Quantum interference induced by multiple Landau-Zener transitions in a strongly driven rf-SQUID qubit

We irradiated an rf superconducting quantum interference device qubit with large-amplitude and high-frequency electromagnetic field. Population transitions between macroscopic distinct quantum states due to multiple Landau-Zener transitions at energy-level avoided crossings were observed. The qubit population on the excited states as a function of flux detuning and microwave power exhibits interference patterns. Some features are found in the interference and a model based on rate equations can well address these features.

Figure 1

(Color online) (a) Optical micrograph of the rf-SQUID qubit together with control and readout circuit. (b) Equivalent circuits of the sample layout. Crosses represent the Josephson junctions. (c) The measurement time sequence.

Figure 2

(Color online) (a) Normalized qubit population in $|L⟩$ state as a function of dc flux detuning ${\Phi }_f^q$. As a guide to eyes, the left dotted line marks the qubit transition step center where the population is 0.5 in the absence of microwave. Four peaks and dips (symmetric about the zero flux detuning position, i.e., the spectrum center) are induced by a $10\mu \text{s}$ microwave pulse at 17.0 GHz and $-16\text{dB}\text{m}$. A resonant peak with strong population inversion was observed around a dc flux detuning $\delta {\Phi }_f^q=3\text{m}{\Phi }_0$, marked with dashed line. (b) Normalized qubit population in $|L⟩$ state versus the nominal microwave power at a fixed dc flux detuning $3\text{m}{\Phi }_0$. (c) An illustration of double-well potential at a fixed flux detuning and the relevant transition processes involved in Fig. 2b. $|0R⟩$ and $|0L⟩$ are the ground states of right well and left well respectively. $|nL⟩$ is the $n\text{th}$ excited state in the left well. $|N⟩$ represents a higher energy level which is not localized in either potential well. The red (blue) lines show the excitation (relaxation) path. The purple lines represent the LZ transitions between $|0R⟩$ and $|nL⟩$. $n=4$ in our following calculation. (d) Calculated qubit population in $|L⟩$ state as a function of microwave amplitude. The relevant parameters used in our theoretical model are: $\omega /2\pi =17\text{GHz}$, $m=8$, $\Delta /2\pi =7\text{MHz}$, ${\Gamma }_2/2\pi =2\text{GHz}$, $\gamma /2\pi =\Gamma /2\pi =2\text{GHz}$, ${\Gamma }_{10}/2\pi =0.6\text{kHz}$, ${\Gamma }_{01}/2\pi =0.1\text{kHz}$, $a/2\pi =5\text{Hz}$, $b=1.4$.

Figure 3

(Color online) (a) Dependence of qubit population in $|L⟩$ state on dc flux detuning $\delta {\Phi }_f^q$ and nominal microwave power, with microwave frequency at 17.0 GHz. Part of two sets of overlapped interference fringes can be observed and population inversion is clearly demonstrated. (b) Energy-level diagram illustrating the multiphoton transition processes. The qubit is initially dc biased at $\delta {\Phi }_f^q$, where the ground state is $|0R⟩$. Microwave consecutively drives the qubit through the energy levels that are approximately linear in flux detuning and LZ transitions occur at the avoided level crossings ${\Delta }_{0R,nL}$ and ${\Delta }_{0R,\left(n+1\right)L}$. The positions of the two level crossings are marked by P1 and P2 ($\text{P}1=-45\text{m}{\Phi }_0$ and $\text{P}2=-56\text{m}{\Phi }_0$ in our simulation). The red and green paths, with different energy-level slope, represent the main processes to generate the first and second interference set, respectively. (c) Numerically simulated qubit population in $|L⟩$ state versus flux detuning and microwave amplitude. The calculated interference patterns show good agreement with the experimental observations [Fig. 3a].