Exact Duality at Low Energy in a Josephson Tunnel Junction Coupled to a Transmission Line

TL;DR

The paper proves an exact low-energy charge–flux duality for a Josephson junction coupled to a finite-length transmission line, showing that the charge-biased and flux-biased circuits share identical spectra under a well-defined dual map of parameters. By deriving Hamiltonians in two gauges, reformulating in a polaron frame, and performing exact diagonalization, the authors show a length-dependent but exact duality that collapses to self-duality at the critical impedance $Z=R_q$ and to a resistively shunted Josephson junction in the infinite-line limit. The duality persists across the full range of $E_J/E_C$, with a conformal-field-theory description of the critical spectrum via mobility parameters and a dual relation $ar{μ}=1-μ$, enabling precise mapping between the two circuits. These insights underpin a robust framework for understanding superconducting–insulating transitions beyond perturbative limits and suggest experimental tests using photonic environmental modes to probe duality signatures. The work also lays groundwork for generalizing exact duality concepts to more complex superconducting circuits and phase transitions.

Abstract

We theoretically explore the low-energy behavior of a Josephson tunnel junction coupled to a finite-length, charge-biased transmission line and compare it to its flux-biased counterpart. For transmission lines of increasing length, we show that the low-energy charge-dependent energy bands of the charge-biased configuration can be exactly mapped onto those of the flux-biased system via a well-defined duality transformation of circuit parameters. In the limit of an infinitely long transmission line, the influence of boundary conditions vanishes, and both circuits reduce to a resistively shunted Josephson junction. This convergence reveals the system's intrinsic self-duality and critical behavior. Our exact formulation of charge-flux duality provides a foundation for generalizations to more complex superconductor-insulator phase transitions.

Figures (12)

• Figure 1: (a) Charge circuit: a Josephson tunnel junction with capacitive energy $E_C=e^2/2C_J$ and Josephson energy $E_J$, coupled to a finite-length transmission line characterized by inductances $L$, capacitances $C$ and number of modes $N_m$ or, equivalently, by impedance $Z=\sqrt{L/C}$, cutoff frequency $\omega_c=2/\sqrt{LC}$, and low energy mode spacing $\Delta=\pi/(N_m\sqrt{LC})$. The circuit includes a superconducting island and can be biased via gate charge $\nu = Q_{ext}/2e = V_g C_g / 2e$. (b) Flux circuit: a shorted transmission line without a superconducting island, forming a superconducting loop, that can biased by an external magnetic flux $\phi = \Phi_{ext}/\Phi_0$, where $\Phi_0$ is the flux quantum for a Cooper pair. (c) Table summarizing the parameter correspondences between the two circuits, such that that their low-energy spectra—specifically, energy band dispersions vs. normalized bias—are identical under the dual mapping. The transformation $\mathcal{F}_{\omega_c,Z,\Delta}$ of $E_J$ is discussed in the main text. (d) In the infinite-length limit, boundary conditions become irrelevant and both circuits reduce to a resistively shunted Josephson junction. (e) Schmid phase diagram showing insulating (I) and superconducting (S) phases. The two heatmaps show parameters $\mu$ and $\bar{\mu}$ that characterize the spectra of the two circuits (defined later in the text) for $\hbar\omega_c=4E_C$ and $\hbar\Delta=E_C$. Points at dual impedances with the same value of these parameters have the same low-energy spectrum, and are hence dual. Arrows indicate this duality transformation for three couples of points.
• Figure 2: Top: Rescaled energy bands of the charge circuit [Fig. 1(a)] vs. gate charge $\nu$, shown for different system lengths (mode spacing $\Delta$) and various $R_q/Z$ values. Here $\widetilde{E}^{ch}(Z,\nu)=E^{ch}(Z,\nu)/E^{ch}_3(R_q,0)$. Bottom: Rescaled energy bands of the flux circuit [Fig. 1(b)] vs. normalized external flux $\phi$, for the same inverse ratios $\bar{Z}/R_q$. Here $\widetilde{E}^{fl}(\bar{Z},\phi)=E^{fl}(\bar{Z},\phi)/E^{fl}_3(R_q,0)$. Parameters: $E_J = E_C$, $\bar{E}_J = E_J$, $\bar{E}_C = E_C$, $\bar{\omega}_c = \omega_c = 4E_C/\hbar$. The four transmission lines considered have lengths of 5, 7, 10, and 16 $LC$ unit cells, respectively.
• Figure 3: (a) Band spectrum of the charge circuit at the critical point $Z = R_q$ for various $E_J/E_C$ values (different colors). (b) Same for the flux circuit at $\bar{Z} = R_q$. Thin dashed white lines show the analytical prediction \ref{['eq:cft-spectrum']}. Parameters: $\hbar \Delta = 0.66 E_C$, $\hbar \omega_c = 4 E_C$. (c) Mobility at criticality vs. $E_J/E_C$ for both circuits. For the flux case, we plot $1 - \bar{\mu}$ to highlight the exact duality between mobilities. (d) Critical self-dual transformation $\mathcal{F}_{\omega_c, R_q}(E_J)$ preserving the band shape. The thick red line is from numerics at $\hbar \omega_c = 4 E_C$; the thin grey line interpolates the $\omega_c \to \infty$ results from lukyanov2007resistively. Intersections with the unit-slope line give the duality center $E_J^*$, where $\mathcal{F}_{\omega_c, R_q}(E_J^*) = E_J^*$ and the spectra coincide.
• Figure 4: Photonic spectral function versus the Josephson-to-charging energy ratio for the two circuits, shown for three impedance values. Top row: results for the charge circuit. Bottom row: results for the flux circuit at the corresponding dual impedances, plotted versus the Josephson energy transformed according to the critical duality relation. Thin dashed orange lines indicate the bare mode frequencies entering Hamiltonians \ref{['eq:charge-hamiltonian']} and \ref{['eq:dual-hamiltonian']}. Parameters: $\bar{\omega}_c = \omega_c = 10E_C/\hbar$, $\bar{\Delta} = \Delta = 0.5E_C/\hbar$.
• Figure 5: Photonic spectral function for the flux circuit, plotted as a function of the original Josephson energy, complementing the results shown in the second panel. Thin dashed orange lines indicate the bare mode frequencies entering Hamiltonians \ref{['eq:charge-hamiltonian']} and \ref{['eq:dual-hamiltonian']}. Parameters: $\bar{\omega}_c = \omega_c = 10E_C/\hbar$, $\bar{\Delta} = \Delta = 0.5E_C/\hbar$.