Quantum optical impurity models in interacting waveguide QED

This paper investigates interacting waveguide QED systems where the competition between attractive Jaynes-Cummings binding and repulsive Kerr nonlinearities leads to a rich phase diagram featuring Mott-like insulating and superfluid phases in periodic arrays of atomic impurities.

The Gist

Imagine a long, narrow hallway made of glass rooms (cavities) connected by open doorways. Inside this hallway, we have two types of characters: Photons (particles of light) and Atoms (tiny quantum magnets).

In a normal hallway, light just zips through the doorways freely. But in this paper, the authors are studying a very special, crowded hallway where two things happen at once:

1. The Atoms are "Magnetic": They love to grab onto the light and hold it close, like a magnet picking up a paperclip.
2. The Light is "Grumpy": The photons don't like being crowded together. If too many try to squeeze into the same room, they push each other away.

The paper explores what happens when these two forces—the Atoms trying to grab the light and the Light trying to push away its neighbors—fight against each other.

The Main Characters and Their Battle

• The Atoms (The Impurities): Think of these as bouncers standing in specific rooms. They have a strong pull (magnetic force) that wants to trap the light right next to them.
• The Photons (The Guests): These are the light particles moving down the hallway.
• The "Kerr" Effect (The Grumpiness): This is the rule that says, "If two photons try to sit in the same chair, they get angry and push each other." This is the repulsive force.

The Story of the Fight

The authors asked: If we have a hallway full of these bouncers and a bunch of light guests, what will the final arrangement look like?

They found that the answer depends on who is stronger: the bouncer's pull or the guests' grumpiness.

1. The "Holding On" Phase (Bound States)

If the bouncers are very strong and the guests aren't too grumpy, the atoms grab the photons and keep them trapped in their immediate rooms. The photons form little "clumps" or "clouds" around the atoms.

• Analogy: Imagine a group of friends (photons) who are slightly annoyed by each other, but they are all so attracted to a famous celebrity (the atom) that they all crowd around them, ignoring their own discomfort.

2. The "Letting Go" Phase (Detachment)

If the photons get too grumpy (too much repulsion), they can't all fit around the celebrity anymore. Some of them are forced to let go and run down the hallway to find empty rooms.

• Analogy: The celebrity tries to hold a hug party, but the guests are so uncomfortable standing shoulder-to-shoulder that some are forced to leave the circle and walk down the hall.

The Big Picture: The "Phase Diagram"

The authors mapped out a "menu" of possible outcomes for a long hallway with many bouncers. They found two main types of "cities" or states the system can settle into:

A. The Mott Insulator (The Frozen City)
In this state, the light is stuck. The bouncers have grabbed exactly the right number of photons, and the "grumpiness" of the light prevents any movement.

• Analogy: Imagine a parking lot where every spot is perfectly filled, and the cars are so angry at each other that they refuse to move even an inch. The traffic is frozen. Nothing flows. This is an "insulator" because light cannot travel through it.

B. The Superfluid (The Flowing River)
In this state, the bouncers are either too weak to hold the light, or the light is so energetic that it breaks free. The photons move freely down the hallway, flowing past the bouncers.

• Analogy: Imagine a river flowing smoothly. The rocks (bouncers) are there, but the water (light) flows around them effortlessly. The light is "superfluid" because it moves without any friction or resistance.

The Twist: The "Chemical Potential" Trick

One of the coolest discoveries in the paper is a new way to control this system. Usually, to make light flow or stop, you need to add or remove light (like opening a valve).

But here, the authors found that by just tuning the strength of the bouncer's pull (how strongly the atom grabs the light), they could control how many photons stay in the hallway.

• Analogy: Imagine a concert hall where you can't let people in or out through the doors. Instead, you control the crowd size by changing how much the VIPs (atoms) want to hug the fans (photons). If the VIPs hug too tight, the fans get stuck. If they hug loosely, the fans can wander the hall. The "hug strength" acts like a volume knob for the crowd density.

Why Does This Matter?

This isn't just about light and atoms; it's about simulating complex physics.

• Real-world use: Scientists can build these "hallways" using superconducting circuits (like tiny super-fast computers) or cold atoms in lasers.
• The Goal: By playing with these light-atom interactions, they can simulate how electrons behave in solid materials (like why some metals conduct electricity and others don't). It's like using a toy train set to figure out how a real train system works, but with light instead of trains.

Summary

The paper is a study of a tug-of-war between attraction (atoms grabbing light) and repulsion (light pushing light away). Depending on who wins, the system either freezes into a solid, blocked state (Mott Insulator) or flows freely like a river (Superfluid). The authors showed that by adjusting the "grab strength" of the atoms, we can control the flow of light, offering a new tool for building future quantum computers and simulating complex materials.

Technical Summary

Here is a detailed technical summary of the paper "Quantum optical impurity models in interacting waveguide QED" by Misselwitz, Luneau, and Rabl.

1. Problem Statement

The paper addresses the physics of interacting waveguide Quantum Electrodynamics (QED) systems. While standard waveguide QED models focus on the interaction between photons and two-level atoms (TLAs) via the Jaynes-Cummings (JC) coupling, this work introduces a critical generalization: intra-cavity photon-photon interactions modeled by a local Kerr nonlinearity ($U$).

The core problem is to understand the competition between two opposing mechanisms in a 1D coupled-cavity array:

1. Attractive Binding: The JC interaction causes photons to localize around atomic impurities, forming atom-photon bound states (polaritons).
2. Repulsive Interaction: The Kerr nonlinearity ($U > 0$) introduces a repulsive energy cost for multiple photons occupying the same cavity mode.

The authors investigate how this interplay affects the formation of few-photon bound states around single impurities and, more significantly, the resulting many-body ground states in periodic arrays with varying impurity densities and filling factors. The goal is to map the phase diagram of this system, specifically looking for transitions between insulating and superfluid phases.

2. Methodology

The authors employ a multi-faceted approach combining analytical approximations and large-scale numerical simulations:

• Model Hamiltonian: They define a generic lattice Hamiltonian (Eq. 1) describing $L$ cavities with nearest-neighbor hopping ($J$), $N_a$ TLAs coupled to specific sites with strength ($g$), and on-site Kerr repulsion ($U$).
• Single-Impurity Analysis:
  • Exact Diagonalization: Used for small systems to determine the exact energy spectrum and bound state energies.
  • Variational/Single-Site Models: In the limit of vanishing hopping ($J \to 0$), they derive analytical conditions for the maximum number of photons ($n$) that can bind to a single atom. This yields a critical coupling strength $g_b(n)$ required to bind $n$ photons against the repulsion $U$.
  • Effective Hamiltonian: For finite $J$, they construct an effective single-particle tight-binding model where the impurity acts as a site with an energy offset ($\epsilon_N$) and modified hopping ($\alpha_N J$). This model captures the "detachment" transitions where photons are ejected from the impurity as $U$ increases.
• Many-Body Analysis (Periodic Arrays):
  • Effective Polariton Model: They derive a generalized Hamiltonian ($H_{pol}$) treating the bound states as effective polaritons with modified hopping and on-site energies.
  • Numerical Simulations: They utilize Matrix Product States (MPS) and the Density Matrix Renormalization Group (DMRG) algorithm (via the TeNPy package) to simulate large 1D chains (40 unit cells) with open boundary conditions.
  • Order Parameters: To distinguish phases, they calculate:
    • The equal-time photon-photon correlation function $g^{(2)}(0)$.
    • Fluctuations in polariton number ($V_{pol}$) and atomic excitation number ($V_{atom}$).
    • Long-range spatial correlations $\langle a^\dagger_x a_{x+r} \rangle$.
  • Mean-Field Theory: Used to analytically estimate critical hopping strengths ($J_c$) for phase transitions.

3. Key Contributions and Results

A. Single Impurity Physics: Photon Capture and Detachment

• Bound State Limits: The study establishes that for a single impurity, there is a finite maximum number of photons that can be bound. As the photon number $n$ increases, the binding energy per photon decreases.
• Detachment Transitions: When the Kerr repulsion $U$ exceeds a threshold relative to the coupling $g$, photons are "detached" from the impurity. The authors identify critical coupling ratios $g_b(n)$ where the system transitions from an $n$-photon bound state to an $(n-1)$-photon bound state plus a free photon.
• Weak vs. Strong Coupling:
  • In the strong coupling regime ($g \gg J$), the binding threshold is determined by the competition between local JC binding and Kerr repulsion.
  • In the weak coupling regime ($g \ll J$), bound states are delocalized over many sites, reducing the impact of local repulsion. The authors derive a scaling law for the binding threshold: $g_b(N=2) \approx \sqrt{2UJ}$.

B. Many-Body Phase Diagrams

The paper reveals a rich phase diagram dependent on impurity density ($d$, the distance between TLAs) and filling factor ($N$, photons per unit cell).

1. Dense Impurities ($d=1$):

• The system resembles the Jaynes-Cummings-Hubbard (JCH) model but with added Kerr repulsion.
• Mott Insulators: Two distinct types of Mott insulating phases emerge:
  • Mott-I: Dominated by JC nonlinearity (equal superposition of atom/photon states).
  • Mott-II: Dominated by Kerr repulsion (photons distributed between atoms and cavities).
• Superfluid Phase: A transition to a superfluid phase occurs at lower $g/J$ as hopping increases. The presence of $U$ modifies the phase boundaries but does not qualitatively change the transition for unit filling.

2. Dilute Impurities ($d > 1$):

• This regime reveals a more complex structure analogous to the Bose-Hubbard model but driven by the atom-photon coupling $g$ rather than a chemical potential.
• Mott Lobes: The phase diagram features distinct "Mott lobes" corresponding to states where each impurity binds a specific integer number of photons ($n$), while the remaining photons are free.
• Correlation Scaling:
  • Inside Mott lobes, correlations decay exponentially (insulating).
  • In the narrow regions between lobes, the impurities become "transparent" to the unbound photons. Here, the system behaves as a dilute gas of hard-core bosons, exhibiting algebraic decay of correlations ($\sim 1/\sqrt{r}$), characteristic of a 1D superfluid.
• Superfluid Transition: For large $J$, all Mott states melt into a global superfluid phase with long-range order.

C. Effective Chemical Potential

A significant conceptual finding is that the atom-photon coupling strength $g$ acts as an effective chemical potential. By tuning $g$, one can control the "capture" or "release" of photons by the impurities, thereby tuning the density of the free photonic fluid in the waveguide without an external reservoir.

4. Experimental Significance and Implementations

The authors argue that while optical frequencies face challenges with decay rates, the predicted physics is accessible in current quantum simulation platforms:

1. Superconducting Circuits (Circuit QED):
   • Transmons: Can realize the model, though the intrinsic Kerr interaction is attractive ($U<0$). The authors propose a unitary transformation to map this to the repulsive case.
   • Fluxoniums: Can naturally provide repulsive nonlinearities ($U>0$) and well-isolated two-level systems, making them ideal for realizing the repulsive waveguide QED model directly.
2. Ultracold Atoms in Optical Lattices:
   • Using state-dependent optical potentials, atoms in one internal state can act as stationary "impurities" (TLAs), while atoms in another state act as mobile "photons."
   • Raman transitions induce the JC coupling, and Feshbach resonances can tune the interaction strengths to satisfy the required hierarchy ($U_{ab} \to 0$, $U_a \sim g \sim J \ll U_b$).

5. Conclusion

This work extends the theory of waveguide QED by integrating strong photon-photon interactions. It demonstrates that the competition between JC binding and Kerr repulsion leads to a complex landscape of few-body bound states and many-body phases. The identification of Mott lobes controlled by coupling strength and the emergence of algebraically decaying correlations in dilute regimes provide new avenues for quantum simulation. The findings suggest that these systems can serve as versatile analogue simulators for isolated bosonic many-body systems where traditional chemical potentials are difficult to implement.