Quantum non-Markovian Hatano-Nelson model

In the quantum world, particles do not always behave like tiny billiard balls bouncing off one another. Sometimes, they exist in a state of constant interaction with their surroundings, a condition physicists call an open system. When these systems lose energy to their environment, a process known as dissipation, they often settle into a predictable rhythm. For decades, scientists have relied on a simplified view of this process, assuming that the environment reacts instantly and without memory, a concept called the Markovian approximation. Under this assumption, the way a particle loses energy does not depend on the specific frequency of its motion. However, in the complex reality of experimental settings, environments often have a memory. They react differently depending on the speed or frequency of the particle, creating a more complicated, non-Markovian behavior that standard theories struggle to describe.

This memory effect becomes particularly fascinating when combined with a phenomenon called nonreciprocity. In a reciprocal system, a particle moving from point A to point B behaves exactly the same as one moving from B to A. In a nonreciprocal system, the rules change depending on the direction of travel; a particle might flow easily in one direction but face a wall in the other. This directional bias is the heart of the Hatano-Nelson model, a theoretical framework that has long intrigued researchers for its exotic properties and potential applications in sensing and information processing. Until now, most attempts to create such models in the lab have relied on the simplified, memory-less view of dissipation. A new study by Sumit Kumar Jana and colleagues challenges this limitation, showing how to build a quantum lattice that retains the memory of its environment, leading to behaviors that were previously thought impossible.

The researchers set out to understand how a quantum lattice behaves when the dissipation it experiences is not constant but depends on the frequency of the particle's motion. To do this, they constructed a detailed microscopic model of a one-dimensional chain of sites, where each pair of neighboring sites is connected to a helper site, forming a triangular unit. These helper sites are then linked to their own environments, which act as a reservoir for energy loss. By mathematically integrating out the helper sites and their environments, the team derived an effective description of the main chain. Crucially, they did this without making the usual shortcuts that assume weak connections or fast time scales. The result was a model where the hopping of particles between sites and the rate at which they lose energy both change depending on the frequency of the motion. This frequency dependence is the hallmark of non-Markovian behavior, transforming the standard Hatano-Nelson model into a version that remembers its past.

When the team analyzed this new model, they discovered that the memory of the environment creates a striking asymmetry in how particles move. In a standard, memory-less system, the nonreciprocity is present but uniform across all frequencies. In their non-Markovian version, the system behaves like a selective filter. For particles moving in one direction, the system becomes highly dissipative at certain frequencies, effectively blocking them, while allowing particles of the same frequency to pass through almost unimpeded in the opposite direction. This phenomenon, which the authors call unidirectional frequency blocking, means that the lattice can act as a one-way gate that is tuned to specific frequencies. It is a feature that simply cannot exist in the simplified, memory-less models that have dominated the field. The researchers confirmed this by calculating the transmission of particles through the chain, showing that while the standard model allows for some flow in both directions at all frequencies, their new model creates a sharp cutoff where flow is completely suppressed in one direction but preserved in the other.

The implications of this discovery extend to how quantum matter behaves at the edge of stability. In a system made of fermions, a type of particle that includes electrons, the researchers found a new kind of phase transition. As they adjusted the energy level of the incoming particles, the current flowing through the lattice did not change smoothly. Instead, at a specific critical point, the behavior of the current shifted abruptly. In one direction, the current remained steady, but in the other, it dropped off exponentially as the chain grew longer. This sudden, non-smooth change, which the authors term a non-equilibrium dissipative quantum phase transition, is a direct consequence of the interplay between the directional bias and the frequency-dependent memory of the environment. It is a phenomenon that has no counterpart in reciprocal systems or in the traditional Markovian models, suggesting that the memory of the environment can fundamentally alter the phase structure of quantum matter.

The work provides a rigorous foundation for understanding and engineering these complex quantum lattices. By deriving the model from first principles without relying on approximations, the team has shown that non-Markovian effects are not just a minor correction but a source of entirely new physics. Their findings suggest that by carefully designing the environment and the connections within a quantum system, scientists can create devices that control the flow of energy and information with unprecedented precision. This could lead to new types of sensors that are sensitive to specific frequencies or quantum circuits that operate with a level of directional control previously unattainable. The study opens a door to a richer landscape of quantum behavior, where the memory of the past shapes the flow of the future, offering a more complete picture of how quantum systems interact with the real world.