Photons in non-classical states are of great importance for both fundamental research and quantum applications. They provide major testbeds for quantum optics experiments^1⇓-3. In the meantime, photons serve as the backbone of quantum networks⁴, which enables various quantum technologies, including quantum communication^{5, 6} and large-scale quantum computation^7-14. Photons in squeezed states, which can be generated through parametric conversion, are essential tools for quantum metrology^{15, 16} and measurement-based quantum computing^{17⇓⇓⇓-21}. Other than that, single photon sources, as an ideal carrier of quantum information, are widely used for entanglement distribution and remote entanglement generation^4,22. In the microwave regime, single photons and propagating photonic qubits can be used to coherently link remote quantum chips to realise modular quantum computing and scale up computing power for superconducting quantum devices^{23⇓⇓⇓-27}. A quantum light source that generates those quantum states of photons is thus a fundamental instrument in the field of quantum information processing^2,16.

Single photons in the optical frequency can be generated with atoms, solid-state quantum dots, or through parametric down-conversion in nonlinear crystals^28⇓⇓-31. In the microwave regime, matter qubits can strongly couple to microwave cavities, which may facilitate a deterministic photon generation process^32⇓⇓-35. Superconducting qubits can be used as ancilla to generate the specific quantum state of photons through either spontaneous emission^36-40 or cavity-assisted Raman transition^41-42. The shape of the emitted photon wave packets can be conveniently controlled by shaping the driving pulse in the latter case. Indeed, spectral control of the quantum light source is of great importance in many circumstances. For instance, when the propagating photons are used for quantum state transfer and entanglement generation, a frequency-tunable quantum light source is necessary to bridge the possible spectral mismatches of distant quantum nodes⁴³. Apart from that, to achieve high transfer fidelity and efficiency, well-controlled bandwidth and shape of the single-photon wave packets are required to fulfill temporal mode matching among different nodes^22,44.

In the current work, a deterministic frequency-tunable microwave single-photon source based on superconducting quantum circuits is demonstrated. The single photon source can also generate various quantum states of propagating photons, including time-bin encoded photonic qubits and qudits. It is also showcased that by encoding the photonic quantum state in the Fock-1 subspace, the fidelity of the prepared photon states is insensitive to the inevitable circuit loss and non-uniform quantum efficiency of the light source, which is particularly important in the microwave regime. As such, this work provides an important building block for future modular quantum computing and microwave quantum optics.

As illustrated in Fig. 1, the device composes a λ/2 coplanar waveguide resonator dispersively coupled to a transmon qubit. To tune the resonator frequency, a superconducting quantum interference device (SQUID) is embedded in the centre of the resonator⁴⁵. The transmon qubit is used to facilitate the generation of the quantum state of light. The transition frequency between the ground state $|g〉$ and the first exited state $|e〉$ of the transmon is 6.1362 GHz, with the charging energy $E_c$ = 217 MHz. The relaxation time for the first and second excited states are $T_1^{\mathrm{ge}}=8.89\mathrm{\mu }\mathrm{s}$ and $T_1^{\mathrm{ef}}=6.99\mathrm{\mu }\mathrm{s}$, respectively. The dephasing time is measured to be $T_2^{\mathrm{ge}}=2.88\mathrm{\mu }\mathrm{s}$ and $T_2^{\mathrm{ef}}=3.16\mathrm{\mu }\mathrm{s}$. In general, the excited-state population of the qubit can be transferred to photons in the resonator via either spontaneous decay^{37, 39} or the resonator-assisted Raman transition⁴², after which the photons escape from the resonator to the transmission line as propagating photons.

The SQUID used for tuning the resonator frequency consists of two parallel Josephson junctions, which behave as a tunable inductor L_j by varying the flux φ through the loop of the Josephson junctions. L_j can be approximated as $\frac{L_{\mathrm{j}0}}{\cos (\pi \phi /{\phi }_0)}$, where L_j0 represents the SQUID inductance when $\phi =0$, ${\phi }_0$ is the flux quanta. The total inductance of the SQUID-embedded resonator is determined by a sum of the resonator inductance L₀ and the flux-dependent SQUID inductance L_j(φ). Correspondingly, the resonance frequency can be written as $\omega =\sqrt{1/(L_0+L_{\mathrm{j}}\left(\phi \right))C_0}$, where C₀ is the resonator capacitance and the capacitance of SQUID is small enough to be neglected. Therefore, it is possible to tune the resonator frequency via an external magnetic field^45,46. In the device, the magnetic field is introduced by applying an analog voltage on the flux line, as shown in Fig. 1, which is linearly related to the magnetic flux seen by the SQUID loop.

The sample was cooled down to about 20 mK in a dilution fridge. As illustrated in Fig. 1, the microwave photons emitted from the quantum light source are first amplified by a Josephson parametric amplifier, and then analyzed with a homodyne setup, where the quadratures of the output photon field are recorded. From the measured quadratures, moments of the propagating photon field can be calculated and the Winger function or density matrix can be correspondingly reconstructed. Details of this method can be found in ref.⁴⁷.

The resonator-assisted Raman process was used to generate propagating single photons, wherein the qubit population in the second excited state $|f〉$ is transferred as a single photon in the resonator via the ‘f0g1’ transition, thus providing a deterministic single-photon generation process²⁸. It has been shown that by using ‘f0g1’ driving pulse with different shapes, lengths and phases, the pulse shape, bandwidth and phase of the single photons can be well controlled⁴², promises that such a single-photon source can be used in scenarios with varied requirements on the properties of single-photon wave packets^43,48.

Fig. 2b shows the histogram of the in-phase and out-of-phase quadratures of the propagating single photon field, by subtracting the measured quadrature histogram of the vacuum field from that when the photon is released from the resonator. A clear signature of the single photon state could be seen, with a circular distribution in the phase space³⁷. From the measured quadrature histograms, the average photon number of the propagating field is determined to be 0.414. Considering that a hanger resonator is used for the single photon generation (see Fig. 1), only half of the photons emitted to the transmission line are sent to the homodyne detector. The quantum efficiency of the single photon source is estimated to be 82.8%. The inefficiency can be mainly attributed to the internal loss of the resonator, which is estimated to be ${\kappa }_{\mathrm{i}}$/(${\kappa }_{\mathrm{c}}+{\kappa }_{\mathrm{i}}$), where ${\kappa }_{\mathrm{i}}$ and ${\kappa }_{\mathrm{c}}$ represents the internal loss rate and the out-coupling rate of the resonator, respectively. Fitting the measured transmission spectra gives ${\kappa }_{\mathrm{c}}+{\kappa }_{\mathrm{i}}$/(${\kappa }_{\mathrm{c}}+{\kappa }_{\mathrm{i}}$) = 17%, with ${\kappa }_{\mathrm{i}}/2\mathrm{\pi }$ = 0.51 MHz and ${\kappa }_{\mathrm{c}}/2\mathrm{\pi }$ = 2.49 MHz, which agrees well with the estimated quantum efficiency of the photon source.

As discussed above, the frequency of the emitted photon is determined by the resonator frequency, which can be tuned by varying bias voltage applied to the flux line. In Fig. 2a we measure the transmission spectra of the resonator as a function of the applied voltage on the flux line with a vector network analyzer, where the resonance frequency can be tuned continuously by more than 200 MHz. It is worth noting that when the resonance frequency is tuned away from the sweep spot, the dip of the transmission spectra becomes smaller, as shown in the inset of Fig. 2a, indicating an increasing internal loss rate. Fig. 2c shows the fitted ${\kappa }_{\mathrm{i}}$ as a function of the resonator frequency, where κi/2π increases from 0.51 MHz at the sweet spot to 1.48 MHz when the resonance frequency is shifted by about 120 MHz. Similar phenomena have been reported in previous works⁴⁶, which is attributed to energy dissipation by the subgap resistance of the SQUID. Since the SQUID inductance increases when the cavity is tuned to a lower frequency, more power will be dissipated by the subgap resistance, thus yielding an increased internal loss rate and a lower quantum efficiency. This problem can be alleviated by using junctions with higher critical current density⁴⁶.

The increased internal loss rate would degrade the quantum efficiency of the single-photon source. In Fig. 2c, we measure average photon numbers of the emitted photon field with varied photon frequencies. As expected, tuning the single-photon frequency away from the sweet spot by about 120 MHz results in a degradation of the quantum efficiency from 82.8% to 38.6%, which is mainly caused by the resonator internal loss-induced inefficiency of 50.5% (with κi/2π = 1.48 MHz and κc/2π = 1.45 MHz). In addition, decreased energy relaxation times (T1) are observed for both $|e〉$ state and $|f〉$ state of the transmon qubit when the resonator frequency is detuned, which may also contribute to the decreased quantum efficiency. Such a limited quantum efficiency of the single-photon source would result in poor fidelity when directly using the single-photon state as propagating qubits.

In the presence of inefficiency of the frequency-tunable single-photon source, time-bin encoding can be used to realize propagating photonic qubits with high fidelity. The qubit state is defined as a single photon that existing in a superposition of two temporal modes⁴¹. The photon that exists in the early or late temporal mode is written as $|E〉\equiv |10〉$ or $|L〉\equiv |01〉$, where $|10〉$ ($|01〉$) represents that the single photon is in the early (late) mode. Since the qubit state is defined in the single-photon subspace, the photon-loss-induced error would leave the photonic state out of the qubit space, and thus can be excluded by projecting the system state to the single-photon subspace^24,41. This is very important for propagating microwave photonic qubits considering the strong propagation loss and insertion loss for photons in the microwave regime. Moreover, it has been demonstrated that the time-bin qubit is robust to phase noise, which makes them an excellent candidate as carriers of quantum information for quantum state transfer and entanglement distribution in microwave quantum networks⁴¹.

The pulse sequence for the generation of the time-bin qubit can be found in Fig. 3a. If the transmon qubit of the photon source is prepared in a superposition state $\alpha |g〉+\beta |e〉$, after the pulse sequence one would have $\alpha |L〉+\beta |E〉$, wherein the quantum state defined with the transmon is transferred to the photonic qubit. In this way, photonic qubits are prepared in different quantum states when the resonator frequency is placed at the sweet spot, and some typical density matrices can be found in Fig. 3b-c. It is worth noting that the density matrices are obtained by projecting the reconstructed two-mode photonic state to the single-photon subspace⁴⁷. Consequently, even though the internal quantum efficiency of the photon source is not unity, the fidelity of the photonic qubit can still be above 92.20% thanks to the time-bin encoding scheme. In such cases, the fidelity of photonic qubit is mainly bottlenecked by the limited energy relaxation rate and dephasing rate of the transmon qubit.

It is worth noting that quantum entanglement can be genetated between the transmon qubit and the propagating photonic qubit. With the pulse sequence shown in Fig. 3a, we prepare the system into a Bell state $(|g〉|L〉+|e〉|E〉/\sqrt{2})$ by initially preparing the transmon in a superposition of $(|g〉+|e〉)/\sqrt{2}$. The reconstructed density matrix is shown in Fig. 3d, with a fidelity of 73.84%±0.42%. The infidelity is mainly contributed by the finite energy relaxation time and coherence time of the superconducting qubit, which can be alleviated by using available qubits with better coherence time. A further improvement on the ‘f0g1’ pulse power, and thus a shorter gate time, is also helpful to improve the prepared state fidelity. Such a qubit-photon entanglement is a crucial resource for the entanglement generation and distribution among remote superconducting qubits.

The frequency of the propagating time-bin qubit can be adjusted by tuning the resonance frequency of the resonator. In the experiment, we measure the fidelity of the time-bin qubit at different frequencies, as presented in Fig. 3e. It can be seen that the fidelity of both $|E〉$ state and $(|E〉+|E〉)/\sqrt{2}$ state show minor dependence on the photon frequency. As discussed above, the time-bin encoding scheme ensures that the fidelity of the propagating qubit is insensitive to the quantum efficiency of the light source. Even though the quantum efficiency of the photon source drops from 82.8% to 38.6% with a detuned photon frequency, the fidelity of the time-bin can still be kept well above 90%. This result promises that the single-photon source can be used to align to spectrally mismatched quantum nodes with high fidelity.

Propagating photonic d-level qudits (d>2) can be effectively synthesized with the single-photon source. As high dimensional carriers of quantum information, photonic qudits are endowed with the advantages of the storage of exponentially greater information, larger channel capacity for quantum communication, and simultaneous generation of multiple entangled pairs^{49⇓⇓⇓⇓⇓⇓-56}. In the experiment, we showcase the generation of photonic qutrits (d=3) by sequentially transferring the transmon coherence to the propagating photonic mode. The pulse sequence is shown in Fig. 4a. We firstly prepare the transmon qubit to a superposition state $\cos \frac{\theta }{2}|e〉+\sin \frac{\theta }{2}|f〉$, then apply the ‘f0g1’ pulse to convert the $|f〉$ state population to a propagating photonic state. After that, we transfer part of the $|e〉$ state population to $|f〉$ and convert it to the second photonic mode. At last, a π pulse of $|e〉-|f〉$ transition are applied and the ‘f0g1’ pulse to convert the residual excitation to the third photonic mode, resulting in disentangled qubit and photonic states. The final photonic state can be written as $\Psi =\cos \frac{\theta }{2}\cos \frac{\varphi }{2}|1_c{0}_b{0}_a〉+\cos \frac{\theta }{2}\sin \frac{\varphi }{2}|0_c{1}_b{0}_a〉+\sin \frac{\theta }{2}|0_c{0}_b{1}_a〉$, which is an arbitrary superposition state of a qutrit. By taking $\varphi =\mathrm{\pi }/2$ and $\theta =\arctan 2\sqrt{2}$, we have $\Psi =(|1_c{0}_b{0}_a〉+|0_c{1}_b{0}_a〉+|0_c{0}_b{1}_a〉)/\sqrt{3}$. In Fig. 4b, we reconstruct the density matrix for this state, obtaining a fidelity of 69.19%±0.93%. The state preparation errors are mainly induced by the qubit decay and dephasing during the pulse sequence. Further improvements in the state preparation fidelity can be realized by improving the qubit coherence and reducing the length of the ‘f0g1’ pulse. Note that by sequentially applying the ‘f0g1’ transition and properly adjusting the state of the superconducting qubit before each ‘f0g1’ pulse, such a scheme is scalable for the generation of arbitrary qudit state. Moreover, such a sequential photon generation process can also be used to prepare photonic cluster states on a chip, which is crucial for one-way quantum computing^57⇓⇓-60.

In conclusion, we have successfully realized a frequency-tunable quantum light source in the microwave regime. By incorporating a SQUID into the cavity and utilizing the resonator-assisted Raman transition, the quantum state defined in the superconducting qubit can be effectively transferred to a frequency-tunable single-photon state. Hereafter, the generation of single photons, time-bin encoded propagating photonic qubits and qutrits were showcased. The frequency of the emitted photons can be tuned by more than 200 MHz, with a sacrifice of internal quantum efficiency. Moreover the results show that with a proper encoding scheme, the fidelity of the emitted quantum state can be well preserved. As such, the work demonstrates a quantum light source in the microwave regime, which has broad applications in future quantum networks, modular quantum computing and one-way quantum computing.

Acknowledgments This work was supported by the Innovation Program for Quantum Science and Technology (2021ZD0301704), the Tsinghua University Initiative Scientific Research Program, and the Ministry of Education of China.

Declaration of Competing Interest The authors declare that there are no competing interests.