Fast generation of Schrödinger cat states using a Kerr-tunable superconducting resonator

Abstract

Schrödinger cat states, quantum superpositions of macroscopically distinct classical states, are an important resource for quantum communication, quantum metrology and quantum computation. Especially, cat states in a phase space protected against phase-flip errors can be used as a logical qubit. However, cat states, normally generated in three-dimensional cavities and/or strong multi-photon drives, are facing the challenges of scalability and controllability. Here, we present a strategy to generate and preserve cat states in a coplanar superconducting circuit by the fast modulation of Kerr nonlinearity. At the Kerr-free work point, our cat states are passively preserved due to the vanishing Kerr effect. We are able to prepare a 2-component cat state in our chip-based device with a fidelity reaching 89.1% under a 96 ns gate time. Our scheme shows an excellent route to constructing a chip-based bosonic quantum processor.

Introduction

Quantum computation has been proven to surpass classical architectures in certain computational tasks¹. Quantum information has been encoded and manipulated in diverse systems such as cold atoms², trapped ions^3,4, superconducting circuits⁵. Especially, superconducting circuit is a promising platform which has shown significant progress on the gate-based quantum computers^1,6. Additional qubit elements are normally required to achieve large-scale error-correctable two-level system-based quantum computation⁷. In contrast, the phase space of a bosonic system inherently provides a larger Hilbert space and thus a larger coding area^8,9,10,11. Therefore, encoding quantum information in continuous variables leads to a significant reduction in hardware overhead on the path towards the fault-tolerance^12,13. The nonclassical states with negative Wigner functions^14,15 can be regarded as a quantum computing resource to obtain quantum computational advantage. Recently, non-classical states including Schrödinger’s cat codes¹⁶ binominal codes¹⁷, GKP states^18,19, and cubic-phase states^18,20, have been demonstrated in cavities coupled to ancillary qubits. However, the ancillary qubit normally has a fixed Kerr nonlinearity which might be detrimental even for the storage of nonclassical state²¹. In previous results, Schrödinger’s cat states were mostly generated by engineering the two-photon losses^22,23 or ancilla-assisted processes²⁴ in two-^25,26 and three-dimensional²³ structures.

In this paper, differently from traditional gate-based cavity control schemes using the dispersive shift of a nominally linear resonator to an ancilla qubit^24,25,26, our cat state preparation scheme is an alternate by applying a displacement followed by a Kerr gate to a nonlinear resonator. The Kerr gate is implemented by quickly tuning the nonlinearity of the resonator terminated by a Superconducting Nonlinear Asymmetric Inductive eLement (SNAIL)^27,28 as shown in Fig. 1a. Moreover, by tuning the flux bias to the Kerr-free point with eliminated four-wave mixing term, we therefore preserve the prepared cat states against the Kerr-induced evolution.

Fig. 1: Structure of the superconducting circuit.

a A microscopic photo of the superconducting circuit. An ancillary qubit in the middle is capacitvely coupled to both a readout resonator (left) and a SNAIL-terminated resonator (right). b Schematic circuit diagram of the system. c Energy structure of the dispersively coupled nonlinear resonator and qubit. $\left|g\right\rangle$ and $\left|e\right\rangle$ are the ground and excited state of qubit respectively. $\left|0\right\rangle,\,\left|1\right\rangle,\,\left|2\right\rangle,\ldots$ represent the energy levels of the resonator. χ is the dispersive shift.

Results

Design of the quantum circuit

The energy of the SNAIL in our circuit with three big junctions and one smaller junction [Fig. 1c] can be written as²⁹

$${U}_{{{{{{{{\rm{SNAIL}}}}}}}}}(\varphi )=-\!\beta {E}_{{{{{{{{\rm{J}}}}}}}}}\cos (\varphi )-3{E}_{{{{{{{{\rm{J}}}}}}}}}\cos \left(\frac{{\varphi }_{{{{{{{{\rm{ext}}}}}}}}}-\varphi }{3}\right),$$

(1)

where the ratio of the Josephson energies of the small and the big junctions of SNAIL, β ≈ 0.095, the Josephson energy E_J/h ≈ 830GHz²⁸ (Details in Methods), φ_ext = 2πΦ_ext/Φ₀ is the phase induced by the external magnetic flux and φ is the phase difference between two ports of the SNAIL. The Hamiltonian of the SNAIL-terminated resonator is^10,27,30:

$${H}_{{{{{{{{\rm{SNAIL-Res}}}}}}}}}=\hslash {\omega }_{{{{{{{{\rm{s}}}}}}}}}{a}^{{{{\dagger}}} }a+{g}_{{{{{{{{\rm{3}}}}}}}}}{(a+{a}^{{{{\dagger}}} })}^{{{{{{{{\rm{3}}}}}}}}}+{g}_{{{{{{{{\rm{4}}}}}}}}}{(a+{a}^{{{{\dagger}}} })}^{{{{{{{{\rm{4}}}}}}}}},$$

(2)

where ω_s is the resonant frequency of the SNAIL-terminated resonator (the tunable range of ω_s/2π is around 4.08-5.00 GHz in our device). a (a^†) is the annihilation (creation) operator. g₃(g₄) is the coupling strength for the three (four)-wave mixing.

Including the coupled ancillary transmon qubit, the total effective Hamiltonian of the system in the dispersive regime is given by³¹

$$\frac{{H}_{{{{{{{{\rm{eff}}}}}}}}}}{\hslash }\,\approx \,\,{\omega }_{{{{{{{{\rm{s}}}}}}}}}{a}^{{{{\dagger}}} }a+K{{a}^{{{{\dagger}}} }}^{{{{{{{{\rm{2}}}}}}}}}{a}^{{{{{{{{\rm{2}}}}}}}}}+\frac{{\omega }_{{{{{{{{\rm{q}}}}}}}}}}{2}{b}^{{{{\dagger}}} }b\\ -\frac{\chi }{2}{a}^{{{{\dagger}}} }a{b}^{{{{\dagger}}} }b-\frac{{K}_{{{{{{{{\rm{q}}}}}}}}}}{2}{{b}^{{{{\dagger}}} }}^{{{{{{{{\rm{2}}}}}}}}}{b}^{{{{{{{{\rm{2}}}}}}}}},$$

(3)

where ω_q is the frequency of the ancillary qubit (around 5.09-5.19 GHz). b (b^†) is the lowering (raising) operator for the ancillary qubit. K is the Kerr nonlinearity of the resonator, defined as the frequency shift per photon, K = K_s + K_qs with the self-Kerr term ${K}_{{{{{{{{\rm{s}}}}}}}}}=12({g}_{{{{{{{{\rm{4}}}}}}}}}-5{g}_{{{{{{{{\rm{3}}}}}}}}}^{{{{{{{{\rm{2}}}}}}}}}/{\omega }_{{{{{{{{\rm{s}}}}}}}}})$ from the SNAIL element and the cross-Kerr term K_qs = χ²/4K_q from the qubit with the dispersive shift χ/2π ≈ 3.5 − 18MHz depending on the flux bias [especially χ/2π ≈ 4.35MHz when the external flux Φ_ext = 0.4026Φ₀ (see Methods)]. The qubit anharmonicity is K_q/2π ≈ − 420MHz. The value of K_s/2π can be tuned from negative to positive with a range up to a few MHz [Fig. 2a], whereas the value of K_qs/2π is always negative on the order of kHz. Therefore, it is possible to cancel the cross-Kerr term from the qubit to obtain K = 0 by tuning K_s with the magnetic flux through the SNAIL³², see details in Table 1. 

Fig. 2: Experimental methods to calibrate the tunable nonlinearity of the resonator.

a Flux bias dependent nonlinearity. The Kerr coefficient K is measured with single-tone and two-tone measurements. b Pulse sequences for the single-tone and two-tone nonlinearity measurements. c–e Results of the single-tone measurement near the Kerr-free point. α is the displacement. ω_p is the frequency of the pump pulse to the SNAIL-terminated resonator.

Characterization and manipulation of the nonlinearity

Firstly, we calibrate the values of the Kerr coefficient K [Fig. 2a] precisely with two approaches, namely single-tone and two-tone measurements [Fig. 2b]. For the single-tone measurement, we sweep the frequency of a displacement pulse D(α) followed by a conditional qubit π-pulse, where the qubit is excited only if the SNAIL-terminated resonator is empty (α is the displacement with photon number N = α²). Therefore, we can observe the resonator frequency shift with the photon number inside as shown in Fig. 2c–e. We can extract the Kerr coefficient K by linearly fitting the relationship between the frequency shift and the photon number (see Methods). This method is valid only for a small K so that the total frequency shift is not larger than the pulse linewidth. For a larger K, we switch to perform a two-tone measurement. In this measurement, we regard the nonlinear resonator as a multi-level system with an anharmonicity (similar to a qubit), where we perform Rabi oscillations on the lowest three levels by applying two pulses on the transition $\left|0\right\rangle \Rightarrow \left|1\right\rangle$ and $\left|1\right\rangle \Rightarrow \left|2\right\rangle$, respectively. Thus, the anharmonicity, corresponding to the value of K, can be obtained as soon as the resonant frequencies are found (see Methods).

As shown in Fig. 2a, the dynamic range of the Kerr coefficient K/2π is approximately from −5 MHz to 6 MHz which is close to the theoretically simulated result²⁹. Particularly, with flux bias Φ_ext = 0.4026Φ₀, we find a working point where K is small, $\left|K/2\pi \right|\, < \,70\,{{{{{{{\rm{kHz}}}}}}}}$ from single-tone measurement. The accuracy is limited by the spectroscopic linewidth of the pump pulse. Therefore, the Kerr-induced dynamic evolution is negligible within a time scale on the order of microseconds. At the more accurate Kerr-free point, the nonlinearity from the qubit K_qs/2π = χ²/4K_q ≈ − 11 kHz can be compensated by K_s, where the dynamic evolution is ideally eliminated. In order to show the merits of preparing the quantum state at the Kerr-free point, as an example, we displace the nonlinear resonator with α = 1.42 at Φ_ext = 0.4026Φ₀(K ≈ 0) and Φ_ext = 0.41Φ₀(K/2π ≈ 0.5 MHz), respectively. Then, we wait for a time duration Δt before performing the Wigner tomography on the states by taking the parity measurements^33,34, where the pulse sequence is shown in Fig. 3a. The results [Fig. 3b–g] clearly illustrate that the quantum states can be preserved well at Kerr-free point whereas the phase of the state collapses when the Kerr nonlinearity is nonzero. Moreover, the frequency shift among energy levels may induce variations in the photon distribution (as what we discussed in the two-tone nonlinearity measurement above). Injection of multiple photons would be much easier at the Kerr-free point because of the simple spectrum²⁸. As a result, a larger Hilbert space of photons provides us with a larger coding area either for error correction^9,35,36 or loss suppression²⁶.

Fig. 3: Time evolution of a coherent state in the SNAIL-terminated resonator.

a The pulse sequence where the pulse width of displacement is 30ns, and the time between two displacement pulses is Δt. The experimental Wigner tomography shows the evolution progress of the coherent states at different time durations as shown in b–d at the Kerr-free point and e–g with K/2π ≈ 0.5MHz where the Kerr effect clearly distorts the state.

Preparation of Schrödinger cat states

Furthermore, the fast tunablity of the Kerr coefficient can also be used to generate non-classical quantum states. The Kerr cat qubits²³ with the related error correction methods³⁷ benefiting from Kerr nonlinearity show the potential of dissipation-insensitive and long lifetime quantum computation in a multidimensional Hilbert space.

For example, we consider the Bloch sphere of a Kerr-cat qubit which is constructed with a group of perpendicular states²³:

$$\left|{\varphi }_{\pm {{{{{{{\rm{X}}}}}}}}}\right\rangle=\left|\pm \alpha \right\rangle,$$

(4)

$$\left|{\varphi }_{\pm {{{{{{{\rm{Z}}}}}}}}}\right\rangle=\left|\alpha \right\rangle \pm \left|-\alpha \right\rangle,$$

(5)

$$\left|{\varphi }_{\pm {{{{{{{\rm{Y}}}}}}}}}\right\rangle=\left|\alpha \right\rangle \pm i\left|-\alpha \right\rangle,$$

(6)

$\left|{\varphi }_{\pm {{{{{{{\rm{X}}}}}}}}}\right\rangle=\left|\pm \alpha \right\rangle$ are the coherent states generated by pumping our nonlinear resonator with coherent pulses. To prepare the cat states $\left|{\varphi }_{\pm {{{{{{{\rm{Y}}}}}}}}}\right\rangle=\left|\alpha \right\rangle \pm i\left|-\alpha \right\rangle$, the Kerr nonlinearity normally plays an important role^23,38. As shown in Fig. 4a, the flux bias pulse (with a pulse width τ) after the first displacement D(α) introduces a flux bias shift, as well as a large Kerr coefficient K. The coherent states evolution under this nonlinear Hamiltonian results in phase shifts among Fock states $\left|N\right\rangle$ (N = 0, 1, 2, …), which means the rotating speed in the phase space is not uniform. Therefore, when we initialize the system with a coherent state $\left|\alpha \right\rangle$, the evolution of the field states (during the flux bias pulse in Fig. 4a) can be written as²³

$$\left|{{\Psi }}(\tau )\right\rangle ={e}^{i\,\frac{K}{2}{({a}^{{{{\dagger}}} }a)}^{{{{{{{{\rm{2}}}}}}}}}\tau }\left|\alpha \right\rangle \\ ={e}^{-{\left|\alpha \right|}^{{{{{{{{\rm{2}}}}}}}}}/2}\mathop{\sum}\limits_{N}\frac{{\alpha }^{N}}{\sqrt{N!}}{e}^{i\,\frac{K}{2}{N}^{{{{{{{{\rm{2}}}}}}}}}\tau }\left|N\right\rangle$$

(7)

Fig. 4: Schrödinger cat states generation through the fast modulation of Kerr nonlinearity.

a Pulse sequence for generating the m-component Poisson distributed cat states. b–d Measured Wigner functions of the m-component cat (m = 2,3,4). e–g Numerical Wigner functions of the m-component cat (m = 2,3,4) obtained through QuTip⁴⁸.

Under certain circumstances, the m-component cat states can be generated when the nonlinear evolution time τ = τ₀/m with τ₀ = 2π/K (m = 1, 2, …). In particular, when

$$\tau=\pi /K\quad {{{{{{{\rm{or}}}}}}}}\quad 3\pi /K,$$

(8)

the final state is

$$\left|{{\Psi }}(\tau )\right\rangle=\left|\alpha \right\rangle \pm i\left|-\alpha \right\rangle .$$

(9)

In our experiment, we chose α = 1.42 and a flux bias with K/2π = 5.21 MHz corresponding to τ₀ = 192 ns. When the pulse width of the flux bias is either τ = τ₀/2 = 96ns or τ = 3τ₀/2 = 288 ns, we get $\left|{\varphi }_{\pm {{{{{{{\rm{Y}}}}}}}}}\right\rangle=\left|\alpha \right\rangle \pm i\left|-\alpha \right\rangle$. By setting τ = τ₀/3(64 ns) and τ = τ₀/4(48 ns), we also implement 3- and 4- component cat states which can be used for quantum error correction^37,39. Wigner functions of 2, 3 and 4- component cat states are measured and shown in Fig. 4b–d.

The fidelity of the above Schrödinger’s cat states can be calculated by comparing the measured Wigner function W_meas [Fig. 4b–d] with the numerical one W_cal [Fig. 4e–g]. The fidelity can be written as⁴⁰

$$F=\pi \int{{{{{{{\rm{d}}}}}}}}{\gamma }^{{{{{{{{\rm{2}}}}}}}}}{W}_{{{{{{{{\rm{meas}}}}}}}}}(\gamma ){W}_{{{{{{{{\rm{cal}}}}}}}}}(\gamma ),$$

(10)

where γ is the displacement vector, integrated through the whole phase space.

Here, the fidelity F_m of the m-component cat from our measurements is

$${F}_{{{{{{{{\rm{2}}}}}}}}}=89.1\%,\,{F}_{{{{{{{{\rm{3}}}}}}}}}=81.3\%,\,{F}_{{{{{{{{\rm{4}}}}}}}}}=83.15\%.$$

(11)

Note that the distortion caused by the imperfect Wigner tomography has not been eliminated. Moreover, the fidelities are currently limited by our device coherent time 1μs (see Supplementary Fig. 1). The dominating dephasing source of cat states is the single photon loss in our system (details in Supplementary Fig. 3). Additionally, we need to mention that the ancillary qubit is a nonnegligible source of the collapse and decoherence of the bosonic quantum states. It is therefore beneficial to reduce the photon loss and the impacts from the qubit (e.g. spectrally isolating the ancillary qubit while not in use).

In previous strategies with a fixed Kerr coefficient^23,38, the m-component cat state is stabilized by applying a squeezing drive continuously. However, in our case, the cat states can be maintained in the Kerr-free system passively. Therefore, after the state generation, the system is immediately tuned back to the Kerr-free point, where our resonator can be described by a linear Hamiltonian⁴¹, leading to a better storage and evolution of the cat states within a desirable lifetime (see Methods).

To verify the feasible controllability of our nonlinear resonator, we successfully generate an odd cat state, $\left|{\varphi }_{{{{{{{{\rm{-Z}}}}}}}}}\right\rangle=\left|\alpha \right\rangle -\left|-\alpha \right\rangle$ by following the ancilla-assisted cat preparation method^42,43. By using the spectral selectivity and different evolution induced by the dispersive shift, the odd cat state is obtained (details see Supplementary Figure 1). It shows the possibilily of constructing a logical qubit with our platform. In conclusion, we have generated nonclassical states through the fast tunable nonlinearity on a SNAIL-terminated resonator where the tunable range is up to 10 MHz. Compared to the cat states in 3D cavities⁴⁴ where the state preparation is based on the ancillary qubit, our method is more straightforward from the fast tuning of the Kerr coefficient of the nonlinear resonator itself. Thus, our scheme is much simpler and has no affect from the imperfect preparation on the ancillary qubit. Moreover, compared to the two-photon driving strategy^23,25,38, the time to prepare the cat states is about 1/K in our method, which is several times faster than the adiabatic case³⁸. Meanwhile, by eliminating the Kerr-induced evolution, the states of light can be stored passively without consecutive pump at the Kerr-free point. Finally, our platform is more compact compared to 3D cavities, and shows the capability to integrate more elements. Therefore, our method shows a possibility of the extensible and low-crosstalk bosonic-based quantum computation in the future.

Our method provides an avenue to achieve continuous-variable quantum information processing. It can be used to achieve universal control of bosonic codes. One direct application of our circuitry is constructing hardware-efficient, loss-tolerable quantum computers with error correction codes^9,44. Furthermore, networks of coupled resonators can be used to achieve quantum annealing architectures⁴⁵ and quantum simulations such as phase transitions⁴⁶, Gaussian boson sampling⁴⁷, etc.

Methods

Photon number calibration

The photon number in the resonator is measured by the spectroscopy of the ancillary qubit. Here, we pump the resonator with a coherent pulse D(α). As shown in Fig. 5a, due to the dispersive coupling to the qubit, the qubit frequency has a Poisson distribution related to the photon pump amplitude V, where the N^th peak (away from the native qubit frequency) corresponds to the probability of the Fock state $\left|N\right\rangle$ with photon number N. By fitting the multi-peak spectra to a Poisson-distribution function, we can extract the average photon number $\overline{N}={\alpha }^{{{{{{{{\rm{2}}}}}}}}}$. Therefore, we can figure out the linear relationship between the photon pump amplitude V and the values of α, namely, α = G ⋅ V, where G is the scale factor between displacement α and amplitude V.

Fig. 5: Two strategies for photon number calibration.

a Qubit spectroscopy under different pump pulse amplitudes. b Poisson fitting with an average photon number 1.92. c Linear relationship between displacement and pump amplitude. d Wigner function of a coherent state with α = 1.42. e Fitting of Wigner function (dashed gray line in (d)).

In addition, to calibrate the photon more precisely under a low photon number ($\overline{N}\, < \,5$), we can also get the relationship α = G ⋅ V by fitting the Wigner distribution of the coherent state with [Fig. 5e]:

$$W= \frac{2}{\pi }\exp (-2{(\alpha -{\alpha }_{0})}^{2})\\= \frac{2}{\pi }\exp (-2{(G\cdot V-{\alpha }_{0})}^{2}),$$

(12)

where α₀ is the initial displacement.

These relationships are used for the nonlinearity, Wigner function and lifetime measurements discussed in the main text. However, because of the flux-dependent nonlinearity, the photon-number distribution does not satisfy Poisson function when the frequency shift N ⋅ K is comparable with the linewidth of the pump pulse (≈2 MHz for the 500 ns pump pulse). The photon number calibration methods above are therefore only available for Φ ~ 0.4Φ₀ with $\left|K/2\pi \right|\, < \,2\,{{{{{{{\rm{MHz}}}}}}}}$.

Nonlinearity characterization

The Hamiltonian of the SNAIL element with three large Josephson junctions (with Josephson energy E_J) and one smaller junction (βE_J) can be written as (same as Eq. (1)):

$$\begin{array}{l}{U}_{{{{{{{{\rm{SNAIL}}}}}}}}}({\varphi }_{{{{{{{{\rm{s}}}}}}}}})=-\beta {E}_{{{{{{{{\rm{J}}}}}}}}}\cos ({\varphi }_{{{{{{{{\rm{s}}}}}}}}})-3{E}_{{{{{{{{\rm{J}}}}}}}}}\cos (\frac{{\varphi }_{{{{{{{{\rm{ext}}}}}}}}}-{\varphi }_{{{{{{{{\rm{s}}}}}}}}}}{3}),\end{array}$$

(13)

where φ_ext is the external magnetic flux induced phase through the SNAIL loop, φ_s is the phase different between the two ports of SNAIL. Coupled with a resonator (represented by capacity C and inductance L):

$$\begin{array}{rl}H&=C\frac{{{{{\Phi }}}_{{{{{{{{\rm{0}}}}}}}}}}^{{{{{{{{\rm{2}}}}}}}}}}{2}{\dot{\varphi }}^{{{{{{{{\rm{2}}}}}}}}}+U(\varphi,\,{\varphi }_{{{{{{{{\rm{s}}}}}}}}}),\end{array}$$

(14)

$$\begin{array}{rl}U(\varphi,\,{\varphi }_{{{{{{{{\rm{s}}}}}}}}})&=\frac{1}{2}{E}_{{{{{{{{\rm{L}}}}}}}}}{(\varphi -{\varphi }_{{{{{{{{\rm{s}}}}}}}}})}^{{{{{{{{\rm{2}}}}}}}}}+{U}_{{{{{{{{\rm{SNAIL}}}}}}}}}({\varphi }_{{{{{{{{\rm{s}}}}}}}}}),\end{array}$$

(15)

where φ is the mode canonical phase coordinate and ${E}_{{{{{{{{\rm{L}}}}}}}}}={{{{\Phi }}}_{{{{{{{{\rm{0}}}}}}}}}}^{{{{{{{{\rm{2}}}}}}}}}/L$ is the inductive energy. After Taylor expansion around the minimum point of potential U, the Hamiltonian of second quantization is^28,29

$${H}_{{{{{{{{\rm{SNAIL-Res}}}}}}}}}=\hslash {\omega }_{{{{{{{{\rm{s}}}}}}}}}{a}^{{{{\dagger}}} }a+{g}_{{{{{{{{\rm{3}}}}}}}}}{(a+{a}^{{{{\dagger}}} })}^{{{{{{{{\rm{3}}}}}}}}}+{g}_{{{{{{{{\rm{4}}}}}}}}}{(a+{a}^{{{{\dagger}}} })}^{{{{{{{{\rm{4}}}}}}}}},$$

(16)

where

$$\hslash {\omega }_{{{{{{{{\rm{s}}}}}}}}}=\sqrt{8{E}_{{{{{{{{\rm{C}}}}}}}}}{E}_{{{{{{{{\rm{J}}}}}}}}}{c}_{{{{{{{{\rm{2}}}}}}}}}},$$

(17)

$$\hslash {g}_{{{{{{{{\rm{3}}}}}}}}}={c}_{{{{{{{{\rm{3}}}}}}}}}\sqrt{{E}_{{{{{{{{\rm{C}}}}}}}}}\hslash {\omega }_{{{{{{{{\rm{s}}}}}}}}}}/6{c}_{{{{{{{{\rm{2}}}}}}}}},$$

(18)

$$\hslash {g}_{{{{{{{{\rm{4}}}}}}}}}={c}_{{{{{{{{\rm{4}}}}}}}}}{E}_{{{{{{{{\rm{C}}}}}}}}}/12{c}_{{{{{{{{\rm{2}}}}}}}}},$$

(19)

$${c}_{{{{{{{{\rm{j}}}}}}}}}=\left.\frac{1}{{E}_{{{{{{{{\rm{J}}}}}}}}}}\frac{{d}^{{{{{{{{\rm{j}}}}}}}}}U}{d{\varphi }^{{{{{{{{\rm{j}}}}}}}}}}\right| {\varphi }_{{{{{{{{\rm{m}}}}}}}}},$$

(20)

$${E}_{{{{{{{{\rm{C}}}}}}}}}={e}^{{{{{{{{\rm{2}}}}}}}}}/2C.$$

(21)

The energy level n is:

$${E}_{{{{{{{{\rm{n}}}}}}}}}/\hslash=n{\omega }_{{{{{{{{\rm{s}}}}}}}}}+6({g}_{{{{{{{{\rm{4}}}}}}}}}-5{{g}_{{{{{{{{\rm{3}}}}}}}}}}^{{{{{{{{\rm{2}}}}}}}}}/{\omega }_{{{{{{{{\rm{s}}}}}}}}})n(n+1).$$

(22)

Thus, the Kerr coefficient of SNAIL-terminated resonator can be written as

$$\hslash K=\frac{{d}^{{{{{{{{\rm{2}}}}}}}}}{E}_{{{{{{{{\rm{n}}}}}}}}}}{d{n}^{{{{{{{{\rm{2}}}}}}}}}}=12\hslash \left({g}_{{{{{{{{\rm{4}}}}}}}}}-\frac{5{{g}_{{{{{{{{\rm{3}}}}}}}}}}^{{{{{{{{\rm{2}}}}}}}}}}{{\omega }_{{{{{{{{\rm{s}}}}}}}}}}\right).$$

(23)

The design parameters β and E_J, also the relationship between injected current and flux φ_ext can be extracted by fitting the flux modulation of the SNAIL-terminated resonator frequency (Fig. 6) with Eq. (22)²⁸.

Fig. 6: Flux modulated spectrum.

The frequencies of the SNAIL-terminated resonator and ancillary qubit are obtained by the reflection coefficient measurement on the readout resonator with different pump frequency.

We developed two methods for the Kerr coefficient characterization, namely, single-tone and two-tone measurements. The single-tone measurement is only suitable for smaller Kerr coefficient (e.g. $\left|K/2\pi \right|\, < \,2\,{{{{{{{\rm{MHz}}}}}}}}$). To contrast, larger K brings more frequency shift with a similar photon number. When the frequency shift is larger than the linewidth of the single pulse, the nonlinearity can be measured by the two-tone method.

In a single-tone measurement, in order to keep the frequency sensitivity, we choose a relatively long pulse with a pulse length up to 500 ns, then we sweep the pulse frequency with following a conditional π-pulse which can excite the qubit only if the cavity is empty. Therefore, it can be regarded as a photon probe. With different pump powers, we can see the shift of resonant frequency Δf which obeys:

$$K/2\pi={f}_{{{{{{{{\rm{N}}}}}}}}}-{f}_{{{{{{{{\rm{N-1}}}}}}}}},$$

(24)

$$K/2\pi={{\Delta }}f/\overline{N},$$

(25)

where $\overline{N}$ is the average photon number. For example, in Fig. 2c, we see the frequency shift with different displacement (i.e. pump amplitude). By linearly fitting the relationship of the frequency shift and the average photon number, we get the Kerr coefficient [Fig. 7a]. If K is too large ($\left|K/2\pi \right|\, > \,2\,{{{{{{{\rm{MHz}}}}}}}}$), however, it is not possible to cover the frequency range with a single tone. Then, we treat the SNAIL-terminated resonator as a three level system with an anharmonicity K. To verify it, Rabi oscillations between energy levels ($\left|0\right\rangle \leftrightarrow \left|1\right\rangle$ and $\left|1\right\rangle \leftrightarrow \left|2\right\rangle$) are measured [Fig. 7b]. Here, the conditional qubit π pulse is also modified to be only valid for $\left|0\right\rangle$, $\left|1\right\rangle$ or $\left|2\right\rangle$, respectively. Thus, K is represented by the frequency difference of the first two pulses as

$$K/2\pi=f(\left|1\right\rangle \leftrightarrow \left|2\right\rangle )-f(\left|0\right\rangle \leftrightarrow \left|1\right\rangle ).$$

(26)

The measurement results from single-tone and two-tone methods agree with the theoretical calculation very well [Fig. 2a].

Fig. 7: Experimental results of 1-tone and 2-tone nonlinearity measurement.

a Relationship between the frequency shift and the photon number in 1-tone measurement with external flux Φ_ext ~ 0.409Φ₀. b Rabi oscillation between the Fock states ($\left|0\right\rangle \leftrightarrow \left|1\right\rangle$ and $\left|1\right\rangle \leftrightarrow \left|2\right\rangle$). c Frequency scan of the SNAIL-terminated resonator under a 2-tone measurement with a conditional π pulse for $\left|1\right\rangle$ (Φ_ext ~ 0.24Φ₀).

Wigner tomography

The Wigner function of the states is obtained by the parity measurement. For definition, the Wigner function can be described as

$$W(\gamma )=\frac{2}{\pi }Tr[D(-\gamma )\rho D(\gamma )P],$$

(27)

$$P={e}^{i\pi {a}^{{{{\dagger}}} }a}={(-1)}^{{{{{{{{\rm{N}}}}}}}}},$$

(28)

By applying a displacement D( − γ) to a density operator ρ, we get a new state with density operator:

$${\rho }^{{\prime} }=D(-\gamma )\rho D(\gamma ).$$

(29)

Thus, the Wigner function W(γ) is proportional to the average of parity operator P which can be measured with the sequence in Fig. 8a. Here, we apply two π/2 pulses to the ancillary qubit. With a time spacing π/χ between two pules, this sequence can be treated as a parity measurement where the state of qubit $\left|g\right\rangle$ ($\left|e\right\rangle$) corresponds to P = − 1(1). The mechanism can be described in the qubit Bloch sphere [Fig. 8e] and signal spectrum [Fig. 8b–d]. The sequence to qubit is π (2π) pulse if photon number N is even (odd). Considering the frequency shift of qubit, we employ a function of ${{{{{{{\rm{sinc}}}}}}}}(t)=\sin (t)/t$ to cover a larger spectrum range uniformly.

Fig. 8: Wigner tomography.

a Pulse sequence for Wigner tomography. b Frequency spectrum of a sinc shaped pulse. c Frequency spectrum of two sinc pulses with a time spacing π/χ. d Enlarged view of (c). e Bloch sphere of the ancillary qubit during the cat state Wigner tomography.

States preservation

After the process of states generation, the cat states are preserved by turning the flux bias off. Here, the Wigner functions are measured after a certain evolution time Δt. As shown in Fig. 9b–d, the Kerr-induced evolution is ideally prevented, the fidelity of 2-component cat ($\left|\alpha \right\rangle \pm i\left|-\alpha \right\rangle$) state is 89.1%, 81.9% and 75.8% after (0, 100, 200 ns).

Table 1 Parameters of the circuit. (ω_q0, ω_s0, and χ₀ are the values at Kerr-free working point.)

Fig. 9: Cat states generation and preservation.

a Pulse sequence for 2-component cat $\left|\alpha \right\rangle \pm i\left|-\alpha \right\rangle$ preparation. After preparation, the states are measured after Δt = (b) 0 ns, (c) 100 ns, (d) 200 ns.