Measurement of Quasiparticle Diffusion in a Superconducting Transmon Qubit

Abstract

Quasiparticles, especially the ones near the Josephson junctions in the superconducting qubits, are known as an important source of decoherence. By injecting quasiparticles into a quantum chip, we characterized the diffusion feature by measuring the energy relaxation time and the residual excited-state population of a transmon qubit. From the extracted transition rates, we phenomenologically modeled the quasiparticle diffusion in a superconducting circuit that contained “hot” nonequilibrium quasiparticles in addition to low-energy ones.

1. Introduction

The superconducting quantum circuit has become a promising candidate for realizing universal quantum computation for rapid development in recent years. At this stage, the prospective superconducting qubits are aimed at longer coherence times and higher readout fidelities. It has been known that some loss channels—such as dielectric loss from two-level defects [1]—and quasiparticles [2,3,4,5,6,7,8] can set limits on qubit coherence. Among them, the quasiparticles that may absorb the qubit energy when tunneling through the junction are regarded as the dominant loss for the microsecond limit on the energy relaxation time $T_1$ [9,10].

While the quasiparticle density, which is assumed to be in thermal equilibrium, is suppressed exponentially as the bath temperature decreases, recently observed experimental results have shown that the normalized nonequilibrium quasiparticle density is in a range of ${10}^{-8}-{10}^{-6}$, which is several orders of magnitude larger than the “cold” equilibrium one at a 20 mK bath temperature [9,11,12,13,14,15,16]. The nonequilibrium quasiparticles are attributed as a source of the loss mechanism and residual excited-state population. As illustrated in Figure 1a, quasiparticles near the junction with energies $E_{qp}$ of all ranges can absorb the qubit-transition energy and transfer the qubit from the excited state to the ground state, and the “hot” ones that have energies greater than $\Delta +E_{ge}$ may release energy and excite the qubit [17,18].

Several techniques have been proposed to depict the nonequilibrium quasiparticles in superconducting qubits, such as treating the transmon qubit as a real-time quasiparticle-tunneling detector [9,19], the charge-parity correlation measurement [10], and direct dispersive monitoring [20]. However, most of these measurements require high-fidelity dispersive readouts or the transmon qubit in a low $E_J/E_C$ regime. In this letter, we propose a simple and practical protocol, which is demonstrated on a normally anharmonic ($E_J/E_C\gtrsim 50$) single-junction transmon qubit. We study the features of quasiparticles in the sample with the contactless quasiparticle injection technique [21] and read out with the routine called “high-power readout” [22], in which the high demand for dispersive readout fidelities is not necessary. In the experiment, the energy relaxation time $T_1$ and the residual excited-state population $P_e^r$ are measured with different parameters of injected quasiparticles. From the down-transition rates of the qubit and the energy distribution of nonequilibrium quasiparticles extracted from the experimental data, we phenomenologically model the quasiparticle diffusion [23,24].

2. Device and Quasiparticle Injection

The single-junction transmon qubit that we used in the experiment was centered in a 3D cavity that was made of aluminum (Al 6061-T6 alloy) and had a size of $35.56\times 5.08\times 17.78$ mm. The bare frequency of the cavity ${\omega }_{bare}/2\pi$ was 9.0524 GHz. The Josephson junction was located in the middle of two thin-film electrodes that had the size of $250\times 500$ $\mathsf{\mu }$m, as illustrated in Figure 1b. The qubit was cooled down to 30 mK in an Oxford DR200 cryogen-free dilution refrigerator with the transition frequencies of ${\omega }_{ge}/2\pi$ = 7.0905 GHz and ${\omega }_{ef}/2\pi$ = 6.748 GHz and the ratio of Josephson coupling energy to charging energy of $E_J/E_C=58$. The average measured energy relaxation time $T_1$ was $7.29$ $\mathsf{\mu }$s. The data were obtained by repeating the measurement 1000 times with a visibility of $50\%$. More details about the sample can be found in Appendix A.

In this work, we modified the technique of quasiparticle injection reported in [21] by changing the injection duration time to 3–5 $\mathsf{\mu }$s from a hundred microseconds. This allowed us to study the diffusion of the quasiparticles in the qubit. The injection pulse was input at the frequency of ${\omega }_{bare}$ with the power [21,22] corresponding to the one on the right side of the red line in Figure 1c. For convenience, we mark the power of 0 dB with the red line. The gap between the injection pulse and the measurement pulse in Figure 1d is the diffusion time of the quasiparticles.

3. Quasiparticle Probe

In the high-frequency regime $\hslash {\omega }_{ge}\gg E_{qp}-\Delta$, the down-transition rate induced by quasiparticles ${\Gamma }_{\downarrow }^{qp}$ for a single-junction transmon qubit is [5,7]:

$${\Gamma }_{\downarrow }^{qp}=\frac{{\omega }_p^2}{{\omega }_{ge}\pi }\sqrt{\frac{2\Delta }{E_{ge}}}{\chi }_{qp},$$

(1)

where ${\omega }_p=\sqrt{8E_C{E}_J}$ is the plasma frequency and ${\chi }_{qp}$ is the quasiparticle density per Cooper pair.

When the qubit is only excited thermally, the residual excited-state population is $P_e^r=e^{-\frac{\hslash {\omega }_{ge}}{k_{B}T}}/(e^{-\frac{\hslash {\omega }_{ge}}{k_{B}T}}+1)=1.2\times {10}^{-5}$. The measured residual excited-state population is $P_e^r=0.1208$, corresponding to an effective qubit temperature of 174 mK, which is much higher than that of the bath environment of T = 30 mK. This phenomenon indicates that the qubit is also partly excited by other sources, such as the “hot” quasiparticles. Hence, despite the uncharted energy distribution of “hot” quasiparticles, it is deemed that the qubit excitation rate can be written in the implicit function of the quasiparticle density ${\chi }_{qp}$, that is, ${\Gamma }_{\uparrow }\left({\chi }_{qp}\right)$.

Consequently, we measure the energy relaxation time $T_1$ and the residual excited-state population $P_e^r$ to study the quasiparticles in the superconducting qubit. Since the longitudinal relaxation rate ${\Gamma }_l=1/T_1$, we can obtain both the up- and down-transition rates ${\Gamma }_{\uparrow }\left({\Gamma }_{\downarrow }\right)$ from the measured $T_1$ and $P_e^r$, which can be written as [25]:

$$\begin{array}{cc}\hfill T_1& =\frac{1}{{\Gamma }_{\downarrow }-{\Gamma }_{\uparrow }}.\hfill \\ \hfill P_e^r& =\frac{{\Gamma }_{\uparrow }}{{\Gamma }_{\uparrow }+{\Gamma }_{\downarrow }}.\hfill \end{array}$$

(2)

In the T = 30 mK environment, the qubit excitation rate in thermal equilibrium can be neglected, since ${\Gamma }_{\uparrow }=e^{-\hslash {\omega }_{ge}/\left(k_{B}T\right)}{\Gamma }_{\downarrow }\approx {10}^{-23}{\Gamma }_{\downarrow }$; as a result, the relaxation rate can simplified as ${\Gamma }_{\downarrow }=1/T_1$.

In this work, we measure $T_1$ and $P_e^r$ with different densities of quasiparticles by adjusting the parameters of the injection pulses. The power and the pulse length of the applied microwave correspond to the density of the quasiparticles, while the time gap between the injection and measurement denotes the diffusion time $t_{diff}$. As shown in Figure 2, the energy relaxation time and the residual excited-state population vary according to different parameters of injection. Figure 2a,b show the experimental data of $T_1$ and $P_e^r$, respectively. We measured the Rabi oscillation between $|1\rangle$ and $|2\rangle$. As shown in Figure 2b, the reference Rabi oscillation, Rabi${}_{ref}$, is measured after a ${\pi }_{g-e}$ pulse is applied to the qubit in the ground state, while the signal one, Rabi${}_{sig}$, is measured with no front operation. With both traces in Figure 2b, we can obtain the accurate $P_e^r$ [11]. In Figure 2c,d, $T_1$ and $P_e^r$ are plotted as a function of diffusion time, from which we can capture some features of the quasiparticle dynamics. With the increase in the injected microwave power, the qubit is affected by more quasiparticles, resulting in a larger decrease in $T_1$. After sufficient diffusion time, $T_1$ converges to the original value, which means that the energy relaxation time is influenced by quasiparticles. The experimental data in Figure 2 indicate that $T_1$ and $P_e^r$ are sensitive to quasiparticles in the superconducting qubit and can be used to characterize the features of quasiparticles.

4. Quasiparticle Diffusion in the Superconducting Qubit

The transition rates ${\Gamma }_{\uparrow }$ and ${\Gamma }_{\downarrow }$ presented in Figure 3a were calculated with Equation (2). Note that ${\Gamma }_{\uparrow }$ are an order of magnitude smaller than ${\Gamma }_{\downarrow }$, which also almost overlap, indicating that the density of “hot” quasiparticles is independent of the injection power. Meanwhile, there is a positive correlation between ${\Gamma }_{\downarrow }$ and the injection power, demonstrating that most of the generated quasiparticles are “cold” in this injection protocol.

Now, we study the phenomenological energy distribution of the quasiparticles injected into the superconducting qubit. To simplify the calculation, the diffusion, which is clearly related to the energy of the quasiparticles, is taken into account to model the quasiparticles near the junction [21]

$$\frac{\partial {\chi }_{qp}}{\partial t_{diff}}=D{\nabla }^2{\chi }_{qp}.$$

(3)

In Equation (4), $D=60v_{qp}$ cm${}^2$/s is the diffusion constant in aluminum, where the normalized velocity $v_{qp}={(1-{\Delta }^2/E_{qp}^2)}^{1/2}$ depends on the quasiparticle’s energy $E_{qp}$ [5,23]. The Gaussian solution of this diffusion equation can be written as

$${\chi }_{qp}(x,t_{diff})=\frac{C_0}{\sqrt{\pi D{t}_{diff}}}exp\left(\frac{-x^2}{4D{t}_{diff}}\right),$$

(4)

where $C_0$ is the initial value of ${\chi }_{qp}$ and $t_{diff}$ represents the diffusion time. x is the distance over which the created quasiparticles diffuse from the outer side of the electrode to the junction [23]. For simplicity, diffusion is considered in only one dimension, and $x=0.5$ mm in our sample. With Equations (1) and (4), we obtain the relationship between the diffusion time and ${\Gamma }_{\downarrow }$.

As shown in Figure 3a, the transition rates of the qubit extracted from the experiment change with the diffusion time. The four colored curves point to the four injection powers of 4, 7, 10, and 13 dB, respectively, and the maximum values, which are denoted as ${\Gamma }_{\downarrow }^{max}$, corresponding to four injection powers are 2.586 $\times {10}^{-4}$, 3.454 $\times {10}^{-4}$, $4.468\times {10}^{-4}$, and $5.22\times {10}^{-4}$. With the increase in the injection power, ${\Gamma }_{\downarrow }$ of the quasiparticle takes more time to reach ${\Gamma }_{\downarrow }^{max}$. This contradicts the previous theoretical model, in which only the “cold” quasiparticles are considered. In this original model, the power is only related to the density of the quasiparticles. With the bath temperature of T = 30 mK, quasiparticles with the average energy $E_{qp}=\Delta +kT$ have the same velocity, resulting in the same diffusion feature. Therefore, the diffusion time corresponding to ${\Gamma }_{\downarrow }^{max}$ should be the same with different injection powers (see the blue line in Figure 3c). Therefore, we modify the original model by introducing the “hot” quasiparticles with the average energy of $E_{qp}=\Delta +E_{ge}$. Considering the phenomenon in which ${\Gamma }_{\uparrow }$ is independent of the injection power, the density of quasiparticles can be phenomenologically written as:

$${\chi }_{qp}(p,t_{diff})={\chi }_{qp}^c(p,t_{diff})e^{-{\Gamma }_d^c·t_{diff}}+{\chi }_{qp}^h\left(t_{diff}\right)e^{-{\Gamma }_d^h·t_{diff}}+{\chi }_{qp}^0.$$

(5)

where p represents the injection power and ${\chi }_{qp}^0$ represents the quasiparticles that are present when no quasiparticles are injected, and the system is in thermal equilibrium. We assume that the “hot” quasiparticles are independent of the injection power. With the presence of “hot” quasiparticles, the fitted diffusion time corresponding to the ${\Gamma }_{\downarrow }^{max}$ changes with different injection powers (see the gray line in Figure 3c), which is consistent with the experimental data in Figure 3a. Furthermore, the decay rates of the quasiparticles ${\Gamma }_d$ are also introduced to fit the sharply descending trace on the right side of ${\Gamma }_{\downarrow }^{max}$. Here, we assume that recombination and vortex trapping [26,27] are important channels for the decay of quasiparticles. From our model, with the measurement errors taken into account, we extracted the decay rates of the “cold” and “hot” quasiparticles, which were ${\Gamma }_d^c=1/140$ and ${\Gamma }_d^h=1/350$$\mathsf{\mu }$s, respectively. With the injection powers of 4, 7, 10, and 13 dB, the fitting $C_0^c$ for “cold” quasiparticles in Equation (4) was $0.75\times C_{ini},1.8\times C_{ini},2.9\times C_{ini}, and 4.1\times C_{ini}$, where $C_{ini}=[2.8\times {10}^{-3},2.95\times {10}^{-3}]$ for the injection durations of 3 and 4.5 $\mathsf{\mu }$s, and $C_0^h$ of the “hot” quasiparticles is $2\times {10}^{-3}$. Figure 3d demonstrates that the fitting results are in good agreement at different injection powers. The deviation between the experimental data and the theoretical fitting could be caused by the following reasons: (1) the transitions between “hot” and “cold” quasiparticles during diffusion; (2) the use of the quasiparticles’ average energies instead of the integral of the energies when calculating the normalized quasiparticle diffusion velocities; (3) recombination and vortex trapping were not taken into account in our model in order to simplify the dynamic features of quasiparticles.

5. Conclusions

The energy decay time $T_1$ and the residual excited-state population $P_e^r$ of a qubit were measured to characterize the features of quasiparticles in a superconducting circuit. Additionally, we phenomenologically modeled quasiparticle diffusion with “hot” and “cold” quasiparticles by using the extracted up- and down-transition rates.