Reading out the state of a flux qubit by Josephson transmission line solitons

We describe the readout process of the state of a Josephson flux qubit via solitons in Josephson transmission lines (JTLs) as they are in use in the standard rapid single flux quantum technology. We consider the situation where the information about the state of the qubit is stored in the time delay of the soliton. We analyze dissipative underdamped JTLs, take into account their jitter, and provide estimates of the measuring time and efficiency of the measurement for relevant experimental parameters.

Figure 1

Setup for the readout of the persistent current qubit based on the delay time of a soliton in the Josephson transmission line (JTL).

Figure 2

(Color online) Energy diagram of the qubit energy levels as a function of the external magnetic flux in the qubit loop. The considered measurement schemes are shown as follows: (a) The qubit is kept away from the symmetry point, (b) the qubit is initially prepared at the symmetry point, but the approaching soliton pushes the qubit far from the symmetry point, and (c) the qubit remains near the symmetry point all the time. Here dots denote the working points of the qubit with no solitons in the JTL, the lines show adiabatic shifts of the working points due to magnetic flux induced in the qubit loop by a passing soliton, $\Delta$ and ${ϵ}_0$ are the tunneling amplitude and energy bias between the two persistent current states of the qubit, and $g$ is the qubit-JTL coupling strength.

Figure 3

Two elementary cells of the discrete JTL.

Figure 4

(Color online) Delay time of a soliton induced by the qubit as a function of initial soliton velocity $u_0$ when the qubit is far from the symmetry point, ${ϵ}_0⪢\Delta$. The velocity and time are measured in units of ${\lambda }_J{\omega }_p$ and ${\omega }_p^{-1}$, respectively. Four curves are obtained numerically from Eqs. (13, 14) for different values of the dissipation strength $\alpha$ (shown in the legend box). The solid line shows the analytical solution (24). The parameters of the JTL and the persistent current qubit are listed in Table . The inset shows the dependence of the delay time on the dissipation strength $\alpha$ for the initial soliton velocity $u_0=0.2$.

Figure 5

(Color online) Fluxon velocity as function of its center coordinate for different values of the dissipation strength $\alpha$ (shown in the legend box).

Figure 6

(Color online) Delay time of a soliton induced by the qubit as function of the initial soliton velocity $u_0$ when the qubit remains at the symmetry point ${ϵ}_0=0$. The velocity and time are measured in units of ${\lambda }_J{\omega }_p$ and ${\omega }_p^{-1}$, respectively. Four curves are numerical solutions of Eqs. (13, 14) for different values of the dissipation strength $\alpha$ (shown in the legend box). The solid curve shows the analytical solution (33). The parameters of the JTL and persistent current qubit are given in Table . The coupling coefficient between the qubit loop and the cell of the JTL is $k=0.05$, which leads to the dimensionless coupling ${\varphi }_1^q=3\times {10}^{-4}$ in the sine-Gordon equation. The value of the coupling coefficient is reduced in order to keep the qubit at the symmetry working point all the time.

Figure 7

(Color online) (a) Schematic of RSFQ confluence buffer and (b) the probability of double switching of output junction J3.

Figure 8

(Color online) Block diagram of a RSFQ time vernier.

Figure 9

Sketch of the distribution functions which describe the propagation time of a soliton. The two functions $P_{\mid 0⟩}\left(t\right)$ and $P_{\mid 1⟩}\left(t\right)$ are peaked around the average values of the propagation times $t_{\mid 0⟩,\mid 1⟩}$ corresponding to the different eigenstates of the qubit ∣0⟩ and ∣1⟩. The standard deviation and peak separation are given by $\sqrt{n}\delta t$ and $n{t}_d$, respectively. The overlap of the distribution functions indicates the regime where the measurement yields an error.

Figure 10

(Color online) Different contributions to the measurement error for $L_r=25{\lambda }_J$, $T=20\mathrm{mK}$, and $\alpha =0.001$. The upper plots correspond to $I_c=0.6\mu \mathrm{A}$ and $k=0.96$, while the lower plots correspond to $I_c=2\mu \mathrm{A}$ and $k=0.53$. The qubit is initially prepared far from the symmetry point. See text for detailed explanations.

Figure 11

(Color online) Different contributions to the measurement error for $L_{\mathit{JTL}}=25{\lambda }_J$, $T=20\mathrm{mK}$, and $\alpha =0.001$. The upper plots correspond to $I_c=0.6\mu \mathrm{A}$ and $k=0.96$, while the lower plots correspond to $I_c=2\mu \mathrm{A}$ and $k=0.53$. The qubit is initially prepared at the symmetry point but is shifted far from it by a moving soliton. See text for detailed explanations.

Figure 12

(Color online) Different contributions to the error of a measurement for $L_{\mathit{JTL}}=25{\lambda }_J$, $T=20\mathrm{mK}$, $\alpha =0.001$, $I_c=0.6\mu \mathrm{A}$, and $k=0.05$. The qubit stays at the symmetry point during the measurement.