Quantum decay rates trigger a phase transition in superradiance

Raphael Holzinger at the University of Innsbruck, Nico S. Bassler at TU Darmstadt, and colleagues have developed a method to model Dicke superradiance with multiple, competing decay channels and to tune the rates of those channels. Their symbolic quantum-trajectory construction yields closed time-domain solutions, predicting the behavior of superradiance as finite combinations of exponential terms regardless of the number of emitters or decay rates involved.

The work unifies existing single-channel dynamics and provides a tool for experiments using engineered reservoirs with multiple decay paths, where the team finds that a ground-state distribution resembles a first-order phase transition when channel-rate ratios equal unity.

Multichannel Dicke Superradiance and Tunable Decay Rates

Researchers have demonstrated a method to predict the behavior of Dicke superradiance, a collective emission of light from multiple excited atoms, when influenced by two or more competing decay pathways, each with independently tunable rates. The work provides closed-form solutions describing the system’s evolution, meaning the populations and observable properties can be calculated at any point in time without complex numerical simulations, regardless of the number of emitting atoms.

The new approach yields solutions expressed as finite combinations of exponential terms, a surprisingly simple mathematical form for a complex quantum system. This scalability is particularly notable; the method applies equally well to systems with a few emitters or those containing a potentially macroscopic number of atoms, and functions with arbitrary decay rates assigned to each channel. This broad applicability stems from the method’s ability to solve the d-channel Dicke master equation, a central equation in understanding collective light-matter interactions.

The researchers detail how these solutions allow for the calculation of emitted power, a key signature of superradiance, using a formula that extends the standard single-channel model to encompass multiple decay pathways. Specifically, a divergence in the system’s susceptibility occurs when the ratio of the two channel decay rates equals unity.

This “winner-takes-all” ground-state selection, as the researchers describe it, is analogous to critical phenomena observed in other driven-dissipative systems, but with the advantage of being controlled by a single, minimal parameter: the ratio of collective decay rates. When decay rates are balanced across all d channels, predictable scaling laws emerge for both the peak time and intensity of the superradiant burst.

The researchers suggest that these findings are particularly relevant to experiments using nanophotonic waveguide quantum electrodynamics, where guided modes can act as permutation-symmetric reservoirs. Polarization or multimode engineering can provide the means to tune the decay rates. Bassler of TU Darmstadt, and colleagues is not merely a theoretical exercise.

The researchers point to the potential for initializing systems near a fully inverted state, or rapidly pumping them and for time-resolved detection of both the superradiant burst and the final ground-manifold distribution. This experimental accessibility is important for validating the predictions of the model and for exploring the full range of behaviors exhibited by multichannel Dicke superradiance.

The team’s work builds on previous efforts to obtain closed-form solutions to the single-channel Dicke problem, extending those solutions to encompass the more realistic scenario of multiple competing decay channels. The method’s ability to provide exact closed solutions for the transient dynamics and stationary state distributions is a significant improvement over previous approaches. In contrast to other methods, the symbolic trajectory construction provides full solutions for any number of emitters, a capability not found in earlier references.

The researchers detail how the system’s evolution can be understood as a series of jumps between states, governed by non-Hermitian dynamics and characterized by a recursive rate equation. This equation, they demonstrate, can be solved analytically, yielding the closed-form solutions that are the hallmark of their approach. The resulting “winner-takes-all” ground-state selection resembles a dissipative many-body phase transition, with a susceptibility that diverges logarithmically with the number of emitters.

Symbolic Quantum-Trajectory Method for Closed-Form Solutions

The ability to predict the complete time-dependent behavior of Dicke superradiance, even with multiple competing decay pathways, now rests on a symbolic quantum-trajectory construction yielding closed-form solutions for system populations and observables. This analytical approach circumvents limitations of previous methods by providing exact solutions applicable to any number of emitters and decay rates.

The resulting expressions are finite combinations of exponential terms, simplifying the complex dynamics of light-matter interactions and offering a new tool for both theoretical and experimental investigations. This method distinguishes itself by directly solving the master equation governing d-channel Dicke superradiance, delivering not only transient dynamics but also precise stationary state distributions.

These scaling laws provide a quantitative understanding of how the superradiant signal changes with the number of emitters, offering a benchmark for experimental verification. The researchers note that generic symmetry-breaking perturbations may round the large-N nonanalytic behavior, so the present solution provides the permutation-symmetric limiting case for assessing the robustness of this transition.

These experiments often engineer multiple collective decay paths through permutation-symmetric reservoirs, making the ability to model multichannel superradiance important for interpreting experimental results. The method’s analytical nature allows for rapid prediction of system behavior, facilitating the design and optimization of experiments aimed at understanding collective light-matter coupling.

Phase Transition in Ground-State Distribution with Balanced Decay

The method reveals that, at this balanced point, the ground-state distribution becomes flat, a stark contrast to the biased distribution expected with unequal decay rates, and a phenomenon not seen in systems with independent decay pathways. For large numbers of emitters, this transition sharpens into a Heaviside step function, a hallmark of first-order phase transitions, as the system’s ground state becomes increasingly sensitive to even infinitesimal imbalances in the decay rates.

This divergence suggests a fundamental connection between seemingly disparate areas of quantum optics and critical phenomena, offering a minimal control knob, the ratio of collective decay rates, to observe this behavior. The symbolic trajectory construction allows for precise calculation of the stationary-state distribution, verifying its agreement with rate equation calculations and revealing a linear expansion around the balanced decay point.

This expansion demonstrates how a small perturbation in the decay-rate ratio amplifies within the ground-state distribution, driving the system towards a defined state. The analysis reveals that the stationary state distribution can be described by a formula that includes a correction term dependent on the deviation from balanced decay, providing a quantitative measure of the transition’s sharpness. The team’s work extends beyond simply modeling the dynamics. It establishes a direct analogy to dissipative quantum many-body phase transitions.

This “winner-takes-all” ground-state selection, where one decay channel dominates, arises purely from the competition between decay channels, constrained by the requirement that the system remains in its ground state. This discovery has direct implications for experimental implementations of Dicke superradiance in nanophotonic waveguide quantum electrodynamics. Guided modes within these waveguides act as permutation-symmetric reservoirs, providing the necessary conditions to engineer multiple collective decay paths with tunable rates.

The method’s ability to accurately predict the stationary-state distribution, even for large numbers of emitters, provides an important benchmark for experimental verification. The researchers note that the cavity setup offers a more faithful implementation of the theoretical model than experiments conducted in free space, where dipolar interactions and short-range photon exchange complicate the dynamics. The resulting insights are not limited to the specific system studied; they offer a pathway towards understanding and manipulating collective light-matter coupling in a wide range of quantum optical platforms.

The detailed analysis of the linear expansion around balanced decay provides a quantitative tool for characterizing the transition and predicting system behavior under varying conditions. The connection to experimentally studied nonequilibrium critical phenomena in driven-dissipative platforms further highlights the broader significance of this work.

The logarithmic divergence of the susceptibility confirms the first-order nature of the transition, providing a clear signature for experimental observation and validation of the theoretical predictions. The researchers’ findings demonstrate that the stationary state distribution becomes analogous to a many-body phase transition in the large-N limit, with the susceptibility diverging at the balanced decay point.

Cavity and Waveguide Implementation of Engineered Decay Paths

The ability to precisely control the pathways through which quantum systems decay is now demonstrated through analysis of engineered reservoirs, specifically lossy cavity modes and guided modes of waveguides. These structures allow for the manipulation of collective decay in multilevel emitters, moving beyond the limitations of single-channel Dicke dynamics and offering a means to explore more complex quantum phenomena. This observation suggests a critical point in the system’s behavior, sensitive to the balance between competing decay pathways.

The work extends beyond theoretical modeling, offering a practical tool for experiments using cavity and waveguide quantum electrodynamics. These platforms engineer multiple collective decay paths by acting as permutation-symmetric reservoirs, enabling researchers to investigate the dynamics of superradiance in systems with multiple stable ground sublevels. The researchers detail how summing over all possible decay paths leads to a compact expression in Laplace space, simplifying the analysis of complex quantum systems.

The analysis reveals scaling laws for the superradiant peak time and intensity when decay channels are balanced, providing quantitative insights into the dynamics of this collective emission process. This approach is particularly useful for understanding the behavior of emitters in the bad-cavity regime, a condition frequently encountered in experiments with cold atoms and artificial atoms in circuit QED. Multichannel Dicke superradiance can be realized in one-dimensional nanophotonic waveguide QED platforms, where guided modes function as the permutation-symmetric reservoirs.

The elimination of multiple cavity modes under the Born-Markov approximation yields an effective system dynamics described by a sum of decay terms, ρ = sum_(α = 1)^$d{\Gamma }_{\alpha }D$[$S_{\alpha }$], where ${\Gamma }_{\alpha }$ represents the decay rate. This simplification allows for a more tractable analysis of the system’s behavior, while still capturing the essential physics of multichannel superradiance.

The symbolic quantum-trajectory construction not only provides closed-form solutions but also offers a means to analyze the influence of individual decay paths on the overall system dynamics. The researchers emphasize that the stationary state refers to the limit of undriven Lindblad dynamics, dependent on the initial state of the system. This scalability is particularly important for studying superradiance in macroscopic collections of emitters, where the complexity of the system can quickly become intractable with traditional analytical methods.

The work unifies and extends single-channel Dicke dynamics to multilevel emitters, providing a more comprehensive framework for understanding collective light-matter interactions. The researchers’ approach offers a pathway toward characterizing and manipulating the complex dynamics of quantum systems.

More like this

PhysicsThermal bosons boost superconductor temperatures, theory shows

Quantum PhysicsCU Boulder Alum Wins Boeing Quantum Prize for Atom Interferometer Work

Quantum Research NewsResearchers map the limits of classical quantum modeling

Quantum Research NewsNSF Grant Backs Columbia’s Work on Quantum Measurement