Coherent population transfer between uncoupled or weakly coupled states in ladder-type superconducting qutrits

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 ladder-type three-level 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.

ARTICLE

Received 8 Jun 2015 | Accepted 12 Feb 2016 | Published 24 Mar 2016

H.K. Xu1, C. Song2, W.Y. Liu1, G.M. Xue1, F.F. Su1, H. Deng1, Ye Tian1, D.N. Zheng1, Siyuan Han1,3, Y.P. Zhong2,H. Wang2, Yu-xi Liu4,5 & S.P. Zhao1,6

Stimulated Raman adiabatic passage offers signicant advantages for coherent population transfer between uncoupled or weakly coupled states and has the potential of realizing efcient quantum gate, qubit entanglement and quantum information transfer. Here we report on the realization of the process in the superconducting Xmon and phase qutritstwo ladder-type three-level 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 efciency 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 signicantly more robust against variations of the experimental parameters compared with that via the conventional resonant p pulse method. Our work opens up a new venue for exploring the process for quantum information processing using the superconducting articial atoms.

DOI: 10.1038/ncomms11018 OPEN

Coherent population transfer between uncoupled or weakly coupled states in ladder-type superconducting qutrits

1 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. 2 Department of Physics, Zhejiang University, Hangzhou 310027, China. 3 Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA.

4 Institute of Microelectronics, Tsinghua University, Beijing 100084, China. 5 Tsinghua National Laboratory for Information Science and Technology (TNList), Beijing 100084, China. 6 Collaborative Innovation Center of Quantum Matter, Beijing, China. Correspondence and requests for materials should be addressed to S.H. (email: mailto:[email protected]

Web End [email protected] ) or to H.W. (email: mailto:[email protected]

Web End [email protected] ) or to S.P.Z. (email: mailto:[email protected]

Web End [email protected] ).

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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 states13. It has been recognized as an important technique in quantum computing and circuit quantum electrodynamics involving superconducting qubits413. For example, qubit rotations can be realized via STIRAP with two computational states plus an auxiliary state forming a three-level L conguration4,5. A scheme for generating arbitrary rotation and entanglement in the three-level L-type ux qutrits is also proposed6, and the experimental feasibility of realizing quantum information transfer and entanglement between qubits inside microwave cavities has been discussed7,8. Unlike the conventional resonant p 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 elds and the environmental noise5,6,11,12.

Recently, multi-level systems (qutrits or qudits) have found important applications in speeding up quantum gates14, realizing quantum algorithms15, simulating quantum systems consisting of spins greater than one half16, implementing full quantum-state tomography1719, testing quantum contextuality20 and mapping to multi-qubit systems21,22. Unlike the highly anharmonic L-type ux qutrits the phase and transmon (or Xmon) qutrits have the ladder-type (X-type) three-level conguration which is weakly anharmonic. The dipole coupling between the ground state |0i

and the second excited state |2i in the phase qutrit is much

weaker than those between the rst excited state |1i and the |0i

state or the |2i state. In the case of the transmon (or Xmon) qutrit

the dipole coupling is simply zero. This unique property makes it difcult to transfer population from |0i to |2i directly using a

single p pulse tuned to their level spacing o20. The usual solution

is to use the high-power resonant two-photon process or to apply two successive p pulses transferring the population rst from |0i

to |1i and then from |1i to |2i (refs 18,19). These methods often

lead to a signicant population in the middle level |1i resulting

in energy relaxation which degrades the transfer process. In contrast, STIRAP transfers the qutrit population directly from |0i

to |2i via the dark state subspace without occupying the middle

level |1i.

In this work, we report on the realization of STIRAP in the X-type superconducting Xmon23 and phase24 qutrits. We demonstrate CPT from the ground state |0i to the second

excited state |2i via STIRAP in the Xmon and phase qutrits in

which population transfer efciency no less than 96% and 67% is achieved, respectively. The experimental results are well described by the numerical simulation of the master equation.

ResultsThe 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 Ic to form a SQUID loop so the potential and level spacing can be tuned via the ux bias. The potential energy and quantized levels |0i, |1i and |2i of the qutrit

are illustrated in Fig. 1b in which the frequencies op,s of the pump

and Stokes elds and their strength Op,s (Rabi frequencies) are also indicated. Since the matrix element between the |1i and |2i

states is a factor of l 2

p larger than that between the |0i and

|1i states for both the Xmon and phase qutrits with weak

anharmonicity2527, applying the rotating-wave approximation in the double-rotating frame the Hamiltonian can be written as26,27:

H

0 gp gse idt 0

gp gseidt Dp l gpeidt gs

0 l gpe idt gs

Dp Ds

2

4

3

5; 1

where the Planck constant : is set to unity, d op os,

Dp o10 op and Ds o21 os are various detunings, gp,s are

c

a

Resonator readout

w bias

s(t )

10(t ) 21(t )

p(t )

C

IC IC

Flux bias

b

d

|2

21, T121

10, T110

s, s

p, p

|1

t

t

|0

Figure 1 | Superconducting Xmon qutrit and measurement pulse sequences. (a) Schematic Xmon qutrit with Josephson critical current Ic and shunt capacitance C. (b) Three bottom energy levels |0i, |1i and |2i of the qutrit with related symbols indicated. Subscripts p and s refer to the pump and Stokes

tones, respectively. (c) Counterintuitive pulse sequence with Os preceding Op for coherent population transfer from |0i to |2i without involving |1i.

(d) Conventional resonant p pulse sequence for successive |0i-|1i-|2i population transfers.

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the qutrit microwave couplings proportional to the amplitudes of the pump and Stokes elds, respectively. In equation (1), the matrix element between the |0i and |2i states is zero, which is

true for the Xmon and is a good approximation for the phase qutrit27. Hence, the Hamiltonian can be used to describe both devices. For d Op;s the fast-oscillating terms in the equation

averages out to zero so the Hamiltonian becomes

H0

0 Op=2 0

Op=2 Dp Os=2

0 Os=2 Dp Ds

a

1.0

300 200 100 0 100 200 300

s,p( t ) / 0

s / 0 p / 0

0.5

2

4

3

0.0

5; 2

in which Op 2gp and Os 2lgs. Equation (2) is the well-known

rotating-wave approximation Raman Hamiltonian1,2. In particular, when the system satises the pump and Stokes two-photon resonant condition:

Dp Ds 0; 3 it has an eigenstate |Di cos Y|0i sin Y|2i, called the dark

state, which corresponds to the eigenvalue of E0. Here

tan Y(t) Op(t)/Os(t). CPT from the ground state |0i to the

second excited state |2i without populating the rst excited state

|1i can therefore be realized via STIRAP by initializing the qutrit in

the ground state |0i (refs 27,28), and then slowly increasing the

ratio Op(t)/Os(t) to innity as long as the following conditions1,2,29,30

d Op;s; R

300 200 100 0 100 200 300

b

1.0

300 200 100 0 100 200 300

P0

P2

P 0,1,2

0.5

P1

0.0

c

1.0

P0

P2

P 0,1,2

0.5

P1

q dt410p 4

are satised so that the qutrit will stay in the dark state subspace spanned by {|0i, |2i}. The rst 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 aluminum-based device23, which is cooled down to TE10 mK in the cryogen-free dilution refrigerator. A dispersive readout scheme with additional gains from a parametric amplier 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 f10 o10/2p 6.101 GHz and

f21 o21/2p 5.874 GHz, and the relative anharmonicity is

a (f10 f21)/f10E3.7%. The measured energy relaxation times

are T1011=G1011:9 ms and T2111=G217:6 ms, respectively,

while the dephasing time determined from Ramsey interference experiment is T10j5:0 ms. To realize STIRAP, a

pair of bell-shaped counterintuitive microwave pulses with the Stokes pulse preceding the pump pulse, as illustrated in Fig. 1c, are used. The pulses are dened by Os(t) O0F(t) cos[pZ(t)/2]

and Op(t) O0F(t)sin[pZ(t)/2] with F t

e t=2Td

1

O2p t

O2s t

0.0

t ( ns )

Figure 2 | Coherent population transfer via STIRAP in the superconducting Xmon qutrit. (a) Stokes and pump microwave pulses

Os(t) and Op(t) with the experimental parameters os/2p f21 5.874 GHz,

op/2p f10 6.101 GHz, O0/2p 30 MHz and Td 100 ns. (b) Measured

level populations P0, P1 and P2 versus time with a maximum value ofP2 85% driven by the STIRAP pulse pair in a in the case of Dp Ds 0.

(c) Experimental level populations with maximum P2 reaching 97% (symbols) after correcting the readout imperfection as described in Methods. The lines are the numerical results calculated using the master equation considering the relaxation and dephasing processes, which agree well with the experimental data after correction. The experimentally determined parameters are used in the calculation: G10 8.4 104 s 1,

G21

1.3 105 s 1, and gj10 2.0 105 s 1. Other parameters in the

master equation are taken as gj20E2gj10 and gj21 gj10.

6 and

Z t

1= 1 e 4t=Td

,

respectively2,30.

Coherent population transfer. Figure 2a shows the two microwave pulses dened by O0/2p 30 MHz and Td 100 ns. As

t increases, Os(t) and Op(t) start to increase and decrease, respectively, across t 0 at which they are equal. The experimentally

measured populations P0, P1, and P2 versus time produced by this counterintuitive pulse sequence in the resonant case Dp Ds 0

are plotted in Fig. 2b. We observe that as time evolves across t 0

the population P2 (P0) increases (decreases) rapidly while P1 remains low, signifying the occurrence of STIRAP via the dark state of the superconducting qutrit system. The experimentally measured maximum P2 is about 85% for the present sample under the resonant condition. The maximum value of P2 can be dened as the transfer efciency or delity of the STIRAP process. As discussed in Supplementary Note 1, the experimentally measured

value is much limited by the state preparation and measurement (SPAM) errors31 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 efciency after correction reaches 97% and the results match very well with the numerical simulations shown in the gure as solid lines. To further check the inuence 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 p pulses (Supplementary Figs 2 and 3), and sequential STIRAP double p pulses (Supplementary Fig. 4) methods and demonstrate that the transfer efciency is no less than 96%, which is close to the value after readout correction indicating that the inuence of the state preparation error is negligible. The calibrations are detailed in Supplementary Note 1.

Notice that in the entire region of tA[ 300, 300] ns, all of the

characteristic features of the experimental data, in particular (i) P1 remaining signicantly lower than P2, (ii) the slight decrease (increase) of P2 (P0) after reaching the maximum (minimum) as well as the slight rising of P1, are reproduced well by the numerical simulations. The simulated temporal proles of the

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populations P0, P1, and P2 are obtained by solving the master equation _

r i=

H; r

L r

; using the measured qutrit

parameters, where L(r) is the Liouvillean containing the relaxation and dephasing processes27 (see Methods section). The numerical results also show that feature (ii) is due primarily to energy relaxation, while the maximum P2 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 satised: d/2p in the resonant case Dp Ds 0 is f10 f21 227

MHz, which is B7.5 times that of O0/2p, and it is easy to verify that the integrated pulse area

R

1 q dtE22p is

greater than 10p. We point out that in addition to the inuence of coherence times, the transfer efciency of the demonstrated STIRAP process can also be improved by increasing the relatively small anharmonicity parameter aE3.7% of the present sample up to, for example, 10% by optimizing device parameters of the

X-type phase32 and transmon (or Xmon)33 qutrits. According to equation (4) larger anharmonicity allows the use of larger O0 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 efciency3,13.

Bright and dark resonances. The STIRAP process is often identied in either the time domain or the frequency domain1,2. The latter is based on equation (3) which species the pump and Stokes two-photon resonance condition. In Fig. 3a,b, we show the corrected experimental level populations P2 and P1 under the variations of the pump and Stokes detunings Dp and Ds, 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 P2 from the top-left to bottom-right corners reecting the resonance condition equation (3), and with a much extended area near Ds,

DpB0. The dark resonance appears as small P1 in areas wherever P2 is large. The other two highly populated areas can also be seen. One is P2 excited by the two-photon 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 Ds 0 can be seen,

which could result from the AutlerTownes splitting of the |2i

level induced by the Stokes microwave tone. The other is the vertical stripes near Dp 0 in Fig. 3b,d originating from the

resonant excitation of P1 by the pump microwave tone. However, the stripes are distorted near Ds 0 due to the dark resonance

from the STIRAP process.In Fig. 3e,f, we compare the populations of the bright (P2) and

dark (P1) states as a function of pump eld detuning Dp when the frequency of the Stokes eld resonates with o21/2p (that is,

Ds 0). While the agreement between the measured and

simulated P1 is pretty well those of P2 differ signicantly in the height of the right-side peak around Dp 115 MHz that results

from the single pump tone two-photon process. At present, it is not clear what is the cause for this discrepancy. However, because the two-photon resonance is located far away from the intended parameter region of STIRAP its effect on the efciency 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 a 2.9% and shorter coherence times on the

order of a few hundred nanoseconds, in which a coherent population transfer efciency 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 |0i to

the second excited state |2i via STIRAP in the X-type

superconducting qutrits. We note that compared with the usual high-power single-tone two-photon process or two non-overlapping successive resonant p pulse excitations shown in Fig. 1d, which involve signicant undesired population in the middle level |1i and require precise single photon resonance and pulse

area11,18, CPT via STIRAP demonstrates simply the opposite. First, in principle CPT between |0i and |2i can be accomplished

without occupying the lossy middle level |1i. More importantly,

the process is much more robust against variations in the frequency, duration and shape of the driving pulses1,2. In fact, in terms of equation (3) and equation (4), we see from Fig. 3a,c,e

1

O2p t

O2s t

a

c

e

80 0.9

0.9

80

0.9

0.9

s / 2 ( MHz )

s / 2 ( MHz )

40

40

40

40

0.8

0.4

0.0

0.0

0.6

0.6

80

40

0

40

80

80

40

0

40

0.6

0.6

s = 0

0

0

P 2

P 2

0.3

0.3

0.3

0.3

0.0

0.0

0.0

0.0 p / 2 ( MHz )

80 100

50

0

50

100 100 50 0 50 100

80 100 50 0 50 100

100 50 0 50 100

b

d

100 50 0 50 100

f

0.3

P 1

P 1

s = 0

80 100

50

0

50

100

p / 2 ( MHz ) p / 2 ( MHz )

Figure 3 | Bright and dark resonances. Level populations P2 and P1 taken at t 100 ns versus detunings Ds and Dp. (a,b) Experimental; (c,d) theoretical.

Bright and dark resonances can be seen clearly in ad, respectively. The right-side peaks in a,c result from the two-photon process of the single pump microwave tone. (e) Bright and (f) dark resonance data plotted with Ds 0. Symbols and lines are, respectively, the experimental results and the results

calculated using the same parameters in Fig. 2.

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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 single-tone two-photon excitation from |0i state to |2i state having a much narrow

peak. On the other hand, although Op,s are limited by the system anharmonicity, their values, together with Td, still have much room for variations while maintaining the transfer efciency. Our simulated results indicate that the transfer efciency of STIRAP is very insensitive to O0, which is limited by systems anharmonicity, and to Td, which should be much smaller than the coherence time. The allowed variations for the present Xmon qutrit are about 20 MHz in O0 and 100 ns in Td for keeping

P2Z96%, which are in sharp contrast to, for example, the case of simple p pulse excitations. The extreme robustness of the STIRAP process is very advantageous and should be useful in various applications such as realizing efcient qubit rotation, entanglement and quantum information transfer in various superconducting qubit and qutrit systems.

DiscussionWe have experimentally demonstrated coherent population transfer between two uncoupled or weakly coupled states, |0i

and |2i, of the superconducting Xmon and phase qutrits having

X-type ladder conguration via STIRAP. The qutrits had small relative anharmonicity around 3% and moderate coherence times ranging from a few hundred ns up to ten ms. We demonstrated that by applying a pair of counterintuitive microwave pulses in which the Stokes tone precedes the pump tone, coherent population transfer from |0i to |2i with efciency no less than

96% and 67% for the two devices can be achieved with a much smaller population in the rst excited state |1i. 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 elds, and the environmental noise over the conventional methods such as using high-power single-tone two-photon excitation and two resonant p pulses tuned to o10 and o21, respectively. Therefore

STIRAP is advantageous for achieving robust coherent population transfer in the ladder-type superconducting articial atoms that play increasingly important roles in various elds ranging from fundamental physics to quantum information processing. With improved qutrit parameters of coherence times up to 40 ms, presently attainable in the Xmon23, transmon33,34 and ux35 type devices, nearly complete transfer above 99% from level |0i to level |2i while keeping the level |1i population below

1% is expected. On the other hand, STIRAP in the L-type systems3 such as superconducting ux 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 on-chip l/4 coplanar waveguide resonator which has a xed resonant frequency at or/2pE6.640 GHz. The qutrit-resonator coupling strength is designed to be about 30 MHz if on-resonance, and the coplanar waveguide resonator is loaded to external circuitry whose microwave response can be probed in terms of its transmission coefcient S. As the Xmon qutrit is far detuned from or, there is a dispersion-induced resonant frequency shift of the resonator, that is, the resulting transmission coefcient S expressed by a complex number I iQ takes different values depending on the exact qutrit state. For

readout we input an 800-ns-long microwave pulse, which is B1 MHz detuned

from or/2p, and the output microwave pulse with the desired resonator information encoded in (I, Q) is sequentially amplied at multiple stages using a Josephson junction parametric amplier36 and other low-noise ampliers before demodulated by room temperature electronics37.

In the perfect absence of noise we would obtain three signal points in the IQ plane for the qutrits three eigenstates |0i, |1i and |2i, 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 800-ns-long 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 |0i, |1i and

|2i 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 delity values. These are SPAM errors related to our specic measurement system38.

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 |ji (j 0, 1 and 2), followed by an immediate qutrit

readout for recording the probability value of correctly measuring the state in |ji and

the other two probability values of incorrectly measuring the state in |ki (kaj). 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 delity 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 populationsP0(t) r00(t), P1(t) r11(t), and P2(t) r22(t) at any given time by solving the

master equation

_

r

i H; r

L r

; 5

where r is the systems 3 3 density matrix, H is the Hamiltonian given by

equation (1), and L(r) 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 f between the two microwaves in the actual calculations39. In this case, the double-rotating reference frame is described by the operator U 0

j i 0

h j 1

j i 1

h jeio t 2

j i 2

h jei o t o t f

,

and the rotating-wave approximation leads to a Hamiltonian in the following form:

H

0 gp gse i dt f 0gp gsei dt f Dp l gpei dt f gs

0 l gpe i dt f gs

Dp Ds

2

64

3

75;

6

where the Liouvillean operator in equation (5) is given by27:L r

1

2

2

4

2G10r11 G10 gj10

r01 G21 gj20

r02

G10 gj10

r10 2G10r11 2G21r22 G10 G21 gj21

r12

G21 gj20

r20 G10 G21 gj21

r21 2G21r22

3

5:

7

In our calculations r(t, f) is obtained by solving equation (5) using the fourth-order RungeKutta method. When the phase difference f of the two microwaves in our experiment is random, we average the result over f and nally arrive at:

r t

1

2p

Z2p

0

r t; f

df: 8

For the Xmon qutrit we use the parameters G10 8.4 104 s 1, G21 1.3 105 s 1, and gj10 2.0 105 s 1 measured directly from experiment, and

we estimate gj20E2gj10 and gj21Egj10 as in the case of phase qutrit (Supplementary Note 2).

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11018

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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, and11161130519). S. Han acknowledges support by the US NSF (PHY-1314861).

Author 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 set-up, 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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How to cite this article: Xu, H. K. et al. Coherent population transfer between uncoupled or weakly coupled states in ladder-type superconducting qutrits. Nat. Commun. 7:11018 doi: 10.1038/ncomms11018 (2016).

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