Inelastic Scattering of a Photon by a Quantum Phase Slip

Spontaneous decay of a single photon is a notoriously inefficient process in nature irrespective of the frequency range. We report that a quantum phase-slip fluctuation in high-impedance superconducting waveguides can split a single incident microwave photon into a large number of lower-energy photons with a near unit probability. The underlying inelastic photon-photon interaction has no analogs in nonlinear optics. Instead, the measured decay rates are explained without adjustable parameters in the framework of a new model of a quantum impurity in a Luttinger liquid. Our result connects circuit quantum electrodynamics to critical phenomena in two-dimensional boundary quantum field theories, important in the physics of strongly correlated systems. The photon lifetime data represent a rare example of verified and useful quantum many-body simulation.

Figure 1

(a) Schematic of a telegraph transmission line terminated by a Josephson junction at the right end and weakly coupled to a measurement port at the left end. Device photographs and microwave setup are shown in Refs. [14, 17]. The quantum field $\phi (x,t)$ represents the superconducting phase difference between the two wires of the line. The lower panel illustrates an inelastic scattering process that splits an incident single photon at $x=0$ in one resonant photon and even number of low-frequency photons. (b) Quantum tunneling of the boundary variable $\phi (x=0)$ in the periodic Josephson potential renders the energy of the first excited level sensitive to quantum fluctuations of the dynamical charge $q$ at the $x=0$ end of the line. The sensitivity of the ground state can be neglected. (c) The measured positions of standing wave resonances as a function of flux through the split-junction loop in device 3a [18]. The color shows probability to lose a single photon in one round-trip time.

Figure 2

The elastic (top) and inelastic parts (bottom) of the reflection amplitude $r(\omega )$ for the devices with progressively larger charging energy $E_C$ indicated on the plots. In each device, the flux $\mathrm{\Phi }$ is tuned such that ${\omega }_0/2\pi \approx 6.5–7.5\mathrm{GHz}$. The blue markers show data at $\mathrm{\Phi }=0$, where the impurity is effectively switched off. The dashed line represents the background dielectric loss inside the transmission line. Device parameters are given in Table S1 of the Supplemental Material [18].

Figure 3

Inelastic scattering rate ${\gamma }_{\mathrm{in}}({\omega }_0)/\mathrm{\Delta }$ (colored markers) for devices with $\alpha >1$ (left panel) and $\alpha <1$ (right panel). The width of theory lines (colored bands) comes from uncertainty in the device parameters. The error bars are the standard errors of ${\gamma }_{\mathrm{in}}/\mathrm{\Delta }$ at the resonance. The color code represents nominally identical values of $E_C$.

Figure 4

An example of the many-body states satisfying energy conservation condition for the decay of the mode $k=47$ in device 2a (${\omega }_0\approx {\omega }_{47}$). The many-body spectrum (right) is obtained by summing all possible combinations of three (blue) and five (green) one-photon frequencies, measured experimentally (left). Each bar’s height indicate the one-photon amplitudes $f_k$ (left; see text) and the relative amplitudes of $f_i{f}_j{f}_k$ and $f_i{f}_j{f}_k{f}_l{f}_m$ of three-photon and five-photon states, respectively. The frequency range in the right panel is equal to the measured half-linewidth of the $k=47$ mode. The visualization in the central panel illustrates the composition of the multiphoton states from the measured one-photon spectrum. Note the higher weight of the decay channels ${\omega }_{47}\to {\omega }_{46}+2{\omega }_1$ and ${\omega }_{47}\to {\omega }_{45}+4{\omega }_1$ involving the lowest frequency mode at ${\omega }_1/2\pi =63\mathrm{MHz}$.