Nonclassical Photon Pair Production in a Voltage-Biased Josephson Junction

We investigate electromagnetic radiation emitted by a small voltage-biased Josephson junction connected to a superconducting transmission line. At frequencies below the well-known emission peak at the Josephson frequency ($2eV/h$), extra radiation is triggered by quantum fluctuations in the transmission line. For weak tunneling couplings and typical Ohmic transmission lines, the corresponding photon-flux spectrum is symmetric around half the Josephson frequency, indicating that the photons are predominately created in pairs. By establishing an input-output formalism for the microwave field in the transmission line, we give further evidence for this nonclassical photon pair production, demonstrating that it violates the classical Cauchy-Schwarz inequality for two-mode flux cross correlations. In connection to recent experiments, we also consider a stepped transmission line, where resonances increase the signal-to-noise ratio.

Figure 1

(a) A biased superconducting transmission line terminated by a small JJ. We consider an Ohmic transmission line ($Z_0=Z_1$) and one with a steplike characteristic impedance ($Z_1\gg Z_0$), supporting the visualized modes of a $\lambda /4$ resonator. (b) The equivalent lumped-element model. The transmission line is characterized by its capacitance $C_0$ and inductance $L_0$ per unit length ($Z_i=\sqrt{L_i/C_i}$). (c) The emitted radiation, either single-photon ($I$) or two-photon ($II$) emission, originates from the electrostatic energy released by a tunneling Cooper pair $2eV$. (d) The output photon-flux density as obtained from the classical treatment [$I$, $k_{B}T{f}_t(\omega )/\hslash ({\omega }_J-\omega )$], from inelastic Cooper-pair tunneling $f_t(\omega )$ [$II$; see Eq. (11)], and from thermal radiation $f_{th}(\omega )$. The first two differ in emission at frequencies near the half Josephson frequency ${\omega }_J/2$, around which $f_t(\omega )$ is symmetric. This indicates emission due to pair production of photons. We use $k_{B}T/2eV=0.017$ and $Z_0{I}_C/V=0.024$.

Figure 2

(a) Violation of classical CS inequality for flux cross correlations between frequencies ${\omega }_a$ and ${\omega }_b$. Negative values are a sign of nonclassicality. The photon pair production is behind the strong violation near the diagonal ${\omega }_a+{\omega }_b={\omega }_J=50\mu \mathrm{eV}/\hslash$. We plot here $\mathcal{V}/|\mathrm{Max}\{\mathcal{V}\}|$, where $\mathcal{V}=\mathrm{Sgn}(v)\ln (1+|v|)$ and $v/(\mu \mathrm{eV}{)}^2=P[\hslash ({\omega }_J-2{\omega }_a)]P[\hslash ({\omega }_J-2{\omega }_b)]-P[\hslash ({\omega }_J-{\omega }_a-{\omega }_b){]}^2$. We use $Z_0=Z_1=50\Omega$, $C_J=10\mathrm{fF}$, and $T=100\mathrm{mK}$. (b) Photon-flux density $f_t(\omega )$ as a function of Josephson frequency ${\omega }_J$ in the neighborhood of ${\omega }_J={\omega }_0+{\omega }_1\approx 140\mu \mathrm{eV}/\hslash$, when $Z_1\to 10Z_0$ and $E_J=(\hslash /2e)I_c=5\mu \mathrm{eV}$. Here, the photon pairs are mainly emitted at ${\omega }_0\approx 35\mu \mathrm{eV}/\hslash$ and at ${\omega }_1\approx 3{\omega }_0$. (c) The measured violation of the CS inequality $-v{f}_t({\omega }_a)f_t({\omega }_b)$ in the diagonal ${\omega }_J={\omega }_a+{\omega }_b$ of (a). (d) The measured violation corresponding to (b) when ${\omega }_a+{\omega }_b={\omega }_0+{\omega }_1={\omega }_J$. The resonance occurs when ${\omega }_a={\omega }_0$.