The ability to create and manipulate quantum number states in a linear resonator is an important task in cavity quantum electrodynamics (QED; ref. 1). Early theory6,7 predicted that parametric frequency conversion could be a way to implement a tunable direct coupling between quantized modes of different energies. Classically, two harmonic oscillators coupled through a time-varying element, modulated at the difference of the resonator frequencies, will periodically exchange energy. At the quantum level, this can be used to swap the quantum states of two harmonic modes. In optics, the efficiency and quantum coherence of frequency up-conversion have been demonstrated using a pumped nonlinear crystal to couple light at different wavelengths12,13,14. However, in these experiments it is challenging to access the state dynamics because strong coupling rates are difficult to obtain15. In hybrid mechanical systems, strong parametric coupling, based on frequency conversion, has been recently achieved16,17, creating the possibility for the manipulation of quantum states of mesoscopic mechanical resonators18. In superconducting circuits, parametric processes have been used mainly to couple superconducting quantum bits (qubits) at their optimal points19, or to make quantum-limited microwave amplifiers20,21,22,23, yet little has been done with frequency conversion. Several circuit designs that enable frequency conversion between linear resonators have been proposed 21,24,25. This particular interaction can be combined with the powerful tools already available in circuit QED (ref. 8), where the creation and the detection of complex quantum states of the field have been demonstrated26. Here we measure the coherent dynamics of the parametric frequency conversion of a single photon between the first three internal resonant modes of a superconducting cavity, the state of which is prepared and read out with a superconducting qubit.

Our circuit consists of a quarter-wave (λ/4) coplanar waveguide (CPW) resonator terminated to ground via a SQUID (Fig. 1a). At the other end, it is coupled to a 50 Ω transmission line through a capacitance Cc to weakly probe the resonator with microwave reflectometry. At this node we also couple it to a flux-biased phase qubit through an effective capacitance Cg. This provides a single-photon source and detector for cavity photons. The SQUID is a superconducting loop asymmetrically intersected by three Josephson junctions. A ‘pump’ line allows both d.c. (Φsqd.c.) and microwave modulation (Φsqμw(t)) of the global flux (Φsq) in the SQUID loop. In the linear current regime (see Supplementary Information for an estimation of the current nonlinearities), and for frequencies lower than the plasma frequency, we can model the SQUID as a flux-dependent lumped-element inductor, Lsq(Φsq), modifying one of the boundary conditions of the cavity. Each cavity mode n (assumed lossless in this model) behaves as a flux-tunable harmonic oscillator (Fig. 1b) with a resonance frequency , where Ln(Φsqd.c.)=Ln+Lsq(Φsqd.c.) and LnCn is the lumped series oscillator describing the modes of the bare cavity23,27,28. Figure 1c shows the spectroscopic data, measured as a function of both probe frequency and d.c. flux Φsqd.c., for the first two modes, 0 and 1. The measurements shown in Figs 2 and 3 are performed at a flux bias point Φsqd.c.=−0.37Φ0 (point ‘A’ in Fig. 1c), where ν0=3.7 GHz and ν1=10.74 GHz (to simplify the notation, the dependence on d.c. flux will not be specified for the cavity resonances). The n=2 mode cannot be directly probed in our setup, but is expected to be around 18 GHz. The extracted loaded quality factors are about 9,000 for both modes 0 and 1. The SQUID inductance can be modulated by applying a small microwave flux Φsqμw(t)=δ Φsqμwcos(2π νpt+φp), with δ ΦsqμwΦ0 at a pump frequency νp. To first order in flux, the SQUID inductance is (the phase of the pump is defined modulo π, depending on the sign of the first derivative of the SQUID inductance with flux)

When νp=νmνn (by convention m>n), frequency conversion is induced between modes n and m (refs 6, 7, 29). In a typical λ/4cavity, the mode frequencies are νnλ/4=(2n+1)ν0λ/4, and therefore a pump at 2ν0λ/4 can also couple other neighbouring modes by conversion, as well as induce degenerate parametric amplification of mode 0 (ref. 29). To couple only two selected modes by frequency conversion (for a given pump frequency), we modify the cavity dispersion relation by slightly varying the characteristic impedance along the CPW resonator. This ensures that the frequencies νn are not equally spaced, satisfying the conditions: |ν2ν1|−|ν1ν0|,|ν1ν0|−2ν0Δν0ν1ν2, where Δνnis the bandwidth of mode n (we measure Δν0,12 MHz). This allows us to restrict our description to a two-mode manifold. Following the usual treatment for two parametrically coupled oscillators29, one can write the classical equations of motion of the two normalized normal mode amplitudes , where Ik is the current in mode k (k{n,m}), coupled by a quasi-resonant pump of frequency νp=νmνnνpνpνm,νn). Keeping the resonant terms for each mode and invoking the rotating wave approximation, one has

where is the parametric coupling frequency. For simplicity, we neglect its dependence on the pump detuning. The quantum model is derived by identifying equation (2) with Heisenberg’s equations of motion for the annihilation operators (the pump field is assumed to be strong enough to be described classically). The state associated with each harmonic mode of the cavity can be described in terms of quantized microwave excitations, namely photons. In the doubly rotating frame defined by the unitary transformation , the Hamiltonian, in the Schrödinger representation, is

When Δνp=0, describes a generalized beam-splitter operation between modes of different frequencies that preserves the total number of photons1. The effective transparency is modulated by the parametric interaction duration Δtp. In particular, the complete swap of a given initial state from one mode to the other is achieved for a pump π-pulse of duration Δtp=1/(4gmn).

Figure 1: Device description and spectroscopy.
figure 1

a, The ‘Superconducting Parametric Beam Splitter’ (SPBS) is a λ/4 CPW resonator with a boundary condition that is varied by a SQUID. The cavity has a total length of 7.57 mm and is composed of two λ/8 sections with characteristic impedances Z1=50 Ω and Z2=46 Ω. The SQUID is intersected by three junctions of critical currents Ic,1≈0.21 μA,Ic,2≈0.41 μA, and Ic,3≈0.88 μA. The flux through the SQUID is d.c.-biased and microwave-modulated via an inductively coupled pump line. To allow reflectometry measurements, the SPBS is coupled to a 50 Ω transmission line through a small capacitance Cc≈1.8 fF. A phase qubit is capacitively coupled to the SPBS with a strength gq1= 32 MHz (Supplementary Information). We manipulate the qubit state with microwave pulses and tune its frequency with a flux bias line (a picture of the device is provided in Supplementary Information). Measurements are performed in a dilution refrigerator operated at 35 mK, which makes thermal noise negligible at our working frequencies. b, SPBS simple model. Each cavity mode nis described by the equivalent LnCn of the λ/4 CPW nth harmonic in series with the SQUID tunable inductor. c, Measured reflection coefficient |S11| on Cc as a function of both d.c.-flux in the SQUID and probe frequency.

Figure 2: Spectroscopy of parametrically coupled cavity modes and one-photon Rabi-swap oscillations.
figure 2

ac, Spectroscopic data of modes 0 and 1, in the presence of a microwave pump drive on the cavity SQUID, at Φsqd.c.=−0.37Φ0 (point ‘A’ on Fig. 1c). |S11| is plotted in colour scale as a function of both probe and pump frequencies, when modes 0 and 1 (a,b), or 1 and 2 (c), are distinctly coupled. Parametric interaction enables a strong coupling between harmonics that are 7 GHz detuned, resulting in an avoided crossing between the two eigenmodes of the system (normal-mode splitting). The dashed black lines outline the asymptotic behaviour. The coupling rate g10 depends linearly on the pump amplitude Ap,μw. For the same pump amplitude, the coupling rate g21between modes 1 and 2 is about . We notice the presence of an unexplained parametrically coupled resonance at νp=7.35 GHz. d,e, Single-photon conversion oscillations. The colour scale encodes the tunnelling probability Pt of the qubit state as a function of both pump pulse duration and frequency. When the qubit is in the ground state, Pt is set to ≈25±2%. This probability increases linearly with the excited state occupancy. When the qubit is prepared in the excited state, Pt is ≈50±2% (the amplitude of the colour scale is chosen as the readout contrast of the qubit). The amplitude of the conversion oscillations is about 15% (see Supplementary Information). f,g, Fourier transform of the time-domain measurements showing the hyperbolic dependence between the coupling rate and the pump detuning. The theoretical curve is superposed with a dotted line.

Figure 3: Single-photon Ramsey interferences.
figure 3

a, Ramsey-fringes experiment with a single photon split between modes 0 and 1, at Φsqd.c.=−0.37Φ0 and Ap,μw=0.39 V at the microwave generator output. The cavity is initially prepared in the state |0011〉. Two phase-coherent ‘π/2’ pump pulses, separated by a variable delay Δtr are then applied. The state of mode 1 is finally measured with the qubit. The tunnelling probability Pt is plotted as a function of both the pump frequency and the delay Δtr. b, Fourier transform of the time-domain data. When starting with a single-photon Fock state, the frequency of the Ramsey oscillations is equal to the pump detuning Δνp.

Figure 2a and b show two separate sets of spectroscopic data probing the parametric coupling between modes n=0 and m=1. They are plotted as a function of νp and driven at three different pump amplitudes Ap,μw (measured at the output of the microwave generator). The probe frequency ν0,1probe is swept through ν0 (Fig. 2a), and ν1 (Fig. 2b). These data show well-resolved normal-mode splittings centred at νp=ν1ν0≈7.043 GHz, characteristic of the strong-coupling regime. In other words, the parametric coupling rate g10 is larger than the bandwidths of the modes. The extracted g10 rate is linearly dependent on pump amplitude, yielding up to 20 MHz coupling for a flux modulation δ Φsqμw/Φ02%. As the SQUID inductance is slightly rectified by the pump modulation, we observe a small shift of the resonance frequencies, leading to a shift of the splitting centres (Supplementary Information). The ‘quasi-resonant pump’ condition, νp=ν1ν0probe (νp=ν0+ν1probe), describes the asymptotic behaviour at small pump detuning. We can also couple modes 1 and 2 (Fig. 2c). As we designed the resonator to shift the harmonic mode frequencies, this requires a pump frequency 250 MHz higher than that needed for the conversion. In addition, the conversion rate is consistent with expectations from our simple model (equation (2)), . As g10,g21are small enough compared with |ν2ν1|−|ν1ν0|, we can, in fact, address these dual-mode manifolds separately.

As an illustration of the circuit efficiency at the quantum level, we realize the conversion oscillations of a one-photon Fock state between modes 0 and 1 (1 and 2). We prepare and measure the state of mode 1 using a superconducting phase qubit, following the method outlined in refs 26, 30, 31. The qubit is well described by a two-level system, with a ground state |g〉 and an excited state |e〉. Their energy difference defines the qubit frequency νq, which we can tune from 7 to 12 GHz by means of an external flux Φq (Supplementary Information). The qubit state is manipulated using microwave pulses and read out with a d.c. SQUID. By applying a fast flux pulse to the qubit, the |e〉 state preferentially tunnels to a different flux state, which is indicated by a shift in the d.c. SQUID critical current. We measure a relaxation time T1q≈100 ns, and a qubit-resonator coupling rate gq1=32 MHz. We load a single-photon Fock state in mode 1 by preparing the qubit in state |e〉 at νq=10.116 GHz, bringing it in resonance with the cavity for a duration Tπ=1/4gq1=8 ns, and shifting it back out of resonance. This prepares the cavity state |0011〉 (the subscript denotes the mode number). After applying a pump pulse of duration Δtp, the state of mode 1 is transferred back to the qubit by bringing it again in resonance with the cavity for a duration Tπ. Finally, we measure the state of the qubit. Figure 2d (2e) shows single photon conversion oscillations between modes 1 and 0 (resp. 1 and 2) as a function of Δtp and νp. At zero pump detuning, the frequency of the oscillations is the lowest, equal to the splitting strength measured in spectroscopies 2(a b c), and their amplitude is at their maxima. The Fourier transforms of these data are computed in Fig. 2f and Fig. 2g and exhibit the expected hyperbolic dependence and , respectively, when the initial state is a single-photon Fock state in one mode. The amplitude of these oscillations follows an exponential decay, with a characteristic time T10Rabi. Its measured value is T10Rabi≈180 ns (T21Rabi≈100 ns), compatible with the harmonic mean of the measured relaxation times T0Decay≈370 ns and T1Decay≈140 ns of both modes.

We measure the coherence of this parametric process by performing a Ramsey interference experiment with a photon ‘shared’ between modes 0 and 1, as a function of the pump detuning, with a pump amplitude corresponding to g10=20 MHz. After preparing |0011〉, we apply two phase-coherent pump pulses of 6 nsduration (corresponding to a π/2 pulse at Δνp=0), separated by a variable delay Δtr, and then measure the final state of mode 1. The first pulse prepares a superposition of states |0011〉 and |1001〉. During the free evolution, these two states acquire opposite phases differing by 2πΔνpΔtr. The second pulse combines these states, resulting in interference oscillations with a period equal to the inverse of the detuning. We verify this experimentally (Fig. 3). The amplitude of the oscillations decays on a characteristic timescale T10Ramsey≈190 ns, which indicates that the decoherence is mainly due to relaxation.

To further confirm the circuit model, we measure the conversion rate (Fig. 4a) as a function of the pump amplitude, for different Φsqd.c. (points ‘A, B, C, D, E’ on Fig. 1c). For the flux bias range that we explore ([−0.4,−0.3]Φ0), the coupling rates g10 are linearly dependent on the experimentally accessible pump amplitudes. The variation of the conversion efficiency g10/ Ap,μw with the flux is also in good agreement with our SQUID inductor model and its simple expansion in equation (1), and is consistent with measured values of the pump line attenuation and d.c. mutual inductance to pump line (Fig. 4b).

Figure 4: Dependence of the coupling rate with the SQUID flux.
figure 4

a, Parametric coupling frequency g10 between modes 0 and 1 as a function of the pump amplitude Ap,μw, for the various operating fluxes A, B, C, D, E shown on Fig. 1c. In the explored flux and pump amplitude ranges, g10 depends linearly on Ap,μw. b, Consistency check of the simple flux expansion of SQUID inductance used in our model (equation (1)). For a given flux, we extract the conversion efficiency g10/ Ap,μw from the measurements in Fig. 4a. This is compared to a theoretical curve of α| LJ/ Φ|, where | LJ/ Φ| is the numerically calculated first derivative with flux of the SQUID inductance, and with , where Msq is the mutual coupling inductance between the pump line and the SQUID, and κ is the attenuation of the pump line. We set the value of the prefactor to the one at point A, and we assume that Msqis equal to the d.c. value (1.2 pH). κ is the only adjustable parameter and is in good agreement, within 2 dB, with the attenuation measured at room temperature.

To conclude, we have demonstrated that parametric frequency conversion is an effective way to coherently manipulate a single microwave-photon Fock state in frequency space. The dynamics are accurately described by a generalized beam-splitter interaction familiar from quantum optics. Combined with a phase shift operation, which can be implemented by fast shifts in SQUID bias flux, this system could be used as a novel linear optical quantum bit based on microwave resonator modes. Furthermore, straightforward technical improvements will enable the manipulation of multi-photon states. Finally, this device also offers the opportunity to explore other parametric interactions, such as amplification, with similarly strong interaction rates.