Relaxation of Josephson qubits due to strong coupling to two-level systems

We investigate the energy relaxation $\left(T_1\right)$ process of a qubit coupled to a bath of dissipative two-level fluctuators (TLFs). We consider the fluctuators strongly coupled to the qubit both in the limit of spectrally sparse single TLFs as well as in the limit of spectrally dense TLFs. We conclude that the avoided level crossings, usually attributed to very strongly coupled single TLFs, could also be caused by many weakly coupled spectrally dense fluctuators.

Figure 1

Level structure of the coupled qubit-TLF system in resonance $\delta \omega =0$. For $v_{\perp }=0$ the middle levels form a degenerate doublet. The coupling lifts the degeneracy and splits the levels by the oscillation frequency ${\omega }_{osc}$.

Figure 2

(Color online) $⟨{\sigma }_z⟩$ as a function of time in units of the inverse TLF relaxation rate ${\gamma }_1^f$ for the case of the qubit exactly in resonance with the TLF (dotted black line). One observes oscillations with frequency ${\omega }_{osc}$. The solid red curve gives the decay averaged over the oscillations, characterized by $a_{av}$ and ${\Gamma }_{av}$ and the dashed blue curve shows the envelope described by $a_{env}$ and ${\Gamma }_{env}$. Parameters in this plot are (in units of ${\gamma }_1^f$): ${ϵ}_q={ϵ}_f=100$ and $v_{\perp }=10$.

Figure 3

(Color online) Level structure of the coupled qubit-fluctuator system in the simplest case where only the TLF couples to a heat bath. The rates ${\Gamma }_{10}^f$ and ${\Gamma }_{20}^f$ in Eq. (8) lead from levels $|1⟩$ and $|2⟩$, respectively, to the ground state $|0⟩$. The excited state $|3⟩$ is not included in this illustration.

Figure 4

(Color online) Amplitudes $a$ and rates $\Gamma$ for the decay of the oscillating (dashed red) and purely decaying (solid blue) part of the qubit’s $⟨{\sigma }_z⟩$ as a function of the detuning $\delta \omega$ between the qubit and fluctuator. The detuning is taken in units of the coupling $v_{\perp }$.

Figure 5

(Color online) Illustration of the relevant transition processes in the general case of arbitrary coupling of the qubit and fluctuator to a heat bath. In addition to the transitions from the central levels $|1⟩$ and $|2⟩$ to the ground state $|0⟩$ with the rates ${\Gamma }_{10}$ and ${\Gamma }_{20}$, we now also have transitions between the two central levels with the rates ${\Gamma }_{12}$ and ${\Gamma }_{21}$. The excited state $|3⟩$ is again omitted in the illustration.

Figure 6

(Color online) ${\Gamma }_{av}$ as a function of the detuning $\delta \omega$ (rates in units of the qubit’s relaxation rate ${\gamma }_1^q$, detuning in units of the coupling strength $v_{\perp }$) for the general case when the TLFs relaxation rate ${\gamma }_1^f$ is higher (solid blue, ${\gamma }_1^f=1.5{\gamma }_1^q$)/lower (dashed red, ${\gamma }_1^f=0.5{\gamma }_1^q$) than the qubit’s relaxation rate ${\gamma }_1^q$. The parameters in this plot are ${\Gamma }_v^q={\Gamma }_v^f=0.3$ and $v_{\perp }=5$ (in units of ${\gamma }_1^q$).

Figure 7

(Color online) Illustration of the performed transformation in the one-excitation subspace of the Hamiltonian for the case of equal couplings $v_{\perp ,1}=v_{\perp ,2}=v_{\perp }$. Applying the rotation [Eq. (25)] we arrive first in a situation where the state $|2⟩$ (the dark state) is completely decoupled from the other two states. The renormalized coupling then splits the remaining two states, giving a situation analogous to the coupling to one TLF.

Figure 8

(Color online) $\Gamma$ as a function of the qubit’s level splitting ${ϵ}_q$ in units of the TLF relaxation rate ${\gamma }_1^f$ for one possible realization of the fluctuator distribution with $\overline{\nu }=0.5$ and $v_{\perp }=5{\gamma }_1^f$. Solid blue line: approximation of independent fluctuators [Eq. (28)]. Dashed red line: with account for correlations, see text. The corrections are most pronounced in areas, where more than one fluctuator is in resonance $\left(\Gamma >\frac{1}{2}{\gamma }_1^f\right)$.

Figure 9

(Color online) (a) Level structure of a qubit coupled to a uniform spectral distribution of fluctuators. The one-excitation subspace is shown. The state, where the qubit is excited, $|e\uparrow \uparrow \uparrow \cdots ⟩$, is coupled with strength $v_{\perp }$ to all other levels in this subspace. The ground state $|g\uparrow \uparrow \uparrow \cdots ⟩$ is energetically well separated from the subspace with one excitation. (b) Overlap of the initial state $|i⟩=|e\uparrow \uparrow \uparrow \cdots ⟩$ with the eigenstates of the coupled system $|{\varphi }_k⟩$ for a uniform distribution of TLFs with an effective density $\overline{\nu }=1$ as depicted above. The energy is counted from the energy of the initial state.

Figure 10

(Color online) (a) Level structure for a nonhomogeneous spectral distribution of TLFs. At energies near the qubit’s level splitting ${ϵ}_q$ the density is increased. The state $|e\uparrow \uparrow \uparrow \cdots ⟩$ is coupled equally to each level in the one-excitation subspace. (b) Overlap of the initial state with the eigenstates of the coupled system with a higher local density ${\overline{\nu }}_{local}=10$ between $ϵ=-10$ and $ϵ=10$ (in units of $v_{\perp }$). Out of resonance the effective density is $\overline{\nu }=1$.