Understanding degenerate ground states of a protected quantum circuit in the presence of disorder

A recent theoretical proposal suggests that a simple circuit utilizing two superinductors may produce a qubit with ground-state degeneracy [Brooks, Phys. Rev. A 87, 052306 (2013)]. We perform a full circuit analysis along with exact diagonalization of the circuit Hamiltonian to elucidate the nature of the spectrum and low-lying wave functions of this $0\text{−}\pi$ device. We show that the ground-state degeneracy is robust to disorder in charge, flux, and critical current as well as insensitive to modest variations in the circuit parameters. Our treatment is nonperturbative, provides access to excited states and matrix elements, and is immediately applicable also to intermediate parameter regimes of experimental interest.

Figure 1

Circuit diagram of the $0\text{−}\pi$ qubit. Each of the four circuit nodes is associated with one phase variable ${\phi }_j$. Additional symbols show the naming of capacitances, inductances, and Josephson-junction parameters, and the magnetic flux ${\Phi }_{\mathrm{ext}}$ that may thread the inner circuit loop.

Figure 2

(a) Potential energy $V(\varphi ,\theta )$, showing the twofold fluxoniumlike potential for cuts along the $\theta =0$ and $\theta =\pi$ ridges. Localization wave function in these ridges occurs when the effective mass along the $\theta$ direction is sufficiently large. (Parameters: $E_{\mathrm{J}}/E_L=165$.) (b) Simplified model with separable potential energy. In this case, wave functions are products $\psi (\varphi ,\theta )={\psi }_{\mathrm{ho}}\left(\varphi \right){\psi }_{\mathrm{dw}}\left(\theta \right)$ of harmonic-oscillator and double-well wave functions along $\varphi$ and $\theta$ direction, respectively.

Figure 3

Density plots of the wave function amplitudes for eigenstates in (a), the full potential $V$ of the $0\text{−}\pi$ qubit and (b), the separable potential $V^{\prime }$. The numbers $n=1,2,...$ enumerate the eigenstates starting from the ground state. Different colors (shades of gray) mark distinct signs of the wave function amplitudes. In the simpler case (b), localization along the two ridges $\theta =0$ and $\theta =\pi$ and pairing of states into doublets of symmetric and antisymmetric states (in $\theta$ direction) are easily visible. Wave functions of the actual $0\text{−}\pi$ circuit in (a) show additional local extrema due to the cosine corrugation of the potential $V$. Overall comparison—in particular, states $n=5$ and 6—shows that delocalization in $\theta$ occurs more easily for the $0\text{−}\pi$ circuit. As a result, the development of nodes in $\varphi$ direction (states 12 and 13) only takes place at higher energies. (Parameter values: $\hslash {\omega }_p/E_L={10}^4,\hslash {\omega }_p/E_{\text{C}\Sigma }=2.2\times {10}^3,\hslash {\omega }_p/E_{\mathrm{J}}=7.9$.)

Figure 4

Wave functions for the ground state and its (nearly) degenerate partner state for two choices of device parameters. While the value of the degeneracy is identical in the two cases ($D=2.7$), it is limited by potential-energy differences in the two ridges in (a), and by tunneling along the $\theta$ direction in (b). For tunneling-induced degeneracy breaking (larger $E_{\text{C}\Sigma }$ and smaller $E_L$), we observe symmetric and antisymmetric superpositions of the states localized in the individual ridges. For degeneracy breaking due to potential offsets between the two ridges (smaller $E_{\text{C}\Sigma }$ and larger $E_L$), we observe localization in the two separate ridges. Figure 9 displays the same eigenfunctions expressed in the charge basis. [Parameter values: (a) $\hslash {\omega }_p/E_L=9.9\times {10}^2,\hslash {\omega }_p/E_{\text{C}\Sigma }={10}^4,\hslash {\omega }_p/E_{\mathrm{J}}=8.3$; (b) same as in Fig. 3, i.e., $\hslash {\omega }_p/E_L={10}^4,\hslash {\omega }_p/E_{\text{C}\Sigma }=2.2\times {10}^3,\hslash {\omega }_p/E_{\mathrm{J}}=7.9$.]

Figure 5

(a) Energy spectrum as a function of external magnetic flux ${\phi }_{\mathrm{ext}}={\Phi }_{\mathrm{ext}}/{\Phi }_0$. For the selected parameters, the lifting of the degeneracy at zero flux is primarily induced by the potential asymmetry and strongly suppressed for ${\phi }_{\mathrm{ext}}=\pi$ where the potential becomes symmetric. (b) Magnetic-flux dependence of the logarithmic degeneracy parameter $D$. (Parameter values: $\hslash {\omega }_p/E_L={10}^3$, $\hslash {\omega }_p/E_{\text{C}\Sigma }={10}^3$, and $\hslash {\omega }_p/E_{\mathrm{J}}=3.95$.)

Figure 6

Maximum value of the degeneracy parameter $D$ for given $E_L$ and $E_{\text{C}\Sigma }$ and optimal $E_{\mathrm{J}}^{*}$. Contours of maximum $D$ are shown in black and show that strong degeneracy requires challengingly small values of $E_L$ and $E_{\text{C}\Sigma }$. Contours for the optimal values $E_{\mathrm{J}}^{*}$ maximizing $D$ are shown as white dashed lines. The two dash-dotted asymptotes show the regime where $D$ is limited by potential offsets [towards larger $E_L$, where wave functions have the form of Fig. 4(a)] and the regime where $D$ is limited by tunnel splitting [towards smaller $E_L$, where wave functions have the form of Fig. 4].

Figure 7

Effect of disorder in the Josephson energies. (a) Dependence of the degeneracy parameter $D$ on relative disorder in the Josephson energy, $\delta E_{\mathrm{J}}/E_{\mathrm{J}}$. The plot shows a comparison of two different parameter sets, both with $\hslash {\omega }_p/E_{\text{C}\Sigma }={10}^3$ and $\hslash {\omega }_p/E_{\mathrm{J}}=7.9$ and zero magnetic flux. The degeneracy is seen to be fairly robust with respect to $E_{\mathrm{J}}$ disorder. The vertical line at $20\%$ disorder marks the worst-case disorder seen in experiments. (b) Density plots of the ground-state wave function showing the expected deformation as $E_{\mathrm{J}}$ disorder is increased.

Figure 8

Density plot of low-lying wave functions with disjoint support. The numbers $n=1,2,...$ enumerate the eigenstates starting from the ground state. [Same parameters as in Fig. 4.]

Figure 9

Charge-basis eigenfunctions for the nearly degenerate ground-state doublet of the symmetric $0\text{−}\pi$ circuit as described in Appendix B. Panels (a) and (b) respectively show the same states shown in Figs. 4 and 4 (there presented in the flux basis). In (b) the small splitting of the doublet is primarily determined by differences in the charging energy of the states. Each wave function has nearly disjoint support in the charge basis, and are located on interpenetrating “even” and “odd” quasicharge lattices (checkerboard pattern). In (a), the wave functions consist of symmetric and antisymmetric combinations of even and odd quasicharge wave functions. In this case, the energy splitting is primarily determined by the inductive energy which mixes states with different quasicharge parity.

Figure 10

In the limit that the inductive energy goes to zero, the eigenfunctions of the symmetric $0\text{−}\pi$ Hamiltonian can be diagonalized within a Bloch wave formalism and assigned a quasicharge $q$. The figure shows the dependence of the energy of the lowest energy band ${\tilde{E}}_{s=0}\left(q\right)$ as a function of $q$. As the charging energy $E_{\text{C}\Sigma }$ decreases, states with quasicharges differing by 1 become nearly degenerate, leading to an effective quasicharge double well. Restoring the inductive energy but maintaining the limit of large inductance leads to a doublet comprised of a state with even quasicharge $[q$ (mod 2) $\approx 0]$ and odd quasicharge $[q$ (mod 2) $\approx 1]$.