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

Introduction.–The control of quantum systems is always an extremely interesting and essential topic among diverse types of quantum information tasks [1, 2]. With the gradual development of experimental techniques, decoherence, mainly resulted from energy relaxation and dephasing, has become a primary barrier lying on the way to experimentally implement the quantum control. To clear this barrier, the researchers choose to take measures to shorten the evolution time and have achieved remarkable successes in the experiments, such as, preparing 20-qubit entanglement [3, 4] and simulating quantum walk on 62-qubit processor [5]. Meanwhile, some researchers attempt on constructing qubits that are insensitive to decoherence [6, 5, 7, 8]. So far, the relaxation time $T_{1}$ and dephasing time $T_{2}$ of superconducting qubit have both been extended to 200 $\mu$s [8]. Indeed, the methods of shortening the evolution time and making more perfect qubits naturally reduce the impact of decoherence rather than avoiding it, which still process quantified errors. For realizing the leap of quantum error rates from $10^{-3}$ to $10^{-15}$ [9, 10] and further accomplishing near-term quantum works, it is necessary to explore the control of open quantum systems, tailoring the decoherence as one of the effective elements involved in the quantum control.

The earlier open-system quantum control theory draws supports from a kind of nuclear magnetic resonance technology, called spin-echo [11], in which a $\pi$ pulse is inserted midway in the evolution time to increase $T_{2}$. However, this inserted $\pi$ pulse will change the systemic dynamics. Several spin-echo-like techniques, such as bang-bang control [12], dynamical decoupling technique [13], and parity-kicks [14], process the same problems with the destruction of dynamics. So these works generally served as techniques for storing coherence of quantum systems [15, 16, 17, 18, 19, 20]. Researchers alternatively develop an open-system adiabatic theorem [21] to fully control the dynamics. This method utilizes super operators to describe the open systems and can approximately predict the density matrix at each moment. Since the adiabatic theorem needs a long evolution time to guarantee the adiabatic approximation, researchers further try to build a shortcut to adiabaticity (STA) (similar to that in the closed systems [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]).

By virtue of super operators, Sarandy $et$ $al$. [36] have proposed a concept about open-system dynamical invariants to build a STA in open systems. They constructed some 4-dimensional invariants for engineering two-level open systems, nevertheless, with several simplifications, such as, either considering energy relaxation or dephasing. Finding proper invariants will be complicated when someone simultaneously takes energy relaxation and dephasing into account, let alone expansion to three-level open systems with 9-dimensional invariants. Therefore other control methods are eagerly in need of discovery.

Recently, Medina and Semião [37] have directly set the density operator with time-dependent parameters and then submitted them to the Markovian master equation [38], which gives appropriate functions of pulses to control the populations in two-level open systems. The error rates of populations control in Ref. [37] under serious decoherence are almost zero, an exciting result. Some follow-up studies [39, 40] have made a lot of additions to the application scenarios, but still remaining in two-level open systems. With the general trend of quantum tasks towards multiplication [41], it is of great significance to explore the dynamics of three (or more)-level open systems.

In this letter, we propose a method for the dynamical control in three-level open systems, and demonstrate it with a superconducting circuit platform [42, 43, 44, 45, 46, 47]. Two time-dependent parameters are presupposed in the Markovian master equation to singly control the populations or coherence, and then the analytical functions of driving pulses are given to complete desired dynamical control. For proving the correctness of this method, we apply the designed driving pulses to a superconducting Xmon qutrit and measure populations of excited states $|1\rangle$ and $|2\rangle$, or coherence between ground and excited states. The experimental results are in good agreement with the theoretical design for the dynamical control. This is the first experiment to precisely control the Markovian dynamics of three-level open systems, where the decoherence is effectively dominated. In addition, an interesting freezing phenomenon of populations is observed. That is, in the late stages of evolutions, the populations stabilize at specific values for a relatively long time ($>$1.8 $\mu s$) in the presence of decoherence. An instant application scenario of this freezing phenomenon is the quantum battery (QB) [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. One can steadily lock the energy of QBs for a comparatively long time ($>$1.2 $\mu s$), resisting the relaxation of excitation, and therefore building prototypes of QBs with more stable energy.

Method.–Assume that a three-level system consists of bases $|0\rangle$, $|1\rangle$, and $|2\rangle$. The systemic Hamiltonian in the interaction picture is ($\hbar=1$ hereafter) $H_{i}(t)=\Omega_{01}(t)|0\rangle\langle 1|+\Omega_{12}(t)|1\rangle\langle 2|+{\rm H.c.}$, where $\Omega_{01}(t)$ and $\Omega_{12}(t)$ respectively represent the Rabi frequencies of the designed driving pulses for controlling transitions $|0\rangle\leftrightarrow|1\rangle$ and $|1\rangle\leftrightarrow|2\rangle$. We assume the density matrix of this three-level system being [basis order {$|2\rangle$,$|1\rangle$,$|0\rangle$} and dropping $(t)$ henceforth]

$\displaystyle\rho=\left(\begin{array}[]{ccc}f_{2}&-ih_{1}&h_{2}\\ ih_{1}&f_{1}&-ih_{3}\\ h_{2}&ih_{3}&1-f_{1}-f_{2}\\ \end{array}\right),$

(4)

in which $f_{1}$, $f_{2}$, $h_{1}$, $h_{2}$, and $h_{3}$ are all time-dependent real functions. Here $f_{1}$ and $f_{2}$ describe the populations of states $|1\rangle$ and $|2\rangle$, respectively. Additionally, $h_{1}$, $h_{2}$, and $h_{3}$ describe the coherence of this three-level system. Equation 4 (5 degrees of freedom) is not the most generalized case for a three-level density operator (8 degrees of freedom), however sufficient as a demonstration. The Markovian master equation [38] here is

$\displaystyle\dot{\rho}$

$\displaystyle=$

$\displaystyle-i[H_{i},\rho]+\sum_{j=1}^{4}\big{[}L_{j}\rho L_{j}^{\dagger}-\frac{1}{2}(L_{j}^{\dagger}L_{j}\rho+\rho L_{j}^{\dagger}L_{j})\big{]},$

(5)

where Lindblad operators are $L_{k}=\sqrt{\gamma_{k}}|k\rangle\langle k|$ and $L_{k+2}=\sqrt{\Gamma_{k}}|k-1\rangle\langle k|$, with $\Gamma_{k}$ and $\gamma_{k}$ being the rates of energy relaxation and dephasing of state $|k\rangle$, respectively ($k=1,2$). By utilizing Eqs. (4, 5), we can derive the Rabi frequencies to control the dynamics of this three-level system, yielding

	$\displaystyle\Omega_{01}=$	$\displaystyle\frac{f_{1}\Gamma_{1}+\dot{f}_{1}+\dot{f}_{2}}{2h_{3}},$		(6a)
	$\displaystyle\Omega_{12}=$	$\displaystyle\frac{f_{2}\Gamma_{2}+\dot{f}_{2}}{2h_{1}}.$		(6b)

It is noteworthy that there are three constraint equations

	$\displaystyle\dot{h}_{1}$	$\displaystyle=-\frac{1}{2}(\gamma_{1}+\gamma_{2}+\Gamma_{1}+\Gamma_{2})h_{1}+(f_{1}-f_{2})\Omega_{12}-h_{2}\Omega_{01},$		(7a)
	$\displaystyle\dot{h}_{2}$	$\displaystyle=-\frac{1}{2}(\gamma_{2}+\Gamma_{2})h_{2}-h_{3}\Omega_{12}+h_{1}\Omega_{01},$		(7b)
	$\displaystyle\dot{h}_{3}$	$\displaystyle=-\frac{1}{2}(\gamma_{1}+\Gamma_{1})h_{3}+h_{2}\Omega_{12}-(-1+f_{2}+2f_{1})\Omega_{01}.$		(7c)

Up to now, the populations or coherence of the three-level system can be effectively controlled by the interaction Hamiltonian $H_{i}$, i.e., by Rabi frequencies $\Omega_{01}$ and $\Omega_{12}$. A specific example is, an evolution from the ground state $|0\rangle$ to a final state with populations respectively being $P_{0}(t_{f})$, $P_{1}(t_{f})$, and $P_{2}(t_{f})$ in states $|0\rangle$, $|1\rangle$, and $|2\rangle$, can be achieved by adjusting $f_{1}=fP_{1}(t_{f})$ and $f_{2}=fP_{2}(t_{f})$. The intermediate function $f$ changing from 0 to 1 in the time interval $[0,t_{f}]$ can be set as $f=[1+e^{-a(t-t_{f}/2)}]^{-1}$ [26, 27, 37], with $a=50/t_{f}$ determining the gradient of the transformation. By submitting $f_{1}$ and $f_{2}$ into Eqs. (6, 7), one can obtain Rabi frequencies $\Omega_{01}$ and $\Omega_{12}$ to accomplish the desired transformation of populations.

Device.–Here we use a frequency-tunable superconducting Xmon qutrit [42, 43, 44, 45, 46, 47] to test the above theory. The original Hamiltonian reads $H=\Omega_{01}e^{i\omega_{01}t}|0\rangle\langle 1|+\Omega_{12}e^{i\omega_{12}t}|1\rangle\langle 2|+{\rm H.c.}+\sum_{i=0,1,2}\omega_{i}|i\rangle\langle i|$, where $\omega_{i}$ and $\omega_{01(12)}$ are the angular frequencies of energy level $|i\rangle$ and microwave pulses coupling $|0\rangle\leftrightarrow|1\rangle\ (|1\rangle\leftrightarrow|2\rangle)$, respectively. We adjust $\omega_{01}/2\pi=(\omega_{1}-\omega_{0})/2\pi=5.9600$ GHz and $\omega_{12}/2\pi=(\omega_{2}-\omega_{1})/2\pi=5.7208$ GHz (in this experiment, frequency accuracy to 4 decimal places is mandatory) to ensure that the pulses can resonantly drive the transitions between adjacent energy levels. Note the fixed frequency of the resonator (not used here) is 5.584 GHz [44, 45, 46, 47], dynamically decoupled with the above system.

An essential process for this experiment is to precisely measure the coefficients of decoherence, $\gamma_{k}$ and $\Gamma_{k}$. According to the above form of Lindblad operators, one can deduce $\Gamma_{1}=1/T_{1}^{01}$, $\gamma_{1}=2/T_{2}^{01}-\Gamma_{1}$, $\Gamma_{2}=1/T_{1}^{12}$, and $\gamma_{2}=2/T_{2}^{12}-\Gamma_{2}-\Gamma_{1}-\gamma_{1}$. Here $T_{1}^{01(12)}$ and $T_{2}^{01(12)}$ are the energy relaxation and dephasing time measured between $|0\rangle$ and $|1\rangle$ ($|1\rangle$ and $|2\rangle$), shown in Figs. 1(a) and 1(b), respectively. From Fig. 1, we find $T_{1}$ and $T_{2}$ fluctuate a lot during the measured period of 20 hours. In addition, several works [59, 15, 16, 17, 18, 19, 20] support that $T_{2}$ will change with the applying of microwave pulses (the spin-echo technique [11] is a strong proof). Therefore the values of $T_{1}$ and $T_{2}$ we utilized to design pulses are a little different from those measured in Fig. 1, specifically, $[T_{1}^{01},T_{1}^{12},T_{2}^{01},T_{2}^{12}]=[9.5,4.6,6,1.9]\ \mu s$, 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 $f_{1}$ and $f_{2}$, there are infeasible zones of the populations transformation, shrinking as $t_{f}$ extending, shown in Fig. 2(a). While the feasible area still takes a large part. As an example, we utilize Eqs. (6, 7) to design $\Omega_{01}$ and $\Omega_{12}$ [see Fig. 2(b)] for the transformation of populations $P_{1}(0)=0\to P_{1}(t_{f})=0.3$ and $P_{2}(0)=0\to P_{2}(t_{f})=0.2$. For the evolution time, we choose $t_{f}=3\ \mu s$ to accumulate enough impact of the decoherence. The corresponding experimental results are shown in Fig. 3, with outer and inner layers indicating the results of applying open-system [see Eqs. (6, 7)] and closed-system [see Eqs. (6, 7), preset $\gamma_{k}=\Gamma_{k}=0$] pulses, respectively. Intuitively, the populations are controlled more precisely in the outer layer, as compared to the inner one. More rigorously, we define a standard deviation to describe the error rate of the population control

$\displaystyle{\rm error}=\sqrt{\sum_{i=0,1,2}[P_{i}(t_{f})^{\rm ideal}-P_{i}(t_{f})^{\rm exp.}]^{2}/3},$

(8)

where $P_{i}(t_{f})^{\rm ideal}$ and $P_{i}(t_{f})^{\rm exp.}$ are the ideal and experimental values of the population in $|i\rangle$, respectively. In the outer layer of Fig. 3, the populations control achieves an error rate of 1.02%, not small [60] because we intentionally extended the evolution time (3 $\mu$s). In contrast, if the control is performed for such a long time by applying closed-system pulses, the error rate reaches 7.49%. This stark difference in error demonstrates the effectiveness of the present control method.

We also measure the results of 29 different controls of populations, 6 of them shown in Fig. 4 and all displayed in the Supplemental Material [61]. The error rates of the controls in Fig. 4 are around 1%, which we believe can be further improved with more stable superconducting qutrits [8].

Another exceptional physical phenomenon is that the populations seem to be frozen after 1.8 $\mu$s in the outer layers of Fig. 3. This freezing phenomenon of populations does not mean that the driving pulses $\Omega_{01}$ and $\Omega_{12}$ have stopped. On the contrary, it is the driving pulses [see the insets of Fig. 2(b)] we applied that caused this freezing phenomenon to occur. Specifically, such a freezing phenomenon arises from the interplay between the dynamics induced by the continuous microwave drives and the two decoherence channels characterized by $T_{1}$ and $T_{2}$ (both available for $|0\rangle\leftrightarrow|1\rangle$ and $|1\rangle\leftrightarrow|2\rangle$ transitions), similar to the cases for generation of steady states in most open systems [62, 63, 64, 65, 36, 66, 67]. But here it differs significantly in that both the energy relaxation and the dephasing are involved in the nonequilibrium dynamical processes and together help freeze the states of three-level systems, as compared to the previous ones which generally consider only one decoherence channel [36, 66, 67].

Results of the coherence control. Since only two free variables, $\Omega_{01}$ and $\Omega_{12}$ in Eqs. (6, 7) can be controlled in the experiment, we can only control two parts of coherence. Here we choose $h_{3}$ and $h_{2}$ as an example for illuminating the coherence control. Corresponding variations of Eqs. (6, 7) and the forbidden zone of coherence control are shown in the Supplemental Material [61]. According to the same preset of parameters and intermediate function $f$, we show 6 groups of results of the coherence control in Fig. 5 (all the 36 groups of results in the Supplemental Material [61]). The error rate here is

$\displaystyle{\rm error}^{\prime}=\sqrt{\sum_{m=2,3}[h_{m}(t_{f})^{\rm ideal}-h_{m}(t_{f})^{\rm exp.}]^{2}/2},$

(9)

where $h_{m}(t_{f})^{\rm ideal}$ and $h_{m}(t_{f})^{\rm exp.}$ are the ideal and experimental values of coherence $h_{m}$, respectively. Figure 5 shows that $h_{2}$ and $h_{3}$ are controlled well and the values of $h_{1}$ are accurately predicted. Similarly, the microwaves protect the coherence (lasting 1.2 $\mu$s) from the influence of dephasing. If there are no microwaves applied, roughly, after the same 1.2 $\mu$s, $h_{3}(1.2\mu{\rm s})/h_{3}(0)\sim\exp(-1.2\mu{\rm s}/T_{2}^{01})\sim 0.8$ and $h_{2}(1.2\mu{\rm s})/h_{2}(0)\sim\exp(-1.2\mu{\rm s}/T_{2}^{12})\sim 0.45$, the coherence will be seriously damaged. In contrast, the error rates are all within 2% when we apply microwaves for the coherence control.

Application.– Quantum battery [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. The freezing phenomenon seems helpful to stabilize the energy of the QB for a fairly long time, especially in the energy-storing stage of the QB. For proving this, we design a pulse for a qutrit in the experiment to simulate the charging, storing, discharging processes of the QB (see Fig. 6). A considerably long period, about 1.2 $\mu$s, of energy [$\epsilon=(\omega_{1}-\omega_{0})P_{1}$] stability was observed. This stability is hard to achieve by applying a pulse designed without considering decoherence because energy relaxation of the superconducting qutrit will exponentially drop the excitation down. Therefore, the present method may contribute to the construction of more stable QBs.

Conclusion.–We have proposed a method to control the dynamics of three-level open systems and realized it in the experiment with a platform of a superconducting Xmon qutrit [42, 43, 44, 45, 46, 47]. The populations and coherence could be singly controlled with error rates around 1% under the influence of decoherence for a relatively long time, 3 $\mu$s, close to the dephasing time $T_{2}^{12}$ of the qutrit. In some situations where the control is more successful, the error rates can even be less than 0.3%. We believe these error rates can be further reduced with more stable superconducting qutrits [8].

Additionally, an interesting freezing phenomenon of populations was observed. The designed microwave pulses just offset the impact of decoherence and visually freeze the populations. We then applied this phenomenon to make more stable prototypes of QBs, whose energy can hold 1.2 $\mu$s with only one charging. Moreover, this freezing phenomenon strongly proves that the Markovian master equation precisely describes the dynamics of three-level open systems, as demonstrated here with a superconducting Xmon qutrit that possesses the specially intrinsic decoherence rates. The present work provides a positive prospect of accurately realizing the dynamical control of three (or more)-level open systems by using the adjustable driving pulses in the Markovian environment.

Note that all the data points of experiments were averaged over 12000 times and the readout errors of the qutrit have been corrected. We thank Ye-Hong Chen and Xin Wang in RIKEN for discussions. This work was supported by the National Natural Science Foundation of China (Grants No. 11874114, and No. 11875108), and the Natural Science Funds for Distinguished Young Scholar of Fujian Province under Grant 2020J06011, Project from Fuzhou University under Grant JG202001-2.