Spectroscopy of flux-driven Kerr parametric oscillators by reflection coefficient measurement

We report the spectroscopic characterization of a Kerr parametric oscillator (KPO) based on the measurement of its reflection coefficient under a two-photon drive induced by flux modulation. The measured reflection spectra show good agreement with numerical simulations in term of their dependence on the two-photon drive amplitude. The spectra can be interpreted as changes in system's eigenenergies, transition matrix elements, and the population of the eigenstates, although the linewidth of the resonance structure is not fully explained. We also show that the drive-amplitude dependence of the spectra can be explained analytically by using the concepts of Rabi splitting and the Stark shift. By comparing the experimentally obtained spectra with theory, we show that the two-photon drive amplitude at the device can be precisely determined, which is important for the application of KPOs in quantum information processing.


Introduction
Dressed states formed in a driven quantum system are typically characterized by spectroscopic measurements.For example, if a two-level system is driven on resonance, the Mollow triplet [1] is observed due to the Rabi splitting of the two energy levels hybridized with the photon number states.If, on the other hand, the drive is far detuned, the shift of the energy level due to the Autler-Townes effect, namely, the ac Stark effect [2] is observed.
The experiments originally done in atomic system have not only been reproduced, but also extended to an unexplored parameter regime using superconducting artificial atoms, referred to as a circuit quantum electrodynamics (c-QED) system [3,4].For example, the ac Stark shift in a c-QED system was first reported in Ref. [5], and extended to the strong dispersive regime, where it is possible to determine the population of each photon number state in the resonator coupled to the qubit [6].
Compared to the experiments using one-photon drive, spectroscopy of a two-photon driven system has been relatively unexplored in c-QED systems.The two-photon drive plays an essential role in a device called a Kerr parametric oscillator (KPO) [7,8], which has been attracting attention for applications in quantum information processing, such as quantum annealing [9,10] and gate-model quantum computing [11][12][13].KPOs are parametric oscillators, in which the Kerr nonlinearity is larger than the photon loss rate, particularly in the single-photon Kerr regime [14].The KPOs can be implemented using Josephson parametric oscillators [15][16][17][18][19] or charge-driven transmons with a superconducting nonlinear asymmetric inductive element (SNAIL) [20][21][22].In both cases, an external field nearly twice the resonance frequency of the KPOs is used for parametric pumping and works as the twophoton drive.Spectroscopic measurement for studying the energy-level structure up to the tenth excited states of the KPO of charge-driven transmons has been reported by Frattini et al. [21].By measuring the excitation energy from the ground states as a function of the two-photon drive amplitude, they revealed the crossover from a nondegenerate spectrum to a pairwise kissing spectrum formed by two excited states in the double-well meta potential of the KPO, which leads to a staircase pattern in the coherent state lifetime.This type of spectroscopic measurement for revealing the energy spectra of KPOs is important not only for understanding its fundamental physics, but also for its application to quantum information processing.For example, in Ref. [23], a gate operation of the cat qubit using excited energy levels was proposed.Moreover, a comparison of the spectroscopic measurement with the theory generally gives the calibration of the drive amplitude at the quantum chip [5,24,25].This is particularly important in KPOs for precisely determining the two-photon drive amplitude to create an optimal schedule to generate Schrödinger's cat state in a short time [13,26,27] Some of the authors of the present study theoretically proposed an alternative method of spectroscopy for revealing the energy level structure of a KPO under two-photon drive [28].
The method is based on the measurement of reflection coefficient of the KPO and is simpler than that used in previous studies [21,22] because it is only performed by continuous wave measurement without using microwave pulses.In the present paper, we experimentally demonstrate the proposed method by performing reflectometry measurements of the KPO under the two-photon drive induced by the flux modulation.We compare the results with numerical simulations on the basis of the theory to interpret the experimental data.We also show that the experimental data can be used to determine the calibration of the two-photon drive amplitude, which is important for the application of KPOs in quantum information processing.

II. Device
The KPO used in this experiment is a lumped-element-type device.An optical microscope image and an equivalent circuit of the KPO are shown in Figs.1(a) and 1(b), respectively.The left end face of the KPO is connected to a signal line via a coupling capacitor  in , from which a probe signal is applied.The bottom end face is connected to a coupler for interaction with other KPOs, which are not used in this experiment.The top end face is connected to the ground plane via two Josephson junctions (JJs) and a SQUID in series, the latter of which is inductively coupled to the pump line.The JJ is placed in series with the SQUID to reduce the magnitude of the Kerr nonlinearity  of the KPO [29].The SQUID contains asymmetric JJs, whose critical currents are designed to be 375 nA and 625 nA, and those of the series junctions are designed to both be 1000 nA.The actual critical current of each JJ was verified by measuring the room-temperature resistance of a test structure.The KPO is placed in a dilution refrigerator and cooled to below 10 mK.
We measure the reflection coefficient of the KPO without the two-photon drive to extract the parameters of the KPO.The reflection coefficient  is expressed by the following equation, where  r is the resonance frequency of the KPO,  ref is the probe frequency, and  e ( i ) is an external (internal) loss rate.1).Note that the internal loss rate obtained from this reflection measurement may include the contribution from pure dephasing; the actual internal loss rate  i * is related to  i as  i * =  i − 2, where  is the pure dephasing rate of the KPO.

III.
Measurement of reflection coefficient under two-photon drive Next, we measure  under the two-photon drive using a vector network analyzer (VNA).
The two-photon drive generated by an additional microwave source at room temperature is applied to the pump line on the chip [Fig.1(b)].The frequency of the two-photon drive  p is approximately twice the resonance frequency of the KPO, and the detuning is defined as the difference between the resonance frequency and half of the two-photon-drive frequency, that is, Δ =  r −  p /2.
The Hamiltonian of the KPO in a frame rotating at  p /2 is expressed as [16] ℋ where  is the photon annihilation operator, and  is the amplitude of the two-photon drive.One of the absorption dips shifts toward higher frequency, and the other toward lower frequency as ̅  is increased.

IV. Numerical simulations
We numerically simulate the reflection coefficient for comparison with the experiments.The reflection coefficients are calculated following Ref.[28] as with where  e ( ̃ ̃) and  i ( ̃ ̃) correspond to nominal external and internal loss rates, respectively.
To extract  RT−KPO , we used  RT−KPO as a fitting parameter minimizing the square of the difference between the measured and calculated transition frequencies.The estimated values of  RT−KPO are -57.0dB, -57.6 dB, and -57.6 dB for Δ 2 ⁄ = +8.20 MHz, +0.05 MHz, and − 8.10 MHz, respectively.From an independent measurement using a chip with a through transmission line, we estimate that the two-photon drive is attenuated by 58 dB from the microwave source to the KPO with an accuracy of ±2 dB, which is consistent with the above result.

VI. Analytical calculation of transition frequencies
The transition frequencies described thus far have been calculated by diagonalizing the Hamiltonian Eq. ( 2), and are in good agreement with the experimental data.As shown in Appendix B, however, some of the transition-frequency shifts for small  can be calculated analytically when Δ~χ/2, in which the frequency of the two-photon drive is in resonance with the energy difference between the states |0⟩ and |2⟩ .Namely, the frequency shifts of  5 shows the comparison of transition frequencies obtained from diagonalizing the Hamiltonian (gray lines) and analytical formula (dashed lines).While they are in good agreement for small β, a discrepancy can be seen in the range of  > 2 × 2 MHz especially for  1 ̃0 ̃ and  0 ̃1 ̃.This is because the contribution of the states with large photon numbers cannot be ignored when  is large.

VII. Comparison of nominal relaxation rates
Thus far, we have investigated the  dependence of the frequency and the amplitude of peaks and dips in || and found good agreement between the experiment and the theory as shown in Figs.3(a)-3(f).However, the overall linewidth of the peaks and dips in the experiments is smaller than the theoretical as seen in the figures, where it should be noted that the color scales are all the same.To examine this difference, we investigate the nominal loss rates  e ( ̃ ̃) and  i ( ̃ ̃) obtained by fitting the measured || to   in Eq. (6).
As shown in Fig. 6(a),  e ( ̃ ̃) agrees well with ̃e ( ̃ ̃) for all ( ̃,  ̃) , except that  e The discrepancy between ̃i ( ̃ ̃) and  i ( ̃ ̃) is not well understood at this time.Because the nominal external photon loss rate, which is related to the single-photon decay to the signal line, is consistent with the theory, we believe that dephasing or losses other than the singlephoton loss can cause the discrepancy between ̃i ( ̃ ̃) and  i ( ̃ ̃).We do not include the phase relaxation effect in Eqs. ( 4) and ( 6), but we found that simply including phase relaxation in the master equation by ℒ =  †  as the Lindblad operator does not reproduce ̃i ( ̃ ̃) .One possibility is the 1/f spectrum of flux noise [31,32], which is not taken into account in the simulation.The phase relaxation and noise spectra will need to be futher investigated in future studies.

VIII. Conclusion
We performed spectroscopic measurements of a KPO under the two-photon drive induced by the magnetic flux modulation and compared the results with theoretical calculations.The transition frequencies are in good agreement with the calculations after adjusting the attenuation values for the microwave transmission line, which turned out to be consistent with the results of independent measurements.We also demonstrated that some transition frequencies can be interpreted as Rabi splitting and the Stark shift and provided a simple analytical formula for them.The magnitude of the reflection coefficients is also in good agreement with the calculation, and we showed that its behavior under the two-photon drive, such as the change from the absorption dip to amplification peak, can be explained by the difference in the population between the initial and final states of the transition.We also investigated the nominal loss rates and found that the external loss is in good agreement with the theoretical calculation, while the internal loss rate shows significant deviation from the theory, which still needs clarification.By using these spectroscopic results, it is possible to precisely determine  , the magnitude of the two-photon drive, which is important for the precise control of the KPOs used as qubits in quantum annealing machines and quantum computers.
as  r () =  r +  cos    , where  r is the static resonance frequency and   is the frequency of the two-photon drive.Then,  is related to the derivative of the resonance frequency with respect to the bias current Therefore, the frequency-modulation amplitude  can be written as  = 4 , which together with Eq. (A1) leads to Eq. ( 5).

Appendix B: Analytical calculation of frequency shifts
In this Appendix, we derive formulae for four of the transition frequencies as a function of the two-photon drive amplitude , which are discussed in Section VI and shown in Fig. 5.We investigate the physical meaning of the frequency shift by deriving its formula analytically.
We consider the situation where   matches the energy difference between |0⟩ and |2⟩.In

Figure 1 (
c) shows the flux-bias dependence of the resonance frequency.The resonance frequency of the KPO changes periodically with the flux bias induced by the DC current applied to the pump line.In the following measurement, we fix the DC flux bias to 0.385Φ 0 , where Φ 0 is the magnetic flux quantum.At this flux bias, the resonance frequency  r 2 ⁄ = 8.9653 GHz, the external loss rate  e /2 = 0.27 MHz and the internal loss rate  i /2 = 0.45 MHz are estimated by fitting the measured  to Eq. (

Figure 2
Figure 2 shows the amplitude of  as a function of  ref and the two-photon drive power at the generator output ̅  for different detunings [(a) Δ 2 ⁄ = +8.20 MHz, (b) + 0.05 MHz, (c) − 8.10 MHz (c) ].Here, ̅  was changed for each scan of  ref using the VNA.In all cases, we observed single dips in || corresponding to the transition from the Figures 3(a)-3(c) show the result of the numerical simulations, which are qualitatively in

Figures 3 (
Figures 3(d)-3(f) show the same experimental data as Fig. 2 plotted as a function of  in

2 , 12 (
, where  amp is the amplitude of the ac current induced by the two-photon drive.Here, we assume that the mutual inductance between the pump line and the SQUID frequency independent.Because the RMS value of the current || is || = 1/√2 amp , the average power ̅ is given by ̅ =  0 || 2 =  0  amp 2 where  0 is a characteristic impedance of the pump line.Therefore, ̅ (in dBm) can be written as ̅ = 10 log 10 [ the two-photon drive  and frequency-modulation amplitude can be determined from the Hamiltonian Eq. (2) in a laboratory frame,  † + )4 + 2 cos( p ) ( † + )2 .(A2)Byexpanding and rearranging the right-hand side of this equation into normally ordered products, the coefficient of  †  becomes  0 −  + 4 cos( p ).The constant terms are the resonance frequency  r =  0 −  of the KPO without the two-photon drive, and the oscillating term represents the frequency modulation caused by the two-photon drive.

FIG. 1 .FIG. 2 .FIG. 3 .FIG. 4 .FIG. 5 .FIG. 7 .
FIG. 1.(a) Optical micrograph of the KPO and (b) its schematic circuit diagram.The probe signal is injected from the signal line, which is capacitively coupled with the KPO.The reflected signal is separated from the injected signal by a circulator.The two-photon drive is applied from the pump line, which is inductively coupled to the SQUID in the KPO.The DC current is also applied to the pump line to induce a static magnetic field in the SQUID loop.(c) Flux-bias dependence of the resonance frequency.The blue dots are experimental data, and the orange line shows the theoretical calculation of the resonance frequency based on the circuit model shown in (b) and fitted to the data.The green diamond represents the operation point used for the spectroscopy measurements with the two-photon drive.