Emission of Photon Multiplets by a dc-Biased Superconducting Circuit

We observe the emission of bunches of $k\ge 1$ photons by a circuit made of a microwave resonator in series with a voltage-biased tunable Josephson junction. The bunches are emitted at specific values $V_k$ of the bias voltage, for which each Cooper pair tunneling across the junction creates exactly $k$ photons in the resonator. The latter is a microfabricated spiral coil which resonates and leaks photons at 4.4 GHz in a measurement line. Its characteristic impedance of $1.97\mathrm{k}\mathrm{\Omega }$ is high enough to reach a strong junction-resonator coupling and a bright emission of the $k$-photon bunches. We show that a rotating-wave approximation treatment of the system accounts quantitatively for the observed radiation intensity, from $k=1$ to 6, and over 3 orders of magnitude when varying the Josephson energy $E_J$. We also measure the second-order correlation function of the radiated microwave to determine its Fano factor $F_k$, which in the low $E_J$ limit confirms with $F_k\simeq k$ the emission of $k$-photon bunches. At larger $E_J$, a more complex behavior is observed in quantitative agreement with numerical simulations.

Popular Summary

When an electron in an atom transitions from one quantum state to another one at lower energy, it radiates a single photon with a frequency given by the energy difference. The photon emission rate is governed by the fine structure constant, which quantifies how strongly charge and light are coupled. Nonlinear optical materials can produce more than one photon in a single quantum event, but the smallness of the fine structure constant makes the rate of these events very low. Here, we operate an artificial electrical circuit specifically designed with a strong charge-light coupling, which boosts the efficiency of emitting photons in bunches of up to six at a time.

Our circuit is a simple battery-biased superconducting tunnel junction in series with a microwave resonator. At discrete values of the battery voltage, a dc current flows through the circuit, with the emission of several photons at the resonator frequency for each superconducting pair of electrons that tunnels across the junction. We measure the total microwave power emitted and characterize the granularity of this emission. We find both in good agreement with a simple theoretical model. In particular, at a small transparency of the tunnel junction, we do find a granularity equal to the size of the photon bunches.

Beyond demonstrating the fundamental phenomenon of bright multiphoton emission, this work provides a particularly pure, controllable, and well-understood test bench for quantum physics, combining strong nonlinearity, strong charge-light coupling, and a steady out-of-equilibrium situation.

Figure 1

Principle of the experiment and first observation of multiphoton emission up to $k=5$ (run 1). (a) A tunable Josephson junction (JJ) with energy $E_J$ (green cross) is connected in series with a dc voltage source $V$ and a microwave resonator (blue) of frequency ${\nu }_R$ and characteristic impedance $Z_R$. Current can flow only at certain values $V_k$ of $V$ for which the energy $2{eV}_k$ of a Cooper pair transferred across the circuit is entirely transformed into an integer number $k$ of photons in the resonator. Is the field leaking out of the resonator quantitatively understood and does it display $k$-photon bunches? (b) Optical micrograph of the sample showing a SQUID (magnetically tunable JJ, main picture and inset) connected to a high inductance coil (resonator). The electrical parameters are indicated and lead to a giant effective fine structure constant $\alpha \sim 1$. The bias and measuring lines are schematized in Fig. 4. (c) Emitted power measured as a function of the bias voltage $V$ for a Josephson energy $E_J$ large enough to observe emission peaks up to $k=5$. The black and red vertical dotted lines indicate the offset voltage and the $V_k$ values, respectively. Note that the small peak visible below $V_6$ is a spurious emission attributed to a high-frequency mode of the circuit.

Figure 2

Multiphoton emission spectra and emitted power for $k=1–6$. (run 2). (a) Examples of measured (dots) power spectral densities (PSD) taken at $E_J/h{\nu }_R\sim 0.142$ around the resonator frequency ${\nu }_R$, for different bias voltages $V_k+\delta V_k$ corresponding to residual frequency detunings ${\delta }_k/2\pi =2e\delta V_k/hk=-39.4$, $-6.0$, $-1.2$, 1.4, $-7.5$, and 0.8 MHz for $k=1,\dots ,6$. A vertical magnification factor with respect to the left axis scale is indicated for each peak. Cyan filled peaks are Lorentzian fits of the PSDs (for $k=6$, a two-Lorentzian fit gives a second spurious mode in violet). The cyan areas are the photon rates ${\mathrm{\Gamma }}_k$ used for comparison with simulations in (b). Bare simulations of the discretized spectral densities at zero detuning (black solid lines) are shown (horizontally shifted) for comparison. (b) Measured (dots) and simulated (solid lines) reduced emission rates $⟨n⟩={\mathrm{\Gamma }}_k/\kappa$ for $k=1–6$ and 12 different $E_J$ values. Dashed curves represent the number of photons obtained from the Purcell rate ${\gamma }_k$ of departure from the vacuum state (see text). Because of magnetic flux jumps in the SQUID, experimental $E_J$ values were not precisely known and were fitted to minimize the difference between simulation and experiment in log scale (see text and Fig. 7 in Appendix pp5). No voltage noise is included in the simulation and no vertical scaling of the data is applied after calibration. The $\pm 5\%$ systematic relative uncertainty on calibration plus the uncertainty on $k$ is about the symbol size. The vertical dashed line corresponds to the dataset in (a).

Figure 3

Noise statistics and Fano factors of the leaking microwave (run 3). (a) Calibration of the $E_J$ values by placing the measured $⟨n⟩$ (dots) on a simulation of the $⟨n⟩(E_J)$ curve including the effect of the observed voltage noise (solid line). Simulation with a noiseless bias voltage (dashed line) is also shown for comparison. The uncertainty on $⟨n⟩$ (vertical error bars) yields the uncertainty on $E_J$ (horizontal error bars). (b) Example of a set of $g^{(2)}$ functions measured at a small $E_J/\hslash {\omega }_R=0.018$. The area between $g^{(2)}$ and 1, integrated over the 0–40 ns time interval (shown in cyan for $k=3$) enters the Fano factor of (c). (c) Experimental Fano factors (dots) and simulated Fano factors including voltage noise (solid curves). Horizontal error bars correspond to those of (a), whereas vertical error bars correspond to $\pm 2$ standard deviations. The vertical dashed line corresponds to the raw data of (a).

Figure 4

Experimental setup. Schematics of the whole measurement circuit used in the different experimental runs (see text).

Figure 5

Characterization of the resonator. Measured total emitted power (dots) as a function of the central frequency of the emitted spectrum. The orange line is a Lorentzian fit yielding the resonator frequency ${\nu }_R=4406.7\pm 0.25\mathrm{MHZ}$ and quality factor $Q=72\pm 1.4$ (run 3).

Figure 6

Calibration of $E_J$ via the field dependence of the emission. (a) Photon emission rate $\mathrm{\Gamma }$ expressed in mega photons per second measured (dots) as a function of the voltage applied to the coil, recorded at ${\nu }_J=5.15\pm 0.01\mathrm{GHz}$, far from the resonance frequency of the resonator. The dashed orange line is a sinusoidal fit. (b) Derivative of the shot-noise signal as a function of the frequency. The orange dot shows the point at ${\nu }_J$ used for dividing $\mathrm{\Gamma }$ (see text).

Figure 7

Magnetic hysteresis and fitted $E_j$ values. Square root of the emitted power $\sqrt{P}$ at 5 GHz (blue points, left axis) versus applied coil voltage $v_{\text{coil}}$, and fitted $E_J$ values (red, right axis) presented in Fig. 2 of the main text. A relative vertical scaling is applied to compare the two datasets.

Figure 8

Fitting of the voltage noise. (a) Example of three measured PSD of emission (blue lines) taken at various voltage biases, below, at, and above the resonator central frequency, as well as their fit (orange) by a product of two Lorentzian lines (see text). The resonator line shape (dashed gray line) is superposed to allow the reader to better locate the bias values. (b) Three examples of emitted power $P$ (solid lines) taken at fixed frequencies $\nu$ as a function of the bias voltage $V$, as well as their Lorentzian fit (dashed lines). (c) FWHM extracted from many $P(V)$ curves as those shown in (b), on a dense bias grid. The orange bar shows the window used to compute the mean value of the emission width and its height indicates the corresponding standard deviation.

Figure 9

Function $g^{(2)}(\tau )$ for various values of $k$ and $E_J$. Raw experimental measurement of $g^{(2)}(\tau )$ from $k=1$ to $k=4$ (from left to right) and for $E_J/h{\nu }_r=\{0.011,0.018,0.025,0.029,0.045,0.065\}$. The left-hand panel each panel pair presents the short time variation of $g^2$ while the right-hand panel presents the noise on the measured $g^{(2)}(\tau )$ fluctuating around 1 at long time. A vertical multiplication factor is indicated in these right-hand panels together with the average value $a$ and the standard deviation $\sigma$ of the noise.

Figure 10

Effects of voltage noise on the correlation function $G^{(2)}(\tau )$ are simulated by averaging over a Gaussian normal distribution $P(\delta V/\mathrm{\Delta }V)$ (gray shaded in the right-hand panel) with the variance extracted from experiment as described in Sec. VI C. The unnormalized correlation functions $G^{(2)}(\tau )$ for three detunings (0,0.2, and $0.4\mathrm{\Delta }V$; see markers) are shown in the left-hand panel, together with the averaged result (black). The areas under the curves (shaded in the left-hand panel) enter the Fano factor according to Eq. (4) of the main text (other parameters are $k=3$, $E_J=0.9E_J^{\mathrm{sat}}=0.07/h\nu$).

Figure 11

Effects of voltage noise on the Fano factor in the weak driving limit. Dependence of the Fano factor on the effective driving strength $E_J/h\nu$ for the resonances $k=1,2,\dots ,6$ simulated (for zero nominal detuning) without noise (dashed lines), and including noise (solid lines) as described by Fig. 10).

Figure 12

Strong emission at multiphoton resonances, $k=1$, 2, 3. Scaled occupation $\alpha ⟨n⟩$ (left) and Fano factor $F$ of photon emission (right). Beyond the weak driving limit, the nonlinear dynamics is complex and shows two transitions (for $k\ge 2$): a saturation transition at $E_J/E_J^{\mathrm{sat}}\equiv 1$ and a parametric transition at lower $E_J$ (see Ref. [14] for derivations and details). While transitions are nonanalytic in the semiclassical limit, $\alpha \to 0$ (dashed), higher values of $\alpha$ progressively smoothen them and yield emission below the parametric threshold according to a rate picture of dynamical Coulomb blockade [cf. Eq. (3) and Fig. 2 of the main text]. The Fano factor starting form the naive bunch size, $F=k$, goes through a peak around the parametric threshold and a dip-peak structure around the saturation transition to settle at $F=1$ for very strong driving. The ranges of drivings shown in Fig. 3 of the main text are marked by arrows.

Figure 13

Simulated intraresonator field Wigner function with the parameters of run 2, at $E_J/h{\nu }_R=0.142$. Amplitude of the Wigner function as a function of the two reduced quadratures of the field, for $k=1–6$. Note that the color scale is bounded by the minimal value 0 indicating the absence of predicted Wigner negativity. The text in each panel indicates the maximal value of the color scale.