From coherent to fermionized microwave photons in a superconducting transmission line

Technical Summary: From Coherent to Fermionized Microwave Photons in a Superconducting Transmission Line

Problem Statement
While quantum fluids of light have emerged as a promising platform for studying many-body physics, most existing experiments operate in weak interaction regimes or are confined to cavity configurations. These cavity-based systems are restricted by small spatial sizes, limited photon lifetimes, and driven-dissipative dynamics involving pumping and losses. Consequently, evidence of a strongly interacting photon gas close to its many-body ground state, exhibiting collective behaviors such as fermionization, has remained elusive. There is a need for a propagating geometry that allows for the conservative dynamics of a macroscopically sized many-body system with strong photon-photon interactions.

Methodology
The authors propose utilizing superconducting nonlinear transmission lines based on Josephson junctions (JJs) as a novel platform. The methodology involves:

1. Theoretical Modeling: The system is modeled using a Lagrangian for a transmission line composed of unit cells with capacitors to ground ($C_0$) and series JJs (characterized by critical current $I_c$ and capacitance $C_J$). By expanding the cosine term of the JJ potential to the fourth order and applying the slowly-varying envelope approximation (SVEA), the authors derive an effective action.
2. Mapping to Bose Gas: Through a change of variables that exchanges the roles of spatial and temporal coordinates ($\zeta = t\bar{v}_g$ and $\tau = z/\bar{v}_g$), the propagation of microwave photons is mapped onto the Hamiltonian evolution of a one-dimensional Bose gas with contact interactions. This results in a Lieb-Liniger (LL) model.
3. Adiabatic Protocol: The authors propose a scheme where the transmission line parameters are tapered (slowly varied) along the propagation direction. This adiabatic ramp is designed to convert an incident monochromatic coherent field into a Tonks-Girardeau (TG) gas of impenetrable photons (the $\gamma \to \infty$ limit of the LL model) close to its ground state.
4. Numerical Simulation: Due to the gapless nature of the LL model, which complicates the direct application of the adiabatic theorem, the authors employ $\tau$-dependent infinite tensor-network (TN) algorithms to numerically simulate the full many-body problem of microwave propagation.

Key Contributions and Results

• Strong Interaction Regime: The study predicts that the intrinsic nonlinearity of Josephson junctions is sufficient to enter a regime of strongly interacting photons for realistic circuit parameters.
• Fermionization Signatures: The primary signature of the many-body ground state is identified in the second-order correlation function ($g_2$) of the transmitted field. The results show:
  • Antibunching: Visible at short time delays, consistent with a two-photon blockade mechanism.
  • Friedel Oscillations: Distinct oscillations at larger time delays, which are a true many-body effect resulting from the quantum-correlated nature of the ground state. The frequency of these oscillations is set by the photon flux (corresponding to the Fermi wave vector), distinguishing them from Rabi-frequency-driven oscillations in resonance fluorescence.
  • Power-Law Decay: The first-order correlation function ($g_1$) exhibits a power-law decay at large time delays, characteristic of the TG ground state.
• Adiabatic Preparation: Numerical simulations demonstrate that for sufficiently long transmission lines, the adiabatic ramp successfully prepares the system in a state with correlation properties matching the exact ground state of the TG gas (with interaction parameter $\gamma \approx 33.5$).
• Light-Cone Propagation: For faster ramps (non-adiabatic regimes), the propagation of excitations follows a light-cone structure. The speed of this expansion is determined by the Lieb-Liniger model ($c_{LL}$) rather than weak-interaction fluid dynamics. Numerical extraction of this speed agrees with theoretical predictions within 2.3%.

Significance and Claims
The paper claims to provide a theoretical framework for realizing a quantum gas of strongly interacting microwave photons close to its ground state in a propagating geometry. The significance of this work lies in:

• Overcoming Spatial Limitations: Unlike cavity configurations, the continuous-wave operation of the propagating scheme allows for the realization of gases with a macroscopic number of photons, overcoming limitations on spatial size.
• Ground State Access: It proposes a method to access the many-body ground state of a strongly correlated system, a regime previously difficult to reach in optical or microwave photonics.
• Platform Versatility: While focused on circuit-QED, the predictions are claimed to extend to generic strongly nonlinear waveguide systems in both microwave and optical regimes, including polaritons and Rydberg-EIT configurations.
• Future Directions: The authors suggest that this approach could enable the exploration of degenerate gases of multiphoton bound states under attractive interactions and strongly interacting Bose gases in multi-waveguide geometries with synthetic magnetic fields, potentially leading to two-dimensional fractional quantum Hall liquids of light.

The work remains modest regarding experimental realization, noting that approaching the TG ground state imposes serious fabrication challenges, particularly regarding the length of the transmission line required for adiabaticity. However, it asserts that signatures of strong quantum correlations are visible even in moderately short lines.