = e iπ |ψ f (3π) .
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 iπ for the case of A 02 = π vs A 02 = 3π.
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 α|0 + β|1 + γ|2 driven by two-photon drive Hamiltonian given in Eq. (3). Here we use the method of adiabatic elimination by assuming Ω 02 << ∆ andβ = 0 as described in Ref. [31]. This leads to an effective Hamiltonian in the 0 − 2 subspace, 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 A 02 occur. This happens because Ω 02 becomes comparable with the anharmonicity ∆, 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 = π, 5π in Fig. 7(a), and A 02 = π, 3π 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.
In Fig. 8(b), results from saSTIRAP driven dynamics is shown as 3D map of p 2 as a function of κ and σ. The map in which p 2 decreases for smaller values of σ and relatively larger t s /σ 2 is acquired with constant phases φ 01 = φ 12 = 0, φ 20 = −π/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 σ 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.
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 ), is driven by the STIRAP and saSTIRAP Hamiltonians with σ = 35 ns, κ = 2.7 and n = ±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 (black lines), |1 (red lines), and |2 (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 − |2 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.