Amplitude and frequency sensing of microwave fields with a superconducting transmon qudit

Experiments with superconducting circuits require careful calibration of the applied pulses and fields over a large frequency range. This remains an ongoing challenge as commercial semiconductor electronics are not able to probe signals arriving at the chip due to its cryogenic environment. Here, we demonstrate how the on-chip amplitude and frequency of a microwave field can be inferred from the ac Stark shifts of higher transmon levels. In our time-resolved measurements, we employ a simple quantum sensing protocol, i.e. Ramsey fringes, allowing us to detect the amplitude of the systems transfer function over a range of several hundreds of MHz with an energy sensitivity on the order of $10^{-4}$. Combined with similar measurements for the phase of the transfer function, our sensing method can facilitate the microwave calibration of high fidelity quantum gates necessary for working with superconducting quantum circuits. Additionally, the potential to characterize arbitrary microwave fields promotes applications in related areas of research, such as quantum optics or hybrid microwave systems including photonic, mechanical or magnonic subsystems.


INTRODUCTION
Implementing a fault-tolerant quantum processor requires gate fidelities far exceeding a threshold of 99% [1][2][3][4]. In superconducting qubits, logical gates are realized by on or near-resonant microwave pulses with defined length and amplitude [5]. However, on the way to the qubit, the shape of these pulses is distorted by multiple passive microwave components such as attenuators, circulators and wires. These distortions negatively affect the gate fidelities if they are not accounted for.
The collective response of all microwave components to an incident signal is described by the transfer function of the system. If the transfer function is known, digital signal processing techniques allow for full control over the shape of applied pulses. However, since superconducting qubits are embedded in a cryogenic environment operated at millikelvin temperatures, the transfer function from pulse source to sample is not accessible with conventional network analyzers. In the past, this problem has been tackled by different calibration methods, which are usually limited to specific pulse shapes [6] or systems [7]. While more general pulse optimization schemes have been proposed theoretically, they have yet to be implemented in a real quantum system [8][9][10].
In recent years, the growing interest in quantum sensors [11] has facilitated a more direct approach, where the signal arriving at the qubit is probed directly. In particular, superconducting qubits have been successfully employed as photon sensors due to their high electrical dipole moment. While sensing based on a variety of physical phenomena such as the cross-Kerr effect [12], occurrence of the Mollow triplet [13] or electromag-netically induced transparency [14,15] has been shown, these methods are limited to the discrete frequencies of the qubit transitions. An alternative approach to operates a qubit as a vector network analyzer only works in the MHz regime [16]. Recently, Schneider et al. demonstrated that the ac Stark effect in anharmonic multi-level quantum systems (qudit) can be used to detect on-chip microwave fields [17]. Here, signals over a range of more than one GHz were measured. When including higher levels [18], this sensor can simultaneously determine the amplitude and frequency of an unknown signal, promoting it as a useful tool for experiments in quantum optics [19][20][21] and quantum microwave photonics [22][23][24], where in-situ frequency detection can be beneficial. However, the spectroscopic measurement techniques employed in these proof of principle experiments offer limited precision for reasonable data acquisition times.
In this work, we investigate the potential of the type of sensor used Ref. [17] to characterize the microwave transmission from source to sample. We use a time-resolved measurement setup to boost the sensor performance by an order of magnitude. Applying a well known microwave signal, we can probe the amplitude of the transfer function over a wide frequency range. Finally, we estimate the errors and limits of our sensing scheme and discuss the potential for further improvement.

RESULTS
The sensor we use in our experiments is a non-tunable superconducting transmon with a concentric architecture [25]. The transmon offers a low anharmonicity (here: which creates the on-chip microwave field to be measured. The combined signals are repeatedly attenuated within the cryostat before reaching the sample mounted at the 25 mK stage. b Graphical representation of the sensor measurement procedure. As the qudit transitions ωi (i = 1, 2) are shifted in the presence of a microwave field, the frequency ωR,i of the corresponding Ramsey oscillations changes, here exemplified for ω1. Ramsey fringes thus reveal the magnitude of these level shifts. Together, the shift of both qudit levels can be used to extract the amplitude and frequency of the microwave field.
280 MHz), which is beneficial for probing higher qudit transitions in our experiments, as well as an enhanced dipole moment, which increases the sensitivity to local ac fields. To allow for manipulation (P G , ω G ) and readout (via a resonator) of the qudit, the sample is connected to a time resolved measurement setup (Fig. 1a). An additional microwave source with frequency ω F and power P F was installed to generate a on-chip field with amplitude A F ∝ √ 10 PF/10 dBm . Neglecting the readout resonator, the Hamiltonian describing our system reads where the anharmonic annihilation and creation operatorb andb † take the different coupling strengths of the transmon levels into account, which are expressed in their Eigenbasis |i . The Eigenenergies E i are calculated from the exact solution of the Transmon Hamiltonian [26]. In the following, we label the qudit transitions ω i = E i − E i−1 and their associated parameters with identical indices.
To detect the amplitude and frequency of an on-chip microwave field we determine the ac Stark shift ∆ i it induces in the first and second qudit transition. A simple but precise way to measure those shifts are Ramsey fringes [27,28]. The overall idea of the measurement scheme is sketched in Fig. 1b. Generally, performing a Ramsey experiment for a specific transition produces oscillations in the population of the associated qudit states. In the absence of an external field, the frequency of these oscillations simply depends on the frequency mismatch between the respective qudit transition and the applied gate tone ω G,i . However, if the qudit is subjected to a microwave field, this mismatch changes due to the ac Stark effect. The shift of any qudit transition can then be calculated from the oscillation frequency ω R,i of the respective Ramsey fringes, as long as the unperturbed qudit frequencies ω i are known. Figure 2 shows Ramsey oscillations of the first and second qudit transition when applying a field with ω F /2π = 5.285 GHz and P F = 4 dBm. In the experiments, a πpulse prior to the Ramsey sequence allows probing the frequency shift of the second excited state. An identical π-pulse after the sequence increases the visibility and removes the spurious signal of the relaxation to the ground state. For these π-pulse to be on resonance with the shifted transition frequencyω 1 = ω 1 − ∆ 1 , knowledge of ∆ 1 is required. Consequently, the order in which the qudit transitions are probed is fixed. To determine the frequencies of the Ramsey oscillations, we fit the data with an exponentially damped sine function, which also accounts for the additional decay channels of higher lying qudit levels [29] via a declining amplitude offset.
Lacking a closed analytical solution, the ac Stark shifts  [17], these lookup tables are generated analytically by modeling the transmon as an anharmonic oscillator. The field dependent level shifts are then calculated from perturbation theory. However, at our level of precision, this simplified model is no longer accurate. We therefore rely on numerical simulations of the exact transmon Hamiltonian (Eq. (1), see Methods for details). The last plot in Fig. 2 shows the lookup table for the first qudit transition, illustrating the dependency of the ac Stark shift on the amplitude and frequency of the microwave field. Here, a black and white line represent the upper and lower limit of the sensor, respectively. These limits originate from the restricted number of measurement points in the Ramsey experiments and will be discussed in detail later. The full sensing scheme proposed in this work can be summarized as a three step process. After measuring the shift of the first and second qudit transition using Ramsey fringes, the field parameters are extracted from the measurement data with the help of pre-calculated lookup tables. To verify the scheme, we apply a well known microwave signal with constant power and gradually increase the frequency over a range of 450 MHz. We probe the arising on-chip field with our sensor and plot the extracted ω F,ex over the applied frequencies ω F,apl (Fig. 3a), finding a good agreement. Plotting A F,ex over the same axis yields the amplitude of the transfer function (Fig. 3c). Here, we observe a strong frequency dependence, demonstrating the significance of calibrating microwave lines.
Shaded areas in Fig. 3 illustrate the uncertainty of our results. The uncertainty is estimated from varying our experimentally determined value of ∆ i by ±σ R,i , where σ R,i is the standard error of ω R,i resulting from the fit. In our case, σ R,i is a consequence of the limited signal to noise ratio (SNR) during the qubit readout and therefore depends on the number of averages N avg used in the experiments. As shown in Fig. 3e, we experimentally observe a decline of σ R,i with 1/ N avg , as expected from the shot noise limit [30]. In the interest of keeping the measurement time reasonable, all experiments were performed at N avg = 3000, fixing the errors at around σ = 10 kHz, see Fig. 3d. On average, this amounts to a relative uncertainty for the amplitude and frequency of ∆A F /A F = 4 % and ∆ω F /ω F = 0.5 %, respectively. This error increases for higher frequencies, as ∆ i decreases for large detuning between microwave field and qudit, while the magnitude of σ R,i remains unchanged. Figure 3b provides a more detailed look on the discrepancy δ between ω F,ex and ω F,apl . While this discrepancy is also governed by the SNR, we find that it sometimes exceeds our estimate for the corresponding sensor uncertainty. We attribute this to natural fluctuations of the qudit transition frequencies, which directly alter the measured Ramsey frequencies [31,32]. Systematic errors, like these fluctuations, are not included in our error estimation.
In the following, we address the limits of our sensor. As discussed in Ref. [17], it is practical to limit the ac Stark qudit sensor to fields that are higher in frequency than the first qudit transition (here: ω 1 /2π = 4.685 GHz). Otherwise, the microwave field is more likely to excite higher qudit states. In this work, using Ramsey fringes results in additional constrains for the range of the sensor. The three parameters defining the total measurement time for a Ramsey experiment are the maximum delay time between the π/2-pulses ∆t max , the number of time steps N R and the passive reset time T rep . To re-FIG. 3. Sensor performance analysis. a Comparison between the frequencies applied with the microwave source (ω F,apl ) and the frequencies extracted from the sensor (ωF,ex). b The magnitude of the discrepancy between ω F,apl and ωF,ex is an indicator for the reliability of our measurements. c Amplitude of the transfer function for a signal with constant power P F,apl = 4 dBm. The shaded area indicates the uncertainty estimated from the standard errors to the Ramsey fits. d Explicit standard error used for the calculation of the uncertainty. The values are extracted from the same fits as the data in (a) and (c). e Standard errors as a function of the number of averages. For this experiment, Navg = 3000 averages were used (indicated by the arrows). duce the measurement time together with the chance of encountering frequency fluctuation [31], it is desirable to minimize these parameters. At the same time, the sampling rate f = N R /∆t max should be large enough to resolve the Ramsey oscillations clearly. Here, we find that values more than five times larger than the minimum value stated by the Nyquist-Shannon theorem [33,34] yields accurate fits. Following the same argumentation, it is also desirable to represent at least one full oscillation period within the measurement interval, which requires a sufficiently large ∆t max .
When operating the sensor with gate pulses that are on resonance with the unperturbed qudit frequency, Eq. (2) simplifies to ∆ i = ω R,i and we can write the boundaries for the detectable frequency shifts as for N R = 80 and ∆t max = 800 ns. Together with T rep = 240 µs and N avg = 3000, all parameters amounts to a total measurement time of ∼ 1 min. Note that the lower boundary is given by ∆ 2 , which is always a stronger constraint than ∆ 1 . The sensor limits corresponding to the boundaries in Eq. (3) are visualized in the lookup table in Fig. 2. Only fields with frequencies and amplitudes within these limits can be reliably detected by our sensor. When a different range is required, these boundaries can be adjusted by changing either t max , N R or ω G,i .

DISCUSSION
We have successfully implemented a sensor for microwave fields based on time-resolved measurements of the ac Stark shift. Employing Ramsey fringes, we harness the high sensitivity of the qudit phase on the frequencies of the first and second qudit transition. Evaluating the measured shifts with numerically generated lookup tables yields the amplitude and frequency of the applied microwave field. Using this sensing scheme, we measure the amplitude of the transfer function over a range of several hundreds of MHz. The results were validated by comparing the frequencies of the applied microwave tone with the sensor output. With regards to the previous implementation by Schneider et al. [17], we were able to increase the precision by an order of magnitude to ∆A F = 3.4 MHz and ∆ω F = 25 MHz for comparable measurement times. While a full pulse calibration requires similar measurements for the phase of the transfer function, these results are already promising for various kinds of experiments with superconducting circuits that require on-chip monitoring of microwave fields.
In the future, employing parametric amplifiers [35,36] and active reset [37,38] could reduce the measurement time of the sensor to a few seconds while simultaneously improving the precision. Additionally, the time resolved measurement setup provides access to advanced quantum sensing protocols that use linear slope detection over an extended dynamic range to push the sensitivities towards the Heisenberg limit [39][40][41].

Experimental setup
We use a standard cQED setup consisting of a transmon qudit (ω 1 /2π = 4.685 GHz and ω 2 /2π = 4.405 GHz) capacitively coupled to a λ/2-wavelength coplanar waveguide resonator (ω r /2π = 6.878 GHz). To fabricate the resonator and the large-scale components of the transmon, thin-film NbTiN is used, whereas the Josephson tunnel junction consists of a conventional Al/AlO x /Al stack [42]. The chip is placed in a copper sample box and cooled down to temperatures below 25mK in a wet dilution refrigerator.
The microwave gate pulses for the Ramsey sequence are generated in a single-sideband mixing scheme, using local oscillators and arbitrary waveform generators (AWG). Combined with the permanent microwave tone generating the on-chip field, these pulses are repeatedly attenuated on different temperature stages of the cryostat before reaching the sample. We use the resonator to dispersively readout the state of the qubit [43,44]. The readout signal is downconverted, digitized, and interpreted by our measurement software (git.io/qkit).

Lookup table calculations
Based on the full system Hamiltonian in Eq. (1), we perform master-equation simulations using the QuTip package [45,46]. Starting with the transmon in the ground (excited) state |0 (|1 ), we compute the full time evolution while applying a Ramsey sequence by temporarily switching on A G (t) in the simulation. After computing each point of the Ramsey fringes, ∆ 1 (∆ 2 ) is determined by fitting the oscillations. This process is repeated for varying field amplitudes A F and frequencies ω F , gradually filling the lookup table.