From coherent to fermionized microwave photons in a superconducting transmission line

TL;DR

The work proposes superconducting nonlinear transmission lines as a platform to realize a quantum fluid of strongly interacting microwave photons in a propagating geometry. By tapering line parameters, a monochromatic input is adiabatically converted into a Tonks-Girardeau gas near its ground state, which maps to a 1D Bose gas with Hamiltonian $\hat{H} = \int_{-\infty}^{\infty} d\zeta \left[ -\frac{\hbar^2}{2m}\hat{\Psi}^\dagger (\zeta) \frac{\partial^2}{\partial \zeta^2} \hat{\Psi}(\zeta) + \frac{g}{2}\hat{\Psi}^{\dagger 2} (\zeta) \hat{\Psi}^2(\zeta) \right]$ and LL parameter $\gamma = m g/(\hbar^2\rho)$ with $\rho = \Phi_{\text{ph}}/\bar{v}_g$. Ground-state signatures include antibunching and Friedel oscillations in $g_2^{TG}(\Delta t) = 1-\left(\frac{\sin(\pi \Phi_{\text{ph}} \Delta t)}{\pi \Phi_{\text{ph}} \Delta t}\right)^2$, and a power-law decay of $g_1^{TG}(\Delta t)$. They validate adiabatic preparation with $\tau$-dependent infinite Tensor Network simulations and analyze light-cone propagation with velocity $c_{LL}$ set by the Lieb-Liniger model, demonstrating potential for macroscopic photon gases in a continuous-wave platform.

Abstract

We investigate superconducting transmission lines as a novel platform for realizing a quantum fluid of microwave photons in a propagating geometry. We predict that the strong photon-photon interactions provided by the intrinsic nonlinearity of Josephson junctions are sufficient to enter a regime of strongly interacting photons for realistic parameters. A suitable tapering of the transmission line parameters allows for the adiabatic conversion of an incident coherent field into a Tonks-Girardeau gas of fermionized photons close to its ground state. Signatures of the strong correlations are anticipated in the correlation properties of the transmitted light.

Figures (5)

• Figure 1: (a) Sketch of the transmission line model, composed of unit cells of length $a$ with a capacitor $C_0$ to ground and a series of JJs characterized by a critical current $I_{c,n}$ and a capacitance $C_{J,n}$, whose values are tapered along the propagation. The monochromatic coherent incident radiation, characterized by a flat second order photon correlation function $g_2(\Delta t)$, is converted into a gas of fermionized photons, with a $g_2(\Delta t)$ showing antibunching and Friedel oscillations. (b) Sketch of the linear dispersion relation of the transmission line (solid blue line), together with the quadratic approximation around the carrier frequency $\omega_0$ (dashed black line). The incident monochromatic field is localized in frequency (red dot), whereas the fermionized photon gas after propagation has a finite spread in frequency (orange area).
• Figure 2: Second (a), (c) and first (b) order correlation function of the microwave field transmitted by the transmission line sketched in Fig. \ref{['fig:pictorial']}. The ramp of the circuit parameters is shown in (d), with a constant plasma frequency of $\omega_p=1/\sqrt{L_JC_J}=2\pi\cdot5$ GHz, a constant ground capacitance $C_0 = 30$ fF and an incident coherent field frequency $\omega_0/2\pi = 4.15$ GHz. The incident photon rate is $\Phi_{\text{ph}}=7\cdot10^{7}~s^{-1}$ in (a) and (b) and varies according to the legend in (c). The length $L$ of the transmission line varies according to the legend in (a) and (b), and is $L = 608$ mm for (c). The dashed black line in (a) and (b) is the numerical ground state prediction computed numerically at the final value $\gamma\approx 33.5$, whereas the dashed-dotted line in (c) is the analytical TG result in the $\gamma=\infty$ limit.
• Figure 3: Propagation of correlations after a moderately fast ramp of length $L=2.03$ mm (same ramp parameters as in Fig. \ref{['fig:slow_ramps']}) followed by a uniform region with constant transmission line parameters of total length $\bar{z}$. (a) Color plot of the second order photon correlation function $g_2(\Delta t)$ as a function of $\bar{z}$. The color of the dashed lines correspond to the different curves in (b), the arrow indicates the end of the ramp. The dashed white line indicates motion at the expected LL speed $2c_{LL}$. (b) Plots of the first order correlation function $g_1(\Delta t)$ at different lengths $\bar{z}$. The gray dashed line is an exponential fit of the short time decay of the longest-$\bar{z}$ curve; the horizontal dashed lines are the large $\Delta t$ asymptotes. For each $\bar{z}$, the intersection between these two lines (colored points) defines $\overline{\Delta t}(\bar{z})$. The excellent agreement of the numerically extracted $\overline{\Delta t}(\bar{z})$ (dots) with the theoretically expected light-cone spreading at the LL speed $c_{LL}$ (solid line) is shown in (c).
• Figure 4: Validity of the approximations. First correction to parabolic dispersion (a), variation of the contact interaction parameter (b), and field amplitude (c). Panel (d) shows the phase diagram of the transmission line as in PhysRevB.101.024518, the solid line represents the chosen parameters for the tapered transmission line. Parameters as in Fig. \ref{['fig:slow_ramps']}(a,b) in the main text.
• Figure 5: First (left) and second (right) order correlation functions in the TG limit (gray dashed-dotted line) and for the finite $\gamma$ (black dashed line) used in the text.