Abstract
Fully controlled coherent coupling of arbitrary harmonic oscillators is an important tool for processing quantum information^{1}. Coupling between quantum harmonic oscillators has previously been demonstrated in several physical systems using a twolevel system as a mediating element^{2,3}. Direct interaction at the quantum level has only recently been realized by means of resonant coupling between trapped ions^{4,5}. Here we implement a tunable direct coupling between the microwave harmonics of a superconducting resonator by means of parametric frequency conversion^{6,7}. We accomplish this by coupling the mode currents of two harmonics through a superconducting quantum interference device (SQUID) and modulating its flux at the difference (∼7 GHz) of the harmonic frequencies. We deterministically prepare a singlephoton Fock state^{8} and coherently manipulate it between multiple modes, effectively controlling it in a superposition of two different ’colours’. This parametric interaction can be described as a beamsplitterlike operation that couples different frequency modes. As such, it could be used to implement linear optical quantum computing protocols^{9,10} onchip^{11}.
Main
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 theory^{6,7} 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 timevarying 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 upconversion have been demonstrated using a pumped nonlinear crystal to couple light at different wavelengths^{12,13,14}. However, in these experiments it is challenging to access the state dynamics because strong coupling rates are difficult to obtain^{15}. In hybrid mechanical systems, strong parametric coupling, based on frequency conversion, has been recently achieved^{16,17}, creating the possibility for the manipulation of quantum states of mesoscopic mechanical resonators^{18}. In superconducting circuits, parametric processes have been used mainly to couple superconducting quantum bits (qubits) at their optimal points^{19}, or to make quantumlimited microwave amplifiers^{20,21,22,23}, 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 demonstrated^{26}. 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 quarterwave (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 C_{c} to weakly probe the resonator with microwave reflectometry. At this node we also couple it to a fluxbiased phase qubit through an effective capacitance C_{g}. This provides a singlephoton 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. (Φ_{sq}^{d.c.}) and microwave modulation (Φ_{sq}^{μw}(t)) of the global flux (Φ_{sq}) 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 fluxdependent lumpedelement inductor, L_{sq}(Φ_{sq}), modifying one of the boundary conditions of the cavity. Each cavity mode n (assumed lossless in this model) behaves as a fluxtunable harmonic oscillator (Fig. 1b) with a resonance frequency , where L′_{n}(Φ_{sq}^{d.c.})=L_{n}+L_{sq}(Φ_{sq}^{d.c.}) and L_{n}C_{n} is the lumped series oscillator describing the modes of the bare cavity^{23,27,28}. Figure 1c shows the spectroscopic data, measured as a function of both probe frequency and d.c. flux Φ_{sq}^{d.c.}, for the first two modes, 0 and 1. The measurements shown in Figs 2 and 3 are performed at a flux bias point Φ_{sq}^{d.c.}=−0.37Φ_{0} (point ‘A’ in Fig. 1c), where ν_{0}=3.7 GHz and ν_{1}=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 Φ_{sq}^{μw}(t)=δ Φ_{sq}^{μw}cos(2π ν_{p}t+φ_{p}), with δ Φ_{sq}^{μw}≪Φ_{0} at a pump frequency ν_{p}. 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 ν_{p}=ν_{m}−ν_{n} (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 ν_{n}^{λ/4}=(2n+1)ν_{0}^{λ/4}, and therefore a pump at 2ν_{0}^{λ/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 ν_{n} are not equally spaced, satisfying the conditions: ν_{2}−ν_{1}−ν_{1}−ν_{0},ν_{1}−ν_{0}−2ν_{0}≫Δν_{0},Δν_{1},Δν_{2}, where Δν_{n}is the bandwidth of mode n (we measure Δν_{0,1}≲2 MHz). This allows us to restrict our description to a twomode manifold. Following the usual treatment for two parametrically coupled oscillators^{29}, one can write the classical equations of motion of the two normalized normal mode amplitudes , where I_{k} is the current in mode k (k∈{n,m}), coupled by a quasiresonant pump of frequency ν_{p}=ν_{m}−ν_{n}+Δν_{p} (Δν_{p}≪ν_{m},ν_{n}). 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 Δν_{p}=0, describes a generalized beamsplitter operation between modes of different frequencies that preserves the total number of photons^{1}. The effective transparency is modulated by the parametric interaction duration Δt_{p}. In particular, the complete swap of a given initial state from one mode to the other is achieved for a pump πpulse of duration Δt_{p}=1/(4g_{mn}).
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 ν_{p} and driven at three different pump amplitudes A_{p,μw} (measured at the output of the microwave generator). The probe frequency ν_{0,1}^{probe} is swept through ν_{0} (Fig. 2a), and ν_{1} (Fig. 2b). These data show wellresolved normalmode splittings centred at ν_{p}=ν_{1}−ν_{0}≈7.043 GHz, characteristic of the strongcoupling regime. In other words, the parametric coupling rate g_{10} is larger than the bandwidths of the modes. The extracted g_{10} rate is linearly dependent on pump amplitude, yielding up to 20 MHz coupling for a flux modulation δ Φ_{sq}^{μw}/Φ_{0}∼2%. 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 ‘quasiresonant pump’ condition, ν_{p}=ν_{1}−ν_{0}^{probe} (ν_{p}=ν_{0}+ν_{1}^{probe}), 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 g_{10},g_{21}are small enough compared with ν_{2}−ν_{1}−ν_{1}−ν_{0}, we can, in fact, address these dualmode manifolds separately.
As an illustration of the circuit efficiency at the quantum level, we realize the conversion oscillations of a onephoton 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 twolevel system, with a ground state g〉 and an excited state e〉. Their energy difference defines the qubit frequency ν_{q}, which we can tune from 7 to 12 GHz by means of an external flux Φ_{q} (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 T_{1}^{q}≈100 ns, and a qubitresonator coupling rate g_{q1}=32 MHz. We load a singlephoton Fock state in mode 1 by preparing the qubit in state e〉 at ν_{q}=10.116 GHz, bringing it in resonance with the cavity for a duration T_{π}=1/4g_{q1}=8 ns, and shifting it back out of resonance. This prepares the cavity state 0_{0}1_{1}〉 (the subscript denotes the mode number). After applying a pump pulse of duration Δt_{p}, the state of mode 1 is transferred back to the qubit by bringing it again in resonance with the cavity for a duration T_{π}. 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 Δt_{p} and ν_{p}. 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 singlephoton Fock state in one mode. The amplitude of these oscillations follows an exponential decay, with a characteristic time T_{10}^{Rabi}. Its measured value is T_{10}^{Rabi}≈180 ns (T_{21}^{Rabi}≈100 ns), compatible with the harmonic mean of the measured relaxation times T_{0}^{Decay}≈370 ns and T_{1}^{Decay}≈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 g_{10}=20 MHz. After preparing 0_{0}1_{1}〉, we apply two phasecoherent pump pulses of 6 nsduration (corresponding to a π/2 pulse at Δν_{p}=0), separated by a variable delay Δt_{r}, and then measure the final state of mode 1. The first pulse prepares a superposition of states 0_{0}1_{1}〉 and 1_{0}0_{1}〉. During the free evolution, these two states acquire opposite phases differing by 2πΔν_{p}Δt_{r}. 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 T_{10}^{Ramsey}≈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 Φ_{sq}^{d.c.} (points ‘A, B, C, D, E’ on Fig. 1c). For the flux bias range that we explore ([−0.4,−0.3]Φ_{0}), the coupling rates g_{10} are linearly dependent on the experimentally accessible pump amplitudes. The variation of the conversion efficiency ∂ g_{10}/∂ A_{p,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).
To conclude, we have demonstrated that parametric frequency conversion is an effective way to coherently manipulate a single microwavephoton Fock state in frequency space. The dynamics are accurately described by a generalized beamsplitter 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 multiphoton states. Finally, this device also offers the opportunity to explore other parametric interactions, such as amplification, with similarly strong interaction rates.
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Acknowledgements
We thank N. Bergren and L. Ranzani for technical help, and J. Park, F. Altomare and L. Spietz for valuable input.
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E.ZB. and F.N. designed the experiment, built the measurement setup and performed the measurements. M.L., R.W.S., J.A. contributed to the experimental design. L.R.V. contributed to the fabrication process development. J.A. conceived the experiment and supervised the project. All authors participated in the sample fabrication, the writing of the manuscript and the data analysis.
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ZakkaBajjani, E., Nguyen, F., Lee, M. et al. Quantum superposition of a single microwave photon in two different ’colour’ states. Nature Phys 7, 599–603 (2011). https://doi.org/10.1038/nphys2035
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DOI: https://doi.org/10.1038/nphys2035
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