Multilevel effects in the Rabi oscillations of a Josephson phase qubit

We present Rabi oscillation measurements of a $\text{Nb}/{\text{AlO}}_{\text{x}}/\text{Nb}$ dc superconducting quantum interference device (SQUID) phase qubit with a $100\mu {\text{m}}^2$ area junction acquired over a range of microwave drive power and frequency detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the qubit dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi oscillations were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed oscillation frequencies with a simplified model constructed from the full phase qubit Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi frequency, and leakage to the higher excited states.

Figure 1

The dc SQUID phase qubit. (a) The qubit junction $J1$ (with critical current $I_{01}$ and capacitance $C_1$) is isolated from the current bias leads by an auxiliary junction $J2$ (with $I_{02}$ and $C_2$) and geometrical inductances $L_1$ and $L_2$. The device is controlled with a current bias $I_b$ and a flux current $I_f$ which generates flux ${\Phi }_a$ through mutual inductance $M$. Transitions can be induced by a microwave current $I_{rf}$, which is coupled to $J1$ via $C_{rf}$. (b) When biased appropriately, the dynamics of the phase difference ${\gamma }_1$ across the qubit junction are analogous to those of a ball in a one-dimensional tilted washboard potential $U$. The metastable state $|n⟩$ differs in energy from $|m⟩$ by $\hslash {\omega }_{nm}$ and tunnels to the voltage state with a rate ${\Gamma }_n$. (c) The photograph shows a $\text{Nb}/{\text{AlO}}_{\text{x}}/\text{Nb}$ device. Not seen is an identical SQUID coupled to this device intended for two-qubit experiments; the second SQUID was kept unbiased throughout the course of this work.

Figure 2

Qubit energy-level spectroscopy and tunneling escape rates. (a) Open circles show the resonance frequency of the transition between the ground and first excited states of the qubit, measured at 20 mK. The scatter in the values is indicative of the uncertainty in the measurement. Also plotted are theoretical values of ${\omega }_{01}$ (solid), ${\omega }_{12}$ (dashed), and ${\omega }_{23}$ (dotted) for $I_{01}=17.930\mu \text{A}$ and $C_1=4.50\text{pF}$. (b) At high bias, the measured background escape rate (open circles) agrees with the predicted ground-state escape rate ${\Gamma }_0$ (solid line) for the junction parameters given above. Calculated ${\Gamma }_1$, ${\Gamma }_2$, and ${\Gamma }_3$ are plotted as dashed, dotted, and dashed-dotted lines. In both plots, the bottom axes show the total bias current $I_b$, while the top axes indicate the normalized barrier height $N_s$, calculated using the extracted junction parameters.

Figure 3

Rabi oscillations in the escape rate $\Gamma$ were induced at $I_b=17.746\mu \text{A}$ by switching on a microwave current at $t=0$ with a frequency of 6.2 GHz (resonant with the $0\to 1$ transition) and source powers $P_S$ between $-12$ and $-32\text{dBm}$, as labeled. The measurements were taken at 20 mK. The solid lines are from a five-level density-matrix simulation with $I_{01}=17.930\mu \text{A}$, $C_1=4.50\text{pF}$, $T_1=17\text{ns}$, and $T_{\varphi }=16\text{ns}$.

Figure 4

Rabi oscillation frequency ${\Omega }_{R,01}$ at fixed bias as a function of microwave current $I_{rf}$. Extracted values from data (including the plots in Fig. 3) are shown as circles, while the rotating wave solution is shown for two- (dashed line) and five- (solid) level simulations, calculated using $I_{01}=17.930\mu \text{A}$ and $C_1=4.50\text{pF}$ with ${\omega }_{rf}/2\pi =6.2\text{GHz}$.

Figure 5

(Color online) Multiphoton, multilevel Rabi oscillations plotted in the time and frequency domains. (a) The escape rate $\Gamma$ (measured at 110 mK) is plotted as a function of the time after which a 6.5 GHz, $-11\text{dBm}$ microwave drive was turned on and the current bias $I_b$ of the qubit; $\Gamma$ ranges from 0 (white) to $3\times {10}^{8}1/\text{s}$ (black). (b) The normalized power spectral density of the time-domain data from $t=1$ to 45 ns is shown with a grayscale plot. The dashed line segments indicate the Rabi frequencies obtained from the rotating wave model for transitions involving (from top to bottom) 1, 2, 3, and 4 photons, evaluated with junction parameters $I_{01}=17.828\mu \text{A}$ and $C_1=4.52\text{pF}$, and microwave current $I_{rf}=24.4\text{nA}$. Corresponding grayscale plots calculated with a seven-level density-matrix simulation are shown in (c) and (d).

Figure 6

(Color online) The (a) on-resonance Rabi oscillation frequencies ${\Omega }_{R,01}^{\text{min}}$ and ${\Omega }_{R,02}^{\text{min}}$ and (b) resonance frequency shifts $\Delta {\omega }_{01}={\omega }_{rf}-{\omega }_{01}$ and $\Delta {\omega }_{02}=2{\omega }_{rf}-{\omega }_{02}$ are plotted as a function of the microwave current, for data taken at 110 mK with a microwave drive of frequency ${\omega }_{rf}/2\pi =6.5\text{GHz}$ and powers $P_S=-23,-20,-17,-15,-10\text{dBm}$. Values extracted from data for the $0\to 1$ $\left(0\to 2\right)$ transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with $I_{01}=17.736\mu \text{A}$ and $C_1=4.49\text{pF}$ are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a two-level system.

Figure 7

Long-time escape rate ${\Gamma }_{\infty }$ as a function of the bare Rabi frequency ${\Omega }_{01}$. (a) ${\Gamma }_{\infty }$ is plotted for a density-matrix simulation of a system with two (dashed-dotted), three (dashed), four (dotted), and five (solid) levels, $I_{01}=17.930\mu \text{A}$, and $C_1=4.50\text{pF}$. (b) ${\Gamma }_{\infty }$ at low power (for a five-level system) is affected by the inclusion of a small microwave current at ${\omega }_{02}$ (dashed) and inhomogeneous broadening in addition to this noise current (solid). One set of experimental data, for source powers between $-68$ and $-11\text{dBm}$ at 6.2 GHz, is plotted in both panels with open circles.

Figure 8

(Color online) Response to a microwave pulse. (a) The measured escape rate $\Gamma$ (circles), due to a 6.2 GHz, $-20\text{dBm}$ microwave pulse nominally 30 ns long, shows Rabi oscillations followed by a decay governed by multiple time constants. The solid line is the result of a five-level density-matrix simulation for a junction with $I_{01}=17.730\mu \text{A}$ and $C_1=4.46\text{pF}$, using the measured microwave pulse amplitude (with a maximum of 11.7 nA) shown in the inset. The simulation also produces (b) the contribution to the escape rate and (c) the normalized occupation for each of the five levels, as labeled.

Figure 9

(Color online) Comparison of possible damping scenarios for multiphoton transitions. The measured escape rates from Fig. 5a for the (a) $0\to 1$, (b) $0\to 2$, and (c) $0\to 3$ transitions are plotted with circles. The solid lines were calculated with the same simulation (with a common dephasing time $T_{\varphi }$) whose results are shown in Fig. 5c. The dotted lines were calculated for $T_1=10\text{ns}$ and no additional dephasing, while the dashed lines correspond to $T_1=17\text{ns}$ and pure dephasing with $T_{\varphi }=16\text{ns}$ due to low frequency noise (see text).