Fluxon-based quantum simulation in circuit QED

Long-lived fluxon excitations can be trapped inside a superinductor ring, which is divided into an array of loops by a periodic sequence of Josephson junctions in the quantum regime, thereby allowing fluxons to tunnel between neighboring sites. By tuning the Josephson couplings and, implicitly, the fluxon tunneling probability amplitudes, a wide class of one-dimensional tight-binding lattice models may be implemented and populated with a stable number of fluxons. We illustrate the use of this quantum simulation platform by discussing the Su-Schrieffer-Heeger model in the one-fluxon subspace, which hosts a symmetry-protected topological phase with fractionally charged bound states at the edges. This pair of localized edge states could be used to implement a superconducting qubit increasingly decoupled from decoherence mechanisms.

Figure 1

Fluxon state preparation. The top row shows the three possible configurations of the superinductor loop: (a) no persistent current $I_p=0$, (b) persistent current under external flux, and (c) $m$ fluxons trapped inside of the loop with zero external field. (d) Circuit setup for insertion of fluxons under external flux and drive. (e) Protocol for drive amplitude ${\epsilon }_d\left(t\right)$ (red) and external flux ${\mathrm{\Phi }}_{\text{ext}}\left(t\right)$ (black) for the insertion of $m$ fluxons. The drive frequency $f_{\text{d}}\approx \frac{1}{2\pi }{\left(L_{\text{J,in}}{C}_{\text{S}}\right)}^{-1/2}$ corresponds to the eigenmode of the radio-frequency resonator consisting of the input junction $E_{\text{J,in}}$ and the shunt capacitance $C_{\text{S}}$. The corresponding persistent currents are represented by the green arrows (see Appendix pp1).

Figure 2

Superconducting circuit implementation of an effective tight-binding model for fluxons. (a) The circuit in Fig. 1 generalizes to a superinductor ring encompassing loops separated by Josephson junctions. The fluxon “input” junction is shown in blue, the “lattice” junctions $E_{\text{J},i}$ are depicted in orange. (b) Circuit representation of the simplified model of Eq. (4); the branch fluxes ${\phi }_i^{-}$ are the degrees of freedom describing fluxon dynamics through the one-dimensional array. (c) The equivalent tight-binding model for fluxons, where every site corresponds to a loop in (b); the on-site and tunneling energy scales are the ones that appear in Eq. (8).

Figure 3

Effective one-dimensional fluxon potential. (a) The classical potential for one-fluxon dynamics has four degenerate minima separated by Josephson energy barriers. The insets show that in $({\phi }_0^{-},{\phi }_1^{-},{\phi }_2^{-})$ space the variable $\tilde{\phi }$ traverses the edges of a hypercube between the four minima [see Eq. (7)]. (b) The one-fluxon state consists of a kink in the superconducting phase. If the kink occurs between junctions $j$ and $j+1$, in the limit $E_{\text{J}}\gg E_{\text{L}}$, the dominant current circulation (green arrows) occurs on the loop delimited by the two junctions. The circulating currents on the neighboring loops are suppressed by a factor of $\sim E_{\text{J}}/E_{\text{L}}$.

Figure 4

Low-lying spectrum and wave functions for the lowest four states (represented in ascending order with respect to their energy by solid black, dashed red, solid blue, and solid black curves) for a three-junction circuit with $E_{\text{C}}=10$ and $E_{\text{J}0}=E_{\text{J}2}=100=\eta E_{\text{J}1}$ in units of $E_{\text{L}}=\frac{{({\mathrm{\Phi }}_0/2\pi )}^2}{2L}$, with (a) $\eta =1$ and (b) $\eta =10$. We show with orange dashed lines the potential energy ${\mathcal{V}}^{-}\left(\tilde{\phi }\right)$ (left vertical axis). The wave functions (in arbitrary units) are offset by their eigenenergies (right vertical axis). For (b), the first and second excited states are intragap boundary-localized excitations of the one-fluxon tight-binding model. The points represent the values obtained from numerical diagonalization, and the lines are direct connections.

Figure 5

(a) Circuit layout illustrating the conventions in the text. (b) Reduced potential function $v\left({\phi }_J\right)$ for the circuit in (a) at ${\phi }_{\text{ext}}=0$ (black solid line) and ${\phi }_{\text{ext}}=2\pi$ (black dashed line). The times $t_1,t_2,t_3,t_4$ correspond to those represented in Fig. 1 of the main text, and the red arrows and circles indicate the four stages of the insertion of one fluxon. The integers $m$ above the three central minima indicate the value of the fluxoid, $2\pi m$, from Eq. (A9) or its equivalent (A10).

Figure 6

$N=3$ junction circuit with open boundaries. The minimum spanning tree (MST) [84] is highlighted in gray. The junctions, not labeled above, are characterized by Josephson energy $E_{\text{J},j}$ and capacitance $C_{\text{J},j}, j=0,1,2$.

Figure 7

Three-dimensional density plots (opaque volumes signify that the absolute value of the probability density exceeds ${10}^{-3}$) for the first four eigenstates of ${\mathcal{H}}^{-}$ in Eq. (9) obtained from the finite-differences solution with $N_p=17$. The low-lying energies are represented in the leftmost panels. We set $E_{\text{C}}=E_{\text{J},0}\times {10}^{-1}, E_{\text{J},0}=\eta \times E_{\text{J},1}=E_{\text{J},2}$ in units of $E_{\text{L}}=\frac{{({\mathrm{\Phi }}_0/2\pi )}^2}{2L}$, with (a) $\eta =1$ and (b) $\eta =10$, corresponding to the values chosen in Fig. 4 of the main text. In (a), the eigenvalues of the first five states, in units of $E_{\text{L}}$, are $189.51,190.10,190.75,191.22,$ and 227.49, respectively. In (b), the first five eigenvalues are $144.55,152.34,152.41,159.01$, and 179.91, respectively.

Figure 8

Typical tunnel rate for a Josephson junction as a function of junction surface area (solid black lines). The black dotted (blue dashed) lines together with the shaded region between them show the spread of the tunnel rate for a $5\%$ ($10\%$) error in the surface area. Thick black dashed lines serve as guides to the eye to illustrate a possible choice of two reference surface areas for the small and large junctions in the dimerized configuration. Inset: Fractional spread of the tunnel rate as a function of junction area for a $5\%$ (black dotted line) and $10\%$ (blue dashed line) tolerance in the junction surface area.

Figure 9

(a) Effect of increasing bond disorder $W_{\mathrm{bond}}$ on the spectrum of the Su-Schrieffer-Heeger model with open boundary conditions. The degenerate boundary states subsist up to strong disorder (the size of the gap). (b) On-site disorder $W_{\mathrm{site}}$ splits the degeneracy of the boundary states, on average by an amount equal to $W_{\mathrm{site}}$.