Bright Side of the Coulomb Blockade

We explore the photonic (bright) side of the dynamical Coulomb blockade (DCB) by measuring the radiation emitted by a dc voltage-biased Josephson junction embedded in a microwave resonator. In this regime Cooper pair tunneling is inelastic and associated with the transfer of an energy $2eV$ into the resonator modes. We have measured simultaneously the Cooper pair current and the photon emission rate at the resonance frequency of the resonator. Our results show two regimes, in which each tunneling Cooper pair emits either one or two photons into the resonator. The spectral properties of the emitted radiation are accounted for by an extension to DCB theory.

Figure 1

The dynamical Coulomb blockade of Josephson tunneling. (a) Generic circuit: a dc voltage-biased Josephson element in series with an electromagnetic environment with impedance $Z(\nu )$, which can be described as a collection of harmonic oscillators. Below the gap voltage, the energy $2eV$ of a tunneling Cooper pair is transferred to the electromagnetic environment in form of one or several photons in its modes. (b) Experimental setup: The sample consists of a SQUID, working as a tunable Josephson junction, in series with a microwave resonator, forming the electromagnetic environment and consisting of two quarter-wave transformers. A bias tee separates the low-frequency voltage bias $V$ and current measurement $I$ from the emitted microwave power $P_{\mu w}$. Three circulators (only one shown) ensure thermalization of the microwave environment. The emitted power is amplified at 4.2 K and then band pass filtered and detected at room temperature.

Figure 2

Fundamental mode ${\nu }_0$ of the resonator and corresponding Cooper pair and photon rates. Top panel: real part of the impedance seen by the junction, calculated from the resonator geometry (black line) and reconstructed (magenta) from a quasiparticle shot noise measurement [10]. Bottom panel: measured Cooper pair rate ${\Gamma }^{\mathrm{Cp}}$ (red) and photon rate ${\Gamma }_0^{\mathrm{ph}}$ (blue) integrated from 5 to 7 GHz. Both rates show a peak around $V=h{\nu }_0/2e\simeq 12\mu \mathrm{V}$. A second small peak in the microwave power at $V=24\mu \mathrm{V}$ (vertical zoom in inset) corresponds to two-photon processes. Solid lines are $P(E)$ theory fits using the reconstructed impedance from 5 to 7 GHz and the calculated impedance elsewhere, an effective temperature of 60 mK, and a single adjustable parameter $E_J=5.1\mu \mathrm{eV}$. Cyan and yellow lines correspond, respectively, to Eq. (1, 5) integrated from 5 to 7 GHz.

Figure 3

Harmonics ${\nu }_n$ of the resonator and corresponding Cooper pair and photon rates. The plotted quantities are the same as in Fig. 2, but for a larger voltage and frequency range and different Josephson coupling. The Cooper pair rate shows peaks at voltages $V=h{\nu }_n/2e$, associated with the emission of photons in the resonator mode at frequency ${\nu }_n$. The power measurement only probes the photons emitted in the fundamental mode of the resonator. It shows a large peak at $V=h{\nu }_0/2e$, and smaller ones at $V=h({\nu }_0+{\nu }_n)/2e$, due to simultaneous emission of one photon at ${\nu }_0$ and one photon at ${\nu }_n$ per tunneling Cooper pair. Here theoretical curves are obtained from the calculated impedance and the fitting parameter $E_J=10.3\mu \mathrm{eV}$.

Figure 4

Main panel: spectral density of the photon emission rate $\gamma$ for the first and second order peaks (${\nu }_0$ and $2{\nu }_0$) as a function of bias voltage and frequency. Data were taken for different Josephson energies and were normalized by the maximum $\gamma$ of each data set. Dotted lines correspond to $2eV=h\nu$ and $2eV=h(\nu +{\nu }_0)$. White dots correspond to power spectra at fixed voltages (12.5 and $24.5\mu \mathrm{eV}$) and solid lines to theoretical predictions Eq. (5) integrated over 100 MHz, the width of the filter used in this experiment, with effective temperature 60 mK and fitting parameters $E_J=5.0\mu \mathrm{eV}$ at $V=12.5\mu \mathrm{V}$ and $E_J=14.3\mu \mathrm{eV}$ at $V=24.5\mu \mathrm{V}$. Right panel: reconstructed dissipative part of the resonator impedance, as in Fig. 2.