Decoherence of a Josephson qubit due to coupling to two-level systems

Noise and decoherence are major obstacles to the implementation of Josephson junction qubits in quantum computing. Recent experiments suggest that two-level systems (TLS) in the oxide tunnel barrier are a source of decoherence. We explore two decoherence mechanisms in which these two-level systems lead to the decay of Rabi oscillations that result when Josephson junction qubits are subjected to strong microwave driving. (A) We consider a Josephson qubit coupled resonantly to a two-level system, i.e., the qubit and TLS have equal energy splittings. As a result of this resonant interaction, the occupation probability of the excited state of the qubit exhibits beating. Decoherence of the qubit results when the two-level system decays from its excited state by emitting a phonon. (B) Fluctuations of the two-level systems in the oxide barrier produce fluctuations and $1∕f$ noise in the Josephson junction critical current $I_0$. This in turn leads to fluctuations in the qubit energy splitting that degrade the qubit coherence. We compare our results with experiments on Josephson junction phase qubits.

Figure 1

Rabi oscillations of a resonantly coupled qubit-TLS system with ${\epsilon }_{\mathrm{TLS}}=\hslash {\omega }_{10}$. There is no mechanism for energy decay. Occupation probabilities of various states are plotted as functions of time. (a) $P_1$ is the occupation probability in the qubit state $\mid 1⟩$; (b) $P_{(0,g)}$ is the occupation probability in the state $\mid 0,g⟩$; (c) $P_{(0,e)}$ is the occupation probability of the state $\mid 0,e⟩$; (d) $P_{\left(1g\right)}$ is the occupation probability of the state $\mid 1,g⟩$; and (e) $P_{\left(1e\right)}$ is the occupation probability of the state $\mid 1,e⟩$. Notice the beating with frequency $2\eta$. Throughout the paper, ${\omega }_{10}∕2\pi =10\mathrm{GHz}$. Parameters are chosen mainly according to the experiment in Ref. 23: $\eta ∕\hslash {\omega }_{10}=0.0005$, $g_{\mathrm{qb}}∕\hslash {\omega }_{10}=0.01$, and $g_{\mathrm{TLS}}=0$. The dotted line in panel (a) shows the usual Rabi oscillations without resonant interaction, i.e., $\eta =0$.

Figure 2

Relaxation of Rabi oscillations due to energy decay of the TLS via phonon emission. Solid lines include the effect of the energy decay of the TLS with ${\tau }_{ph}=10\mathrm{ns}$ and the dotted lines have ${\tau }_{ph}=\infty$. These results are averaged over 100 runs in this figure and in all of the following figures. (a) No coupling between the TLS and the microwaves. (b) Direct coupling between the TLS and the microwaves with $g_{\mathrm{TLS}}∕{\omega }_{10}=0.008$. The rest of the parameters are the same as in Fig. 1. We note that the coupling between the TLS and microwaves degrades the qubit coherence.

Figure 3

The qubit-TLS system starts in its ground state at $t=0$. A microwave $\pi$ or $3\pi$ pulse (from 0 to 5 ns) puts the qubit-TLS system in the qubit excited state $\mid 1⟩$ that is a superposition of the two entangled states $\mid {\psi }_1^{\prime }⟩\equiv (\mid 0,e⟩+\mid 1,g⟩)∕\sqrt{2}$ and $\mid {\psi }_2^{\prime }⟩\equiv (\mid 0,e⟩-\mid 1,g⟩)∕\sqrt{2}$. After the microwaves are turned off, the occupation probability starts oscillating coherently. Values of $g_{\mathrm{TLS}}$ indicated in the figure are normalized by $\hslash {\omega }_{10}$. The rest of the parameters are the same as in Fig. 1. (a) No energy decay of the excited TLS, i.e., ${\tau }_{ph}=\infty$. Coherent oscillations with various values of $g_{\mathrm{TLS}}$. (b) Oscillations following a $\pi$ pulse with ${\tau }_{ph}=40\mathrm{ns}$ and various values of $g_{\mathrm{TLS}}$. (c) Oscillations following a $\pi$ pulse and a $3\pi$ pulse with ${\tau }_{ph}=40\mathrm{ns}$ and $g_{\mathrm{TLS}}=0.004$. The dip in the dotted-dashed line is one and one-half Rabi cycles.

Figure 4

Solid lines show the Rabi oscillation decay due to qubit level fluctuations caused by a single fluctuating two-level system trapped inside the insulating tunnel barrier. The TLS produces random telegraph noise in $I_0$ that modulates the qubit energy level splitting ${\omega }_{10}$. $⟨{\omega }_{10}⟩∕2\pi =10\mathrm{GHz}$. Calculation used Eq. (12) with $g_{\mathrm{qb}}=h{f}_R$. (a) The level fluctuation $\delta {\omega }_{10}∕⟨{\omega }_{10}⟩=0.001$. The characteristic fluctuation rate $t_{\mathrm{TLS}}^{-1}=0.6\mathrm{GHz}$. The Rabi frequency $f_R=0.1\mathrm{GHz}$. The dotted lines show the usual Rabi oscillations without any noise source. (b) $\delta {\omega }_{10}∕⟨{\omega }_{10}⟩=0.006$, $t_{\mathrm{TLS}}^{-1}=0.6\mathrm{GHz}$, and $f_R=0.1\mathrm{GHz}$. The dotted lines show the usual Rabi oscillations without any noise source. (c) $\delta {\omega }_{10}∕⟨{\omega }_{10}⟩=0.006$, $t_{\mathrm{TLS}}^{-1}=0.6\mathrm{GHz}$, and $f_R=0.5\mathrm{GHz}$. (d) $\delta {\omega }_{10}∕⟨{\omega }_{10}⟩=0.006$, $t_{\mathrm{TLS}}^{-1}=0.06\mathrm{GHz}$, and $f_R=0.5\mathrm{GHz}$. Note that the scales of the horizontal axes in (a)–(c) are the same. They are different from that in (d).

Figure 5

Solid line represents Rabi oscillations in the presence of both TLS decoherence mechanisms, resonant interaction between the TLS and the qubit, and low frequency qubit energy level fluctuations caused by a single fluctuating TLS. The TLS couples to microwaves $[g_{\mathrm{TLS}}∕\left(\hslash {\omega }_{10}\right)=0.008]$ and the energy decay time for the TLS is ${\tau }_{ph}=10\mathrm{ns}$, the same as in Fig. 2b. The size of the qubit level fluctuations is $\delta {\omega }_{10}∕⟨{\omega }_{10}⟩$, the same as in Fig. 4b. $\eta$ and $q_{\mathrm{qb}}$ are the same as in Fig. 1. $⟨{\omega }_{10}⟩∕2\pi =10\mathrm{GHz}$. The dotted line shows the unperturbed Rabi oscillations.

Figure 6

(a) and (b) Rabi oscillations in the presence of $1∕f$ noise in the qubit energy level splitting. Dotted curves show the Rabi oscillations without the influence of noise. Panel (c) shows the two noise power spectra $S\left(f\right)\equiv {\mid \delta {\omega }_{10}\left(f\right)∕⟨{\omega }_{10}⟩\mid }^2$ of the fluctuations in ${\omega }_{10}$ that were used to produce the solid curves in panels (a) and (b). $g_{\mathrm{qb}}=h{f}_R$ where the Rabi frequency $f_R=0.1\mathrm{GHz}$. $⟨{\omega }_{10}⟩∕2\pi =10\mathrm{GHz}$.