Dark state spectroscopy in nonlinear waveguide quantum electrodynamics

This paper proposes using weakly squeezed light in nonlinear waveguides to perform spectroscopy on completely dark states in emitter arrays, thereby overcoming the fundamental trade-off between long coherence times and measurability to enable robust quantum technologies.

The Gist

Imagine you have a group of tiny, invisible dancers (quantum emitters) inside a long hallway (a waveguide). These dancers are trying to perform a special routine where they move in perfect sync so that their combined sound cancels itself out completely. In physics, we call this a "dark state."

The problem with being a "dark state" is a classic catch-22:

1. The Good News: Because they cancel each other out perfectly, they don't lose any energy to the hallway. They can hold their pose forever without getting tired (infinite lifetime). This is great for storing information.
2. The Bad News: Because they cancel out perfectly, they are completely silent. If you try to listen to them or take a picture, you can't. They are invisible to standard tools. To see them, scientists usually have to make the dancers slightly imperfect, but that makes them lose their energy quickly.

The Paper's Big Idea: The "Whispering Amplifier"

The researchers (Shay Nadel, Amir Sivan, and Aviv Karnieli) found a clever way to listen to these silent dancers without making them imperfect. They propose using a special "magic hallway" made of nonlinear material.

Here is how their solution works, using a simple analogy:

1. The Setup: The Magic Hallway

Imagine the hallway isn't just a plain tube; it's lined with a special material (like a $\chi^{(2)}$ crystal) that acts like a whispering amplifier.

• Normally, if you shout in a hallway, the sound travels straight through.
• In this magic hallway, if you pump it with a specific "pump" light (twice the frequency of the dancers), the hallway itself starts generating a special kind of "squeezed" light. Think of this as the hallway humming a low, constant background noise that is perfectly tuned to the dancers.

2. The Trick: Breaking the Silence

In a normal hallway, the dancers' perfect cancellation (silence) keeps them hidden. But in this magic hallway, the background hum (the squeezed light) does something unexpected:

• It acts like a matchmaker. It connects the dancers to the hallway in a way they couldn't connect before.
• It breaks the "perfect symmetry" of their dance. Suddenly, the dancers aren't perfectly silent anymore; they start to "talk" to the hallway, but only in very specific, controlled ways.

3. The Result: Hearing the Invisible

Because the dancers are now talking to the hallway, they emit tiny flashes of light (photons) that escape to the end of the hallway.

• The Spectrum: When the scientists measure the color (frequency) of these escaping flashes, they don't just see random noise. They see a specific pattern of peaks.
• The Map: These peaks act like a fingerprint. They reveal the exact energy differences between the different "dark states" the dancers were holding. It's like being able to hear the specific notes of a song that was previously played in total silence.

Why This Matters (According to the Paper)

The paper claims this method allows scientists to:

• Measure the Unmeasurable: You can now look at "completely dark" states without ruining their perfect silence by making them imperfect.
• Switch On and Off: The magic only happens when the "pump" light is on. If you turn the pump off, the dancers go back to being perfectly silent and invisible, preserving their infinite lifetime. If you turn it on, you can read their state.
• Work in Real Life: The researchers checked if this works even if the hallway isn't perfect (if some sound leaks out the walls). They found that even with small leaks, the "fingerprint" of the dark states is still clear enough to be seen.

In Summary:
The paper proposes using a special, non-linear "magic hallway" to gently nudge invisible, perfectly silent quantum dancers just enough so they whisper their secrets to us, without making them lose their super-power of staying silent forever. This opens the door to reading and controlling these hidden states for future quantum computers and memory.

Technical Summary

Technical Summary: Dark State Spectroscopy in Nonlinear Waveguide Quantum Electrodynamics

Problem Statement
Quantum technologies, particularly those relying on many-body entanglement, face a fundamental trade-off: systems must remain decoupled from the environment to preserve long coherence times, yet they require interaction with the environment to be accessible for measurement and control. In Waveguide Quantum Electrodynamics (WQED), arrays of quantum emitters coupled to a one-dimensional waveguide exhibit collective behaviors where destructive interference can suppress radiation into the waveguide, creating "dark states" with theoretically infinite lifetimes. However, this perfect decoupling renders completely dark states invisible to standard optical probes. Current methods to detect subradiant states rely on system non-idealities (such as imperfect emitter spacing or free-space leakage) that break perfect interference, or on indirect measurements that limit coherence times. Completely dark states remain fundamentally hidden in linear waveguide setups.

Methodology
The authors propose a solution utilizing a $\chi^{(2)}$ nonlinear waveguide acting as a traveling-wave parametric amplifier. The system consists of an array of $N$ identical two-level emitters spaced by a wavelength ($\lambda$) within a nonlinear section of length $L$. This section is pumped by coherent light at frequency $2\omega$ (twice the emitter transition frequency), generating weakly squeezed vacuum modes at frequency $\omega$ within the waveguide.

The theoretical framework employs the following steps:

1. Input-Output Theory: The authors derive input-output relations for the photonic field propagating through the nonlinear medium containing the emitter array. This connects the temporal correlations of the output light to the emitter correlations.
2. System Dynamics: Using the SLH (Schrödinger-Lindblad-Hamiltonian) formalism, they derive the interaction Hamiltonian ($H_{\text{int}}$) and jump operators ($L_s$) governing the emitter array. The interaction Hamiltonian describes long-range interactions mediated by the parametric gain, where the interaction strength increases with the distance between emitters. Crucially, unlike standard WQED, this Hamiltonian involves the creation and annihilation of excitation pairs ($\sigma_i \sigma_j + \sigma_i^\dagger \sigma_j^\dagger$).
3. Spectroscopic Analysis: The authors calculate the steady-state incoherent power spectrum of the emitted light. They project the dynamics onto the dark-state subspace, identifying that the squeezed vacuum induces transitions between dressed dark eigenstates.

Key Contributions

• Breaking Permutation Symmetry: The work demonstrates that the parametric gain from the nonlinear pump breaks the permutation symmetry of the emitter array. This allows the system to couple to the waveguide modes even when the emitters are perfectly spaced, a condition where linear waveguides would yield zero coupling for dark states.
• Selection Rules: The authors identify a strict $j$-parity selection rule. The jump operators, which are linear combinations of individual emitter operators, permit transitions exclusively between dark eigenstates possessing different $j$-parity (even vs. odd).
• Spectroscopic Readout: The paper establishes that the incoherent fluorescence spectrum of the system directly encodes the transition frequencies between the dressed dark eigenstates. This provides a method to spectroscopically probe completely dark states without relying on system non-idealities.

Results

• Ideal Systems ($N=4, 6$): Numerical simulations for ideal systems (where free-space decay $\gamma_0 = 0$) show that the incoherent power spectrum exhibits distinct peaks corresponding exactly to the allowed transition frequencies between dark eigenstates. For $N=4$, the spectrum resembles a Mollow triplet, while for $N=6$, the spectrum becomes significantly richer with 21 distinct parity-allowed transition frequencies.
• Effect of Squeezing: Increasing the squeezing parameter $r$ broadens the spectral peaks and separates them further, though it may reduce the distinctness of overlapping features in larger systems.
• Non-Ideal Systems: The authors analyze the impact of free-space decay ($\gamma_0$), characterized by the coupling efficiency $\beta = \gamma/(\gamma + \gamma_0)$. Even with realistic coupling efficiencies (e.g., $\beta = 0.995$), the spectral signatures of dark-manifold transitions remain highly distinguishable. While free-space decay introduces peak broadening and Fano-like frequency shifts, the core spectroscopic capability is preserved.

Significance and Claims
The paper claims to pave the way for the measurement and control of completely dark states in WQED. By utilizing weakly squeezed light generated in $\chi^{(2)}$ nonlinear waveguides, the authors propose a mechanism to access the decoherence-free manifold of dark states. The authors suggest that this approach enables:

• Robust Quantum Memories and Computation: Accessing these states allows for the manipulation of many-body entangled states that are naturally protected from decoherence.
• Precise Metrology: The ability to probe these states could enhance precision measurement techniques.
• Subradiance-Protected Communication: Pushing the limits of communication protocols that rely on the protection offered by subradiance.

The work suggests that the underlying mechanism could be realized experimentally using superconducting circuits (e.g., transmon qubits coupled to 1D microwave transmission lines) and Josephson junction arrays acting as nonlinear metamaterials, which can provide the necessary distributed parametric gain and $\chi^{(2)}$ mixing. The authors emphasize that the pump acts as an "activation switch," coupling dark states for readout when on and decoupling them to restore infinite lifetimes when off.