Measurements of Quasiparticle Tunneling Dynamics in a Band-Gap-Engineered Transmon Qubit

We have engineered the band gap profile of transmon qubits by combining oxygen-doped Al for tunnel junction electrodes and clean Al as quasiparticle traps to investigate energy relaxation due to quasiparticle tunneling. The relaxation time $T_1$ of the qubits is shown to be insensitive to this band gap engineering. Operating at relatively low-$E_J/E_C$ makes the transmon transition frequency distinctly dependent on the charge parity, allowing us to detect the quasiparticles tunneling across the qubit junction. Quasiparticle kinetics have been studied by monitoring the frequency switching due to even-odd parity change in real time. It shows the switching time is faster than $10\mu \mathrm{s}$, indicating quasiparticle-induced relaxation has to be reduced to achieve $T_1$ much longer than $100\mu \mathrm{s}$.

Figure 1

(a) Low energy–level structure of a transmon qubit. The parity state with larger separation between $|0⟩$ and $|1⟩$ qubit states is assigned to odd without loss of generality. ${\delta }_0$ and ${\delta }_1$: Energy difference between odd-even parities for the ground and first excited state, respectively, ${\delta }_1\gg {\delta }_0$. ${\omega }_{01}^{\mathrm{odd}}-{\omega }_{01}^{\mathrm{even}}={\omega }_{\mathrm{oe}}\approx {\delta }_1$ is the charge dispersion. ${\Gamma }_{\mathrm{oe},i}^{\mathrm{qp}}$ is the switching rate between qubit states of different parities. ${\Gamma }_{1o\to 0e}^{\mathrm{qp}}\approx {\Gamma }_{1e\to 0o}^{\mathrm{qp}}={\Gamma }_{1\to 0}^{\mathrm{qp}}$ is the qubit relaxation rate because ${\omega }_{01}^{\mathrm{odd}}\approx {\omega }_{01}^{\mathrm{even}}={\omega }_{01}\gg {\delta }_1$. (b) SEM image of a typical device fabricated by three-angle evaporation to engineer the band gap profile. (c) Expected band gap profile near the tunnel junction.

Figure 2

(a) Measured qubit relaxation time $T_1$ as a function of temperature for two transmon qubits: one qubit is fabricated with clean Al only with a transition frequency $f_{01}=4.25\mathrm{GHz}$ and the other with oxygen-doped Al, but without the third layer of quasiparticle traps, with $f_{01}=5.16\mathrm{GHz}$. The device layout is identical to Fig. 1(a) in Ref. 1. Dashed black and magenta lines are theory for relaxation due to thermal quasiparticles, assuming $\Delta \approx 180$ and $280\mu \mathrm{eV}$, respectively [8]. The solid green and cyan lines represent the sum of the theoretical expectations for thermal-equilibrium quasiparticles and the best-fit saturated $T_1$. (b) $T_1$ vs frequency on qubits fabricated with oxygen-doped Al and quasiparticle traps. The device layout is identical to Fig. 1(a) in Ref. 2. Dashed red line: Purcell-induced relaxation time; solid blue line: $Q=65,000$; solid black line: a combination of the Purcell effect and a constant $Q$. Neither oxygen-doped Al nor quasiparticle traps improve $T_1$ qualitatively. The black arrow indicates the cavity frequency.

Figure 3

(a) Spectroscopy of a qubit with engineered band gap as a function of gate-induced charge $n_g$. The gate voltage is applied to the qubit through a bias tee at the cavity input port. For each pixel, a Gaussian pulse ($\sigma =20\mathrm{ns}$, corresponding to a $\pi$ pulse on resonance) is applied at the indicated frequency and the qubit is immediately measured. Each pixel is an average of 5000 repetitions (50 ms). Darker pixels correspond to higher homodyne readout voltages that are proportional to the probability of the qubit in the excited state. An “eye”-shaped pattern indicates charge-$e$ jumps associated with the tunneling of nonequilibrium quasiparticles. (b) Cross section of (a) at $n_g=0.5$ as indicated by the black arrow in (a). The charge dispersion is 60 MHz. We refer to the lower (upper) frequency branch as the even (odd) parity peak.

Figure 4

(a) Schematic of the measurement. A selective $\pi$ pulse is applied to one of the two parity peaks in Fig. 3, followed by an immediate readout of the qubit state at about 7 GHz, where readout fidelity is improved. Lines: The flux pulse sequence. The process is repeated every $10\mu \mathrm{s}$. (b) Histogram of the readout at the odd parity peak in Fig. 3b. A threshold $V_{\mathrm{th}}=19\mathrm{mV}$ is chosen to digitize the readout. Inset: Schematic of an imperfect readout with false positives (negatives) ${ϵ}_0$ (${ϵ}_1$).

Figure 5

(a) Power spectral densities of control experiments using a $\pi$ mask with different time constants ${\tau }_{\pi }={\tau }_{\pi 0}+{\tau }_{\pi 1}$. ${\tau }_{\pi 0}={\tau }_{\pi 1}$ are the time constants associated with the $\pi$ mask. The color curves are data and black curves are theoretical predictions. Inset: Schematic of the control experiment with a $\pi$ mask, an RTS with specified time constants and generated by an FPGA. The expected RTS amplitude is reduced to $A_{\mathrm{con}}=A_{\pi }-A_0=P_{\mathrm{odd}}F$, where $A_{\pi }$ and $A_0$ are the statistical averages of the readout during the $\pi$ mask assuming parity switches fast compared to ${\tau }_{\pi }$, $P_{\mathrm{odd}}$ is the probability of the qubit to be in the odd parity state, and $F$ is the readout fidelity. (b) Measured PSDs and theoretical predictions for the direct experiment on ${\Gamma }_{\mathrm{oe}}^{\mathrm{qp}}$ without the $\pi$ mask. Blue curve is the measured Fourier spectrum and the color curves are theoretical predictions for different qubit parity switching times ${\tau }_{oe}={\tau }_o+{\tau }_e$, where ${\tau }_o$ and ${\tau }_e$ are the odd-to-even and even-to-odd switching times, respectively. The red dashed line is the white-noise spectrum in the limit ${\tau }_o$, ${\tau }_e<t_s$, the sampling time. The near absence of any deviation from white noise at high frequencies indicates ${\tau }_o$, ${\tau }_e<10\mu \mathrm{s}$. Inset: The PSD at low frequency does not show a Lorentzian plateau.