Superadiabatic population transfer by loop driving and synthetic gauges in a superconducting circuit

Superadiabatic population transfer by loop driving and synthetic gauges in a superconducting circuit

Abstract

The achievement of fast and error-insensitive control of quantum systems is a primary goal in quantum information science. Here we use the first three levels of a transmon superconducting circuit to realize a loop driving scheme, with all three possible pairs of states coupled by pulsed microwave tones. In this configuration, we implement a superadiabatic protocol for population transfer, where two couplings produce the standard stimulated Raman adiabatic passage, while the third is a counterdiabatic field which suppresses the nonadiabatic excitations. We demonstrate that the population can be controlled by the synthetic gauge-invariant phase around the loop as well as by the amplitudes of the three pulses. The technique enables fast operation, with transfer times approaching the quantum speed limit, and it is remarkably robust against errors in the shape of the pulses.

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supplementary_material

I Introduction

Quantum control - the manipulation of a system such that it reaches a target state or that it follows a given path in the Hilbert space - is an essential tool of modern quantum information processing. Two main paradigms of quantum control originate from the early days of quantum physics: resonant (Rabi) pulses - which can be fast but sensitive to errors in pulse parameters, and adiabatic pulses - which are more error-parameter robust but inherently slow. Both approaches have an enormous range of applications: Rabi pulses are used to implement all standard quantum gates (1), while adiabatic protocols have been applied successfully in chemical reaction dynamics, cooling of atomic ensembles, interferometry (2), and circuit QED (3).

At the turn of the century, around seven decades after the seminal work of Rabi in 1937 (4) and after the proof of the adiabatic theorem by Born and Fock in 1928 (5), Berry (6) and separately Demirplak and Rice (7); (8); (9) made a simple but powerful observation: a system can follow exactly the adiabatic state by using an additional counterdiabatic Hamiltonian tailored to cancel the nonadiabatic excitations. This type of evolution is called superadiabatic or transitionless, and several variations have been explored theoretically (10). Superadiabatic methods form a bridge between the two paradigms of quantum control, and allow one to exploit the advantages of both. The combination of robustness under parameter fluctuations and drive errors, together with fast operation times would make superadiabatic protocols especially advantageous for reducing the effects of decoherence and increasing the signal-to-noise ratio. For adiabatic quantum computers (11), quantum-annealing processors (12); (13), and holonomic quantum computing (14); (15); (16) this would be one important route to achieve quantum advantage (17). In quantum thermodynamics, the suppression of interlevel transitions during the adiabatic cycle of a quantum engine could lead to superadiabatic engines with increased power (18), while at the same time providing novel insights into the foundations of the third law of thermodynamics (19); (20); (21).

However, a major difficulty in implementing these protocols is that the counterdiabatic drive typically needs complex couplings between energy levels, with externally-controlled and stable Peierls phases (22). In optical setups this would require lasers with exquisitly low phase noise. This is why so far superadiabatic protocols have been demonstrated only in simple configurations, involving either two levels (23); (24) or two control fields (25); (26); (27).

Here we show that the required phase stability can be achieved by working in the microwave regime and using circuit quantum electrodynamics as a platform for demonstrating superadiabatic population transfer. We use the first three states of a superconducting transmon circuit (28); (29) to transfer population between the ground state and the second excited state. This is a generic task in quantum control of multi-level systems, where fast and efficient state-preparation serves as an initial step for more complicated algorithms. We achieve the population transfer by using three microwave pulses: two of them realize a stimulated Raman adiabatic passage (STIRAP), while the third is a two-photon process creating the counterdiabatic Hamiltonian. This type of driving, called loop configuration, has been discussed theoretically (30) but never implemented on any experimental platform. It results in the creation of a synthetic gauge potential with a gauge-invariant Aharonov-Bohm phase, which can be controlled externally. This contrasts to the simpler case of two-field drive, where the phases of the driving fields can be eliminated by a gauge transformation, and also with the case of two-level systems, where again the phase of the counterdiabatic pulse is irrelevant.

Our experiment is a paradigmatic example for the multi-level, multi-field complex systems envisioned as future quantum processors. As quantum mechanical objects, qutrits are fundamentally different from qubits: indeed, the qutrit is the simplest quantum system that shows noncontextuality (31). In quantum cryptography, entangled qutrits provide a higher level of security and increased coding density (32); (33); (34); (35); (36); (37); (38), while in quantum computing they bring in the benefit of operation on a larger Hilbert space (39); (40); (41). Our work demonstrates that a qutrit can be controlled by three fields, allowing the realization of superadiabatic protocols and synthetic gauge potentials. The results open up new perspectives in circuit quantum electrodynamics, for example toward the realization of qubits immune to phase noise in loop configurations with detuning (42), and the realization of synthetic gauge potentials similar to those recently studied in nanoelectronics (43) and in ultracold gases (44); (45).

Ii Results

 Figure 1: Schematic of the energy levels and pulse sequence. a) Loop driving: a counterdiabatic drive with effective Rabi frequency (dashed blue arrow) is applied in parallel with the standard STIRAP sequence consisting of pulses and , which are resonant with the respective transitions 0 - 1 and 1 - 2. The counterdiabatic drive is a two-photon process realized by an off-resonant pulse (detuning with respect to the first transition) which couples with strengths and into the corresponding transitions. b) Schematic of the timings and shapes of the pulses. The last pulse is the measurement pulse applied to the resonator. c) Schematic and optical image of the transmon. d) In the synthetic space, the Hamiltonian describes a three-site plaquette with Peierls hopping and with total magnetic flux penetrating the plaquette .

Counterdiabatic driving. To set forth our conventions, let us consider a generic three-level system with energy eigenstates , and . The three transitions can be driven resonantly by external fields with Rabi couplings and phases , with . In the rotating wave approximation, the Hamiltonian can be expressed in a compact form by using the symmetric and anti-symmetric Gell-Mann matrices (see Supplementary Note 1 and 2), defined as and . We also introduce and define as a unit two-dimensional vector .

Using these notations, the STIRAP Hamiltonian takes the form (3)

(1)

In the STIRAP protocol, the system adiabatically follows the instantenous eigenstate of the above Hamiltonian, called the dark state, , where is varied slowly from to . However, if this change is too fast, the system gets diabatically excited away from the state , reducing the transferred population. To accelerate STIRAP in a three-level ladder system the technique of suppressing nonadiabatic excitations (6); (7); (8); (9) requires the addition of a counterdiabatic field

(2)

In this protocol, which we will refer to as superadiabatic STIRAP (saSTIRAP), the Rabi coupling is varied such that

(3)

while the phase of the counterdiabatic field satisfies the relation . This specific form of the pulse is found by reverse Hamiltonian engineering (see Methods and Supplementary Note 3). Thus, the counterdiabatic field requires the creation of a complex Peierls matrix element with the amplitude and phase dependent on the other two tones used.

Experimental realization. The experiment employs a transmon driven by microwave fields, see Figure 1a), b), and c). To create the matrix elements for the STIRAP sequence in Eq. (1) we employ two microwave tones with externally-controlled phases and , which drive resonantly the corresponding transitions with Rabi couplings and . By using frequency mixers we can shape these signals into Gaussian pulses controlled by an arbitrary waveform generator (see Figure 1 b) and c)), giving

(4)
(5)

This results in matrix elements and .

The Gaussian pulses are not the only possible choice for the STIRAP pulse shape, but they are experimentally and theoretically convenient without sacrificing performance (30). In this parametrization is the width of the pulses, and the counterintuitive sequence is realized at negative pulse separation times . To realize the counterdiabatic Hamiltonian in Eq. (2) we use a two-photon process generated by a third microwave drive field with phase and Rabi couplings and into the corresponding transitions. The two-photon drive is operated so that it is detuned from both the and transitions by , which is sufficiently large to avoid parasitic excitations to state . This generates an effective matrix element , where the Rabi coupling is obtained from the perturbation theory (46) as and . The value of is fixed by the two-photon resonance condition, which gives .

The coupling is realized using the two-photon process because in the transmon the direct transition is forbidden in the first order due to its almost harmonic energy level structure. Overall, these couplings create a Hamiltonian with the desired structure ,

(6)

This Hamiltonian realizes the so-called loop driving configuration for three-level systems (30), with complex couplings between each pair of states.

Using this Hamiltonian we demonstrate the saSTIRAP method for population transfer between the states and . The transmon (see Figure 1c) was operated at a flux bias point where the frequencies of the first two transitions were GHz and GHz. This corresponds to GHz and a transmon anharmonicity MHz. For this device (see Methods and Supplementary Note 4) the decoherence was dominated by relaxation with rates 5.0 MHz and MHz. In the experiments below, the values and used were significantly smaller than the qubit anharmonicity, in order to minimize the effects of cross-coupling (3). For qubits with higher anharmonicity it would be advantageous to use even higher values for and in order to improve the transfer efficiency.

 Figure 2: Experimental proof of gauge invariance. Under loop driving, the phase is a gauge-invariant quantity, in analogy with lattice gauge theories, where it is typically produced by an applied magnetic field. The plot shows lines of constant population on the state , in the planes (with constant), (with constant) and (with constant). The gauge-invariance relation can be imagined visually as tilted planes that intersect the axes. Note also that the periodicity along the axis is double that of the periodicity along the axes and as a result of two-photon driving. In the experiment we had MHz, MHz, ns, and ns.

Synthetic gauge-invariant phase. We start by analyzing the notrivial gauge structure induced by the counterdiabatic term, as anticipated in Figure 1d). We note that the Hamiltonian in Eq. (6) describes three simultaneous rotations in the three subspaces , , and around the vectors . In each of the subspaces , the action of the Hamiltonian is analogous to that of a spin-1/2 particle in a magnetic field of magnitude and direction . In one subspace it is possible to rotate arbitrarily the axis to align one of them along . It is possible to do this in two subspaces simultaneously, but crucially, one cannot rotate arbitrarily all the three vectors . Formally, by applying a unitary local gauge transformation of the form where , , and are arbitrary phases, one obtains a Hamiltonian with a similar structure to Eq. (6) with different angles (see Supplemetary Note 2); however, these new angles are not independent of each other, but they must satisfy the constraint . Thus, by performing this transformation we can always eliminate two of the phases but the third one will be constrained by the value of the gauge-invariant quantity .

In Figure 2 we demonstrate experimentally that the dynamics of the system is determined by the gauge-invariant phase . We present the population transferred to state when one of the angles , , or is kept fixed, while the other two are varied. The populations are measured at a time ns after the maximum of the 0 – 1 drive pulse and the two-photon pulse is set to satisfy Eq. (3). The experiment shows clearly that the transferred population to state depends only on and not on each phase separately.

A convenient choice of gauge is , , and , which leads to the following structure for Eq. (6),

(7)

where . This form puts in evidence the role of the phase as a parameter in the Hamiltonian, which in this gauge becomes and can be controlled externally along with the Rabi frequencies , , and . From the experiment we can see that maximum transfer of population occurs at certain optimal values of , which we choose as the operating point for saSTIRAP. These values are not exactly at , as found by reverse Hamiltonian engineering (47); (30); (48); (49), due to the existence of ac Stark shifts in the energy levels of the driven system, which are not included in the ideal Hamiltonian (7). These produce an accumulated phase shift over the entire duration of the process (see Methods and Supplementary Note 5). However, the values of the optimal phases are reproduced very well by our numerical simulations based on the full Hamiltonian (see Methods) with an accuracy better than .

 Figure 3: Complete parameter mapping of saSTIRAP. The upper panel a) shows the population in the state for the STIRAP process with MHz and MHz as a function of the pulse width and the normalized pulse separation . The lower panel b) shows the corresponding saSTIRAP process. The left plots are experiments, while the right ones are the corresponding simulation results. The solid black lines show the transfer time in nanoseconds in either STIRAP or saSTIRAP to achieve the population , calculated by applying Eq. (21) (Methods) to the simulation data. In a) the transfer times are calculated only for those STIRAP parameters for which reaches values higher than 0.8, marked with dashed blue line.

Efficient transfer of population. To further demonstrate that the superadiabatic process succeeds in cancelling the non-adiabatic excitations, we compare the saSTIRAP method to STIRAP for a wide range of different STIRAP parameters. Here the STIRAP amplitudes and are kept constant at MHz and MHz, and the parameter space is explored by varying the STIRAP pulse width and the normalized STIRAP pulse separation , as shown in Figure 3a). At each point, the algorithm searches for and selects the optimal value of . The experiment can be compared to a numerical simulation, which replicates the experimental result accurately (for details, see Methods). From Figure 3a) we can see that typically STIRAP works well when the pulses are relatively close to each other, corresponding to . The lower panel Figure 3b) demonstrates that by adding the counterdiabatic drive with an optimal phase we are able to counteract the diabatic losses for all the STIRAP parameters. The population transferred by saSTIRAP, shown in Figure 3b), typically reaches values over 0.9. The experimental fidelity suffers mostly from the relatively short lifetime of the qutrit, but still reaches values higher than STIRAP, because the protocol can be driven faster. However, the transfer speed of saSTIRAP is limited by the maximum achievable coupling created by the two-photon pulse, which ultimately depends on the anharmonicity of the qutrit.

The performance of the protocol can be further characterized by comparing its transfer speed to the quantum speed limit at maximum coupling. We follow a convention where the duration of the saSTIRAP protocol is defined as the time lapse beween an initial state with population 0.99 in the ground state and a final state with population 0.8 in the second excited state (see (30) and Methods). This corresponds to mixing angles of and , respectively. For calculating the quantum speed limit we use the Bhattacharyya bound (50) for the two-level subspace spanned by the states and under a two-photon Rabi drive of MHz. This is the experimental value of in saSTIRAP at ns and ns (the upper left corner in Figure 3), resulting in a quantum speed limit of ns. The overlayed solid lines in Figure 3 represent constant-value transfer times for the STIRAP and saSTIRAP protocols, and the dashed lines in the STIRAP simulation show . In STIRAP, this population level is reached only in the area delineated by the dashed line while in saSTIRAP the value is everywhere higher than 0.8.

 Figure 4: Phase and pulse area control a) Shows the population of state () as a function of the area of the counterdiabatic pulse and the gauge-invariant phase. The experimental result is shown in the left panel with the corresponding simulation in the right panel. The parameters used in the experiment are ns, 25 ns, and . Note that corresponds to pure STIRAP. The blue dashed-line ellipses represent the areas where saSTIRAP is robust against changes in parameters and . b) Shows the phase dependence of the superadiabatic process as a function of the STIRAP amplitude with the experiment in the left panel and the simulation in the right panel. corresponds to the case without STIRAP and is thus completely phase independent. The experiment was performed with ns, ns, and .

Phase and pulse area control: robustness properties. STIRAP is known to be insensitive to changes in the amplitudes of the drive fields. We now show that this robustness property extends to the saSTIRAP protocol. For practical applications of the protocol, the robustness of saSTIRAP is a critical feature distinguishing it from the non-adiabatic methods. We introduce the area of the counterdiabatic pulse

(8)

and we define STIRAP pulse area as

(9)

which is a measure of adiabaticity of the STIRAP process. In Figure 4a) we show the population of state as a function of the counterdiabatic pulse area and its phase. The saSTIRAP process reveals its useful properties for the parameter values inside the area outlined with the blue dashed-line ellipses, where the pulse areas are close to , as expected from Eq. (3). For the parameters inside the ellipses, is a rather slow-varying function of , rendering the saSTIRAP process robust against errors in the area of the counterdiabatic pulse. In contrast, population transfer can take place also for values outside the ellipses, but without robustness against variations of . The right panel shows a corresponding numerical simulation, which matches the experimental results quite accurately. As noted before, the maximum transfer occurs around some optimal phases which are shifted from the ideal due to the ac Stark effect. These optimal values of are well reproduced in the simulations. Even though the operation of saSTIRAP strongly depends on the correct phase , the population transfer is not very sensitive to small variations around the optimal phase.

We can also examine the dependence of the population on the STIRAP area , while keeping the counterdiabatic pulse area constant at . The value for is chosen based on Figure 4a) so that is not sensitive to either increase or decrease in counterdiabatic pulse area. As seen in Figure 4b), high population transfer is achieved for a large set of values. There is no phase dependence for , as expected when only the two-photon pulse is applied, while in the other extreme case, at large values , STIRAP dominates and the phase dependence becomes again weaker.

 Figure 5: Robustness of the protocol with respect to the areas of the STIRAP and of the counter-diabatic pulse a) Transferred population (experiment) as a function of the STIRAP pulse area and the two-photon pulse area . We also show iso-population lines (from 0.1 to 0.7 in steps of 0.1 and from 0.7 to 0.75 in steps of 0.01) obtained from simulations, showing agreement with the data and delineating the same region of high transfer as that obtained from experiment. In this experiment, the STIRAP Rabi frequencies were increased from zero to a maximum value of MHz and MHz while keeping their ratio constant. Similarly, the two-photon pulse was varied from zero to maximum values of MHz. The separation time was ns and the width was ns. b) Transfer efficiency of saSTIRAP when the STIRAP pulse width and the counterdiabatic pulse area are varied. The orange circles are values obtained in 5 different saSTIRAP experiments where the value was obtained, while the solid lines are simulations (with that for thickened). The yellow circle is another saSTIRAP experiment, which yielded . The dashed line (simulation) corresponds to the case where is obtained by applying only the counter-adiabatic pulse. The amplitudes of the STIRAP pulses were MHz and MHz.

In order to explicitly compare saSTIRAP with the direct non-adiabatic process we show in Figure 5a) the transferred population as a function of the area of the STIRAP pulses and of the counterdiabatic pulse. The phase is tuned to yield the maximum population in state at each value of the STIRAP area according to Figure 4b). In the presence of only the counterdiabatic pulse (along the horizontal axis where ) the population transfer, as expected, occurs in a rather narrow range of values around . When STIRAP starts to work properly (at approximately ), the range of values of where the transfer occurs enlarges significantly (see also Supplementary Note 6 and Supplementary Figure 6). This demonstrates that saSTIRAP offers advantage over both the direct pulse and STIRAP: it has better fidelity than in STIRAP while being less sensitive to the variation in than the pulse.

A slightly different perspective on robustness with respect to pulse shapes and amplitudes is provided in Figure 5b), where we map the expected population transfer in saSTIRAP with respect to and . The thicker curve corresponding to a transferred population is used as a convenient delineation of the region where the transfer is effective. We checked this in five different experiments where by adjusting the parameters and we got : indeed, the experimental points correspond well to the simulation. The dotted line represents the iso-population line with only the counterdiabatic pulse acting. As increases from 10 ns to about 28 ns, the duration of the counterdiabatic pulse increases as well, and decoherence further reduces the population that can be transferred only by the counterdiabatic pulse. Surprisingly, turning on the STIRAP pulses enables population transfer with for up to 38 ns, and also in a much wider range of values for . In this sense, saSTIRAP provides an advantage in counteracting the detrimental effects of decoherence. For example, a saSTIRAP experiment with parameters corresponding to the yellow dot in Figure 5b) yielded a population (well matched by the simulation lines), while with the same parameters for and we can reach only a population of 0.45 using the two-photon pulse alone.

Iii Conclusions

We have shown that it is possible to speed up the adiabatic population transfer by introducing an additional counterdiabatic control pulse that cancels the errors resulting from imperfect adiabaticity. The counterdiabatic pulse acts on the transition, which is a forbidden direct transition in a transmon. We circumvent the problem by using a two-photon process, which effectively drives the desired transition. We have carefully characterized the robustness of the process with respect to the counterdiabatic field, and evaluate the trade-off between the speed of the process and the insensitivity to control parameters. The superadiabatic method enables a continuos interpolation between these two competing features, allowing one to select the optimal values for a given experimental process. This is in strong contrast with STIRAP, where the trade-off is between the speed of the protocol and the population transfer fidelity.

Iv Methods

Superadiabatic (transitionless) driving. Given a time-dependent but slow-changing Hamiltonian , the adiabatic theorem allows us to approximate the state of the system at each point by the instantaneous eigenvectors of the Hamiltonian . To make this evolution exact also when adiabaticity is broken, one can add a counterdiabatic Hamiltonian designed such that it quenches the transitions to states other than . This can be identified by reverse Hamiltonian engineering (see Supplementary Note 3), and has the general form (6); (7); (8); (9)

(10)

We can express the eigenstates of the STIRAP Hamiltonian in Eq. (1) as

(11)

where and After substitution of the eigenstates into Eq. (10) we get the Hamiltonian Eq. (2), with the Rabi coupling as given in Eq. (3) and the phase .

Numerical simulations. It is convenient to write the Hamiltonian in an interaction picture with respect to the undriven Hamiltonian by applying the transformation , resulting in . We separate the resulting total Hamiltonian into a part that corresponds to couplings via the fields and used in STIRAP, and a part produced by two-photon driving.

For the STIRAP part we have

(12)
(13)

Here the factors of in the cross-coupling terms are due to the increase by of the matrix elements as we go from the first to the second transition. In the rotating wave approximation, by neglecting terms oscillating at frequencies , we find

(14)

This eventually leads to the Hamiltonian of Eq. (1) under the gauge transformation described in the text.

To drive the two-photon transition we use a single microwave field with frequency and phase , such that the energy conservation condition holds. This tone is detuned from the and transitions by . We denote by and the Rabi couplings corresponding to this field into the and transitions respectively, noting again that in the weak anharmonicity approximation for the transmon . The two-photon field results in the Hamiltonian

(15)

We neglect the fast rotating terms at and and obtain

(16)

which produces (46) a two-photon complex coupling with Rabi frequency and phase ,

(17)

allowing us to use this coupling as a counterdiabatic Hamiltonian as in Eq. (2). Note that the relative phase between the counterdiabatic two-photon pulse and the STIRAP pulses is fixed during the evolution: once defined at one time, it will remain the same at any other time due to the frequency matching relation (see also Suppplementary Note 2).

In the simulations we use the full Hamiltonian given in Eq. (13) and Eq. (15), which incorporates all cross-couplings of the fields into the transmon transitions. To include decoherence we use the standard Lindblad formalism, with a three-level superoperator , see e.g. (51); (52) for details.

All the off-detuned drivings produce parasitic ac Stark shifts of the energy levels, which are the main source of fidelity loss in our experiment besides decoherence. The largest ac Stark shifts are produced by the two-photon pulse, which effectively displaces the energy levels of the qutrit as seen by the STIRAP pulses. We can reduce these erros by operating the STIRAP slightly off-resonance with the bare qutrit frequencies but on-resonance with the new (shifted) energy levels, effectively canceling the ac Stark shifts in the region where most of the population transfer occurs (see Supplementary Figure 4, Supplementary Note 5). More sophisticated methods have been proposed, such as time-dependent frequency corrections or exact cancellations using an additional two-photon drive, designed with a detuning with opposite sign and a phase in one of the drives, such that these shifts are canceled exactly (42). A more in-depth analysis of these errors is delegated to Supplementary Note 5.

Experimental methods. The qutrit and the readout resonator were realized by two-angle deposition of aluminum on high-resistivity Si substrate. The dependence of the 0 – 1 transition frequency on flux was determined from the spectroscopy measurements, where one microwave tone was sent to the resonator, and another tone was used to excite the qutrit. By increasing the qutrit excitation tone power we could also identify the two-photon transition frequency to the second excited state. The pulses for all three transitions (0 – 1, 0 – 2, and 1 – 2) were deduced from the standard Rabi experiments. The relaxation rate MHz was determined by first exciting the transmon to the state with a pulse and recording the exponential decay traces to the state as a function of time. To find the rate MHz we excited the transmon to the state with a two-photon pulse and observed its decay to the ground state. was found by fitting the numerical three-level exponential decay model to the measured data. Additional sources of noise exist in the experimental setup, which lead to energy level shifts and dephasing. However, for this particular sample the large relaxation rates provide the dominant decoherence mechanism, making the precise determination of the additional pure dephasing noise difficult. We have verified this by performing Ramsey measurements, where two pulses with a variable time separation are applied to the system. We estimate that the pure dephasing rates (53) are at most of the order of MHz.

Special precautions are taken to ensure the stability of the relative phases between the pulses: to achive this, we use a single microwave signal which is split in three parts and mixed in three IQ mixers with the waveforms produced by a high-sampling rate multichannel waveform generator (see Supplementary Note 4 and Supplementary Figure 1 for more details). The state of the qutrit is obtained by three-level quantum tomography, where the diagonal elements of the density matrix are obtained from the averaged IQ traces of the cavity response (54). The measured trace

(18)

is a linear combination of calibration traces corresponding to states , and with weight factors , , and , which give the occupation probability of each state. Using the least squares fit of the calibration traces to the measured trace, we can extract the most likely occupation probabilities for the three level system.

The calibration traces inevitably include the effect of relaxation, which, if left uncompensated, can lead to an artificial overestimation of the state population in both STIRAP and saSTIRAP. However, since we know the relaxation rates, we can correct for this effect by modifying the calibration trajectories to include some contribution from the lower states, described by errors with . The measured trajectory is then given by

(19)

with describing the ideal responses of state . Using the already corrected responses for the lower states, we can iteratively correct the response of the next state by substracting the contributions for all the lower states, finally yielding

(20)

We use , , and , which are obtained from the relaxation experiments with three levels.

Quantum speed limit. In an ideal saSTIRAP protocol, the tails of the Gaussians forming the STIRAP part extend to infinitely large times. Thus, to get a measure of the time over which population transfer occurs one has to resort to introducing a convention. Here we define the transfer time between an initial dark state and a final state as

(21)

A convenient choice for a dissipative system is to take an initial state with 99 % population in and a final state with 80 % population in (for dissipationless systems the latter is usually taken 90%, see (30)). This corresponds to STIRAP mixing angles and . In Figure 3 we plot the resulting for both the STIRAP and the saSTIRAP protocols.

To find the quantum speed limit in the subspace spanned by , we use the Bhattacharyya bound (50) applied to the initial and final states and yielding

(22)

To connect it with the real experimental situation we take the initial and final states with the same populations as above, while for we use the maximum value of the 0 – 2 Rabi frequency accessible experimentally . This results in

(23)

In the experiment, we reach a maximum two-photon Rabi frequency MHz (upper left corner in Figure 3), yielding ns.

Acknowledgements
This work used the cryogenic facilities of the Low Temperature Laboratory at Aalto University. We acknowledge financial support from FQXi, Väisalä Foundation, the Academy of Finland (project 263457), and the Center of Excellence “Low Temperature Quantum Phenomena and Devices” (project 250280).

Author Contributions
AV wrote the codes for the simulations. SD fabricated the sample. SD and AV took the measurements. GSP. initialized and supervised the project and contributed to the theoretical analysis. AV and GSP wrote the manuscript with additional contributions from SD.

Additional information
Supplementary Information
accompanies this paper. Competing financial interests: The authors declare no competing financial interests.

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