Research ArticlePHYSICS

Superadiabatic population transfer in a three-level superconducting circuit

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Science Advances  08 Feb 2019:
Vol. 5, no. 2, eaau5999
DOI: 10.1126/sciadv.aau5999

Abstract

Adiabatic manipulation of the quantum state is an essential tool in modern quantum information processing. Here, we demonstrate the speedup of the adiabatic population transfer in a three-level superconducting transmon circuit by suppressing the spurious nonadiabatic excitations with an additional two-photon microwave pulse. We apply this superadiabatic method to the stimulated Raman adiabatic passage, realizing fast and robust population transfer from the ground state to the second excited state of the quantum circuit.

INTRODUCTION

The ability to accurately manipulate the state of quantum systems is one of the prerequisites for high-fidelity quantum information processing (1). The adiabatic control of quantum states is based on slowly modifying the energy eigenstates of gapped systems; if the condition for adiabatic following is satisfied, the system remains in its instantaneous eigenstate at any moment in time. Techniques that are generically referred to as shortcuts to adiabaticity (2) aim at achieving faster operation times through a guided evolution of the system toward the desired final state, bypassing the restriction of the adiabatic theorem.

For adiabatic quantum computing (3), quantum annealing (4, 5), and holonomic quantum computing (68), shortcuts to adiabaticity would be one important route to quantum advantage (9). In quantum thermodynamics, the suppression of interlevel transitions during adiabatic cycles could lead to engines with increased efficiency (10), providing novel insights into the foundations of the third law of thermodynamics (1113). Furthermore, in multilevel quantum information processing (14), shortcuts to adiabaticity can be used for robust gates (15) and efficient initial state preparation.

Superadiabatic protocols (also called transitionless driving) (1619) are a type of shortcut to adiabaticity based on counterdiabatic driving—designed such that they suppress nonadiabatic excitations; in consequence, the system follows the instantaneous Hamiltonian eigenstate at any time during evolution. These protocols are universal, and the robustness against errors is inherited from the corresponding adiabatic process. However, a major difficulty in implementing them stems from the fact that the superadiabatic control drive uses complex couplings with externally controlled and stable Peierls phases (20). In optical setups, this would require lasers with exquisitely low phase noise. This is why so far superadiabatic protocols have been demonstrated only in simple configurations, involving either two levels (21, 22) or two control fields (2325).

Here, we show that the required phase stability can be achieved in the microwave regime using circuit quantum electrodynamics as the experimental platform (26). We use the first three states of a superconducting transmon circuit (27) to transfer population between the ground state and the second excited state. This is an important task in quantum control of multilevel systems, where fast and efficient state preparation serves as an initial step for many algorithms (28, 29). We achieve this by using three microwave pulses: Two of them realize the stimulated Raman adiabatic passage (STIRAP) (3032), while the third is a two-photon process creating the counterdiabatic Hamiltonian, which forces the system to follow its instantaneous eigenstate even though the adiabatic condition is violated. This type of driving, called loop 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 resonant STIRAP Hamiltonian can be written asEmbedded Image(1)where Ω01(t) and Ω12(t) describe the Rabi coupling of the microwave drive pulses to the transmon in the frame rotating with the drive frequencies. The drives have a Gaussian shape (32)Embedded Image(2)where ts is the lag between the two pulses. In the experiment, we use two intermediate frequency microwave tones with externally controlled phases, ϕ01 and ϕ12, which are digitally mixed with the pulse envelopes Ω01(t) and Ω12(t) using an arbitrary waveform generator (see the Supplementary Materials for details). The pulses are further mixed in an analog IQ mixer with a local oscillator tone ωLO/(2π) = 6.92 GHz to produce two signals that resonantly drive the 0–1 and 1–2 transitions of the three-level system at frequencies ω01/(2π) = 6.99 GHz and ω12/(2π) = 6.62 GHz (see Fig. 1).

Fig. 1 Schematic of the experiment.

(A) Loop driving for saSTIRAP: A counterdiabatic drive with effective Rabi frequency Ω02 (dashed purple arrow) is applied in parallel with a STIRAP sequence consisting of pulses Ω01 and Ω12, 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 Δ with respect to the first transition), which couples with strengths Ω2ph and Embedded Image 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 of the transmon. (D) Geometric representation of the Hamiltonian on a three-site plaquette with Peierls hopping and resulting gauge-invariant phase Φ = ϕ01 + ϕ12 + ϕ20.

In the STIRAP protocol, the system follows adiabatically one of the instantaneous eigenstates of the above Hamiltonian, called the dark state, Embedded Image, where Θ(t) = tan−101(t)/Ω12(t)] changes slowly from 0 to π/2. This implies that the pulse driving the 1–2 transition is counterintuitively applied before the 0–1 pulse, enabling the population to be transferred directly to the second excited state without exciting the intermediate state |1〉 at any time in between. However, if the change in the amplitudes of the control signals is too abrupt, the system gets diabatically excited away from the state |D(t)〉, reducing the transferred population and therefore limiting the fidelity of the process.

The spurious excitations of STIRAP can be canceled using the superadiabatic method (1619). The idea is to design a new control Hamiltonian, which evolves the system through the adiabatic states given by the STIRAP Hamiltonian in Eq. 1, even when the adiabatic condition is not fully satisfied (16). The form of the counterdiabatic Hamiltonian can be found by reverse Hamiltonian engineering (19, 34, 35) (see the Supplementary Materials for the derivation), requiring the addition of a third control pulse given byEmbedded Image(3)with Rabi couplingEmbedded Image(4)and a phase ϕ20 that must satisfy the relation ϕ01 + ϕ12 + ϕ20 = − π/2 (36). For the STIRAP pulse amplitudes given in Eq. 2 and assuming Ω01 = Ω12, the shape of the counterdiabatic pulse can be evaluated as (35)Embedded Image(5)

To experimentally create the microwave pulse implementing the counterdiabatic Hamiltonian, we use a two-photon process generated by a third microwave drive field with frequency ω2ph = (ω01 + ω12)/2 and phase ϕ2ph, which couples into the 0–1 and 1–2 transitions with respective Rabi couplings Ω2ph and Embedded Image. The factor Embedded Image is a consequence of the almost harmonic energy level structure of the transmon circuit, which results in a higher dipole coupling for higher transitions (27). The low anharmonicity also leads to selection rules that prevent us from using a direct 0–2 drive to implement the counterdiabatic Hamiltonian. The chosen drive frequency results in detunings ± Δ from both the 0–1 and 1–2 transitions, Δ = ω01 − ω2ph = (ω01 − ω12)/2, thus satisfying the two-photon resonance condition. The two-photon driving generates an effective Rabi coupling Embedded Image and phase ϕ20 = − 2ϕ2ph − π, which can be obtained from perturbation theory (15, 37). In addition, two-photon driving creates small ac-Stark shifts to all the energy levels, which appear as dynamic detunings of the drive frequencies from the transitions. We compensate for this effect by slightly tuning the phases of all the drive pulses during the evolution (see Methods for details).

RESULTS

Efficient transfer of population

To demonstrate that the superadiabatic protocol corrects for the nonadiabatic losses even when the adiabaticity condition for STIRAP is not satisfied, we experimentally compare the two methods in Fig. 2. Here, the peak STIRAP amplitudes Ω01 and Ω12 were chosen as Ω01/(2π) = Ω12/(2π) = 25.5 MHz, the separation of the two STIRAP pulses is ts/σ = −1.5, and the widths of the Gaussian pulse shapes are σ = 20 ns. During STIRAP, there is a significant population in state |1〉 due to the violation of the adiabatic condition, which results in transitions between the instantaneous eigenstates of the system. Consequently, the population p2 is only 0.8 after the pulses. In the saSTIRAP experiment, there is almost no population in state |1〉 and p2 reaches 0.96, which is very close to the ideal performance, demonstrating the power of the superadiabatic method. The result is supported by the numerical simulation, shown with solid lines (see Methods for details). The dashed lines show a simulation with the same parameters but without decoherence, resulting in p2 = 0.9997 and confirming that most of the remaining losses in the saSTIRAP experiment are caused by the energy relaxation of the qutrit (with rates Γ01 = 0.6 MHz and Γ12 = 0.83 MHz, obtained by independent measurements).

Fig. 2 Comparison between STIRAP and saSTIRAP.

Time evolution of the populations p0, p1, and p2 during STIRAP (diamonds) and saSTIRAP (circles). The solid lines show the corresponding simulation, which includes decoherence. A simulation for the ideal case without decoherence is presented with dashed lines. The experiment was performed with the parameters Ω01 = Ω12 = 25.5 MHz, ts/σ = − 1.5, and σ = 20 ns.

In Fig. 3, we show the performance of the superadiabatic protocol for a wide range of STIRAP parameters. We explore the parameter space (ts, σ) by varying the STIRAP pulse width σ and the normalized STIRAP pulse separation |ts|/σ, as shown in Fig. 3A. The optimal pulse separation for STIRAP is ts/σ = −1.5 (31). In the upper part of the plot, the STIRAP fidelity is low because the separation of the pulses is too large, whereas for small σ the adiabatic condition is not satisfied. STIRAP also fails for too small pulse separations; some high-fidelity population transfer seen around ts = 0 in the experiment is not due to STIRAP but is driven by the holonomic gate studied in (8, 38). The experiment can be compared to a numerical simulation, which replicates the results accurately (right panel in the figure). Figure 3B demonstrates that, by adding the counterdiabatic drive, we are able to counteract the diabatic losses for almost all the STIRAP parameters.

Fig. 3 Correction of the nonadiabatic losses with the saSTIRAP protocol.

(A) Population p2 in the state |2〉 for the STIRAP process with Ω01/(2π) = Ω12/(2π) = 25.5 MHz as a function of the pulse width σ and the normalized pulse separation |ts|/σ. (B) Population p2 for 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 Embedded Image in nanoseconds to achieve the population p2 = 0.9 in saSTIRAP.

The performance of the protocol can be further characterized by comparing its transfer speed to the quantum speed limit at the maximum counter-diabatic pulse coupling. We follow a convention where the duration of the saSTIRAP protocol is defined as the time lapse between an initial state with population 0.99 in the ground state and a final state with population 0.9 in the second excited state (35). This corresponds to initial and final mixing angles of Θi = 0.03 π and Θf = 0.4 π, respectively. For calculating the quantum speed limit, we use the Bhattacharyya bound (39) for the two-level subspace spanned by the states |0〉 and |2〉 under the maximal experimentally accessible two-photon Rabi drive Embedded Image MHz. We take the initial and final states with the same populations as above, which results in Embedded Image. The quantum speed limit can be compared to the transfer times for the saSTIRAP protocol, which are shown by the overlaid solid lines in Fig. 3B. The transfer times are the fastest (TsaSTIRAP ≈ 2.0TQSL) in the upper left corner of the panels corresponding to σ = 10 ns and |ts/σ| = 3. However, as we approach that point, the STIRAP fidelity is also reduced and, in consequence, the population transfer occurs predominantly due to the counterdiabatic driving. Thus, the population transfer will start to be increasingly sensitive to the amplitudes of the pulses. To improve the robustness, the strength of the STIRAP part must be increased by reducing ts/σ or by increasing σ, which leads to a reduction of transfer speed. The trade-off is important for the potential applications of the superadiabatic method and will be analyzed later in greater detail.

Gauge-invariant phase

Loop driving with complex couplings between each pair of states results in a nontrivial synthetic gauge structure on the triangular plaquette formed by the three states, previously studied theoretically in (36, 40); related schemes have been proposed for cold atom lattices in (41). See Fig. 1D for a simple illustration.

In Fig. 4, we demonstrate experimentally that, in a three-level transmon, the dynamics of the system is determined by the gauge-invariant phase Φ = ϕ01 + ϕ12 + ϕ20. We present the population transferred to state |2〉, when one of the phases ϕ01, ϕ12, or ϕ2ph is kept fixed, while the other two are varied. The populations are measured at a time t = 20 ns after the maximum of the 0–1 drive pulse, and the two-photon pulse is set to satisfy Eq. 4. The experiment shows that the transferred population to state |2〉, p2, depends only on ϕ01 + ϕ12 + ϕ20 = Φ and not on each phase separately (36). This allows us to choose the gauge ϕ01 = ϕ12 = 0 and use ϕ20 = Φ = −2ϕ2ph − π as the externally controlled gauge-invariant phase.

Fig. 4 Control of the system dynamics with the gauge-invariant phase.

Under loop driving, the phase Φ is a gauge-invariant quantity, in analogy with lattice gauge theories, where it is typically produced by an applied magnetic field. The three-dimensional plot shows lines of constant population p2 in the state |2〉, in the orthogonal planes (ϕ12, ϕ2ph) (with ϕ01 constant), (ϕ01, ϕ2ph) (with ϕ12 constant), and (ϕ12, ϕ01) (with ϕ2ph constant). The gauge-invariance relation ϕ01 + ϕ12 − 2ϕ2ph − π = Φ corresponds to tilted planes that intersect the axes. Note also that the periodicity along the ϕ2ph axis is twice that of the periodicity along the axes ϕ01 and ϕ12 as a result of two-photon driving. In the experiment, we had Ω01/(2π) = Ω12/(2π) = 25.5 MHz, ts = −30 ns, and σ = 20 ns.

In this gauge, the full Hamiltonian of the system readsEmbedded Image(6)thus simplifying the problem significantly (see also the Supplementary Materials).

Robustness properties

STIRAP is known to be insensitive to changes in the amplitudes of the drive fields. The crucial question is whether this robustness extends to the amplitude of the counterdiabatic field, as for the practical applications of the protocol its resilience to errors is a critical feature distinguishing it from the nonadiabatic methods. First, we introduce the area of the counterdiabatic pulseEmbedded Image(7)and we define STIRAP pulse area asEmbedded Image(8)which is the measure of adiabaticity of STIRAP according to the global adiabatic condition Embedded Image (31). In Fig. 5, we show the population of state |2〉, p2, 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 blue dashed-line ellipses, where the pulse areas Embedded Image are close to π, as expected from Eq. 5. For the parameters Embedded Image inside the ellipses, p2 is a rather slow-varying function of Embedded Image, indicating that saSTIRAP is robust against errors in the area of the counterdiabatic pulse. In contrast, population transfer can also take place for values Embedded Image outside the ellipses, but without robustness against variations of Embedded Image. The right panel shows a corresponding numerical simulation, which matches the pattern seen in the experiment quite well. From the simulation, we can also see that the maximum transfer occurs around an optimal phase, which is very close to the ideal ϕ2ph = −π/4 + nπ. In the experiment, a small shift exists in the phases due to the phase imbalance of the IQ mixer (see Fig. 1C) used to combine the driving pulses (more details available in the Supplementary Materials).

Fig. 5 Robustness of saSTIRAP against variations in the counterdiabatic pulse parameters.

Population p2 in the state |2〉 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 ts = −30 ns, σ = 20 ns, and Embedded Image. Note that Embedded Image corresponds to pure STIRAP. The blue dashed-line ellipses represent the areas where saSTIRAP is robust against changes in parameters Embedded Image and ϕ2ph.

To explicitly compare saSTIRAP with the direct nonadiabatic process, we show in Fig. 6 the transferred population as a function of the area A of the STIRAP pulses and Embedded Image of the counterdiabatic pulse. The phase ϕ2ph is tuned to yield the maximum population in state |2〉 at each value of the STIRAP area A. In the presence of only the counterdiabatic pulse (along the horizontal axis where Embedded Image), the population transfer, as expected, occurs in a rather narrow range of Embedded Image values around π. When the area of the STIRAP pulses is increased (at approximately Embedded Image), the range of values of Embedded Image where the transfer occurs enlarges significantly. This demonstrates the advantage that the superadiabatic method offers: It has better fidelity than STIRAP while being less sensitive to the variation in Embedded Image than a raw π pulse. Theoretically, the fidelity of STIRAP approaches unity only in the limit of infinite pulse area (the adiabatic condition is fully satisfied), whereas ideal saSTIRAP has unit fidelity for all the values of the STIRAP pulse area.

Fig. 6 Comparison between saSTIRAP and nonadiabatic population transfer.

Transferred population p2 (experiment) as a function of the STIRAP pulse area A defined in Eq. 8 and the two-photon pulse area Embedded Image from Eq. 7. We also show isopopulation lines (from 0.1 to 0.8 in steps of 0.1 and from 0.8 to 1.0 in steps of 0.01) obtained from the simulations, showing agreement with the data and delineating the same region of high transfer as that obtained from the experiment. In this experiment, the peak STIRAP Rabi frequencies were increased from zero to Ω01/(2π) = Ω12/(2π) = 40 MHz. Similarly, the two-photon pulse amplitude was varied from zero to Ω2ph/(2π) = 77 MHz. The horizontal axis with Embedded Image corresponds to two-photon Rabi driving, whereas the vertical axis with Embedded Image corresponds to standard STIRAP. In the experiment, the STIRAP pulse separation was ts = −30 ns and the pulse width was σ = 20 ns.

Figure 6 also demonstrates that, even though the maximal effective two-photon coupling is smaller than the direct 0–1 and 1–2 couplings, it does not severely restrict the speed of the method, because the optimal two-photon pulse area Embedded Image is usually significantly smaller than the STIRAP area A required to provide the demanded robustness. This is also an advantage over rapid adiabatic passage (42, 43), where a much stronger two-photon pulse on the 0–2 transition would be needed.

METHODS

Three-level quantum tomography

The state of the qutrit was obtained by three-level quantum tomography, where the diagonal elements of the density matrix were calculated from the averaged IQ traces of the cavity response (44). The measured traceEmbedded Image(9)is a linear combination of calibration traces corresponding to states |0〉, |1〉, and |2〉 with weight factors p0, p1, and p2, which give the occupation probability of each state. Here, τ is the time from the beginning of the measurement pulse. 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 ζij with i < j. The measured trajectory rj is then given byEmbedded Image(10)with Embedded Image describing the unknown ideal responses of state |i〉. From the above equation, the ideal responses can be solved iteratively, yieldingEmbedded Image(11)

We used ζ01 = 0.01, ζ12 = 0.01, and ζ02 = 0.02, which are obtained by comparing a reference Rabi experiment to a corresponding simulation with known energy relaxation rates.

Dynamical phase correction

The off-resonant two-photon driving produced parasitic ac-Stark shifts of the energy levels, which we compensated for by using dynamically adjusted phases. Following (15), the ac-Stark shifts can be calculated from the second-order perturbation theory as Embedded Image, where V(t) consists of the off-diagonal elements of the two-photon drive Hamiltonian Embedded Image in the frame rotating with the drive. The energies En are the detunings of the drive from the 0–1 and 1–2 transitions, E0 = 0, E1 = ℏΔ, and E2 = 0. The resulting ac-Stark shifts εn,k = Embedded ImagekEmbedded Imagen − (EkEn) are ε01(t) = ℏ|Ω2ph|2/Δ, ε12 = − 5ℏ|Ω2ph|2/(4Δ), and ε0,2 = − ℏ|Ω2ph|2/(4Δ). To compensate for the shifts in the energy levels, we dynamically modified the phases of all the three drives as Embedded Image. As a result, the frequencies of the drives matched the ac-Stark shifted qutrit transition frequencies at all instants of time.

Numerical simulations

The system was modeled with the HamiltonianEmbedded Image(12)in the frame rotating with the STIRAP drives. Here, H0 is the STIRAP Hamiltonian given in Eq. 1, and the evolution of the system was solved from the Lindblad master equation Embedded Image, where ρ(t) is the density matrix of the system and Γi,i + 1 are the energy relaxation rates (obtained by independent qubit characterization measurements).

CONCLUSIONS

We have demonstrated a speedup of population transfer in STIRAP by introducing an additional counterdiabatic two-photon control pulse that corrects for nonadiabaticity. The process was controlled by the pulse amplitudes and by a gauge-invariant phase. We have characterized the robustness of the process with respect to the counterdiabatic field and evaluated the trade-off between the speed of the process and the insensitivity to control parameters.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/2/eaau5999/DC1

Experimental setup and sample

Reverse engineering of the counteradiabatic drive

Synthetic Peierls couplings on the triangular plaquette

Fig. S1. Electronics, cryogenics, and sample schematic.

Fig. S2. Pulse sequence for saSTIRAP.

References (4547)

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

REFERENCES AND NOTES

Acknowledgments: This work used the cryogenic facilities of the Low Temperature Laboratory 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 “Finnish Center 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 the results. A.V. wrote the manuscript together with G.S.P., with additional contributions from S.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. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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