This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Powered by TCPDF (www.tcpdf.org) This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Vepsäläinen, Antti; Danilin, Sergey; Sorin Paraoanu, Gheorghe Superadiabatic population transfer in a three-level superconducting circuit Published in: Science Advances DOI: 10.1126/sciadv.aau5999 Published: 08/02/2019 Document Version Publisher's PDF, also known as Version of record Please cite the original version: Vepsäläinen, A., Danilin, S., & Sorin Paraoanu, G. (2019). Superadiabatic population transfer in a three-level superconducting circuit. Science Advances, 5(2), 1-6. [eaau5999]. https://doi.org/10.1126/sciadv.aau5999
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This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.
Powered by TCPDF (www.tcpdf.org)
This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user.
Vepsäläinen, Antti; Danilin, Sergey; Sorin Paraoanu, GheorgheSuperadiabatic population transfer in a three-level superconducting circuit
Published in:Science Advances
DOI:10.1126/sciadv.aau5999
Published: 08/02/2019
Document VersionPublisher's PDF, also known as Version of record
Please cite the original version:Vepsäläinen, A., Danilin, S., & Sorin Paraoanu, G. (2019). Superadiabatic population transfer in a three-levelsuperconducting circuit. Science Advances, 5(2), 1-6. [eaau5999]. https://doi.org/10.1126/sciadv.aau5999
Superadiabatic population transfer in a three-levelsuperconducting circuitAntti Vepsäläinen, Sergey Danilin, Gheorghe Sorin Paraoanu*
Adiabaticmanipulation of the quantum state is an essential tool inmodern quantum information processing. Here, wedemonstrate the speedup of the adiabatic population transfer in a three-level superconducting transmon circuit bysuppressing the spurious nonadiabatic excitations with an additional two-photon microwave pulse. We apply thissuperadiabatic method to the stimulated Raman adiabatic passage, realizing fast and robust population transfer fromthe ground state to the second excited state of the quantum circuit.
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INTRODUCTIONThe ability to accuratelymanipulate the state of quantum systems is oneof the prerequisites for high-fidelity quantum information processing(1). The adiabatic control of quantum states is based on slowly modify-ing the energy eigenstates of gapped systems; if the condition for adia-batic following is satisfied, the system remains in its instantaneouseigenstate at any moment in time. Techniques that are generically re-ferred to as shortcuts to adiabaticity (2) aimat achieving faster operationtimes through a guided evolution of the system toward the desired finalstate, bypassing the restriction of the adiabatic theorem.
For adiabatic quantum computing (3), quantum annealing (4, 5),and holonomic quantum computing (6–8), shortcuts to adiabaticitywould be one important route to quantum advantage (9). In quantumthermodynamics, the suppression of interlevel transitions during adia-batic cycles could lead to engines with increased efficiency (10),providing novel insights into the foundations of the third law of ther-modynamics (11–13). Furthermore, inmultilevel quantum informationprocessing (14), shortcuts to adiabaticity can be used for robust gates(15) and efficient initial state preparation.
Superadiabatic protocols (also called transitionless driving) (16–19)are a type of shortcut to adiabaticity based on counterdiabatic driving—designed such that they suppress nonadiabatic excitations; in conse-quence, the system follows the instantaneous Hamiltonian eigenstate atany time during evolution. These protocols are universal, and the ro-bustness against errors is inherited from the corresponding adiabaticprocess. However, a major difficulty in implementing them stems fromthe fact that the superadiabatic control drive uses complex couplingswith externally controlled and stable Peierls phases (20). In optical set-ups, this would require lasers with exquisitely low phase noise. This iswhy so far superadiabatic protocols have been demonstrated only insimple configurations, involving either two levels (21, 22) or two controlfields (23–25).
Here, we show that the required phase stability can be achieved inthe microwave regime using circuit quantum electrodynamics as theexperimental platform (26). We use the first three states of a super-conducting transmon circuit (27) to transfer population between theground state and the second excited state. This is an important task inquantum control of multilevel systems, where fast and efficient statepreparation serves as an initial step for many algorithms (28, 29). Weachieve this by using three microwave pulses: Two of them realize thestimulated Raman adiabatic passage (STIRAP) (30–32), while the third
is a two-photon process creating the counterdiabatic Hamiltonian,which forces the system to follow its instantaneous eigenstate eventhough the adiabatic condition is violated. This type of driving, calledloop configuration (33), results in an externally controlled gauge-invariant phase and implements the superadiabatic STIRAP (saSTIRAP)protocol (34, 35).
For a three-level system in the ladder configuration, the resonantSTIRAP Hamiltonian can be written as
H0ðtÞ ¼ ℏ
2W01ðtÞeif01 j0� �
1j þ W12ðtÞeif12 j1� �
⟨2j þ h:c:� ð1Þ
where W01(t) and W12(t) describe the Rabi coupling of the microwavedrive pulses to the transmon in the frame rotating with the drive fre-quencies. The drives have a Gaussian shape (32)
where ts is the lag between the two pulses. In the experiment, we use twointermediate frequency microwave tones with externally controlledphases, f01 and f12, which are digitally mixed with the pulse envelopesW01(t) and W12(t) using an arbitrary waveform generator (see the Sup-plementary Materials for details). The pulses are further mixed in ananalog IQ mixer with a local oscillator tone wLO/(2p) = 6.92 GHzto produce two signals that resonantly drive the 0–1 and 1–2 transi-tions of the three-level system at frequencies w01/(2p) = 6.99 GHz andw12/(2p) = 6.62 GHz (see Fig. 1).
In the STIRAP protocol, the system follows adiabatically one of theinstantaneous eigenstates of the above Hamiltonian, called the darkstate, jDðtÞi ¼ cosQðtÞeif12 j0i � sinQðtÞe�if01 j2i , where Q(t) =tan−1[W01(t)/W12(t)] changes slowly from 0 to p/2. This implies thatthe pulse driving the 1–2 transition is counterintuitively applied beforethe 0–1 pulse, enabling the population to be transferred directly to thesecond excited statewithout exciting the intermediate state |1⟩ at any timein between. However, if the change in the amplitudes of the controlsignals is too abrupt, the system gets diabatically excited away fromthe state |D(t)⟩, reducing the transferred population and thereforelimiting the fidelity of the process.
The spurious excitations of STIRAP can be canceled using the super-adiabaticmethod (16–19).The idea is todesign anewcontrolHamiltonian,which evolves the system through the adiabatic states given by the STIRAPHamiltonian in Eq. 1, even when the adiabatic condition is not fullysatisfied (16). The form of the counterdiabatic Hamiltonian can be
found by reverse Hamiltonian engineering (19, 34, 35) (see the Sup-plementary Materials for the derivation), requiring the addition of athird control pulse given by
HcdðtÞ ¼ ℏ
2W02ðtÞe�if20 j0� �
⟨2j þ h:c:� ð3Þ
with Rabi coupling
W02ðtÞ ¼ 2 _QðtÞ ð4Þ
and a phase f20 that must satisfy the relation f01 + f12 + f20 = − p/2 (36).For the STIRAPpulse amplitudes given inEq. 2 and assumingW01 =W12,the shape of the counterdiabatic pulse can be evaluated as (35)
W02ðtÞ ¼ � tss2
1
cosh � tss2 ðt � ts=2Þ
� � ð5Þ
To experimentally create the microwave pulse implementing thecounterdiabatic Hamiltonian, we use a two-photon process generated bya third microwave drive field with frequency w2ph = (w01 + w12)/2 andphase f2ph, which couples into the 0–1 and 1–2 transitions with re-spective Rabi couplings W2ph and
ffiffiffi2
pW2ph. The factor
ffiffiffi2
pis a conse-
quence of the almost harmonic energy level structure of the transmoncircuit, which results in a higher dipole coupling for higher transitions(27). The low anharmonicity also leads to selection rules that preventus from using a direct 0–2 drive to implement the counterdiabaticHamiltonian. The chosen drive frequency results in detunings ± D fromboth the 0–1 and 1–2 transitions, D = w01 − w2ph = (w01 − w12)/2, thus
Vepsäläinen et al., Sci. Adv. 2019;5 : eaau5999 8 February 2019
satisfying the two-photon resonance condition. The two-photon driv-ing generates an effective Rabi coupling W02ðtÞ ¼
ffiffiffi2
pW2
2ph=ð2DÞ andphase f20 = − 2f2ph − p, which can be obtained from perturbationtheory (15, 37). In addition, two-photon driving creates small ac-Starkshifts to all the energy levels, which appear as dynamic detunings of thedrive frequencies from the transitions.We compensate for this effect byslightly tuning the phases of all the drive pulses during the evolution (seeMethods for details).
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RESULTSEfficient transfer of populationTo demonstrate that the superadiabatic protocol corrects for the non-adiabatic losses even when the adiabaticity condition for STIRAP is notsatisfied, we experimentally compare the two methods in Fig. 2. Here,the peak STIRAP amplitudes W01 and W12 were chosen as W01/(2p) =W12/(2p) = 25.5MHz, the separation of the two STIRAP pulses is ts/s =−1.5, and the widths of the Gaussian pulse shapes are s = 20 ns. DuringSTIRAP, there is a significant population in state |1⟩ due to the violationof the adiabatic condition, which results in transitions between the in-stantaneous eigenstates of the system. Consequently, the population p2is only 0.8 after the pulses. In the saSTIRAP experiment, there is almostno population in state |1⟩ and p2 reaches 0.96, which is very close to theideal performance, demonstrating the power of the superadiabaticmethod. The result is supported by the numerical simulation, shownwith solid lines (see Methods for details). The dashed lines show asimulation with the same parameters but without decoherence, result-ing in p2 = 0.9997 and confirming that most of the remaining losses inthe saSTIRAP experiment are caused by the energy relaxation of thequtrit (with rates G01 = 0.6 MHz and G12 = 0.83 MHz, obtained byindependent measurements).
In Fig. 3, we show the performance of the superadiabatic protocolfor a wide range of STIRAP parameters. We explore the parameterspace (ts, s) by varying the STIRAP pulse width s and the normalizedSTIRAP pulse separation |ts|/s, as shown in Fig. 3A. The optimal pulseseparation for STIRAP is ts/s = −1.5 (31). In the upper part of theplot, the STIRAP fidelity is low because the separation of the pulses
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A
C D
B
Fig. 1. Schematic of the experiment. (A) Loop driving for saSTIRAP: A counter-diabatic drive with effective Rabi frequency W02 (dashed purple arrow) is applied inparallel with a STIRAP sequence consisting of pulses W01 and W12, which are res-onant with the respective transitions 0–1 and 1–2. The counterdiabatic drive is atwo-photon process realized by an off-resonant pulse (detuning D with respect tothe first transition), which couples with strengths W2ph and
ffiffiffi2
pW2ph 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(including the IQ mixers used for driving and measurement) and optical image ofthe transmon. (D) Geometric representation of the Hamiltonian on a three-siteplaquette with Peierls hopping and resulting gauge-invariant phase F = f01 +f12 + f20.
Fig. 2. Comparison between STIRAP and saSTIRAP. Time evolution of the popula-tions p0, p1, and p2 during STIRAP (diamonds) and saSTIRAP (circles). The solid linesshow the corresponding simulation, which includes decoherence. A simulation forthe ideal case without decoherence is presented with dashed lines. The experimentwas performed with the parametersW01 =W12 = 25.5 MHz, ts/s = − 1.5, and s = 20 ns.
is too large, whereas for small s the adiabatic condition is not satisfied.STIRAP also fails for too small pulse separations; some high-fidelitypopulation transfer seen around ts = 0 in the experiment is not due toSTIRAP but is driven by the holonomic gate studied in (8, 38). Theexperiment can be compared to a numerical simulation, which repli-cates the results accurately (right panel in the figure). Figure 3B de-monstrates that, by adding the counterdiabatic drive, we are able tocounteract the diabatic losses for almost all the STIRAP parameters.
The performance of the protocol can be further characterized bycomparing its transfer speed to the quantum speed limit at the maxi-mum counter-diabatic pulse coupling. We follow a convention wherethe duration of the saSTIRAP protocol is defined as the time lapse be-tween an initial state with population 0.99 in the ground state and afinal state with population 0.9 in the second excited state (35). Thiscorresponds to initial and final mixing angles of Qi = 0.03 p and Qf =0.4 p, respectively. For calculating the quantum speed limit, we usethe Bhattacharyya bound (39) for the two-level subspace spanned bythe states |0⟩ and |2⟩ under the maximal experimentally accessibletwo-photon Rabi drive Wmax
02 =ð2pÞ ¼ 48 MHz. We take the initial andfinal states with the same populations as above, which results inT0:9
QSL ¼2arccosj⟨DðqiÞjDðqf Þ⟩j=Wmax
02 ≈ 7:7 ns. The quantum speed limit canbe compared to the transfer times for the saSTIRAP protocol, whichare shown by the overlaid solid lines in Fig. 3B. The transfer times arethe fastest (TsaSTIRAP ≈ 2.0TQSL) in the upper left corner of the panelscorresponding to s = 10 ns and |ts/s| = 3. However, as we approach thatpoint, the STIRAP fidelity is also reduced and, in consequence, the pop-ulation transfer occurs predominantly due to the counterdiabatic driv-ing. Thus, the population transfer will start to be increasingly sensitiveto the amplitudes of the pulses. To improve the robustness, the strengthof the STIRAP part must be increased by reducing ts/s or by increasings, which leads to a reduction of transfer speed. The trade-off is impor-
Vepsäläinen et al., Sci. Adv. 2019;5 : eaau5999 8 February 2019
tant for the potential applications of the superadiabaticmethod andwillbe analyzed later in greater detail.
Gauge-invariant phaseLoop driving with complex couplings between each pair of states resultsin a nontrivial synthetic gauge structure on the triangular plaquetteformed by the three states, previously studied theoretically in (36, 40);related schemes have been proposed for cold atom lattices in (41). SeeFig. 1D for a simple illustration.
In Fig. 4, we demonstrate experimentally that, in a three-level trans-mon, the dynamics of the system is determined by the gauge-invariantphaseF = f01 + f12 + f20.We present the population transferred to state|2⟩, when one of the phases f01, f12, or f2ph is kept fixed, while the othertwo are varied. The populations are measured at a time t = 20 ns afterthe maximum of the 0–1 drive pulse, and the two-photon pulse is setto satisfy Eq. 4. The experiment shows that the transferred popula-tion to state |2⟩, p2, depends only on f01 + f12 + f20 = F and not oneach phase separately (36). This allows us to choose the gauge f01 =f12 = 0 and use f20 = F = −2f2ph − p as the externally controlledgauge-invariant phase.
In this gauge, the full Hamiltonian of the system reads
HðtÞ ¼ ℏ
2½W01ðtÞj0i⟨1j þW12ðtÞj1⟩ 2jh
þ W02ðtÞe�iFj0i 2j þ h:c:h � ð6Þ
thus simplifying the problem significantly (see also the Supplemen-tary Materials).
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10 20 30 400
1
2
3
0
1
2
3
20
30
5070
130
10 20 30 400
1
(ns)
A
B
Fig. 3. Correction of the nonadiabatic losses with the saSTIRAP protocol.(A) Population p2 in the state |2⟩ for the STIRAP process with W01/(2p) = W12/(2p) =25.5 MHz as a function of the pulse width s and the normalized pulse separation|ts|/s. (B) Population p2 for the corresponding saSTIRAP process. The left plots areexperiments, while the right ones are the corresponding simulation results. Thesolid black lines show the transfer time t0:9tr in nanoseconds to achieve the pop-ulation p2 = 0.9 in saSTIRAP.
Fig. 4. Control of the system dynamics with the gauge-invariant phase. Un-der loop driving, the phase F is a gauge-invariant quantity, in analogy with latticegauge theories, where it is typically produced by an applied magnetic field. Thethree-dimensional plot shows lines of constant population p2 in the state |2⟩, inthe orthogonal planes (f12, f2ph) (with f01 constant), (f01, f2ph) (with f12 constant), and(f12, f01) (with f2ph constant). The gauge-invariance relation f01 + f12 − 2f2ph − p = Fcorresponds to tilted planes that intersect the axes. Note also that the periodicity alongthe f2ph axis is twice that of the periodicity along the axes f01 and f12 as a result oftwo-photon driving. In the experiment, we hadW01/(2p) =W12/(2p) = 25.5 MHz, ts =−30 ns, and s = 20 ns.
Robustness propertiesSTIRAP is known to be insensitive to changes in the amplitudes of thedrive fields. The crucial question is whether this robustness extends tothe amplitude of the counterdiabatic field, as for the practical applica-tions of the protocol its resilience to errors is a critical featuredistinguishing it from the nonadiabatic methods. First, we introducethe area of the counterdiabatic pulse
which is the measure of adiabaticity of STIRAP according to the globaladiabatic conditionA ≫ p=2(31). In Fig. 5, we show the population ofstate |2⟩, p2, as a function of the counterdiabatic pulse area and its phase.The saSTIRAP process reveals its useful properties for the parametervalues inside the area outlined with blue dashed-line ellipses, wherethe pulse areasA02 are close to p, as expected fromEq. 5. For the param-eters ðA02; f2phÞ inside the ellipses, p2 is a rather slow-varying functionofA02, indicating that saSTIRAP is robust against errors in the area ofthe counterdiabatic pulse. In contrast, population transfer can also takeplace for values ðA02; f2phÞ outside the ellipses, but without robustnessagainst variations ofA02. The right panel shows a corresponding nu-merical simulation, whichmatches the pattern seen in the experimentquite well. From the simulation, we can also see that the maximumtransfer occurs around an optimal phase, which is very close to theideal f2ph = −p/4 + np. In the experiment, a small shift exists in thephases due to the phase imbalance of the IQmixer (see Fig. 1C) used tocombine the driving pulses (more details available in the SupplementaryMaterials).
To explicitly compare saSTIRAP with the direct nonadiabatic pro-cess, we show in Fig. 6 the transferred population as a function of theareaA of the STIRAP pulses andA02 of the counterdiabatic pulse. Thephase f2ph is tuned to yield themaximumpopulation in state |2⟩ at eachvalue of the STIRAP areaA. In the presence of only the counterdiabatic
Vepsäläinen et al., Sci. Adv. 2019;5 : eaau5999 8 February 2019
pulse (along the horizontal axis whereA ¼ 0), the population transfer,as expected, occurs in a rather narrow range of A02 values around p.When the area of the STIRAP pulses is increased (at approximatelyA ≈ 2p), the range of values ofA02 where the transfer occurs enlargessignificantly. This demonstrates the advantage that the superadiabaticmethod offers: It has better fidelity than STIRAP while being less sen-sitive to the variation inA02 than a raw p pulse. Theoretically, the fidelityof STIRAP approaches unity only in the limit of infinite pulse area (theadiabatic condition is fully satisfied), whereas ideal saSTIRAP has unitfidelity for all the values of the STIRAP pulse area.
Figure 6 also demonstrates that, even though the maximal effec-tive two-photon coupling is smaller than the direct 0–1 and 1–2couplings, it does not severely restrict the speed of the method, be-cause the optimal two-photon pulse area A02 ¼ p is usually signif-icantly smaller than the STIRAP area A required to provide thedemanded robustness. This is also an advantage over rapid adiabat-ic passage (42, 43), where a much stronger two-photon pulse on the0–2 transition would be needed.
METHODSThree-level quantum tomographyThe state of the qutrit was obtained by three-level quantum tomogra-phy, where the diagonal elements of the density matrix were calculated
– – /2 0 /20
1
– – /2 0 /20
/2
3 /2
2
Fig. 5. Robustness of saSTIRAP against variations in the counterdiabatic pulseparameters. Population p2 in the state |2⟩ as a function of the area of the counter-diabatic pulse and the gauge-invariant phase. The experimental result is shown inthe left panel with the corresponding simulation in the right panel. The parametersused in the experiment are ts = −30 ns, s = 20 ns, andA ¼ 4:2p. Note thatA02 ¼ 0corresponds to pure STIRAP. The blue dashed-line ellipses represent the areas wheresaSTIRAP is robust against changes in parameters A02 and f2ph.
0 20
2
4
6
0
1
0.1
0.1
0.3
0.3
0.5
0.5
0.7
0.7
0.9
0.9
0 1
Fig. 6. Comparison between saSTIRAP and nonadiabatic population transfer.Transferred population p2 (experiment) as a function of the STIRAP pulse areaA defined in Eq. 8 and the two-photon pulse area A02 from Eq. 7. We also showisopopulation lines (from 0.1 to 0.8 in steps of 0.1 and from 0.8 to 1.0 in steps of0.01) obtained from the simulations, showing agreement with the data and deli-neating the same region of high transfer as that obtained from the experiment. Inthis experiment, the peak STIRAP Rabi frequencies were increased from zero to W01/(2p) = W12/(2p) = 40 MHz. Similarly, the two-photon pulse amplitude was varied fromzero toW2ph/(2p) = 77MHz. The horizontal axis withA ¼ 0corresponds to two-photonRabi driving, whereas the vertical axis withA02 ¼ 0 corresponds to standard STIRAP.In the experiment, the STIRAP pulse separation was ts = −30 ns and the pulse widthwas s = 20 ns.
from the averaged IQ traces of the cavity response (44). The mea-sured trace
rmeasðtÞ ¼ ∑i¼0;1;2
piriðtÞ ð9Þ
is a linear combination of calibration traces corresponding to states |0⟩,|1⟩, and |2⟩withweight factors p0, p1, and p2, which give the occupationprobability of each state. Here, t is the time from the beginning of themeasurement pulse. Using the least squares fit of the calibration tracesto the measured trace, we can extract the most likely occupation prob-abilities 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 statepopulation in both STIRAP and saSTIRAP. However, since we know therelaxation rates, we can correct for this effect bymodifying the calibrationtrajectories to include some contribution from the lower states, describedby errors zij with i < j. The measured trajectory rj is then given by
rjðtÞ ¼ 1� ∑i<jzij
� �~r jðtÞ þ ∑
i<jzij~r iðtÞ ð10Þ
with ~r iðtÞ describing the unknown ideal responses of state |i⟩. From theabove equation, the ideal responses can be solved iteratively, yielding
We used z01 = 0.01, z12 = 0.01, and z02 = 0.02, which are obtainedby comparing a reference Rabi experiment to a correspondingsimulation with known energy relaxation rates.
Dynamical phase correctionThe off-resonant two-photon driving produced parasitic ac-Stark shiftsof the energy levels, which we compensated for by using dynamicallyadjusted phases. Following (15), the ac-Stark shifts can be calculated fromthe second-order perturbation theory as ~EnðtÞ ¼ En þ ⟨njVðtÞjn⟩ þ∑k≠n
⟨kjVðtÞjn⟩En�Ek
, where V(t) consists of the off-diagonal elements ofthe two-photon drive Hamiltonian V ¼ ℏW2phðtÞðj0i⟨1jeif2phþffiffiffi2
p j1⟩ 2jeif2ph þ h:c:Þ=2�in the frame rotating with the drive. The en-
ergies En are the detunings of the drive from the 0–1 and 1–2 transi-tions, E0 = 0, E1 = ℏD, and E2 = 0. The resulting ac-Stark shifts en,k =~Ek − ~En − (Ek − En) are e01(t) = ℏ|W2ph|
2/D, e12 = − 5ℏ|W2ph|2/(4D), and
e0,2 =−ℏ|W2ph|2/(4D). To compensate for the shifts in the energy levels,we
dynamically modified the phases of all the three drives as fnkðtÞ → fnk þ∫t�∞dtenkðtÞ=ℏ.As a result, the frequencies of the drivesmatched the ac-Stark shifted qutrit transition frequencies at all instants of time.
Numerical simulationsThe system was modeled with the Hamiltonian
HsimðtÞ ¼ H0 þ ℏW2phðtÞ=2ðj0i⟨1jeiðf2phðtÞ�DtÞ
þffiffiffi2
pj1⟩⟨2jeiðf2phðtÞþDtÞ þ h:c:Þ ð12Þ
in the frame rotating with the STIRAP drives. Here, H0 is the STIRAPHamiltonian given in Eq. 1, and the evolution of the system was solved
Vepsäläinen et al., Sci. Adv. 2019;5 : eaau5999 8 February 2019
from the Lindbladmaster equation:rðtÞ ¼ �i½HsimðtÞ; rðtÞ�=ℏþ ∑i¼0;1
Gi;iþ1ðjii⟨iþ 1jrðtÞjiþ 1⟩ ij � 12 ðji
� �⟨ijrðtÞ þ rðtÞji⟩ ijÞÞh , where r(t)
is the densitymatrix of the systemandGi,i + 1 are the energy relaxation rates(obtained by independent qubit characterization measurements).
CONCLUSIONSWe have demonstrated a speedup of population transfer in STIRAP byintroducing an additional counterdiabatic two-photon control pulsethat corrects for nonadiabaticity. The process was controlled by thepulse amplitudes and by a gauge-invariant phase. We have character-ized the robustness of the process with respect to the counterdiabaticfield and evaluated the trade-off between the speed of the process andthe insensitivity to control parameters.
SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/2/eaau5999/DC1Experimental setup and sampleReverse engineering of the counteradiabatic driveSynthetic Peierls couplings on the triangular plaquetteFig. S1. Electronics, cryogenics, and sample schematic.Fig. S2. Pulse sequence for saSTIRAP.References (45–47)
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Acknowledgments: This work used the cryogenic facilities of the Low TemperatureLaboratory at Aalto University. Funding: We acknowledge financial support from FQXi,Väisalä Foundation, the Academy of Finland (project 263457), the Center of Excellence“Low Temperature Quantum Phenomena and Devices” (project 250280), and the “FinnishCenter of Excellence in Quantum Technology” (project 312296). Author contributions:A.V. and S.D. performed the experiments. S.D. fabricated the sample, and A.V. analyzed theresults. A.V. wrote the manuscript together with G.S.P., with additional contributions fromS.D. G.S.P. supervised the project. All authors contributed to the planning of the experiments.Competing interests: The authors declare that they have no competing interests. Dataand materials availability: All data needed to evaluate the conclusions in the paper arepresent in the paper and/or the Supplementary Materials. Additional data related to this papermay be requested from the authors.
Submitted 4 July 2018Accepted 20 December 2018Published 8 February 201910.1126/sciadv.aau5999
Citation: A. Vepsäläinen, S. Danilin, G. S. Paraoanu, Superadiabatic population transfer in athree-level superconducting circuit. Sci. Adv. 5, eaau5999 (2019).
