Quantum Impurity Models coupled to Markovian and Non Markovian Baths

This paper develops a novel theoretical framework combining an exact real-time hybridization expansion and a Non-Crossing-Approximation resummation technique to compute the evolution of quantum impurities linearly coupled to non-Markovian environments while simultaneously interacting with additional Markovian baths via generic non-linear couplings.

The Gist

Here is an explanation of the paper, translated into everyday language using analogies.

The Big Picture: A Noisy Room and a Sticky Floor

Imagine you have a tiny, delicate quantum machine (let's call it the Impurity). This machine is the star of the show. It has its own internal gears and springs (interactions), but it doesn't exist in a vacuum. It's surrounded by two very different types of "noise" or environments.

1. The "Memory" Room (Non-Markovian Bath): Imagine the Impurity is in a room filled with a thick, sticky fluid. If the Impurity moves, it pushes the fluid aside. But here's the catch: the fluid doesn't just disappear; it swirls around and pushes back later. The environment "remembers" what the Impurity did a moment ago. This creates complex, wobbly, and unpredictable motion. In physics, this is called Non-Markovian behavior (it has a memory).
2. The "Forgetful" Room (Markovian Bath): Now, imagine the Impurity is also being poked by a crowd of people who are very forgetful. They poke the machine, it reacts, and then they immediately forget they ever poked it. They don't push back later. This is Markovian behavior (no memory). In the real world, this happens when things lose energy as heat or light instantly (dissipation).

The Problem:
Scientists have been great at studying the "Sticky Fluid" (Non-Markovian) and the "Forgetful Crowd" (Markovian) separately. But what happens when your tiny machine is stuck in both at the same time?

• The sticky fluid makes the machine wobble in complex ways.
• The forgetful crowd tries to dampen the wobble or change the machine's state randomly.
• The Challenge: Calculating exactly how the machine moves when both forces are fighting each other is incredibly hard. The math usually breaks down.

The Solution: A New Way to Count the Steps

The authors of this paper (Marco Schiro and Orazio Scarlatella) developed a new mathematical "recipe" to solve this problem.

1. The "Hybridization Expansion" (The Recipe Book)

Think of the machine's movement as a story. To understand the story, you have to look at every possible way the machine could interact with the fluid and the crowd.

• The Old Way: Scientists usually wrote down a list of every possible interaction, but it got too messy to read.
• The New Way: The authors created a "Hybridization Expansion." Imagine this as a master recipe book. Instead of trying to solve the whole movie at once, they break the story down into tiny "scenes" (diagrams).
  • Scene A: The machine moves, the fluid pushes back.
  • Scene B: The crowd pokes the machine, then the fluid pushes back.
  • Scene C: The crowd pokes, the fluid pushes, the crowd pokes again...
    They wrote a formula that lists all these possible scenes. This is the "exact" solution, but it's a huge list.

2. The "Non-Crossing Approximation" (The Shortcut)

The list of scenes is so long that even supercomputers can't read it all. So, the authors used a clever shortcut called the Non-Crossing Approximation (NCA).

• The Analogy: Imagine the interactions are like lines of people shaking hands.
  • Crossing: Person A shakes hands with Person B, while Person C shakes hands with Person D, but their arms cross over each other. This is complicated and messy.
  • Non-Crossing: Person A shakes hands with Person B, then Person B shakes hands with Person C. The lines never cross.
• The Magic: The authors realized that for many physical situations, the "crossing" handshakes don't matter as much as the "non-crossing" ones. By ignoring the messy, crossed lines and only counting the clean, non-crossing ones, they could solve the equation much faster without losing the most important physics.

What Did They Find? (The Results)

They tested their new method on a simple model: a single electron (the Impurity) sitting in a quantum dot, being poked by a "forgetful" crowd (losses, pumping, and dephasing) while sitting in a "sticky" fluid (a cold quantum bath).

Here are the cool things they discovered:

1. The "Oscillation" Effect:

   • If you only have the "forgetful crowd" (Markovian), the electron just settles down smoothly, like a ball rolling to the bottom of a hill.
   • But when you add the "sticky fluid" (Non-Markovian), the electron starts bouncing and wobbling before it settles. It's like the fluid is pushing it back up the hill every time it tries to stop.
   • Why it matters: This shows that the environment's "memory" can make quantum systems behave in rhythmic, oscillating ways that pure noise cannot explain.

2. The "Dephasing" Surprise:

   • Usually, "dephasing" (random noise that scrambles quantum information) just makes things messy.
   • But the authors found that when you mix the "forgetful crowd" with the "sticky fluid," the dephasing actually changes how many electrons are in the machine.
   • Analogy: Imagine a crowded dance floor. If people just bump into each other randomly (Markovian), the dancers stay in their spots. But if the floor is sticky (Non-Markovian) and people are bumping into each other, the random bumps actually push the dancers into a different part of the room. The "noise" changed the population count.

Why Does This Matter?

This paper is like building a new bridge between two islands that were previously separated.

• Island 1: Quantum Optics (lasers, atoms, where noise is usually "forgetful").
• Island 2: Condensed Matter Physics (solids, metals, where environments have "memory").

By creating a tool that handles both types of noise at once, this research helps us understand:

• Future Quantum Computers: How to keep quantum bits (qubits) stable when they are surrounded by both sticky materials and noisy electronics.
• New Materials: How to design materials that use "noise" to their advantage, perhaps to create new states of matter or to engineer specific quantum states.

In Summary:
The authors built a new mathematical toolkit to describe a tiny quantum object being squeezed by two different types of environments: one that remembers the past and one that forgets it instantly. They found that when these two forces mix, the object doesn't just settle down; it dances, wobbles, and changes its state in surprising ways that we couldn't predict before.

Technical Summary

Here is a detailed technical summary of the paper "Quantum Impurity Models coupled to Markovian and Non-Markovian Baths" by Marco Schiró and Orazio Scarlatella.

1. Problem Statement

The paper addresses the challenge of modeling open quantum systems that are simultaneously coupled to two distinct types of environments:

1. A Non-Markovian Bath: A structured environment (e.g., a fermionic or bosonic reservoir) with frequency-dependent correlations and memory effects. The system couples bilinearly to this bath.
2. A Markovian Bath: An environment causing dissipation, dephasing, or pumping, characterized by memoryless dynamics. The system couples non-linearly to this bath, and its effect is described by a Lindblad master equation.

The Core Challenge: Standard methods for quantum impurity models (like Diagrammatic Monte Carlo) typically handle unitary dynamics or single-bath non-Markovian effects. Conversely, standard Lindblad approaches handle Markovian dissipation but often fail to capture the complex memory effects of structured reservoirs. There is a lack of unified theoretical frameworks to treat the interplay between local many-body interactions, non-Markovian memory, and Markovian dissipation/dephasing on equal footing.

2. Methodology

The authors develop a rigorous formalism to compute the real-time evolution operator ($V$) of the reduced density matrix of the impurity after tracing out both environmental degrees of freedom.

A. Hybridization Expansion

The authors derive an exact real-time hybridization expansion for the evolution operator $V(t, 0)$.

• Formalism: They utilize the Schwinger/Keldysh contour formalism to handle the non-equilibrium evolution of the density matrix.
• Tracing the Non-Markovian Bath: By expanding in powers of the impurity-bath coupling ($H_{I\bar{M}}$) and using Wick's theorem, they integrate out the non-Markovian bath. This results in a series of terms involving the hybridization function $\Delta(t, t')$, which encodes the bath's memory.
• Tracing the Markovian Bath: Crucially, they treat the Markovian environment using super-operator formalism. Assuming the impurity-Markovian bath subsystem obeys a Lindblad master equation, they show that the partial trace over the Markovian bath can be performed exactly. This converts the standard unitary evolution operators into Markovian evolution super-operators ($V_0$).
• Result: The final expansion (Eq. 21) is a series of super-operators acting on the initial density matrix, connected by hybridization lines ($\Delta$) and dressed by the Markovian propagator ($V_0$). This generalizes standard real-time hybridization expansions to include non-unitary Markovian dynamics.

B. Diagrammatic Resummation (NCA)

To make the problem tractable, the authors formulate a Dyson equation for the full evolution operator $V$:
$$\partial_t V(t, t') = \mathcal{L} V(t, t') + \int_{t'}^t dt_1 \Sigma(t, t_1) V(t_1, t')$$
where $\mathcal{L}$ is the Lindbladian generator and $\Sigma$ is the self-energy arising from the non-Markovian bath.

They apply the Non-Crossing Approximation (NCA):

• Approximation: The self-energy $\Sigma$ is approximated by summing only those Feynman diagrams where hybridization lines do not cross.
• Self-Energy Expression: They derive an analytic expression for the NCA self-energy (Eq. 27), which involves the hybridization function $\Delta$ and the dressed propagator $V$.
• Properties: The authors prove that the resulting NCA map preserves trace and Hermiticity. They discuss the preservation of complete positivity (noting it is a difficult task to prove generally but likely holds in specific limits).

3. Key Contributions

1. Unified Formalism: The derivation of an exact hybridization expansion that simultaneously accounts for non-Markovian memory effects (via $\Delta$) and Markovian dissipation (via Lindblad super-operators).
2. Super-Operator Generalization: The introduction of a super-operator notation for the hybridization expansion, allowing the Markovian bath to be traced out exactly without approximations, while retaining the diagrammatic structure necessary for Monte Carlo sampling.
3. NCA for Mixed Baths: The development of a self-consistent NCA scheme for this mixed environment. The resulting Dyson equation allows for the numerical solution of the impurity dynamics in regimes where exact diagonalization is impossible (due to the bath) and standard perturbation theory fails (due to strong coupling).
4. Physical Insight into Dephasing: The demonstration that Markovian dephasing, when combined with non-Markovian processes, can alter population dynamics (occupation numbers), a phenomenon that does not occur in purely Markovian systems where dephasing only affects coherences.

4. Results (Case Study)

The authors applied their method to a spin-less fermionic impurity (Resonant Level Model) coupled to:

• A zero-temperature, particle-hole symmetric fermionic bath (Non-Markovian).
• Markovian losses, particle pumping, and dephasing.

Key Findings:

• Spectral Properties: The eigenvalues of the evolution super-operator $V(t)$ show one eigenvalue equal to 1 (representing the stationary state) and others decaying to zero. In the non-Markovian case, the decay is non-exponential and exhibits oscillations, unlike the pure exponential decay of purely Markovian systems.
• Population Dynamics:
  • Increasing the non-Markovian coupling strength ($\eta$) induces oscillations in the impurity population.
  • Increasing the bandwidth of the non-Markovian bath suppresses these oscillations, recovering the unitary Resonant Level Model behavior in the wide-band limit.
  • Dephasing Effect: In a purely Markovian setting, dephasing does not affect the steady-state population (as the dephasing operator commutes with the number operator). However, in the presence of a non-Markovian bath, dephasing significantly alters the steady-state occupation. This is explained by a mechanism where non-Markovian processes convert populations to coherences, which are then damped by Markovian dephasing, effectively feeding back into the population dynamics.
• Stationary State: The steady-state occupation depends strongly on the energy level position when coupled to the non-Markovian bath, whereas it is energy-independent in the purely Markovian limit (leading to an infinite-temperature state).

5. Significance

• Theoretical Bridge: This work bridges the gap between condensed matter physics (quantum impurity models, Kondo physics) and quantum optics/information (driven-dissipative systems, Lindblad dynamics).
• Computational Tool: The derived expansion provides a natural starting point for Diagrammatic Monte Carlo (DiagMC) simulations of dissipative quantum systems, potentially allowing for numerically exact solutions in the future.
• Physical Phenomena: It reveals that the interplay between memory (non-Markovian) and noise (Markovian) creates emergent many-body effects, such as the modification of steady-state populations by dephasing, which is crucial for understanding quantum devices coupled to electromagnetic resonators or for engineering specific quantum states via dissipation.
• Future Applications: The framework is applicable to more complex models (e.g., Anderson Impurity Model, bosonic extensions) and can serve as an impurity solver within Dynamical Mean-Field Theory (DMFT) for driven-dissipative systems.