Abstract
Stimulated Raman adiabatic passage offers significant advantages for coherent population transfer between uncoupled or weakly coupled states and has the potential of realizing efficient quantum gate, qubit entanglement and quantum information transfer. Here we report on the realization of the process in the superconducting Xmon and phase qutrits—two laddertype threelevel systems in which the ground state population is coherently transferred to the second excited state via the dark state subspace. We demonstrate that the population transfer efficiency is no less than 96% and 67% for the two devices, which agree well with the numerical simulation of the master equation. Population transfer via stimulated Raman adiabatic passage is significantly more robust against variations of the experimental parameters compared with that via the conventional resonant π pulse method. Our work opens up a new venue for exploring the process for quantum information processing using the superconducting artificial atoms.
Introduction
Stimulated Raman adiabatic passage (STIRAP), which combines the processes of stimulated Raman scattering and dark state adiabatic passage, is a powerful tool used for coherent population transfer (CPT) between uncoupled or weakly coupled quantum states^{1,2,3}. It has been recognized as an important technique in quantum computing and circuit quantum electrodynamics involving superconducting qubits^{4,5,6,7,8,9,10,11,12,13}. For example, qubit rotations can be realized via STIRAP with two computational states plus an auxiliary state forming a threelevel Λ configuration^{4,5}. A scheme for generating arbitrary rotation and entanglement in the threelevel Λtype flux qutrits is also proposed^{6}, and the experimental feasibility of realizing quantum information transfer and entanglement between qubits inside microwave cavities has been discussed^{7,8}. Unlike the conventional resonant π pulse method STIRAP is known to be much more robust against variations in experimental parameters, such as the frequency, amplitude and interaction time of microwave fields and the environmental noise^{5,6,11,12}.
Recently, multilevel systems (qutrits or qudits) have found important applications in speeding up quantum gates^{14}, realizing quantum algorithms^{15}, simulating quantum systems consisting of spins greater than one half^{16}, implementing full quantumstate tomography^{17,18,19}, testing quantum contextuality^{20} and mapping to multiqubit systems^{21,22}. Unlike the highly anharmonic Λtype flux qutrits the phase and transmon (or Xmon) qutrits have the laddertype (Ξtype) threelevel configuration which is weakly anharmonic. The dipole coupling between the ground state 0〉 and the second excited state 2〉 in the phase qutrit is much weaker than those between the first excited state 1〉 and the 0〉 state or the 2〉 state. In the case of the transmon (or Xmon) qutrit the dipole coupling is simply zero. This unique property makes it difficult to transfer population from 0〉 to 2〉 directly using a single π pulse tuned to their level spacing ω_{20}. The usual solution is to use the highpower resonant twophoton process or to apply two successive π pulses transferring the population first from 0〉 to 1〉 and then from 1〉 to 2〉 (refs 18, 19). These methods often lead to a significant population in the middle level 1〉 resulting in energy relaxation which degrades the transfer process. In contrast, STIRAP transfers the qutrit population directly from 0〉 to 2〉 via the dark state subspace without occupying the middle level 1〉.
In this work, we report on the realization of STIRAP in the Ξtype superconducting Xmon^{23} and phase^{24} qutrits. We demonstrate CPT from the ground state 0〉 to the second excited state 2〉 via STIRAP in the Xmon and phase qutrits in which population transfer efficiency no less than 96% and 67% is achieved, respectively. The experimental results are well described by the numerical simulation of the master equation.
Results
The STIRAP concept
For clarity, our results will be mainly presented for the Xmon qutrit, which has longer coherence times and thus better performance, while those for the phase qutrit will be discussed as a comparison showing the effect of system decoherence. As is shown schematically in Fig. 1a, the Xmon qutrit has a shunt capacitance C and two Josephson junctions each with critical current I_{c} to form a SQUID loop so the potential and level spacing can be tuned via the flux bias. The potential energy and quantized levels 0〉, 1〉 and 2〉 of the qutrit are illustrated in Fig. 1b in which the frequencies ω_{p,s} of the pump and Stokes fields and their strength Ω_{p,s} (Rabi frequencies) are also indicated. Since the matrix element between the 1〉 and 2〉 states is a factor of larger than that between the 0〉 and 1〉 states for both the Xmon and phase qutrits with weak anharmonicity^{25,26,27}, applying the rotatingwave approximation in the doublerotating frame the Hamiltonian can be written as^{26,27}:
where the Planck constant ħ is set to unity, δ=ω_{p}−ω_{s}, Δ_{p}=ω_{10}−ω_{p} and Δ_{s}=ω_{21}−ω_{s} are various detunings, g_{p,s} are the qutrit microwave couplings proportional to the amplitudes of the pump and Stokes fields, respectively. In equation (1), the matrix element between the 0〉 and 2〉 states is zero, which is true for the Xmon and is a good approximation for the phase qutrit^{27}. Hence, the Hamiltonian can be used to describe both devices. For the fastoscillating terms in the equation averages out to zero so the Hamiltonian becomes
in which Ω_{p}=2g_{p} and Ω_{s}=2λg_{s}. Equation (2) is the wellknown rotatingwave approximation Raman Hamiltonian^{1,2}. In particular, when the system satisfies the pump and Stokes twophoton resonant condition:
it has an eigenstate D〉=cos Θ0〉−sin Θ2〉, called the dark state, which corresponds to the eigenvalue of . Here tan Θ(t)=Ω_{p}(t)/Ω_{s}(t). CPT from the ground state 0〉 to the second excited state 2〉 without populating the first excited state 1〉 can therefore be realized via STIRAP by initializing the qutrit in the ground state 0〉 (refs 27, 28), and then slowly increasing the ratio Ω_{p}(t)/Ω_{s}(t) to infinity as long as the following conditions^{1,2,29,30}
are satisfied so that the qutrit will stay in the dark state subspace spanned by {0〉, 2〉}. The first condition is required to reduce equation (1) to equation (2) leading to the existence of the dark state solution, while the second ensures the adiabatic state following.
Sample parameters and measurements
The Xmon qutrit used in this work is an aluminumbased device^{23}, which is cooled down to T≈10 mK in the cryogenfree dilution refrigerator. A dispersive readout scheme with additional gains from a parametric amplifier is used to detect the qutrit states (see Methods). For the present experiment, the lowest three levels used as the qutrit states have the relevant transition frequencies of f_{10}=ω_{10}/2π=6.101 GHz and f_{21}=ω_{21}/2π=5.874 GHz, and the relative anharmonicity is α=(f_{10}−f_{21})/f_{10}≈3.7%. The measured energy relaxation times are μs and μs, respectively, while the dephasing time determined from Ramsey interference experiment is μs. To realize STIRAP, a pair of bellshaped counterintuitive microwave pulses with the Stokes pulse preceding the pump pulse, as illustrated in Fig. 1c, are used. The pulses are defined by Ω_{s}(t)=Ω_{0}F(t) cos[πη(t)/2] and Ω_{p}(t)=Ω_{0}F(t)sin[πη(t)/2] with and , respectively^{2,30}.
Coherent population transfer
Figure 2a shows the two microwave pulses defined by Ω_{0}/2π=30 MHz and T_{d}=100 ns. As t increases, Ω_{s}(t) and Ω_{p}(t) start to increase and decrease, respectively, across t=0 at which they are equal. The experimentally measured populations P_{0}, P_{1}, and P_{2} versus time produced by this counterintuitive pulse sequence in the resonant case Δ_{p}=Δ_{s}=0 are plotted in Fig. 2b. We observe that as time evolves across t=0 the population P_{2} (P_{0}) increases (decreases) rapidly while P_{1} remains low, signifying the occurrence of STIRAP via the dark state of the superconducting qutrit system. The experimentally measured maximum P_{2} is about 85% for the present sample under the resonant condition. The maximum value of P_{2} can be defined as the transfer efficiency or fidelity of the STIRAP process. As discussed in Supplementary Note 1, the experimentally measured value is much limited by the state preparation and measurement (SPAM) errors^{31} for the Xmon qutrit. In Fig. 2c, we show the corrected experimental data (symbols) assuming that SPAM errors are mostly due to the readout imperfection (see Methods section). The transfer efficiency after correction reaches 97% and the results match very well with the numerical simulations shown in the figure as solid lines. To further check the influence of the state preparation error ignored in the readout correction, we carry out a series of rigorous calibrations using the standard randomized benchmarking (Supplementary Fig. 1), sequential double π pulses (Supplementary Figs 2 and 3), and sequential STIRAP double π pulses (Supplementary Fig. 4) methods and demonstrate that the transfer efficiency is no less than 96%, which is close to the value after readout correction indicating that the influence of the state preparation error is negligible. The calibrations are detailed in Supplementary Note 1.
Notice that in the entire region of t∈[−300, 300] ns, all of the characteristic features of the experimental data, in particular (i) P_{1} remaining significantly lower than P_{2}, (ii) the slight decrease (increase) of P_{2} (P_{0}) after reaching the maximum (minimum) as well as the slight rising of P_{1}, are reproduced well by the numerical simulations. The simulated temporal profiles of the populations P_{0}, P_{1}, and P_{2} are obtained by solving the master equation using the measured qutrit parameters, where L(ρ) is the Liouvillean containing the relaxation and dephasing processes^{27} (see Methods section). The numerical results also show that feature (ii) is due primarily to energy relaxation, while the maximum P_{2} reachable would mainly be limited by dephasing, which can be seen more clearly for the phase qutrit (Supplementary Fig. 5) having shorter coherence times as presented and discussed in Supplementary Note 2.
In our experiment the conditions imposed by equation (4) are satisfied: δ/2π in the resonant case Δ_{p}=Δ_{s}=0 is f_{10}−f_{21}=227 MHz, which is ∼7.5 times that of Ω_{0}/2π, and it is easy to verify that the integrated pulse area ≈22π is greater than 10π. We point out that in addition to the influence of coherence times, the transfer efficiency of the demonstrated STIRAP process can also be improved by increasing the relatively small anharmonicity parameter α≈3.7% of the present sample up to, for example, 10% by optimizing device parameters of the Ξtype phase^{32} and transmon (or Xmon)^{33} qutrits. According to equation (4) larger anharmonicity allows the use of larger Ω_{0} which would proportionally reduce the duration of the pump and Stokes pulses when the pulse area is kept unchanged to satisfy the adiabatic condition. Shorter pulses also reduce the negative effect of decoherence on the transfer efficiency^{3,13}.
Bright and dark resonances
The STIRAP process is often identified in either the time domain or the frequency domain^{1,2}. The latter is based on equation (3) which specifies the pump and Stokes twophoton resonance condition. In Fig. 3a,b, we show the corrected experimental level populations P_{2} and P_{1} under the variations of the pump and Stokes detunings Δ_{p} and Δ_{s}, respectively. The results are accompanied by the numerical simulations via the master equation (Fig. 3c,d) with fair agreement. Bright and dark resonances can be seen clearly in Fig. 3a,c and Fig. 3b,d, respectively. The bright resonance manifests itself as a stretched line with large P_{2} from the topleft to bottomright corners reflecting the resonance condition equation (3), and with a much extended area near Δ_{s}, Δ_{p}∼0. The dark resonance appears as small P_{1} in areas wherever P_{2} is large. The other two highly populated areas can also be seen. One is P_{2} excited by the twophoton process from the single pump microwave tone, appearing as a thin vertical line on the right side in Fig. 3a,c. A split of the line near Δ_{s}=0 can be seen, which could result from the Autler–Townes splitting of the 2〉 level induced by the Stokes microwave tone. The other is the vertical stripes near Δ_{p}=0 in Fig. 3b,d originating from the resonant excitation of P_{1} by the pump microwave tone. However, the stripes are distorted near Δ_{s}=0 due to the dark resonance from the STIRAP process.
In Fig. 3e,f, we compare the populations of the bright (P_{2}) and dark (P_{1}) states as a function of pump field detuning Δ_{p} when the frequency of the Stokes field resonates with ω_{21}/2π (that is, Δ_{s}=0). While the agreement between the measured and simulated P_{1} is pretty well those of P_{2} differ significantly in the height of the rightside peak around Δ_{p}=115 MHz that results from the single pump tone twophoton process. At present, it is not clear what is the cause for this discrepancy. However, because the twophoton resonance is located far away from the intended parameter region of STIRAP its effect on the efficiency and robustness of the coherent population transfer can be ignored.
Uniqueness and robustness
Similar results are obtained for the phase qutrit (Supplementary Figs 6 and 7) with a relative anharmonicity of α=2.9% and shorter coherence times on the order of a few hundred nanoseconds, in which a coherent population transfer efficiency as high as 67% is achieved, consistent with the numerical simulations using the experimentally determined sample parameters listed in Supplementary Table 1 (see discussions in Supplementary Note 2). All these results demonstrate clearly CPT from the ground state 0〉 to the second excited state 2〉 via STIRAP in the Ξtype superconducting qutrits. We note that compared with the usual highpower singletone twophoton process or two nonoverlapping successive resonant π pulse excitations shown in Fig. 1d, which involve significant undesired population in the middle level 1〉 and require precise single photon resonance and pulse area^{11,18}, CPT via STIRAP demonstrates simply the opposite. First, in principle CPT between 0〉 and 2〉 can be accomplished without occupying the lossy middle level 1〉. More importantly, the process is much more robust against variations in the frequency, duration and shape of the driving pulses^{1,2}. In fact, in terms of equation (3) and equation (4), we see from Fig. 3a,c,e that the pump and Stokes tones resonance condition is greatly relaxed due to a much wider peak width of the STIRAP process as compared, for example, with the singletone twophoton excitation from 0〉 state to 2〉 state having a much narrow peak. On the other hand, although Ω_{p,s} are limited by the system anharmonicity, their values, together with T_{d}, still have much room for variations while maintaining the transfer efficiency. Our simulated results indicate that the transfer efficiency of STIRAP is very insensitive to Ω_{0}, which is limited by systems anharmonicity, and to T_{d}, which should be much smaller than the coherence time. The allowed variations for the present Xmon qutrit are about 20 MHz in Ω_{0} and 100 ns in T_{d} for keeping P_{2}≥96%, which are in sharp contrast to, for example, the case of simple π pulse excitations. The extreme robustness of the STIRAP process is very advantageous and should be useful in various applications such as realizing efficient qubit rotation, entanglement and quantum information transfer in various superconducting qubit and qutrit systems.
Discussion
We have experimentally demonstrated coherent population transfer between two uncoupled or weakly coupled states, 0〉 and 2〉, of the superconducting Xmon and phase qutrits having Ξtype ladder configuration via STIRAP. The qutrits had small relative anharmonicity around 3% and moderate coherence times ranging from a few hundred ns up to ten μs. We demonstrated that by applying a pair of counterintuitive microwave pulses in which the Stokes tone precedes the pump tone, coherent population transfer from 0〉 to 2〉 with efficiency no less than 96% and 67% for the two devices can be achieved with a much smaller population in the first excited state 1〉. Using the measured qutrit parameters, including coherence times, we simulated the STIRAP process by numerically solving the master equation. The results agreed well with the experimental data.
Coherent population transfer via STIRAP is much more robust against variations of the experimental parameters, including the amplitude, detuning and time duration of the microwave fields, and the environmental noise over the conventional methods such as using highpower singletone twophoton excitation and two resonant π pulses tuned to ω_{10} and ω_{21}, respectively. Therefore STIRAP is advantageous for achieving robust coherent population transfer in the laddertype superconducting artificial atoms that play increasingly important roles in various fields ranging from fundamental physics to quantum information processing. With improved qutrit parameters of coherence times up to 40 μs, presently attainable in the Xmon^{23}, transmon^{33,34} and flux^{35} type devices, nearly complete transfer above 99% from level 0〉 to level 2〉 while keeping the level 1〉 population below 1% is expected. On the other hand, STIRAP in the Λtype systems^{3} such as superconducting flux qutrits, in which the initial and target states locate in different potential wells representing circulating currents in opposite directions, is important in various applications and its experimental implementation remains to be explored. Our work paves the way for further progress in these directions.
Method
Dispersive readout of Xmon qutrit and SPAM errors
The Xmon qutrit is capacitively coupled to an onchip λ/4 coplanar waveguide resonator which has a fixed resonant frequency at ω_{r}/2π≈6.640 GHz. The qutritresonator coupling strength is designed to be about 30 MHz if onresonance, and the coplanar waveguide resonator is loaded to external circuitry whose microwave response can be probed in terms of its transmission coefficient S. As the Xmon qutrit is far detuned from ω_{r}, there is a dispersioninduced resonant frequency shift of the resonator, that is, the resulting transmission coefficient S expressed by a complex number I+iQ takes different values depending on the exact qutrit state. For readout we input an 800nslong microwave pulse, which is ∼1 MHz detuned from ω_{r}/2π, and the output microwave pulse with the desired resonator information encoded in (I, Q) is sequentially amplified at multiple stages using a Josephson junction parametric amplifier^{36} and other lownoise amplifiers before demodulated by room temperature electronics^{37}.
In the perfect absence of noise we would obtain three signal points in the I–Q plane for the qutrit’s three eigenstates 0〉, 1〉 and 2〉, respectively. However, unavoidable noise in the measurement system gives rise to random scattering of the signal points around the ideal values, resulting in effectively three circular clouds corresponding to the three eigenstates. For a single measurement event in which a point (I, Q) is demodulated from an 800nslong microwave pulse, we categorize the qutrit state according to the minimum distance between this point (I, Q) and the three cloud centres. We repeat the sequence several hundred or thousand times for many points of (I, Q)s, from which the occupation probabilities for 0〉, 1〉 and 2〉 can be counted. Obviously, slight overlaps between clouds or unexpected transitions between eigenstates during the preparation of the initial state and/or the readout stage give errors and reduce the relevant fidelity values. These are SPAM errors related to our specific measurement system^{38}.
Assuming that SPAM errors are mostly related to the readout imperfection, which can then be corrected, we perform a preliminary readout correction of the raw data. We prepare the state in j〉 (j=0, 1 and 2), followed by an immediate qutrit readout for recording the probability value of correctly measuring the state in j〉 and the other two probability values of incorrectly measuring the state in k〉 (k≠j). The resulting 9 probability values can be used to construct the readout correction matrix. We note that this method may not be accurate since the state preparation error, though likely small, is ignored in constructing the correction matrix. However, the corrected experimental data agree well with the estimation from the full calibration of the STIRAP fidelity via concatenated pulses, as detailed in Supplementary Note 1, and with the calculated results using the master equation.
Numerical simulations
We numerically calculate the level populations P_{0}(t)=ρ_{00}(t), P_{1}(t)=ρ_{11}(t), and P_{2}(t)=ρ_{22}(t) at any given time by solving the master equation
where ρ is the system’s 3 × 3 density matrix, H is the Hamiltonian given by equation (1), and L(ρ) is the Liouvillean containing various relaxation and dephasing processes. Considering the general situation that the pump and Stokes microwaves are not correlated, we introduce a phase difference ϕ between the two microwaves in the actual calculations^{39}. In this case, the doublerotating reference frame is described by the operator , and the rotatingwave approximation leads to a Hamiltonian in the following form:
where the Liouvillean operator in equation (5) is given by^{27}:
In our calculations ρ(t, ϕ) is obtained by solving equation (5) using the fourthorder Runge–Kutta method. When the phase difference ϕ of the two microwaves in our experiment is random, we average the result over ϕ and finally arrive at:
For the Xmon qutrit we use the parameters Γ_{10}=8.4 × 10^{4} s^{−1}, Γ_{21}=1.3 × 10^{5} s^{−1}, and =2.0 × 10^{5} s^{−1} measured directly from experiment, and we estimate ≈ and ≈ as in the case of phase qutrit (Supplementary Note 2).
Additional information
How to cite this article: Xu, H. K. et al. Coherent population transfer between uncoupled or weakly coupled states in laddertype superconducting qutrits. Nat. Commun. 7:11018 doi: 10.1038/ncomms11018 (2016).
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Acknowledgements
We thank J.M. Martinis (UCSB) for providing us with the samples used in this work. This work was supported by the Ministry of Science and Technology of China (Grant Nos. 2011CBA00106, 2014CB921202, and 2015CB921104) and the National Natural Science Foundation of China (Grant Nos. 91321208, 11222437, and 11161130519). S. Han acknowledges support by the US NSF (PHY1314861).
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Contributions
H.K.X., S.H. and S.P.Z. designed the experiment. H.K.X., W.Y.L., G.M.X. and F.F.S. performed the experiment in phase qutrit and analysed data with S.H. and S.P.Z. providing supervision. H.K.X., W.Y.L., and G.M.X. performed numerical simulation. C.S. and Y.P.Z. performed the measurement in Xmon qutrit with H.W. in supervision. Y.T., H.D. and D.N.Z. contributed to the experimental setup, sample mounting and characterization. Y.X.L. provided theoretical support. S.P.Z., S.H. and H.W. wrote the manuscript in cooperation with all the authors.
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Correspondence to Siyuan Han or H. Wang or S. P. Zhao.
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Supplementary Information
Supplementary Figures 17, Supplementary Table 1, Supplementary Notes 12 and Supplementary References (PDF 899 kb)
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Xu, H., Song, C., Liu, W. et al. Coherent population transfer between uncoupled or weakly coupled states in laddertype superconducting qutrits. Nat Commun 7, 11018 (2016) doi:10.1038/ncomms11018
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