Circuit QED with fluxonium qubits: Theory of the dispersive regime

In circuit QED, protocols for quantum gates and readout of superconducting qubits often rely on the dispersive regime, reached when the qubit-photon detuning $\Delta$ is large compared to the mutual coupling strength. For qubits including the Cooper-pair box and transmon, selection rules dramatically restrict the contributions to dispersive level shifts $\chi$. By contrast, in the absence of selection rules many virtual transitions contribute to $\chi$ and can produce sizable dispersive shifts even at large detuning. We present theory for a generic multilevel qudit capacitively coupled to one or multiple harmonic modes and give general expressions for the effective Hamiltonian in second- and fourth-order perturbation theory. Applying our results to the fluxonium system, we show that the absence of strong selection rules explains the surprisingly large dispersive shifts observed in experiments and leads to the prediction of a two-photon vacuum Rabi splitting. Quantitative predictions from our theory are in good agreement with experimental data over a wide range of magnetic flux and reveal that fourth-order resonances are important for the phase modulation observed in fluxonium spectroscopy.

Figure 1

(a) General scheme of circuit quantum electrodynamics with capacitive coupling between one (or several) resonant mode(s) and a superconducting circuit acting as a qudit. Examples of superconducting qudits: (b) the Cooper pair box in the charging and transmon regime and (c) the fluxonium circuit.

Figure 2

Second-order ladder diagram showing the contribution of harmonic mode $j$ to the eigenenergy correction for state $|\mathbf{n},l\rangle$. The two interfering paths correspond to virtual transitions which intermediately decrease ($V_j^{+}{V}_j^{-}$) and increase ($V_j^{-}{V}_j^{+}$) the photon number in mode $j$. Without selection rules, summation includes all qudit levels $l_1$. ${\mathbf{e}}_j$ denotes the unit vector with “direction” $j$.

Figure 3

Level diagram illustrating the second-order dispersive shifts for a multilevel qudit coupled to a single photon mode. For each column (definite photon number), solid lines show the bare energy levels with $l=0,1,\mathrm{and}2$ qudit excitations. Diagonal dashed lines show the relevant couplings between the $l=0,1$ levels with $n_a$ photons (middle column) to states with $n_a\pm 1$ photons (red lines, coupling $l=0,1$ states; blue lines, coupling to higher states). Note that higher levels (e.g., $l=2$) participate in the couplings and affect the dispersive shifts. The shifted levels for the $l=0,1$ states are shown as horizontal dashed lines. In this particular example, the qudit transition ${\epsilon }_{21}$ is nearly resonant with the photon frequency ${\omega }_a$ leading to a large shift of the $l=1$ level.

Figure 4

(a) Single-mode ladder diagram involving only virtual excitations of mode $j$. (b) The corresponding table specifies terms according to Eq. (20). An additional prime on $E_2$ signals exclusion of the term $l_2=l$ from the sum over $l_2$.

Figure 5

(a)–(c) Dual-mode ladder diagrams contributing to fourth-order eigenenergy corrections. For a given selection of two modes, labeled $a$ and $b$, each diagram has an associated diagram obtained by interchanging the roles of $a$ and $b$. Only for (a) one finds that label interchange ($a\leftrightarrow b$) yields an identical expression for the resulting contribution. (d) Table detailing the resulting contributions according to Eq. (21).

Figure 6

(a) Magnitude of the charge matrix elements $|\langle l|\mathsf{N}|l^{\prime }\rangle |$ between fluxonium states $l$ and $l^{\prime }$, for external magnetic flux ${\Phi }_{\mathrm{ext}}=0.4{\Phi }_0$. The parameters used match the experimental values.[42] Note the absence of a nearest-neighbor selection rule. (b) The magnitude of the charge matrix elements between ground state and low-lying excited states, plotted as a function of external magnetic flux. An even/odd selection rule holds for zero and half-integer flux quantum due to parity. Away from these special flux values, simple selection rules are absent. The vertical line marks the flux value $0.4{\Phi }_0$ used in (a).

Figure 7

(a) Dispersive shift ${\chi }_{a;0}$ of the resonator frequency for fluxonium ground state, obtained by second-order perturbation theory. For convenient comparison with panel (b), the negative shift $-{\chi }_{a;0}$ is shown. (b) Comparison between experimental data for direct homodyne detection of the reflected amplitude (red/light gray curve; data as published in Ref. 27) with theory fit according to Eq. (26) (blue/dark gray curve). The fit parameters are determined as $A=0.0094,B=0.0887{\mathrm{GHz}}^{-1},C=10.3052,{\theta }_0=-4.543$.

Figure 8

Homodyne voltage of a reflected microwave tone: comparison between experimental data from Ref. 27 and fourth-order theory for dispersive shifts, with thermal averaging over fluxonium levels according to Eq. (27) included ($T=20\mathrm{mK}$). (a) Theory curves show dispersive shifts of the resonator frequency for occupation numbers $n_a=n_b=0$ and $n_a=1,n_b=0$, as indicated. The position of the additional pole at ${\Phi }_{\mathrm{ext}}/{\Phi }_0\approx \pm 0.15$ is in good agreement with that of the peak-dip structure observed in the experiment. (b) For mean photon number ${\overline{n}}_a=0.05$ the theory prediction based on Eq. (29) also reasonably matches the amplitude of the peak-dip feature. (c) Additional poles are predicted if the $\mathsf{b}$ mode (associated with the array) is occupied. In each panel (a)–(c), significant deviations persist close to half-integer ${\Phi }_{\mathrm{ext}}/{\Phi }_0$. (d) Fluxonium transition and harmonic mode frequencies, as specified by labels. Vertical dashed lines show alignment of resonances with corresponding poles in dispersive shifts and experimental features.

Figure 9

Level diagram illustrating a two-photon vacuum Rabi splitting. In this diagram, horizontal solid lines represent bare fluxonium levels which are coupled via a single-photon mode. Here, the bare states with $n_a=0$, $l=4$ and $n_a=2$, $l=0$ are degenerate and coupled by an effective two-photon transition. The resulting two-photon vacuum Rabi splitting is represented by the dashed lines.

Figure 10

(a) Two-tone spectrum. The vertical axis represents the frequency of the sweeping tone, ${\omega }_{d2}$, and the horizontal axis represents the external magnetic flux. The color scale represents the phase difference between the ground state and the state $l$ excited by the sweeping tone, namely ${\theta }_l-{\theta }_0$, where red represents positive value and blue negative. The three types of color changing, I, II, and III are illustrated by magnified insets. The color change marked by the dashed box corresponds to a spurious phase wrapping (see text). (b) Plot of the nonlinear dispersive shifts ${\mu }_{a;1}(n_a,n_b)-{\mu }_{a;0}(n_a,n_b)$, where $n_b=0$ for all three curves, representing shifts with photon numbers $n_a=0,4,9$, respectively. (c) Plot of the nonlinear dispersive shifts ${\mu }_{a;2}(n_a,n_b)-{\mu }_{a;0}(n_a,n_b)$ with photon numbers $n_a$ and $n_b$ as in (b). (d) Plot of qudit transition frequencies and harmonic mode frequencies. Horizontal dashed lines show harmonic frequencies. Solid curves represent qudit frequency differences and coloring distinguishes transitions starting from $l=0$, 1, and 2. Circles and squares mark zero points, resonances (and corresponding poles), and their alignment with experimental data.