Abstract
We have designed a tunable nonlinear resonator terminated by a SNAIL (Superconducting Nonlinear Asymmetric Inductive eLement). Such a device possesses a Kerr-free point in which the external magnetic flux allows to suppress the Kerr interaction. We have excited photons near this Kerr-free point and characterized the device using a transmon qubit. The excitation spectrum of the qubit allows to observe photon-number-dependent frequency shifts about nine times larger than the qubit linewidth. Our study demonstrates a compact integrated platform for continuous-variable quantum processing that combines large couplings, considerable relaxation times and excellent control over the photon mode structure in the microwave domain.
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Find the latest articles, discoveries, and news in related topics.Introduction
Encoding quantum information in the infinite Hilbert space of a harmonic oscillator is a promising avenue for quantum computing. Recently, significant progress has been made by using three-dimensional (3D) microwave cavities1,2,3,4,5,6,7. Thanks to the strong-dispersive coupling, quantum states such as cat states8,9, GKP states10 and the cubic phase state11 have been engineered. However, currently, the scalability and connectivity is difficult, limited by the size of the cavity. Another simpler method is to use coplanar microwave resonators where resonators and qubits can be fabricated together in a single chip12,13,14. The drawback is the shorter relaxation time of coplanar resonators compared to 3D cavities.
Both 2D and 3D microwave resonators as well as acoustic resonators15,16 host linear modes. Therefore, an ancillary qubit is customarily used to introduce nonlinearity for state preparation and operation. However, the limited coherence of the ancillary qubit and the imperfect operations on it will decrease the fidelity of the actual states11,17,18. To avoid operations on ancillary qubits, a Superconducting QUantum Interference Device (SQUID) can be used to terminate a coplanar resonator. This not only provides the tunability of the mode frequency by changing the external magnetic flux through the loop19,20,21,22,23,24, it also induces sufficient nonlinearity. Using this nonlinearity, experiments in waveguide quantum electrodynamics have demonstrated the generation of entangled microwave photons by the parametrical pumping of a symmetrical SQUID loop25,26. Non-Gaussian states, regarded as a resource for quantum computing, have also been realized27,28,29. Therefore, state preparation and operations can be implemented in a nonlinear resonator with a nonlinearity about 0.3%, much weaker than in a typical transmon qubit (~4%). However, in those experiments, the generated states can not be stored for a long time since the resonators are directly coupled to the waveguides.
Theoretically, it has been shown that pulsed operations on a novel tunable nonlinear resonator can be used to achieve a universal gate set for continuous-variable (CV) quantum computation30. The proposed device is similar to a parametric amplifier where the Josephson junction or SQUID loop is replaced by an asymmetric Josephson device known as the SNAIL (Superconducting Nonlinear Asymmetric Inductive eLement)31,32, is applied. Using a SNAIL or an asymmetrically threaded SQUID loop33,34, it is possible to realize three-wave mixing free of residual Kerr interactions by biasing the element at a certain external magnetic flux. In this work, we refer to this flux spot as the Kerr-free point.
Therefore, it is meaningful to investigate a SNAIL-terminated resonator in circuit quantum electrodynamics (cQED) where the nonlinear resonator, decoupled from the waveguide, can be used for state preparation and storage. Strong dispersive coupling between oscillators and qubits was achieved a decade ago for photons12 and recently for phonons35,36,37, in linear resonators. Here, we observed the very well-resolved photon-number splitting up to the 9th-photon Fock state in a SNAIL-terminated resonator coupled to a qubit. Our nonlinear resonator has a considerable relaxation time up to T1 = 8 μs at the single-photon level, intrinsically limited by Two-Level Systems (TLSs). Our study opens the door to implementing, operating and storing quantum states on this scalable platform in the future. Moreover, compared to a linear resonator, our resonator has a non-negligible dephasing rate from its high sensitivity to the magnetic flux noise due to the SNAIL loop.
Results
Characterization of the SNAIL-terminated resonator
To characterize the parameters for the SNAIL-terminated resonator directly, we first fabricated a SNAIL-terminated λ/4 resonator capacitively coupled to a coplanar transmission line [not shown], which is similar to the nonlinear resonator shown (in blue) in Fig. 1. By measuring the transmission coefficient through a vector network analyzer similar to measuring conventional resonators38,39,40,41, at the 10 mK stage of a dilution refrigerator, we can extract the nonlinear resonator frequency at different external magnetic fluxes [see Methods]. For our SNAIL-terminated resonator, the inductive energy can be written as30,31,32
where β is the ratio of the Josephson energies of the small and the big junctions of the SNAIL, ϕ is the superconducting phase across the small junction, ϕext = 2πΦext/Φ0 is the reduced external magnetic flux, and EJ, related to the Josephson inductance LJ, is the Josephson energy of the big junctions in the SNAIL. Upon quantization, the Hamiltonian for the SNAIL-terminated resonator becomes
where g3 (g4) is the three (four)-wave mixing coupling strength, and ωs is the frequency of the SNAIL-terminated resonator, which follows the relation30
where ωr0 describes the bare resonance frequency of the resonator without the SNAIL, Zr = 58.7 Ω for the characteristic impedance of the resonator, and c2 is a numerically determinable coefficient for the linear coupling whose specific value depends on β and ϕext in our case.
A superconducting qubit with a cross-shaped island (green) and a single Josephson junction (JJ, light blue in the middle inset), capacitively coupled to both a coplanar read-out resonator (red) and the nonlinear resonator (blue). The nonlinear resonator is formed by a linear coplanar resonator terminated by a superconducting nonlinear asymmetric inductive element (SNAIL) that has three big Josephson junctions and one small junction (orange in right inset). The charge line (pink) is used to operate the qubit and drive the nonlinear resonator. The qubit state read-out is implemented through the transmission of the feedline. The on-chip flux line is not used in this work, instead, a superconducting coil on the top of the chip [not shown] is used to generate the external magnect flux Φext through the SNAIL. See more fabrication details and the measurement setup for this device in Methods. The lengths of scale bars for the figure and the inserts are 200 μm and 10 μm, respectively.
We measure the transmission coefficient which results from a very weak probe (the mean number of photons on the resonator due to the probe is smaller than one). From this measurement we extract the internal Qs values at Φext = 0 and Φext = 0.386 Φ0 close to the Kerr-free point, where the magnetic flux is calibrated using the flux period Φ0 of the SNAIL-terminated resonator frequency. As shown in Table 1, we find that the coherence time in both cases is above one microsecond, especially, Ts ≈ 7 μs at Φext = 0. When we tune the resonator to Φext = 0.386 Φ0, the coherence time is reduced by a factor of five due to the pure dephasing from the magnetic flux noise through the SNAIL.
A SNAIL-terminated resonator coupled to a qubit
In order to verify the nonlinear resonator performance in cQED in Fig. 1, the nonlinear resonator is dispersively coupled to a fixed-frequency superconducting qubit [shown in green in Fig. 1] with the bare qubit frequency ωq0/2π ≈ 5.222 GHz [see more details on estimating this value in Methods]. The effective Hamiltonian that contains the leading order corrections to the rotating wave approximation for describing the coupled SNAIL-terminated resonator and qubit in the dispersive regime (g0 ≪ Δ0)42 is
where the bare SNAIL-terminated resonator is dispersively shifted to \({\omega }_{{{{\rm{c}}}}}={\omega }_{{{{\rm{s}}}}}-{g}_{0}^{2}/{{{\Delta }}}_{0}\), and similarly, the bare frequency ωq0 of the qubit mode b is dispersively shifted to \({\omega }_{{{{\rm{q}}}}}={\omega }_{{{{\rm{q0}}}}}+{g}_{0}^{2}/{{{\Delta }}}_{0}\) with g0 the coupling strength between the nonlinear resonator and the qubit, and Δ0 = ωq0 − ωs the detuning between the resonator and the bare qubit frequencies. The dispersive shift is \({\chi }_{0}=\frac{{g}_{0}^{2}{\alpha }_{{{{\rm{q}}}}}}{{{{\Delta }}}_{0}({{{\Delta }}}_{0}-{\alpha }_{{{{\rm{q}}}}})}\) with the qubit anharmonicity αq/2π ≈ 450 MHz. Furthermore, near the Kerr-free point where the Kerr nonlinearity (strength K) is cancelled in the leading order, the residual Kerr nonlinearity is due to the interplay of four- and three-wave mixing processes30 as well as the qubit-induced nonlinearity.
To experimentally find the qubit frequency, we sweep the frequency of a qubit-excitation pulse with the pulse length τp = 1 μs (spectroscopy pulse, see the pulse generation in Methods). When the nonlinear resonator frequency is changed by the external flux Φext, the values of Δ0 and the qubit frequency are also changed. In Fig. 2a, at a fixed Φext, we sweep the pulse frequency, and when the pulse is on resonance with the qubit, the qubit is excited. Then, we measure the qubit state by sending a readout pulse to obtain S(ρee) ∝ ρee, which is manifested as bright dots in Fig. 2a. After finding the qubit frequency and calibrating the corresponding π-pulses at different values of Φext, we send a continuous coherent drive to the nonlinear resonator while driving the qubit on resonance. However, when the drive is on resonance with the nonlinear resonator, the resonator is populated, leading to a change of the qubit frequency due to the dispersive interaction. Thus, the π-pulse will not excite the qubit anymore, resulting in a smaller signal S(ρee) (Fig. 2b).
a The qubit frequency vs. the external magnetic flux through the SNAIL. b The SNAIL-terminated resonator frequency vs. the external magnetic flux through the SNAIL. The nonlinear resonator is dispersively coupled to the qubit as shown in Fig. 1. S(ρee) ∝ ρee is the measured signal from the read-out pulse of the qubit. ωp is the probe frequency.
With the values of ωq0 and the dressed-qubit frequencies, we can obtain the values of the Stark shift at different external magnetic fluxes. Therefore, we can compensate the Stark shift from the qubit onto the SNAIL-terminated resonator to obtain the bare resonator frequency ωs [blue dots in Fig. 3a from Fig. 2b]. With the parameters extracted from Fig. 3a, we can calculate the Kerr coefficient as a function of external magnetic flux [pink curve in Fig. 3b]. Moreover, at the device operation point Φext = 0.386 Φ0 [green dot in Fig. 3b], close to the Kerr-free point Φext = 0.392 Φ0, we estimate the residual Kerr interaction strength K/2π = − 310 ± 40 kHz with g3/2π = − 11.6 ± 0.4 MHz, and g4/2π = − 0.128 ± 0.004 MHz, where the residual Kerr nonlinearity is still mainly from the nonlinearity of the SNAIL.
a The frequency of the nonlinear resonator vs. the external magnetic flux through the SNAIL. Blue dots (red curve) are (is) the experimental (fitting) data. By fitting the data to Eq. (3) [red curve], we obtain β = 0.09993 ± 0.00005, well matched with the resistance ratio of the big junction over the small junction 0.104, measured at room temperature, LJ = 629 ± 8 pH and ωr0/2π = 8.87 ± 0.07 GHz. The crossed point from two blacklines indicate the corresponding resonator frequency at the Kerr-free point. b The Kerr-frequency shift per photon, K, vs. the external magnetic flux through the SNAIL calculated from the parameters in (a). The crossed point from two black lines indicates that the Kerr coefficient is zero at Φext = 0.392 Φ0. The green dot at Φext = 0.386 Φ0 is the operating flux point in our work.
Due to the dispersive coupling to the qubit, the qubit can be used as a very efficient probe to determine the frequency of the SNAIL-terminated resonator. In our case, the external coupling strength between the nonlinear resonator and charge line is only γc/2π = 2.07 kHz. Thus, it will be extremely difficult to find the resonator frequency based on the reflection coefficient measurement43,44,45,46 through the charge line. However, since our qubit is coupled to the resonator with a dispersive shift up to a few MHz as shown in Fig. 5b, a few photons inside the resonator will shift the qubit frequency significantly. Consequently, even though the frequency detuning between the probe frequency and the resonator frequency is much larger than the instrinsic linewidth of the resonator, as soon as the intensity of the probe is strong enough to inject a few photons inside the resonator, we can perform qubit spectroscopy to roughly find the resonator frequency. Afterwards, we can find the resonator frequency more accurately by decreasing the probe intensity (Fig. 4a).
a The signal S(ρee) is from the readout pulse, depending on the qubit population ρee where ρee is determined by the nonlinear resonator photon population. ωp is the frequency of the continuous probe where we vary the probe power to the sample from −142 dBm to −124 dBm. The dip indicates the resonator frequency. Each experimental spectroscopy (dots) is fitted to a Gaussian function (solid curves) to obtain the resonator frequencies and the linewidths at different probe powers. At the lowest probe power, we find that the qubit-dressed resonator frequency is ωc/2π ≈ 4.296 GHz. b The frequency shift, Δωc = ωc,n − ωc, is obtained from the fittings in (a), where ωc,n is the resonator frequency with the average photon number n in the resonator. Due to the residual Kerr effect, we obtain K/2π = − 236 ± 65 kHz by fitting the data to a linear function according to Δωc = K〈n〉. The average photon number is estimated from probe powers according to Eq. (5). c The linewidth of the dip is reduced from about 1.5 MHz to 0.3 MHz, where the linewidth is fitted to a function \(\xi \sqrt{\langle n\rangle }+{\gamma }_{{{{\rm{s}}}}}\)47, with a parameter ξ related to K and the value of γs/2π = 112 kHz from Table 1.
We estimate the photon number according to the relation
where we have the charge-line impedance Zc = 46 Ω, the loaded quality factor of the resonator Ql ≈ Qs = 3.86 × 104, the external quality factor Qc = 2.08 × 106 from the numerical simulation, and the probe power P. Due to the residual Kerr coefficient, the spectroscopy is therefore of Gaussian shape from the photon-number fluctuation in the resonator47. Thus, we can extract both the resonator frequency shift and the linewidth with the estimated photon number, as shown in Fig. 4b and c with the residual Kerr coefficient K/2π = − 236 ± 65 kHz, agreeing well with the value extracted from Fig. 3b.
Photon-number splitting near the Kerr-free point
Once we have determined the nonlinear resonator frequency, we perform a pump-probe measurement consisting of a short pulse (50 ns) to excite the resonator followed by a qubit excitation with τp = 2 μs, Rabi frequency Ω/2π = 250 kHz while sweeping the probe frequency ωp, along with a read-out pulse at the end to infer the qubit excited-state population. Due to the weak hybridization between the qubit and the nonlinear resonator near the Kerr-free point, the short coherent pulse drives the resonator into an approximate coherent state with amplitude ∣α∣. We observed the qubit frequency splitting with the Fock states up to 9 photons where each peak has a Gaussian shape due to the quantum fluctuation of the photon number inside the resonator47 [Fig. 5a]. The separation of each peak is about 3 MHz, ten times larger than the qubit linewidth \(\frac{{\gamma }_{{{{\rm{q}}}}}}{2\pi }\approx 280\,{{{\rm{kHz}}}}\), leading to well-resolved peaks, where the qubit linewidth is mainly dominated by the Rabi-broadening. The pulse length satisfies τp ≫ 1/γq, resulting in a good frequency resolution.
a Normalized transmitted readout signal S(ωq) where the envelope is the qubit-frequency distribution P(ωq), satisfying \(1/2\pi \int\nolimits_{0}^{\infty }P({\omega }_{{{{\rm{q}}}}})\,d{\omega }_{{{{\rm{q}}}}}=1\), with the average photon number ∣α∣2 ≈ 6, of the coherent displacement in the nonlinear resonator. The envelope of the distribution matches well with a Poisson distribution with the corresponding coherent-state amplitude ∣α∣ ≈ 2.4 [red solid line]. b Dispersive shift χ over different photon numbers n inside the nonlinear resonator. The data is fitted to the equation \(\chi ={\chi }_{0}+{\chi }^{{\prime} }(n-1)/2\) with χ0/2π = 3.143 ± 0.021 MHz and the higher order correction \({\chi }^{{\prime} }/2\pi =35\pm 11\,{{{\rm{kHz}}}}\). c the qubit population ρee depending on the photon number 〈n〉 = ∣α∣2 in the nonlinear resonator. τ is the waiting time after the short displacement before the qubit read-out. d The relaxation time T1 of the nonlinear resonator vs. different photon number 〈n〉 inside the resonator. The error bars are for the standard deviation.
From the peak difference of the qubit spectroscopy, we can extract the dispersive shift which increases with the photon number shown in Fig. 5b [see Methods]. The coupling strength g0/2π ≈ 53 MHz follows from the dispersive shift χ0 and the qubit anharmonicity αq. Moreover, near the Kerr-free point, the self-Kerr Ksq of the SNAIL-resonator induced by the qubit-resonator coupling42,48 is \({K}_{{{{\rm{sq}}}}}=-24{g}_{4}{g}_{0}^{2}/{{{\Delta }}}_{0}^{2}\approx -2\pi \times 9\,{{{\rm{kHz}}}}\), which is more than three hundred times smaller than the dispersive shift and more than 10 times smaller than γs. Thus, the residual Kerr coefficient from the qubit has a small influence on the states stored in our device, which can be also completely suppressed by slightly shifting the external flux Φext.
Lifetime of a single photon in the nonlinear resonator near the Kerr-free point
Even though a statistical study can be done by measuring multiple resonators coupled to waveguides to infer the average lifetime, it is still important to study the relaxation time directly on a specific device in cQED since in reality the device performance varies over different devices due to the imperfect fabrication process.
Here, for the device shown in Fig. 1, utilizing the dispersive coupling to the qubit, we can determine the energy lifetime of the nonlinear resonator T1 at the single-photon level. Again, we send a 50 ns pulse to displace the resonator with the initial amplitude ∣α(τ = 0)∣, and after a time delay τ, we apply a conditional π-pulse where the frequency is equal to the qubit frequency when the resonator is empty. In the experiment, the spectral width of the excitation pulse is σ/2π ≈ 80 kHz, more than one order of magnitude smaller than the dispersive shift χ, making the pulse selectivity \(1-\exp (\frac{-{\chi }^{2}}{2{\sigma }^{2}})\, >\, 0.999\)49. Therefore, the qubit-excitation population strongly depends on the photon number 〈n〉. In Fig. 5c at τ ~ 0, when 〈n〉 < 1, the qubit frequency is still mainly centered at ωq/2π = 5.228 GHz, leading to the qubit population ρee ≈ 1. However, when 〈n〉 > 1, the qubit frequency starts to split, leading to a decreased ρee where the population goes to zero smoothly and reaches ρee ≈ 0 with 〈n〉 = 5.8. However, with increasing τ, the coherent field in the resonator decreases as \(\alpha (\tau )=\alpha (0)\exp (-\tau /(2{T}_{1}))\) and the qubit frequency moves back. Thus, the conditional pulse can excite the qubit again.
Two-level systems (TLSs) are investigated extensively using conventional coplanar resonators39,50,51,52,53,54,55 along waveguides, where the photon number inside the resonator can be only roughly estimated according to the attenuation of the setup. Here, thanks to the accurate calibration of the photon number inside the nonlinear resonator, it is possible to directly study the effects of the TLSs on our nonlinear resonator. The value of T1 in Fig. 5d, extracted from the data in Fig. 5c [see Methods], increases from 8 μs to 20 μs, which can be explained by saturation of two-level systems (TLS). According to the TLS model52,55,56, the resonator internal Q1 is given by
where F is the filling factor describing the ratio of electrical field in the TLS host volume to the total volume. δTLS is the TLS loss tangent of the dielectric hosting the TLSs, nc is the critical photon number within the resonator to saturate one TLS, and δother is the contribution from non-TLS loss mechanisms. A fit to Eq. (6) gives FδTLS ≈ 4.5 × 10−6, δother ≈ 1.3 × 10−6 and nc ≈ 0.1 photons [red solid curve in Fig. 5d], which is several orders of magnitude smaller than for normal linear resonators38,39,40,52,54,55,57. Here, it is reasonable to consider that the TLSs are nearly resonant with the resonator, and therefore, they lead to its energy dissipation. Thus, the Rabi frequency induced by the resonator electrical field on the TLSs is \({{{\Omega }}}_{{{{\rm{TLS}}}}}={g}_{{{{\rm{TLS}}}}}\sqrt{\langle n\rangle }\) with a single-photon coupling strength gTLS and the average number of photons 〈n〉 in the resonator. According to \(\langle n\rangle /{n}_{{{{\rm{c}}}}}={{{\Omega }}}_{{{{\rm{TLS}}}}}^{2}{T}_{1,{{{\rm{TLS}}}}}{T}_{2,{{{\rm{TLS}}}}}\)56, with T1,TLS and T2,TLS as the average relaxation time and the decoherence time of the TLSs, respectively, we obtain
due to T2,TLS ≤ 2T1,TLS. Since there is no anti-crossing in the spectroscopy measurement on the resonator, it is very possible that the TLSs are distributed on the surface of the resonator instead of the junction barrier due to no strong coupling between our resonator and TLSs. Thus, according to the numerical simulation for the surface TLSs where in our case the maximal coupling strength is estimated as \({g}_{{{{\rm{TLS}}}}}^{\max }/2\pi =222\,{{{\rm{kHz}}}}\), we obtain T1,TLS ≥ 1.60 μs [See Methods]. In addition, we also analyse the non-TLS loss where the loss could be possible from both the circuit design and chip modes [See Methods].
Compared to the coherence time of the SNAIL-terminated resonator coupled to a waveguide near the Kerr-free point, the relaxation time here is significantly larger, indicating that the coherence time is limited by the pure dephasing. Assuming that the resonator coupled to the waveguide has the same internal relaxation time as the nonlinear resonator coupled to a qubit, according to 1/Ts = 1/(2T1) + 1/Tϕ with T1 ≈ 8 μs at 〈n〉 ≈ 0 from Fig. 5d, we can infer the pure dephasing time Tϕ ≈ 12 μs and 2 μs at Φext = 0 and Φext = 0.386 Φ0, respectively. Especially, at Φext = 0.386 Φ0, the resonator becomes much more sensitive to the magnetic flux noise, leading to a shorter coherence time which could be increased by reducing the SNAIL parameter β or adding more magnetic shielding to suppress the magnetic-flux noise or decrease the loop size.
Discussion
In this study, we developed an efficient method to characterize a SNAIL-terminated nonlinear resonator and observed well-resolved photon-number splitting up to nine photons, where the relaxation time can be up to 8 μs, limited by the TLSs. It is important to demonstrate the possibility to reach the strong dispersive regime necessary for the resonator readout at the Kerr-free point.
The parameters of our device are suitable for implementing universal gate sets for quantum computing30 [See Methods]. Compared to linear modes coupled to qubits, in our case, the Kerr effect from the ancilla qubit can be suppressed, leading to better storage of bosonic modes where the Kerr coefficient makes the quantum state collapse49. In order to estimate the capability of storing bosonic states for quantum information in our nonlinear resonator, the maximal coherent state which can be stored in the resonator is limited by the cavity critical photon number \({n}_{{{{\rm{c}}}}}\equiv {({{{\Delta }}}_{0}/g)}^{2}/4\approx 77\) with the pure dephasing time roughly given by Tϕ/〈n〉2 [See Methods]. Due to a limited value of Tϕ ≈ 2 μs for our current device, in the future, the main problem is to reduce the pure dephasing rate of the resonator to improve the coherence time of the bosonic states stored in our system. Moreover, by taking the advantage of the nonlinearity of our resonator, we could also generate and store non-Gaussian entanglement in different modes28. The dispersive coupling between the qubit and the resonator could also enable us to make quantum state tomography very efficiently. Finally, compared to 3D cavities, our chip-scale structure is more compact for integration and scalability required by the large-scale CV quantum processing in the future.
Methods
Experimental setup
The complete experimental setup is shown in Fig. 6. The sample, as shown in Fig. 1, is wire-bonded in a nonmagnetic oxygen-free cooper sample box, mounted to the 10 mK stage of a dilution refrigerator, and shielded by a μ-metal can which is used for suppressing the static and fluctuating magnetic field. The signal to the qubit and the SNAIL-terminated resonator, with heavy attenuation, is combined to the charge line on the sample. A pulse is sent to the feedline on the sample to read out the state of the qubit with amplification using a traveling-wave parametric amplifier (TWPA) at the 10 mK stage, a high electron mobility transistor (HEMT) amplifier at 3 K and a room-temperature amplifier.
Complete wiring diagram and room temperature setup for the nonlinear resonator coupled to a qubit. The qubit operation and the pulsed displacement are made by analog upconversion of the pulsed generated by an AWG (blue and orange). The dashed line in the blue box means that sometimes the microwave generator is connected to the combiner to pump the resonator continuously. The qubit readout (yellow) is controlled by the up- and down-conversion of a pulse generated by an AWG, and then digitized by an analog to digital converter (ADC).
Sample fabrication
We first clean a high-resistance (10 kΩ) 2-inch silicon wafer by using hydrofluoric acid with 2% concentration. Then, we evaporate aluminum on top of the silicon substrate, followed by direct laser writing, and etching using wet chemistry to obtain all the sample details except the Josephson junction. The Josephson junctions for the qubit and the SNAIL are defined in a bi-layer resist stack using electron-beam lithography. Later, we deposit aluminum again by using a two-angle evaporation technique. In order to ensure a superconducting contact between the junctions and the rest of circuit, an argon ion mill is used to remove native aluminium oxide before the junction aluminium deposition. Finally, the wafer is diced into individual chips and cleaned properly using both wet and dry chemistry.
Device parameters for the SNAIL-terminated resonator coupled to the waveguide
By sending a microwave probe to the SNAIL-terminated resonator coupled to the waveguide [the device used for Table 1], we measure the transmission coefficient of a SNAIL-terminated resonator to obtain the bare resonator frequency [blue stars in Fig. 7a]. Then, we get β = 0.095 and \({E}_{{{{\rm{J}}}}}/(2\pi \hslash )=\frac{\hslash }{4{e}^{2}{L}_{{{{\rm{J}}}}}}\approx 830\,{{{\rm{GHz}}}}\) with the Josephson inductance LJ = 600 pH by fitting the data to Eq. (3) [solid curve]. In Fig. 7b, based on the values of EJ and β, we numerically calculate the Kerr coefficient K due to the four-wave mixing coupling, in which K can be suppressed to zero at the flux point Φext/Φ0 = 0.39. When we tune the SNAIL-terminated resonator to the minima frequency by the external magnetic flux, the frequency detuning between the qubit and the resonator is maximized, large enough to obtain Δ0 = ωq0 − ωs ≈ ωq − ωs ≈ 2π × 1.211 GHz with ωq/2π ≈ 5.227 GHz and ωs/2π = 4.016 GHz. With the value of αq/2π ≈ 450 MHz and χ0/2π = 3.143 MHz, we therefore obtain \(\frac{{g}_{0}^{2}}{{{{\Delta }}}_{0}}=\frac{{\chi }_{0}({{{\Delta }}}_{0}-{\alpha }_{{{{\rm{q}}}}})}{{\alpha }_{{{{\rm{q}}}}}}\approx 2\pi \times 5.35\,{{{\rm{MHz}}}}\). Thus, we have the bare qubit frequency \({\omega }_{{{{\rm{q0}}}}}={\omega }_{{{{\rm{q}}}}}-{g}_{0}^{2}/{{{\Delta }}}_{0}\approx 2\pi \times 5.222\,{{{\rm{GHz}}}}\).
a The frequency of the nonlinear resonator vs. the external magnetic flux through the SNAIL. Stars are the experimental data determined from the transmission coefficient of the vector network analyzer. The solid blue curve is calculated numerically. b The Kerr-frequency shift per photon, K, vs. the external magnetic flux through the SNAIL. The Kerr-free point occurs at Φext = 0.39 Φ0.
Pulse generation and calibration
The microwave control for the qubit and the nonlinear resonator is achieved by up-converting the in-phase (I) and the quadrature (Q) components of a low-frequency pulse generated by four AWG channels (two each for the qubit and the resonator). The qubit read-out pulse is up-converted to the sample and then down-converted after the amplification with the same local oscillator (yellow area). The pump pulse for the displacement is generated by an IQ mixer with an input pulse from an Arbitrary Waveform Generator (AWG) with amplitude. In Fig. 8, we vary the amplitude of the pump pulse by changing the voltage amplitude A, and then obtain the corresponding displacement in the resonator according to the photon-number splitting. The data (black dots) is fitted to the equation ∣α∣ = kA with k ≈ 8. This measurement is useful to calibrate the pump pulse with different amplitudes. We also notice that the residual thermal population with A = 0 is ∣α∣ = 0.16 (〈n〉 = ∣α∣2 ≈ 0.03 photons). The thermal and Poisson distributions are almost the same when the thermal photon number is small. In this case, we only take into account the single photon P1 and the vacuum P0 populations.Thus, we can calculate the corresponding thermal temperature as \({T}_{{{{\rm{res}}}}}=\hslash {\omega }_{{{{\rm{c}}}}}/\left[{k}_{{{{\rm{B}}}}}\ln \left(1+\frac{1}{\langle n\rangle }\right)\right]\approx 58\,{{{\rm{mK}}}}\).
Kerr coefficient estimation
According the extracted parameters β and LJ from Fig. 3a, we can calculate the coefficients g3 and g4 for the three-wave mixing and four-wave mixing terms, respectively (Fig. 9). Then, we can infer the coupling strength for the Kerr coefficient K according to \(K=12({g}_{4}-5{g}_{3}^{2}/{\omega }_{{{{\rm{s}}}}})\), as shown in Fig. 3b.
Qubit spectroscopy
In order to increase the signal-to-noise ratio, here, we average the qubit spectroscopy with different resonator-frequency driven voltages from Fig. 10a to obtain Fig. 10b. Afterwards, each peak corresponding to a specific photon number is fitted to a Gaussian function to extract the dispersive shift as shown in Fig. 5b.
The lifetime T 1 of the resonator
In Fig. 11, The data trace [black stars] is from the top trace in Fig. 5c with the average photon number 〈n〉 = 5.8. Here, we show an example to obtain the value of T1 from Fig. 5c. The qubit excitation probability is \({\rho }_{{{{\rm{ee}}}}}=\exp [-| \alpha (0){| }^{2}\exp (-\tau /{T}_{1})]\). By fitting the rising curve (Fig. 11) with the calibrated \(| \alpha (0)| =\sqrt{5.8}\,\)(〈n〉 = 5.8), we extract T1 = 19.2 ± 0.2 μs, corresponding to a quality factor of Q1 = ωcT1 ≈ 5.18 × 105 which is close to the Q1 value for a coplanar linear resonator in the few-photon limit40,57.
Estimation on the value of g TLS
The capacitance of the SNAIL-terminated resonator is from the bare resonator. Therefore, according to \({\omega }_{{{{\rm{r0}}}}}/2\pi =\frac{1}{\sqrt{{L}_{l}{C}_{l}}}\frac{1}{4l}=\frac{1}{4\sqrt{{L}_{{{{\rm{r0}}}}}{C}_{{{{\rm{r0}}}}}}}\) where Ll, Cl and l are the inductance per length, the capacitance per length and the length of the resonator with Lr0 = Ll × l and Cr0 = Cl × l, we obtain \({C}_{{{{\rm{r0}}}}}=\frac{\pi }{2{\omega }_{{{{\rm{r0}}}}}{Z}_{{{{\rm{r}}}}}}\) with \({Z}_{{{{\rm{r}}}}}=\sqrt{\frac{{L}_{{{{\rm{r0}}}}}}{{C}_{{{{\rm{r0}}}}}}}\). The root mean square of the resonator electric field is given by58:
where Tosc is the oscillation time period and \({U}_{{{{\rm{s}}}}}(t)=\sqrt{\hslash {\omega }_{{{{\rm{s}}}}}/{C}_{{{{\rm{r0}}}}}}\sin ({\omega }_{{{{\rm{s}}}}}t)\). For our resonator with a gap width of tgap = 12 μm, we have \({U}_{{{{\rm{s}}}}}^{{{{\rm{rms}}}}}\approx 1.72\,{{{\rm{\mu V}}}}\) with \({E}_{{{{\rm{s}}}}}^{{{{\rm{rms}}}}}\approx 0.14\,{{{\rm{V/m}}}}\). In order to obtain the boundary of the value of gTLS, we perform a numerical simulation with COMSOL. The result in Fig. 12 shows that the electric field is concentrated around the edge of the resonator with the maximal available numerical value of \({E}_{{{{\rm{q}}}}}^{\max }=23\,{{{\rm{V/m}}}}\) at the bottom corner. Therefore, according to the dipole moment of TLSs with d = 0.2 − 0.4 eÅ58, we can estimate the maximum value of \({g}_{{{{\rm{TLS}}}}}^{\max }/2\pi =222\,{{{\rm{kHz}}}}\).
The black dot from the numerical simulation is the electric field E over the distance r, starting from bottom corner of the resonator towards the ground plane indicated by the red arrow in the inset. The data is fitted to a function of E = h/rd with h = 4.76 and d = 0.42 (red curve), where the fit indicates the polynomial decay law of the field strength51,58,59. Inset: the numerical simulation on the electric field by COMSOL. AlOx and SiOx denote the oxidized layers on the top of resonator and the silicon substrate, respectively. In the simulation, for the oxidized layer, the maximum (minimum) mesh size is 0.02 (0.015) nm. For the Aluminum electrode, the maximum mesh size is 0.1 nm with the minimum size 0.02 nm.
The losses of the nonlinear resonator
In order to improve T1 further in the future, we also analyse the non-TLS loss. The non-TLS loss limits the lifetime to be Tother = 1/(δotherωc) ≈ 28 μs comparable to our qubit lifetime T1,q ≈ 20 μs. In our sample, the nonlinear resonator couples to the qubit, charge line and a flux line where they contribute to the non-TLS loss as \({\gamma }_{{{{\rm{q}}}}}/2\pi ={({g}_{0}/{{{\Delta }}}_{0})}^{2}\frac{1}{{T}_{1,{{{\rm{q}}}}}}\approx 170\,{{{\rm{Hz}}}}\), γc/2π ≈ 2.07 kHz and γf/2π ≈ 477 Hz, respectively (the values of γc and γf are from the simulation). In total, the loss from the chip design is γtotal = γq + γc + γf ≈ 2π × 2.717 kHz corresponding to Ttotal = 1/γtotal ≈ 59 μs ≈ 2Tother, meaning that other unknown losses such as the chip modes are similar to the loss from the chip design.
Coherent states in the resonator
The current in the resonator should be much smaller than the critical current of the Josephson junctions of the SNAIL in order to make the element linear. Here we take the currents through the junctions 10 times smaller than the critical currents (I = 0.1Ic). According to Ic = Φ0/(2πLJ) with LJ = 629 pH, we obtain Ismall,c ≈ 52 nA and Ibig,c ≈ 520 nA for the small and big junctions, respectively. Therefore, the energy of the big (small) Josephson junction is Ebig,J = LJI2/2 ≈ 9 × 10−25 J (Esmall,J ≈ 9 × 10−26 J). Meanwhile, the linear inductance of the bare resonator is Lr0 = πZr/2ωr0 ≈ 1.63 nH, leading to an inductive energy \({E}_{{{{\rm{r0}}}}}={L}_{{{{\rm{r0}}}}}{({I}_{{{{\rm{big,c}}}}}+{I}_{{{{\rm{small,c}}}}})}^{2}/2\approx 3.0\times 1{0}^{-22}\,{{{\rm{J}}}}\). Thus, the corresponding photon number is (Er0 + Esmall,J + 3Ebig,J)/ℏωc ≈ 107 with ℏωc ≈ 2.8 × 10−24 J. Meanwhile, it is also important to avoid to excite the qubit, therefore, the photon number should be much smaller than the cavity critical photon number \({n}_{{{{\rm{c}}}}}\equiv {({{{\Delta }}}_{0}/g)}^{2}/4\approx 77\)47. In summary, the current device has a limited maximal photon number that can be stored due to the dispersive coupling to the qubit.
Moreover, since our nonlinear resonator mainly suffers the pure dephasing, at the Kerr-free point in the rotating frame of the resonator frequency, we estimate the dephasing rate for a coherent state according to the Lindblad master equation,
where ρ is the density matrix of the state in the resonator, and κϕ = 1/Tϕ is the instrinc pure dephasing rate. Therefore, we can obtain \(\frac{d{\rho }_{nm}}{dt}=-{\kappa }_{\phi }{(n-m)}^{2}{\rho }_{nm}\), where n and m indicate the photon number in the Fock space. Considering that the photon-number variation of a coherent state is 〈n〉, the pure phasing rate is therefore about κϕ〈n〉2 = 〈n〉2/Tϕ.
Summary of sample parameters for the cubic-phase state
Our device is possible for implementing some non-Gaussian states for the continuous-variable quantum computing. For implementing the cubic phase state which is the key element for the universal quantum set for the continuous-variable quantum computing, the required device parameters are summarized in Table 2. Based on these parameters, we numerically obtain a cubic phase state with a fidelity up to 93% in Fig. 13a. (See more details on the simulation in ref. 30.)
a The numerical cubic phase state based on the parameters from Table 2. b The ideal cubic phase state.
Data availability
The data that supports the findings of this study is available from the corresponding authors upon reasonable request.
Code availability
The code that supports the findings of this study is available from the corresponding authors upon reasonable request
References
Gao, Y. Y. et al. Entanglement of bosonic modes through an engineered exchange interaction. Nature 566, 509 (2019).
Grimm, A. et al. Stabilization and operation of a kerr-cat qubit. Nature 584, 205 (2020).
Gertler, J. M. et al. Protecting a bosonic qubit with autonomous quantum error correction. Nature 590, 243 (2021).
Gao, Y. Y. et al. Programmable interference between two microwave quantum memories. Phys. Rev. X 8, 021073 (2018).
Ma, Y. et al. Error-transparent operations on a logical qubit protected by quantum error correction. Nat. Phys. 16, 827 (2020).
Hu, L. et al. Quantum error correction and universal gate set operation on a binomial bosonic logical qubit. Nat. Phys. 15, 503–508 (2019).
Reinhold, P. et al. Error-corrected gates on an encoded qubit. Nat. Phys. 16, 822 (2020).
Vlastakis, B. et al. Deterministically encoding quantum information using 100-photon schrödinger cat states. Science 342, 607 (2013).
Wang, C. et al. A schrödinger cat living in two boxes. Science 352, 1087 (2016).
Campagne-Ibarcq, P. et al. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature 584, 368 (2020).
Kudra, M. et al. Robust preparation of wigner-negative states with optimized snap-displacement sequences. PRX Quantum 3, 030301 (2022).
Schuster, D. et al. Resolving photon number states in a superconducting circuit. Nature 445, 515 (2007).
Wang, H. et al. Measurement of the decay of fock states in a superconducting quantum circuit. Phys. Rev. Lett. 101, 240401 (2008).
Hofheinz, M. et al. Generation of fock states in a superconducting quantum circuit. Nature 454, 310 (2008).
Chu, Y. et al. Creation and control of multi-phonon fock states in a bulk acoustic-wave resonator. Nature 563, 666 (2018).
Andersson, G., Suri, B., Guo, L., Aref, T. & Delsing, P. Non-exponential decay of a giant artificial atom. Nat. Phys. 15, 1123 (2019).
Heeres, R. W. et al. Cavity state manipulation using photon-number selective phase gates. Phys. Rev. Lett. 115, 137002 (2015).
Heeres, R. W. et al. Implementing a universal gate set on a logical qubit encoded in an oscillator. Nat. Commun. 8, 1 (2017).
Wallquist, M., Shumeiko, V. & Wendin, G. Selective coupling of superconducting charge qubits mediated by a tunable stripline cavity. Phys, Rev. B 74, 224506 (2006).
Mahashabde, S. et al. Fast tunable high-q-factor superconducting microwave resonators. Phys. Rev. Appl. 14, 044040 (2020).
Kennedy, O. et al. Tunable Nb superconducting resonator based on a constriction nano-squid fabricated with a Ne focused ion beam. Phys. Rev. Appl. 11, 014006 (2019).
Palacios-Laloy, A. et al. Tunable resonators for quantum circuits. J. Low Temperature Phys. 151, 1034 (2008).
Vissers, M. R. et al. Frequency-tunable superconducting resonators via nonlinear kinetic inductance. Appl. Phys. Lett. 107, 062601 (2015).
Sandberg, M. et al. Tuning the field in a microwave resonator faster than the photon lifetime. Appl. Phys. Lett. 92, 203501 (2008).
Schneider, B. H. et al. Observation of broadband entanglement in microwave radiation from a single time-varying boundary condition. Phys. Rev. Lett. 124, 140503 (2020).
Sandbo Chang, C. W. et al. Generating multimode entangled microwaves with a superconducting parametric cavity. Phys. Rev. Appl. 10, 044019 (2018).
Chang, C. W. S. et al. Observation of three-photon spontaneous parametric down-conversion in a superconducting parametric cavity. Phys. Rev. X 10, 011011 (2020).
Agustí, A. et al. Tripartite genuine non-gaussian entanglement in three-mode spontaneous parametric down-conversion. Phys. Rev. Lett. 125, 020502 (2020).
Wang, Z. et al. Quantum dynamics of a few-photon parametric oscillator. Phys. Rev. X 9, 021049 (2019).
Hillmann, T. et al. Universal gate set for continuous-variable quantum computation with microwave circuits. Phys. Rev. Lett. 125, 160501 (2020).
Frattini, N. E., Sivak, V. V., Lingenfelter, A., Shankar, S. & Devoret, M. H. Optimizing the nonlinearity and dissipation of a snail parametric amplifier for dynamic range. Phys. Rev. Appl. 10, 054020 (2018).
Sivak, V. et al. Kerr-free three-wave mixing in superconducting quantum circuits. Phys. Rev. Appl. 11, 054060 (2019).
Lescanne, R. et al. Exponential suppression of bit-flips in a qubit encoded in an oscillator. Nat. Phys. 16, 509 (2020).
Miano, A. et al. Frequency-tunable kerr-free three-wave mixing with a gradiometric snail. Appl. Phys. Lett. 120, 184002 (2022).
Sletten, L. R., Moores, B. A., Viennot, J. J. & Lehnert, K. W. Resolving phonon fock states in a multimode cavity with a double-slit qubit. Phys. Rev. X 9, 021056 (2019).
Arrangoiz-Arriola, P. et al. Resolving the energy levels of a nanomechanical oscillator. Nature 571, 537 (2019).
von Lüpke, U. et al. Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics. https://arxiv.org/abs/2110.00263 (2021).
Verjauw, J. et al. Investigation of microwave loss induced by oxide regrowth in high-q niobium resonators. Phys. Rev. Appl. 16, 014018 (2021).
Calusine, G. et al. Analysis and mitigation of interface losses in trenched superconducting coplanar waveguide resonators. Appl. Phys. Lett. 112, 062601 (2018).
Kowsari, D. et al. Fabrication and surface treatment of electron-beam evaporated niobium for low-loss coplanar waveguide resonators. Appl. Phys. Lett. 119, 132601 (2021).
Probst, S. et al. Efficient and robust analysis of complex scattering data under noise in microwave resonators. Rev. Sci. Instrum. 86, 024706 (2015).
Noguchi, A. et al. Fast parametric two-qubit gates with suppressed residual interaction using the second-order nonlinearity of a cubic transmon. Phys. Rev. A 102, 062408 (2020).
Lu, Y. et al. Propagating wigner-negative states generated from the steady-state emission of a superconducting qubit. Phys. Rev. Lett. 126, 253602 (2021).
Lu, Y. et al. Characterizing decoherence rates of a superconducting qubit by direct microwave scattering. Npj Quantum Inf. 7, 35 (2021).
Lu, Y. et al. Steady-state heat transport and work with a single artificial atom coupled to a waveguide: Emission without external driving. PRX Quantum 3, 020305 (2022).
Lin, W.-J. et al. Deterministic loading and phase shaping of microwaves onto a single artificial atom. Nano Lett. 22, 8137 (2022).
Gambetta, J. et al. Qubit-photon interactions in a cavity: Measurement-induced dephasing and number splitting. Phys. Rev. A 74, 042318 (2006).
Hillmann, T. & Quijandría, F. Designing kerr interactions for quantum information processing via counterrotating terms of asymmetric josephson-junction loops. Phys. Rev. Appl. 17, 064018 (2022).
Kirchmair, G. et al. Observation of quantum state collapse and revival due to the single-photon kerr effect. Nature 495, 205 (2013).
Woods, W. et al. Determining interface dielectric losses in superconducting coplanar-waveguide resonators. Phys. Rev. Appl. 12, 014012 (2019).
Wenner, J. et al. Surface loss simulations of superconducting coplanar waveguide resonators. Appl. Phys. Lett. 99, 113513 (2011).
Burnett, J. et al. Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator. Nat. Commun. 5, 4119 (2014).
de Graaf, S. et al. Two-level systems in superconducting quantum devices due to trapped quasiparticles. Sci. Adv. 6, eabc5055 (2020).
Brehm, J. D. et al. Transmission-line resonators for the study of individual two-level tunneling systems. Appl. Phys. Lett. 111, 112601 (2017).
McRae, C. R. H. et al. Materials loss measurements using superconducting microwave resonators. Rev. Sci. Instrum. 91, 091101 (2020).
Gao, J. The physics of superconducting microwave resonators (PhD thesis, California Institute of Technology, 2008). https://www.proquest.com/docview/1080814168?pq-origsite=gscholar&fromopenview=true.
Burnett, J., Bengtsson, A., Niepce, D. & Bylander, J. Noise and loss of superconducting aluminium resonators at single photon energies. J. Phys.: Conf. Ser., 969, 012131 (2018).
Lisenfeld, J. et al. Electric field spectroscopy of material defects in transmon qubits. Npj Quantum Inf. 5, 105 (2019).
Barends, R. et al. Coherent josephson qubit suitable for scalable quantum integrated circuits. Phys. Rev. Lett. 111, 080502 (2013).
Acknowledgements
The authors acknowledge the use of the Nano fabrication Laboratory (NFL) at Chalmers. We also acknowledge IARPA and Lincoln Labs for providing the TWPA used in this experiment. We wish to express our gratitude to Lars Jönsson and Xiaoliang He for help and we appreciate the fruitful discussions with Axel Eriksson, Dr. Jürgen Lisenfeld, Dr. Alexander Bilmes, Prof. Simone Gasparinetti and Prof. Zhirong Lin. This work was supported by the Knut and Alice Wallenberg Foundation via the Wallenberg Center for Quantum Technology (WACQT) and by the Swedish Research Council.
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Open access funding provided by Chalmers University of Technology.
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Y.L. planned the project. Y.L. performed the measurements with help from M.K. and J.Y. Y.L. designed and fabricated the sample. T.H. and F.Q. helped to make the numerical calculation. Y.L. and H.-X.L. did the Comsol simulation. Y.L. wrote the manuscript with input from all the authors. Y.L. analyzed the data. P.D. and Y.L. supervised this project.
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Lu, Y., Kudra, M., Hillmann, T. et al. Resolving Fock states near the Kerr-free point of a superconducting resonator. npj Quantum Inf 9, 114 (2023). https://doi.org/10.1038/s41534-023-00782-w
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DOI: https://doi.org/10.1038/s41534-023-00782-w
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