Quantum behavior of a dc SQUID phase qubit

We analyze the behavior of a dc superconducting quantum interference device phase qubit in which one junction acts as a phase qubit and the rest of the device provides isolation from dissipation and noise in the bias leads. Ignoring dissipation, we find the two-dimensional Hamiltonian of the system and use numerical methods and a cubic approximation to solve Schrödinger’s equation for the eigenstates, energy levels, tunneling rates, and expectation value of the currents in the junctions. Using these results, we investigate how well this design provides isolation while preserving the characteristics of a phase qubit. In addition, we show that the expectation value of current flowing through the isolation junction depends on the state of the qubit and can be used for nondestructive read out of the qubit state.

Figure 1

(a) Schematic of a phase qubit. (b) Tilted washboard potential of the phase qubit. (c) Schematic of dc SQUID phase qubit. The circuit contains two Josephson junctions $J1$ and $J2$, with junction capacitance $C_1$ and $C_2$, respectively. The inductances of the two arms of the SQUID are $L_1$ and $L_2$. The current source $I$ biases the two junctions, while an external flux ${\Phi }_a$ is provided by the current source $I_f$ through the mutual inductance $M$. The currents flowing through the right and left arms of the SQUID are $I_1$ and $I_2$, respectively.

Figure 2

Potential $U$ for the dc SQUID phase qubit with the parameters in Table and $I=17.00\mu \text{A}$. (a) 2D surface plot of the potential. (b) Cross section of $U$ along ${\gamma }_2$ for ${\gamma }_1=0$. The numbers label the individual wells. (c) Cross section of $U$ along ${\gamma }_1$ for ${\gamma }_2=0$. (d) Cross section of $U$ along ${\gamma }_1$ for ${\gamma }_2=0$ showing a single well.

Figure 3

(a) Critical current $I_c$ vs well number along ${\gamma }_2$. (b) Plasma frequency ${\omega }_2/2\pi$ vs well number along ${\gamma }_2$. For both graphs the well numbers are from Fig. 2b.

Figure 4

(a) Energy levels and (b) tunneling rates for different metastable states for well $k=0$ as a function of bias current $I$. The zero of the vertical axis is at the bottom of the metastable well. Symbols are from numerical calculation of the full 2D Hamiltonian using complex scaling, while the solid curves show analytical calculation where coupling between the two junctions is ignored. Dashed curves in panel (b) show results from second-order perturbation theory. States are labeled by the ket $|n,m⟩$, where the first index represents the qubit junction state and the second index represents the isolation junction state.

Figure 5

(a) Energy levels and (b) tunneling rates for various metastable states for well $k=8$ as a function of bias current $I$. The meaning of the symbols etc. is as in Fig. 4.

Figure 6

Absolute value of the response functions $G_c$ and $G_q$ as a function of current for well number $k=0$ and 8. Points are obtained by exact numerical calculation, solid lines are from perturbation theory (see Sec. 6) and dashed line shows predictions from classical analysis.

Figure 7

Expectation value of the current $⟨I_2⟩$ flowing through the isolation junction as a function of $I$ for well number 0. The points are numerical calculations based on the exact 2D Hamiltonian, whereas the solid lines correspond to an analytical calculation with Hamiltonian $\overline{H}$ without the coupling term [see Eq. (A1)].