Strong coupling between a single-photon and a two-photon Fock state

Abstract

The realization of strong nonlinear coupling between single photons has been a long-standing goal in quantum optics and quantum information science, promising wide impact applications, such as all-optical deterministic quantum logic and single-photon frequency conversion. Here, we report an experimental observation of the strong coupling between a single-photon and a two-photon Fock state in an ultrastrongly-coupled circuit-QED system. This strong nonlinear interaction is realized by introducing a detuned flux qubit working as an effective coupler between two modes of a superconducting coplanar waveguide resonator. The ultrastrong light--matter interaction breaks the excitation number conservation, and an external flux bias breaks the parity conservation. The combined effect of the two enables the strong one--two-photon coupling. Quantum Rabi-like avoided crossing is resolved when tuning the two-photon resonance frequency of the first mode across the single-photon resonance frequency of the second mode. Within this new photonic regime, we observe the thresholdless second harmonic generation for a mean photon number below one. Our results represent a key step towards a new regime of quantum nonlinear optics, where individual photons can deterministically and coherently interact with each other in the absence of any stimulating fields.

Figures (4)

• Figure 1: $|$ Setup.a, Schematic of the device. A flux qubit embedded in a ${\lambda}/{2}$ coplanar waveguide resonator, working as a nonlinear coupler between two modes of the resonator. The dashed cyan lines represent the vacuum current distribution of the $n=1$ (${\lambda}/{2}$) and $n=2$ ($\lambda$) modes of the resonator ${\kappa}_{\mathrm{out}}=3{\kappa}_{\mathrm{in}}$, where ${\kappa}_{\mathrm{in(out)}}$ is the loss rate from the input (output) port of the resonator, b, Optical image of the flux qubit. c, Effective coupling mechanism between the bare states $|2,0,g \rangle$ and $|0,1,g \rangle$ via virtual transitions involving intermediate states of the circuit-QED system. Here the first two entries in the kets denote the number of photons in the first two modes of the resonator, and the third indicates the qubit state ($|g\rangle$ is the ground state). d, Scheme of the energy levels at the flux offset corresponding to the minimum gap of the $|2,0,g \rangle-|0,1,g \rangle$ anticrossing, resulting in an effective coupling between these two states. A key feature of this configuration is that, at the minimum anticrossing gap, all the transitions shown by the arrows have comparable large efficiency, with transition matrix elements $|{X}_{1,0}| \simeq |{X}_{+,1}| \simeq |{X}_{+,0}| \sim 1$ (see Supplementary Fig. 2). Moreover, the transition energy $\tilde{\omega}_3$ is almost equal to $2 \tilde{\omega}_1$, so that the second harmonic generation (with only two initial photons) and degenerate down conversion (with only one initial photon) can occur efficiently at sub-photon input levels.
• Figure 2: $|$ Quantum Rabi-like splitting. Quantum Rabi-like splitting between a single-photon and a two-photon Fock state. The left part in each panel (negative flux offset) reports the measured spectra. The corresponding calculated spectra are reported for positive flux offset (right), using the parity symmetry with respect to the flux offset of the theoretical calculations. a, Measured and calculated transmission spectra of the one-photon line of the $n=1$ mode of the resonator as a function of the flux offset. b, Measured and calculated transmission spectra showing the excitation of both the two-photon state of the $n=1$ mode and the one-photon state of the $n=2$ mode of the resonator. An avoided-level crossing between these two lines is clearly visible. From the fitting, we obtain the Rabi frequency for the one--two-photon coupling $g_{\mathrm{eff}}/2\pi = 59 \ \mathrm{MHz}$, the loss rate due to the input--output ports $({\kappa}_{\mathrm{in}} + \kappa_\mathrm{out})/2\pi = 2.6 \ \mathrm{MHz}$, the internal loss rate $\kappa_\mathrm{int} / 2\pi = 10.4 \ \mathrm{MHz}$ (total loss rate of the resonator $\kappa_\mathrm{tot} / 2\pi = (\kappa_\mathrm{in} + \kappa_\mathrm{out} + \kappa_\mathrm{int}) / 2\pi = 13 \ \mathrm{MHz}$), the intrinsic loss rate of the qubit ${\kappa}_{q}/2\pi = 200\ \mathrm{MHz}$, and the pure dephasing rate of the qubit ${\kappa}_{q, \mathrm{dep}}/2\pi = 200\ \mathrm{MHz}$ (the last two have a very weak influence on the calculated spectra).
• Figure 3: $|$ Second harmonic generation.a, The amplitude of the SHG versus the external flux bias $\delta {\mathrm{\Phi}}_{\mathrm{ext}}$ and the SHG frequency. The efficiency $\eta \equiv S_{21}^{(2\omega)} / S_{21}^{(\omega)}$ of the SHG is about 0.1 at the point where the SHG amplitude is maximized (${\omega}_1/2\pi=4.9\ \mathrm{GHz}$ and ${\delta \mathrm{\Phi}}_{\mathrm{ext}}=-45~{\rm m \Phi_0}$), which qualitatively agrees with the theoretical result in Supplementary Fig. 2. b, Theoretical calculation corresponding to the plot in a. c, The amplitude of the SHG versus the average photon number ${\overline{n}}_1$ in the resonator. The signal frequency applied at the $n=1$ mode of the resonator is ${\omega}_1/2\pi=4.9\ \mathrm{GHz}$ (at ${\delta \mathrm{\Phi}}_{\mathrm{ext}}=-45~{\rm m \Phi_0}$). The red solid curve is the theoretical fit. The average photon number in c is determined by contrasting the experimental data---plotting the maximum second-harmonic generation (SHG) amplitude against input power---with the theoretical simulation outlined in Supplementary Eq. (35), as depicted in c. The power applied in a and b corresponds to an average photon number ${\overline{n}}_1 \simeq 0.25$ in the resonator.
• Figure 4: $|$ Interference between two probe tones.a(c), Gain of the transmitted amplitude of the signal field in the $n=1$ ($n=2$) mode of the resonator versus the average photon number ${\overline{n}}_{\mathrm{2(1)}}$ and the phase of the control field in the $n=2$ ($n=1$) mode of the resonator. The average photon numbers of the signal fields in the resonator's $n=1$ and $n=2$ modes are about 0.25 and 0.13, respectively. The frequencies of the probe tones are $\omega/ 2\pi =4.905$ GHz and $2 \omega/ 2\pi$ (at ${\delta \mathrm{\Phi}}_{\mathrm{ext}}=-46~{\rm m \Phi_0}$). b (d), Theoretical calculations corresponding to the plots in a (c). The gain is defined as the ratio between the output signal amplitude at $\omega$ ($2 \omega$) when a secondary signal is applied at $2 \omega$ ($\omega$), and the output amplitude of the same signal at $\omega$ ($2 \omega$) in the absence of the secondary signal applied at $2 \omega$ ($\omega$). The phase shifts in the experimental data in a (c) are an artifact due to an electronic delay.