Inelastic Decay from Integrability

A hallmark of integrable systems is the purely elastic scattering of their excitations. Such systems possess an extensive number of locally conserved charges, leading to the conservation of the number of scattered excitations, as well as their set of individual momenta. In this work, we show that inelastic decay can nevertheless be observed in circuit-QED realizations of integrable boundary models. We consider the scattering of microwave photons off impurities in superconducting circuits implementing the boundary sine-Gordon and Kondo models, which are both integrable. We show that not only is inelastic decay possible for the microwave photons, in spite of integrability, and due to a nonlinear relation between them and the elastically scattered excitations, but also that integrability in fact provides powerful analytical tools allowing us to obtain exact expressions for response functions describing the inelastic decay. Using the framework of form factors, we calculate the total inelastic decay rate and elastic phase shift of the microwave photons, extracted from a two-point response function. We then go beyond linear response and obtain the exact energy-resolved inelastic decay spectrum, using a novel method to evaluate form-factor expansions of three-point response functions, which could prove useful in other applications of integrable quantum field theories. Our results could be relevant to several recent photon-splitting experiments and, in particular, to recent experimental works that provide evidence for the elusive Schmid-Bulgadaev dissipative quantum phase transition.

Popular Summary

Integrability has become synonymous with purely elastic scattering. Integrable systems are defined by a set of quasiparticles that, because of a particularly large set of symmetries, maintain their individual energies and momenta in any collision process. A particular class of such systems, integrable quantum impurity models, that are composed of a waveguide coupled to an atomlike object, provide a great platform to probe those collisions directly. Current experiments have studied the scattering of single microwave photons off impurities in nonintegrable models, showing they can be split into multiple photons at a high rate; in integrable impurity models, one might be tempted to conclude that any trace of such inelastic behavior should be abolished by the strong symmetries. Strikingly, we explicitly show how the purely elastic scattering of the quasiparticles builds up an inelastic scattering of the photon.

This apparently contradictory result is fundamentally enabled by a nonlinear relation between the photon and the quasiparticles. In fact, we use this nonlinear relation, which is encoded in the matrix elements of the bosonic (photonic) operator in the basis of the quasiparticles, to calculate the exact scattering rates, both elastic and inelastic, of a single incoming photon. We then devise a novel method for the calculation of multipoint correlation functions in integrable systems and use it to obtain the exact distribution of the outgoing photons. The rates provide a much-needed quantitative description of ongoing experiments and in particular reveal the emergence of the elusive Schmid-Bulgadaev superconductor-to-insulator quantum phase transition, which has been the center of intense debate over the past few years.

Our findings, as well as our technical achievements, should pave the way for tackling other integrable systems, from cold atoms and strongly correlated electrons to high-energy physics.

Figure 1

(a) An incoming photon with frequency $\omega$, injected from the antenna on the left and propagating through the transmission line in the center, may decay inelastically as it scatters off the impurity on the right, in spite of the purely elastic reflection of the fundamental excitations of the integrable models. Each photon may be represented as a combination of eigenstates composed of excitations ${|\overrightarrow{\lambda }⟩}_{{ϵ}_{\lambda }}$ with types ${ϵ}_{\lambda }$ and rapidities $\overrightarrow{\lambda }$, where the weights are determined by the form factors, $f_{{ϵ}_{\lambda }}(\overrightarrow{\lambda })$, and the total energy of the excitations in each eigenstate, ${\nu }_{\lambda }\sim \sum _i{e}^{{\lambda }_i}$, is equal to the photon frequency $\omega$ (for notation and definitions, see Sec. 3). The excitations scatter elastically off the boundary (quantum impurity), picking up phases determined by the reflection matrix ${\mathsf{R}}_{{ϵ}_{\lambda }}^{{ϵ}_{\lambda }^{\mathrm{\prime }}}(\overrightarrow{\lambda })$, such that the outgoing combination no longer represents a single-photon state but, rather, a multiphoton state. The measured observables—the total inelastic decay rate $\gamma \left(\omega \right)$ and the energy-resolved inelastic decay spectrum $\gamma ({\omega }^{\mathrm{\prime }}|\omega )$, as well as the elastic phase shift $\delta \left(\omega \right)$ (not depicted here)—all shed light on the fundamental properties of the impurity models. (b) Implementation of Eq. (1) with the bsG and Kondo impurities [Eqs. (3) and (5), respectively] in a cQED setup. The array of Josephson junctions and capacitors implements a high-impedance transmission line, due to the kinetic inductance of the Josephson junctions.

Figure 2

The total inelastic rate $\gamma \left(\omega \right)$ and phase shift $\delta \left(\omega \right)$ for the bsG and Kondo models and several values of $z$. The power laws at low and high frequencies are denoted on the plots. In the phase-shift panels, we plot both $\delta \left(\omega \right)$ and $\pi /2-\delta \left(\omega \right)$ or $\pi -\delta \left(\omega \right)$ (for the bsG and Kondo models, respectively). We use Eq. (48) to evaluate $r\left(\omega \right)$ at integer $p=1/z$ and $r\left(\omega \right)=r_{+-}\left(\omega \right)$ at noninteger $z>1/2$.

Figure 3

(a) The energy-resolved inelastic decay spectrum as a function of ${\omega }^{\mathrm{\prime }}$, at several fixed values of $\omega /T_B$, for the bsG and Kondo models and $z=1/3,1/2$. The diagrams used to evaluate the spectrum are listed in Appendix pp6. (b) The ratio between the left-hand side and right-hand side of Eq. (53) for both models and several values of $z$. Note that the power laws of $\gamma \left(\omega \right)$ are recovered by the sum rule for all $z$.

Figure 4

The contributions of $f_m^{\mathcal{R}},f_{+-}^{\mathcal{R}},f_{12}^{\mathcal{R}},f_{+-1}^{\mathcal{R}}$ (the latter evaluated only for integer $p=1/z$) to the reflection coefficient of the free theory, $r^0=1$, given in Eqs. (44)–(47) with $\omega \to \mathrm{\infty }$. Here, $\sum _{{ϵ}_{\lambda }}^{\mathrm{\prime }}{r}_{{ϵ}_{\lambda }}^0=\sum _{m=1}^{\lceil 1/z\rceil -2}{r}_m^0+r_{+-}^0+r_{12}^0+r_{+-1}^0$; the dips of $1-\sum _{{ϵ}_{\lambda }}^{\mathrm{\prime }}{r}_{{ϵ}_{\lambda }}^0$ are at integer $p=1/z$, where $r_{+-1}^0$ is evaluated.

Figure 5

The weight of the contributions to the $z=1/3$ spectrum, evaluated using the sum rule.