Energy Relaxation Time between Macroscopic Quantum Levels in a Superconducting Persistent-Current Qubit

We measured the intrawell energy relaxation time ${\tau }_d\simeq 24\mu \mathrm{s}$ between macroscopic quantum levels in the double well potential of a Nb persistent-current qubit. Interwell population transitions were generated by irradiating the qubit with microwaves. Zero population in the initial well was then observed due to a multilevel decay process in which the initial population relaxed to lower energy levels during the driven transitions. The decoherence time, estimated from ${\tau }_d$ within the spin-boson model, is about $20\mu \mathrm{s}$ for this configuration with a Nb superconducting qubit.

Figure 1

(a) Schematic of the PC qubit surrounded by a readout dc SQUID. (b) Schematic of the qubit’s double-well potential with energy levels for an applied frustration close to $0.485{\Phi }_0$. Microwaves pump the qubit from the lowest level of $|1⟩$ ($|1{⟩}_0$) to the third excited level of $|0⟩$ ($|0{⟩}_3$), then decay to the second excited level of $|0⟩$ ($|0{⟩}_2$) with a rate ${\gamma }_d$.

Figure 2

Time profiles of (a) bias frustration, (b) microwave amplitude, (c) SQUID bias current, and (d) SQUID voltage for one measurement trial. $\mathbf{0}$ and $\mathbf{1}$ indicate that the qubit states ($|0⟩$ or $|1⟩$) result in different $I_{\mathrm{sw}}$.

Figure 3

Contour plots of the switching current distribution (a) without microwaves and with microwaves at (b) $\nu =6.77$, (c) 7.9, and (d) 9.66 GHz. In each plot, the leftmost tip of the upper branch corresponds to a fixed frustration point $f_q\approx 0.484{\Phi }_0$. Without microwave irradiation, the population in the upper branch (state $|1⟩$) decreased continuously to zero as the frustration decreased from ${\Phi }_0/2$. Microwaves pumped the population from state $|1⟩$ to state $|0⟩$ at the resonant frustration, the bias point at which the microwave frequency matched the energy level spacing between two states. The white arrows indicate that the resonant frustration moves toward ${\Phi }_0/2$ with increasing microwave frequency, in agreement with the qubit energy structure [Ref. 14].

Figure 4

The amplitude of the microwave resonant dip as a function of microwave duration $t_{\mathrm{pul}}$. The microwave frequency $\nu =9.66\mathrm{GHz}$ and nominal power $P_{\mathrm{rf}}=31.3\mu \mathrm{W}$. The solid squares are experimental data and the line is a best fit to an exponential decay. The inset shows the resonant dips at $t_{\mathrm{pul}}=0.2$, 0.5, 0.8, and 1 ms, from top to bottom.

Figure 5

${\tau }^{\prime }$ vs microwave power for $\nu =9.66\mathrm{GHz}$. The solid line is a best fit to Eq. (5).