Building a bigger Hilbert space for superconducting devices, one Bloch state at a time

Superconducting circuits for quantum information processing are often described theoretically in terms of a discrete charge, or equivalently, a compact phase/flux, at each node in the circuit. Here we revisit the consequences of lifting this assumption for transmon and Cooper-pair box circuits, which are constituted from a Josephson junction and a capacitor, treating both the superconducting phase and charge as noncompact variables. The periodic Josephson potential gives rise to a Bloch band structure, characterized by the Bloch quasicharge. We analyze the possibility of creating superpositions of different quasicharge states by transiently shunting inductive elements across the circuit and suggest a choice of eigenstates in the lowest Bloch band of the spectrum that may support an inherently robust qubit encoding.

Figure 1

(a) We use the Zak basis to analyze the circuit for a transmon or Cooper-pair box (CPB), consisting of a Josephson junction and a capacitor, which is characterized by a flux $\widehat{\mathrm{\Phi }}$ and a conjugate charge $\widehat{Q}$. An external voltage source $V_x$ may be capacitively coupled to the node of the circuit. (b) An inductive element (box) is transiently switched into the circuit for the time interval $0<t<t_{\mathrm{hold}}$. The inductive element may be a linear inductor, which we analyze in Sec. 3a or a nonlinear $4\pi$-periodic element, which we analyze in Sec. 3b.

Figure 2

The spectrum of the Hamiltonian in Eq. (11), as a function of the quasicharge, $\kappa$, with $E_C/E_J=1$, and $n_x=0$. Note that the regime $E_J/E_C=1$ is intermediate [13] between the transmon and CPB ones and is used here for illustrative purpose only. Bands are labeled by the band index $b$. In this regime, the lowest band is quite narrow relative to the gap. Inset figures depict selected Bloch wave functions $F_{\kappa ,b}^{\mathsf{tmon}}\left(\varphi \right)$ in the noncompact phase basis, given in Eq. (19), for the lowest two bands, at the center and at the edge of the Brillouin zone. The vertical axis in the insets is arbitrary, but scaled identically for all. The restriction of $F_{\kappa ,b}^{\mathsf{tmon}}$ to the compact domain $(-\pi ,\pi ]$ yields the form on the RHS of Eq. (18). The inset modes at the center and edge of the lowest Bloch band are comblike states, with periodicity $2\pi$ and $4\pi$, respectively.

Figure 3

[(a)–(e)] Snapshots of the time evolution of the wave function probability density, $|{\psi }^{\mathsf{ind}}{(t;k,\phi )|}^2$, of a transmon transiently shunted by an inductor (contours). The hue shows the local phase of the wave function, $arg\left({\psi }^{\mathsf{ind}}\right)$, relative to the phase at the origin, enumerated in the scale bar. The initial state is a (nearly) separable, $\mathrm{\Delta }$-broadened state given in Eq. (28), and we have set the (dimensionless) broadening parameter to $\mathrm{\Delta }=0.2$. For these plots, other parameters are $E_C/E_J=1, E_L/E_J=1/{\left(2\pi \right)}^2$, and $n_x=0$. For numerics, we have generated the evolution using the first 100 eigenmodes.

Figure 4

Time-evolution of the return probability of the inductively shunted transmon (solid-red), initialized in the $\mathrm{\Delta }$-broadened state ${|{\psi }_{0,0}^{\mathsf{tmon}}\rangle }_{\mathrm{\Delta }}$, Eq. (28). Also shown is the probability of the $\mathrm{\Delta }$-broadened state localized near the Brillouin zone boundary, ${|{\psi }_{\frac{1}{2},0}^{\mathsf{tmon}}\rangle }_{\mathrm{\Delta }}$ (dashed-blue), along with the residual probability in other states (dotted-shaded). The parameter values used in the simulation are $E_C/E_J=1, E_L/E_J=1/{\left(2\pi \right)}^2$, and $\mathrm{\Delta }=0.2$.

Figure 5

(a) The lowest band wave functions, $b=0$, for the unshunted transmon ground state and the Brillouin zone edge (respectively, $\kappa =0$, solid red and $\kappa =\frac{1}{2}$, dashed blue), shown relative to the cosine potential in the extended Zak basis, Eq. (33). The system is initialized in the ground state of the transmon Hamiltonian, $|{\psi }_{\kappa =0,0}^{\mathsf{tmon}}\rangle$ (solid red). Parameters are $E_C/E_J=1$ and $E_{\mathsf{4}\pi }=0$. (b) The lowest three eigenstates of the shunted system with $\tilde{\kappa }=0$, in the shunted $4\pi$-periodic potential with $E_{\mathsf{4}\pi }/E_J=1/2$. The initial wave function, $F_{0,0}^{\mathsf{tmon}}$, has large and approximately equal overlap with the ground and first excited wave functions of the $4\pi$-periodic system, $F_{\tilde{\kappa }=0,b=0}^{\mathsf{4}\pi }$ and $F_{0,1}^{\mathsf{4}\pi }$. (c) The system evolves from the $2\pi$-periodic initial state $F_{\kappa =0,0}^{\mathsf{tmon}}$, through to a state at $t=t_{2\pi }/2$ that has very large overlap with the $4\pi$-periodic $F_{\kappa =\frac{1}{2},0}^{\mathsf{tmon}}$ pictured in the top panel (dashed blue). At $t=t_{2\pi }$, the system returns to a state close to the initial state. The scale bar shows the complex phase of the wave function.

Figure 6

Time evolution of the return probability of a transmon initialized in the state $|{\psi }_{0,0}^{\mathsf{tmon}}\rangle$ and then shunted by a the $4\pi$-periodic element (solid red). Also shown is the probability of the state $|{\psi }_{\frac{1}{2},0}^{\mathsf{tmon}}\rangle$ localized near the Brillouin zone boundary (dashed blue), along with the residual probability in other states (dotted) which is less than 1% for the chosen parameters, $E_C/E_J=1$ and $E_{\mathsf{4}\pi }/E_J=1/2$.

Figure 7

[(a)–(g)] The first seven complex-valued eigenmodes of the inductively shunted transmon (i.e., the fluxonium), in the Zak basis, ${\psi }_j^{\mathsf{ind}}\left(k,\phi \right)$. The corresponding, calculated eigenenergies $E_j^{\mathsf{ind}}/E_J$ are shown above each plot, for the parameter values $E_C/E_J=1$ and $E_L/E_J=1/{\left(2\pi \right)}^2$. For the time evolution shown in Fig. 3, modes $j=0$ and 2 have large overlap with the initial state (shown at the left of Fig. 3). Colors indicate the phase of the wave function at each point, as specified in the legend (bottom right).

Figure 8

Probabilities for each mode in the inductively shunted transmon used to calculate the time evolution in Fig. 3. Odd-numbered modes have zero amplitude. The initial state has overwhelming support on the ground and second excited state, making the evolution approximately that of a two-level system.