High-Sensitivity ac-Charge Detection with a MHz-Frequency Fluxonium Qubit

Owing to their strong dipole moment and long coherence times, superconducting qubits have demonstrated remarkable success in hybrid quantum circuits. However, most qubit architectures are limited to the GHz frequency range, severely constraining the class of systems they can interact with. The fluxonium qubit, on the other hand, can be biased to very low frequency while being manipulated and read out with standard microwave techniques. Here, we design and operate a heavy fluxonium with an unprecedentedly low transition frequency of 1.8 MHz. We demonstrate resolved sideband cooling of the “hot” qubit transition with a final ground state population of 97.7%, corresponding to an effective temperature of $23\mathrm{\mu }\mathrm{K}$. We further demonstrate coherent manipulation with coherence times $T_1=34\mathrm{\mu }\mathrm{s}$, $T_2^{*}=39\mathrm{\mu }\mathrm{s}$, and single-shot readout of the qubit state. Importantly, by directly addressing the qubit transition with a capacitively coupled waveguide, we showcase its high sensitivity to a radio-frequency field. Through cyclic qubit preparation and interrogation, we transform this low-frequency fluxonium qubit into a frequency-resolved charge sensor. This method results in a charge sensitivity of $33\mathrm{\mu }\mathrm{e}/\sqrt{\mathrm{Hz}}$, or an energy sensitivity (in joules per hertz) of $2.8\hslash$. This method rivals state-of-the-art transport-based devices, while maintaining inherent insensitivity to dc-charge noise. The high charge sensitivity combined with large capacitive shunt unlocks new avenues for exploring quantum phenomena in the 1–10 MHz range, such as the strong-coupling regime with a resonant macroscopic mechanical resonator.

Popular Summary

Akin to artificial atoms, superconducting qubits are electrical circuits that can jump between discrete states when excited at just the right frequency. Usually, these transitions occur in the gigahertz range, close to the cell phone signal frequency. In this work, we extend the operational boundaries of such an artificial atom below 2 MHz, lower than most radio stations. This is an important breakthrough since this qubit can be used to probe quantum phenomena occurring near their transition frequency.

The “heavy-fluxonium” circuit that we employ is made of an array of hundreds of Josephson junctions arranged in a loop, which supports persistent current states flowing either clockwise or counterclockwise. The energy difference between these two states depends on the precise value of the magnetic field threading the loop. This arrangement, as demonstrated in previous work, can be used to lower the transition frequency, which is directly proportional to the energy difference between the qubit states. However, operating the qubit at such a reduced frequency comes with unique challenges, notably the coupling to external factors such as thermal excitations and magnetic noise, potentially compromising its quantum behavior.

This work overcomes such challenges, demonstrating the operation of a qubit with a transition frequency substantially lower than the blackbody radiation in our experiment, while retaining a state-of-the-art qubit’s quantum behavior. Moreover, we demonstrate that at a specific magnetic field value, where the qubit states are Schrödinger’s cat–like superpositions of the persistent current states, the qubit exhibits a giant sensitivity to applied electric signals. This remarkable feature could be used to detect the quantum vibrations of a nearly resonant tuning fork, or even to prepare such a massive object in a quantum superposition where it occupies two distinct positions simultaneously.

Figure 1

Circuit diagram and implementation of the fluxonium qubit, controls, and readout. Optical micrograph (a) and circuit diagram (c) of the fluxonium qubit composed of a capacitor (green), an array of 360 Josephson junctions (blue), and a single junction (red). The qubit is capacitively coupled to the readout resonator (purple). The magnetic flux through the superconducting loop can be rapidly tuned via the current passing through the flux line (orange), and the qubit is capacitively coupled to a charge line (cyan). (b) Scanning electron micrograph of the fluxonium single junction (red) and four junctions of the array (blue). (d) Two-tone spectrum centered on the flux frustration point ${\phi }_{\mathrm{ext}}/2\pi =0.5$. The color scale in the image represents the phase of the reflected probe pulse. The fit to the data (dotted lines) yields the qubit parameters $E_J/h=5.178\mathrm{GHz}$, $E_C/h=0.4144\mathrm{GHz}$, and $E_L/h=0.18\mathrm{GHz}$. Central inset: detailed view of the avoided crossing near the frustration point. The left- and right-hand insets represent the energy-level diagrams of the four lowest eigenstates, with the potential in gray and the wave functions in colors for an external flux of ${\phi }_{\mathrm{ext}}/2\pi =0.5$ and ${\phi }_{\mathrm{ext}}/2\pi =0.6$, respectively.

Figure 2

Sideband preparation of the fluxonium qubit. (a) Level diagram illustrating the sideband reset protocol: The qubit is reset to either $|g⟩$ or $|e⟩$ by driving one of the sideband transitions: $|e0⟩\to |g1⟩$ or $|g0⟩\to |e1⟩$. The subsequent rapid decay of the cavity photon ensures a directional transition toward the desired qubit state. (b) Pulse sequence for the reset protocol. The flux bias (blue) is set to the target value within $2\mathrm{\mu }\mathrm{s}$. A reset tone (purple) is applied to the readout resonator port for $10\mathrm{\mu }\mathrm{s}$. The flux is then reset to ${\phi }_{\mathrm{ext}}=\pi$ within $2\mathrm{\mu }\mathrm{s}$. The qubit state is read out by transferring the population from $|g⟩$ to $|h⟩$ (orange $\pi$ pulse) followed by single-shot dispersive readout in the $eh$ manifold (yellow pulse). (c) Qubit population (color scale) as a function of flux bias ($x$ axis) and reset pulse detuning ($y$ axis). The reset pulse power is adjusted to maintain a constant intracavity field at ${\omega }_{\mathrm{R}}\pm {\omega }_{ge}({\phi }_{\mathrm{ext}})$. The horizontal dashed line indicates the readout frequency. The nominal working points for $|e⟩$ and $|g⟩$ preparation are denoted by the orange and blue dots, respectively. The orange and blue dot-dashed lines represent the predicted frequencies based on the Hamiltonian parameters in Fig. 1. (d) Final qubit population as a function of reset pulse duration [other parameters are the nominal parameters indicated in (c)]. The population extracted from single-shot readout distributions is shown on the left-hand axis, while the right-hand axis displays the population corrected for decay and mislabeling during readout (see Appendix pp4). Exponential fits to the data (yellow and blue lines) yield preparation times of $1.9\mathrm{\mu }\mathrm{s}$ ($|e⟩$) and $1.7\mathrm{\mu }\mathrm{s}$ ($|g⟩$) with final occupations of 97.7% and 97.7%, respectively.

Figure 3

Qubit coherence. (a) Energy relaxation time ($T_1$) measured at the flux point ${\phi }_{\mathrm{ext}}=\pi$. The raw qubit population is plotted as a function of delay time after preparation in $|e⟩$ (yellow dots) or $|g⟩$ (blue dots) as the qubit relaxes toward a thermal mixture. A common exponential fit yields $T_1=34\mathrm{\mu }\mathrm{s}$. The apparent imbalance in the equilibrium population (dashed line) is due to the residual decay of the intermediate state $|f⟩$ used for readout. (b) Ramsey experiment after state preparation in $|g⟩$. The exponential decay of the Ramsey fringes (black line) yields $T_2^{*}=39.7\mathrm{\mu }\mathrm{s}$ and a transition frequency of 1.8 MHz. (c) $T_2^{*}$ (yellow curve, right-hand axis) and transition frequency (blue curve, left-hand axis) as a function of flux bias. The points highlighted in red correspond to the data presented in (b). The blue solid line is the predicted qubit frequency for the Hamiltonian parameters given in Fig. 1.

Figure 4

Direct Rabi manipulation of the radio-frequency qubit transition. After initial preparation in $|g⟩$, the flux is reset to ${\phi }_{\mathrm{ext}}=\pi$, and the qubit is driven via the charge port with a MHz pulse of variable duration and frequency. The final qubit population is read out using the technique described in Fig. 2. The negative frequency part of the graph is here to highlight the validity range of the rotating-wave approximation. The inset shows the Rabi frequency for a resonant drive at 1.8 MHz, extracted from a sinusoidal fit, as a function of the drive voltage amplitude (upper horizontal axis). A linear fit of Eq. (5) to the data provides the lower horizontal axis calibration, where the drive amplitude is expressed in Cooper pairs on the fluxonium electrode. The red dot is obtained for the parameters of the main figure.

Figure 5

ac-charge sensing. (a) A weak monochromatic charge drive (also referred to as calibration tone) is detected thanks to a repeated pulse sequence: The qubit is prepared in $|g⟩$ [black arrow in the Bloch sphere (b)]. After interacting for a time ${\tau }_I$ with the tone, a partial information on the qubit state is obtained by performing a $\pi /2$ pulse in one of the four directions $+X,+Y,-X,-Y$, followed by a qubit state readout in the $eg$ basis. From the measurement samples $m_k\in \{0,1\}$, a complex telegraphic signal ${\sigma }_k=i^k(m_k-1/2)$ is constructed. The noise spectrum centered around the qubit frequency is estimated by the Bartlett’s method, with periodograms of 1000 nonoverlapping consecutive samples. (c) The estimated noise spectrum presents a residual-bandwidth-limited peak at the calibration tone frequency ${\nu }_{\mathrm{cal}}=1.853\mathrm{MHz}$. Red inset: enlargement on the calibration peak and sinus-cardinal fit (solid line). Left- and right-hand insets: signal-to-noise-ratio (SNR) for the calibration peak as a function of interrogation time ${\tau }_I$ and calibration peak amplitude, respectively. The red dots in the insets correspond to the parameters used in the main graph of (c). The solid lines are the results of an analytic model taking into account the evolution of the qubit during the interrogation time. Signal cancellation occurs when the calibration tone amplitude is a multiple of that of a $\pi$ pulse. The spectrum in (c) is calibrated using the known variance of the calibration tone. RO stands for readout.

Figure 6

Room-temperature rf and dc circuitry. IR mixer: Image rejection mixer; IF: intermediate frequency; RT amplifier: room temperature amplifier; and TWPA: traveling wave parametric amplifier.

Figure 7

Cryogenic rf and dc circuitry. The color code matches that of Fig. 1. VLFX 500: 500 MHz low-pass filter from Minicircuits. K&L: 12 GHz low-pass filter (model 6L250-12000) from K&L. SLP 1.9+: 1.9 MHz low-pass filter from Minicircuits. JTWPA: Josephson traveling wave parametric amplifier.

Figure 8

(a) Histograms of the real ($I$) and imaginary ($Q$) parts of the cavity reflection coefficient, as obtained by demodulating and integrating a 600 ns pulse. The $x$ and $y$ axes have been rescaled by the standard deviation of the Gaussian envelope. Top plot is the histogram obtained when the system is initially prepared in its thermal state. Even though $|g⟩$ and $|e⟩$ are almost equally populated, the dispersive shift is insufficient to separate the two distributions. Bottom plot is the histogram obtained after a $|g⟩\to |h⟩$ $\pi$ pulse. The left-hand blob corresponds to the $|e⟩$ population, unaffected by the $\pi$ pulse, while the right-hand blob is due to the population transferred in $|h⟫$. Setting a threshold at $I=0$ (white dashed line) implements a single-shot readout. (b) Calibration of the state-preparation fidelity. The qubit is prepared through sideband cooling in $|e⟩$ (yellow), $|g⟩$ (blue), or a thermal state (red). We apply a 64 ns pulse at the $|g⟩–|h⟩$ transition frequency with a varying amplitude. The dynamics is then fitted using Eq. (d3) to extract the preparation fidelity (see main text for details).

Figure 9

Temperature dependence of the decoherence rate. (a) Two-dimensional histogram of the I, Q quadratures of the readout reflection coefficients for a qubit at thermal equilibrium with the environment. (b) Histogram of the I quadrature. The population in the manifolds $\{|g⟩,|e⟩\}$, $|f⟩$, $|h⟩\}$ are determined by fitting the various peaks with Gaussian functions (see text for details). Qubit effective temperature $T_{\mathrm{circ}}$, as determined from Eq. (e1), as a function of cryostat temperature $T_{{\mathrm{RuO}}_2}$. (d) Energy decay rate $2\mathrm{\Gamma }$ in the $\{|g⟩,|e⟩\}$ manifold measured via $T_1$ relaxometry [see Fig. 3], as a function of the effective temperature $T_{\mathrm{circ}}$. The dashed line is a linear fit to the data yielding $2\mathrm{\Gamma }=\xi ·T_{\mathrm{circ}}+\zeta$, with $\xi =0.27{\mathrm{mK}}^{-1}{\mathrm{ms}}^{-1}$ and $\zeta =16{\mathrm{ms}}^{-1}$. The filled point in (c) and (d) is the one extracted from the histograms in (a) and (b).

Figure 10

Aliasing diagram of the charge spectrum analyzer. (a) The blue lines indicate the position of the peaks in discrete frequency space ${\mathrm{\Delta }}_n$ as a function of the continuous frequency $\mathrm{\Delta }$ of the applied tone, $\mathrm{\Delta }\equiv {\mathrm{\Delta }}_n(\text{mod}2{\mathrm{\Omega }}_{\mathrm{Ny}})$ and $\mathrm{\Delta }\equiv {\mathrm{\Omega }}_{\mathrm{Ny}}-{\mathrm{\Delta }}_n(\text{mod}2{\mathrm{\Omega }}_{\mathrm{Ny}})$ (see main text for details). (b) Experimental spectrogram with an applied tone of fixed frequency. The detuning $\mathrm{\Delta }$ is swept by adjusting the qubit local oscillator frequency. In this particular instance, the repetition time of the experiment was set to $\tau =14.4\mathrm{\mu }\mathrm{s}$, corresponding to a Nyquist frequency ${\mathrm{\Omega }}_{\mathrm{Ny}}/2\pi =35\mathrm{kHz}$. The frequency axis in the Fig. 5 has been cropped to only display frequencies from $-{\mathrm{\Omega }}_{\mathrm{Ny}}/2\le {\mathrm{\Delta }}_n\le {\mathrm{\Omega }}_{\mathrm{Ny}}/2$.

Figure 11

Sensitivity of the heavy-fluxonium charge spectrum analyzer. (a) Charge sensitivity as a function of interrogation time ${\tau }_I$, for various qubit lifetimes $T_1$ (see legend). The dashed line corresponds to an ideal scenario where the duty cycle ${\tau }_I/\tau =1$ [see Eq. (f22)]. The full line corresponds to a fixed preparation or readout time ${\tau }_{\mathrm{prep}}=13\mathrm{\mu }\mathrm{s}$. (b) Minimal sensitivity (obtained for an optimal value of ${\tau }_I$) as a function of $T_1$. The two scenarios considered in (a) are still represented by dashed and full lines, respectively.