Time-Dependent Magnetic Flux in Devices for Circuit Quantum Electrodynamics

Recent theoretical work has highlighted that quantizing a superconducting circuit in the presence of time-dependent flux $\mathrm{\Phi }(t)$ generally produces Hamiltonian terms proportional to $d\mathrm{\Phi }/dt$ unless a special allocation of the flux across inductive terms is chosen. Here, we present an experiment probing the effects of a fast flux ramp applied to a heavy-fluxonium circuit. The experiment confirms that naïve omission of the $d\mathrm{\Phi }/dt$ term leads to theoretical predictions inconsistent with experimental data. Experimental data are fully consistent with recent theory that includes the derivative term or equivalently uses “irrotational variables” that uniquely allocate the flux to properly eliminate the $d\mathrm{\Phi }/dt$ term.

Figure 1

The fluxonium circuit is formed by a Josephson junction and an inductor with respective branch variables ${\phi }_J,{\phi }_L$. The bottom node is chosen as the reference ground while the top node is associated with the node variable $\phi$. The junction and inductor form a loop which may be threaded by an external flux $\mathrm{\Phi }$.

Figure 2

(a) Fluxonium energy spectrum as a function of external flux, showing the first three energy levels. Vertical lines indicate flux bias points used in the experiment. Starting flux biases are close to half-flux, ${\mathrm{\Phi }}_a/{\mathrm{\Phi }}_0\in [0.498,0.503]$. The final flux bias reached after the pulse is ${\mathrm{\Phi }}_b/{\mathrm{\Phi }}_0=0.812$. (b) External flux as a function of time for the sudden approximation experiment. The dc flux of the external magnet can be precisely tuned to set the starting point near half-flux. Then, using the QICK qubit controller, we generate a step function with a $1\mathrm{ns}$ rising edge to drive external flux via the on-chip flux bias line. We calibrate the drive power such that we always drive to ${\mathrm{\Phi }}_b/{\mathrm{\Phi }}_0=0.812$. By tuning the starting flux point we are able to change the initial state and thus the final state occupation probability.

Figure 3

Fluxonium eigenstates as a function of flux. (a) The two lowest-energy fluxonium eigenstates at the half-flux sweet spot. (b),(c) The two lowest-energy fluxonium eigenstates at $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.812$ using the (b) inductor and (c) junction allocation of the flux. The potential and eigenstates for half-flux are transparently overlaid. While the eigenenergies do not depend on the allocation of the external flux to different terms in the fluxonium Hamiltonian, the shifts of the potential minima and thus the relative wave function locations in $\phi$ space do depend on it. Since final occupation probability depends on wave function overlaps, the incomplete junction allocation and inductor allocation produce different predictions. Circuit parameters used are $E_J/h=6.49\mathrm{GHz}$, $E_C/h=0.755\mathrm{GHz}$, $E_L/h=0.445\mathrm{GHz}$. Wave function plots are generated using scQubits [27].

Figure 4

Occupation probabilities $p_m=|⟨m_b|0_a⟩{|}^2$ after a sudden flux ramp from initial flux ${\mathrm{\Phi }}_a\in [0.498,0.503]$ to final flux ${\mathrm{\Phi }}_b=0.812{\mathrm{\Phi }}_0$, starting in state $|0_a⟩$. (a) Representative single-shot data of ground state (red) and first excited state (blue) occupation probability when pulsed from ${\mathrm{\Phi }}_a$ = 0.498, 0.5, 0.502 ${\mathrm{\Phi }}_0$ to ${\mathrm{\Phi }}_b=0.812{\mathrm{\Phi }}_0$. (b) Occupation probability as a function of initial flux point ${\mathrm{\Phi }}_a$. Points represent measured data, solid lines signify inductor allocation, and dashed lines the incomplete junction allocation. The ground (first excited) state is represented in red (blue). Experimental error bars are plotted on the data points, but are small enough to be enclosed within the size of the markers. Simulation curves are plotted accounting for measurement infidelity and state preparation errors as described in Sec. 3, with $\alpha =0.05$ as the center and shaded region showing range $\alpha \in [0.0,0.1]$. For the inductor allocation the occupation probability primarily remains in the qubit subspace, while for incomplete junction allocation most of the occupation probability would leak into higher-lying states (not shown).

Figure 5

Optical microscope image of the device. (b) The device has three ports for the readout resonator rf input, output, and on-chip flux bias line for driving fast flux. The dashed-outline rectangle indicates the device flux loop. (a) Dark-field optical image zoom-in on qubit flux loop. The qubit flux loop is formed by a large array of Josephson junctions in parallel with a single junction. The on-chip flux bias line used to source external flux runs near the loop and dumps current to the ground plane of the chip.

Figure 6

Internal wiring of the dilution refrigerator. Four input lines were used: an external magnet line, an rf input to the device which was used for driving both the qubit and the cavity, a fast flux drive line to the on-chip flux bias line, and a TWPA pump input. In addition, one output line was used to read out the qubit.

Figure 7

Measurement equipment. Time-domain measurements were performed using the QICK ZCU216 board [37]. The QICK was used to generate all drive pulses and to read out the device. A dc voltage supply was used to control the external magnet.