arX

iv:2

108.

1296

7v1

[qu

ant-

ph]

30

Aug

202

1

Demonstration of dynamical control of three-level open systems with a

superconducting qutrit

Ri-Hua Zheng,∗ Wen Ning,∗ Zhen-Biao Yang,† Yan Xia,‡ and Shi-Biao Zheng§

Fujian Key Laboratory of Quantum Information and Quantum Optics,

College of Physics and Information Engineering,

Fuzhou University, Fuzhou, Fujian 350108, China

We propose a method for the dynamical control in three-level open systems and realize it inthe experiment with a superconducting qutrit. Our work demonstrates that in the Markovianenvironment for a relatively long time (3 µs), the systemic populations or coherence can still strictlyfollow the preset evolution paths. This is the first experiment for precisely controlling the Markoviandynamics of three-level open systems, providing a solid foundation for the future realization ofdynamical control in multiple open systems. An instant application of the techniques demonstratedin this experiment is to stabilize the energy of quantum batteries.

Introduction.–The control of quantum systems is al-ways an extremely interesting and essential topic amongdiverse types of quantum information tasks [1, 2]. Withthe gradual development of experimental techniques, de-coherence, mainly resulted from energy relaxation anddephasing, has become a primary barrier lying on theway to experimentally implement the quantum control.To clear this barrier, the researchers choose to take mea-sures to shorten the evolution time and have achieved re-markable successes in the experiments, such as, prepar-ing 20-qubit entanglement [3, 4] and simulating quan-tum walk on 62-qubit processor [5]. Meanwhile, someresearchers attempt on constructing qubits that are in-sensitive to decoherence [5–8]. So far, the relaxation timeT1 and dephasing time T2 of superconducting qubit haveboth been extended to 200 µs [8]. Indeed, the methodsof shortening the evolution time and making more per-fect qubits naturally reduce the impact of decoherencerather than avoiding it, which still process quantified er-rors. For realizing the leap of quantum error rates from10−3 to 10−15 [9, 10] and further accomplishing near-termquantum works, it is necessary to explore the control ofopen quantum systems, tailoring the decoherence as oneof the effective elements involved in the quantum control.

The earlier open-system quantum control theory drawssupports from a kind of nuclear magnetic resonance tech-nology, called spin-echo [11], in which a π pulse is in-serted midway in the evolution time to increase T2. How-ever, this inserted π pulse will change the systemic dy-namics. Several spin-echo-like techniques, such as bang-bang control [12], dynamical decoupling technique [13],and parity-kicks [14], process the same problems withthe destruction of dynamics. So these works generallyserved as techniques for storing coherence of quantumsystems [15–20]. Researchers alternatively develop anopen-system adiabatic theorem [21] to fully control thedynamics. This method utilizes super operators to de-scribe the open systems and can approximately predictthe density matrix at each moment. Since the adiabatictheorem needs a long evolution time to guarantee the

adiabatic approximation, researchers further try to builda shortcut to adiabaticity (STA) (similar to that in theclosed systems [22–35]).

By virtue of super operators, Sarandy et al. [36] haveproposed a concept about open-system dynamical invari-ants to build a STA in open systems. They constructedsome 4-dimensional invariants for engineering two-levelopen systems, nevertheless, with several simplifications,such as, either considering energy relaxation or dephas-ing. Finding proper invariants will be complicated whensomeone simultaneously takes energy relaxation and de-phasing into account, let alone expansion to three-levelopen systems with 9-dimensional invariants. Thereforeother control methods are eagerly in need of discovery.

Recently, Medina and Semiao [37] have directly set thedensity operator with time-dependent parameters andthen submitted them to the Markovian master equation[38], which gives appropriate functions of pulses to con-trol the populations in two-level open systems. The errorrates of populations control in Ref. [37] under seriousdecoherence are almost zero, an exciting result. Somefollow-up studies [39, 40] have made a lot of additions tothe application scenarios, but still remaining in two-levelopen systems. With the general trend of quantum taskstowards multiplication [41], it is of great significance toexplore the dynamics of three (or more)-level open sys-tems.

In this letter, we propose a method for the dynam-ical control in three-level open systems, and demon-strate it with a superconducting circuit platform [42–47].Two time-dependent parameters are presupposed in theMarkovian master equation to singly control the popu-lations or coherence, and then the analytical functionsof driving pulses are given to complete desired dynami-cal control. For proving the correctness of this method,we apply the designed driving pulses to a superconduct-ing Xmon qutrit and measure populations of excitedstates |1〉 and |2〉, or coherence between ground and ex-cited states. The experimental results are in good agree-ment with the theoretical design for the dynamical con-

2

trol. This is the first experiment to precisely control theMarkovian dynamics of three-level open systems, wherethe decoherence is effectively dominated. In addition,an interesting freezing phenomenon of populations is ob-served. That is, in the late stages of evolutions, the pop-ulations stabilize at specific values for a relatively longtime (>1.8 µs) in the presence of decoherence. An in-stant application scenario of this freezing phenomenon isthe quantum battery (QB) [48–58]. One can steadily lockthe energy of QBs for a comparatively long time (>1.2µs), resisting the relaxation of excitation, and thereforebuilding prototypes of QBs with more stable energy.Method.–Assume that a three-level system consists of

bases |0〉, |1〉, and |2〉. The systemic Hamiltonian inthe interaction picture is (h = 1 hereafter) Hi(t) =Ω01(t)|0〉〈1|+Ω12(t)|1〉〈2|+H.c., where Ω01(t) and Ω12(t)respectively represent the Rabi frequencies of the de-signed driving pulses for controlling transitions |0〉 ↔ |1〉and |1〉 ↔ |2〉. We assume the density matrix of thisthree-level system being [basis order |2〉,|1〉,|0〉 anddropping (t) henceforth]

ρ =

f2 −ih1 h2

ih1 f1 −ih3

h2 ih3 1− f1 − f2

, (1)

in which f1, f2, h1, h2, and h3 are all time-dependentreal functions. Here f1 and f2 describe the populationsof states |1〉 and |2〉, respectively. Additionally, h1, h2,and h3 describe the coherence of this three-level system.Equation 1 (5 degrees of freedom) is not the most gen-eralized case for a three-level density operator (8 degreesof freedom), however sufficient as a demonstration. TheMarkovian master equation [38] here is

ρ = −i[Hi, ρ] +

4∑

j=1

[

LjρL†j −

1

2(L†

jLjρ+ ρL†jLj)

]

, (2)

where Lindblad operators are Lk =√γk|k〉〈k| and

Lk+2 =√Γk|k− 1〉〈k|, with Γk and γk being the rates of

energy relaxation and dephasing of state |k〉, respectively(k = 1, 2). By utilizing Eqs. (1, 2), we can derive theRabi frequencies to control the dynamics of this three-level system, yielding

Ω01 =f1Γ1 + f1 + f2

2h3, (3a)

Ω12 =f2Γ2 + f2

2h1. (3b)

It is noteworthy that there are three constraint equations

h1 = −1

2(γ1 + γ2 + Γ1 + Γ2)h1 + (f1 − f2)Ω12 − h2Ω01,

(4a)

h2 = −1

2(γ2 + Γ2)h2 − h3Ω12 + h1Ω01, (4b)

h3 = −1

2(γ1 + Γ1)h3 + h2Ω12 − (−1 + f2 + 2f1)Ω01. (4c)

Up to now, the populations or coherence of the three-level system can be effectively controlled by the interac-tion Hamiltonian Hi, i.e., by Rabi frequencies Ω01 andΩ12. A specific example is, an evolution from the groundstate |0〉 to a final state with populations respectivelybeing P0(tf ), P1(tf ), and P2(tf ) in states |0〉, |1〉, and|2〉, can be achieved by adjusting f1 = fP1(tf ) andf2 = fP2(tf ). The intermediate function f changingfrom 0 to 1 in the time interval [0, tf ] can be set asf = [1 + e−a(t−tf/2)]−1 [26, 27, 37], with a = 50/tf de-termining the gradient of the transformation. By sub-mitting f1 and f2 into Eqs. (3, 4), one can obtain Rabifrequencies Ω01 and Ω12 to accomplish the desired trans-formation of populations.Device.–Here we use a frequency-tunable supercon-

ducting Xmon qutrit [42–47] to test the above theory.The original Hamiltonian reads H = Ω01e

iω01t|0〉〈1| +Ω12e

iω12t|1〉〈2| + H.c. +∑

i=0,1,2 ωi|i〉〈i|, where ωi andω01(12) are the angular frequencies of energy level |i〉 andmicrowave pulses coupling |0〉 ↔ |1〉 (|1〉 ↔ |2〉), respec-tively. We adjust ω01/2π = (ω1 − ω0)/2π = 5.9600 GHzand ω12/2π = (ω2−ω1)/2π = 5.7208 GHz (in this exper-iment, frequency accuracy to 4 decimal places is manda-tory) to ensure that the pulses can resonantly drive thetransitions between adjacent energy levels. Note the fixedfrequency of the resonator (not used here) is 5.584 GHz[44–47], dynamically decoupled with the above system.An essential process for this experiment is to precisely

measure the coefficients of decoherence, γk and Γk. Ac-cording to the above form of Lindblad operators, one candeduce Γ1 = 1/T 01

1 , γ1 = 2/T 012 − Γ1, Γ2 = 1/T 12

1 , and

γ2 = 2/T 122 − Γ2 − Γ1 − γ1. Here T

01(12)1 and T

01(12)2

are the energy relaxation and dephasing time measuredbetween |0〉 and |1〉 (|1〉 and |2〉), shown in Figs. 1(a)and 1(b), respectively. From Fig. 1, we find T1 and T2

fluctuate a lot during the measured period of 20 hours.In addition, several works [15–20, 59] support that T2

will change with the applying of microwave pulses (thespin-echo technique [11] is a strong proof). Therefore thevalues of T1 and T2 we utilized to design pulses are a lit-tle different from those measured in Fig. 1, specifically,[T 01

1 , T 121 , T 01

2 , T 122 ] = [9.5, 4.6, 6, 1.9] µs, which are fixed

and utilized for all the experimental control (Figs. 3, 4,and 5).Results of the populations control.–Due to the choice of

f1 and f2, there are infeasible zones of the populationstransformation, shrinking as tf extending, shown in Fig.2(a). While the feasible area still takes a large part. Asan example, we utilize Eqs. (3, 4) to design Ω01 andΩ12 [see Fig. 2(b)] for the transformation of populationsP1(0) = 0 → P1(tf ) = 0.3 and P2(0) = 0 → P2(tf ) = 0.2.For the evolution time, we choose tf = 3 µs to accumu-late enough impact of the decoherence. The correspond-ing experimental results are shown in Fig. 3, with outerand inner layers indicating the results of applying open-system [see Eqs. (3, 4)] and closed-system [see Eqs. (3,

3

1 5 10 15 20

10

20

30

40

50

0 100

1

1 5 10 15 200

5

10

15

20

0 50

1

(a) (b)

FIG. 1. (Color online) Measurement results of (a) relaxationtime T1 and (b) dephasing time T2. The data are measured12 groups per hour for a total of 20 hours. The insets, asexamples, are the fitting processes for T1 and T2 in (a) and(b), respectively.

0 10

1

0 30

2

Rab

i fre

quen

cy(2

MH

z)

2 30

0.010.020.03IFA

FA

(a) (b)

FIG. 2. (Color online) (a) Feasible area (FA, surrounded bycurves and x-axis) and infeasible area (IFA, the outside zone)of the populations control in the experiment when the evolu-tion time is 3, 5, and 10 µs. Condition P1(tf ) + P2(tf ) ≤ 1triangulates the boundary. (b) A sample point marked aspurple plus sign in (a): Rabi frequencies for the controlof populations P1(0) = 0 → P1(tf ) = 0.3 and P2(0) =0 → P2(tf ) = 0.2. The used parameters of decoherence are[T 01

1 , T 121 , T 01

2 , T 122 ] = [9.5, 4.6, 6, 1.9] µs and the inset shows

the magnification picture of the Rabi frequencies.

4), preset γk = Γk = 0] pulses, respectively. Intuitively,the populations are controlled more precisely in the outerlayer, as compared to the inner one. More rigorously, wedefine a standard deviation to describe the error rate ofthe population control

error =

√

∑

i=0,1,2

[Pi(tf )ideal − Pi(tf )exp.]2/3, (5)

where Pi(tf )ideal and Pi(tf )

exp. are the ideal and experi-mental values of the population in |i〉, respectively. In theouter layer of Fig. 3, the populations control achieves anerror rate of 1.02%, not small [60] because we intention-ally extended the evolution time (3 µs). In contrast, ifthe control is performed for such a long time by applyingclosed-system pulses, the error rate reaches 7.49%. Thisstark difference in error demonstrates the effectiveness ofthe present control method.

FIG. 3. (Color online) Experimental results of the controlof populations P1(0) = 0 → P1(tf ) = 0.3 and P2(0) = 0 →

P2(tf ) = 0.2. The experimental results of applying open-system and closed-system pulses are shown (with circles, tri-angles, and pentagrams) in the outer and inner layers, re-spectively. The ideal results of P0(t), P1(t), and P2(t) areindicated by solid, dashed and dotted dashed curves, respec-tively.

We also measure the results of 29 different controls ofpopulations, 6 of them shown in Fig. 4 and all displayedin the Supplemental Material [61]. The error rates of thecontrols in Fig. 4 are around 1%, which we believe canbe further improved with more stable superconductingqutrits [8].

Another exceptional physical phenomenon is that thepopulations seem to be frozen after 1.8 µs in the outerlayers of Fig. 3. This freezing phenomenon of popula-tions does not mean that the driving pulses Ω01 and Ω12

have stopped. On the contrary, it is the driving pulses[see the insets of Fig. 2(b)] we applied that caused thisfreezing phenomenon to occur. Specifically, such a freez-ing phenomenon arises from the interplay between thedynamics induced by the continuous microwave drivesand the two decoherence channels characterized by T1

and T2 (both available for |0〉 ↔ |1〉 and |1〉 ↔ |2〉 transi-tions), similar to the cases for generation of steady statesin most open systems [36, 62–67]. But here it differssignificantly in that both the energy relaxation and thedephasing are involved in the nonequilibrium dynamicalprocesses and together help freeze the states of three-levelsystems, as compared to the previous ones which gener-ally consider only one decoherence channel [36, 66, 67].

Results of the coherence control. Since only two freevariables, Ω01 and Ω12 in Eqs. (3, 4) can be controlledin the experiment, we can only control two parts of co-herence. Here we choose h3 and h2 as an example forilluminating the coherence control. Corresponding vari-ations of Eqs. (3, 4) and the forbidden zone of coherencecontrol are shown in the Supplemental Material [61]. Ac-

4

FIG. 4. (Color online) Six groups of experimental results(error bars) of the controls of populations. The ideal resultsof P0(t), P1(t), and P2(t) are represented by solid, dashed anddotted dashed curves, respectively. The error rates are shownin corresponding subgraphs.

FIG. 5. (Color online) Six groups of experimental results(error bars) of the controls of coherence. The ideal results ofh0(t), h1(t), and h2(t) are represented by solid, dashed anddotted dashed curves, respectively. The error rates are shownin corresponding subfigures.

cording to the same preset of parameters and intermedi-ate function f , we show 6 groups of results of the coher-ence control in Fig. 5 (all the 36 groups of results in theSupplemental Material [61]). The error rate here is

error′ =

√

∑

m=2,3

[hm(tf )ideal − hm(tf )exp.]2/2, (6)

where hm(tf )ideal and hm(tf )

exp. are the ideal and ex-perimental values of coherence hm, respectively. Fig-ure 5 shows that h2 and h3 are controlled well andthe values of h1 are accurately predicted. Simi-

0 1 2 30

0.2

0.4

0.6

0.8

1

Pop

ulat

ions

char

ging

storing discharging

FIG. 6. (Color online) Experimental verification of the QB.We plot the experimental and ideal results by diamond pointsand solid curves, respectively. The inserted circle depicts theBloch sphere of the ideal evolution and the density of pointsrepresents the speed of evolution, one point per 15 ns. Theparameters of decoherence are [T 01

1 , T 012 ] = [9.5, 6] µs.

larly, the microwaves protect the coherence (lasting1.2 µs) from the influence of dephasing. If there areno microwaves applied, roughly, after the same 1.2µs, h3(1.2µs)/h3(0) ∼ exp(−1.2µs/T 01

2 ) ∼ 0.8 andh2(1.2µs)/h2(0) ∼ exp(−1.2µs/T 12

2 ) ∼ 0.45, the coher-ence will be seriously damaged. In contrast, the errorrates are all within 2% when we apply microwaves forthe coherence control.

Application.– Quantum battery [48–58]. The freezingphenomenon seems helpful to stabilize the energy of theQB for a fairly long time, especially in the energy-storingstage of the QB. For proving this, we design a pulsefor a qutrit in the experiment to simulate the charg-ing, storing, discharging processes of the QB (see Fig.6). A considerably long period, about 1.2 µs, of energy[ǫ = (ω1 − ω0)P1] stability was observed. This stabilityis hard to achieve by applying a pulse designed withoutconsidering decoherence because energy relaxation of thesuperconducting qutrit will exponentially drop the exci-tation down. Therefore, the present method may con-tribute to the construction of more stable QBs.

Conclusion.–We have proposed a method to controlthe dynamics of three-level open systems and realized itin the experiment with a platform of a superconduct-ing Xmon qutrit [42–47]. The populations and coherencecould be singly controlled with error rates around 1%under the influence of decoherence for a relatively longtime, 3 µs, close to the dephasing time T 12

2 of the qutrit.In some situations where the control is more successful,the error rates can even be less than 0.3%. We believethese error rates can be further reduced with more stablesuperconducting qutrits [8].

Additionally, an interesting freezing phenomenon of

5

populations was observed. The designed microwavepulses just offset the impact of decoherence and visu-ally freeze the populations. We then applied this phe-nomenon to make more stable prototypes of QBs, whoseenergy can hold 1.2 µs with only one charging. More-over, this freezing phenomenon strongly proves that theMarkovian master equation precisely describes the dy-namics of three-level open systems, as demonstrated herewith a superconducting Xmon qutrit that possesses thespecially intrinsic decoherence rates. The present workprovides a positive prospect of accurately realizing thedynamical control of three (or more)-level open systemsby using the adjustable driving pulses in the Markovianenvironment.

Note that all the data points of experiments were av-eraged over 12000 times and the readout errors of thequtrit have been corrected. We thank Ye-Hong Chenand Xin Wang in RIKEN for discussions. This work wassupported by the National Natural Science Foundationof China (Grants No. 11874114, and No. 11875108),and the Natural Science Funds for Distinguished YoungScholar of Fujian Province under Grant 2020J06011,Project from Fuzhou University under Grant JG202001-2.

∗ These two authors contributed equally to this work.† zbyang@fzu.edu.cn‡ xia-208@163.com§ t96034@fzu.edu.cn

[1] N. V. Vitanov, A. A. Rangelov, B. W. Shore,and K. Bergmann, Stimulated raman adia-batic passage in physics, chemistry, and beyond,Rev. Mod. Phys. 89, 015006 (2017).

[2] C. P. Koch, M. Lemeshko, and D. Sugny,Quantum control of molecular rotation,Rev. Mod. Phys. 91, 035005 (2019).

[3] C. Song, K. Xu, H. Li, Y.-R. Zhang, X. Zhang, W. Liu,Q. Guo, Z. Wang, W. Ren, J. Hao, H. Feng, H. Fan,D. Zheng, D.-W. Wang, H. Wang, and S.-Y. Zhu, Gen-eration of multicomponent atomic schrodinger cat statesof up to 20 qubits, Science 365, 574 (2019).

[4] A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T.Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pich-ler, S. Choi, J. Cui, M. Rossignolo, P. Rembold,S. Montangero, T. Calarco, M. Endres, M. Greiner,V. Vuletic, and M. D. Lukin, Generation and manipu-lation of schrodinger cat states in rydberg atom arrays,Science 365, 570 (2019).

[5] M. Gong, S. Wang, C. Zha, M.-C. Chen, H.-L. Huang,Y. Wu, Q. Zhu, Y. Zhao, S. Li, S. Guo, et al., Quan-tum walks on a programmable two-dimensional 62-qubitsuperconducting processor, Science 372, 948 (2021).

[6] M. Gong, M.-C. Chen, Y. Zheng, S. Wang, C. Zha,H. Deng, Z. Yan, H. Rong, Y. Wu, S. Li, et al., Gen-uine 12-qubit entanglement on a superconducting quan-tum processor, Phys. Rev. Lett. 122, 110501 (2019).

[7] Y. Wu, W.-S. Bao, S. Cao, F. Chen, M.-C.

Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du,D. Fan, et al., Strong quantum computational ad-vantage using a superconducting quantum processor,arXiv:2106.14734 (2021).

[8] Qiskit, An open-source framework for quantum comput-ing, Online (2021).

[9] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N.Cleland, Surface codes: Towards practical large-scalequantum computation, Phys. Rev. A 86, 032324 (2012).

[10] Google-Quantum-AI, Exponential suppression ofbit or phase errors with cyclic error correction,Nature 595, 383 (2021).

[11] E. L. Hahn, Spin echoes, Phys. Rev. 80, 580 (1950).[12] L. Viola and S. Lloyd, Dynamical suppression

of decoherence in two-state quantum systems,Phys. Rev. A 58, 2733 (1998).

[13] L. Viola, E. Knill, and S. Lloyd, Dynam-ical decoupling of open quantum systems,Phys. Rev. Lett. 82, 2417 (1999).

[14] D. Vitali and P. Tombesi, Using parity kicks for decoher-ence control, Phys. Rev. A 59, 4178 (1999).

[15] J. Du, X. Rong, N. Zhao, Y. Wang, J. Yang, and R. B.Liu, Preserving electron spin coherence in solids by opti-mal dynamical decoupling, Nature 461, 1265 (2009).

[16] M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga,W. M. Itano, and J. J. Bollinger, Optimized dy-namical decoupling in a model quantum memory,Nature 458, 996 (2009).

[17] G. de Lange, Z. H. Wang, D. Riste, V. V. Do-brovitski, and R. Hanson, Universal dynamical decou-pling of a single solid-state spin from a spin bath,Science 330, 60 (2010).

[18] T. van der Sar, Z. H. Wang, M. S. Blok, H. Bernien, T. H.Taminiau, D. M. Toyli, D. A. Lidar, D. D. Awschalom,R. Hanson, and V. V. Dobrovitski, Decoherence-protected quantum gates for a hybrid solid-state spinregister, Nature 484, 82 (2012).

[19] M. Zhong, M. P. Hedges, R. L. Ahlefeldt, J. G.Bartholomew, S. E. Beavan, S. M. Wittig, J. J.Longdell, and M. J. Sellars, Optically addressable nu-clear spins in a solid with a six-hour coherence time,Nature 517, 177 (2015).

[20] Q. Guo, S.-B. Zheng, J. Wang, C. Song, P. Zhang, K. Li,W. Liu, H. Deng, K. Huang, D. Zheng, X. Zhu, H. Wang,C.-Y. Lu, and J.-W. Pan, Dephasing-insensitive quantuminformation storage and processing with superconductingqubits, Phys. Rev. Lett. 121, 130501 (2018).

[21] M. S. Sarandy and D. A. Lidar, Adia-batic quantum computation in open systems,Phys. Rev. Lett. 95, 250503 (2005).

[22] M. Demirplak and S. A. Rice, Adiabaticpopulation transfer with control fields,J. Phys. Chem. A 107, 9937 (2003).

[23] M. V. Berry, Transitionless quantum driving,J. Phys. A: Math. Theor. 42, 365303 (2009).

[24] X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo,D. Guery-Odelin, and J. G. Muga, Fast optimal friction-less atom cooling in harmonic traps: Shortcut to adia-baticity, Phys. Rev. Lett. 104, 063002 (2010).

[25] X. Chen, I. Lizuain, A. Ruschhaupt, D. Guery-Odelin, and J. G. Muga, Shortcut to adia-batic passage in two- and three-level atoms,Phys. Rev. Lett. 105, 123003 (2010).

[26] N. V. Golubev and A. I. Kuleff, Control of populations

6

of two-level systems by a single resonant laser pulse,Phys. Rev. A 90, 035401 (2014).

[27] N. V. Golubev and A. I. Kuleff, Control of chargemigration in molecules by ultrashort laser pulses,Phys. Rev. A 91, 051401(R) (2015).

[28] Y.-H. Chen, W. Qin, X. Wang, A. Miranow-icz, and F. Nori, Shortcuts to adiabaticity for thequantum rabi model: Efficient generation of gi-ant entangled cat states via parametric amplification,Phys. Rev. Lett. 126, 023602 (2021).

[29] Y.-H. Chen, Z.-C. Shi, J. Song, and Y. Xia, Invariant-based inverse engineering for fluctuation transfer be-tween membranes in an optomechanical cavity system,Phys. Rev. A 97, 023841 (2018).

[30] R.-H. Zheng, Y.-H. Kang, D. Ran, Z.-C. Shi, and Y. Xia,Deterministic interconversions between the greenberger-horne-zeilinger states and the w states by invariant-basedpulse design, Phys. Rev. A 101, 012345 (2020).

[31] R.-H. Zheng, Y.-H. Kang, S.-L. Su, J. Song, and Y. Xia,Robust and high-fidelity nondestructive rydberg paritymeter, Phys. Rev. A 102, 012609 (2020).

[32] Y.-H. Kang, Y.-H. Chen, Z.-C. Shi, B.-H. Huang,J. Song, and Y. Xia, Complete bell-state analy-sis for superconducting-quantum-interference-devicequbits with a transitionless tracking algorithm,Phys. Rev. A 96, 022304 (2017).

[33] Y.-H. Kang, Y.-H. Chen, Z.-C. Shi, B.-H. Huang, J. Song,and Y. Xia, Pulse design for multilevel systems by utiliz-ing lie transforms, Phys. Rev. A 97, 033407 (2018).

[34] J.-L. Wu, Y. Wang, J. Song, Y. Xia, S.-L. Su, andY.-Y. Jiang, Robust and highly efficient discriminationof chiral molecules through three-mode parallel paths,Phys. Rev. A 100, 043413 (2019).

[35] F.-Q. Guo, J.-L. Wu, X.-Y. Zhu, Z. Jin, Y. Zeng,S. Zhang, L.-L. Yan, M. Feng, and S.-L. Su, Completeand nondestructive distinguishment of many-body ryd-berg entanglement via robust geometric quantum opera-tions, Phys. Rev. A 102, 062410 (2020).

[36] M. S. Sarandy, E. I. Duzzioni, and M. H. Y. Moussa, Dy-namical invariants and nonadiabatic geometric phases inopen quantum systems, Phys. Rev. A 76, 052112 (2007).

[37] I. Medina and F. L. Semiao, Pulse engineering forpopulation control under dephasing and dissipation,Phys. Rev. A 100, 012103 (2019).

[38] M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information

(Cambridge University, Cambridge, England, 2004) p.386.

[39] D. Ran, W.-J. Shan, Z.-C. Shi, Z.-B. Yang, J. Song, andY. Xia, Pulse reverse engineering for controlling two-levelquantum systems, Phys. Rev. A 101, 023822 (2020).

[40] D. Ran, B. Zhang, Y.-H. Chen, Z.-C. Shi, Y. Xia,R. Ianconescu, J. Scheuer, and A. Gover, Effective pulsereverse-engineering for strong field–matter interaction,Opt. Lett. 45, 3597 (2020).

[41] S. Paesani, J. F. F. Bulmer, A. E. Jones, R. San-tagati, and A. Laing, Scheme for universal high-dimensional quantum computation with linear optics,Phys. Rev. Lett. 126, 230504 (2021).

[42] R. Barends, J. Kelly, A. Megrant, A. Veitia,D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G.Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro,A. Dunsworth, C. Neill, P. O’Malley, P. Roushan,A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cle-

land, and J. M. Martinis, Superconducting quantum cir-cuits at the surface code threshold for fault tolerance,Nature 508, 500 (2014).

[43] J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jef-frey, T. C. White, D. Sank, J. Y. Mutus, B. Camp-bell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth,I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana,P. Roushan, A. Vainsencher, J. Wenner, A. N. Cle-land, and J. M. Martinis, State preservation by repeti-tive error detection in a superconducting quantum cir-cuit, Nature 519, 66 (2015).

[44] C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang,Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang,D. Zheng, X. Zhu, and H. Wang, Continuous-variable geometric phase and its manipulation forquantum computation in a superconducting circuit,Nature Commun. 8, 1061 (2017).

[45] W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B.Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B.Zheng, Deterministic entanglement swapping in a super-conducting circuit, Phys. Rev. Lett. 123, 060502 (2019).

[46] K. Xu, W. Ning, X.-J. Huang, P.-R. Han, H. Li, Z.-B.Yang, D. Zheng, H. Fan, and S.-B. Zheng, Demonstra-tion of a non-abelian geometric controlled-NOT gate ina superconducting circuit, Optica 8, 972 (2021).

[47] Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li,K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, Experimentaldemonstration of entanglement-enabled universal quan-tum cloning in a circuit, npj Quantum. Inf. 7, 44 (2021).

[48] R. Alicki and M. Fannes, Entanglement boost for ex-tractable work from ensembles of quantum batteries,Phys. Rev. E 87, 042123 (2013).

[49] F. Campaioli, F. A. Pollock, F. C. Binder, L. Celeri,J. Goold, S. Vinjanampathy, and K. Modi, En-hancing the charging power of quantum batteries,Phys. Rev. Lett. 118, 150601 (2017).

[50] D. Ferraro, M. Campisi, G. M. Andolina,V. Pellegrini, and M. Polini, High-power collec-tive charging of a solid-state quantum battery,Phys. Rev. Lett. 120, 117702 (2018).

[51] F. Barra, Dissipative charging of a quantum battery,Phys. Rev. Lett. 122, 210601 (2019).

[52] F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold,Quantacell: powerful charging of quantum batteries,New J. Phys. 17, 075015 (2015).

[53] D. Farina, G. M. Andolina, A. Mari, M. Polini,and V. Giovannetti, Charger-mediated energy trans-fer for quantum batteries: An open-system approach,Phys. Rev. B 99, 035421 (2019).

[54] T. P. Le, J. Levinsen, K. Modi, M. M. Parish, and F. A.Pollock, Spin-chain model of a many-body quantum bat-tery, Phys. Rev. A 97, 022106 (2018).

[55] L. Peng, W.-B. He, S. Chesi, H.-Q. Lin, andX.-W. Guan, Lower and upper bounds of quan-tum battery power in multiple central spin systems,Phys. Rev. A 103, 052220 (2021).

[56] S.-Y. Bai and J.-H. An, Floquet engineeringto reactivate a dissipative quantum battery,Phys. Rev. A 102, 060201(R) (2020).

[57] F. Tacchino, T. F. F. Santos, D. Gerace, M. Campisi, andM. F. Santos, Charging a quantum battery via nonequi-librium heat current, Phys. Rev. E 102, 062133 (2020).

[58] F.-Q. Dou, Y.-J. Wang, and J.-A. Sun, Closed-loop three-level charged quantum battery, EPL 131, 43001 (2020).

7

[59] J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang, andC. Urbina, Decoherence of a superconducting qubit dueto bias noise, Phys. Rev. B 67, 094510 (2003).

[60] M. Kjaergaard, M. E. Schwartz, J. Braumuller,P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D.Oliver, Superconducting qubits: Current state of play,Annu. Rev. Conden. Ma. Phys. 11, 369 (2020).

[61] See supplemental material, including experimental setupand details, variation of eqs. (3, 4), and supplementaryexperimental data.

[62] S. Koyu, A. Dodin, P. Brumer, and T. V.Tscherbul, Steady-state fano coherences in a v-type system driven by polarized incoherent light,Phys. Rev. Research 3, 013295 (2021).

[63] J. Cui, J. I. Cirac, and M. C. Banuls, Variational ma-trix product operators for the steady state of dissipative

quantum systems, Phys. Rev. Lett. 114, 220601 (2015).[64] D.-W. Luo, P. V. Pyshkin, C.-H. Lam, T. Yu, H.-Q.

Lin, J. Q. You, and L.-A. Wu, Dynamical invariantsin a non-markovian quantum-state-diffusion equation,Phys. Rev. A 92, 062127 (2015).

[65] X. L. Huang, X. X. Yi, C. Wu, X. L. Feng,S. X. Yu, and C. H. Oh, Effective hamiltonianapproach to open systems and its applications,Phys. Rev. A 78, 062114 (2008).

[66] M.-F. Chen, L.-T. Shen, R.-X. Chen, and Z.-B.Yang, Driving to the steady ground-state su-perposition assisted by spontaneous emission,Phys. Rev. A 92, 033403 (2015).

[67] J. Jing, T. Yu, C.-H. Lam, J. Q. You, and L.-A. Wu, Con-trol relaxation via dephasing: A quantum-state-diffusionstudy, Phys. Rev. A 97, 012104 (2018).

arX

iv:2

108.

1296

7v1

[qu

ant-

ph]

30

Aug

202

1

Supplemental Material for “Demonstration of dynamical control of three-level open

systems with a superconducting qutrit”

Ri-Hua Zheng,∗ Wen Ning,∗ Zhen-Biao Yang,† Yan Xia,‡ and Shi-Biao Zheng§

Fujian Key Laboratory of Quantum Information and Quantum Optics,

College of Physics and Information Engineering,

Fuzhou University, Fuzhou, Fujian 350108, China

I. EXPERIMENTAL SETUP

The entire control and readout layout of the experi-ment is shown in Fig. S1. Electronic devices for qutritreadout, Josephson parametric amplifier (JPA) control,and qutrit control are displayed from top to bottom. Thereadin and XY control signals are the mixtures of low-frequency signals [from two independent digital-to-analogconverter (DAC) channels I and Q] and high-frequencysignals (from microwave sources) to achieve nanosecondfast tuning. On the other hand, the qutrit Z control sig-nal is sent directly from the DAC without mixing, whosefrequency can also be slowly tuned by the direct cur-rent (DC) bias line. Additionally, the output signal ofthe readout feeder is amplified by the JPA, high electronmobility transistor (HEMT), and room temperature am-plifier, and then demodulated by the analog-to-digitalconverter (ADC). Four cryogenic unidirectional circula-tors are inserted between the JPA and the 4K HEMT toblock the reflection and noise from outside. As for theJPA, it is pumped by an independent microwave signalsource and a DC bias, which is converged by the bias tee.At different temperature stages of the dilution refrigera-tor, each control line is balanced with some attenuatorsand filters (intuitively shown in Fig. S1) to prevent un-wanted noise from affecting the equipment.

II. QUTRIT READOUT AND CALIBRATION

Here we show the results of the qutrit readout in Fig.S2, where blue, orange, and yellow points represent theresults of applying I, X01, and X01X12 to the qutrit, i.e.,depicting the results of preparing |0〉, |1〉, |2〉, respec-tively. The readout pulses are 1 µs-long and the repeti-tion is 3000. It can be seen that these three states areclearly separated, though with several error transitionscaused by the readout errors. Like that in Refs. [1, 2],the calibration matrix can be defined as

F =

F00 F01 F02

F10 F11 F12

F20 F21 F22

, (S1)

∗ These two authors contributed equally to this work.† zbyang@fzu.edu.cn‡ xia-208@163.com§ t96034@fzu.edu.cn

where Fii′ (i, i′ = 0, 1, 2) is the probability of measuring

the qutrit in |i〉 when it is prepared in |i′〉. Such that themeasured probability is given by Pm

i =∑

i′ Fii′Pci′ , with

P ci′ being the calibrated probability of |i′〉, viz.,

P c0

P c1

P c2

= F−1

Pm0

Pm1

Pm2

. (S2)

We can derive, for instance, F01 = NOL/3000, in whichNB,O,YL,R,T is the number of blue, orange, yellowpoints in the left, right, top zones in Fig. S2. In ad-dition, we repeat the measurement for 10 times to takethe average values, yielding

F =

0.974 0.102 0.0410.017 0.885 0.1410.009 0.013 0.818

. (S3)

Note that all the experimental data have been calibratedby the matrix in Eq. (S3).

III. VARIATION OF EQS. (3, 4) IN THE MAINTEXT FOR THE COHERENCE CONTROL

Equations (3, 4) in the main text are used to controlthe populations, with designable fk (k = 1, 2) and itera-tive hi+1 (i = 0, 1, 2). If we alternatively aim to controltwo parts of the coherence (e.g., h2 and h3) of systems,we need to alter Eqs. (3, 4) in the main text as

Ω01 =h3[(γ + Γ) + 2h3] + h2[(γ2 + Γ2) + 2h2]

2h1h2 − 2h3(2f1 + f2 − 1),

Ω12 =(2f1 + f2 − 1)[h2(γ2 + Γ2) + 2h2] + h1[h3(γ + Γ) + 2h3]

2h1h2 − 2h3(2f1 + f2 − 1),

h1 = Ω12(f1 − f2)−1

2h1(γ + Γ+ γ2 + Γ2)− h2Ω01,

f1 = −Γf1 + Γ2f2 + 2h3Ω01 − 2h1Ω12,

f2 = −Γ2f2 + 2h1Ω12, (S4)

in which hk+1 is designable and fk is iterative. Thereforeone can design suitable h2 and h3 to accomplish desiredcoherence control.

IV. FEASIBLE AND INFEASIBLE AREAS OFTHE COHERENCE CONTROL

For the control of h2 and h3, similarly, there are fea-sible area (FA) and infeasible area (IFA), which mainly

2

Q1 Z

Mixing Chamber4 K

Readout

Q1 XY

HEMT

DC

DAC

7.5 GHz

7.5 GHz

7.5 GHz80 MHz

ADC

DAC

JPA13.1 GHz20 dB 10 dB

RC

20 dB 20 dB 20 dB 20 dB

10 dB 20 dB 20 dB 10 dB

RC

80 MHz

Vacuum Cavity

IQ mixer Attenuator Bias tee

Circulator

RC Filter

AmplifierSignal Generator Low-pass Filter

Band-pass

FilterRC

Josephson

Parametric

AmplifierJPA

50 ohm terminator

Sample

DC

DAC

0.5 GHz

10 dB20 dB

Readin

FIG. S1. (Color online) The wiring diagram of the experimental setup. Green top channel: readin and readout of the Xomnqutrit; blue bottom channel: XY and Z control of the Xomn qutrit. The detailed description is placed in the text.

-600 -400 -200 0 200 400I (a.u.)

-400

-200

0

200

400

600

Q (

a.u.

)

FIG. S2. (Color online) Results of the qutrit readout in theI-Q plane. Blue, orange and yellow points are the results of|0〉, |1〉, |2〉, respectively. The geometric centers of the threecluster points (3000 points per cluster) are marked by blackcross signs.

depend on the intermediate function f and decoherence.For f = [1 + e−a(t−tf/2)]−1, we plot FA and IFA of thecoherence control in Fig. S3. The total area is con-structed by conditions [h2, h3] ≤ 0.5 and h2

2 + h23 ≤

[1− f21 − f2

2 − (1 − f1 − f2)2 − 2h2

1]/2 ≤ 1/3.

V. TOMOGRAPHY OF THE COHERENCECONTROL

The populations of three-level systems can be directlymeasured by the readout cavity. (In the experiments,only diagonal elements of the density matrix can be di-rectly measured.) While the coherence can be indirectlymeasured by a simplified tomography, including 4 stepsas follows,(i) U1 = I,

ρ1 = U1ρU†1 =

f2 −ih1 h2

ih1 f1 −ih3

h2 ih3 1− f1 − f2

, (S5)

(ii) U2 = (X/2)01,

ρ2 = U2ρU†2

=

f2 − i(h1−h2)√2

h1+h2√2

i(h1−h2)√2

− f22+ h3 +

12

12i(2f1 + f2 − 1)

h1+h2√2

− 12i(2f1 + f2 − 1) 1

2(−f2 − 2h3 + 1)

,

3

-0.5 0 0.5-0.5

0

0.5 infeasible area

feasible area

FIG. S3. (Color online) Feasible area (surrounded by solid,dashed, and dotted dashed curves) and infeasible area (theoutside zone surrounded by dotted curves) of the coherencecontrol in the experiment when the evolution time is 3, 5, and10 µs.

(S6)

(iii) U3 = (X/2)12,

ρ3 = U3ρU†3

=

f1+f22

+ h1−i2(f1 − f2)

1√2(h2 − h3)

i2(f1 − f2)

12(f1 + f2 − 2h1)

−i√2(h2 + h3)

1√2(h2 − h3)

i√2(h2 + h3) 1− f1 − f2

,

(S7)

(iv) U4 = U3U2,

ρ4 = U4ρU†4

=

14

(

f2 + 2√2h1 − 2

√2h2 + 2h3 + 1

)

14i(3f2 − 2h3 − 1) 2f1+f2+

√2h1+

√2h2−1

2√

2

− 14i(3f2 − 2h3 − 1) 1

4

(

f2 − 2√2h1 + 2

√2h2 + 2h3 + 1

)

− i(−2f1−f2+√

2h1+√2h2+1)

2√

2

2f1+f2+√

2h1+√2h2−1

2√

2

i(−2f1−f2+√

2h1+√2h2+1)

2√

2

12(−f2 − 2h3 + 1)

, (S8)

where I represents identity gate and (X/2)01(12) denotesa π/2 rotation over the X axis of Bloch sphere in basis|0〉,|1〉 (|1〉,|2〉). For all the experimental data pointsof the coherence (i.e., hi+1), we measure three diagonal

elements of ρp (p = 1, 2, 3, 4), signed as ρ(ii)p (i = 0, 1, 2),

and further deduce hi+1, i,e.,

h1 = ρ(22)3 − ρ

(22)1 + ρ

(11)1

2, h3 = ρ

(11)2 − 1− ρ

(22)1

2,

h2 = −ρ(22)1 − 2

√2h1 + 2h3 − 4ρ

(11)1 + 1

2√2

. (S9)

By now, a simplified tomography for reading out coher-ence hi+1 of three-level systems is completed.

VI. SUPPLEMENTARY EXPERIMENTALDATA FOR THE DYNAMICAL CONTROL

Here we show 30 and 36 groups of experimental datafor the populations and coherence control in Figs. S4 andS5, respectively, whose average error rates are 1.02% and1.39%, respectively, demonstrating the feasibility of thedynamical control in three-level open systems.

[1] Y. Zheng, C. Song, M.-C. Chen, B. Xia, W. Liu,Q. Guo, L. Zhang, D. Xu, H. Deng, K. Huang,Y. Wu, Z. Yan, D. Zheng, L. Lu, J.-W. Pan, H. Wang,

C.-Y. Lu, and X. Zhu, Solving systems of linearequations with a superconducting quantum processor,Phys. Rev. Lett. 118, 210504 (2017).

4

0 1 2 30

0.5

1

Pop

ulat

ions

0.36% 0.41% 0.36% 1.36% 0.74%

1.29% 0.24% 1.71% 2.14% 0.47%

1.01% 1.02% 0.49% 0.58% 1.99%

0.87% 0.67% 0.85% 1.16% 2.34%

0.88% 1.01% 0.39% 1.43% 0.65%

1.41% 1.30% 1.60% 0.61% 1.42%

FIG. S4. (Color online) Experimental results (error bars) of the controls of populations. The ideal results of P0(t), P1(t),and P2(t) are represented by solid, dashed and dotted dashed curves, respectively. The error rates are shown in correspondingsubgraphs.

0 1 2 3

0

Coh

eren

ce

0.06% 1.66% 1.09% 0.94% 0.77% 0.56%

0.57% 0.35% 1.32% 1.87% 0.28% 0.72%

1.40% 0.92% 2.17% 0.23% 1.85% 1.36%

2.43% 1.36% 2.18% 2.06% 1.32% 2.22%

2.63% 1.86% 2.73% 1.83% 1.79% 1.27%

1.97% 1.13% 1.32% 0.59% 1.63% 1.70%

FIG. S5. (Color online) Experimental results (error bars) of the controls of coherence. The ideal results of h0(t), h1(t), andh2(t) are represented by solid, dashed and dotted dashed curves, respectively. The error rates are shown in correspondingsubfigures.

5

[2] W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B.Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B.

Zheng, Deterministic entanglement swapping in a super-conducting circuit, Phys. Rev. Lett. 123, 060502 (2019).